General equilibrium theory
Lecture notes
Alberto Bisin
Dept. of Economics
NYU
November 18, 2010
Contents
1 Introduction
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Demand theory: A quick review
2.1 Consumer theory . . . . . . . .
2.1.1 Duality . . . . . . . . .
2.1.2 Aggregate demand . . .
2.2 Producer theory . . . . . . . . .
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3 Arrow-Debreu exchange economies
3.1 The economy . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Pareto e¢ ciency . . . . . . . . . . . . . . . . . . . . . . .
3.3 Competitive equilibrium . . . . . . . . . . . . . . . . . .
3.3.1 Uniqueness . . . . . . . . . . . . . . . . . . . . .
3.3.2 Local uniqueness . . . . . . . . . . . . . . . . . .
3.3.3 Di¤erentiable approach: A rough primer . . . . .
3.3.4 Competitive equilibrium in production economies
3.4 Mathematical appendix . . . . . . . . . . . . . . . . . . .
3.4.1 References . . . . . . . . . . . . . . . . . . . . . .
3.5 Strategic foundations . . . . . . . . . . . . . . . . . . . .
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4 Two-period economies
4.1 Arrow-Debreu economies . . . . . . .
4.2 Financial market economies . . . . .
4.2.1 The stochastic discount factor
4.2.2 Arrow theorem . . . . . . . .
4.2.3 Existence . . . . . . . . . . .
4.2.4 Constrained Pareto optimality
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vi
CONTENTS
4.2.5
4.2.6
4.2.7
4.2.8
Aggregation . . . . . . . . . . . . . . . . .
Asset pricing . . . . . . . . . . . . . . . .
Some classic representation of asset pricing
Production . . . . . . . . . . . . . . . . .
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6 In…nite-horizon economies
6.1 Asset pricing . . . . . . . . . . . . . . . . . . . .
6.1.1 Arrow-Debreu economy . . . . . . . . . . .
6.1.2 Financial markets economy . . . . . . . .
6.1.3 Conditional asset pricing . . . . . . . . . .
6.1.4 Predictability or returns . . . . . . . . . .
6.1.5 Fundamentals-driven asset prices . . . . .
6.2 Bubbles . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 (Famous) Theoretical Examples of Bubbles
6.3 Double In…nity . . . . . . . . . . . . . . . . . . .
6.3.1 Overlapping generations economies . . . .
6.4 Default . . . . . . . . . . . . . . . . . . . . . . . .
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5 Asymmetric information
5.1 The moral hazard economy . . . . . . . . . . .
5.1.1 The Symmetric information benchmark
5.1.2 The moral hazard economy . . . . . .
5.1.3 References . . . . . . . . . . . . . . . .
7 To add
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115
Chapter 1
Introduction
These notes constitute the material for the second half of the Micro I (…rst
year) graduate course at NYU. The topic of this section of the course is
general equilibrium theory.
The standard approach to graduate teaching of general equilibrium theory involves introducing a series of theorems on existence, characterization,
and welfare properties of competitive equilibria under weaker and weaker
assumptions in larger and larger commodity spaces. Such an approach introduces the students to precise rigorous mathematical analysis and invariably
impresses them with the elegance of the theory. Various textbooks take this
approach, in some form or another:
A. Mas-Colell, M. Whinston, and J. Green (1995): Microeconomic Theory,
Oxford University Press, ch. ??, is the main reference; it also contains
a short introduction to two-period economies.
L. McKenzie (2002), Classical General Equilibrium Theory, MIT Press, is a
beautiful modern treatment of the classical theory.
K. Arrow and F. Hahn (1971): General Competitive Analysis, North Holland, is the classical treatment of the classical theory.1
B. Ellickson (1994): Competitive Equilibrium: Theory and Applications,
Cambridge University Press. It contains a useful chapter on non-convex
economies.
1
And so is G. Debreu (1972), Theory of Value: An Axiomatic Analysis of Economic
Equilibrium, Cowles Foundation Monographs Series, Yale University Press, an invaluable
little book for several generations of theorists.
1
2
CHAPTER 1 INTRODUCTION
M. Magill and M. Quinzii (1996): Theory of Incomplete Markets, MIT Press,
takes a classical theory approach on …nancial market equilibrium in
two-period economies.
The approach adopted in these notes aims instead at introducing general equilibrium theory per se but also as the canonical microfoundation for
macroeconomics and …nance. To this end, the standard theory of general
equilibrium is introduced in its rigour and elegance, but only under restrictive assumptions, allowing some shortcuts in analysis and proofs. On the
other hand, we shall be able to introduce …nancial market equilibria in twoperiod economies after only a few classes, exposing students to fundamental conceptual notions like complete and incomplete markets, no-arbitrage
pricing, constained e¢ ciency, equilibria in moral hazard and adverse selection economies, and more. The course ends with a treatment of dynamic
economies and recursive competitive equilibria. Pedagogically, from twoperiod to fully dynamic economies the step is rather short, so that we can
concentrate on purely dynamic concepts, like bubbles.
1.1
Notation
We shall use the following notation regarding vectors in RL+ :
y > x if y
x and yl > xl for at least one element l:
Chapter 2
Demand theory: A quick
review
The consumption set X is the set of admissible levels of consumption of L
existing commodities. In this section we shall assume
X = RL+ ; with generic element x:
X is then a convex set, bounded below.
Assumption 1 The consumer has a utility function
U : RL+ ! R
which satis…es the following properties,
Strong Monotonicity: y > x ) U (y) > U (x); for any x; y 2 RL+ ;
Strict Convexity: U : RL+ ! R is strictly quasi-concave,
U ( x + (1
U ( x + (1
) y)
) y) >
U (x) + (1
U (x) + (1
) U (y); for any x; y 2 RL+ and
) U (y); if x =
6 y and 2 (0; 1)
Di¤erentiability: U : RL+ ! R is C 2 on the interior of its domain.
Common properties often required in applications include:
Homotheticity: U ( x) = U (x); for any
3
2 R++ ;
2 (0; 1); and
4
CHAPTER 2 DEMAND THEORY: A QUICK REVIEW
Quasi-linearity: U (x) = x1 + (x2 ; :::; xL ); for some : RL+ 1 ! R satisfying
strong monotonicity, strict convexity, and di¤erentiability.
Di¤erentiability is violated, for instance, by Leontief preferences:
U (x) = min f:::; l xl;::: g ; for some
2 RL++ :
Continuity (hence di¤erentiability) is violated, for instance, by lexicographic pref erences:
U (x)
2.1
U (y) whenever either x1 > y1 or x1 = y1 and xl
yl ; l = 2; :::; L:
Consumer theory
Markets are competitive, that is, the consumer takes market prices as given,
independent of his decisions (each agent is a price taker). In addition we
consider the case where:
prices are linear: unit price pl of each commodity l is …xed, independent of
level of individual trades (and the same for all agents);
prices are non-negative: this is justi…ed under free disposal, that is, when
agents can freely dispose of any amount of any commodity;
markets are complete: for each commodity l in X there is a market where
the commodity can be traded.
Given wealth level m, the budget set is:
(
B(p; m) =
x2X:p x=
L
X
l=1
pl xl
m
)
The budget set is convex, compact, and non-empty for p
0; m
B(p; m) = B( p; m) for all > 0:
Consider the consumer’s utility maximization problem,
max U (x):
x2B(p;m)
0:Furthermore,
5
2.1 CONSUMER THEORY
For any p
0; m
0; under our assumptions on U (x), by the Maximum
theorem, a solution of the consumer’s problem exists. Furthermore, the solution of the consumer’s problem for every p; m induces the consumer’s demand
correspondence x : RL++ R+ ! RL+ ; x(p; m): By the Maximum theorem ,
x(p; m) is a in fact a continuous function.
Proposition 1 The individual consumer’s demand x(p; m) satis…es the following properties:
Homogeneity of degree zero in p; m:
x(p; m) = x( p; m) for all p
0;
> 0;
Continuity: x(p; m) is a continuous function in p; m;
WARP: for any (p; m); (p0 ; m0 ) 2 RL++ R+ such that x = x(p; m) 6= x0 =
x(p0 ; m0 ):
if px0 m; then p0 x > m0 ;
Walras Law: p x(p; m) = m.
Proof. These properties are straightforward consequences of the assumptions. Homogeneity of degree zero is a consequence of homogeneity of degree
one of the budget set. Continuity follows from the Maximum theorem under
strict quasi-concavity of U (x) (weak quasi-concavity would induce an upperhemi-continuous, convex valued correspondence x : RL++ R+ ! RL+ : WARP
and Walras law follow easily from strict monotonicity.
Suppose, e.g., that preferences are homothetic, that is, they are represented by
U (x) = g(u(x)); where g : R ! R has g 0 > 0 and u : RL+ ! R is homogenous of degree 1:
In this case, marginal rates of substitution,
@U (x)
@xl
@U (x)
@xl0
; for any l; l0 ; are invariant
with respect to expansions along rays from the origin, that is:
x(p; m) = x(p; m) for all m; p
Examples of homothetic utility functions include:
0;
> 0:
6
CHAPTER 2 DEMAND THEORY: A QUICK REVIEW
Cobb Douglas: U (x) = x1 x21
;
CES: U (x) = x1 + x2 ;
Quasi linear: U (x) = x1 + u(x2 ); for u : RL+ ! R.
Any solution of the consumer’s problem satis…es the following system of
Kuhn Tucker necessary and su¢ cient conditions, stated here for the case of
interior solutions,
rU
p = 0
m p x = 0;
for some > 0, the Lagrange multiplier associated to the budget constraint.
Local properties of x(p; m) can then be obtained using the Implicit Function
theorem on the previous …rst order conditions (foc’s):
3
2
3
2
32
3 2
2
m
0
rU p
dx
5 dp + 4
5 dm
4
54
5=4
xT
1
pT 0
d
Note that the second order conditions (soc’s) guarantee that
2
3
2
rU p
4
5 is invertible
T
p
0
when evaluated at an (x; ; p) satisfying foc’s. A simple proof of this statement follows for completeness.
Proof. We show that @(y; z) 2 RL+1 , (y; z) 6= 0 such that
2
32 3 2
3
2
2
rU p
y
r U y pz
4
54 5 = 4
5 = 0:
pT 0
z
py
By contradiction, suppose such a (y; z) exists. Then pre-multiplying r2 U y
pz by y, yields:
yr2 U y = ypz:
But ypz 0; and hence yr2 U y 0; a contradiction with soc’s.
We …nally state an important implication of utility maximization on the
Slutsky matrix of compensated price e¤ects, de…ned as
S(p; m) = Dp x(p; m) + Dm x x:
7
2.1 CONSUMER THEORY
Proposition 2 S(p; m) is negative semi de…nite.
Proof. Consider the following compensated price change:
dp; dm : dm = x dp
The induced demand change is
dx = Dp x(p; m) dp + Dm x(p; m) (x dp)
and, by WARP
dp dx
2.1.1
0 for any dp:
Duality
Let V : RL++
R+ ; V (p; m), be de…ned by
V (p; m) = U (x(p; m)):
V (p; m) is the indirect utility function.
Proposition 3 The indirect utility function V (p; m) satis…es the following
properties:
@V (p; m)=@m = > 0: the consumer’s marginal utility of wealth equals the
shadow value of relaxing the budget constraint;
V (p; m) is homogenous of degree zero in p; m;
@V (p; m)=@pl
0 for all l; p >> 0;
V (p; m) is quasi-convex in p: the lower contour set
convex.
p : V (p; m)
V
is
Proof. The properties are straighforward consequences of the assumptions
on U (x) and the properties of x(p; m): We leave them to the reader, except
quasi-convex in p; which is proved as follows. Take any pair p0 ; p00 such that
V (p0 ; m); V (p00 ; m) V and consider p^ = p0 + (1
)p00 for 2 [0; 1]. Note
that for all x such that p^ x
m we must have either p0 x
m and/or
00
p x m; thus U (x) v .
8
CHAPTER 2 DEMAND THEORY: A QUICK REVIEW
Consider the consumer’s cost minimization problem,
min p x
x2X
s.t:U (x)
u
A solution exists for all u U (0) and p
0: The solution h(p; u); with
L
L
h : R++ R ! R+ ; is typically referred to as compensated - or Hicksian demand. By the soc’s,
Dp h(p; u) is symmetric, negative semi-de…nite.
We say that Hicksian demands h(p; u) satisfy the law of demand.
Let e : RL++ R ! R+ ; e(p; u) = p h(p; u) de…ne the expenditure function.
Proposition 4 The expenditure function e(p; u) has the following properties:
@e(p; u)=@u > 0;
@e(p; u)=@pl
0 for all l = 1; ::; L;
e(p; u) is homogeneous of degree one in p;
e(p; u) is concave in p:
Proof. The properties are straighforward consequences of the Maximum
theorem and the assumptions on U (x). We leave them to the reader, except
quasi-concavity in p; which is proved as follows. For any pair p0 ; p00 , consider
p^ = p0 + (1
)p00 for 2 [0; 1]. Then, for any u, e(^
p; u) = p^ h(^
p; u) =
0
p h(^
p; u) + (1
)p00 h(^
p; u); which is in turn
e(p0 ; u) + (1
)e(p00 ; u):
For all p
0, m > 0; u > U (0), the following identities hold:
x(p; m) = h(p; u)
(2.1)
for u = V (p; m) and e(p; u) = m:
By the envelope theorem, the compensated demand can be obtained from
the expenditure function:
h(p; u) = Dp e(p; u):
9
2.1 CONSUMER THEORY
Figure 2.1: Brief summary of duality relations
Hence the properties of h(p; u) can also be obtained from those of e(p; u).
Similarly, di¤erentiating the equation de…ning the indirect utility,
V (p; m) = U (x(p; m))
with respect to p, we obtain Roy’s identity:
rp V (p; m) =
x(p; m) =
rm V (p; m) x(p; m):
A brief summary of the duality relationships can be helpful:
x(p; m) ! h(p; u) Slutsky:
Dp x(p; u)
V (p; m) ! x(p; m) Roy’s identity:
x(p; m) =
Dm x xT = Dp h(p; u)
1
rp V (p; m)
rm V (p; m)
10
CHAPTER 2 DEMAND THEORY: A QUICK REVIEW
e(p; u) ! h(p; u) Expenditure function properties:
h(p; u) = rp e(p; u)
V (p; m) $ e(p; u) Utility-expenditure duality:
V (p; m) = e 1 (p; m)
e(p; u) = V 1 (p; u)
x(p; m) $ h(p; u) Marshallian-Hicks demand duality:
x(p; m) = h(p; V (p; m))
h(p; u) = x(p; e(p; u))
2.1.2
Aggregate demand
It is useful to study in detail the properties of aggregate demand, as a function of wealth. With some abuse of notation, let introduce the notation to
index agents, i = 1; :::; I. Let the wealth of agent i be denoted mi : Fix the
distribution of wealth as follows,
X
i
mi = i m for some given i 0, for all i,
= 1:
i
The Marshallian demand of any agent i is then denoted xi (p; mi ) and
X
xi (p; mi )
x(p; m) =
i2I
is the aggregate demand. Does aggregate demand x(p; m) satisfy WARP?
Not, typically. Take (p; m) and (p0 ; m0 ) such that x (p0 ; m0 ) 6= x (p; m) and
p x (p0 ; m0 )
m:
It is immediate to see that the following inequality can still hold:
p0 x (p; m)
m0
(because p x (p0 ; m0 ) m does not imply p xi (p0 ; mi0 ) mi for all i; similarly
for p0 x (p; m) m0 .)
A su¢ cient condition for aggregate demand x (p; m) to satisfy WARP
is that individual Marshallian demands satisfy the law of demand, that is,
substitution e¤ects prevail over income e¤ects.
11
2.2 PRODUCER THEORY
Proposition 5 Proposition 6 Suppose xi (p; mi ) satis…es the law of demand, that is,
Dp xi (p; mi ) is negative semide…nite
for all i: Then x (p; m) satis…es WARP.
Proof. It is convenient to proof the statement in terms of discrete changes,
letting the limit argument to the reader. Let (p; m) ; (p0 ; m0 ): x(p; m) 6=
x(p0 ; m0 ); p x(p0 ; m0 )
m. Take p00
(m=m0 )p0 ,; x(p00 ; m) = x(p0 ; m0 ) by
the homogeneity of degree zero of demand. By the law of demand (p00 p)
[x(p00 ; m) x(p; m)] < 0. Using Walras law and the property p x(p00 ; m) m;
we obtain p00 x(p; m) > m ,.p0 x(p; m) > m0 .
It is immediate to verify that a su¢ cient condition for individual demand
to satisfy the law of demand is that preferences of every agent are homothetic.
2.2
Producer theory
We shall study the production activity of …rms operating in competitive
markets. A production plan is a y 2 RL , interpreted as the net output of the
L goods:
yi < 0 : input
yi > 0 : output
The production set is the set of (technologically feasible) production plans:
Y
RL :
Whenever the commodities which are outputs in the production set are …xed,
O f1; ::; Lg (and hence also the complementary set of those which are inputs), the outer boundary of Y can typically be represented by a (continuous)
production function, describing the maximal output level attainable for any
level of inputs. In the case where O = f1g ; e.g.,
y1 = f (z) i¤
(y1 ; z) 2 Y
0
@y1 > y1 : (y10 ; z) 2 Y
We assume that Y and and f : RL+
tions:
1
! R+ satisfy the following assump-
12
CHAPTER 2 DEMAND THEORY: A QUICK REVIEW
Regularity Y is nonempty, closed, Y \ RL+ = f0g (no free lunch and
possibility of inaction), and it satis…es free disposal:
y 2 Y and y 0
Correspondingly, f : RL+
1
y ) y0 2 Y
! R+ is monotonically increasing.
Convexity Y is strictly convex:
y; y 0 2 Y ) y + (1
)y 0 2 intY
Correspondingly, f : RL+ 1 ! R+ is strictly concave (has decreasing
returns to scale). (Y is convex,
y 2 Y ) y 2 Y for all
corresponds to f : RL+
to scale.)
1
2 [0; 1]
! R+ is concave - has non-increasing returns
Any …rm chooses a production plan so as to maximize pro…ts:
max p y
s.t. y 2 Y
Existence of a solution requires conditions ensuring that Y is bounded
above. For every p
0, a solution induces a net supply correspondence
y(p):
Proposition 7 The net supply correspondence y(p) satis…es the following
properties:
y(p) is homogeneous of degree 0 in p;
y(p) is a convex-valued correspondence (a single valued function if f :
RL+ 1 ! R+ is strictly concave).
The value of the solution is a pro…t function
(p) = p y(p):
Proposition 8 The pro…t function (p) satis…es the following properties:
(p) is homogeneous of degree 1 in p;
13
2.2 PRODUCER THEORY
(p) is a convex function.
Proof. The properties are straighforward consequences of the properties
of y(p): We leave them to the reader, except convexity; which is proved as
follows. Take any pair p0 ; p00 and consider p^ = p0 + (1
)p00 for 2 (0; 1).
Note that (^
p) = y(^
p) ( p0 + (1
)p00 )
y(p0 ) p0 + (1
)y(p00 ) p00
0
00
= (p ) + (1
) (p ):
Suppose Y is a convex cone,
y 2 Y ) y 2 Y for all
> 0;
that is, the technology has constant returns to scale: Correspondingly, the
production function f : RL+ 1 ! R+ is homogeneous of degree 1 in its arguments and,
- y 2 y(p) ) y 2 y(p) for all
-
> 0;
(p) = 0 for all p:
Whenever f is di¤erentiable, any solution of the …rm’s problem satisfy
the following system of foc’s (stated here for the case of interior solutions):
p1 Df = w:
Focs are also su¢ cient if f is concave.
Assume f is continuously di¤erentiable and strictly concave. Applying
the Implicit Function theorem to foc’s, we obtain:
Dw z =
1
D2 f
p1
1
is symmetric, negative de…nite
Furthermore, by the envelope theorem, y(p) = Dp , so that Dp y = D2
is:
- symmetric,
- positive - recall z =
(y2 ; ::; yL ) ! - semide…nite (by the convexity of ) and
- such that Dp y p = 0 (by the homogeneity of y(p)).
14
CHAPTER 2 DEMAND THEORY: A QUICK REVIEW
The input level z which solves the …rm’s choice problem also solves the
following problem:
min C = w z
s:t: f (z)
y1
This is perfectly analogous to expenditure minimization problem of consumer. Hence we know that:
- C(w; y1 ) is concave in w and such that @C=@y1 > 0 and @C=@wl
l = 1; ::; L 1;
0;
- z(w; y1 ) = Dw C exhibits the properties of a compensated demand function.
Chapter 3
Arrow-Debreu exchange
economies
3.1
The economy
The economy is polupated by I consumers with preferences described by
U i : RL+ ! R and resources ! i 2 RL+ , i 2 I = f1; ::; Ig :1 An allocation is an
array (x1 ; ::; xI ) 2 RLI
+ , and it is feasible if it satis…es
I
X
x
I
X
i
i=1
!i
i=1
In the special case in which L = 2, I = 2, feasible allocations can be graphically represented using the Edgeworth Box.
Unless otherwise noted, we shall impose the following strong (but not
outrageous) assumptions.
Assumption 2 U i : RL+ ! R is C2 in any open subset of RL+ ;strictly monotonic
in its arguments and strictly quasi-concave.2 Furthermore, ! 2 RLI
++ :
1
2
We are conscious of the notational abuse.
Strict Monotonicity: for any x; x0 2 RL
+;
x
x0 ) U (x)
with > if x 6= x0
15
U (x0 );
16 CHAPTER 3 ARROW-DEBREU EXCHANGE ECONOMIES
3.2
Pareto e¢ ciency
An allocation x 2 RLI
+ is Pareto e¢ cient if it is feasible and there is no other
feasible allocation x
^ 2 RLI
+ which Pareto dominates it:
U i (^
xi )
U i (xi ) for all i, > for at least one i.
Pareto dominance de…nes a (social) preference relation over the set of allocations RLI
+ , which is however incomplete.
PE allocations are solutions of the problem:
s.t.
0
maxx U i (xi )
P i P i
ix
i!
0
U i (xi )
0
U i for all i0 6= i
0
for some given U i i0 6=i :
The foc’s (for an interior solution) of this problem are:
rU i =
0
rU i = for all i0 6= i
X
X
xi =
!i
i0
i
i
0
where i i0 6=i and are the Lagrange multipliers of the two sets of constraints above. Thus
rU i =
i0
0
rU i for all i0 6= i
and utility gradients are co-linear for all agents (Marginal rates of substitution are equalized across agents).
0
Varying the values of U i for i0 6= i we obtain the set Pareto e¢ cient
allocations, the Pareto frontier.
Strict quasi-concavity: for any x; x0 2 RL
+ and any t 2 [0; 1]
U (tx + (1
t)x0 )
tU (x) + (1 t)U (x0 )
with > if t 2 (0; 1) and x; x0 2 RL
+:
17
3.3 COMPETITIVE EQUILIBRIUM
Let the Utility possibility set be the image of the feasible allocations in
the space of utility levels
(
)
X
X
I
U P = U 2 RI U = U i (xi ) i=1 , for some x 2 RLI
xi
!i
+ such that
i
i
Proposition 9 U P is closed, convex, bounded above.
Theorem 10 (Negishi) Let x 2 RLI
+ be a Pareto e¢ cient allocation. Then
I
there exist a 2 R+ , 6= 0, such that
x = arg max
s.t.
X
x
X
i
i
X
U i (xi )
i2I
i
!:
i
Furthermore,
i
_ rU i ; for any i 2 I:
Proof. The proof exploits a separating hyperplane theorem.
PE allocations support points on the outer boundary of U P .
3.3
Competitive equilibrium
Agents trade in perfectly competitive markets.
A competitive equilibrium is an allocation x = (:::; xi ; ::) 2 RLI
+ and a
L
price p 2 R++ such that
xi 2 arg max U i (xi )
s.t. pxi
p! i ;
for any i 2 I; and
X
i
xi
X
!i:
i
Trade is voluntary, hence equilibrium allocations will satisfy individual rationality:
U i (xi ) U i (! i ) for all i = 1; :::; I:
18 CHAPTER 3 ARROW-DEBREU EXCHANGE ECONOMIES
Letting mi = p! i we can derive agent I’s Marshallian demand from his
consumer problem, as in the previous chapter. It is convenient, however,
represent demand functions in terms of endowments rather than wealth:
xi : RL++
RL++ ! RL+ :
Let z i (p; ! i ) denote agent i 2 I’s excess demand: z i (p; ! i ) = xi (p; ! i ) ! i :
L
Finally, the aggregate excess demand z : RL++ RLI
++ ! R+ is de…ned as
X
z (p; !) =
z i p; ! i :
i2I
Proposition 11 For any economy ! 2 RLI
++ the aggregate excess demand
z (p; !) satis…es the following properties:
smoothness: z (p; !) is C 1 ;
homogeneity of degree 0: z ( p; !) = z (p; !) ;for any
> 0;
Walras Law: pz (p; !) = 0; 8p >> 0;
lower boundedness: 9s such that zl (p; !) >
s, 8l 2 L;
boundary property:
pn ! p 6= 0; with pl = 0 for some l; ) max fz1 (pn ; !); :::; zL (pn ; !)g ! 1:
Let’s …rst study the welfare properties of competitive equilibrium allocations.
Theorem 12 (First welfare) All competitive equilibrium allocations are
Pareto e¢ cient.
Proof. Suppose x = (::; xi ; ::) is a competitive equilibrium allocation for
some price p and it is not Pareto e¢ cient. Then there must be another
allocation x^ = (::; x^i ; ::) which is feasible and Pareto dominates x. But since
xi is the optimal choice of consumer i at prices p and preferences are strongly
monotone, U i (^
xi ) U i (xi ) implies p x^i p xi for all i and also, p x^i p ! i ,
with all the previous inequalities being
forPat least some i. Summing
P strict
i
i
the latter inequality over i yields p
x
^
>
p
i
i ! which contradicts the
feasibility of x^:
19
3.3 COMPETITIVE EQUILIBRIUM
Theorem 13 (Second welfare) For any Pareto
x 2
P ie¢ cient allocation
LI
LI
LI
R+ , there exist transfers t 2 R , such that i t = 0 and x 2 R+ is a
L
) of the economy
competitive equilibrium allocation (for some prices p 2 R++
LI
with initial endowment ! + t 2 R+ .
Proof. Let ti = xi ! i for all i. The theorem is an implication of the
separating hyperplane theorem.
Let ! = (! i )i2I denote the vector of aggregate endowments:Competitive
equilibrium prices are solutions of:
X
X
z(p; !)
xi (p; p ! i )
! i = 0;
i
i
a system of L equations in L unknowns, the prices p. By Walras law
!
X
X
p
xi (p; p ! i )
! i = 0; for all p
i
i
and hence at most L 1 equations are independent (the market clearing
equation for one market can be o,itted without loss of generality). By homogeneity of degree 0 in p of xi (p; p ! i ); for any i 2 I; prices can always be
normalized, e.g., restricted without loss of generality to
)
(
X
pl = 1 ;
p2 L 1
p 2 RL+ :
l
the L-simplex, a compact ad convex set.
The equilibrium equations can thus be always reduced to L 1 equations
in L 1 unknowns. Since equations are typically nonlinear, having number
of unknowns less or equal than number of independent equations does not
ensure a solution exists.
Theorem 14 (Existence) Let z : L 1 ! RL be a continuous function,
such that p z(p) = 0 for all p. Then there exists p such that z(p ) 0:
Sketch of the proof.
pl + max f0; zl (p)g
, l = 1; ::L
Let 'l (p) = PL
[p
+
max
f0;
z
(p)g]
j
l
l=1
20 CHAPTER 3 ARROW-DEBREU EXCHANGE ECONOMIES
Note that L 1 is convex and compact, and ' : L 1 !
the Brouwer Fixed Point theorem there is a …xed point p :
pl = PL
pl + max f0; zl (p )g
l=1
Then
zl (p )pl
L
X
j=1
L 1
. Hence by
, l = 1; ::L.
pj + max f0; zl (p )g
pj + max f0; zj (p )g = zl (p )pl + zl (p ) max f0; zl (p )g
and summing over l yields:
X
0=
zl (p ) max f0; zl (p )g ) zl (p )
0 for all l = 1; :::; L:
Since p satis…es z(p )
z(p ) = 0,
l
0 and Walras law, p
zl (p ) < 0 i¤ pl = 0:
But this is impossible under strong monotonicity and hence:
z(p ) = 0:
To properly claim existence of a competitive equilibrium need to face one last
problem: the consumers’demand is not de…ned for prices on the boundary of
L 1
(when the price of some good is zero). We need
P to use a limit argument,
L
L
L 1
L 1
p 2 R+ : l pl = 1, pl " for all l ,
considering z : " ! R , for "
and taking limits as " ! 0.
3.3.1
Uniqueness
Existence of a competitive equilibrium can be proved under quite general conditions.3 Equilibria are however unique only under very strong restrictions.
Several examples of such restrictions are listed in the following.
The endowment distribution is Pareto e¢ cient, and hence the only equilibrium is autarchic: xi = ! i for all i 2 I:
3
We shall leave this statement essentially unsubstantiated. General equilibrium theory,
for more than half a century, has considered this as one of its main objectives.
21
3.3 COMPETITIVE EQUILIBRIUM
Preferences satisfy an aggregation condition (so that a representative consumer exists) for all p and for given (mh )H
h=1 :
Aggregate demand satis…es WARP and equilibria are locally isolated.
Aggregate demand satis…es the gross substitution property:
p0l > pl and p0j = pj ; for all j 6= l;
=)
0
zj (p ) > zj (p).
Note that gross substitution implies that the law of demand holds at
any equilibrium price. Gross substitution holds for instance for Cobb
Douglas and CES utility functions (provided < 1).
3.3.2
Local uniqueness
Let an economy be parametrized by the endowment vector ! 2 RLI
++ keeping
preferences (ui )i2I …xed. Furthermore, normalize pL = 1 and eliminate the
L-th component of the excess demand. Then
L 1
z : R++
L
RLI
++ ! R
1
represents an aggregate excess demand for an exchange economy ! = (! i )i2I 2
RLI
++ .
De…nition 15 A p 2 RL++ such that z (p; !) = 0 is regular if Dp z (p; !)
has rank L 1:
De…nition 16 An economy ! 2 RLI
++ is regular if Dp z (p; !) has rank L
for any p 2 RL++1 such that z (p; !) = 0.
1
De…nition 17 An equilibrium price p 2 RL++1 is locally unique if 9 an
open set P such that p 2 P and for any p0 6= p 2 P , z (p0 ; !) 6= 0:
Proposition 18 A regular equilibrium price p 2 RL++1 is locally unique. Furthermore, the set of equilibrium prices of an economy ! 2 RLI
++ is a smooth
manifold (see Math Appendix) of dimension LI:
22 CHAPTER 3 ARROW-DEBREU EXCHANGE ECONOMIES
Proof. Fix an arbitrary ! 2 RLI
1; by
++ : Since Dp z (p; !) has rank L
regularity of p, the Inverse function theorem - Local (see Math Appendix)
applied to the map z : RL++1 ! RL 1 , directly implies local uniqueness of p 2
RL++1 . Since Dp z (p; !) has rank L 1; by regularity of p, the Inverse function
L 1
;
theorem - Global (see Math Appendix) applied z : RL++1 RLI
++ ! R
1
LI
directly implies that the set p 2 z (0); as a function of ! 2 R++ , is a
smooth manifold of dimension LI:
Proposition 19 Any economy ! in a full measure Lebesgue subset of RLI
++
is regular.
We say then that regularity is a generic property in RLI
++ , as it holds in a
LI
full measure Lebesgue subset of R++ .
Proof. The statement follows by the Transversality theorem (see Math
Appendix), if z t 0: We now show that z t 0: Pick an arbitrary agent
i 2 I: It will be su¢ cient to show that, for any (p; !) 2 RL++1 RLI
++ such
that z(p; !) = 0; we can …nd a perturbation d! i 2 RL such that dz =
D!i z(p; !)d! i , for any dz 2 RL 1 . Consider any perturbation d! i such that
L
d! i1 + pd! i 1 = 0; for d! i 1 = (! il )l=2 : Any such perturbation, leaves each
agent i 2 I demand unchanged and hence it implies D!i z(p; !)d! i = d! i 1 ,
for any arbitrary d! i 1 2 RL :
3.3.3
Di¤erentiable approach: A rough primer
The di¤erential techniques exploited to study generic local uniqueness can
be expanded to provide a general characterization of competitive equilibria
as a manifold parametrized by endowments. This characterization implies an
existence result. We sketch some of the analysis, just to provide the reader
with the ‡avor of the arguments.
De…nition 20 The index i(p; !) of a price p 2 RL++1 such that z (p; !) = 0
is de…ned as
i(p; !) = ( 1)L 1 sign jDp z (p; !)j :
The index i(!) of an economy (! i )i2I is de…ned as
i(!) =
X
p:z(p;!)=0
i(p; !):
3.3 COMPETITIVE EQUILIBRIUM
23
Theorem 21 (Index) For any regular economy ! 2 RLI
++ , i(!) = 1:
Proof. The theorem is a deep mathematical result whose proof is clearly
beyond the scope of this class. Let it su¢ ce to say that the proof relies
crucially on the boundary property of excess demand. Adventurous reader
might want to look at Mas Colell (1985), section 5,6, p. 201-15.
Corollary 22 Any regular economy ! 2 RLI
++ has an odd number of equilibria. In particular, any regular economy ! 2 RLI
++ has at least one equilibrium.
Corollary 23 Any economy ! 2 RLI
++ has at least one equilibrium price p 2
L 1
R++ :
Proof. By contradiction. Suppose there exist an economy ! 2 RLI
++ with
no equilibrium. Then, ! 2 RLI
is
regular,
by
de…nition
of
regularity
- a
++
contradiction with previous corollary.
The existence result is then a corollary of the Index theorem. We can
try and give a sense of the arguments, o¤ of the proof of the Index theorem,
required for this approach to the existence question. Let ! 2 RLI
++ be an
such
that
there
exist
arbitrary regular economy. Pick an economy ! 0 2 RLI
++
L 1
a unique price p 2 R++ such that z(p) = 0; and Dp z(p) has rank L 1: One
such economy can always be constructed by choosing ! 0 2 RLI
++ to be a Pareto
optimal allocation. In fact, [we can show that] generic regularity holds in the
subset of economies with Pareto optimal endowments. Let t! + (1 t)! 0 ; for
0 t 1; represent a 1-dimensional subset of economies. Let Z(p; t) be the
map Z : RL++1 [0; 1] ! RL 1 induced by Z(p; t) = z(p; t! + (1 t)! 0 ) for
given (!; ! 0 ): We say that Z(p; t) is an homotopy, or that z(p; !) and z(p; ! 0 )
are homotopic to each other. [We can show that] DZ(p; t) has rank L 1 in its
domain. It follows from the Corollary of the Inverse function theorem - Global
(see Math Appendix) that the set (p; t) 2 Z 1 (0); is a smooth manifold of
dimension 1: [We can show that] prices p can, without loss of generality,
be restricted to a compact set P such that Z 1 (0) \ P [0; 1] = ?:4 As a
consequence Z 1 (0) is a compact smooth manifold of dimension 1: By the
Classi…cation theorem (see Math Appendix), Z 1 (0) is then homeomorphic
4
This is a consequence of the boundary conditions of excess demand systems. In other
words, we could adopt the alternative normalization, restricting prices in the simplex ;
a compact set, and show that equilibrium prices are never on @ :
24 CHAPTER 3 ARROW-DEBREU EXCHANGE ECONOMIES
Figure 3.1: Characterization of Z
1
(0) - impossible
to a countable set of segments in R and of circles S.5 Regularity of Z 1 (0) at
the boundary, t = 0 and t = 1 and the property that Z 1 (0) \ P [0; 1] = ?
imply that at least one component of Z 1 (0) is homeomorphic to a line with
boundary at t = 0 and t = 1: It looks confusing, but it’s easier with a few
…gures.
The only possible representation of Z 1 (0) is then as in the following
…gure. Therefore: an equilibrium price p 2 P exists, for any t 2 [0; 1];
and furthermore, for a full-measure Lebesgue subset of [0; 1] the number of
equilibria is odd.
3.3.4
Competitive equilibrium in production economies
[...]
5
Along a component of Z
manifold folds.
1
(0) (a line or a circle), a change in index occurs when the
25
3.3 COMPETITIVE EQUILIBRIUM
Figure 3.2: Characterization of Z
1
(0) - impossible
26 CHAPTER 3 ARROW-DEBREU EXCHANGE ECONOMIES
Figure 3.3: Characterization of Z
1
(0) - impossible
27
3.3 COMPETITIVE EQUILIBRIUM
Figure 3.4: Characterization of Z
1
(0)
28 CHAPTER 3 ARROW-DEBREU EXCHANGE ECONOMIES
Figure 3.5: Parametrization of a 1-manifold X
3.4
Mathematical appendix
Theorem 24 (Inverse function theorem - Local). Let f : Rn ! Rn be C 1 :
If Df has rank n; at some x 2 Rn , there exist an open set V
Rn and
1
n
1
a function f
: V ! R such that f (x) 2 V and f (f (z)) = z in a
neighborhood of x:
De…nition 25 A subset X Rm is a smooth manifold of dimension n if for
any x 2 X there exist a neighborhood U X and a C 1 function f : U ! Rm
such that Df has rank n in the whole domain.
Let f (U ) = V: A smooth manifold of dimension n is then locally parametrized by a restriction of the function f 1 on the open set V \ Rn f0gm n ,
in the sense that f 1 maps V \ Rn f0gm n onto U; a neighborhood of x
on X.
3.4 MATHEMATICAL APPENDIX
29
Example 26 An example of a 1-manifold of R2 is S = x 2 R2 (x1 )2 + (x2 )2 = 1 ,
the circle. An explicit parametrization for S can be constructed as follows.
Seeing a restriction of f 1 on the open set V \ Rn
f0gm n as a map
n
m
i : R ! R ; the following four maps are su¢ cient to parametrize S:
1 (x1 )
=
2 (x1 )
=
3 (x2 )
=
4 (x2 )
=
q
x1 ; 1 (x1 )2 if x2 > 0
q
x1 ;
1 (x1 )2 if x2 < 0
q
1 (x2 )2 ; x2 if x1 > 0
q
1 (x2 )2 ; x2 if x1 < 0
De…nition 27 Let f : Rm ! Rn ; m
n; be C 1 : f is transversal to 0,
denoted f t 0; if Df has rank n for any x 2 Rm such that f (x) = 0:
n
Theorem 28 (Transversality). Let f : Rm
n; be C 1 and
++ ! R++ ; m
x1
transversal to 0, f t 0: Decompose any vector x 2 Rm
++ as x = x2 ; with
n
n
x 1 2 Rm
++ ; x2 2 R++ : Then Dx2 f (x) has rank n for all x in a Lebesgue
measure-1 subset of Rm
++ :
De…nition 29 A subset X
Rm is a smooth manifold with boundary of
dimension n if for any x 2 X there exist a neighborhood U
X and a C 1
m 1
function f : U ! R
R+ such that Df has rank n in the whole domain.
The boundary @X of X is de…ned by @X = f 1 (f0gm 1 R+ )\X: It can
be shown that, if X Rm is a smooth manifold with boundary of dimension
n; then @X is a smooth manifold (without boundary) of dimension n 1:
Example 30 An example of a smooth manifold with boundary of dimension
2 in R2 is S = x 2 R2 (x1 )2 + (x2 )2 1 , the sphere. A parametrization
for can be constructed S, by means of a series of maps i : R2 ! R2+ ; along
the lines of the parametrization of the circle, S: Furthermore, @S =S:
Theorem 31 (Inverse function theorem - Global) Let f : Rm ! Rn ;
m n; be C 1 : Suppose that Dx f has rank n for any x 2 Rm . Then f 1 (0) =
fx 2 Rm jf (x) = 0g is a smooth manifold of dimension m n:
30 CHAPTER 3 ARROW-DEBREU EXCHANGE ECONOMIES
Figure 3.6: Parametrization of a 2-manifold with boundary
3.5 STRATEGIC FOUNDATIONS
31
Corollary 32 Let f : Rn [0; 1] ! Rn be C 1 : Suppose that Dx f has rank n
for any (x; t) 2 Rn [0; 1]. Then f 1 (0) = f(x; t) 2 Rn [0; 1] jf (x; t) = 0g
is a smooth 1 manifold with boundary:
Theorem 33 (Classi…cation) Every compact smooth 1 manifold is homeomorphic to a disjoint union of countably many copies of segments in R and
of S.
Furthermore, @ f
3.4.1
1
(0) = fx 2 Rn jf (x; 0) = 0g [ fx 2 Rn jf (x; 1) = 0 g :
References
The main reference is:
A. Mas-Colell (1985): The Theory of General Economic Equilibrium: A
Di¤erentiable Approach, Econometric Society Monograph, Cambridge
University Press.
But, as Andreu told me once, "I do not hate students so much that I
would give them this book to read." Similar comments hold, in my opinion,
for
Y. Balasko (1988): Foundations of the Theory of General Equilibrium, Academic Press.
You are then left with:
A. Mas-Colell, M. Whinston, and J. Green (1995): Microeconomic Theory,
Oxford University Press, ch. 17.D.
A. Mas-Colell, Four Lectures on the Di¤erentiable Approach to General
Equilibrium Theory, in A. A. Ambrosetti, F. Gori, and R. Lucchetti
(eds.), Lecture Notes in Mathematics, No. 1330, Springer-Verlag, Berlin,
1986.
3.5
[...]
Strategic foundations
Chapter 4
Two-period economies
In a two-period pure exchange economy we study …nancial market equilibria. In particular, we study the welfare properties of equilibria and their
implications in terms of asset pricing.
In this context, as a foundation for macroeconomics and …nancial economics, we study su¢ cient conditions for aggregation, so that the standard
analysis of one-good economies is without loss of generality, su¢ cient conditions for the representative agent theorem, so that the standard analysis of
single agent economies is without loss of generality.
The No-arbitrage theorem and the Arrow theorem on the decentralization of equilibria of state and time contingent good economies via …nancial
markets are introduced as useful means to characterize …nancial market equilibria.
4.1
Arrow-Debreu economies
Consider an economy extending for 2 periods, t = 0; 1. Let i 2 f1; :::; Ig
denote agents and l 2 f1; :::; Lg physical goods of the economy. In addition,
the state of the world at time t = 1 is uncertain. Let f1; :::; Sg denote the
state space of the economy at t = 1. For notational convenience we typically
identify t = 0 with s = 0, so that the index s runs from 0 to S:
n
De…ne n = L(S +1): The consumption space is denoted then by X R+
.
i
i
i
i
i
L
Each agent is endowed with a vector ! = (! 0 ; ! 1 ; :::; ! S ), where ! s 2 R+ ;
for any s = 0; :::S. Let ui : X ! R denote agent i’s utility function. We
will assume:
33
34
CHAPTER 4 TWO-PERIOD ECONOMIES
n
Assumption 1 ! i 2 R++
for all i
Assumption 2 ui is continuous, strongly monotonic, strictly quasiconcave
and smooth, for all i (see Magill-Quinzii, p.50 for de…nitions and details).
Furthermore, ui has a Von Neumann-Morgernstern representation:
ui (xi ) = ui (xi0 ) +
S
X
probs ui (xis )
s=1
Suppose now that at time 0, agents can buy contingent commodities.
That is, contracts for the delivery of goods at time 1 contingently to the
realization of uncertainty. Denote by xi = (xi0 ; xi1 ; :::; xiS ) the vector of all
L
;
such contingent commodities purchased by agent i at time 0, where xis 2 R+
1
I
for any s = 0; :::; S: Also, let x = (x ; :::; x ):
L
Let = ( 0 ; 1 ; :::; S ); where s 2 R+
for each s; denote the price of
state contingent commodities; that is, for a price ls agents trade at time 0
the delivery in state s of one unit of good l:
Under the assumption that the markets for all contingent commodities
are open at time 0, agent i’s budget constraint can be written as1
i
0 (x0
! i0 )
+
S
X
i
s (xs
! is ) = 0
(4.1)
s=0
De…nition 34 An Arrow-Debreu equilibrium is a (x ;
1: x
I
X
2:
x
i
i
2 arg max ui (xi ) s.t.
i
0 (x0
! i0 ) +
S
X
) such that
i
s (xs
! is ) = 0; and
s=0
! is = 0, for any s = 0; 1; :::; S
i=1
Observe that the dynamic and uncertain nature of the economy (consumption occurs at di¤erent times t = 0; 1 and states s 2 S) does not
manifests itself in the analysis: a consumption good l at a time t and state s
is treated simply as a di¤erent commodity than the same consumption good
l at a di¤erent time t0 or at the same time t but di¤erent state s0 . This is
1
We write the budget constraint with equality. This is without loss of generality under
monotonicity of preferences, an assumption we shall maintain.
35
4.1 ARROW-DEBREU ECONOMIES
the simple trick introduced in Debreu’s last chapter of the Theory of Value.
It has the fundamental implication that the standard theory and results of
static equilibrium economies can be applied without change to our dynamic)
environment. In particular, then, under the standard set of assumptions on
preferences and endowments, an equilibrium exists and the First and Second
Welfare Theorems hold.2
De…nition 35 Let (x ; ) be an Arrow-Debreu equilibrium. We say that x
is a Pareto optimal allocation if there does not exist an allocation y 2 X I
such that
1: u(y i )
u(x i ) for any i = 1; :::; I (strictly for at least one i), and
I
X
2:
y i ! is = 0, for any s = 0; 1; :::; S
i=1
Theorem 36 Any Arrow-Debreu equilibrium allocation x is Pareto Optimal.
Proof. By contradiction. Suppose there exist a y 2 X such that 1) and 2)
in the de…nition of Pareto optimal allocation are satis…ed. Then, by 1) in
the de…nition of Arrow-Debreu equilibrium, it must be that
i
0 (y0
! i0 ) +
S
X
i
s (ys
! is )
i
s (ys
! is ) > 0;
0;
s=0
for all i; and
i
0 (y0
! i0 ) +
S
X
s=0
for at least one i.
Summing over i, then
0
I
X
i=1
(y0i
! i0 )
+
S
X
s=0
s
I
X
(ysi
! is ) > 0
i=1
which contradicts requirement 2) in the de…nition of Pareto optimal allocation.
The proof exploits strict monotonicity of preferences. Where?
2
Having set de…nitions for 2-periods Arrow-Debreu economies, it should be apparent
how a generalization to any …nite T -periods economies is in fact e¤ectively straightforward.
In…nite horizon will be dealt with in successive notes.
36
CHAPTER 4 TWO-PERIOD ECONOMIES
4.2
Financial market economies
Consider the 2-period economy just introduced. Suppose now contingent
commodities are not traded. Instead, agents can trade in spot markets and
in j 2 f1; :::; Jg assets. An asset j is a promise to pay ajs 0 units of good
l = 1 in state s = 1; :::; S.3 Let aj = (aj1 ; :::; ajS ): To summarize the payo¤s
of all the available assets, de…ne the S J asset payo¤ matrix
0
a11
aJ1
1
:::
B
C
B
C
A = B :::
::: C :
@
A
1
J
aS ::: aS
It will be convenient to de…ne as to be the s-th row of the matrix. Note that
it contains the payo¤ of each of the assets in state s.
L
Let p = (p0 ; p1 ; :::; pS ), where ps 2 R+
for each s, denote the spot price
vector for goods. That is, for a price pls agents trade one unit of good l
in state s: Recall the de…nition of prices for state contingent commodities in
Arrow-Debreu economies, denoted : Note the di¤erence. Let good l = 1
at each date and state represent the numeraire; that is, p1s = 1, for all
s = 0; :::; S.
Let xisl denote the amount of good l that agent i consumes in good s. Let
J
q = (q1 ; :::; qJ ) 2 R+
, denote the prices for the assets.4 Note that the prices
of assets are non-negative, as we normalized asset payo¤ to be non-negative.
Given prices (p; q) and the asset structure A, any agent i picks a consumption vector xi 2 X and a portfolio z i 2 RJ to
max ui (xi )
s.t.
p0 (xi0
ps (xis
3
! i0 ) = qz i
! is ) = As z i ; for s = 1; :::S:
The non-negativity restriction on asset payo¤s is just for notational simplicity.
Quantities will be row vectors and prices will be column vectors, to avoid the annoying
use of transposes.
4
37
4.2 FINANCIAL MARKET ECONOMIES
De…nition 37 A Financial markets equilibrium is a (x ; z ; p ; q ) such that
1: x i
p0 (xi0 ! i0 )
ps (xis ! is )
I
X
2:
x i ! is
i=1
2 arg max ui (xi ) s.t.
= qz i ; and
= as z i ; for s = 1; :::S; and furthermore
I
X
= 0, for any s = 0; 1; :::; S; and
zi
i=1
Financial markets equilibrium is the equilibrium concept we shall care
about. This is because i) Arrow-Debreu markets are perhaps too demanding a
requirement, and especially because ii) we are interested in …nancial markets
and asset prices q in particular. Arrow-Debreu equilibrium will be a useful
concept insofar as it represents a benchmark (about which we have a wealth
of available results) against which to measure Financial markets equilibrium.
Remark 38 The economy just introduced is characterized by asset markets
in zero net supply, that is, no endowments of assets are allowed for. It is
straightforward to extend the analysis to assets in positive net supply, e.g.,
stocks. In fact, part of each agent i’s endowment (to be speci…c: the projection
of his/her endowment on the asset span, < A >= f 2 RS : = Az; z 2
RJ g) can be represented as the outcome of an asset endowment, zwi ; that is,
letting ! i1 = (! i11 ; :::; ! i1S ), we can write
! i1 = w1i + Azwi
and proceed straightforwardly by constructing the budget constraints and the
equilibrium notion.
No Arbitrage
Before deriving the properties of asset prices in equilibrium, we shall invest
some time in understanding the implications that can be derived from the
milder condition of no-arbitrage. This is because the characterization of noarbitrage prices will also be useful to characterize …nancial markets equilbria.
For notational convenience, de…ne the (S + 1) J matrix
2
3
q
5:
W =4
A
38
CHAPTER 4 TWO-PERIOD ECONOMIES
De…nition 39 W satis…es the No-arbitrage condition if there does not exist
a
there does not exist a z 2 RJ such that W z > 0:5
The No-Arbitrage condition can be equivalently formulated in the following way. De…ne the span of W to be
< W >= f 2 RS+1 :
= W z; z 2 RJ g:
This set contains all the feasible wealth transfers, given asset structure A.
Now, we can say that W satis…es the No-arbitrage condition if
\
S+1
<W >
R+
= f0g:
Clearly, requiring that W = ( q; A) satis…es the No-arbitrage condition is
weaker than requiring that q is an equilibrium price of the economy (with
asset structure A). By strong monotonicity of preferences, No-arbitrage is
equivalent to requiring the agent’s problem to be well de…ned. The next
result is remarkable since it provides a foundation for asset pricing based
only on No-arbitrage.
Theorem 40 (No-Arbitrage theorem)
\
S+1
S+1
<W >
R+
= f0g () 9^ 2 R++
such that ^ W = 0:
First, observe that there is no uniqueness claim on the ^ , just existence.
Next, notice how ^ W = 0 implies ^ = 0 for all 2< W > : It then provides
a pricing formula for assets:
0
1 0 1
:::
:::
B
C B C
B
C B C
^ W = B ^ 0 q j + ^ 1 aj1 + ::: + ^ S ajS C = B 0 C
@
A @ A
:::
:::
Jx1
and, rearranging, we obtain for each asset j,
qj =
5
j
1 a1
+ ::: +
j
S aS ;
for
W z > 0 requires that all components of W z are
s
=
^s
^0
(4.2)
0 and at least one of them > 0:
4.2 FINANCIAL MARKET ECONOMIES
39
Note how the positivity of all components of ^ was necessary to obtain
(4.2).
P
S+1
S+1
Proof. =) De…ne the simplex in R+
as = fT 2 R+
: Ss=0 s = 1g.
Note that by the No-arbitrage condition, < W >
is empty. The proof
hinges crucially on the following separating result, a version of Farkas Lemma,
which we shall take without proof.
Lemma 41 Let X be a …nite dimensional vector space. Let K be a nonempty, compact and convex subset of X. Let M be a non-empty, closed and
convex subset of X. Furthermore, let K and M be disjoint. Then, there
exists ^ 2 Xnf0g such that
sup ^ < inf ^ :
2M
2K
S+1
Let X = R+
, K = and M =< W >. Observe that all the required
properties hold and so the Lemma applies. As a result, there exists ^ 2
Xnf0g such that
sup ^ < inf ^ :
(4.3)
2<W >
2
S+1
It remains to show that ^ 2 R++
: Suppose, on the contrary, that there
is some s for which ^ s 0. Then note that in (4.3 ), the RHS 0: By (4.3),
then, LHS < 0: But this contradicts the fact that 0 2< W > .
We still have to show that ^ W = 0, or in other words, that ^ = 0 for
all 2< W >. Suppose, on the contrary that there exists 2< W > such
that ^ 6= 0: Since < W > is a subspace, there exists
2 R such that
2< W > and ^
is as large as we want. However, RHS is bounded
above, which implies a contradiction.
S+1
(= The existence of ^ 2 R++
such that ^ W = 0 implies that ^ = 0
for all 2< W > : By contradiction, suppose 9 2< W > and such that
S+1
2 R+
nf0g: Since ^ is strictly positive, ^ > 0; the desired contradiction.
A few …nal remarks to this section.
Remark 42 An asset which pays one unit of numeraire in state s and nothing in all other states (Arrow security), has price s according to (4.2). Such
asset is called Arrow security.
40
CHAPTER 4 TWO-PERIOD ECONOMIES
Remark 43 Is the vector ^ obtained by the No-arbitrage theorem unique?
Notice how (??) de…nes a system of J equations and S unknowns, represented
by . De…ne the set of solutions to that system as
S
R(q) = f 2 R++
: q = Ag:
Suppose, the matrix A has rank J 0 J (that it, A has J 0 linearly independent
column vectors and J 0 is the e¤ective dimension of the asset space). In
general, then R(q) will have dimension S J 0 . It follows then that, in this
case, the No-arbitrage theorem restricts ^ to lie in a S J 0 + 1 dimensional
set. If we had S linearly independent assets, the solution set has dimension
zero, and there is a unique vector that solves (??). The case of S linearly
independent assets is referred to as Complete markets.
Remark 44 Let preferences be Von Neumann-Morgernstern:
X
ui (xi ) = ui (xi0 ) +
probs ui (xis )
s=1;:::;S
where
X
probs = 1: Let then ms =
s
probs
. Then
s=1;:::;S
q j = E (mAj )
S
In this representation of asset prices the vector m 2 R++
is called Stochastic
discount factor.
4.2.1
The stochastic discount factor
In the previous section we showed the existence of a vector that provides the
basis for pricing assets in a way that is compatible with equilibrium, albeit
milder than that. In this section, we will strengthen our assumptions and
study asset prices in a full-‡edged economy. Among other things, this will
allow us to provide some economic content to the vector
Recall the de…nition of Financial market equilibrium. Let M RSsi (xi )
denote agent i’s marginal rate of substitution between consumption of the
numeraire good 1 in state s and consumption of the numeraire good 1 at
date 0:
4.2 FINANCIAL MARKET ECONOMIES
M RSsi (xi )
=
41
@ui (xis )
@xi1s
@ui (xi0 )
@xi10
Let M RS i (xi ) = (: : : M RSsi (xi ) : : :) denote the vector of marginal rates
of substitution for agent i, an S dimentional vector. Note that, under the
S
assumption of strong monotonicity of preferences, M RS i (xi ) 2 R++
:
By taking the First Order Conditions (necessary and su¢ cient for a maximum under the assumption of strict quasi-concavity of preferences) with
respect to zji of the individual problem for an arbitrary price vector q, we
obtain that
j
q =
S
X
probs M RSsi (xi )ajs = E M RS i (xi ) aj ;
(4.4)
s=1
for all j = 1; :::; J and all i = 1; :::; I; where of course the allocation xi is the
equilibrium allocation. At equilibrium, therefore, the marginal cost of one
j
more unit of asset j, qP
, is equalized to the marginal valuation of that agent
for the asset’s payo¤, Ss=1 probs M RSsi (xi )ajs .
Compare equation (4.4) to the previous equation (4.2). Clearly, at any
equilibrium, condition (4.4) has to hold for each agent i. Therefore, in equilibrium, the vector of marginal rates of substitution of any arbitrary agent i
can be used to price assets; that is any of the agents’vector of marginal rates
of substitution (normalized by probabilities) is a viable stochastic discount
factor m:
In other words, any vector (: : : probs M RSsi (xi ) : : :) belongs to R(q) and is
hence a viable for the asset pricing equation (4.2). But recall that R(q) is of
dimension S J 0 ; where J 0 is the e¤ective dimension of the asset space. The
higher the the e¤ective dimension of the asset space (sloppily said, the larger
…nancial markets) the more aligned are agents’marginal rates of substitution
at equilibrium (sloppily said, the smaller are unexploited gains from trade
at equilibrium). In the extreme case, when markets are complete (that is,
when the rank of A is S), the set R(q) is in fact a singleton and hence the
M RS i (xi ) are equalized across agents i at equilibrium: M RS i (xi ) = M RS;
for any i = 1; :::; I:
Problem 45 Write the Pareto problem for the economy and show that, at
any Pareto optimal allocation, x; it is the case that M RS i (xi ) = M RS; for
42
CHAPTER 4 TWO-PERIOD ECONOMIES
any i = 1; :::; I: Furthermore, show that an allocation x which satis…es the
feasibility conditions (market clearing) for goods and is such that M RS i (xi ) =
M RS; for any i = 1; :::; I: is Pareto optimal.
We conclude that, when markets are Complete, equilibrium allocations
are Pareto optimal. That is, the First Welfare theorem holds for Financial
market equilibria when markets are Complete.
Problem 46 (Economies with bid-ask spreads)Extend our basic two-period
incomplete market economy by assuming that, given an exogenous vector 2
J
:
R++
the buying price of asset j is qj + j
while
the selling price of asset j is qj
for any j = 1; :::; J:and exogenous. Write the budget constraint and the
First Order Conditions for an agent i’s problem. Derive an asset pricing
equationfor qj in terms of intertemporal marginal rates of substitution at
equilibrium.
4.2.2
Arrow theorem
The Arrow theorem is the fondamental decentralization result in …nancial
economics. It states su¢ cient conditions for a form of equivalence between
the Arrow-Debreu and the Financial market equilibrium concepts. It was
essentially introduced by Arrow (1952). The proof of the theorem introduces
a reformulation of the budget constraints of the Financial market economy
which focuses on feasible wealth transfers across states directly, on the span
of A,
< A >=
2 RS : = Az; z 2 RJ
in particular. Such a reformulation is important not only in itself but as a
lemma for welfare analysis in Financial market economies.
Proposition 47 Let (x ; ) represent an Arrow-Debreu equilibrium. Suppose rank(A) = S (…nancial markets are Complete). Then (x ; z ; p ; q ) is
a Financial market equilibrium, where
s
=
q =
s ps ;
S
X
for any s = 1; :::; S: and
probs M RSsi (xi )As
s=1
43
4.2 FINANCIAL MARKET ECONOMIES
Futhermore, the converse also holds: if (x ; z ; p ; q ) is a Financial market
equilibrium of an economy with rank(A) = S, (x ; ) represents an ArrowDebreu equilibrium, where s = s ps ; for any s = 1; :::; S:
Proof. Financial market equilibrium prices of assets q satisfy No-arbitrage.
S+1
There exists then a vector ^ 2 R++
such that ^ W = 0; or q = A. The
budget constraints in the …nancial market economy are
p0 xi0
! i0 + q z i = 0
! is = As z i ; for s = 1; :::S:
ps xis
Substituting q = A; expanding the …rst equation, and writing the constraints at time 1 in vector form, we obtain:
p0 xi0
! i0 +
S
X
s ps
xis
! is = 0
s=1
2
:
6
6
6
:
6
6
6 ps (xis
6
6
6
:
4
:
(4.5)
3
7
7
7
7
i 7
!s) 7 2 < A >
7
7
7
5
(4.6)
2
:
6
6
6
:
6
6
But if rank(A) = S; it follows that < A >= RS ; and the constraint 6 ps (xis
6
6
6
:
4
:
A > is never binding. Each agent i’s problem is then subject only to
p0
xi0
! i0
+
S
X
s ps
xis
! is = 0;
s=1
the budget constraint in the Arrow-Debreu economy with
s
=
s ps ;
for any s = 1; :::; S:
3
7
7
7
7
i 7
! s ) 7 2<
7
7
7
5
44
CHAPTER 4 TWO-PERIOD ECONOMIES
Furthermore, by No-arbitrage
q =
S
X
probs M RSsi (xi )As :
s=1
Finally, using s = probs M RSsi (xi ), for any s = 1; :::; S; proves the result. (Recall that, with Complete markets M RS i (xi ) = M RS; for any
i = 1; :::; I.)
The converse is straightforward.
4.2.3
Existence
We do not discuss here in detail the issue of existence of a …nancial market
equilibrium when markets are incomplete (when they are complete, existence
follows from the equivalence with Arrow-Debreu equilibrium provided by
Arrow theorem). A sketch of the proof however follows.
The proof is a modi…cation of the existence proof for Arrow-Debreu equilibrium. By Arrow theorem, fact, we can reduce the equilibrium system to an
excess demand system for consumption goods; that is, we can solve out for the
asset portfolios z i ’s. The only conceptual problem with the proof is that the
boundary condition on the excess demand system
2 might not3be guaranteed as
:
7
6
7
6
7
6
:
7
6
7
6
each agent’s excess demand is restricted by 6 ps (xis ! is ) 7 2< A > : This
7
6
7
6
7
6
:
5
4
:
is where the Cass trick comes in handy. It is in fact an important Lemma.
Cass trick. For any Financial 2
market economy,
3 consider a modi…ed econ:
6
7
6
7
6
7
:
6
7
6
7
i
i
omy where the constraint 6 ps (xs ! s ) 7 2< A > is imposed on all
6
7
6
7
6
7
:
4
5
:
4.2 FINANCIAL MARKET ECONOMIES
45
agents i = 2; :::; I but not on agent i = 1: Any equilibrium of the
Financial Market economy is an equilibrium of the modi…ed economy,
and any equilibrium of the modi…ed economy is a Financial market
equilibrium.
Proof. Consider an P
equilibrium of the modi…ed economy
statePI in the
I
i
i
i
i
!
)
=
0:
Therefore,
p
(x
!
ment. At equilibrium,
p
(x
s) =
s
s
s3
i=2 s
i=1 s
2
:
6
7
6
7
6
7
:
6
7
6
7
1
1
i
i
ps (xs ! s ) : But 6 ps (xs ! s ) 7 2< A >; for any i = 2; :::; I; and hence
6
7
6
7
6
7
:
4
5
:
P
PI
i
i
! s ) 2< A > : Since Ii=2 ps (xis ! is ) = ps (x1s ! 1s ) ; it foli=2 ps (xs
lows that ps (x1s ! 1s ) 2< A >; and hence that ps (x1s ! 1s ) 2< A > :
Therefore, the constraint ps (x1s ! 1s ) 2< A > must necessarily hold at an
equilibrium of the modi…ed economy. In other words, the constraint ps (x1s ! 1s ) 2<
A > is not binding at a Financial market equilibrium. The equivalence
between the modi…ed economy and the Financial Market economy is now
straightforward.
4.2.4
Constrained Pareto optimality
Under Complete markets, the First Welfare Theorem holds for Financial
market equilibrium. This is a direct implication of Arrow theorem.
Proposition 48 Let (x ; z ; p ; q ) be a Financial market equilibrium of an
economy with Complete markets (with rank(A) = S): Then x is a Pareto
optimal allocation.
However, under Incomplete markets Financial market equilibria are generically ine¢ cient in a Pareto sense. That is, a planner could …nd an allocation
that improves some agents without making any other agent worse o¤.
Theorem 49 At a Financial Market Equilibrium (x ; z ; p ; q ) of an incomplete …nancial market economy, that is, of an economy with rank(A) < S,
46
CHAPTER 4 TWO-PERIOD ECONOMIES
the allocation x is generically6 not Pareto Optimal.
Proof. From the proof of Arrow theorem, we can write the budget constraints of the Financial market equilibrium as:
p0
xi0
! i0
+
S
X
s ps
xis
s=1
2
:
6
6
6
:
6
6
6 ps (xis
6
6
6
:
4
:
! is = 0
(4.7)
3
7
7
7
7
i 7
!s) 7 2 < A >
7
7
7
5
(4.8)
S
: Pareto optimality of x requires that there does not exist
for some 2 R++
an allocation y such that
1: u(y i )
u(x i ) for any i = 1; :::; I (strictly for at least one i), and
I
X
2:
y i ! is = 0, for any s = 0; 1; :::; S
i=1
Reproducing the proof of the
2 First Welfare3theorem, it is clear that, if such
:
7
6
7
6
7
6
:
7
6
7
6
i
i
= A >; for some i = 1; :::; I;
a y exists, it must be that 6 ps (ys
! s ) 7 2<
7
6
7
6
7
6
:
5
4
:
otherwise the allocation y would be budget feasible for all agent i at the
6
We say that a statement holds generically when it holds for a full Lebesgue-measure
subset of the parameter set which characterizes the economy. In these notes we shall
assume that the an economy is parametrized by the endowments for each agent, the asset
payo¤ matrix, and a two-parameter parametrization of utility functions for each agent;
see Magill-Shafer, ch. 30 in W. Hildenbrand and H. Sonnenschein (eds.), Handbook of
Mathematical Economics, Vol. IV, Elsevier, 1991.
4.2 FINANCIAL MARKET ECONOMIES
47
equilibrium prices. Generic Pareto sub-optimality of x follows then directly
from the following Lemma, which we leave without proof.7
Lemma 50 Let (x ; z ; p ; q ) represent a Financial Market Equilibrium of
an economy
2 with rank(A)
3 < S: For a generic set of economies, the con:
6
7
6
7
6
7
:
6
7
6
7
i
i
straints 6 ps (xs
! s ) 7 2< A > are binding for some i = 1; :::; I.
6
7
6
7
6
7
:
4
5
:
Remark 51 The Lemma implies a slightly stronger result than generic Pareto
sub-optimality of Financial market equilibrium for economies with incomplete
markets. It implies in fact that a Pareto improving allocation can be found
locally around the equilibrium, as a perturbation of the equilibrium.
Pareto optimality might however represent too strict a de…nition of social
welfare of an economy with frictions which restrict the consumption set, as in
the case of incomplete markets. In this case, markets are assumed incomplete
exogenously. There is no reason in the fundamentals of the model why they
should be, but they are. Under Pareto optimality, however, the social welfare
notion does not face the same contraints. For this reason, we typically de…ne a
weaker notion of social welfare, Constrained Pareto optimality, by restricting
the set of feasible allocations to satisfy the same set of constraints on the
consumption set imposed on agents at equilibrium. In the case of incomplete
markets, for instance, the feasible wealth vectors across states are restricted
to lie in the span of the payo¤ matrix. That can be interpreted as the
economy’s “…nancial technology”and it seems reasonable to impose the same
technological restrictions on the planner’s reallocations. The formalization
S
SL
of an e¢ ciency notion capturing this idea follows. Let xit=1 = (xis )s=1 2 R+
;
S
SL
and similarly pt=1 = (ps )s=1 2 R+
7
The proof can be found in Magill-Shafer, ch. 30 in W. Hildenbrand and H. Sonnenschein (eds.), Handbook of Mathematical Economics, Vol. IV, Elsevier, 1991. It requires
mathematical tecniques from di¤erential topology which are not appropriate to be introduced in this course.
48
CHAPTER 4 TWO-PERIOD ECONOMIES
De…nition 52 (Diamond, 1968; Geanakoplos-Polemarchakis, 1986) Let (x ; z ; p ; q )
represent a Financial market equilibrium of an economy whose consumption
set at time t = 1 is restricted by
xit=1 ; 2 B(pt=1 ); for any i = 1; :::; I
In this economy, the allocation x is Constrained Pareto optimal if there does
not exist a (y; ) such that
2:
1:
I
X
u(y i )
ysi
u(x i ) for any i = 1; :::; I, strictly for at least one i
! is = 0, for any s = 0; 1; :::; S
i=1
and
3:
i
yt=1
2 B(gt=1 (!; )); for any i = 1; :::; I
where gt=1 (!; ) is a vector of equilibrium prices for spot markets at t = 1
opened after each agent i = 1; :::; I has received income transfer A i :
The constraint on the consumption set restricts only time 1 consumption allocations. More general constraints are possible but these formulation
is consistent with the typical frictions we encounter in economics, e.g., on
…nancial markets. It is important that the constraint on the consumption
set depends in general on gt=1 (!; ), that is on equilibrium prices for spot
markets opened at t = 1 after income transfers to agents. It implicit identi…es income transfers (besides consumption allocations at time t = 0) as the
instrument available for Constrained Pareto optimality; that is, it implicitly
constrains the planner implementing Constraint Pareto optimal allocations
to interact with markets, speci…cally to open spot markets after transfers.
On the other hand, the planner is able to anticipate the spot price equilibrium map, gt=1 (!; ); that is, to internalize the e¤ects of di¤erent transfers
on spot prices at equilibrium.
Proposition 53 Let (x ; z ; p ; q ) represent a Financial market equilibrium
of an economy with complete markets (rank(A) = S) and whose consumption
set at time t = 1 is restricted by
xit=1 2 B
SL
R+
; for any i = 1; :::; I
In this economy, the allocation x is Constrained Pareto optimal.
49
4.2 FINANCIAL MARKET ECONOMIES
Crucially, markets are complete and B is independent of prices. The
proof is then a straightforward extension of the First Welfare theorem combined with Arrow theorem.8 Constraint Pareto optimality of Financial market equilibrium allocations is guaranteed as long as the constraint set B is
exogenous.
Proposition 54 Let (x ; z ; p ; q ) represent a Financial market equilibrium
of an economy with Incomplete markets (rank(A) < S). In this economy,
the allocation x is not Constrained Pareto optimal.
Proof. By the decomposition of the budget constraints in the proof of Arrow
theorem, this economy is equivalent to one with Complete markets whose
consumption set at time t = 1 is restricted by
xit=1 2 B(gt=1 (!; z)); for any i = 1; :::; I; for any i = 1; :::; I
with the set B(gt=1 (!; z)) de…ned implicitly by
3
2
:
7
6
7
6
7
6
:
7
6
7
6
i
i
6 gs (! s ; z) (xs ! s ) 7 2< A >; for any i = 1; :::; I
7
6
7
6
7
6
:
5
4
:
Note …rst of all that, by construction, ps 2 gs (! s ; z ): Following the proof of
Pareto sub-otimality of Financial market equilibrium allocations,
it then
2
3 fol:
6
6
6
:
6
6
lows that if a Pareto-improving y exists, it must be that 6 ps (ysi
6
6
6
:
4
:
8
7
7
7
7
i 7
=
! s ) 7 2<
7
7
7
5
To be careful, we need to guarantee that monotonicity of preferences on RSL
++ results
in monotonicity on B
RSL
:
This
is
the
case
for
B
open.
Thanks
to
the
students
for
++
noticing this.
50
CHAPTER 4 TWO-PERIOD ECONOMIES
2
3
:
6
7
6
7
6
7
:
6
7
6
7
i
i
A >; for some i = 1; :::; I; while 6 gs (! s ; ) (ys ! s ) 7 = A i , for all i =
6
7
6
7
6
7
:
4
5
:
1; :::; I: Generic Constrained Pareto sub-optimality of x follows then directly
from the following Lemma, which we leave without proof.9
Lemma 55 Let (x ; z ; p ; q ) represent a Financial Market Equilibrium of
an economy
with rank(A) < S: For
3 a generic set of economies, the con2
:
7
6
7
6
7
6
:
7
6
7
6
straints 6 gs (! s ; z + dz) (ysi ! is ) 7 = A (z i + dz i ), for some dz 2 RJI n f0g
7
6
7
6
7
6
:
5
4
:
X
dz i = 0; are weakly relaxed for all i = 1; :::; I, strictly for at
such that
i=1I
10
least one.
There is a fundamental di¤erence between incomplete market economies,
which have typically not Constrained Optimal equilibrium allocations, and
economies with constraints on the consumption set, which have, on the contrary, Constrained Optimal equilibrium allocations. It stands out by comparing the respective trading constraints
gs (! s ; )(xis
! is ) = As i ; for all i and s,
vs.
xit=1 2 B, for all i:
The trading constraint of the incomplete market economy is determined at
equilibrium, while the constraint on the consumption set is exogenous. Another way to re-phrase the same point is the following. A planner choosing
9
The proof is due to Geanakoplos-Polemarchakis (1986). It also requires di¤erential
topology techniques.
10
Once again, note that the Lemma implies that a Pareto improving allocation can be
found locally around the equilibrium, as a perturbation of the equilibrium.
4.2 FINANCIAL MARKET ECONOMIES
51
(y; ) will take into account that at each (y; ) is typically associated a different trading constraint gs (! s ; )(xis ! is ) = As i ; for all i and s; while any
agent i will choose (xi ; z i ) to satisfy ps (xis ! is ) = As z i ; for all s, taking as
given the equilibrium prices ps :
The constrained ine¢ ciency due the dependence of constraints on equilibrium prices is sometimes called a pecuniary externality.11 Several examples of
such form of externality/ine¢ ciency have been developed recently in macroeconomics. Some examples are:
- Thomas, Charles (1995): "The role of …scal policy in an incomplete markets
framework," Review of Economic Studies, 62, 449–468.
- Krishnamurthy, Arvind (2003): "Collateral Constraints and the Ampli…cation Mechanism,"
Journal of Economic Theory, 111(2), 277-292.
- Caballero, Ricardo J. and Arvind Krishnamurthy (2003): "Excessive Dollar Debt: Financial Development and Underinsurance," Journal of Finance, 58(2), 867-94.
- Lorenzoni, Guido (2008): "Ine¢ cient Credit Booms," Review of Economic
Studies, 75 (3), 809-833.
- Kocherlakota, Narayana (2009): "Bursting Bubbles: Consequences and
Causes,"
http://www.econ.umn.edu/~nkocher/km_bubble.pdf.
- Davila, Julio, Jay Hong, Per Krusell, and Victor Rios Rull (2005): "Constrained E¢ ciency in the Neoclassical Growth Model with Uninsurable
Idiosyncractic Shocks," mimeo, University of Pennsylvania.
Remark 56 Consider an economy whose constraints on the consumption set
depend on the equilibrium allocation:
xit=1 2 B(xt=1 ; z ); for any i = 1; :::; I
This is essentially an externality in the consumption set. It is not hard to
extend the analysis of this section to show that this formulation introduces
ine¢ ciencies and equilibrium allocations are Constraint Pareto sub-optimal.
11
The name is due to Joe Stiglitz (or is it Greenwald-Stiglitz?).
52
CHAPTER 4 TWO-PERIOD ECONOMIES
Corollary 57 Let (x ; z ; p ; q ) represent a Financial market equilibrium of
a 1-good economy (L = 1) with Incomplete markets (rank(A) < S). In this
economy, the allocation x is Constrained Pareto optimal.
Proof. The constraint on the consumption set implied by incomplete markets, if L = 1, can be written
(xis
! is ) = As z i :
It is independent of prices, of the form xit=1 2 B.
Remark 58 Consider an alternative de…nition of Constrained Pareto optimality, due to Grossman (1970), in which constraints 3 are substituted by
3
2
:
7
6
7
6
7
6
:
7
6
7
6
i
0
i
i
3 : 6 ps (xs
! s ) 7 = Az ; for any i = 1; :::; I
7
6
7
6
7
6
:
5
4
:
where p is the spot market Financial market equilibrium vector of prices.
That is, the planner takes the equilibrium prices as given. It is immediate to
prove that, with this de…nition of Constrained Pareto optimality, any Financial market equilibrium allocation x of an economy with Incomplete markets
is in fact Constrained Pareto optimal, independently of the …nancial markets
available (rank(A) S):
Problem 59 Consider a Complete market economy (rank(A) = S) whose
feasible set of asset portfolios is restricted by:
z i 2 Z ( RJ ; for any i = 1; :::; I
A typical example is borrowing limits:
zi
b; for any i = 1; :::; I
Are equilibrium allocations of such an economy Constrained Pareto optimal
(also if L > 1)?
53
4.2 FINANCIAL MARKET ECONOMIES
Problem 60 Consider a 1-good (L = 1) Incomplete market economy (rank(A) <
S) which lasts 3 periods. De…ne an Financial market equilibrium for this
economy as well as Constrained Pareto optimality. Are Financial market
equilibrium allocations of such an economy Constrained Pareto optimal?
Problem 61 Extend our basic two-period incomplete market economy by asJ
:
suming that, given an exogenous vector 2 R++
the buying price of asset j is qj +
j
while
the selling price of asset j is qj
for any j = 1; :::; J:and exogenous. 1) Suppose the asset payo¤ matrix is
full rank. Do you expect …nancial market equilibria to be Pareto e¢ cient?
Carefully justify your answer. (I am not asking for a formal proof, though
you could actually prove this.) 2) Continue to suppose the asset payo¤ matrix
is full rank. How would you de…ne Constrained Pareto e¢ ciency in this
economy? Do you expect …nancial market equilibria to be Constrained Pareto
e¢ cient? Carefully justify your answer. (I am really not asking for a formal
proof.)
4.2.5
Aggregation
Agent i’s optimization problem in the de…nition of Financial market equilibrium requires two types of simultaneous decisions. On the one hand, the
agent has to deal with the usual consumption decisions i.e., she has to decide
how many units of each good to consume in each state. But she also has
to make …nancial decisions aimed at transferring wealth from one state to
the other. In general, both individual decisions are interrelated: the consumption and portfolio allocations of all agents i and the equilibrium prices
for goods and assets are all determined simultaneously from the system of
equations formed by (??) and (??). The …nancial and the real sectors of
the economy cannot be isolated. Under some special conditions, however,
the consumption and portfolio decisions of agents can be separated. This
is typically very useful when the analysis is centered on …nancial issue. In
order to concentrate on asset pricing issues, most …nance models deal in fact
with 1-good economies, implicitly assuming that the individual …nancial decisions and the market clearing conditions in the assets markets determine
54
CHAPTER 4 TWO-PERIOD ECONOMIES
the …nancial equilibrium, independently of the individual consumption decisions and market clearing in the goods markets; that is independently of the
real equilibrium prices and allocations. In this section we shall identify the
conditions under which this can be done without loss of generality. This is
sometimes called "the problem of aggregation."
The idea is the following. If we want equilibrium prices on the spot
markets to be independent of equilibrium on the …nancial markets, then
the aggregate spot market demand for the L goods in each state s should
must depend only on the incomes of the agents in this state (and not in
other states) and should be independent of the distribution of income among
agents in this state.
Theorem 62 Budget Separation. Suppose that each agent i’s preferences are separable across states, identical, homothetic within states, and
von Neumann-Morgenstern; i.e. suppose that there exists an homothetic
u : RL ! R such that
i
i
u (x ) =
u(xi0 )
+
S
X
probs u(xis ); for all i = 1; ::; I:
s=1
Then equilibrium spot prices p are independent
of asset prices q and oof the
n
L(S+1) PI
i
i
income distribution; that is, constant in ! 2 R++
i=1 ! given :
Proof. Normalize all spot prices of good 1: p10 = p1s = 1; for any s 2 S:
The consumer’s maximization problem in the de…nition of Financial market
equilibrium can be decomposed into a sequence of spot commodity allocation problems and an income allocation problem as follows. The spot commodity allocation problems. Given the current and anticipated spot prices
p = (p0 ; p1 ; :::; pS ) and an exogenously given stream of …nancial income
S+1
y i = (y0i ; y1i ; :::; ySi ) 2 R++
in units of numeraire, agent i has to pick a
L(S+1)
i
consumption vector x 2 R+
to
max ui (xi )
s.t.
p0 xi0 = y0i
ps xis = ysi ; f or s = 1; :::S:
4.2 FINANCIAL MARKET ECONOMIES
55
Let the L(S + 1) demand functions be given by xils (p; y i ), for l = 1; :::; L;
s = 0; 1; :::S. De…ne now the indirect utility function for income by
v i (y i ; p) = ui (xi (p; y i )):
The Income allocation problem. Given prices (p; q); endowments ! i , and the
asset structure A, agent i has to pick a portfolio z i 2 RJ and an income
S+1
stream y i 2 R++
to
max v i (y i ; p)
s:t:
p0 ! i0
qz i = y0i
ps ! is + as z i = ysi ; f or s = 1; :::S:
By additive separability across states of the utility, we can break the consumption allocation problem into S +1 ‘spot market’problems, each of which
yields the demands xis (ps ; ysi ) for each state. By homotheticity, for each
s = 0; 1; :::S; and by identical preferences across all agents,
xis (ps ; ysi ) = ysi xis (ps ; 1);
and since preferences are identical across agents,
ysi xis (ps ; 1) = ysi xs (ps ; 1)
Adding over all agents and using the market clearing condition in spot markets s, we obtain, at spot markets equilibrium,
xs (ps ; 1)
I
X
ysi
i=1
I
X
! is = 0:
i=1
Again by homothetic utility,
xs (ps ;
I
X
i=1
ysi )
I
X
! is = 0:
(4.9)
i=1
Recall from the consumption allocation problem that ps xis = ysi ; for s =
0; 1; :::S: By adding over all agents, and using market clearing in the spot
56
CHAPTER 4 TWO-PERIOD ECONOMIES
markets in state s,
I
X
ysi
= ps
i=1
= ps
I
X
i=1
I
X
xis ; for s = 0; 1; :::S
(4.10)
! is ; for s = 0; 1; :::S:
i=1
By combining (4.9) and (4.10), we obtain
xs (ps ; ps
I
X
! is )
i=1
=
I
X
! is :
(4.11)
i=1
Note how we have passed from the aggregate demand of all agents in the
economy to the demand of an agent owning the aggregate endowments. Observe also how equation (4.11) is a system of L equations with L unknowns
that determines spot prices ps for each state s independently of asset prices
i
q: Note also
PI thati equilibrium spot prices ps de…ned by (4.11) only depend !
through i=1 ! s :
The Budget separation theorem can be interpreted as identifying conditions under which studying a single good economy is without loss of generality. To this end, consider the income allocation problem of agent i, given
equilibrium spot prices p :
max
i
J
y i 2RS+1
++ ;z 2R
v i (y i ; p )
s.t. y0i = p0 ! i0 qz i
ysi = ps ! is + as z i ; for s = 1; :::S
If preferences ui (xi ) are identical, homothetic within states, and von NeumannMorgenstern, that is, if they satisfy
ui (xi ) = u(xi0 ) +
S
X
probs u(xis ); with u(x) homothetic, for all i = 1; ::; I
s=1
it is straightforward to show that indirect preferences v i (y i ; p ) are also
identical, homothetic within states, and von Neumann-Morgenstern:
i
i
v (y ; p ) =
v(y0i ; p
S
X
)+
probs v(ysi ; p ); with v(y; p ) homothetic, for all i = 1; ::; I:
s=1
4.2 FINANCIAL MARKET ECONOMIES
57
Let w0i = p0 ! i0 ; wsi = ps ! is ; for any s = 1; :::; S; and disregard for notational
simplicity the dependence of v(y; p ) on p : The income allocation problem
can be written as:
S
X
i
probs v(ysi )
v(y0 ) +
max
i
J
y i 2RS+1
++ ;z 2R
s.t.
y0i
ysi
w0i
wsi
s=1
i
= qz
= As z i ; for s = 1; :::S
which is homeomorphic to any agent i’s optimization problem in the de…nition of Financial market equilibrium with l = 1. Note that ysi gains the
interpretation of agent i’s consumption expenditure in state s, while wsi is
interpreted as agent i’s income endowment in state s:
The representative agent theorem
A representative agent is the following theoretical construct.
De…nition 63 Consider a Financial market equilibrium (x ; z ; p ; q ) of an
economy populated by i = 1; :::; I agents with preferences ui : X ! R and
endowments ! i : A Representative agent for this economy is an agent with
preferences U R : X ! R and endowment ! R such that the Financial market
equilibrium of an associated economy with the Representative agent as the
only agent has prices (p ; q ).
In this section we shall identify assumptions which guarantee that the
Representative agent construct can be invoked without loss of generality.
This assumptions are behind much of the empirical macro/…nance literature.
Theorem 64 Representative agent. Suppose preferences satisfy:
i
i
u (x ) =
u(xi0 )
+
S
X
s=1
probs u(xis ); with homothetic u(x); for all i = 1; ::; I:
2
:
3
6
7
6
7
6 : 7
6
7
6
7
i
Let p denote equilibrium spot prices. If 6 ps ! s 7 2< A >; then there exist
6
7
6
7
6 : 7
4
5
:
58
CHAPTER 4 TWO-PERIOD ECONOMIES
S+1
a map uR : R+
! R such that:
!
R
=
I
X
! is ;
i=1
R
R
U (x) = u (y0 ) +
S
X
R
probs u (ys ); where ys = p
s=1
I
X
xis ; s = 0; 1; :::; S
i=1
constitutes a Representative agent.
Since the Representative agent is the only agent in the economy, her
consumption allocation and portfolio at equilibrium, x R ; z R ; are:
x
R
z
R
R
= ! =
I
X
!i
i=1
= 0
If the Representative agent’s preferences can be constructed independently of the equilibrium of the original economy with I agents, then equilibrium prices can be read out
agent’s marginal rates of
P
P of the Representative
substitution evaluated at Ii=1 ! i . Since Ii=1 ! i is exogenously given, equilibrium prices are obtained without computing the consumption allocation
and portfolio for all agents at equilibrium, (x ; z ):
Proof. The proof is constructive. Under the assumptions on preferences in
the statement, we need to show
agents i = 1; :::;
o I, equilibrium asn that, for allP
L(S+1)
I
i
i
set prices q are constant in ! 2 R++
i=1 ! given .If preferences satP
isfy ui (xi ) = u(xi0 ) + Ss=1 probs u(xis ); for all i = 1; ::; I, with an homothetic
u(x); then by the Budget separation ntheorem, equilibrium spot prices
o p are
L(S+1) PI
i
i
independent of q and constant in ! 2 R++
i=1 ! given : There2
3
:
6
7
6
7
6 : 7
6
7
6
7
i
fore, 6 ps ! s 7 2< A > can be written as an assumption on fundamentals;
6
7
6
7
6 : 7
4
5
:
59
4.2 FINANCIAL MARKET ECONOMIES
in particular on ! i : Furthermore, we can restrict our analysis to the single
good economy, whose agent i’s optimization problem is:
max
i
J
y i 2RS+1
++ ;z 2R
s.t.
y0i
ysi
v(y0i )
+
S
X
probs v(ysi )
s=1
i
w0 = qz
ws = As z i ; for s = 1; :::S
where v(y) is homothetic.
P
We show next that uR (y) = v(y) and ! R = Ii=1 ! is constitute a Representative agent. By Arrow theorem„we can write budget constraints as
y0i
w0i
+
S
X
s=1
2
3
s
2
ysi
wsi = 0
3
:
7
6
7
6
7
6
:
7
6
7
6 i
i
6 ys ws 7 2 < A >
7
6
7
6
7
6
:
5
4
:
2
3
:
:
7
7
6
6
7
7
6
6
6 : 7
6 : 7
7
7
6
6
7
7
6
6
But, 6 wsi 7 2< A > implies that there exist a zwi such that 6 wsi 7 = Azwi :
7
7
6
6
7
7
6
6
6 : 7
6 : 7
5
4
5
4
:
:
2
3
:
6
7
6
7
6 : 7
6
7
6 i7
Therefore, 6 ws 7 2< A > implies that ysi = As (z i + zwi ), for any s 2 S: We
6
7
6
7
6 : 7
4
5
:
60
CHAPTER 4 TWO-PERIOD ECONOMIES
can then write each agent i’s optimizationPproblem in terms P
of (y0i ; z i ); and
the value of agent i0 s endowment is w0i + Ss=1 s wsi = w0i + Ss=1 s as zwi =
w0i + qzwi :
In summary, we write the budget constraint as:
y0i
+
S
X
i
s ys
= w0i +
i
s As zw
= w0i + qzwi :
s=1
By the fact that preferences are identical across agents and by homotheticity of v(y); then we can write
y0i q; w0i = w0i + qzwi y0 (q; 1)
ysi q; w0i = w0i + qzwi ys (q; 1) ; for any s 2 S
At equilibrium then
y0 (q; 1)
X
w0i
+
qzwi
= y0
q;
i2I
ys (q; 1)
X
X
w0i
+
qzwi
i2I
w0i + qzwi = ys
i2I
and prices q only depend on
q;
X
w0i + qzwi
i2I
PI
i=1
w0i and
PI
!
!
=
X
w0i
i2I
= As
i
i=1 zw :
X
i2I
zwi ; for any s 2 S
2
:
3
7
6
7
6
6 : 7
7
6
7
6
i
Make sure you understand where we used the assumption 6 p ! s 7 =
7
6
6
7
6 : 7
5
4
:
2
3
:
6
7
6
7
6 : 7
6
7
6 i7
6 ws 7 2< A > . Convince yourself that the assumption is necessary in the
6
7
6
7
6 : 7
4
5
:
proof.
61
4.2 FINANCIAL MARKET ECONOMIES
The Representative agent theorem, as noted, allows us to obtain equilibrium prices without computing the consumption
and portfolio for
PI allocation
i
all agents at equilibrium, (x ; z ):Let w = i=1Pw : Under the assumptions
P
of the Representative agent theorem, let w0 = i2I w0i ; and ws = i2I wsi ;
for any s 2 S: Then
q=
S
X
s=1
probs M RSs (w)As ; for M RSs (w) =
@v(ws )
@ws
@v(w0 )
@w0
That is, asset prices can be computed from agents’ preferences uR = v :
R ! R and from the aggregate endowment (w0 ; :::; ws ; :::) : This is called the
Lucas’trick for pricing assets.
Problem 65 Note that, under
the3 Complete markets assumption, the span
2
:
7
6
7
6
6 : 7
7
6
7
6
restriction on endowments,6 p ! is 7 2< A >; for all agents i; is trivially sat7
6
7
6
6 : 7
5
4
:
is…ed. Does this assumption imply Pareto optimal allocations in equilibrium?
Problem 66 Assume all agents have identical quadratic
2
3 preferences. Derive
:
6
7
6
7
6 : 7
6
7
6
7
individual demands for assets (without assuming 6 p ! is 7 2< A >) and show
6
7
6
7
6 : 7
4
5
:
that the Representative agent theorem is obtained.
Another interesting but misleading result is the "weak" representative
agent theorem, due to Constantinides (1982).
62
CHAPTER 4 TWO-PERIOD ECONOMIES
Theorem 67 Suppose markets are complete (rank(A) = S) and preferences
ui (xi ) are von Neumann-Morgernstern (but not necessarily identical nor homothetic). Let (x ; z ; p ; q ) be a Financial markets equilibrium. Then,
!
R
=
I
X
!i;
i=1
R
U (x) = max
(xi )Ii=1
where
i
= ( i)
I
X
1
i i
i
u (x ) s.t.
i=1
and
I
X
xi = x;
i=1
i
@ui (xi )
=
@xi10
constitutes a Representative agent.
Clearly, then,
q=
S
X
probs M RSs (w)As ; for M RSs (w) =
s=1
@U R (ws )
@ws
@U R (w0 )
@w0
:
Proof. Consider a Financial market equilibrium (x ; z ; p ; q ). By complete
markets, the First welfare theorem holds and x is a Pareto optimal allocation. Therefore, there exist some weights that make x the solution to the
planner’s problem. It turns out that the required weights are given by
i
=
@ui (xi )
@xi10
1
:
This is left to the reader to check; it’s part of the celebrated Negishi theorem.
This result is certainly very general, as it does not impose identical homothetic preferences, however, it is not as useful as the “real”Representative
agent theorem to …nd equilibrium asset prices. The reason is that to de…ne
the speci…c weights for the planner’s objective function, ( i )Ii=1 ; we need to
know what the equilibrium allocation, x ; which in turn depends on the whole
distribution of endowments over the agents in the economy.
4.2.6
Asset pricing
Relying on the aggregation theorem in the previous section, in this section
we will abstract from the consumption allocation problems and concentrate
63
4.2 FINANCIAL MARKET ECONOMIES
on one-good economies. This allows us to simplify the equilibrium de…nition
as follows.
4.2.7
Some classic representation of asset pricing
Often in …nance, especially in empirical …nance, we study asset pricing representation which express asset returns in terms of risk factors. Factors are
to be interpreted as those component of the risks that agents do require a
higher return to hold.
How do we go from our basic asset pricing equation
q = E(mA)
to factors?
Single factor beta representation
Consider the basic asset pricing equation for asset j;
qj = E(maj )
Let the return on asset j, Rj , be de…ned as Rj =
equation becomes
1 = E(mRj )
Aj
.
qj
Then the asset pricing
1
This equation applied to the risk free rate, Rf , becomes Rf = Em
. Using the
fact that for two random variables x and y, E(xy) = ExEy + cov(x; y), we
can rewrite the asset pricing equation as:
ERj =
1
Em
cov(m; Rj )
= Rf
Em
cov(m; Rj )
Em
or, expressed in terms of excess return:
Rf =
ERj
cov(m; Rj )
Em
Finally, letting
j
=
cov(m; Rj )
var(m)
64
CHAPTER 4 TWO-PERIOD ECONOMIES
and
=
var(m)
Em
we have the beta representation of asset prices:
ERj = Rf +
(4.12)
j m
We interpret j as the "quantity" of risk in asset j and m (which is the
same for all assets j) as the "price" of risk. Then the expected return of
an asset j is equal to the risk free rate plus the correction for risk, j m .
Furthermore, we can read (4.12) as a single factor representation for asset
prices, where the factor is m, that is, if the representative agent theorem
holds, her intertemporal marginal rate of substitution.
Multi-factor beta representations
A multi-factor beta representation for asset returns has the following form:
j
f
ER = R +
F
X
jf
(4.13)
mf
f =1
where (mf )Ff=1 are orthogonal random variables which take the interpretation
of risk factors and
cov(mf ; Rj )
jf =
var(mf )
is the beta of factor f , the loading of the return on the factor f .
Proposition 68 A single factor beta representation
ERj = Rf +
j m
is equivalent to a multi-factor beta representation
f
ERj = R +
F
X
f =1
jf
mf
with m =
F
X
f =1
bf m f
65
4.2 FINANCIAL MARKET ECONOMIES
In other words, a multi-factor beta representation for asset returns is
consistent with our basic asset pricing equation when associated to a linear
statistical model for the stochastic discount factor m, in the form of m =
P
F
f =1 bf mf .
cov(m;R )
j
Proof. Write 1 = E(mRj ) as Rj = Rf
and then to substitute
Em
PF
m = f =1 bf mf and the de…nitions of jf , to have
mf
=
var(mf )bf
Emf
The CAPM
The CAPM is nothing else than a single factor beta representation of the
following form:
ERj = Rf +
jf
mf
where
mf = a + bRw
the return on the market portfolio, the aggregate portfolio held by the investors in the economy.
It can be easily derived from an equilibrium model under special assumptions.
For example, assume preferences are quadratic:
u(xio ; xi1 )
=
1 i
(x
2
# 2
x )
S
1 X
probs (xis
2 s=1
x# )2
P
Moreover, assume agents have no endowments at time t = 1. Let Ii=1 xis =
P
xs ; s = 0; 1; :::; S; and Ii=1 w0i = w0 . Then budget constraints include
xs = Rsw (w0
x0 )
Then,
ms =
xs
x0
which is the CAPM for a =
x#
(w0 x0 ) w
=
R
x#
(x0 x# ) s
x#
x0 x#
and b =
x#
x0 x#
(w0 x0 )
:
(x0 x# )
66
CHAPTER 4 TWO-PERIOD ECONOMIES
#
Note however that a = x0 x x# and b = (x(w0 0 xx#0)) are not constant, as they
do depend on equilibrium allocations. This will be important when we study
conditional asset market representations, as it implies that the CAPM is
intrinsically a conditional model of asset prices.
Bounds on stochastic discount factors
Write the beta representation of asset returns as:
cov(m; Rj )
(m; Rj ) (m) (Rj )
=
Em
Em
where 0
(m; Rj )
1 denotes the correlation coe¢ cient and
standard deviation. Then
ERj
Rf =
j
ERj Rf
j
(Rj )
(:), the
(m)
Em
The left-hand-side is the Sharpe-ratio of asset j.
The relationship implies a lower bound on the standard deviation of any
stochastic discount factor m which prices asset j. Hansen-Jagannathan are
responsible for having derived bounds like these and shown that, when the
stochastic discount factor is assumed to be the intertemporal marginal rate
of substitution of the representative agent (with CES preferences), the data
does not display enough variation in m to satisfy the relationship.
A related bound is derived by noticing that no-arbitrage implies the existence of a unique stochastic discount factor in the space of asset payo¤s,
denoted mp , with the property that any other stochastic discount factor m
satis…es:
m = mp +
where is orthogonal to mp .
The following corollary of the No-arbitrage theorem leads us to this result.
Corollary 69 Let (A; q) satisfy No-arbitrage. Then, there exists a unique
2< A > such that q = A :
S
Proof. By the No-arbitrage theorem, there exists 2 R++
such that q = A:
We need to distinguish notationally a matrix M from its transpose, M T : We
write then the asset prices equation as q T = AT T . Consider p :
T
p
= A(AT A) 1 q:
4.2 FINANCIAL MARKET ECONOMIES
67
Clearly, q T = AT Tp ; that is, Tp satis…es the asset pricing equation. Furthermore, such Tp belongs to < A >, since Tp = Azp for zp = (AT A) 1 q: Prove
uniqueness.
We can now exploit this uniqueness result to yield a characterization of
the “multiplicity”of stochastic discount factors when markets are incomplete,
and consequently a bound on (m). In particular, we show that, for a given
(q; A) pair a vector m is a stochastic discount factor if and only if it can
be decomposed as a projection on < A > and a vector-speci…c component
orthogonal to < A >. Moreover, the previous corollary states that such a
projection is unique.
S
Let m 2 R++
be any stochastic discount factor, that is, for any s =
s
1; : : : ; S, ms = probs and qj = E(mAj ); for j = 1; :::; J: Consider the orthogonal projection of m onto < A >, and denote it by mp . We can then write any
stochastic discount factors m as m = mp + ", where " is orthogonal to any
vector in < A >; in particular to any Aj . Observe in fact that mp + " is also
a stochastic discount factors since qj = E((mp + ")aj ) = E(mp aj ) + E("aj ) =
E(mp aj ), by de…nition of ". Now, observe that qj = E(mp aj ) and that we
just proved the uniqueness of the stochastic discount factors lying in < A > :
In words, even though there is a multiplicity of stochastic discount factors,
they all share the same projection on < A >. Moreover, if we make the economic interpretation that the components of the stochastic discount factors
vector are marginal rates of substitution of agents in the economy, we can
interpret mp to be the economy’s aggregate risk and each agents " to be the
individual’s unhedgeable risk.
It is clear then that
(m)
(mp )
the bound on (m) we set out to …nd.
4.2.8
Production
Assume for simplicity that L = 1, and that there is a single type of …rm in the
economy which produces the good at date 1 using as only input the amount
k of the commodity invested in capital at time 0:12 The output depends on
12
It should be clear from the analysis which follows that our results hold unaltered
if the …rms’ technology were described, more generally, by a production possibility set
Y
RS+1 .
68
CHAPTER 4 TWO-PERIOD ECONOMIES
k according to the function f (k; s), de…ned for k 2 K, where s is the state
realized at t = 1. We assume that
- f (k; s) is continuously di¤erentiable, increasing and concave in k;
; K are closed, compact subsets of R+ and 0 2 K.
-
In addition to …rms, there are I types of consumers. The demand side of
the economy is as in the previous section, except that each agent i 2 I is also
endowed with i0 units of stock of the representative …rm. Consumer i has
von Neumann-Morgernstern preferences over consumption in the two dates,
represented by ui (xi0 ) + Eui (xi ), where ui ( ) is continuously di¤erentiable,
strictly increasing and strictly concave.
Competitive equilibrium
Let the outstanding amount of equity be normalized
to 1: the initial distriP i
bution of equity among consumers satis…es i 0 = 1. The problem of the
…rm consists in the choice of its production plan k:.
Firms are perfectly competitive and hence take prices as given. The
…rm’s cash ‡ow, f (k; s); varies with k. Thus equity is a di¤erent “product”
for di¤erent choices of the …rm. What should be its price when all this
continuum of di¤erent “products”are not actually traded in the market? In
this case the price is only a “conjecture.”It can be described by a map Q(k)
specifying the market valuation of the …rm’s cash ‡ow for any possible value
of its choice k.13 The …rm chooses its production plan k so as to maximize
its value. The …rm’s problem is then:
max k + Q(k)
k
(4.14)
When …nancial markets are complete, the present discounted valuation of
any future payo¤ is uniquely determined by the price of the existing assets.
This is no longer true when markets are incomplete, in which case the prices
of the existing assets do not allow to determine unambiguously the value of
any future cash ‡ow. The speci…cation of the price conjecture is thus more
problematic in such case. Let k denote the solution to this problem.
At t = 0, each consumer i chooses his portfolio of …nancial assets and of
equity, z i and i respectively, so as to maximize his utility, taking as given
13
These price maps are also called price perceptions.
69
4.2 FINANCIAL MARKET ECONOMIES
the price of assets, q and the price of equity Q. In the present environment a
consumer’s long position in equity identi…es a …rm’s equity holder, who may
have a voice in the …rm’s decisions. It should then be treated as conceptually
di¤erent from a short position in equity, which is not simply a negative
holding of equity. To begin with, we rule out altogether the possibility of
short sales and assume that agents can not short-sell the …rm equity:
i
0; 8i
(4.15)
The problem of agent i is then:
max i ui xi0 + Eui xi
(4.16)
xi0 ;xi ;z i ;
subject to (4.15) and
xi0 = ! i0 + [ k + Q]
xi (s) = ! i (s) + f (k; s)
i
0
i
Q
i
q zi
+ A(s)z i ; 8s 2 S
(4.17)
(4.18)
Let xi0 ; xi ; z i ; i denote the solution to this problem.
In equilibrium, the following market clearing conditions must hold, for
the consumption good:14
X
X
xi0 + k
! i0
i
X
i
i
x (s)
i
X
! i (s) + f (k; s); 8s 2 S
X
zi = 0
(4.19)
i
(4.20)
i
or, equivalently, for the assets:
i
X
= 1
i
In addition, the equity price map faced by …rms must satisfy the following
consistency condition:
14
We state here the conditions for the case of symmetric equilibria, where all …rms take
the same production and …nancing decision, so that only one type of equity is available
for trade to consumers. They can however be easily extended to the case of asymmetric
equilibria as, for instance, in the example of Section ??.
70
CHAPTER 4 TWO-PERIOD ECONOMIES
i) Q(k ) = Q;
This condition requires that, at equilibrium, the price of equity conjectured by …rms coincides with the price of equity, faced by consumers in the
market: …rms’conjectures are “correct”in equilibrium.
We also restrict out of equilibrium conjectures by …rms, requiring that
they satisfy:
ii) Q(k) = maxi E [M RS i f (k)], 8k, where M RS i denotes the marginal rate
of substitution between consumption at date 0 and at date 1 in state
s for consumer i; evaluated at his equilibrium consumption allocation
(xi0 ; xi ).
Condition ii) says that for any k (not just at equilibrium!) the value of
the equity price map Q(k) equals the highest marginal valuation - across all
consumers in the economy - of the cash ‡ow associated to k. The consumers’
i
marginal rates of substitutions M RS (s) used to determine the market valuation of the future cash ‡ow of a …rm are taken as given, una¤ected by
the …rm’s choice of k. This is the sense in which, in our economy, …rms are
competitive: each …rm is “small”relative to the mass of consumers and each
consumers holds a negligible amount of shares of the …rm.
To better understand the meaning of condition ii), note that the consumers with the highest marginal valuation for the …rm’s cash ‡ow when
the …rm chooses k are those willing to pay the most for the …rm’s equity in
that case and the only ones willing to buy equity - at the margin - when
its price satis…es ii). Given i) such property is clearly satis…ed for the …rms’
equilibrium choice k . Condition ii) requires that the same is true for any
other possible choice k: the value attributed to equity equals the maximum
any consumer is willing to pay for it. Note that this would be the equilibrium
price of equity of a …rm who were to “deviate” from the equilibrium choice
and choose k instead: the supply of equity with cash ‡ow corresponding to
k is negligible and, at such price, so is its demand.
In this sense, we can say that condition ii) imposes a consistency condition on the out of equilibrium values of the equity price map; that is, it
corresponds to a "re…nement" of the equilibrium map, somewhat analogous
to bacward induction. Equivalently, when price conjectures satisfy this condition, the model is equivalent to one where markets for all the possible types
4.2 FINANCIAL MARKET ECONOMIES
71
of equity (that is, equity of …rms with all possible values of k) are open, available for trade to consumers and, in equilibrium all such markets - except the
one corresponding to k - clear at zero trade.15
It readily follows from the consumers’…rst order conditions that in equilibrium the price of equity and of the …nancial assets satisfy:
Q = max E M RS i
i
q = E M RS i
f (k )
(4.21)
A
The de…nition of competitive equilibrium is stated for simplicity for the
case of symmetric equilibria, where all …rms choose the same production plan.
When the equity price map satis…es the consistency conditions i) and ii) the
…rms’choice problem is not convex. Asymmetric equilibria might therefore
exist, in which di¤erent …rms choose di¤erent production plans. The proof
of existence of equilibria indeed requires that we allow for such asymmetric
equilibria, so as to exploit the presence of a continuum of …rms of the same
type to convexify …rms’choice problem. A standard argument allows then
to show that …rms’ aggregate supply is convex valued and hence that the
existence of (possibly asymmetric) competitive equilibria holds.
Proposition 70 A competitive equilibrium always exist.
Objective function of the …rm
Starting with the initial contributions of Diamond (1967), Dreze (1974),
Grossman-Hart (1979), and Du¢ e-Shafer (1986), a large literature has dealt
with the question of what is the appropriate objective function of the …rm
when markets are incomplete.The issue arises because, as mentioned above,
…rms’production decisions may a¤ect the set of insurance possibilities available to consumers by trading in the asset markets.
If agents are allowed in…nite short sales of the equity of …rms, as in the
standard incomplete market model, a small …rm will possibly have a large
e¤ect on the economy by choosing a production plan with cash ‡ows which,
when traded as equity, change the asset span. It is clear that the price
15
An analogous speci…cation of the price conjecture has been earlier considered by
Makowski (1980) and Makowski-Ostroy (1987) in a competitive equilibrium model with
di¤erentiated products, and by Allen-Gale (1991) and Pesendorfer (1995) in models of
…nancial innovation.
72
CHAPTER 4 TWO-PERIOD ECONOMIES
taking assumption appears hard to justify in this context, since changes in
the …rm’s production plan have non-negligible e¤ects on allocations and hence
equilibrium prices. The incomplete market literature has struggled with this
issue, trying to maintain a competitive equilibrium notion in an economic
environment in which …rms are potentially large.
In the environment considered in these notes, this problem is avoided
by assuming that consumers face a constraint preventing short sales, (4.15),
which guarantees that each …rm’s production plan has instead a negligible
(in…nitesimal) e¤ect on the set of admissible trades and allocations available
to consumers. Evidently, for price taking behavior to be justi…ed a no short
sale constraint is more restrictive than necessary and a bound on short sales
of equity would su¢ ce; see Bisin-Gottardi-Ruta (2009).
When short sales are not allowed, the decisions of a …rm have a negligible
e¤ect on equilibrium allocations and market prices. However, each …rm’s decision has a non-negligible impact on its present and future cash ‡ows. Price
taking can not therefore mean that the price of its equity is taken as given
by a …rm, independently of its decisions. However, as argued in the previous
section, the level of the equity price associated to out-of-equilibrium values
of k is not observed in the market. It is rather conjectured by the …rm. In a
competitive environment we require such conjecture to be consistent, as required by condition ii) in the previous section. This notion of consistency of
conjectures implicitly requires that they be competitive, that is, determined
by a given pricing kernel, independent of the …rm’s decisions.16 But which
pricing kernel? Here lies the core of the problem with the de…nition of the objective function of the …rm when markets are incomplete. When markets are
incomplete, in fact, the marginal valuation of out-of-equilibrium production
plans di¤ers across di¤erent agents at equilibrium. In other words, equity
holders are not unanimous with respect to their preferred production plan
for the …rm. The problem with the de…nition of the objective function of the
…rm when markets are incomplete is therefore the problem of aggregating
equity holders’marginal valuations for out-of-equilibrium production plans.
The di¤erent equilibrium notions we …nd in the literature di¤er primarily in
the speci…cation of a consistency condition on Q (k), the price map which
the …rms adopts to aggregate across agents’marginal valuations.17
16
Independence of the kernel is guarantee by the fact that M RS i (s); for any i, is
evaluated at equilibrium.
17
A minimal consistency condition on Q (k) is clearly given by i) in the previous section,
which only requires the conjecture to be correct in correspondence to the …rm’s equilibrium
4.2 FINANCIAL MARKET ECONOMIES
73
Consider for example the consistency condition proposed by Dreze (1974):
"
#
X
i
QD (k) = E
M RS i f (k) ; 8k
(4.22)
i
Such condition requires the price conjecture for any plan k to equal the
pro rata marginal valuation of the agents who at equilibrium are the …rm’s
equity holders (that is, the agents who value the most the plan chosen by
…rms in equilibrium). It does not however require that the …rm’s equity
holders are those who value the most any possible plan of the …rm, without
contemplating the possibility of selling the …rm in the market, to allow the
new equity buyers to operate the production plan they prefer. Equivalently,
the value of equity for out of equilibrium production plans is determined using
the - possibly incorrect - conjecture that the …rms’equilibrium shareholders
will still own the …rm out of equilibrium.
Grossman-Hart (1979) propose another consistency condition and hence
a di¤erent equilibrium notion. In their case
"
#
X
i
i
QGH (k) = E
0 M RS f (k) ; 8k
i
We can interpret such notion as describing a situation where the …rm’s plan
is chosen by the initial equity holders (i.e., those with some predetermined
stock holdings at time 0) so as to maximize their welfare, again without
contemplating the possibility of selling the equity to other consumers who
value it more. Equivalently, the value of equity for out of equilibrium production plans is again derived using the conjecture belief that …rms’initial
shareholders stay in control of the …rm out of equilibrium.
Unanimity
Under the de…nition of equilibrium proposed in these notes, equity holders
unanimously support the …rm’s choice of the production and …nancial decisions which maximize its value (or pro…ts), as in (4.14). This follows from
the fact that, when the equity price map satis…es the consistency conditions
i) and ii), the model is equivalent to one where a continuum of types of equity
choice. Du¢ e-Shafer (1986) indeed only impose such condition and …nd a rather large
indeterminacy of the set of competitive equilibria.
74
CHAPTER 4 TWO-PERIOD ECONOMIES
is available for trade to consumers, corresponding to any possible choice of
k the representative …rm can make, at the price Q(k). Thus, for any possible value of k a market is open where equity with a payo¤ f (k; s) can be
traded, and in equilibrium such market clears with a zero level of trades for
the values of k not chosen by the …rms.
For any possible choice k of a …rm, the (marginal) valuation of the …rm
by an agent i is
E M RS i f (k ) ;
and it is always weakly to the market value of the …rm, given by
max E M RS i
i
f (k ) :
Proposition 71 At a competitive equilibrium, equity holders unanimously
support the production k ; that is, every agent i holding a positive initial
amount i0 of equity of the representative …rm will be made - weakly - worse
o¤ by any other choice k 0 of the …rm.
E¢ ciency
A consumption allocation (xi0 ; xi )Ii=1 is admissible if:18
1. it is feasible: there exists a production plan k such that
X
X
xi0 + k
! i0
i
X
i
i
x (s)
X
i
(4.23)
i
! i (s) + f (k; s); 8s 2 S
(4.24)
2. it is attainable with the existing asset structure: for each consumer i,
there exists a pair z i ; i such that:
xi (s) = ! i (s) + f (k; s)
i
+ A(s)z i ; 8s 2 S
(4.25)
Next we present the notion of e¢ ciency restricted by the admissibility
constraints:
18
To keep the notation simple, we state both the de…nition of competitive equilibria and
admissible allocations for the case of symmetric allocations. The analysis, including the
e¢ ciency result ,extends however to the case where asymmetric allocations are allowed are
admissible; see also the next section.
75
4.2 FINANCIAL MARKET ECONOMIES
Constrained e¢ ciency. A competitive equilibrium allocation is constrained
Pareto e¢ cient if we can not …nd another admissible allocation which
is Pareto improving.
The validity of the First Welfare Theorem with respect to such notion
can then be established by an argument essentially analogous to the one used
to establish the Pareto e¢ ciency of competitive equilibria in Arrow-Debreu
economies.
First welfare theorem. Competitive equilibria are constrained Pareto ef…cient.
Modigliani-Miller
We examine now the case where …rms take both production and …nancial
decisions, and equity and debt are the only assets they can …nance their
production with. The choice of a …rm’s capital structure is given by the
decision concerning the amount B of bonds issued. The problem of the …rm
consists in the choice of its production plan k and its …nancial structure B.
To begin with, we assume without loss of generality that all …rms’debt is risk
free. The …rm’s cash ‡ow in this context is then [f (k; s) B] and varies with
the …rm’s production and …nancing choices, k; B. Equity price conjectures
have the form Q(k; B), while the price of the (risk free) bond is independent
of (k; B); we denote it p. The …rm’s problem is then:
(4.26)
max k + Q(k; B) + p B
k;B
The consumption side of the economy is the same as in the previous
section, except that now agents can also trade the bond. Let bi denote the
bond portfolio of agent i; and let continue to impose no-short sales contraints:
i
b
i
0
0; 8i:
Proceeding as in the previous section, at equilibrium we shall require that
Q(k) = max E M RS i
i
p = max E M RS i
i
[f (k)
B] ; 8k;
76
CHAPTER 4 TWO-PERIOD ECONOMIES
where M RS i denotes the marginal rate of substitution between consumption
at date 0 and at date 1 in state s for consumer i; evaluated at his equilibrium
consumption allocation (xi0 ; xi ). Suppose now that …nancial markets are
complete, that is rank(A) = S: At equilibrium then M RS i = M RS , 8i.
Therefore, in this case
Q(k; B) + p B = E [M RS
f (k)] ; 8k;
and the value of the …rm, Q(k; B) + p B; is independent of B: This proves
the celebrated
Modigliani-Miller theorem. If …nancial markets are complete the …nancing decision of the …rm, B; is indeterminate.
It should be clear that when …nancial markets are not complete and agents
are restricted by no-short sales constraints, the Modigliani-Miller theorem
does not quite necessarily hold.
Chapter 5
Asymmetric information
Do competitive insurance markets function orderly in the presence of moral
hazard and adverse selection? What are the properties of allocations attainable as competitive equilibria of such economies? And in particular, are
competitive equilibria incentive e¢ cient?
For such economies the interaction between the private information dimension (e.g., the unobservable action in the moral hazard case, the unobservable type in the adverse selection case) and the observability of agents’
trades plays a crucial role, since trades have typically informational content over the agents’ private information. In particular, to decentralize incentive e¢ cient Pareto optimal allocations the availability of fully exclusive
contracts, i.e., of contracts whose terms (price and payo¤) depend on the
transactions in all other markets of the agent trading the contract, is generally required. The implementation of these contracts imposes typically the
very strong informational requirement that all trades of an agent need to be
observed.
The fundamental contribution on competitive markets for insurance contracts is Prescott and Townsend (1984). They analyze Walrasian equilibria
of economies with moral hazard and with adverse selection when exclusive
contracts are enforceable, that is, when trades are fully observable.1 ”In these
1
The standard strategic analysis of competition in insurance economies, due to
Rothschild-Stiglitz (1976), considers the Nash equilibria of a game in which insurance
companies simultaneously choose the contracts they issue, and the competitive aspect of
the market is captured by allowing the free entry of insurance companies. Such equilibrium concept does not perform too well: equilibria in pure strategies do not exist for
robust examples (Rothschild-Stiglitz (1976)), while equilibria in mixed strategies exist
77
78
CHAPTER 5 ASYMMETRIC INFORMATION
notes we concentrate on the simpler case of moral hazard. We refer to BisinGottardi (2006) for the case of adverse selection.
Remark 72 In the context of these economies M. Harris and R. Townsend
(1981) prove a version of the Revelation principle. Formulate the statement
and sketch the proof.
5.1
The moral hazard economy
Agents live two periods, t = 0; 1; and consume a single consumption good
only in period 1. Uncertainty is purely idiosyncratic and all agents are exante identical. In particular, each agent faces a (date 1) endowment which
is an identically and independently distributed random variable ! on a …nite
support S.2
Moral hazard (hidden action) is captured by the assumption that the
probability distribution of the period 1 endowment that each agent faces
depends from the value taken by a variable e 2 E, an unobservable level of
e¤ort which is chosen by the agent.
P Let s (e) be the probability of the realization ! s given e. Obviously
s2S s (e) = 1; for any e 2 E; a compact, convex set: By the Law of
Large Numbers, s (e) is also the fraction of agents who have chosen e¤ort e
for which state s is realized. Agents’ preferences are represented by a von
Neumann-Morgenstern utility function of the following form:
X
v(e)
s (e)u(xs )
s2S
where v(e) denotes the disutility of e¤ort e. We assume the following
regularity conditions:
The utility function u(x) is strictly increasing, strictly concave, twice
continuously di¤erentiable, and limx!0 u0 (x) = 1: The cost function v(e)
(Dasgupta-Maskin (1986)) but, in this set-up, are of di¢ cult interpretation. Even when
equilibria in pure strategies do exist, it is not clear that the way the game is modelled is
appropriate for such markets, since it does not allow for dynamic reactions to new contract
o¤ers (Wilson (1977) and Riley (1979); see also Maskin-Tirole (1992)). Moreover, once
sequences of moves are allowed, equilibria are not robust to ‘minor’perturbations of the
extensive form of the game (Hellwig (1987)).
2
Measurability issues arise in probability spaces with a continuum of indipendent random variables. We adopt the usual abuse of the Law of Large Numbers.
79
5.1 THE MORAL HAZARD ECONOMY
is strictly increasing, strictly convex, twice continuously di¤erentiable, and
supe2E v 0 (e) = 1:
Essentially without loss of generality, let the state space S be ordered so
that ! s > ! s 1 ; for all s = 2; :::; S: We then impose the following standard
restriction.
Single-crossing property. The odds ratio
e, for any s = 2; :::; S:
5.1.1
s (e)
s 1 (e)
is strictly increasing in
The Symmetric information benchmark
Consider now the benchmark case of symmetric information, in which e is
commonly observed. An allocation (x; e) 2 RS+ E of consumption and e¤ort
is optimal under symmetric information if it solves:
X
max
v(e);
(5.1)
s (e)u(xs )
x;e
s.t.
s2S
X
s (e)(xs
!s) = 0
s2S
Let qs (e) denote the (linear) price of consumption in state s for agents
who chose e¤ort e. By allowing the prices of the securities whose payo¤ is
contingent on the idiosyncratic uncertainty to depend on e, we e¤ectively
are introducing price conjectures: we read qs (e) as the price of consumption
contingent to state s if the agent chooses e¤ort e, for any e 2 E; not just
at the equilibrium e. This is the same problem we found in production
economies with incomplete markets, where the price faced by the …rm was a
whole map Q(k); interpreted as a price conjecture.
At a competitive equilibrium each agent solves
X
max
v(e);
(5.2)
s (e)u(xs )
x;e
s.t.
s2S
X
qs (e)(xs
!s) = 0
s (e)(xs
!s) = 0
s2S
and markets clear
X
s2S
(5.3)
80
CHAPTER 5 ASYMMETRIC INFORMATION
We impose the following consistency condition on the price conjecture:
qs (e) =
s (e)u
0
(xs ):
Note that, at equilibrium agents fully insure and hence u0 (xs ) = u0 (x ):
Modulo a normalization the consistency condition therefore takes the form:
qs (e) =
s (e):
Under the consistency condition it is now straightforward to prove the
First and Second Welfare theorems for this economy under symmetric information.
5.1.2
The moral hazard economy
Consider now the case of asymmetric information, in which his choice of
e¤ort e is private information of each agent. In this context, an allocation
(x; e) 2 RS+ E of consumption and e¤ort is incentive constrained optimal if
it solves:
X
max
v(e);
(5.4)
s (e)u(xs )
x;e
s.t.
s2S
X
s (e)(xs
!s) = 0
s2S
and
e 2 e(x) = arg max
X
s (e)u(xs )
v(e); given x
s2S
The last constraint, called incentive constraint, requires that the allocation
(x; e) must be such that the agent prefers (x; e) to any other allocation (x; e0 ),
for any e; e0 2 E.
At a Prescott-Townsend competitive equilibrium prices cannot have the
form qs (e), as the agent’s choice for e is not observed. What is observed,
however is the consumption allocation x demanded by the agent in the market. (Note that the exclusivity assumption guarantees that this is the case,
that x is observable). Price conjectures can then be de…ned as a function of
x as
qs (x) = qs (e(x)):
81
5.1 THE MORAL HAZARD ECONOMY
At a competitive equilibrium then each agent solves
max
x;e
s.t.
X
s (e)u(xs )
v(e);
(5.5)
s2S
X
qs (e(x))(xs
!s) = 0
s2S
and markets clear
X
s (e)(xs
(5.6)
!s) = 0
s2S
At an equilibrium (x ; e ), we impose the following consistency condition
on the price conjecture:
qs (e(x)) =
s (e(x))
The First and Second Welfare theorems, in its incentive constrained versions,
hold straightforwardly for the moral hazard economy.
Remark 73 An equivalent notion of equilibrium is possible, which is equivalent to the one just proposed. Suppose at a competitive equilibrium each agent
solves
X
max
v(e);
(5.7)
s (e)u(xs )
x;e
s2S
s.t.
X
X
qs (e)(xs
! s ) = 0; and
s2S
s (e)u(xs )
v(e)
s2S
and markets clear
X
0
s (e )u(xs )
s2S
X
s (e)(xs
!s) = 0
v(e0 ); for any e0 2 E
(5.8)
s2S
Prices depend on e¤ort e; q(e): As e¤ort is not observed, this is to be interpreted that prices depend on the (implicit) declaration of e¤ort on the part of
the agent; for instance di¤erent markets could be present with di¤erent prices
82
CHAPTER 5 ASYMMETRIC INFORMATION
- and by choosing one of these where to trade each agent implicitly declares
his e¤ort choice. The condition
X
X
0
(e)u(x
)
v(e)
v(e0 ); for any e0 2 E
s
s
s (e )u(xs )
s2S
s2S
ther requires that in the market with prices q(e) only incentive compatible allocations are o¤ered, that is, only allocations which induce the agent to choose
e¤ort e: It is straightforward to see that this equilibrium notion is equivalent
to the one with rational price conjectures we have proposed. The interpretation is di¤erent however: with rational conjectures it is the conjectures
on prices of non incentive compatible allocations which are restricted, while
with the equilibrium notion in this remark it is tradable allocations which are
restricted to exclude non incentive compatible ones.
As we noted, the exclusivity assumption guarantees that x is observable.
Suppose it is not (this is the non-exclusivity case). The next problem deals
with this case in a simple but instructive example.
Problem 74 Characterize Incentive constrained optima and competitive equilibria (with the consistency condition on price maps) of the moral hazard
economy specialized so that
S = 1; 2; e = h; l; v(h) > v(l);
1 (h)
>
1 (l);
!1 > !2:
Set-up and characterize equilibria of this economy under the constraint that
prices are linear (independent of x):
qs (e(x)) = qs :
Are equilibrium allocations in this case Incentive constraint e¢ cient?
5.1.3
References
Bennardo, A. and P.A. Chiappori (2003): ‘Bertrand and Walras Equilibria
under Moral Hazard’, Journal of Political Economy, 111(4), 785-817.
Bisin, A. and P. Gottardi (2006): ‘E¢ cient Competitive Equilibria with
Adverse Selection,’Journal of Political Economy, 114(3), 485-516.
5.1 THE MORAL HAZARD ECONOMY
83
Bisin, A. and P. Gottardi (1999): ‘Competitive Equilibria with Asymmetric
Information’, Journal of Economic Theory, 87, 1-48.
Dubey, P., J. Geanakoplos (2004): ‘Competitive Pooling: Rothschild-Stiglitz
Re-considered,’Quarterly Journal of Economics.
Gale, D. (1992): ‘A Walrasian Theory of Markets with Adverse Selection’,
Review of Economic Studies, 59, 229-55.
Gale, D. (1996): ‘Equilibria and Pareto Optima of Markets with Adverse
Selection’, Economic Theory, 7, 207-36.
Grossman, S. and O. Hart (1983): ‘An Analysis of the Principal-Agent
Problem,’Econometrica, Vol. 51, No. 1.
Hellwig, M. (1987): ‘Some Recent Developments in the Theory of Competition in Markets with Adverse Selection’, European Economic Review,
31, 319-325.
Prescott, E.C. and R. Townsend (1984): ‘Pareto Optima and Competitive
Equilibria with Adverse Selection and Moral Hazard’, Econometrica,
52, 21-45; and extended working paper version dated 1982.
Prescott, E.C. and R. Townsend (1984b): ‘General Competitive Analysis
in an Economy with Private Information’, International Economic Review, 25, 1-20.
Rothschild, M. and J. Stiglitz (1976): ‘Equilibrium in Competitive Insurance Markets: An Essay in the Economics of Imperfect Information’,
Quarterly Journal of Economics, 80, 629-49.
Wilson, C. (1977): ‘A Model of Insurance Markets with Incomplete Information’, Journal of Economic Theory, 16, 167-207.
Chapter 6
In…nite-horizon economies
Note that the commodity space is in…nite dimensional. Good discussion
of the spaces we typically study is in Lucas-Stokey with Prescott (1989),
Recursive economic dynamics, Harvard Univ. Press. Existence and …rst
welfare theorems are straightforwardly extended as long as the number of
agents is …nite. See Zame, "Competitive equilibria in production economies
with an in…nite dimensional commodity space," Econometrica, 55(5), 10751108.
6.1
Asset pricing
Assume a representative-agent economy with one good. Let time be indexed
by t = 0; 1; 2; :::: Uncertainty is captured by a probability space represented
by a tree. Suppose that there is no uncertainty at time 0 and call s0 the
root of the tree. Without much loos of generality, we assume that each node
has a constant number of successors, S. At generic node at time t is called
st 2 S t . Note that the dimensionality of S t increases exponentially with time
t (abusing notation it is in fact S t ).
When a careful speci…cation of the underlying state space process is not
needed, we will revert to the usual notation in terms of stochastic processes.
Let x := fxt g1
t=0 denote a stochastic process for an agent’s consumption,
t
where xt : S ! R+ is a random variable on the underlying probability
space, for each t. Similarly, let ! := f! t g1
t=0 be a stochastic processes describing an agent’s endowments. Let 0 < < 1 denote the discount factor.
85
86
6.1.1
CHAPTER 6 INFINITE-HORIZON ECONOMIES
Arrow-Debreu economy
Suppose that at time zero, the agent can trade in contingent commodities.
t
Let p := fpt g1
!
t=0 denote the stochastic process for prices, where pt : S
R+ , for each t.
Then ((x i )i ; p ) is an Arrow-Debreu Equilibrium if
x i 2 arg maxfu(x0 ) + E0 [
i. given p ; s:t:
P1
t=0
ii. and
P
i
x
i
pt (xt
P1
t=1
t
u(xt )]g
!t) = 0
! i = 0:
The notation does not make explicit that the agent chooses at time 0 a
whole sequence of time and state contingent consumption allocations, that
is, the whole sequence of x(st ) for any st 2 S t and any t 0.
6.1.2
Financial markets economy
Suppose that throughout the uncertainty tree, there are J assets. We shall
allow assets to be long-lived. In fact we shall assume they are and let the
reader take care of the straightforward extension in which some of the assets
pay o¤ only in a …nite set of future times. Let z := fzt g1
t=1 denote the sequence of portfolios of the representative agent, where zt : St ! RJ . Assets’
payo¤s are captured at each time t by the S J matrix At . Furthermore,
capital gains are qt qt 1 , and returns are Rt = Aqtt+q1 t .
In a …nancial market economy agents do not trade at time 0 only. They
in fact, at each node st receive endowments and payo¤s from the portfolios
they carry from the previous node, they re-balance their portfolios and choose
state contingent consumption allocations for any of the successor nodes of st ,
which we denote st+1 j st .
De…nition 75 f(x i ; z i )i ; q g is a Financial Markets Equilibrium if
87
6.1 ASSET PRICING
i. given q ; at each time t
0
P
(x i ; z i ) 2 arg maxfu(xt ) + Et [ 1=1
j
u(xt+
jst )]g
s:t:
xt+ + qt+ zt+ = ! t+ + At+ zt+
for
= 0; 1; 2; :::; with z
1;
=0
1
some no-Ponzi scheme condition
De…nition 76 ii.
6.1.3
P
i
x
i
! i = 0 and
P
i
z i = 0:
Conditional asset pricing
From the FOC of the agent’s problem, we obtain
qt = Et
u0 (xt+1 )
At+1
u0 (xt )
= Et (mt+1 At+1 )
(6.1)
or,
1 = Et
u0 (xt+1 )
Rt+1 :
u0 (xt )
(6.2)
Example 1. Consider a stock. Its payo¤ at any node can be seen as the
dividend plus the capital gain, that is,
Rt+1 =
qt+1 + dt+1
;
qt
for some exogenously given dividend stream d. By plugging this payo¤ into
equation (6.1), we obtain the price of the stock at t.
Example 2. For a call option on the stock, with strike price k at some
future period T > t, we can de…ne
At = 0; t < T; and AT = maxfqT
De…ne now
mt;T =
T t 0
u (xT )
t)
u0 (x
k; 0g
88
CHAPTER 6 INFINITE-HORIZON ECONOMIES
and observe that the price of the option is given by
qt = Et (mt;T maxfqT
k; 0g) ;
Note how the conditioning information drives the price of the option: the
price changes with time, as information is revealed by approaching the execution period T .
Example 3. The risk-free rate is know at time t and therefore, equation
(6.2) applied to a 1-period bond yields
1
f
Rt+1
= Et
u0 (xt+1 )
u0 (xt )
:
Once again, note that the formula involves the conditional expectation at
time t. Therefore, while the return of a risk free 1-period bond paying at
t+1 is known at time t, the return of a risk free 1-period bond paying at t+2
is not known at time t. [.... relationship between 1 and period bonds....
from the red Sargent book]
Conditional versions of the beta representation hold in this economy:
Et (Rt+1 )
Covt (mt+1 ; Rt+1 )
=
Et (mt+1 )
Covt (mt+1 ; Rt+1 )
V art (mt+1 )
=
V art (mt+1 )
Et (mt+1 )
f
Rt+1
=
(6.3)
=:
t t:
Unconditional moment restrictions
Recall that our basic pricing equation is a conditional expectation:
qt = Et (mt+1 At+1 );
(6.4)
In empirical work, it is convenient to test for unconditional moment restrictions.1 However, taking unconditional expectations of the previous equation
1
Otherwise, speci…c parametric assumptions need be imposed on the stochastic
process of the economy underlying the asset pricing equation. For instance this is
the route taken by the literature on autoregressive conditionally heteroschedastic (that
is, ARCH - and then ARCH-M, GARCH,...) models. See for instance the work by
??rengle/http://pages.stern.nyu.edu/ rengle/‘=
˜ 13 .
89
6.1 ASSET PRICING
implies in principle a much weaker statement about asset prices than equation (6.4):
E(qt ) = E(mt+1 At+1 );
(6.5)
where we have invoked the law of iterated expectations. It should be clear
that equation (6.4) implies but it is not implied by (6.5).
The theorem in this section will tell us that actually there is a theoretical
way to test for our conditional moment condition by making a series of tests
of unconditional moment conditions.
De…ne a stochastic process fit g1
t=0 to be conformable if for each t, it
belongs to the time-t information set of the agent. It then follows that for
any such process, we can write
it qt = Et (mt+1 it At+1 )
and, by taking unconditional expectations,
E(it qt ) = E(mt+1 it At+1 ):
This fact is important because for each conformable process, we obtain an
additional testable implication that only involves unconditional moments.
Obviously, all these implications are necessary conditions for our basic pricing
equation to hold. The following result states that if we could test these
unconditional restrictions for all possible conformable processes then it would
also be su¢ cient. We state it without proof.
Theorem 77 If E(xt+1 it ) = 0 for all it conformable then Et (xt+1 ) = 0:
By de…ning xt+1 = mt+1 At+1
6.1.4
qt ; the theorem yields the desired result.
Predictability or returns
Recall the asset-pricing equation for stocks:
qt = Et (mt+1 (qt+1 + dt+1 )) :
It is sometimes argued that returns are predictable unless stock prices to
follow a random walk. (Where in turn predictability is interpreted as a
property of e¢ cient market hypothesis, a fancy name for the asset pricing
theory exposed in these notes). Is it so? No, unless strong extra assumptions
90
CHAPTER 6 INFINITE-HORIZON ECONOMIES
are imposed Assume that no dividends are paid and agents are risk neutral;
then, for values of close to 1 (realistic for short time periods), we have
qt = Et (qt+1 ):
That is, the stochastic process for stock prices is in fact a martingale. Next,
for any f"t g such that Et ("t+1 ) = 0 at all t, we can rewrite the previous
equation as
qt+1 = qt+1 + "t+1 :
This process is a random walk when vart ("t+1 ) = is constant over time.
A more important observation is the fact that marginal utilities times
asset prices (a risk adjusted measure of asset prices) follow approximately
a martingale (a weaker notion of lack of predictability). Again under no
dividends,
u0 (ct )qt = Et (u0 (ct+1 )(qt+1 + dt+1 ));
which is a supermartingale and approximately a martingale for
6.1.5
close to 1.
Fundamentals-driven asset prices
For assets whose payo¤ is made of a dividend and a capital gain, FOC dictate
qt = Et (mt+1 (qt+1 + dt+1 )) ;
where
u0 (ct+1 )
:
u0 (ct )
By iterating forward and making use of the Law of Iterated Expectations,
!
!
T
T
X
X
qt = lim Et
mt;t+j dt+j + lim Et
mt;t+j qt+j ;
mt+1 =
T !1
T !1
j=1
j=1
As we shall see, in…nite horizon models (with in…nitely lived agents) usually satisfy the no-bubbles condition, or
!
T
X
lim Et
mt;t+j qt+j = 0:
T !1
j=1
In that case, we say that asset prices are fully pinned down by fundamentals
since
!
1
X
qt = Et
mt;t+j dt+j :
j=1
91
6.2 BUBBLES
Conditional factor models and the conditional CAPM
[...from Cochrane...]
Frictions
He-Modest
Luttmer
6.2
Bubbles
1
S t be the set of nodes of the tree. Recall we denoted with s0
Let2 N = Xt=0
denote the root of the tree and with st an arbitrary node of the tree at time
t. Use st+ jst to indicate that st+ is some successor of st , for > 0:
At each node, there are J securities traded. Also, it is important that
the notation include Overlapping Generation economies. We need therefore
to account for …nitely lived agents. Let I(st ) be the set of agents which are
active at node st . Let N i be the subset of nodes of the tree at which agent i
i
is allowed to trade. Also, denote by N the terminal nodes for agent i.
The following assumptions will not be relaxed.
1. If an agent i is alive at some non-terminal node st , she is also alive at
i
all the immediate successor nodes. That is, st N i nN =) fst+1 N :
st+1 jst g N i :
2. The economy is connected across time and states: at any state there is
some agent alive and non-terminal. Formally,
i
8st ; 9i : st 2 N i nN :
Assets are long lived. Let q : N ! RJ be the mapping de…ning the
vector of security prices at each node st . Similarly, let d : N ! RJ denote
the vector-valued mapping that de…nes the dividends (in units of numeraire)
that are paid by the assets at node st . We assume that d(st ) 0 for any st :
Each of the households alive at s0 enters the markets with an initial
endowment of securities z!i
0: Therefore, the initial net supply of assets is
2
In this section we follow closely Santos-Woodford (1997), Econometrica.
92
CHAPTER 6 INFINITE-HORIZON ECONOMIES
given by
z! =
X
z!i .
i2I(s0 )
As assets are long lived, a supply z! of assets is available at any st :
At each node st , each households in I(st ) has an endowment of numeraire
good of ! i (st )
0. We shall assume that the economy has a well-de…ned
aggregate endowment
!(st ) =
X
! i (st )
0
i2I(st )
at each node st . This is the case, e.g., if I(st ) is …nite, for any st : Taking into
account the dividends paid by securities in units of good, the aggregate good
supply in the economy is given by
!
e (st ) = !(st ) + d(st )z!
0:
The utility function of any agent i is written
i
0
U (x) = u (x(s ) +
1
X
t=1
t
X
prob(st ) ui (x(st )):
st js0
De…ne the 1-period payo¤ vector (in units of numeraire) at node st by
A(st ) = d(st ) + q(st ):
Any agent i faces a sequence of budget constraints, one for every node st 2
N i : Let xi (st ) denote the consumption of agent i at node st 2 N i and z i (st )
his a J-dimensional portfolio vector at st 2 N i : The agent budget constraint
at any st js0 2 N i is:
xi (st ) + q(st )z i (st )
! i (st ) + A(st )z i (st
with
xi (st )
q(st )z i (st )
0
B i (st );
1);
93
6.2 BUBBLES
where B i : N ! R+ indicates an exogenous and non-negative household
speci…c borrowing limit at each node. We assume households take the borrowing limits as given, just as they take security prices as given. At s0 ; if
s0 2 N i , the budget constraint is:
xi (s0 ) + q(s0 )z i (s0 )
! i (s0 ) + q(s0 )z!i ;
At equilibrium, markets clear: that is, at each st 2 N;
X
xi (st ) = !
e (st )
i2I(st )
X
z i (st ) = z!
i2I(st )
Given the price process q, we say that no arbitrage opportunities exist at
st if there is no z 2 RJ such that
A(st+1 )z
q(st )z
0; for all st+1 jst ;
0;
with at least one strict inequality.
Lemma 78 When q satis…es the no-arbitrage condition at st 2 N; there
exists a set of state prices f (st+1 jst )g with (st+1 jst ) > 0 for all st+1 jst ,
such that the vector of asset prices at st 2 N can be written as
X
q(st ) =
(st+1 jst )A(st+1 jst ):
(6.6)
st+1 jst
Proof. As usual, proof follows from applying an appropriate separation
theorem.
Applying the Lemma at any st 2 N we can construct a stochastic process
S
: N ! R++
recursively as follows:
(st+2 jst ) = (st+2 jst+1 ) (st+1 jst ); for t = 0; 1; :::
Let (st ) denote the set of such processes for the subtree with root st .
Only under complete markets is the set (st ) a singleton.
As a remark, note that …nancial market completeness is an endogenous
property in this economy since one-period payo¤s A contain asset prices.
94
CHAPTER 6 INFINITE-HORIZON ECONOMIES
Therefore, the rank property which de…nes completeness can only be assessed
at each given equilibrium.
For any state-price process 2 (st ); de…ne the J vector of fundamental
values for the securities traded at node st by
t
f (s ; ) =
1
X
X
(sT jst )d(sT jst ):
T =t+1 sT jst
(6.7)
Observe that the fundamental value of a security is de…ned with reference
to a particular state-price process, however the following properties it displays
are true regardless of the state prices chosen.
Proposition 79 At each st 2 N , f (st ; ) is well-de…ned for any
and satis…es
0 f (st ; ) q(st ):
2
(st )
Proof. First of all, 0
f (st ; ) follows directly from non-negativity of
, the dividend process d, and the price process q. We therefore turn to
f (st ; ) q(st ): From equation (6.6), we have
X
X
q(st ) =
(st+1 jst )d(st+1 jst ) +
(st+1 jst )q(st+1 )
st+1 jst
st+1 jst
and, iterating on this equation we obtain
t
q(s ) =
Tb
X
X
T =t+1 sT jst
(sT jst )d(sT jst ) +
X
sTb jst
b
b
(sT jst )q(sT )
b
for any Tb > t: Since by construction, q(sT ) is non-negative and 2 (st )
is a positive state-price vector, the second term on the right-hand-side is
non-negative. So,
t
q(s )
Tb
X
X
T =t+1 sT jst
and
t
q(s )
(sT )d(sT jst ); for any Tb > t:
1
X
X
T =t+1 sT jst
(sT jst )d(sT jst )
95
6.2 BUBBLES
We can correspondingly de…ne the vector of asset pricing bubbles as
(st ; ) = q(st )
for any
2
f (st ; );
(6.8)
(st ) for the J securities. It follows from the proposition that
(st ; )
0
q(st );
for any 2 (st ): This corollary is known as the impossibility of negative
bubbles result. Substituting (6.8) and (6.7) into (6.6) yields
X
(st ; ) =
(st+1 jst ) (st+1 ):
st+1 jst
This is known as the martingale property of bubbles: if there exists a (nonzero)
price bubble on any security at date t, there must exist a bubble as well on
the security at date T , with positive probability, at every date T > t. Furthermore, if there exists a bubble on any security at node st , then there must
have existed a bubble as well on some security at every predecessor of the
node st .
Remark 80 Suppose our economy is deterministic. Some of the earliest
analyses on bubbles dealt with this case, e.g., Samuelson’s and Bewley’s models of money. In this case, let t denote the bubble at time t: The martingale
property of bubbles implies that
t
t
=
=
t t+1
i
M RSt+1;t
(xit )
We shall get back to this example when we’ll study the Overlapping Generation economy at the end of this section.
What does this imply for securities with …nite maturity? Your answer
must depend on how you de…ne securities with …nite maturity in this context.
In an economy with incomplete markets, the fundamental value need not
be the same for all state-price processes consistent with no arbitrage. But
even in this case, we can de…ne the range of variation in the fundamental
value, given the restrictions imposed by no-arbitrage.
96
CHAPTER 6 INFINITE-HORIZON ECONOMIES
Let x : N ! R+ denote a non-negative stream of consumption goods.
For any st , pick any 2 (st ) and de…ne the present value at st of x with
respect to by
1
X
X
t
Vx (s ; ) =
(sT jst )x(sT ).
T =t+1 sT jst
Since this present value depends on the stochastic discount factor , let
us now de…ne the bounds for the present value at st of dividends x.
For any st , de…ne
x (s
t
x (s
t
) =
) =
inf fVx (st ; )g
2 (st )
sup fVx (st ; )g:
2 (st )
A few remarks follow from these de…nitions. First note that these de…nitions are conditional on a given price process q since the set of no-arbitrage
stochastic discount factors are de…ned with respect to q. Next observe that,
for any security with dividend process dj ,
dj (s
t
)
f j (st ; )
dj (s
t
)
q j (st ); for all
2
(st );
furthermore, dj (st ) < q j implies that there is a pricing bubble for the security with payo¤ process xj .
For any a non-negative stream of consumption goods x : N ! R+ it
can be shown that xj (st ) = sup q (st ) z (st ) where the sup is taken over all
plans fz (sr jst ) ; r tg such that
x sr jsr
1
+ d sr jsr
1
z sr
1
q (sr ) z (sr ) ; for any sr jst and any r
t
and
9T such that q (sr ) z (sr )
0; for any sr jst with r
T:
In words, xj (st ) is the least upper bound for the amount that an agent
can borrow at node st if the endowment of the agent is x (sr jsr 1 ) for any
sr jst and any r t; under the constraint that the agent has to maintain a
non-negative wealth after some time T:
Recall that to rule out Ponzi schemes when agents are in…nitely lived, a
lower bound on individual wealth is needed. Let us de…ne a particular type
of borrowing limit.
97
6.2 BUBBLES
An agent’s borrowing ability is only limited by her ability to repay out of
her own future endowment if
B i (st ) =
!
~ i (s
t
(6.9)
);
i
for each st N i nN .
It can be shown that these borrowing limits never bind at any …nite
date (see Magill-Quinzii, Econometrica, 94), but rather only constrain the
asymptotic behavior of a household’s debt.3
This constraint is to the e¤ect that each agent can e¤ectively trade his
own future endowment stream any node st : An important consequence of
this speci…cation is the following.
Proposition 81 Suppose that agent i has borrowing limits of the form (6.9).
Then the existence of a solution to the agent’s problem for given prices q
implies that !~ i (st ) < 1; at each st 2 N i ; so that the borrowing limit is
…nite at each node.
This is because, if agent i can borrow o¤ of the value of !
~ i ; this value must
be …nite at equilibrium prices for the agent’s problem to be well-de…ned. If
it is …nite at equilibrium prices, it must be …nite at all implied no-arbitrage
state prices. If
dj (s
t
)
f j (st ; )
dj (s
t
)
q j (st ); for all
2
(st );
t
t
in fact, then x (st )
x (s ) and x (s ) is less than or equal to the equilibrium
price of the portfolio strategy supporting x:
We can now prove the following fundamental lemma.
Lemma 82 Consider an equilibrium fx i ; z i ; qg. Suppose that the (supremum of the) value of aggregate wealth is …nite, i.e., !~ (st ) < 1. Suppose
also that there exists a bubble on some security in positive net supply at st
so that (st )z(st ) > 0. Then, 8K > 0; there exists a time T and sT jst such
that
(sT )z(sT ) > K !
e (sT ):
3
They are equivalent to requiring that the consumption process lies in the space of
measurable bounded sequences. In the case of …nitely lived agents, these borrowing limits
are equivalent to imposing no-borrowing at all nonterminal nodes.
98
CHAPTER 6 INFINITE-HORIZON ECONOMIES
Proof. The martingale property of pricing bubbles implies,
X
(st )z(st ) =
(sT jst ) (sT )z(sT )
sT jst
P
and hence sT jst (sT jst ) (sT )z(sT ) is constant for any T > t: On the other
P
hand, sT jst (sT jst )e
! (sT ) must converge to 0 in T ! 1 to guarantee that
P
P
t
T t
! (sT ) < 1:
!
~ (s ) =
T
sT jst (s js )e
That is, there is a positive probability that the total size of the bubble
on the securities becomes an arbitrarily large multiple of the value of the
aggregate supply of goods in the economy. A contradiction. The proof
exploits crucially the martingale property of bubbles. It follows from this
result that some agent must accumulate, at some node sT , a wealth whose
value is larger than the value of the aggregate endowment.
We already learned that no bubbles can arise in securities with …nite
maturity. The next theorem shows that no bubbles can arise to securities
in positive net supply as long as we are at equilibria with …nite aggregate
wealth. The proof uses the nonoptimality of the behavior implied by the
previous lemma.
Theorem 83 Consider an equilibrium fx i ; z i ; qg. Suppose that at each
node st 2 N; there exists 2 (st ) such that !~ (st ) < 1: Then
q j (sT ) = f j (sT ; );
for all sT jst and 2
net supply, z!j > 0.
(st ), for each security j traded at sT that has positive
This is a crucial: if an agent’s endowment can be traded, then its value
is on the right hand side of the present value budget constraint of the agent,
and hence it must be …nite.
The next two corollaries to the theorem provide conditions on the primitives of the model that guarantee that the value of aggregate wealth is …nite
at any equilibrium.
J
Corollary 84 Suppose that there exists a portfolio zb R+
such that
d(st js0 )b
z
!
~ (st ); 8st 2 N:
Then the theorem holds at any equilibrium.
99
6.2 BUBBLES
Intuitively, if the existing securities allow such a portfolio zb to be formed,
it must have a …nite price at any equilibrium. But since the dividends paid
by this portfolio are higher at every state than the aggregate endowment, the
equilibrium value of the aggregate endowment is bounded by a …nite number.
Corollary 85 Suppose that there exists an (in…nitely lived) agent i and an
" > 0 such that i) ! i (st ) "!(st ); 8st 2 N and ii) B i (st ) = !i (st ); 8st 2 N
and for all i. Then the theorem holds at any equilibrium.
Again, the result follows because in equilibrium, "!(st ) must have a …nite
value as it appears on the right hand side of agent i’s budget set. If a positive
fraction of aggregate wealth has a …nite value in equilibrium, then aggregate
wealth has …nite value.
As a remark, note that these two corollaries share the same spirit and
somewhat imply that bubbles are not a robust equilibrium phenomenon (for
securities in positive net supply).
6.2.1
(Famous) Theoretical Examples of Bubbles
Recall that …at money is a security that pays no dividends. Its only return
comes from paying one unit of itself in the next period. Therefore if …at
money is in positive net supply and has a positive price in equilibrium, that
is a bubble. The following two models have equilibria with such a property.
Samuelson (1958)’s OLG model
[
I(st ) is (countably) in…nite, even though
Consider an economy in which
st 2N
I(s ) is …nite for any s 2 N: In this case even if !~ i (st ) < 1, it is still
possible that !~ (st ) = 1 (and hence that !~ (st ) = 1). The theorem does
not apply and bubbles are possible.
t
t
Bewley (1980)’s turnpike model
Consider the case in which stringent borrowing limits are imposed on trading,
i.e., B i (st ) < !~ i (st ). In this case, nothing will exclude the possibility that
t
!
~ i (s ) = 1. The theorem does not apply and bubbles are possible. For a
detailed but simple treatment, see L. Ljungqvist and T. Sargent, Recursive
macroeconomic theory, MIT Press, 2004; Ch. 25.
100
CHAPTER 6 INFINITE-HORIZON ECONOMIES
More recent examples of bubbles
Many papers have studied bubbles recently. This is so even though the
previous analysis leaves relatively little space for bubbles in the classes of
models we are studying. Nonetheless, see, e.g.,
- Boyan Jovanovic (2007), Bubbles in Prices of Exhaustible Resources, NBER
Working Paper No. 13320, http://www.nber.org/papers/w13320.
Others have studied models with either behavioral agents (e.g., overcon…dent). See, e.g.,
- Harrison Hong, Jose Scheinkman, Wei Xiong (2006), Asset Float and Speculative Bubbles, Journal of Finance, American Finance Association,
vol. 61(3), pages 1073-1117.
- Dilip Abreu and Markus Brunnermeier (2003), Bubbles and Crashes, Econometrica, 71(1), 173-204.
Yet others have studied economies where the rational expectations assumption is relaxed (more precisely, the common knowledge assumption of
rational expectation models). See, e.g.,
- Franklin Allen, Stephen Morris, and Hyun Song Shin (2003), Beauty Contests, Bubbles and Iterated Expectations in Asset Markets, Cowles
Foundation Discussion Paper 1406, http://cowles.econ.yale.edu/P/cd/d14a/d1406.pdf
6.3
Double In…nity
In this section we consider in…nite horizon economies with an in…nite number of agents (hence double in…nity). The classic example is an overlapping
generation economy, where at each time t = 0; 1; 2; :::1 a …nitely-lived generation is born.
We consider …rst the case of an Arrow-Debreu economy with a complete
set of time and state contingent markets. Assume a representative-agent
economy with one good. Let time be indexed by t = 0; 1; 2; :::: Uncertainty is
captured by a probability space represented by a tree with root s0 and generic
node, at time t, st 2 S t . Let `1
+ denote the space of non-negative bounded
101
6.3 DOUBLE INFINITY
sequences endowed with the sup norm.4 Let xi := fxit g1
t=0 2
i
stochastic process for an agent i’s consumption, where xt : S t
random variable on the underlying probability space, for each
1
let ! i = f! it g1
t=0 2 `+ be a stochastic processes describing
endowments. Each agent i0 s preferences , U i : `1
+ ! R; satisfy:
1
X
U i (x) = ui0 (x0 ) + E0 [
`1
+ denote a
! R+ is a
t. Similarly,
an agent i’s
t i
ut (xt )]
t=1
where uit : R+ ! R satisfy the standard di¤erentiability, monotonicity, and
concavity properties, for any i > 0 and any t > 0. Obviously, 0 < < 1
denotes the discount factor.
We say that an agent i is alive at st 2 N if ! i (st ) > 0 and uit (xt ) > 0
for some xt 2 R+ : We assume that ! i (st ) > 0 implies uit (xt ) > 0 for some
xt 2 R+ and conversely, uit (xt ) > 0 for some xt 2 R+ implies ! i (st ) > 0 for
all st 2 S t :We maintain the previous section assumptions that if an agent i
is alive at some non-terminal node st , she is also alive at all the immediate
successor nodes, and that the economy is connected across time and states
(at any state there is some agent alive and non-terminal). We …nally assume
that,
i) each agent is …nitely-lived: ! i > 0 for …nitely many times t
0;
ii) at each node st 2 S t ; the set of agents alive (with positive endowments
and preferences for consumption), I( st ) is …nite.
Suppose that at time zero, the agent can trade in contingent commodities.
1
Let p := fpt g1
t=0 2 `+ denote the stochastic process for prices, where pt :
S t ! R+ , for each t. The de…nition of Arrow-Debreu equilibrium in this
economy is exactly as for the one with …nite agents i 2 I: Let x = (xi )i 0 :
De…nition 86 (x ; p ) is an Arrow-Debreu Equilibrium if
x i 2 arg maxfu(x0 ) + E0 [
i. given p ; s:t:
P1
t=0
4
pt (xt
P1
t=1
t
u(xt )]g
! it ) = 0
See Lucas-Stokey with Prescott (1989), Recersive methods in economic dynamics, Harvard University Press, for de…nitions.
102
CHAPTER 6 INFINITE-HORIZON ECONOMIES
ii. and
P
i
x
i
! i = 0:
Note that, under our assumptions,
1
X
t=0
pt ! it < 1; for any i
X
!i
0
1:
i
While the de…nition of Arrow-Debreu equilibrium is unchanged once an
in…nite number of agents is allowed for, its welfare properties change substantially.
As always, we say that x is a Pareto optimal allocation if there does not
exist an allocation y 2 `1
+ such that
U i (y i )
I
X
yi
U i (x i ) for any i
0 (strictly for at least one i), and
! i = 0,
i=1
Does the First welfare theorem hold in this economy? Is any Arrow-Debreu
equilibrium allocation x Pareto Optimal? Well, a quali…cation is needed.
First welfare theorem for double in…nity economies. Let (x ; p ) be an
Arrow-Debreu equilibrium. If at equilibrium aggregate wealth is …nite,
!
1
X
X
pt
! it < 1;
t=0
i
then the equilibrium allocation x Pareto Optimal.
The important result
is that the converse does not hold. An equiP1 here P
librium with in…nite t=0 pt ( i ! it ) might not display Pareto optimal allocation. In other words, with double
P1 in…nity
P iof agents the proof of the First
welfare theorem breaks unless t=0 pt ( i ! t ) < 1: Let’s see this.
Proof. By contradiction. Suppose there exist a y 2 `1
+ which Pareto dominate x :Then it must be that
1
X
t=0
pt (yti
! it )
0; strictly for one i
0:
103
6.3 DOUBLE INFINITY
Summing over i
0, even though
X
yi;
i 0
X
! i are …nite,
i 0
might not be. In this case, we cannot conclude that
!
!
1
1
X
X
X
X
pt
yti >
pt
! it :
t=0
P1
t=0
i
P1
t=0
P
pt ( i ! it )
i
P i
P1
P i
The
quali…cation
p
(
!
)
<
1
is
however
su¢
cient
to
obtain
p
(
t
t
t
t=0
i
t=0
i yt ) >
P i
P1
)
and
hence
the
contradiction
(
!
p
i t
t=0 t
X
i
yti >
X
! it , for some t
0:
i
Importantly, the proof has
for the converse. In other
P i
P no implication
p
(
!
)
<
1
being necessary for Pareto
words, the proof is silent on 1
t=0 t
i t
optimality. We shall show by example that:
-
P
pt ( i ! it ) < 1 is not necessary for Pareto optimality; that is, there
exist economies which have Arrow-Debreu
equilibria
P i whose allocations
P
p
(
are Pareto optimal and nonetheless 1
t=0 t
i ! t ) is in…nite;
P1
t=0
- there exist economies which have Arrow-Debreu equilibria whose allocations are NOT Pareto optimal.
Remark 87 The double in…nity is at the root of the possibility of ine¢ cient
equilibria in these economies. The First welfare theorem in fact holds with
…nite agents i 2 I even if the economy has an in…nite horizon, t 0. This we
know already. But it follows trivially (check this!) from the proof above that
the First welfare theorem holds for …nite horizon economies, t = 0; 1; :::; T
even if populated by an in…nite number of agents i
0; provided of course
P
i
1:
i!
6.3.1
Overlapping generations economies
We will construct in this section a simple overlapping generations economy
which displays i) Arrow-Debreu equilibria with Pareto ine¢ cient allocations
(aggregate wealth is necessarily in…nite in this case, ii) Arrow-Debreu equilibria with in…nite aggregate wealth whose allocations are Pareto e¢ cient.
104
CHAPTER 6 INFINITE-HORIZON ECONOMIES
The economy is deterministic and is populated by two-period lived agents.
An agent’s type i 0 indicates his birth date (all agent born at a time t ae
identical, for simplicity). Therefore, the stochastic process for the endowment
of an agent i 0; ! i , satis…es
! it > 0 for t = i; i + 1 and = 0 otherwise.
We assume there is also an agent i =
are as follows:
U i (xi ) = u(xii ) + (1
U 1 (x 1 ) = u(x0 1 ):
1 with ! 0 1 > 0. The utility functions
)u(xii+1 ); for any i
0; 5
Arrow-Debreu equilibria are easily characterized for this economy.
Autarchy The economy has a unique Arrow-Debreu equilibrium (x ; p )
which satis…es:
xtt = ! tt ; xtt+1 = ! tt+1 ; for any t
x0 1 = ! 0 1
and
p0 = 1; and
(1
pt+1
=
pt
0;
)u0 (! tt+1 )
; for t
u0 (! tt )
0:
The restriction p0 = 1 is the standard normalization due to the homogeneity of the Arrow-Debreu budget constraint.
Proof. First of all,
x0 1 = ! 0 1
follows directly from agent i = 1’s budget constraint. Then, market clearing
at time t = 0 requires
x0 1 + x00 = ! 0 1 + ! 00 :
Substituting x0 1 = ! 0 1 ; we obtain x00 = ! 00 : We can now proceed by induction to show that xtt = ! tt implies xtt+1 = ! tt+1 (using the budget constraint
t+1
of agent t), which in turn implies xt+1
t+1 = ! t+1 (using the market clearing
condition at time t + 1:
The characterization of equilibrium prices then follows trivially from the
…rst order conditions of each agent’s maximization problems.
105
6.3 DOUBLE INFINITY
It is convenient to specialize this economy to a simple stationary example
where
! 0 1 = ; ! tt = 1
u(x) = ln x:
; ! tt+1 = ; for any t
0;
Then, at the Arrow-Debreu equilibrium,
x0 1 =
pt =
; xtt = 1
1
1
; xtt+1 = ; for any t
0;
t 1
:
A symmetric Pareto optimal allocation is as follows (it is straightforward
to derive, once symmetry is imposed): at the Arrow-Debreu equilibrium,
x0 1 = ; xtt = 1
; xtt+1 = ; for any t
0:
It follows then that,
The Arrow-Debreu equilibrium (autarchy) is Pareto e¢ cient if and only if
:
Proof. The case
=
is obvious. If
<
all agents i
0 prefer
i
i
the allocation xi = 1
; xi+1 =
to autarchy, but agent i = 1 prefers
1
1
x0 = to x0 = . Autarchy is then Pareto e¢ cient. If otherwise >
all agents i 0 prefer the allocation xii = 1
; xii+1 =
to autarchy, and
1
1
agent i = 1 as well prefers x0 = to x0 = . Autarchy is then NOT
Pareto e¢ cient.
P
Furthermore, note that in this economy, the aggregate endowment i 0 ! it =
1; for any t 0; and hence the value of aggregate wealth is:
!
1
1
1
t 1
X
X
X
X
1
i
pt
!t =
pt =
:
1
t=0
t=0
t=0
i
It follows that
The value of aggregate wealth is …nite when
<a
106
CHAPTER 6 INFINITE-HORIZON ECONOMIES
and autarchy is e¢ cient. On the other hand, the value of aggregate
wealth is in…nite when
a:
In this case, autarchy is e¢ cient if
= a and ine¢ cient when
> a:
Note that we have in fact shown what we were set to from the beginning
of this section: i) Arrow-Debreu equilibria with Pareto ine¢ cient allocations
(aggregate wealth is necessarily in…nite in this case, ii) Arrow-Debreu equilibria with in…nite aggregate wealth whose allocations are Pareto e¢ cient.
Bubbles
We have seen previously that, when the value of the aggregate endowment is
in…nite, bubbles in in…nitely-lived positive net supply assets might arise. We
shall now show that this is the case in the overlapping generation economy we
have just studied. Interestingly, it will turn out that bubbles, in this economy
might restore Pareto e¢ ciency. (This is special to overlapping generations
economies, by no means a general result).
Suppose agent i = 1 is endowed with a in…nitely-lived asset, in amount
m: The asset pays no dividend ever: it is then interpreted to be …at money.Any
positive price for money, therefore, must be due to a bubble. Let the price of
money, in units of the consumption good at time t = 0; at time t; be denoted
qtm : We continue to normalize p0 = 1: The Arrow-Debreu budget constraints
in this economy for agent i = t 0 can be written as follows:
1
X
pt (xt
! it ) = pt (xtt
! tt ) + pt+1 (xtt+1
! tt+1 ) = 0;
t=0
or, denoting st the demand for money of agent i = t
pt xtt + qtm st = ! tt
m
pt+1 (xtt+1 ! tt+1 ) = qt+1
st :
The budget constraint for agent i =
1 is:
x0 1 = ! 00 + q0m m:
0:
107
6.3 DOUBLE INFINITY
Turning back to the stationary economy example where
! 0 1 = ; ! tt = 1
u(x) = ln x;
; ! tt+1 = ; for any t
0;
we can easily characterize equilibria. Suppose in particular that
>
and autarchy is ine¢ cient. We restrict the analysis to following stationarity
restriction:
qtm = q m for any t 0:
The autarchy allocation is obtained as an Arrow-Debreu equilibrium of this
economy, for
p
1
q m = 0 and t+1 =
:
pt
1
The Pareto optimal allocation
x0 1 = ; xtt = 1
; xtt+1 = ; for any t
0
is also obtained an Arrow-Debreu equilibrium of this economy, for
qm =
m
and
pt+1
= 1:
pt
It is straightforward to check that these are actually Arrow-Debreu equilibria of the economy. Note that Pareto optimality is obtained at equilibrium
for q m > 0; that is, when money has positive value and hence a bubble exist.
At this equilibrium, prices pt are constant over time and hence the value of
the aggregate wealth is in…nite.
Remark 88 The stationary overlapping generation example introduced in
this section has been studied by Samuelson (1958). The fundamental intuition
for the role of money in this economy is straightforward: money allows any
agent i 0 to save by acquiring money (in exchange for goods) at time t = i
from agent i 1 and then transfering the same amount of money (in exchange
for the same amount of goods) to agent i + 1 at time t = i + 1: Money, in
other words, serves the purpose of a pay-as-you-go social security system;
and in fact such a system, implemented by an in…nitely lived agent (like a
benevolent government), could substitute for money in this economy. (Try
and set it up formally!) Furthermore, note that, for money to have value in
108
CHAPTER 6 INFINITE-HORIZON ECONOMIES
this economy, we need > . Consistently with the interpretation of money
as a social security mechanism, the condition > is interpreted to require
that i) the endowment of any agent i 0 at time t = i + 1, ; be relatively
small and that ii) any agent i 0 discounts relatively little the future, that
is, = 1 is high.
Remark 89 Large-square economies, with a continuum of agents and commodities, are studied by Kehoe-Levine-Mas Colell-Woodford (1991), "Grosssubstitutability in large-square economies," Journal of Economic Theory, 54(1),
1-25.
Problem 90 Consider our basic Overlapping Generation economy with no
money, when specialized to specialize this economy to a simple stationary
example where
! 0 1 = ; and ! tt = 1
; ! tt+1 = ; for any t 0;
U 1 (x 1 ) = ln x0 1 ; and U t (xt ) = ln xtt + ln xtt+1 ; for any t
0.
Assume however that population grows at rate n > 0: at time t = 0 the
economy is populated by a mass 1 of agents of generation i = 1 and a mass
1 + n of agents of generation i = 0; and so on. Suppose also that a social
security system can be imposed on the economy. It works as follows: any
generation i
0 pays to the social security system 0
1
units of
the consumption good at time t = i and receive b = (1 + n) units of the
consumption good at time t = i + 1:
1. For given 0
and prices).
1
solve for a competitive equilibrium (quantities
2. Derive a condition on which guarantees that the equilibrium allocation
Pareto dominates autarchy (with no social security system).
3. When this conditions is satis…ed, characterize the subset of ’s which
induce equilibrium allocations which are Pareto optimal. Does aggregate
wealth have …nite value at such Pareto optimal allocations.
6.4
Default
Consider the following economy, populated by a …nite set of in…nitely-lived
1
agents, i 2 I: Let N = Xt=0
S t be the set of nodes of the tree, s0 the root of
109
6.4 DEFAULT
the tree, st an arbitrary node of the tree at time t. Use st+ jst to indicate
that st+ is some successor of st , for > 0:
At each node, there are S securities in zero-net supply traded with linearly independent payo¤s: …nancial markets are complete. Without loss of
generality we let the …nancial assets be a full set of Arrow securities: A = IS
(the S dimensional identity matrix).
Let q : N ! RS be the mapping de…ning the vector of asset prices at
each node st .
At each node st , each households has an endowment of numeraire good
of ! i (st ) > 0; aggregate endowment is then
X
! i (st ) 0
!(st ) =
i2I
at each node st . At the root of the tree the expected utility of agent i 2 I is:
U i (x; s0 ) = ui (x(s0 ) +
1
X
t=1
t
X
prob(st s0 ) ui (x(st s0 )):
st js0
Any agent i 2 I; at any node st 2 N can default (more precisely: cannot
commit not to default). If he does default at a node st , he is forever kept out
of …nancial markets and is therefore limited to consume his own endowment
! i (st+ jst ) at any successor node st+ jst 2 N:
An agent i 2 I; therefore, will default at a node st on allocation xi if
U i (xi ; st ) < U i (! i ; st ):
The notion of equilibrium we adopt for this economy will be the one used
by Prescott and Townsend for economies with asymmetric information where
a no-default constraint
U i (xi ; st )
U i (! i ; st ); for any st 2 N;
takes the place of the incentive compatibility constraint. In particular we
shall choose the notion of equilibrium introduced in the remark, where these
constraints are interpreted as constraints on the set of tradable allocation
(rather than rationality restrictions on price conjectures).
Let ((x i )i ; p ) be an Arrow-Debreu Equilibrium if x i ; for any agent
i 2 I; solves
max
U i (xi ; s0 )
i
x
110
CHAPTER 6 INFINITE-HORIZON ECONOMIES
s.t.
p (xi ! i ) = 0; and
U i (xi ; st )
U i (! i ; st ); for any st 2 N
and markets clear:
X
x
i
! i = 0:
i
This problem is formulated and studied by T. Kehoe and D. Levine,
Review of Economic Studies, 1993. Note that
i) the value of aggregate wealth for each agent i 2 I is necessarily …nite at
equilibrium:
p !i < 1
(and I is …nite by assumption)
ii) the set of constraints which guarantee no-default at equilibrium do not
depend on equilibrium prices
U i (xi ; st )
U i (! i ; st ); for any st 2 N:
Let now incentive constrained Pareto optimal allocations be those which
solve
X
i i i 0
U (x ; s )
max
i
(x )i2I
s.t.
X
xi
i2I
! i = 0; and
i2I
i
for some
U (xi ; st )
U i (! i ; st ); for any st 2 N;
P
2 RI+ such that i2I i = 1:
It follows easily then that
Any Arrow-Debreu Equilibrium allocation of this economy is constraint
Pareto optimal.6
6
In fact, in the Kehoe and Levine paper, L > 1 commodities are traded at each node
and the conquence of default is that trade is restricted to spot markets at any future node.
In this economy the non-default constraints depend on spot prices and Arrow-Debreu
equilibrium allocations are not incentive constrained optimal.
111
6.4 DEFAULT
Consider now a …nancial market equilibrium for this economy. The objective is to capture the no-default constraints by means of appropriate borrowing constraints. In other words we want to set borrowing constraints as
loose as possible provided no agent would ever default. To this end we need
to write exploit the recursive structure of the economy. Let i (st+1 jst ) denote the portfolio of Arrow security paying o¤ in node st+1 acquired at the
predecessor node st : The borrowing constraints an agent i 2 I will face at
node st will then take the form
i
(st+1 st )
B i (st+1 st ); for any st+1 st ;
and B i (st+1 jst ) must be chosen to be as loose as possible provided it induces
agent i 2 I not to default at any node st+1 jst : Formally, the value function
of the problem of agent i 2 I at any node st 2 N when i) the agent enters
state st 2 N with i units of the Arrow security which pays at st 2 N; and
the agent faces borrowing limits B i (st+1 jst ), can be constructed as follows:
P
t+1 t
js )V i ( i (st+1 jst ); st )
V i ( i ; st ) = maxxi ; i (st+1 jst ) ui (xi ) +
st+1 jst prob(s
s:t:
xi +
i
P
st+1 jst
i
(st+1 jst )q(st+1 jst ) =
i
+ ! i (st+1 jst ); and
(st+1 jst ) B i (st+1 jst ):
The condition that the borrowing limits B i (st+1 jst ) be as loose as possible
is then endogenously determined as
V i (B i (st+1 st ; st ) = U i (! i ; st+1 ).
It can be shown that V i (B i (st+1 jst ; st ) is monotonic (decreasing) in its …rst
argument, for any st 2 N: Borrowing limits B i (st+1 jst ) are then uniquely
determined at any node. Note that they are determined at equilibrium,
however, as they depend on the value function V i (:; st ):
Market clearing brings no surprises:
X
xi ! i = 0:
i
We can now show the following.
The autarchy allocation
x i (st ) = ! i (st ); for any st 2 N;
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CHAPTER 6 INFINITE-HORIZON ECONOMIES
can always be supported as a …nancial market equilibrium allocation with
borrowing constraints
B i (st+1 st ) = 0; for any st 2 N:
Note in particular that V i (B i (st+1 jst ; st ) = U i (! i ; st+1 ) is satis…ed at this
equilibrium.
Let
q(st+2 jst ) = q(st+2 jst+1 )q(st+1 jst ); for t = 0; 1; :::
We can then show the following.
Any …nancial market equilibrium allocation (x i )i2I whose supporting prices
q (st+1 jst ) satisfy
! i s0 +
X X
t 0 st+1 js0
q (st+1 s0 )! i (st+1 s0 ) < 1
is an Arrow-Debreu equilibrium allocation supported by prices
p (s0 ) = 1; p (st ) = q (st s0 ):
This can be proved by repeatedly solving forward the …nancial market
equilibrium budget constraints, using the relation between Arrow-Debreu
and …nancail market equilibrium prices in the statement. The solution is
necessarily the Arrow-Debreu budget constraint only if limT !1 q (sT js0 ) =
0, which is the case if the value of aggregate wealth is …nite at the …nancial
market equilibrium.
In this case, then the …nancial market equilibrium allocation (x i )i2I is
constrained Pareto optimal. When (x i )i2I is not autarchic (easy to show by
example that robustly, such …nancial market equilibrium allocations exist)
it Pareto dominates autarchy (as autarchy is always budget feasible and
satis…es the no-default constraint). In this case the autarchy allocation is not
constrained Pareto optimal and the value of some agent’s wealth is in…nite
at the autarchy equilibrium.7
7
A?? by Gaetano Bloise and Pietro Reichlin (2009) proves that in this economy …nancial
market equilibria with in…nite value of aggregate wealth other than autrachy exist. Some
of these equilibria are e¢ cient and some are not. The machinery used to prove these
results is related to the classical analysis of Overlapping Generations and Bewley models.
Not surprisingly, bubbles also arise.
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6.4 DEFAULT
Problem 91 Show that an the autarchy allocation asset prices must satisfy
q (st+1 st )
max
i2I
ui0 (! i (st+1 ))
:
ui0 (! i (st ))
Problem 92 Write down the …nancial market equilibrium notion in which
rational price conjecture substitute borrowing constraints. How do prices look
like?
Problem 93 Agents live two periods, t = 0; 1; and consume a single consumption good only in period 1. Uncertainty is purely idiosyncratic. Each
agent faces a (date 1) endowment which is an identically and independently
distributed random variable ! = (! H ; ! L ) ; with ! H > ! L : Agents come in 2
types, e = h; l; which are di¤erentiated in terms of the probability distribution
of their endowments: es is the probability of state s for agents of type e: Let
l
h
H > H : (Notation is set so that H is the "high state" (the state where endowment is high) and h is the "high type" (the type with a higher probability
of the high state). The fraction of h types in the economy is 21 :Types are only
privately observable. (Yes, this is an adverse selection economy). Agents’
preferences are represented by a von Neumann-Morgenstern utility function
of the following form:
X
e
s ln xs
s2S
1. Write the incentive constrained optimal planning problem for the planner (with equal weight across types).
2. Write the de…nition of competitive equilibrium with rational conjectures
(where prices are function of the allocations). Anything "strange" in
these prices?
Chapter 7
To add
1. aggregation in Angeletos (two periods and in…nite horizon)
2. Citanna-Siconol… on decentralization of adverse selection
3. papers by Townsend with Weerachart Kilenthong <[email protected]>;
http://cier.uchicago.edu/papers/working/Moral%20Hazard_Submitted.pdf
4. Papeers by Meier-Minelli-Polemarchakis, Correia-da-Silva on two-sided
asymmetric information; write them in the context of a …nancial market
economy (see also Tirelli et al. on ET); see also Acharya-Bisin
5. Section on assignment problem, equilibrium in matching and search
6. Section on monopolistic competition
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