Economics 603 — Problem Set #2
Due date: Thursday, February 10, 2000, at the beginning of the lecture.
1. An economy is made up of two people. The utility functions are u1 (x1 ) = x11 x12 and u2 (x) =
2x21 + 2x22 − x11 . The initial bundles are ω1 = (1, 0), ω2 = (0, 1). Although 2 suffers from 1’s
consumption of good 1, he cannot control it, nor does he realize that the total quantity of good 1
available is 1.
A Calculate a competitive equilibrium from ω. Draw an Edgeworth box diagram to illustrate
your answer.
Using familiar properties of the Cobb-Douglas function, the demand of the first individual is
( 21 , 12 pp12 ). When p1 > p2 , the demand of the second individual is (0, 1), and when p1 < p2 it
is ( pp12 , 0). Clearly, markets will not clear in either case. Thus, p1 = p2 , and the equilibrium
consumptions of both individuals are ( 12 , 12 ).
B Find the locus of interior Pareto optimal points.
One way of approaching the problem is to derive the first order conditions for the Pareto
optimality. Calculations are simplified if the utility of the first individual is monotonely transformed to be ũ1 = ln(x11 ) + ln(x1 2). Noting that the endowment constraints imply that
x21 = 1 − x11 and x22 = 1 − x12 , the problem may be expressed as,
max 2(1 − x11 ) + 2(1 − x12 ) − x11
x11 ,x12
s.t. , ln(x11 ) + ln(x12 ) ≥ u ,
for any u ∈ (−∞, 0]. Denoting the Lagrange multiplier by λ, the first order conditions are
−3 = xλ11 and −2 = xλ12 , what leads to x12 = 32 x11 . Points in the Edgeworth box that
satisfy this condition constitute the set of all interior Pareto allocations, and correspond to
u ≤ ln( 32 ). For u > ln( 23 ), the optimum is on the border of the Edgeworth box — these are
points (x11 , x12 ) = (α, 1) for α ∈ ( 23 , 1].
C Calculate prices p1 and p2 , a per unit subsidy s or tax t, and lump sum cash transfers T1 and
T2 to bring the economy to the allocation y1 = ( 31 , 12 ), y2 = ( 23 , 12 ).
The marginal rates of substitution of both individuals at this point are different, so that
they will never choose this point if they face the same prices. So consider imposing per unit
tax t on consumption of good 1 by individual 1. For individual 1 to choose this point for
some endowment
level, the marginal rate of substitution must be equal to the relative price
.
p1 +t
∂u1
∂u1
3
=
=
p2
∂x1
∂x2
2 . For individual 2 to choose an interior solution, it must be that p1 = p2 .
Solving for t yields t = p22 . Normalizing p2 = 1, we get that p1 = 1 and t = 12 . The issue
now is to find transfers that will make this point feasible at the prices just found. The value
of endowment of individual 1 is 1p1 = 1, and the cost of his allocation is (p1 + t) 13 + p2 12 = 1.
Thus the transfer of T1 = 1 − 1 = 0 will do. For individual 2, the value of endowments is also
1, and the cost of his consumption is p1 32 + p2 12 = 76 . Thus T2 = 16 . It is straightforward to
verify that government collects no revenue.
2. We have n agents with identical strictly concave utility functions. There is some initial bundle of
goods ω. Show that equal division is a Pareto efficient allocation.
Pn
Suppose not. Then, there is another allocation (x1 , . . . , xn ), such that
i=1 xn ≤ ω, and u(xi ) ≥
P
n
i, and u(xi ) > u( ωnP
) for some i. It must be then that n1 i=1 u(xi ) > u( ωn ). By strict
u( ωn ) for anyP
n
n
concavity, n1 i=1 u(xi ) < u( n1 i=1 xi ) ≤ u( ωn ) — contradiction. Thus, there is no allocation that
Pareto dominates equal division.
3. Consider a pure exchange economy with free disposal as described in section 17.B. Assume that for
L
every consumer i, Xi = R+
and i is continuous, strictly convex and locally non-satiated. Suppose
1
P
also that ω =
ωi 0. Does this economy always have a Walrasian equilibrium? (Notice that
proposition 17.C.1 makes the stronger assumption that preferences are strongly monotone.) If the
answer is yes, verify explicitly that proposition 17.C.1 goes through with the weaker assumption. If
not, provide a counterexample and point out what step of the proof of the proposition 17.C.1 is not
valid.
The picture below presents a counterexample. Let the endowment be given by ω. An Indifference
curve of consumer 1 is tangent to the vertical axis at ω. Note that preferences of individual 2
are not monotone, but that preferences of both individuals are strictly convex, locally non-satiated
and clearly they can also be continuous. The prices must be nonnegative, because preferences of
individual 1 are monotone. If p2 was positive, there would be no equilibrium because individual 2
would choose ω while individual 1 would select an interior allocation. When p2 = 0, individual 1
demands an infinite amount of good 2. Thus, no equilibrium exists.
ω
Direction of Increasing
preferences for Consumer 1
Direction of Increasing
Preferences for Consumer 2
Assumptions imposed on the preferences in proposition 17.C.1 implied conditions (i) to (v) of the
proposition 17.B.2, and in fact these conditions are necessary and sufficient for the existence of a
Walrasian equilibrium. Thus one of them needs to fail in our example. Preferences in the example
were continuous, so that z(p) is continuous, clearly it must also be homogeneous of degree 1 and
satisfy the Walras law (by local non-satiation). The lower bound condition (property iv) is also
trivial to verify. Thus, the property (v) needs to fail. Indeed, fix p1 > 0 and consider a positive
sequence {pn2 }, such that lim pn2 = 0. Second individual always demands ω2 , and the demand of
n→∞
the first one converges to ω1 as p2 tends to zero. Thus, lim z(pn2 ) = ω1 + ω2 − ω = 0, violating
n→∞
property (v). This property plays a role in the step 4 of the proof of the proposition 17.B.2. Without
it, it is not possible to demonstrate that the fixed point correspondence constructed in the proof
is upper hemicontinuous, and the Kakutani’s fixed point theorem may not be applied. Although
under the assumption of local non-satiation a Walrasian equilibrium does not have to exist, it is
possible to demonstrate that a quasiequilibrium must exist. See appendix B in chapter 17 of MWG
for extensive discussion.
4. Consider a standard growth model. There are N infinitely-lived identical consumers endowed with
an initial capital stock `0 and utility function u. There is a single consumption good, and if consumer
2
i consumes cit in periods t = 0, 1, 2, . . . , ∞, his total utility is
∞
X
β t u(cit ),
t=0
where β ∈ (0, 1) is the discount factor. Assume u : R+ → R is strictly increasing and strictly concave. There are three goods in every period: capital at the beginning of the period, the consumption
good, and capital at the end of the period. There are N identical firms (j = 1, . . . , N ) that use
capital at the beginning of the period (Kjt ) to produce the consumption good (Yjt ) and capital at the
end of the period (Ftj ). The production function is
Yjt
f (Kjt )
=
, Kjt ≥ 0
Fjt
(1 − δ)Kjt
where δ ∈ [0, 1] is the depreciation rate, and f (K) is an increasing and concave function satisfying
f (K) > 0 and lim f 0 (K) > δ. There is also a firm that produces capital for the beginning of
K→0
next period (Lt+1 ) from capital (Et ) at the end of the current period and consumption good (Xt ),
according to the production function Lt+1 = Et + Xt . All firms are jointly owned by the consumers
in equal shares.
A Provide a precise definition of a Walrasian equilibrium for this economy.
A Walrasian equilibrium is an allocation {(c∗t , (Kt∗ , Yt∗ , Ft∗ ), (Et∗ , Xt∗ , L∗t+1 )}t≥0 together with
a price vector {(qt , pt , rt )}t≥0 , where c∗t = (c∗1t , . . . , c∗N t ) and (qt , pt , rt ) are the prices of capital
at the beginning of the period, the consumption good, and capital at the end of the period,
respectively. WLOG we can make capital at the beginning of period 0 the numeraire: q0 = 1.
The allocation and price vector must satisfy:
(1) For each consumer i, the sequence {c∗it } solves
P
max P t≥0 β t u(cit )
s.t.
t≥0 pt ct ≤ `0 .
∗
∗
∗
(2) For each firm j and index t, (Yjt∗ , Kjt
, Fjt
) belongs to the production set, pt Yjt∗ + rt Fjt
−
∗
qt Kjt ≥ pt Yjt +rt Fjt −qt Kjt for all Kjt , Yjt , Fjt with Kjt ≥ 0, Yjt ≤ f (Kjt ) and Fjt = (1−
δ)Kjt . Note that no “intertemporal” conditions are present, as the production technologies
in different periods are unrelated.
(3) qt+1 L∗t+1 − rt Ee∗t − pt Xt∗ ≥ qt+1 Lt+1 − rt Et − pt Xt for all (Et , Xt , Lt+1 ) with Et ≥ 0,
Xt ≥ 0, and Lt+1 = Et + Xt .
PN
(4) Kt∗ = L∗t , i=1 c∗it + Xt∗ = Yt∗ , and Et∗ = Ft∗ .
Condition (1) states that each consumer’s consumption plan is optimal. Conditions (2) and (3)
state that the firms choose optimal production plans. Finally, (4) contains the market clearing
conditions. More precisely, a consumption plan for a consumer should include the amount of
every good he consumes. But, because the consumers only care about the consumption good,
they don’t buy any of the capital goods.
B
Argue that although this economy has an infinite number of commodities (three for each
period), the allocation of any Walrasian equilibrium is efficient.
We can directly verify that the standard proof still applies. Suppose the allocation is not Pareto
efficient, i.e. there are consumption plans cit such that everybody is at least weakly better off
with some individual strictly better off, and supporting production plans {(Kjt , Yjt , Fjt )}t≥0 ,
j = 1, . . . , N , {(Et , Xt , Lt )}t≥0 . By individual optimization, for the individuals who are strictly
better off the new consumption plan couldn’t have been available, i.e. pci > pc∗i (note that pc∗i
3
is finite because pc∗i ≤ `0 ). For other individuals, by local non satiation, this new consumption
plan may not be in the interior of the budget set: pci ≥ pc∗i . All firms were maximizing
their profits, thus the aggregate profits
the P
new allocations may not exceed those in the
Punder
I
I
original one π ∗ ≥ π(p). As a result, i=1 pci > i=1 pc∗i = qo I`0 + π(p). But this may not
PI
be, because by the market clearing, for any prices i=1 pci = qo I`0 + π(p).
C Find the Walrasian equilibrium for the case where u(c) = log(c), f (K) = AK, A > 1 and
`0 = 1.
Since all the firms exhibit constant returns-to-scale, they make no profits, and assuming an
interior solution, conditions (2) and (3) are equivalent to
Apt + (1 − δ)rt = qt ,
qt+1 = rt ,
and qt+1 = pt .
That is,
qt+1 = rt = pt
and pt = ρpt−1
for all t ≥ 1,
where ρ = [1 − δ + A]−1 . In addition, since q0 = 1, we must have p0 = ρ. Hence, the solution
to the last difference equation is pt = ρt+1 , t ≥ 0.
Obviously, for any price vector, there is only one solution for the consumer’s optimization
problem and c∗it = c∗1t for i = 2, . . . , N and all t ≥ 0. Each individual maximizes,
P
P
t
t
max
s.t.
t≥0 β ln(ct )
t≥0 ρ ct ≤ `0 .
The first order conditions (Euler equation) lead to ρu0 (ct ) = βu0 (ct+1 ), i.e. ct+1 = ct βρ . Thus,
ct = c0 ( βρ )t . Subsituting it into budget constraint and solving for c0 yields c0 = `0 (1 − β) (note
that β < 1 so that this step is valid).
Because all technologies are constant returns to scale we can concentrate on the case when
just one firm of each kind produces output (so that we can forget about index j). The market
clearing condition implies that K0 = n`0 . Since Ft = (1 − δ)Kt , the market clearing condition
and the production technology of the firm producing capital for the beginning of the period
implies that Lt+1 = (1 − δ)Kt + Yt − nct . Substituting for Yt from the production function
yields Lt+1 = (1 + A − δ)Kt − nct . Consequently, Kt+1 = (1 + A − δ)Kt − n`0 (1 − β)( βρ )t . This
Pt
leads to, Kt+1 = (1 + A − δ)t+1 K0 − n`0 (1 − β) i=0 (1 + A − δ)t−i ( βρ )i . Substituting for ρ
Pt
and K0 , it simplifies to Kt+1 = n`0 (1 + A − δ)t (1 + A − δ) − (1 − β) i=1 β i , yielding the
solution of Kt = n`0 (1 + A − δ)t−1 (A − δ + β t ). In particular, our assumptions guarantee that
Kt > 0, so we do not need to worry about the nonnegativity constraint.
All other components of the production plans may be expressed in terms of Kt : Yt = AKt ,
Ft = (1 − δ)Kt , Lt = Kt , ET = (1 − δ)Kt , and Xt = AKt − n(1 − β)`0 ( βρ )t . This completes a
description of the Walrasian equilibrium.
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