Graduate Macroeconomic Theory
Joe Haslag
Department of Economics, University of Missouri
E-mail address: [email protected]
URL: http://www.
The author thanks students for years of honing the topics covered in this
text..
Abstract. Replace this text with your own abstract.
Contents
Introduction
v
Chapter 1. A Static Decision Problem
1
1. A One-Period Model
2
2. Competitive equilibrium
7
3. Pareto optimum
9
4. Comparative statics
11
5. Government
14
6. Problems
18
Chapter 2. Intertemporal models
21
1. Consumers
21
2. Firm
25
3. Competitive equilibrium
25
4. Problems
28
Chapter 3. Overlapping generations
1. Problems
31
44
iii
Introduction
\chapter*{Preface}The purpose of this book is to develop a one-semester
course that covers the essential topics for a first-year graduate course in
macroeconomic theory. The material is also suitable for an advanced undergraduate course.
v
CHAPTER 1
A Static Decision Problem
Because the questions are essentially ones about aggregate economic
behavior, the analytical framework will deal with the simultaneous solution
of activities in several markets. In short, a model economy in which two
or more markets—quantities and prices—are determined simulataneously is
a general equilibrium model. For our purposes, we begin with the simplest
possible general equilibrium model; that is one with three markets and three
prices. As we proceed, it will be convenient to normalize the price of one
good and that Walrasian economies will have one market that is dependent
on what is going on in the other markets. For our purposes, this means we
will have two independent markets and two relative prices.
The tools learned in this chapter will form the backbone of our anlaysis.
Indeed, the reader will see that modifications to this basic structure permit
us to study more complicated, and interesting, questions. But the same
basic tools will be applied to these setups.
Before specifying the model economy, it is important to present the key
features common to most descriptions of general equilibrium models. The
four features are:
(1) Technologies and endowments
(2) Preferences
(3) Trades
(4) Equilibrium concept
The first three pieces define the structure of the model economy while
the fourth piece governs how these three pieces fit together in our analysis.
1
2
1. A STATIC DECISION PROBLEM
1. A One-Period Model
Consider a model economy in which all trades take place in a single period. Imagine after that period that the economy ends and market participation is not permitted. Though perhaps unrealistic, such an environment
permits us to see how a general equilibrium is constructed. The economy
has two types of participants: consumers and firms. We now turn to a
description of consumers and firms.
1.1. Consumers. Consumers are endowed with one unit of time and
some quantity of physical capital. The total amount of capital endowed
is represented by k0 . Consumers decide how to divide their time between
leisure, which is enjoyable, and labor, which is not. When consumers provide
labor services, they receive wages. Capital is owned by consumers and can be
rented to firms or be permitted to lay idle. Consumers are compensated for
the quantity of capital they rent to firms. In addition to leisure, consumers
also enjoy eating quantities of the single, perishable consumption good.1
The quantities of the consumption good are nonnegative.
We assume there are N of these consumers populating this economy.
Each consumer has preferences over the consumption good, denoted by c,
and leisure, denoted by l. Consumers are identical in the sense that they
each have the same endowment and the same preferences. Formally, each
consumer is endowed with one unit of time and the same quantity of capital,
k0
N.
Preferences are captured by a utility function, represented as u (c, l). We
assume that both arguments in this function are goods; that is, each consumer prefers more of each item to less. We further assume that the additional utility generated by an additional quantity of each good is decreasing.
1For readers who want something more concrete, think of the single perishable good
as apples.
1. A ONE-PERIOD MODEL
3
Formally, we assume the utility function is strictly increasing in each argument and strictly concave. This feature is captured as: uc (c, l) , ul (c, l) > 0
with ucc (c, l) , ull (c, l) < 0 such that ucc (c, l) ull (c, l) − [ucl (c, l)]2 > 0.2 To
ensure that we obtain an interior solution, we further assume that Inada
conditions hold; specifically, limc→0 uc (c, l) = ∞ and limc→∞ uc (c, l) = 0.
Likewise, for leisure, we have liml→0 ul (c, l) = ∞ and liml→1 ul (c, l) = 0.
Because each consumer is identical, we can solve the problem for a representative consumer. Formally, the problem is represented as:
(PC)
max u (c, l)
c,l
c ≤ w (1 − l) + rks
0 ≤ ks ≤
k0
N
0≤l≤1
c≥0
where k s is the quantity of capital rented to a firm, w is the wage rate
and r is the rental rate paid per unit of capital. Note that w and r are both
measured in units of the consumption good.3 In other words, consumption
is picked as the numeraire so that its price is set to one. The prices of other
goods, capital and labor, for instance, are measured relative the price of the
numeraire good.
For a constrained optimization problem, we apply the Kuhn-Tucker Theorem. The Lagrangean is
2Here, the notation is u (c, l) = du for i = c, l and u (c, l) = d2 u for i, j = c, l.
i
ij
di
didj
3More concretely, trade for one unit of labor will cost w units of the consumption
good.
4
1. A STATIC DECISION PROBLEM
(1.1)
¶
µ
k0
= u (c, l) + λ w + r − wl − c
N
where λ is the Lagrange multiplier.
Note that we have taken some shortcuts. If we applied the Kuhn-Tucker
Theorem literally, there would be a multiplier for each constraint; that is,
there should be four constraints. The inequality constraint on capital is
solved by the following argument. Since capital is an endowment, as long as
r > 0, the consumer would rent thier entire endowment because idle capital
means less income and therefore, less consumption. The Inada conditions
ensure that the conditions on leisure and consumption will hold as strict
inequalities.
It is easy to show that the budget constraint will hold as a strict equality. The intuition is straightforward. If consumption is less than income, it
means that consumers are leaving goods on the table; in other words, units
of the consumption good received as factor payments are not consumed.
Free disposal is, therefore, an option. Because the marginal utility of consumption is positive for any finite level of consumption, the shadow price
of consumption, λ, will be positive. In other words, the consumer will always prefer to eat any units of consumption good provided as income to the
alternative.4 The complementary slack condition implies that the budget
4The first-order conditions for the general structure are given by:
∂
= uc (c, l) − λ = 0
∂c
∂
= ul (c, l) − λw = 0
∂l
¸
k0
−c =0
λ w (1 − l) + r
N
∙
With uc > 0, then λ > 0, which further implies that w (1 − l) + r kN0 − c = 0.
1. A ONE-PERIOD MODEL
5
constraint holds as a strict equality. Formally, c = w + r kN0 − wl. If we substitute for consumption, the problem can be rewritten as an unconstrained
optimization problem; that is,
¸
∙
k0
max u w + r − wl, l
l
N
At the maximum, the following condition is satisfied:
(1.2)
¶
¶
µ
µ
k0
k0
−w uc w + r − wl, l + ul w + r − wl, l = 0
N
N
1.2. Firms. Firms can be thought of as being endowed with a production technology. In other words, the firm is the only entity that knows how
to combine labor and capital to produce units of the consumption good.
Firms then pay the factors of production.
The technology used to combine labor and capital to produce the consumption good is captured by the production function.
Let the quan-
tity of the consumption good produced by firms be denoted by y. Then
y = zf (k, n), where k is the quantity of capital rented by firms and n is
the quantity of labor time employed by firms. Here, z > 0 captures total factor productivity. The production function yields more units of the
consumption good as more inputs are added to the process.
Formally,
fk (k, n) , fn (k, n) > 0 and fkk (k, n) , fnn (k, n) < 0. To ensure that both
inputs are used, we assume f (0, 0) = f (0, n) = f (k, 0) = 0. Some positive
quantity of both inputs are necessary to obtain any output. Lastly, we assume the production technology exhibits constant returns to scale; formally,
for any ϕ > 0, zf (ϕk, ϕn) = ϕy.
There are M firms in the economy. The constant returns to scale assumption greatly simplifies the analysis. To see this, consider the expression
that defines a constant returns to scale function; that is, zf (ϕk, ϕn) = ϕy.
Next, differentiate this expression with respect to ϕ, obtaining
6
1. A STATIC DECISION PROBLEM
(1.3)
y = zfk (ϕk, ϕn) k + zfn (ϕk, ϕn) n.
We evaluate (1.3) at ϕ = 1, resulting in y = zfk (k, n) k + zfn (k, n) n
which implies that zf (k, n) = zfk (k, n) k + zfn (k, n) n. To proceed, we
need the conditions under which a profit-maximizing firm will operate. The
consumption good is used as the numeraire so that its price is set equal to
one. Thus, proifts are expressed as
(1.4)
max zf (k, n) − rk − wn
k,n
where r is the rental rate on capital and w is the wage rate. Both
the rental rate and wage are measured in units of the consumption good.
Each firm takes the rental rate and wage rate as given. Profit maximum is
identified by differentiating the profit function with respect to k and n and
setting the derviatives equal to zero.
(1.5)
zfk (k, n) − r = 0
(1.6)
zfn (k, n) − w = 0
It follows from (1.5) and (1.6) that zfk (k, n) k + zfn (k, n) n = rk + wn.
In a competitive environment, no one firm will earn positive profits. If
profits were positive, production could expand until zero profits are realized.
Or, zf (k, n) − rk − wn = 0, which implies that zf (k, n) − zfk (k, n) k −
zfn (k, n) n = 0. Let n = ϕn∗ and k = ϕk ∗ for any ϕ. Insofar as ϕ represents
the scale of the representative firm, and because the scale is indeterminate.
Thus, without loss of generality M = 1. A representative firm is sufficient
to characterize the firm’s behavior in our model economy.
2. COMPETITIVE EQUILIBRIUM
7
2. Competitive equilibrium
We define a competitive equilibrium as an allocation, {c, l, n, k}, and
prices, {w, r}, such that
(i) consumers choose the quantity of the consumption good and leisure
to maximize 1.1, taking wages and rental rates as given;
(ii) firms choose the quantity of labor and capital to employ to maximize
1.4, taking wages and rental rates as given;
(iii) markets clear: formally, k0 = k, y = Nc, N (1 − l) = n;
The necessary and sufficient condition for the consumers maximization
problem are provided by equation (1.2). The necessary and sufficient condition for the firm’s maximization problem is given by equations (1.5) and
(1.6). Combined with the market clearing conditions, we have six equations
and six unknowns.
We next illustrate how one would solve for the equilibrium values. Because the consumer is a representative consumer, we can assume that N = 1
without loss of generality. Thus, 1 − l = n. We substitute for wages and the
rental rate, applying market clearing conditions for the capital stock and for
employment, obtaining
(2.1)
−zfn (k0 , 1 − l) uc [zf (k0 , 1 − l) , l] + ul [zf (k0 , 1 − l) , l] = 0
Note that we have rearranged the expression so that there is one unknown. For strictly concave utility, there is one value of leisure that satisifes
equation (2.1), which is denoted as l∗ . Plug l∗ into equation (1.5) to obtain the equilibrium value of the rental rate; that is, zfk (k0 , 1 − l∗ ) =
r∗ . Similarly, the equilibrium wage rate is determined by the equation
zfn (k0 , 1 − l∗ ) = w∗ . The equilibrium quantity of labor is determined in
the market clearing condition for labor; that is, 1 − l∗ = n∗ and the equilibrium quantity of capital is determined by the endowment of capital; k = k0 .
8
1. A STATIC DECISION PROBLEM
Finally, the equilibrium quantity of consumption determined by the consumer’s budget constraint; that is, w∗ (1 − l∗ ) + r∗ kN0 = c∗ .
The intuition is familiar. Consumers choose the quantity of labor to
supply and firms choose the quantity of labor to employ and the wage rate
is determined so that these quantities are equal. Likewise, the quantity
of capital rented by firms is equal to the quantity of capital supplied by
consumers and the rental rate ensures that these two quantities are equal.
Consumers demand the consumption good and supply labor and capital,
firms demand labor and capital and supply the consumption good, and
prices adjust so that the quantites demanded equal the quantities supplied.
Note that there are three market clearing conditions. Only two of these
equations are linearly independent. To show this, we multiply the price
of each good by the excess demand for each item. Formally, (c − y) +
w [n − (1 − l)] + r (k − k0 )
Because the consumer’s budget constraint holds with equality—c = w (1 − l)+
rk0 —and because the firm has zero profits—zf (k, n) = y = rk + wn, we combine the two, implying that
(2.2)
(c − y) + w [n − (1 − l)] + r (k − k0 ) = 0.
This expression is Walras’ Law. In words, the sum of excess demands in
an economy are always equal to zero.5 Thus, w [n − (1 − l)] + r (k − k0 ) =
− (c − y). The most important implication of this result is that there exists
an interdependence among the excess-demand equations. More concretely,
if we know that there is excess demand in the markets for the consumption
good and the market for labor services (that is, c > y and n > (1 − l)),
equation (2.2) implies that there must be an excess supply in the market
5To make this point explicit, c − w (1 − l) − rk = y − rk − wn. After subtracting
0
the terms on the right-hand-side of the equation from both sides of the expression and
rearranging, we have (c − y) + w [n − (1 − l)] + r (k − k0 ) = 0.
3. PARETO OPTIMUM
9
for capital. Or, if the markets for consumption goods and capital clear—
c = y and n = (1 − l)— it follows that k = k0 . We use this interdependence
to ignore one equation in our model economy. Only two of the excess demands are independent. At the point at which we have six equations and
six unknowns, the linear dependence implies that we drop one market clearing condition. For example, if drop c = y − w [n − (1 − l)] , we have five
equations and five unknowns.
3. Pareto optimum
We begin with the definition of an allocation as a production plan and
a distribution of goods. An allocation is Pareto optimum if there exists no
other allocation which is strictly preferred by some agents but does not make
any other agent worse off.
To illustrate this point, consider a fictious social planner that can costly
acquire all the production and factors of production. In our simple static
economy, the social planner then chooses the quantity of capital and labor
that each agent will supply to the production process and the distribution
of consumption good received by each agent. Since all our agents are identical, the social planner’s problem reduces to solving the problem for one
representative agent. Formally,
max u (c, l)
c,l
(SP)
c = zf (k0 , 1 − l)
Thus, the social planner is benevolent in the sense that the objective is
to maximize the welfare of the representative agent subject to the boundary
of the feasible set. One can think of the feasible set as being the budget constraint faced by the omniscient, benevolent social planner. We assume the social planner can freely dispose, but since the marginal utility
10
1. A STATIC DECISION PROBLEM
of the consumption good is positive, the planner will exhaust any production that is available. In other words, we are concentrating on cases that
lie on the frontier of the production possibilities curve. The upshot is that
we can substitute for consumption in the planner’s problem, rewriting it as
maxl u [zf (k0 , 1 − l) , l].
The necessary condition for solving this unconstrained maximization
problem is
(3.1)
−uc [zf (k0 , 1 − l) , l] ∗ [zfn (k0 , 1 − l)] + ul [zf (k0 , 1 − l) , l] = 0
Upon rearranging, we obtain
(3.2)
zfn (k0 , 1 − l) =
ul [zf (k0 , 1 − l) , l]
.
uc [zf (k0 , 1 − l) , l]
Note that the left-hand-side of equation (3.2) is the marginal rate of
social transformation and the right-hand-side is the marginal rate of substitution. In other words, the left-hand side is the rate at which foregone
leisure—labor—is transformed into units of the consumption good by the social planner while the right-hand side is rate at which consumers marginally
value leisure relative to their marginal value of the consumption good. In
short, this is the condition that satisifes Pareto efficiency.
The solution for the social planner’s problem is straightforward. Note
that (3.2) is one equation in unknown so that the solution for the leisure
allocation is obtained. With strictly concave utility and production, there
is one, unique solution to this expression. If we denote the solution as lSP ,
¡
¢
then labor is represented by nSP = 1 − lSP . Lastly, cSP = zf k0 , 1 − lSP .
The condition for Pareto efficiency is identical the condition in equation
(2.1). Since the latter was derived in our efforts to derive the competitive
equilibrium and the former was the solution to the social planner’s problem. With the planner’s allocation being Pareto optimal, this equivalence
suggests a general result: namely: (i) A competitive equilirium in which
4. COMPARATIVE STATICS
11
there are no externalities, markets are complete and there are no distorting
taxes is Pareto optimal; and (ii) Any Pareto optimum can be supported as
a competitive equilibrium with an appropriate choice of endowments. Condition (i) is the First Welfare Theorem and Condition (ii) is the Second
Welfare Theorem. A connection between the two Welfare Theorems and
the Kuhn-Tucker Theorem is presented in the Apprendix.
4. Comparative statics
In this section, our aim is to find how changes in the exogenous variables affect the equilibrium prices and quantities. To assess the effect on
quantities, it is convenient to use the allocation determined by the social
planner and rely on the Second Welfare Theorem is to ensure the effects we
find from the solution to the social planner’s problem will be the same as
the solution in the competitive equilibrium allocation.
We begin by looking at the effect of change in technology on lesiure. We
obtain this by totally differentiating (3.1), setting dk0 = 0, yielding
−uc [zf (k0 , 1 − l) , l] ∗ fn (k0 , 1 − l) dz − zfn (k0 , 1 − l) ∗ f (k0 , 1 − l) ∗ ucc [zf (k0 , 1 − l) , l] dz
+f (k0 , 1 − l) ∗ ulc [zf (k0 , 1 − l) , l] dz + zfnn (k0 , 1 − l) ∗ uc [zf (k0 , 1 − l) , l] dl
+ucc [zf (k0 , 1 − l) , l] ∗ [zfn (k0 , 1 − l)]2 dl
−zfn (k0 , 1 − l) ucl
∗ [zf (k0 , 1 − l) , l] dl − zfn (k0 , 1 − l) ucl [zf (k0 , 1 − l) , l] dl + ull [zf (k0 , 1 − l) , l] dl
= 0
After rearranging, we get,
uc fn + zfn f ucc − fucl
dl
=
dz
zfnn uc + (zfn )2 ucc − 2zfn ucl + ull
The denominator is negative because we assume that the utility function
is strictly concave. We assume that consumption and leisure are normal
12
1. A STATIC DECISION PROBLEM
goods. With uc , ucl > 0, ull < 0, however, the sign of the numerator is
indeterminate.
4.1. On income and substitution effects. The Slutzky equation
tells us that we can decompose the total effect that a change in total factor
productivity has on leisure into two components: the income effect and the
substitution effect. The decomposition rests on the ability to assess the
impact of the parameter, holding utility constant. To illustrate this point,
start with the following expression:
(4.1)
u (c, l) = h
combined with the equation (3.1), we can proceed with deriving the
substitution effect. Totally differentiate (4.1) and (3.1), setting dh = 0.
From (3.1), one obtains the following expression (note that terms inside
parethenses are omitted)
−fn uc dz − zfn ucc dc − zfn ucl dl + ucl dc + ull dl = 0
Note that uc dc + ul dl = 0 is what holds utility constant in this exercise. For constant utility, we substitute, using dc =- uucl dl, to obtain fn uc dz =
zfn ucc uucl dl−zfn ucl dz −ucl uucl dl +ull dl. Since this expression is conditioned
on welfare held constant, we adopt the notation
dl
dz |subst
to distinguish be-
tween the substitution effect and the total effect. Next, we use the fact that
-zfn uc + ul = 0, which implies that zfn = − uucl , which yields the following
expression
(4.2)
dl
fn uc
|subst =
< 0.
2
dz
zfnn uc + (zfn ) ucc − 2zfn ucl + ull
dl
dl
dl
dz = dz |subst + dz |inc (the
zfn f ucc −f ucl
> 0.
zfnn uc +(zfn )2 ucc −2zfn ucl +ull
It is clear that the numerator is positive. By,
dl
dz |inc =
dl
for dz
<0
Slutzky equation), we know that
Therefore, a sufficient condition
(an increase in total factor pro-
ductivity will result in a decline in equilibrium quantity of lesiure) if the
4. COMPARATIVE STATICS
13
substitution effect is larger in absolute value (dominates) the income effect.
Conversely,
dl
dz
> 0 if the income effect dominates the substitution effect.
With n = 1 − l, it follows that
dn
dz
dl
= − dz
. The change in the equilibrium
quantity of labor depends on whether the income or the substitution effect
dominates. If the substitution effect dominates, labor increases, for instance,
when total factor productivity increases. The other equilibrium quantity is
consumption and the budget constraint is c = zf (k0 , n). Totally differentiating the budget constraint results in dc = fdz + zfn dn ⇒
dc
dz
= f + zfn dn
dz .
From this expression, we can tell that equilibrium consumption increases,
for instance, if the substitution effect dominates.
To illustrate the underlying economic intuition, consider a case in which
total factor productivity increases. Such a positive, unexpected increase
in total factor productivity results in greater income and change in the
marginal product—the relative return—the leisure. Because of higher income,
the consumer will elect to enjoy more leisure. However, the relative return to
work induces the consumer to enjoy less leisure. The latter is the substitution
effect. So, if the substitution effect dominates, lesiure will decline with an
increase in total factor productivity.
To see the effect on equilibrium prices, we begin with the impact on
wages. By the firms’ first-order condition, w = zfn . Totally differentiating
the expression for wages, yields dw = fn dz +zfnn dn ⇒
dw
dz
= fn +zfnn dn
dz . If
the substitition effect dominates, the second term is negative. In words, an
increase in total factor productivity, for example, will result in two countervailing forces. The first term captures the direct effect on wages, reflecting
the gain in marginal productivity. The second term captures the impact
on the quantity of labor; if labor increases, it reduces the wage owing to
diminsihing marginal product of labor.
14
1. A STATIC DECISION PROBLEM
5. Government
In this section, we extend the model economy to consider a role for fiscal
policy. The modification involves a government that collects goods from
consumers by a lump-sum tax. These units of the consumption good are
transformed into a government good at a one-for-one rate. We assume that
the government goods provide some utility to the representative consumer.
We further assume that any such utility is separable in the sense that the
marginal utility of leisure and the consumption good is independent of the
quantity of government goods that are consumed. The level of lump-sum
taxes are set exogenously and consequently, the level of government goods
is exogenously determined. The upshot is that any utility derived from the
government good is akin to a constant level added to the consumer’s welfare
level.
Formally,
u (c, l) + ϕ (g)
s.t. c = w (1 − l) − τ
where τ denotes the quantity of goods collected in the form of lump-sum
taxes. The government budget constraint is represented by the expression,
g = τ.
We proceed along the same lines as we did in the economy without
government. Specifically, substitute for consumption and solve the following
unconstrained maximization problem:
max u [w (1 − l) − τ , l] + ϕ (g)
l
The necessary condition for the maximum is
−wuc [w (1 − l) − τ , l] + ul [w (1 − l) − τ , l] = 0
5. GOVERNMENT
15
which is one equation in unknown.
Meanwhile, for simplicity we consider an economy in which the representative firm has a production technology that is linear in labor and that
capital is excluded from the production process. Let y = zn. Thus, the firm
will maximize
max zn − wn
n
where z = w.6
A competitive equilibrium is defined as an allocation {c, l, n, τ } and a
price {w} which satisfies the following conditions:
(i) the representative consumer chooses c and l to maximize utility, taking w and τ as given;
(ii) the representative firm chooses n to maximize profits, taking w as
given;
(iii) markets for the consumption good and labor clear;
(iiia) the government budget constraint is satisfied.
In the absence of any externality, the Second Welfare theorem will hold,
implying that we can employ the solution to the planner’s problem to determine the quantities. Formally,
u (c, l)
s.t. c + g = z (1 − l)
where the constraint is intrepreted as the economy’s resource cosntraint.
After substitution, the first-order condition for the planner’s maximization
6This condition ensures that the firm will satisfy the zero-profit condition. If z > w,
the firm would maximize profits by employing the full amount of labor. If z < w, the
shutdown condition applies.
16
1. A STATIC DECISION PROBLEM
problem is
(5.1a)
−zuc [z (1 − l) − g, l] + ul [z (1 − l) − g, l] = 0
Following the methods we employed above, the unique solution to this
problem with yield l∗ , which is then plugged into the time constraint to
obtain n∗ = 1 − l∗ and into the representative agent’s budget constraint and
taking g as given to obtain c∗ = z (1 − l∗ ) − g.
Consider the effect that a change in government purchases will have on
the equilibrium values. Totally differentiating (5.1a) yields
zucc dg − ucl dg + z 2 ucc dl − 2zucl dl + ull dl = 0
After rearranging, we get
−zucc + ucl
dl
= 2
dg
z ucc − 2zucl + ull
(5.2)
If leisure is a normal good, the denominator is negative and the numerator is positive, implying that
dl
dg
< 0. In words, the equilibrium quantity
of leisure will decrease, for instance, in response to an exogenous increase
in government purchases. The intuition is straightforward. In this case,
we have a simple income effect. In order to finance larger government purchases, there must be higher taxes. With higher taxes, the representative
consumer sees a reduction in after-tax resources. The income contraction
results in less leisure demanded by the representative consumer.
The effect on equilibrium consumption is determined by totally differentiating the resource constraint. Thus, dc = −z dl − dg ⇒
Upon substituting for
dl
dg
dc
dg
= −z
dl
dg
− 1.
and rearranging terms, we get
zucl − ull
dc
= 2
<0
dg
z ucc − 2zucl + ull
so that the increase in government purchases, for instance, crowds out
purchases of private consumption. Lastly, we study the effect on equilibrium
5. GOVERNMENT
output. With y = z (1 − l), the total derivative is
for
dl
dg ,
17
dy
dg
= −z
dl
dg .
Substitute
expand terms and rearrange, leaving
dy
z 2 ucc − zucl
= 2
.
dg
z ucc − 2zucl + ull
Note that 0 <
dy
dg
< 1 since the numerator is smaller (and the same sign)
as the denominator. The interpretation is for a balanced-budget multiplier.
If we think of government purchases as contributing to the demand side
of the resource constraint, then we are simply asking how an exogenous
increase in demand affects the equilibrium quantity of output. In this simple
economy, we find that the income effect induces some additional work effort
in equilibrium, but not enough to result in a one-for-one (or more) increase
in output. Overall, the increase in government purchases increases output.
However, the overall impact on output is that private consumers have a
smaller share while the government has a larger share. Even though prices—
read wages—are flexible, they do not respond to a change in government
purchases. So the driving force in this simple economy is the reduction
in private wealth that accompanies an increase in government purchases.
While consumers are willing to work a little harder to offset the deleterious
wealth effect, it is not enough to raise output so that both private and public
spending can increase.
Overall, this exercise points to a significant difference between the static general equilibrium model and the textbook IS-LM model. In particular, when general equilibrium effects are properly accounted for, welfaremaximizing consumers will respond to incentives associated with government
policies in a way that renders the policies less attractive than in the sense
that IS-LM models typically deliver a balanced-budget multiplier that is
greater than one.
18
1. A STATIC DECISION PROBLEM
6. Problems
(1) Consider the following representative agent model. The representtive consumer has preferences given by
u (c, l) = c + βl
where c is consumption, l is leisure, and β > 0. The consumer has an
endowment of one unit of time and k0 units of capital. The representative
firm has a technology for producing consumption goods, given by
y = zk α n1−α
where y is output, z is total factor productivity, k is the capital input, n
is the labor input, and 0 < α < 1. The market real wage is w and r denotes
the rental rate on capital.
a. : solve for all prices and quantities in a competitive equilibrium
(there are two cases to consider).
b.: determine the effects that a change in z would have consumption,
output, employment, the real wage, and the rental rate on capital.
Explain your results.
2. Consider an economy with a continuum of consumers, and normalize the total mass of consumers to one. Each consumer has
preferences given by
U (c, l, c̄) = u (c, l) + v (c̄)
where c and l are the individual’s consumption and leisure, respectively,
and c̄ is the average consumption across the population (note that, because
any individual is very small relative to the population, each consumer will
treat c̄ as given). Assume that u (c, l) has standard properties and that v (c̄)
is strictly increasing, strictly concave, and twice differentiable. There is an
6. PROBLEMS
19
externality in consumption in that any individual is better off when others
consume more. The production technology is given by
y=n
where y is output and n is the labor input.
a.: Determine the Pareto optimum (confine attention to allocations
where all consumers consume the same quantities).
b.: Determine the competitive equilibrium, and show that is not
Pareto optimal.
c.: Now suppose that the government subsidizes each individual’s
consumption. that is, for each unit he or she consumes, a consumers receives s units of consumption from the government. the
government finances subsidies to consumers by imposing a lumpsum tax τ on each consumer. Show that, if the government sets the
subsidiy appropriately, then the competitive equilibirum is Pareto
optimal. Determine the optimal subsidy, and explain your results.
CHAPTER 2
Intertemporal models
The purpose of this chapter is two fold. First, we extend the basic static
model to include decisions that explicitly take decisions across time into
account. Second, we develop a model that distinguishes between complete
and incomplete markets. In doing so, we can see how incomplete markets
invalidates the Second Welfare Theorem.
1. Consumers
The consumer’s problem changes in one important aspect. In this model
economy, the consumer is infinitely lived. We continue with the assumption
that all consumers are identical. Their preferences also depend on the quantity of the consumption good and quantity of leisure in a specific time period.
Time is indexed by t = 0, 1, 2, ... We further assume that the utility function is separable across time periods. We formalize the consumer’s lifetime
preferences as
U=
∞
X
β t u (ct , lt )
t=0
where xt denotes the quantity of the good consumer’s enjoy at date t,
for x = c, l. Note that there are now an infinite quantity of goods the
consumer can enjoy over this infinite horizon. To ensure that the problem is
well defined, we need a construct that will guarantee that the infinite sum
of utilities is not infinity. It is difficult to choose a utility maximum when
the value of utility is infinity. Here, we introduce the notion of discounting.
More specifically, 0 < β < 1, is included in the consumer’s problem for a
technical reason and it has an intuitive appeal. Technically, discounting is
21
22
2. INTERTEMPORAL MODELS
a means to ensure that lifetime utility is finite. The intuitive appeal is that
the future requires patience. Suppose c0 = c1 and l0 = l1 . With discounting,
we are saying that future quantities do not yield as much date-0 utility as
current quantities do, holding everything else constant. The time that one
has to wait to enjoy the future quantities is captured by the discount factor,
β.
At each date t, the consumer faces a budget constraint represented as
(1.1)
ct = wt (1 − lt ) − τ t − st+1 + (1 + rt ) st
f or t = 0, 1, 2, ...
where all terms have the same meaning as in the static model. Note
that we have introduced s to stand for the stock of government bonds that
consumers possess. To be more concrete, think of this as consisting of the
quantity of the perishable good that traded to the government. At date t,
st+1 denotes the the quantity of the consumption good traded for one-period
bonds, i.e., bonds that mature in one period. Here, st stands for the quantity
of bonds that mature this period. We assume that bonds acquired at date
t − 1 (that is, st ) will yield 1 + rt units of the consumption good at date
t. Hence, the last term on the right-hand-side (hereafter, rhs) of equation
(1.1), combined with wage income (the first term on the rhs) represents the
resources available for consumption at date t after taxes and newly acquired
government bonds are subtracted.
For now, we will assume the production technology employs only labor.
For simplicity, let the technology be a linear function of the quantity of labor
employed. Formally, yt = zt nt .
The government faces a budget constraint. We permit the government
to issue one-period bonds. At any date, the quantity of government bonds
can be either positive or negative. In each period, the government’s budget
constraint is represented as
1. CONSUMERS
(1.2)
gt + (1 + rt ) bt = τ t + bt+1
23
f or t = 0, 1, 2, ...
where bonds issued at date t − 1 mature, paying 1 + rt units of the
consumption good at date t. Here, bt+1 stands the quantity of bonds issued
by the government at date t. The government budget constraint says that
at each date, the amount of resources spent by the government must be
collected by the government in the form of taxes or bonds issued. Bonds
and storage are perfect substitutes in this environment as indicated by the
fact that both offer the same gross rate of return, 1+rt . For initial conditions
in the bond market, assume that b0 = 0.
There is a looming problem associated with a government that can borrow. Namely, infinitely far out into the future, the government can neither a borrower nor a lender be. So that the government cannot run a
pyramid scheme by paying off current consumers by borrowing from future
versions of the same consumers, we impose a no Ponzi condition: that is,
limT →∞
bT
−1
ΠT
i=1 (1+ri )
= 0. One can crudely translate this condition as saying
that as the economy approaches a limit that is infinitely far into the future,
the present value of outstanding government bonds will be equal to zero.
The counterpart for consumers is that the present value of government
bonds, as one looks out infinitely far into the future, will also equal zero
because of the no-Ponzi condition. Formally, limT →∞
sT
−1
ΠT
i=1 (1+ri )
= 0. For
the consumer, the intuition is borrowed from finite horizon problems. The
idea is essentially as follows: if the economy ends at date T , a consumer
would have no incentive to store goods at date T . Rather, the consumer
would gain utility from eating the consumption good since the marginal
utility of the consumption is positive for any finite quantity of the good.
With the no-Ponzi condition, it is possible to restate the sequence of
budget constraint into a single budget constraint. To do so, note that s1 =
c1 +s2
1+r1
−
w1 (1−l1 )−τ 1
.
1+r1
Repeat this process for s2 =
c2 +s3
(1+r1 )(1+r2 )
−
w2 (1−l2 )−τ 2
(1+r1 )(1+r2 )
24
2. INTERTEMPORAL MODELS
and so on. Because the limiting condition stipulates that the present value
of saving will equal zero, we can substitute for government bonds in the
consumer’s budget constraint, rewriting as
(1.3)
c0 +
∞
X
t=1
∞
X
ct
wt (1 − lt ) − τ t
=
w
(1
−
l
)
−
τ
+
0
0
0
t
Πi=1 (1 + ri )
Πti=1 (1 + ri )
t=1
where the consumer’s budget constraint says that the present value of
goods consumed equals the present value of after-tax resources paid to the
consumer. This representation of the budget constraint establishes a subtle
form of equivalence; that is, there is no difference between the sequence of
budget constraints corresponding a markets meeting at each date t and the
charaxterization of an economy in which all markets meet at the beginning
of time and all goods—present and future—are traded at that Arrow-Debreu
spot market. I am not suggesting that these perishable goods are literally
traded at date t = 0. Rather, it is equivalent to think of the date-0 market
as trading claims against future work and consumption goods.
The first-order conditions for the consumer’s constrained optimization
problem is represented as
λ
(1.4)
β t uc (t) −
Πti=1 (1
(1.5)
β t ul (t) −
λwt
t
Πi=1 (1 + ri )
+ ri )
=0
for t = 1, 2, 3, ...
=0
f or t = 1, 2, 3, ...
(1.6)
uc (0) − λ = 0
(1.7)
ul (0) − λw0 = 0
3. COMPETITIVE EQUILIBRIUM
25
where I adopt the notation that ui (ct , lt ) = ui (t) for i = c, l. Equations
(1.4) and (1.6) say that the discounted marginal utility of consumption is
equal to the present value of the shadow price in the date-0 spot market.
Similarly, equations (1.5) and (1.7) say that the discounted marginal utility
of leisure is equal to the present value of the shadow wage. In all cases, there
is a price for all goods in this economy; the spot price that the consumer
faces depends on the product of the gross real interest rates.
We can rerrange the first-order conditions to obtain:
ul (t)
= wt
uc (t)
and
1
βuc (t + 1)
=
uc (t)
1 + rt+1
2. Firm
The representative firm maximizes profits at each date t, where profits
are represented as
max (zt − wt ) nt
nt
where nt denotes labor demand. Note that labor demand is perfectly
elastic at zt = wt .
3. Competitive equilibrium
A competitive equilibrium consists of quantities, {ct , lt , nt , st+1 , bt+1 , τ t }∞
t=0
and prices, {wt , rt+1 }∞
t=0 that satisfy the following:
(1) consumers choose {ct , lt , st+1 }∞
t=0 taht maximize lifetime utility,
∞
taking {τ t }∞
t=0 and {wt , rt+1 }t=0 as given;
∞
(2) firms choose {nt }∞
t=0 to maximzie profits, taking {wt }t=0 as given;
∞
(3) given {gt }∞
t=0 , {bt+1 , τ t }t=0 satisfy the sequence of government bud-
get constraints;
26
2. INTERTEMPORAL MODELS
(4) markets for the consumption good, for labor, and for government
bonds clear.
By Walras’ Law we can eliminate one market. We choose the market for
the consumption good, leaving us with
st+1 = bt+1
f or t = 0, 1, 2, ...
1 − lt = nt
f or t = 0, 1, 2, ...
and
So the basic intertemporal model can be written in either of two equivalent ways. The first way is to solve it as a sequence of markets each meeting
at a different point of time. Alternatively, each date market is a date good;
there is an infinite variety of goods available at one date. The trade can
occur in a spot market just as Arrow and Debreu and MacKenzie developed the model. The implication is that there is a complete set of ArrowDebreu markets for an infinite dimensional variety of goods. Moreover, we
have prices for these different goods; a date-t consumption good sells for
1
Πti=1 (1+ri )
date-0 goods. Similarly, date-t labor sells for
wt
Πti=1 (1+ri )
units of
the date-0 consumption good.
It is possible to construct an intertemporal government budget constraint. Follow the same methodology that we did to constuct the consumer’s
intertemporal budget constraint; that is, solve for bt+1 and repeatedly substitute. With b0 = 0, we get
(3.1)
g0 +
∞
X
t=1
∞
X
gt
τt
=
τ
+
0
t
t
Πi=1 (1 + ri )
Πi=1 (1 + ri )
t=1
The present value of government purchases is exactly equal to the present
value of taxes.
Now suppose that the sequence of wages and rental rates are those obtained in a competitive equilibrium.Those equilibrium prices are invariant
3. COMPETITIVE EQUILIBRIUM
27
to any sequence of taxes that satisfies (3.1). In other words, taxes can rise
today and fall in the future, or vice versa and the equilibrium prices will be
the same. It further follows that consumer’s allocation and firm’s allocation
are also invariant to the timing of taxes. To illustrate the consumer’s invariance, substitute the government budget constraint into the consumer’s
intertemporal budget constraint, yielding
(3.2)
c0 +
∞
X
t=1
∞
X
ct
wt (1 − lt ) − gt
=
w
.
(1
−
l
)
−
g
+
0
0
0
t
Πti=1 (1 + ri )
Π
(1
+
r
)
i
i=1
t=1
Equation (3.2) indicates that the timing of taxes does not matter since
taxes do not enter into the expression.
This invariance is known as Ricardian Equivalence. For a given present
value of government purchases and taxes, the timing of the government’s
actions do not affect the equilibrium allocations.
Ricardo mentioned to something like this in his analysis. An increase
in government spending today is offset by an increase in future taxes. If
the present value of government purchases is constant, this pattern has no
impact on consumption, labor supply, wages, or interest rates. The key
feature of this model is that there exist a complete set of markets on which
consumers trade. These complete set of markets rest on the notion that
taxes are nondistortionary, consumers are infinitely lived, private firms and
consumers can borrow or lend at the send interest rate (capital markets
are perfect), consumers and firms are identical in the sense that there is no
distributional effects associated with the government actions. In the next
chapter, we examine an economy in which consumers are not infinitely lived.
The upshot is that some consumers cannot trade with future consumers,
rendering markets incomplete.
Thus, one initial result is that if markets are complete, the timing of
consumption is invariant to movements in the nondistortionary taxes. The
consumer has access to markets that permit consumption smoothing. More
concretely, borrowing and lending markets are perfect so that in periods in
28
2. INTERTEMPORAL MODELS
which disposable income is low, the consumer can borrow and repay the
loan when disposable income is high.
4. Problems
(1) Consider the following representative agent model. There is a representative consumer with preferences given by the utility function
u (c, l), where c is the consumption good and l is leisure. Moreover,
the utility function has the properties that we assumed in class.
The representative consumer is endowed with one unit of time and
k0 units of capital. Let the production technology be given by
y = zf (k, n) where y is output, z is total factor productivity, k is
the capital input, n is the labor input. Assume that f (k, n) has
the properties we have assumed in class. Finally, the government
purchases g units of the consumption and finances these purchases
by imposing a lump-sum tax, denoted τ , on consumers.
a.: Determine the equilibrium effects of a change in government purchases on consumption, employment, the real wage, and output.
Assume that consumption and leisure are normal goods for the
representative consumer. Explain your results.
b.: Determine the equilibrium effects of a change in total factor productivity on consumption, employment, the real wage, and output.
Show that your results depend on income and substitution effects
and, where possible, determine the income and substitution effects.
Explain your results
2. Consider a representative agent model where the representative
consumer has preferences given by:
E0
∞
X
t=0
β t [ln (ct ) + ln (lt )]
4. PROBLEMS
29
where 0 < β < 1 is the consumer’s subjective time rate of preference, ct
is consumption, and lt is leisure. The consumer is endowed with one unit of
time each period. The production technology is given by
yt = zt ktα n1−α
t
where y is output, z is a technology shock, k is the capital input, and n
is the labor input. We assume 0 < α < 1. The capital stock depreciates at
a 100% rate each period. In period t, one unit of the consumption good can
be transformed into one unit of capital and this capital becomes productive
in date t + 1. Let zt+1 = ztρ ²t where ln ²t is an i.i.d. random variable with
mean zero and 0 < ρ < 1.
a.: Solve for the competitive equilibrium.
b.: How does employment vary with the technology shock zt ? Is this
model capable of explaining observed fluctuations in employment?
Explain.
c.: How does persistence in the technology shock (ρ > 0) affect consumption, investment, and output over time? Which of these properties do you think are special to this example? Explain.
CHAPTER 3
Overlapping generations
In this chapter, we develop an economic environment in which physical restrictions keep some markets from being available. The overlapping
generations economy is an environemtn in which agents are born and die.
The overlapping part comes from the fact that at any particular date, multiple generations coexist. For simplicity, we focus on an economy in which a
consumer lives for two periods. Thus, two generations are alive at any one
point in time. Here, market incompleteness owes to the physical inability
for agents born at date t to be unable to enter into a market trade with
consumers born at date t + 2 or later. More concretely, Abraham Lincoln
cannot trade with Michael Jordan. At least in the model economy populated
with infinitely-lived households, the decendents of Abraham Lincoln could
trade with Michael Jordan.
There is an infinite sequence of dates, indexed by t = 0, 1, 2, ... The
physical environment initially focuses on the description of the factors of
production. We assume that the initial aggregate stock of capital is K0
and the economy is endowed with this quantity. The population follows a
simple path over time, growing geometrically. Let Lt denote the number of
consumers born at date t growth, then Lt = L0 (1 + n)t , where L0 denotes
the number of consumers at date t = 1 that live for only one period. We
refer to this group as the initial old.
Consumers born at date t ≥ 1 are endowed with one of productive
time when young and nothing when old. Here, young refers to the first
period of the consumers life and old refers to the second period of their life.
Preferences are such that consumers want to eat in both periods of their
31
32
3. OVERLAPPING GENERATIONS
life. Formally, U (c1t , c2t+1 ) where c1t is the quantity of goods consumed
when young and c2t+1 is the quantity of good consumed when old. Further,
we assume that M RS1,2 =
∂U (.,.)/∂c1
∂U (.,.)/∂c2
= ∞ as c1 → 0 and M RS1,2 = 0 as
c1 → ∞. Note that since leisure is not valued, it is straightforward to show
that consumers will work their entire endowment. The inelastic supply of
labor can be thought of as a vertical labor supply curve.
Aggregate production uses capital and labor to produce units of the
consumption good. The technology exhibits constant returns to scale. Formally, we write production as Yt = F (Kt , Lt ). Note that capital consists of
the aggregate quantity of goods accumulated as capital by date t − 1.
There are several things about this environment that are worth noting.
First, there is limited, indeed, no communication across generations that do
not coexist. In other words, a consumer born at date t cannot write a debt
contract that any future generation. The contracts cannot be written when
young because future generations are not born and therefore cannot enter
into contracts. Nor will the date t old accept an iou from the young because
the old will be gone before they get repaid. From the perspective of issuing
debt, such contracts cannot be issued when old because by the time the debt
matures, usually one period later, the old person is gone from the market
and there is no way for the curent young to get repaid.
Second, note that all consumers have the same lifetime preferences. It
will be convenient to start with the aggregate resource constraint. In this
way, we can begin to analyze the planner’s problem. The resource constraint
is
(0.1)
F (Kt , Lt ) + Kt = Kt+1 + Lt c1t + Lt−1 c2t .
The left-hand-side of equation (0.1) represents the total value of resources that are available for this economy, while the right-hand-side talleys
up the potential uses. In words, total output plus the value of the existing
3. OVERLAPPING GENERATIONS
33
(undepreciated) capital stock is used for (gross) investment, consumption
by those born at date t and consumption by those born at date t − 1.
Since all consumers have identical preferences, we start with the supposition that a social planner seeks to maximize the lifetime welfare of
the representative two-period life consumer. Therefore, it simplifies out
analysis to convert the resource constraint into quantities that are specified in per-young-person terms. Divide (0.1) by Lt and using the fact that
Lt = (1 + n) Lt−1 , we obtain
(0.2)
f (kt ) + kt = (1 + n) kt+1 + c1t +
c2t
1+n
We turn now to some definitions presented in welfare economics; namely,
we are interested in Pareto optimality.
©
ª∞
∗
Definition 1. An allocation, c∗1t , c∗2t , kt+1
is Pareto optimal if it
o∞
nt=1
such that
is feasible and there exists no other allocation ĉ1t , ĉ2t , k̂t+1
t=1
¢
¡
ĉ20 ≥ c∗20 and U (ĉ1t , ĉ2t+1 ) ≥ U c∗1t , c∗2t+1 for all t ≥ 1 with at least one
inequality that is strict.
With this definition of inequality, we focus on steady states. Specifically,
c1t = c1t+1 = c1 , c2t = c2t+1 = c2 and kt = kt+1 = k for all t ≥ 1.
After substituting for the steady state value in the resource constraint, the
planner’s problem can be written as:
max U (c1 , c2 )
c1 ,c2 ,k
s.t. f (k) − nk = c1 +
c2
.
1+n
It is possible to further simplify the planner’s problem, substituting the
steady-state representation of the resource constraint for c2 , obtaining the
following unconstrained optimization problem:
34
3. OVERLAPPING GENERATIONS
(0.3)
max U {c1 , (1 + n) [f (k) − nk − c1 ]} .
c1 ,k
The first-order necessary conditions for the optimum are:
(0.4)
U1 − (1 + n) U2 = 0
(0.5)
f 0 (k) − n = 0
Equation (0.5) says that the marginal product of capital must equal
the economy’s net propulation growth rate. Equation (0.4) says that the
consumer is will substitute an infinitesmial amount of consumption when
young provided the utility lost is offset by the marginal utility of the extra
utility that can be gained by consuming when old. Because of the population
growth, every unit of the consumption good that is foregone at date t will
be transformed into 1 + n units of the date-t + 1 consumption good.
It is useful to make two points in order to ease intrepretation later. First,
the two first-order conditions for the planner’s problem can be rearranged,
yielding
(0.6)
U1
= 1 + n = 1 + f 0 (k)
U2
which says that the marginal rate of substitution for the two consumption goods—consumption when young and consumption when old is equal for
all consumers. This condition is one of two necessary conditions for Pareto
optimality. Equation (0.6) further states that the marginal rate of substitution is equal to the marginal rate of transformation. Second, the allocation
that satisfies the first-order conditions is efficient in the sense that all resources are used in their most highly valued fashion, as consumption for
young or old and for investment. There is free disposal in this economy, but
3. OVERLAPPING GENERATIONS
35
consumers would never choose to dispose of goods when either consumption
or investment is an option.
It is straigtforward to solve the for the planner’s allocation. Because
the production technology is strictly concave, equation (0.5) indicates that
there will be
f 0 (k sp ) = n
exactly one value of k, denoted k sp , that satisfies this first-order condition.
With the unique value ksp , we solve for the unique value of csp
1 that satisfies
sp
sp
sp
sp
sp
sp
sp
U1 {csp
1 , (1 + n) [f (k ) − nk − c1 ]}−(1 + n) U2 {c1 , (1 + n) [f (k ) − nk − c1 ]} =
sp
sp
sp
0. It follows that csp
2 = (1 + n) [f (k ) − nk − c1 ]. Thus, we have the al-
location for that solves the planner’s problem.
0.1. Competitive equilibrium. In this section, we consider a decentralized economy. Our aim is to determine whether the competitive equilibrium will yield the same allocation as the planner would choose.
The consumer seeks to maximize lifetime utility. We assume that consumers supply saving, denoted st , when young. The consumer’s program is
written as
max
c1t ,c2t+1 ,st
U (c1t , c2t+1 )
(0.7)
s.t. c1t = wt − st
(0.8)
c2t+1 = (1 + rt+1 ) st .
Note that each unit saved at date t yields 1 + rt+1 goods at date t + 1.
The consumer receives wages, wt units of the consumption good when young.
In a competitive market, the consumer takes w and r as given.We substitute for consumption when young and consumption when old, rewriting the
consumer’s program as an unconstrained maximization problem. Formally,
36
3. OVERLAPPING GENERATIONS
(0.9)
max U [wt − st , (1 + rt+1 ) st ]
st
The first-order necessary condition for the consumer’s program is
(0.10) −U1 [wt − st , (1 + rt+1 ) st ]+(1 + rt+1 ) U2 [wt − st , (1 + rt+1 ) st ] = 0.
Thus, we have one equation in one unknown. We solve equation (0.10)
for st as a function of wages and the real interest rate. Formally, st =
s (wt , rt+1 ). Note that the marginal rate of substitution for the consumer is
U1 (.)
U2 (.)
= 1 + rt+1 .
The firm seeks to maximize profits. Profits are written as the difference
between sales of output produced and expenses, with the latter consisting
of wages and rental rates; formally
F (Kt , Lt ) − wt Lt − rt Kt .
With constant returns to scale, F (Kt , Lt ) = Lt f (kt ). We can rewrite
the profit function, after dividing by Lt , as
max f
kt
µ
Kt
Lt
¶
− wt − rt
Kt
Lt
implying that profit maximization is given by the following two conditions:
(0.11)
f 0 (kt ) − rt = 0
(0.12)
f (kt ) − f 0 (kt ) kt − wt = 0
Together, there are two first-order conditions. The first implies that
the marginal product of the capital-labor ratio equals the rental rate on
3. OVERLAPPING GENERATIONS
37
capital and the zero-profit condition implies that output minus the expense
on capital equals the wage rate.
With the ooptimizing conditions for the two market participants, we can
specify the following definition.
Definition 2. A competitive equilibrium is a sequence of quantities
∞
{kt+1 , st }∞
t=0 and prices {wt , rt }t=0 such that: (i) consumer chooses st to
maximize utility; (ii) firm chooses kt to maximize profit; (iii) markets clear,
given k0 .
The market clearing conditions amount to ensuring that the supply of
capital is equal to the demand for capital. In the aggregate, we can write
Kt+1 = Lt s (wt , rt+1 )
which says that the total quantity of capital demanded is equal to the
total volume of saving. Divide this expression in order to put this market
clearing into per-young-person terms; that is,
Kt+1 Lt+1
Lt+1 Lt
= s (wt , rt+1 ). After
rearranging, we have the following first-order nonlinear difference equation:
(0.13)
£
¤
(1 + n) kt+1 = s f (kt ) − f 0 (kt ) kt , f 0 (kt+1 )
where we substitute for wages and rental rates from the first-order conditions for the firm’s maximization problem. Given k0 , it is possible for
solve sequentially for the entire path of the capital-labor ratio. Once we
have the path for the capital-labor ratio, we can solve for the sequence of
wages and rental rates.1 With these prices, it is straightforward to solve for
saving, for consumption when young and consumption when old.2 Indeed,
1Note that the first-order difference equation for capital is obtained by satisfying
equilibrium conditions. Therefore, it is appropriate to refer to (0.13) as the equilibrium
law of motion.
2With {k}∞ , wages are determined by f (k ) − f 0 (k ) k and the rental rate is det
t
t
t=0
termined by f 0 (kt ). Plug these values into the saving function, st = s (wt , rt+1 ), implying
that c1t = wt − s (wt , rt+1 ) and c2t+1 = (1 + rt+1 ) s (wt , rt+1 ).
38
3. OVERLAPPING GENERATIONS
the rental rate is determined by (0.11) and the wage rate by (0.12). With
the rental rate and wage rate, we compute the level of saving from (0.13).
Consumption when young and when old are determined by equations (0.7)
and (0.8), respectively. Thus, the equilibrium values are obtained.
Next, we turn to a comparison of the optimal allocations under the social
planner’s and the decentralized market ones. One comparison is with respect
to the first-order conditions depicting the trade-off between consumption
when young and consumption when old. Recall that the social planner’s
problem yielded
U1
U2
= 1 + n; in contrast, the representative young person
solves a problem in which
U1
U2
= 1 + r. In the happy coincidence in which
r = n, these conditions are identical and the First Welfare Theorem is
satisfied.
To show how a government can become involved to achieve the firstbest allocation—the one chosen by the social planner—consider a particular
example of an overlapping generations economy in which r 6= n. Suppose
preferences at log and the production function is Cobb-Douglas. Formally,
the person born at date t ≥ 1, maximizes
(0.14)
max [ln (wt − st )] + β ln [(1 + rt+1 ) st ]
st
where β is a parameter that indicates the extent to which the consumer
discounts future utility. We assume that 0 < β < 1. Solving this problem
we find that
(0.15)
st =
β
wt .
1−β
With production technology in intensive form given by
γktα
3. OVERLAPPING GENERATIONS
39
where γ > 0 denotes total factor productivity. This implies that rt =
αγktα−1 and wt = (1 − α) γktα . The goods market clears when the demand
for saving equals the supply:
(0.16)
(1 + n) kt+1 =
β
(1 − α) γktα .
1−β
Focus on a steady state equilibrium, defined as kt+1 = kt = k∗ . Equation
(0.16), reduces to
(1 + n) k∗ =
We solve for k∗ , obtaining
(0.17)
∗
k =
∙µ
β
(1 − α) γ (k ∗ )α .
1−β
β
1+β
¶µ
γ (1 − α)
1+n
¶¸
1
1−α
By plugging in the value of k ∗ into the equilibrium expressions for the
rental rate and the wage rate, we obtain
∙
(1 + β) (1 + n)
r =α
β (1 − α)
∗
and
∙
¸
γβ (1 − α)
w = (1 − α)
(1 + β) (1 + n)
∗
¸
α
1−α
.
Steady state consumption over the the representative consumer’s life is
given by
c∗1 =
w∗
1+β
and
c∗2
∗
= (1 + r )
µ
β
1+β
¶
w∗ .
Our first comparison is between the rental rate and the population
growth rate. In doing so, we are making a comparison between the allocations obtained in the decentralized economy and those obtained by the
40
3. OVERLAPPING GENERATIONS
social planner. With r∗ = α
h
(1+β)(1+n)
β(1−α)
i
, it is only a happy coincidence
that r∗ = n. In general, this condition will not hold. Let ksp denote the
stationary value of the capital stock under the social planner’s program.
¡ ¢ 1
1−α . In words, the long-run
For our setup, γα (ksp )α−1 = n, or ksp = γα
n
steady state value of the capital-labor ratio in the competitive equilibrium
is not equal to the one chosen by the social planner.
Thus, the results indicate that, in general, the competitive equilibrium
is not socially optimal. Without picking parameter values, it is not possible
to determine whether the capital-labor ratio in the competitive equilibrium
is greater than or less than the social planner’s capital-labor ratio. With
k ∗ 6= k sp , we identify a case of dynamic inefficiency. If the capital-labor ratio in the competitive equilibrium is greater than the socially optimal value,
then consumption when young could be greater. If the capital-labor ratio
in the competitive equilibrium is less than the socially optimal value, then
consumption when old could be greater. The bottom line is that lifetime
welfare of the two-period lived consumers is lower in the competitive equilibrium than in the social planner setting.
The source of the dynamic inefficiency in the overlapping generations
economy is market incompleteness. The inability of the current generations
to trade with unborn generations results in the ”wrong” price for future
goods in the overlapping generations economy. The price of old-age consumption, from the perspective of the young person, is 1/r. The rate at
which society can trade one unit of consumption when young for one unit
of consumption when old is 1/n. The price is not equal to the marginal
rate of technical substitution. Based on this (inverse of the) rental rate,
consumers will choose too little (too much) consumption when young when
the rental rate is greater than (less than) the population growth rate. The
wedge between these prices exists because of the restrictions on trades that
is inherent to the overlapping generations model.
3. OVERLAPPING GENERATIONS
41
The purpose of the next extension is to describe a mechanism that permits transfers between the young and the old. In a lump-sum form, these
intergenerational transfers can eliminate the wedge between the marginal
rate of technical substitution and the rental rate determined in the competitive equilibrium. As such, the mechanism designed originally by Diamond
(1965), demonstrates a more general characteristic; there exists a mechanism that guarantees that restores the equality between the competitive
equilibrium allocations and those determined by the social planner.
0.2. The Diamond economy. In this section, we include government
debt as a means of executing intergenerational transfers. The government,
who has a role in economies in which the First Welfare Theorem breaks
down, will choose the size of the transfer so that the allocations in the competitive equilibrium is equal to those chosen by the social planner. Here, the
government’s chief activities is to issue debt to young consumers, execute
a transfer to young consumers, and then tax future young consumers to
pay the interest and principal on this debt. In short, our aim is to demonstrate that a mechanism designed to execute intergenerational transfers can
fix the dynamic inefficiency present in the baseline overlapping generations
economy.
Let Bt+1 denote the aggregate quantity of government debt issued at
date t. The subscript reflects the maturity structure of our government debt;
specifically, all government debt matures one period after issue. For each
one unit of the consumption good traded for government debt at date t, the
bearer of the debt will receive 1+rt=1 units of the consumption good at date
t+1. Note that government debt and capital offer the same gross real return.
The upshot is that government debt and capital are perfect substitutes. I
further assume that the quantity of government debt is fixed in per capita
terms; that is, Bt+1 = bLt , where b is the quantity of government debt per
young consumer.
42
3. OVERLAPPING GENERATIONS
In this setup, suppose the government issues bonds and collects taxes
in order to meets its principal and interest expenses. Taxes are lump-sum
payments made by young consumers. Formally, the government budget
constraint is
Bt+1 + Tt = (1 + rt ) Bt
where Tt = τ t Lt . To represent the government budget constraint in
per-young-consumer terms, we divide by Lt to get
b + τt =
(0.18)
1 + rt
b
1+n
After collecting terms and rearranging to solve for the tax, we get
(0.19)
τt =
µ
rt − n
1+n
¶
b.
The two-period lived consumer solves the following maximization problem
max U [wt − st − τ t , (1 + rt+1 ) st ]
st
taking wages, the real interest rate and taxes as given. The first-order
condition yields a saving function that is written as st = s (wt − τ t , rt+1 ).
Thus, the market clearing condition in the asset market is (in per-youngperson terms): kt+1 (1 + n) + b = s (wt − τ t , rt+1 ), where the left hand side
is interpreted as the supply of asset and the right hand side is the demand.
Because government bonds and capital are perfect substitutes, we do not
need to distinguish between the two on the demand side of the marketclearing expression.
We can further substitute for equilibrium values of wages, the rental and
lump-sum taxes, representing the market-clearing expression as
½
∙ 0
¾
¸
f (kt ) − n
(0.20) kt+1 (1 + n) + b = s f (kt ) − f 0 (kt ) kt −
b, f 0 (kt+1 )
1+n
3. OVERLAPPING GENERATIONS
43
which represents the market-clearing condition as an equilibrium law of
motion for the capital-labor ratio. Indeed, equation (0.20) is a nonlinear
first-order difference equation in the capital-labor ratio. For our purposes,
note that there exists a stationary, or steady state, value of the capital-labor
ratio that satisfies k ∗ (b) = kt = kt+1 . Thus, (0.20) becomes
(0.21)
¸
½
∙ 0 ∗
¾
f [k (b)] − n
∗
0 ∗
∗
0 ∗
k (b) (1 + n)+b = s f [k (b)] − f [k (b)] k (b) −
b, f [k (b)]
1+n
∗
We return to the question that initiated this section; specifically, does
there exists a value of the steady state capital stock such that the stationary
allocation in the decentralized economy is identical to the planner’s allocation. More precisely, is there a value of k ∗ (b) such that f 0 [k ∗ (b)] = n?
From the stationary representation of the equilibrium law of motion, (0.21)
we know that
¸
½
∙ 0 ∗
¾
f [k (b)] − n
∗
0 ∗
∗
0 ∗
b, f [k (b)]
b = −k (b) (1 + n)+s f [k (b)] − f [k (b)] k (b) −
1+n
∗
Note that b can be either positive or negative. A positive value would
correspond to a case in which the government borrows from private citizens
and a negative value would correspond to a government that loans resources
to consumers. The function k∗ (b) is continuous in the bond-per-youngconsumer ratio. Thus, there exists a value of b such that f 0 [k ∗ (b)] = n.
Two additional results follow in our decentralized economy. First, note
³ 0 ∗
´
(b)]−n
that τ t = f [k1+n
b from equation (0.19). It follows immediately that
lump-sum taxes will equal zero since the numerator of this expression vanishes.
Second, there is an intergenerational transfer operating. By assumption, b is a constant that is interpreted as the quantity of bonds issued per
young consumer. Thus, the aggregate quantity of bonds must grow at the
same rate as the population grows. If b > 0, there is a transfer of goods
44
3. OVERLAPPING GENERATIONS
from young consumers to old consumers. Remember that we do not know
whether the decentralized economy chooses a capital-labor ratio that is too
large or too small relative to the Pareto optimum; that is, k ∗ > kSP or
k ∗ < kSP . If k ∗ > kSP , then young consumers are saving ”too much.” Corresondingly, the market rental rate is too low relative to the marginal rate
of technical substitution. In order to reduce the capital-labor ratio, the government issues bonds that are purchased by young consumers. In practice,
young consumers are giving up goods to the government that are then used
to repay old bondholders. It is in this sense that there is an operational
intergenerational transfer. The young consumer’s portfolio is thereby restructured so that the dynamic inefficiency is eliminated, yielding k∗ = kSP
and the rental rate is equal to one plus the population growth rate.
Conversely, if k∗ < kSP , the government sets b < 0. By lending to
young consumers and receiving goods from old consumers, the government is
executing a transfer between old consumers to young consumers. The notion
of an intergenerational transfer that occurs when the government gives goods
to young consumers, using the proceeds from principal and interest paid
by old consumers. Thus, the Diamond model shows that there exists a
market economy, augmented by government paper, that will eliminate the
dynamic inefficiency. The dynamic inefficiency owes to the existence of the
market incompleteness. The First Welfare Theorem ensures that we could
have eliminated the dynamic inefficiency by a series of lump-sum taxes and
transfers. Diamond shows that the dynamic inefficiency can be undone by
issuing government paper.
1. Problems
(1) Consider the Diamond economy.
a. Verify that function k∗ (b) is a continuous function in the bond-peryoung-consumer ratio.
1. PROBLEMS
45
b. Derive the derviative of the function k ∗ (b) with respect to the bondper-young-consumer ratio. State sufficient conditions under which
the derivative is increasing; that is, k ∗0 (b) > 0.