Graduate Macroeconomic Theory Joe Haslag

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 31

1. Problems 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 under-

graduate 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 pe-

riod. Imagine after that period that the economy ends and market partici-

pation 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,k0N .

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 con-

sumer prefers more of each item to less. We further assume that the addi-

tional 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 argu-

ment 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 Toensure 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 rep-

resentative consumer. Formally, the problem is represented as:

(PC) maxc,l

u (c, l)

c ≤ w (1− l) + rks

0 ≤ ks ≤ k0N

0 ≤ l ≤ 1

c ≥ 0

where ks 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 The-

orem. The Lagrangean is

2Here, the notation is ui (c, l) = dudifor i = c, l and uij (c, l) = d2u

didjfor i, j = c, l.

3More concretely, trade for one unit of labor will cost w units of the consumption

good.


(1.1) = u (c, l) + λ

µw + r

k0N− wl − c

¶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 equal-

ity. 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 con-

sumption is positive for any finite level of consumption, the shadow price

of consumption, λ, will be positive. In other words, the consumer will al-

ways 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:

∂

∂c= uc (c, l)− λ = 0

∂

∂l= ul (c, l)− λw = 0

λ

∙w (1− l) + r k0

N− c¸= 0

With uc > 0, then λ > 0, which further implies that w (1− l) + r k0N− c = 0.

1. A ONE-PERIOD MODEL 5

constraint holds as a strict equality. Formally, c = w+ r k0N −wl. If we sub-stitute for consumption, the problem can be rewritten as an unconstrained

optimization problem; that is,

maxl

u

∙w + r

k0N− wl, l

¸At the maximum, the following condition is satisfied:

(1.2) −wucµw + r

k0N− wl, l

¶+ ul

µw + r

k0N− wl, l

¶= 0

1.2. Firms. Firms can be thought of as being endowed with a produc-

tion 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 con-

sumption 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 to-

tal 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 as-

sume 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 as-

sumption 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


(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) maxk,n

zf (k, n)− rk − wn

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}, andprices, {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 con-

dition 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. Be-

cause 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 therental 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 un-

known. 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 ob-

tain 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 equilib-rium quantity of capital is determined by the endowment of capital; k = k0.


Finally, the equilibrium quantity of consumption determined by the con-

sumer’s budget constraint; that is, w∗ (1− l∗) + r∗ k0N = 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 com-

bine 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 existsan 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) − rk0 = y − rk − wn. After subtracting

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 interdependenceto ignore one equation in our model economy. Only two of the excess de-

mands are independent. At the point at which we have six equations and

six unknowns, the linear dependence implies that we drop one market clear-

ing condition. For example, if drop c = y − w [n− (1− l)] , we have fiveequations 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 iden-

tical, the social planner’s problem reduces to solving the problem for one

representative agent. Formally,

maxc,l

u (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 bud-

get constraint faced by the omniscient, benevolent social planner. We as-

sume the social planner can freely dispose, but since the marginal utility


of the consumption good is positive, the planner will exhaust any produc-

tion 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 sub-

stitution. 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 so-

cial 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 prob-

lem. 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. Con-

dition (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 vari-

ables 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,

dl

dz=

ucfn + zfnfucc − fuclzfnnuc + (zfn)

2 ucc − 2zfnucl + ullThe denominator is negative because we assume that the utility function

is strictly concave. We assume that consumption and leisure are normal


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)

−fnuc dz − zfnucc dc− zfnucl dl + ucl dc+ ull dl = 0

Note that uc dc + ul dl = 0 is what holds utility constant in this exer-

cise. For constant utility, we substitute, using dc =- ulucdl, to obtain fnuc dz =

zfnucculucdl−zfnucl dz−ucl uluc dl+ull dl. Since this expression is conditioned

on welfare held constant, we adopt the notation dldz |subst to distinguish be-

tween the substitution effect and the total effect. Next, we use the fact that

-zfnuc + ul = 0, which implies that zfn = − uluc , which yields the followingexpression

(4.2)dl

dz|subst =

fnuc

zfnnuc + (zfn)2 ucc − 2zfnucl + ull

< 0.

It is clear that the numerator is positive. By, dldz =dldz |subst +

dldz |inc (the

Slutzky equation), we know that dldz |inc =

zfnfucc−fuclzfnnuc+(zfn)

2ucc−2zfnucl+ull> 0.

Therefore, a sufficient condition for dldz < 0 (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, dldz > 0 if the income effect dominates the substitution effect.

With n = 1− l, it follows that dndz = −dldz . 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 differenti-

ating the budget constraint results in dc = fdz + zfndn⇒ dcdz = f + zfn

dndz .

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 = fndz+zfnndn⇒ dwdz = fn+zfnn

dndz . If

the substitition effect dominates, the second term is negative. In words, an

increase in total factor productivity, for example, will result in two counter-

vailing 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.


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:

maxlu [w (1− l)− τ , l] + ϕ (g)

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 repre-

sentative 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

maxn

zn− wn

where z = w.6

A competitive equilibrium is defined as an allocation {c, l, n, τ} and aprice {w} which satisfies the following conditions:

(i) the representative consumer chooses c and l to maximize utility, tak-

ing 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 de-

termine 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.


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 andtaking 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 + z2ucc dl − 2zucl dl + ull dl = 0

After rearranging, we get

(5.2)dl

dg=

−zucc + uclz2ucc − 2zucl + ull

If leisure is a normal good, the denominator is negative and the numer-

ator is positive, implying that dldg < 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 pur-

chases, 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 differ-

entiating the resource constraint. Thus, dc = −z dl− dg ⇒ dcdg = −z

dldg − 1.

Upon substituting for dldg and rearranging terms, we get

dc

dg=

zucl − ullz2ucc − 2zucl + ull

< 0

so that the increase in government purchases, for instance, crowds out

purchases of private consumption. Lastly, we study the effect on equilibrium

5. GOVERNMENT 17

output. With y = z (1− l), the total derivative is dydg = −z

dldg . Substitute

for dldg , expand terms and rearrange, leaving

dy

dg=

z2ucc − zuclz2ucc − 2zucl + ull

.

Note that 0 < dydg < 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 sta-

tic general equilibrium model and the textbook IS-LM model. In partic-

ular, when general equilibrium effects are properly accounted for, welfare-

maximizing 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.


6. Problems

(1) Consider the following representative agent model. The represent-

tive 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 = zkαn1−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 nor-

malize the total mass of consumers to one. Each consumer has

preferences given by

U (c, l, c) = u (c, l) + v (c)

where c and l are the individual’s consumption and leisure, respectively,

and c is the average consumption across the population (note that, because

any individual is very small relative to the population, each consumer will

treat c as given). Assume that u (c, l) has standard properties and that v (c)

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 con-

sumers receives s units of consumption from the government. the

government finances subsidies to consumers by imposing a lump-

sum 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 quan-

tity 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 func-

tion is separable across time periods. We formalize the consumer’s lifetime

preferences as

U =∞Xt=0

βtu (ct, lt)

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 for 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 datet. 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 = ztnt.

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 23

(1.2) gt + (1 + rt) bt = τ t + bt+1 for t = 0, 1, 2, ...

where bonds issued at date t − 1 mature, paying 1 + rt units of theconsumption 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 bor-

row. Namely, infinitely far out into the future, the government can nei-

ther 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

ΠT−1i=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

ΠT−1i=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+s21+r1

− w1(1−l1)−τ11+r1

. Repeat this process for s2 = c2+s3(1+r1)(1+r2)

− w2(1−l2)−τ2(1+r1)(1+r2)


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 +∞Xt=1

ctΠti=1 (1 + ri)

= w0 (1− l0)− τ0 +∞Xt=1

wt (1− lt)− τ tΠti=1 (1 + ri)

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

(1.4) βtuc (t)−λ

Πti=1 (1 + ri)= 0 for t = 1, 2, 3, ...

(1.5) βtul (t)−λwt

Πti=1 (1 + ri)= 0 for t = 1, 2, 3, ...

(1.6) uc (0)− λ = 0

(1.7) ul (0)− λw0 = 0


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)

uc (t)= wt

andβuc (t+ 1)

uc (t)=

1

1 + rt+1

2. Firm

The representative firm maximizes profits at each date t, where profits

are represented as

maxnt

(zt − wt)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=0and 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;


(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 for t = 0, 1, 2, ...

and

1− lt = nt for t = 0, 1, 2, ...

So the basic intertemporal model can be written in either of two equiva-

lent 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 devel-

oped the model. The implication is that there is a complete set of Arrow-

Debreu markets for an infinite dimensional variety of goods. Moreover, we

have prices for these different goods; a date-t consumption good sells for1

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 con-

straint. Follow the same methodology that we did to constuct the consumer’s

intertemporal budget constraint; that is, solve for bt+1and repeatedly sub-

stitute. With b0 = 0, we get

(3.1) g0 +∞Xt=1

gtΠti=1 (1 + ri)

= τ0 +∞Xt=1

τ tΠti=1 (1 + ri)

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 ob-

tained in a competitive equilibrium.Those equilibrium prices are invariant


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 in-

variance, substitute the government budget constraint into the consumer’s

intertemporal budget constraint, yielding

(3.2) c0 +∞Xt=1

ctΠti=1 (1 + ri)

= w0 (1− l0)− g0 +∞Xt=1

wt (1− lt)− gtΠti=1 (1 + ri)

.

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


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 rep-

resentative 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 pur-

chases 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 pro-

ductivity 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

∞Xt=0

βt [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 = ztkαt n

1−αt

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 = zρt 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 con-

sumption, investment, and output over time? Which of these prop-

erties do you think are special to this example? Explain.

CHAPTER 3

Overlapping generations

In this chapter, we develop an economic environment in which physi-

cal 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, mul-

tiple 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 MRS1,2 =∂U(.,.)/∂c1∂U(.,.)/∂c2

= ∞ as c1 → 0 and MRS1,2 = 0 as

c1 →∞. Note that since leisure is not valued, it is straightforward to showthat 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. For-

mally, 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 + Ltc1t + Lt−1c2t.

The left-hand-side of equation (0.1) represents the total value of re-

sources 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 sup-

position 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 speci-

fied 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 +c2t1 + n

We turn now to some definitions presented in welfare economics; namely,

we are interested in Pareto optimality.

Definition 1. An allocation,©c∗1t, c

∗2t, k

∗t+1

ª∞t=1

is Pareto optimal if it

is feasible and there exists no other allocationnc1t, c2t, kt+1

o∞t=1

such that

c20 ≥ c∗20 and U (c1t, c2t+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:

maxc1,c2,k

U (c1, c2)

s.t. f (k)− nk = c1 +c21 + 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:


(0.3) maxc1,k

U {c1, (1 + n) [f (k)− nk − c1]} .

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)U1U2= 1 + n = 1 + f 0 (k)

which says that the marginal rate of substitution for the two consump-

tion 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 substitu-

tion is equal to the marginal rate of transformation. Second, the allocation

that satisfies the first-order conditions is efficient in the sense that all re-

sources 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


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 (ksp) = n

exactly one value of k, denoted ksp, that satisfies this first-order condition.

With the unique value ksp, we solve for the unique value of csp1 that satisfies

U1 {csp1 , (1 + n) [f (ksp)− nksp − csp1 ]}−(1 + n)U2 {c

sp1 , (1 + n) [f (k

sp)− nksp − csp1 ]} =0. It follows that csp2 = (1 + n) [f (ksp)− nksp − csp1 ]. Thus, we have the al-location for that solves the planner’s problem.

0.1. Competitive equilibrium. In this section, we consider a decen-

tralized economy. Our aim is to determine whether the competitive equilib-

rium will yield the same allocation as the planner would choose.

The consumer seeks to maximize lifetime utility. We assume that con-

sumers supply saving, denoted st, when young. The consumer’s program is

written as

maxc1t,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 substi-

tute for consumption when young and consumption when old, rewriting the

consumer’s program as an unconstrained maximization problem. Formally,


(0.9) maxstU [wt − st, (1 + rt+1) 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 isU1(.)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)− wtLt − rtKt.

With constant returns to scale, F (Kt, Lt) = Ltf (kt). We can rewrite

the profit function, after dividing by Lt, as

maxkt

f

µKtLt

¶− wt − rt

KtLt

implying that profit maximization is given by the following two condi-

tions:

(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


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 = Lts (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+1Lt

= 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 con-

ditions 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}∞t=0, wages are determined by f (kt) − f

0 (kt) kt and the rental rate is de-

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).


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 U1U2= 1 + n; in contrast, the representative young person

solves a problem in which U1U2= 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 first-

best 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) maxst[ln (wt − st)] + β ln [(1 + rt+1) 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

1− βwt.

With production technology in intensive form given by

γkαt


where γ > 0 denotes total factor productivity. This implies that rt =

αγkα−1t and wt = (1− α) γkαt . The goods market clears when the demand

for saving equals the supply:

(0.16) (1 + n) kt+1 =β

1− β(1− α) γkαt .

Focus on a steady state equilibrium, defined as kt+1 = kt = k∗. Equation

(0.16), reduces to

(1 + n) k∗ =β

1− β(1− α) γ (k∗)α .

We solve for k∗, obtaining

(0.17) k∗ =

∙µβ

1 + β

¶µγ (1− α)

1 + n

¶¸ 11−α

By plugging in the value of k∗ into the equilibrium expressions for the

rental rate and the wage rate, we obtain

r∗ = α

∙(1 + β) (1 + n)

β (1− α)

¸and

w∗ = (1− α)

∙γβ (1− α)

(1 + β) (1 + n)

¸ α1−α

.

Steady state consumption over the the representative consumer’s life is

given by

c∗1 =w∗

1 + β

and

c∗2 = (1 + r∗)

µβ

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 allo-

cations obtained in the decentralized economy and those obtained by the


social planner. With r∗ = αh(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.

For our setup, γα (ksp)α−1 = n, or ksp =¡γαn

¢ 11−α . In words, the long-run

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= ksp, we identify a case of dynamic inefficiency. If the capital-labor ra-tio 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 equi-

librium 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 con-

sumption, 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.


The purpose of the next extension is to describe a mechanism that per-

mits 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 compet-

itive equilibrium. As such, the mechanism designed originally by Diamond

(1965), demonstrates a more general characteristic; there exists a mecha-

nism 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 com-

petitive 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 demon-

strate 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.


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 tLt. To represent the government budget constraint in

per-young-consumer terms, we divide by Lt to get

(0.18) b+ τ t =1 + rt1 + n

b

After collecting terms and rearranging to solve for the tax, we get

(0.19) τ t =

µrt − n1 + n

¶b.

The two-period lived consumer solves the following maximization prob-

lem

maxst

U [wt − st − τ t, (1 + rt+1) 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-young-

person 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 market-

clearing expression.

We can further substitute for equilibrium values of wages, the rental and

lump-sum taxes, representing the market-clearing expression as

(0.20) kt+1 (1 + n) + b = s½f (kt)− f 0 (kt) kt −

∙f 0 (kt)− n1 + n

¸b, f 0 (kt+1)

¾


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)

k∗ (b) (1 + n)+b = s

½f [k∗ (b)]− f 0 [k∗ (b)] k∗ (b)−

∙f 0 [k∗ (b)]− n

1 + n

¸b, f 0 [k∗ (b)]

¾We 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 alloca-

tion. 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

b = −k∗ (b) (1 + n)+s½f [k∗ (b)]− f 0 [k∗ (b)] k∗ (b)−

∙f 0 [k∗ (b)]− n

1 + n

¸b, f 0 [k∗ (b)]

¾Note 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-young-

consumer 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

that τ t =³f 0[k∗(b)]−n

1+n

´b from equation (0.19). It follows immediately that

lump-sum taxes will equal zero since the numerator of this expression van-

ishes.

Second, there is an intergenerational transfer operating. By assump-

tion, 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


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.” Cor-

resondingly, the market rental rate is too low relative to the marginal rate

of technical substitution. In order to reduce the capital-labor ratio, the gov-

ernment 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 re-

structured 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-per-

young-consumer ratio.

1. PROBLEMS 45

b. Derive the derviative of the function k∗(b) with respect to the bond-

per-young-consumer ratio. State sufficient conditions under which

the derivative is increasing; that is, k∗0(b) > 0.

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Graduate Macroeconomic Theory Joe Haslag