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Notes on Macroeconomic Theory
Chapter 1
Simple Representative Agent
Models
This chapter deals with the most simple kind of macroeconomic model,
which abstracts from all issues of heterogeneity and distribution among
economic agents. Here, we study an economy consisting of a representative
¯rm and a representative consumer. As we will show, this is
equivalent, under some circumstances, to studying an economy with
many identical ¯rms and many identical consumers. Here, as in all the
models we will study, economic agents optimize, i.e. they maximize
some objective subject to the constraints they face. The preferences of
consumers, the technology available to ¯rms, and the endowments of
resources available to consumers and ¯rms, combined with optimizing
behavior and some notion of equilibrium, allow us to use the model to
make predictions. Here, the equilibrium concept we will use is competitive
equilibrium, i.e. all economic agents are assumed to be price-takers.
1.1 A Static Model
1.1.1 Preferences, endowments, and technology
There is one period and N consumers, who each have preferences given
by the utility function u(c; `); where c is consumption and ` is leisure.
Here, u(¢; ¢) is strictly increasing in each argument, strictly concave, and
1
2 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
twice di®erentiable. Also, assume that lim c!0u1(c; `) = 1; ` > 0; and
lim`!u (c; `) = 1; c > 0: Here, (c; `) is the partial derivative with
respect to argument i of u(c; `): Each consumer is endowed with one
unit of time, which can be allocated between work and leisure. Each
consumer also owns k0
N units of capital, which can be rented to ¯rms.
There are M ¯rms, which each have a technology for producing
consumption goods according to
y = zf(k; n);
where y is output, k is the capital input, n is the labor input, and z is
a parameter representing total factor productivity. Here, the function
f(¢; ¢) is strictly increasing in both arguments, strictly quasiconcave,
twice di®erentiable, and homogeneous of degree one. That is, production
is constant returns to scale, so that
¸y = zf(¸k; ¸n); (1.1)
for ¸ > 0: Also, assume that limk!f1(k; n) = 1; lk!11(k; n) = 0;
limn!02(k; n) = 1; and lin!1f(k; n) = 0:
1.1.2 Optimization
In a competitive equilibrium, we can at most determine all relative
prices, so the price of one good can arbitrarily be set to 1 with no loss of
generality. We call this good the numeraire. We will follow convention here by treating the consumption good as the numeraire. There are
markets in three objects, consumption, leisure, and the rental services
of capital. The price of leisure in units of consumption is w; and the
rental rate on capital (again, in units of consumption) is r:
Consumer's Problem
Each consumer treats w as being ¯xed, and maximizes utility subject
to his/her constraints. That is, each solves
max
c;`sk
u(c; `)
1.1. A STATIC MODEL 3
subject to
c · w(1 ¡ `) + s(1.2)
0 · ks·
k0
N
(1.3)
0 · ` · 1 (1.4)
c ¸ 0 (1.5)
Here, ks is the quantity of capital that the consumer rents to ¯rms, (1.2)
is the budget constraint, (1.3) states that the quantity of capital rented
must be positive and cannot exceed what the consumer is endowed
with, (1.4) is a similar condition for leisure, and (1.5) is a nonnegativity
constraint on consumption.
Now, given that utility is increasing in consumption (more is preferred
to less), we must have k s= k0
N ; and (1.2) will hold with equality.
Our restrictions on the utility function assure that the nonnegativity
constraints on consumption and leisure will not be binding, and in equilibrium
we will never have ` = 1; as then nothing would be produced,
so we can safely ignore this case. The optimization problem for the consumer
is therefore much simpli¯ed, and we can write down the following
Lagrangian for the problem.
L = u(c; `) + ¹(w + r
k0
N ¡ w` ¡ c);
where ¹ is a Lagrange multiplier. Our restrictions on the utility function
assure that there is a unique optimum which is characterized by
the following ¯rst-order conditions.
@L
@c
= u1 ¡ ¹ = 0
@L
@`
= u2 ¡ ¹w = 0
@L
@¹
= w + r
k0
N ¡ w` ¡ c = 0 Here, u iis the partial derivative of u(¢; ¢) with respect to argument i:
The above ¯rst-order conditions can be used to solve out for ¹ and c
to obtain
wu 1(w + r
k0
N ¡ w`; `) ¡ 2(w + r
k0
N ¡ w`; `) = 0; (1.6)
4 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
which solves for the desired quantity of leisure, `; in terms of w; r; and
k0
N : Equation (1.6) can be rewritten as
u2
u1
= w;
i.e. the marginal rate of substitution of leisure for consumption equals
the wage rate. Diagrammatically, in Figure 1.1, the consumer's budget
constraint is ABD, and he/she maximizes utility at E, where the budget
constraint, which has slope ¡w; is tangent to the highest indi®erence
u1rve, where an indi®erence curve has slope ¡ 2
:
Firm's Problem
Each ¯rm chooses inputs of labor and capital to maximize pro¯ts, treating
w and r as being ¯xed. That is, a ¯rm solves
max
k;n
[zf(k; n) ¡ rk ¡ wn];
and the ¯rst-order conditions for an optimum are the marginal product
conditions
zf1= r; (1.7)
zf2= w; (1.8)
where f idenotes the partial derivative of f(¢; ¢) with respect to argument
i: Now, given that the function f(¢; ¢) is homogeneous of degree
one, Euler's law holds. That is, di®erentiating (1.1) with respect to ¸;
and setting ¸ = 1; we get
zf(k; n) = z1k + zf2n: (1.9)
Equations (1.7), (1.8), and (1.9) then imply that maximized pro¯ts
equal zero. This has two important consequences. The ¯rst is that we
do not need to be concerned with how the ¯rm's pro¯ts are distributed
(through shares owned by consumers, for example). Secondly, suppose
k¤ and n ¤are optimal choices for the factor inputs, then we must have
zf(k; n) ¡ rk ¡ wn = 0 (1.10)
1.1. A STATIC MODEL 5
Figure 1.1:
6 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
for k = k¤and n = n ¤: But, since (1.10) also holds for k = ¸k ¤and
n = ¸n¤ for any ¸ > 0; due to the constant returns to scale assumption,
the optimal scale of operation of the ¯rm is indeterminate. It therefore
makes no di®erence for our analysis to simply consider the case M = 1
(a single, representative ¯rm), as the number of ¯rms will be irrelevant for determining the competitive equilibrium.
1.1.3 Competitive Equilibrium
A competitive equilibrium is a set of quantities, c; `; n; k; and prices
w and r; which satisfy the following properties.
1. Each consumer chooses c and ` optimally given w and r:
2. The representative ¯rm chooses n and k optimally given w and r:
3. Markets clear.
Here, there are three markets: the labor market, the market for
consumption goods, and the market for rental services of capital. In a
competitive equilibrium, given (3), the following conditions then hold.
N(1 ¡ `) = n (1.11)
y = Nc (1.12)
k0 = k (1.13)
That is, supply equals demand in each market given prices. Now, the
total value of excess demand across markets is
Nc ¡ y + w[n ¡ N(1 ¡ `)] + r(k ¡ k0);
but from the consumer's budget constraint, and the fact that pro¯t
maximization implies zero pro¯ts, we have
Nc ¡ y + w[n ¡ N(1 ¡ `)] + r(k ¡ k0) = 0: (1.14)
Note that (1.14) would hold even if pro¯ts were not zero, and were distributed
lump-sum to consumers. But now, if any 2 of (1.11), (1.12),
1.1. A STATIC MODEL 7
and (1.13) hold, then (1.14) implies that the third market-clearing condition
holds. Equation (1.14) is simply Walras' law for this model.
Walras' law states that the value of excess demand across markets is
always zero, and this then implies that, if there are M markets and
M ¡ 1 of those markets are in equilibrium, then the additional market
is also in equilibrium. We can therefore drop one market-clearing
condition in determining competitive equilibrium prices and quantities.
Here, we eliminate (1.12).
The competitive equilibrium is then the solution to (1.6), (1.7),
(1.8), (1.11), and (1.13). These are ¯ve equations in the ¯ve unknowns
`; n, k; w; and r; and we can solve for c using the consumer's budget
constraint. It should be apparent here that the number of consumers,
N; is virtually irrelevant to the equilibrium solution, so for convenience
we can set N = 1, and simply analyze an economy with a single representative
consumer. Competitive equilibrium might seem inappropriate
when there is one consumer and one ¯rm, but as we have shown, in this
context our results would not be any di®erent if there were many ¯rms
and many consumers. We can substitute in equation (1.6) to obtain an
equation which solves for equilibrium `:
zf2(k0; 1 ¡ `)1(zf(k0; 1 ¡ `); `) 2(zf(k0; 1 ¡ `); `) = 0 (1.15)
Given the solution for `; we then substitute in the following equations
to obtain solutions for r; w; n; k, and c:
zf1(k0; 1 ¡ `) = r (1.16)
zf2(k0; 1 ¡ `) = w (1.17)
n = 1 ¡ `
k = k0
c = zf(k0; 1 ¡ `) (1.18)
It is not immediately apparent that the competitive equilibrium exists and is unique, but we will show this later.
8 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
1.1.4 Pareto Optimality
A Pareto optimum, generally, is de¯ned to be some allocation (an allocation
being a production plan and a distribution of goods across
economic agents) such that there is no other allocation which some
agents strictly prefer which does not make any agents worse o®. Here,
since we have a single agent, we do not have to worry about the allocation
of goods across agents. It helps to think in terms of a ¯ctitious
social planner who can dictate inputs to production by the representative
¯rm, can force the consumer to supply the appropriate quantity of
labor, and then distributes consumption goods to the consumer, all in a
way that makes the consumer as well o® as possible. The social planner
determines a Pareto optimum by solving the following problem.
max
c;`
u(c; `)
subject to
c = zf(k0; 1 ¡ `) (1.19)
Given the restrictions on the utility function, we can simply substitute
using the constraint in the objective function, and di®erentiate with
respect to ` to obtain the following ¯rst-order condition for an optimum.
zf2(k0; 1 ¡ `)1[zf(k; 1 ¡ `); `] ¡2[zf(0; 1 ¡ `); `] = 0 (1.20)
Note that (1.15) and (1.20) are identical, and the solution we get for
c from the social planner's problem by substituting in the constraint
will yield the same solution as from (1.18). That is, the competitive
equilibrium and the Pareto optimum are identical here. Further, since
u(¢; ¢) is strictly concave and f(¢; ¢) is strictly quasiconcave, there is a
unique Pareto optimum, and the competitive equilibrium is also unique.
Note that we can rewrite (1.20) as
zf2=
u2
u1
;
where the left side of the equation is the marginal rate of transformation,
and the right side is the marginal rate of substitution of consumption
for leisure. In Figure 1.2, AB is equation (1.19) and the Pareto
1.1. A STATIC MODEL 9
optimum is at D, where the highest indi®erence curve is tangent to
the production possibilities frontier. In a competitive equilibrium, the
representative consumer faces budget constraint AFG and maximizes
at point D where the slope of the budget line, ¡w; is equal to ¡ u2
u1
:
In more general settings, it is true under some restrictions that the
following hold.
1. A competitive equilibrium is Pareto optimal (First Welfare Theorem).
2. Any Pareto optimum can be supported as a competitive equilibrium
with an appropriate choice of endowments. (Second Welfare
Theorem).
The non-technical assumptions required for (1) and (2) to go through include the absence of externalities, completeness of markets, and absence
of distorting taxes (e.g. income taxes and sales taxes). The First
Welfare Theorem is quite powerful, and the general idea goes back as
far as Adam Smith's Wealth of Nations. In macroeconomics, if we can
successfully explain particular phenomena (e.g. business cycles) using
a competitive equilibrium model in which the First Welfare Theorem
holds, we can then argue that the existence of such phenomena is not
grounds for government intervention.
In addition to policy implications, the equivalence of competitive
equilibria and Pareto optima in representative agent models is useful for
computational purposes. That is, it can be much easier to obtain competitive
equilibria by ¯rst solving the social planner's problem to obtain
competitive equilibrium quantities, and then solving for prices, rather
than solving simultaneously for prices and quantities using marketclearing
conditions. For example, in the above example, a competitive
equilibrium could be obtained by ¯rst solving for c and ` from the social
planner's problem, and then ¯nding w and r from the appropriate marginal
conditions, (1.16) and (1.17). Using this approach does not make
much di®erence here, but in computing numerical solutions in dynamic
models it can make a huge di®erence in the computational burden.
10 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
Figure 1.2:
1.1. A STATIC MODEL 11
1.1.5 Example
Consider the following speci¯c functional forms. For the utility function,
we use
u(c; `) =
c1¡°¡ 1
1 ¡ °
+ `;
where ° > 0 measures the degree of curvature in the utility function
with respect to consumption (this is a \constant relative risk aversion"
utility function). Note that
lim
°!1
c1¡°¡ 1
1 ¡ °
= lim
d!1
d°[e(1¡°) lo¡ 1]
d
d°(1 ¡ °)
= log c;
using L'Hospital's Rule. For the production technology, use
f(k; n) = ® n1¡®;
where 0 < ® < 1: That is, the production function is Cobb-Douglas.
The social planner's problem here is then
max
`
(
[zk® 0(1 ¡ `)1¡®1¡°¡ 1
1 ¡ °
+ `
)
;
and the solution to this problem is
` = 1 ¡ [(1 ¡ ®)(zk®
0)1¡°]
1
®+(1¡®(1.21)
As in the general case above, this is also the competitive equilibrium
solution. Solving for c; from (1.19), we get
c = [(1 ¡ ®) 1¡(zk ®
0)]
1
®+(1¡®; (1.22)
and from (1.17), we have
w = [(1 ¡ ®) 1¡®(zk ®
0)]
°
®+(1¡®(1.23)
From (1.22) and (1.23) clearly c and w are increasing in z and k 0: That
is, increases in productivity and in the capital stock increase aggregate
consumption and real wages. However, from equation (1.21) the e®ects
12 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
on the quantity of leisure (and therefore on employment) are ambiguous.
Which way the e®ect goes depends on whether ° < 1 or ° > 1:
With ° < 1; an increase in z or in k 0 will result in a decrease in leisure,
and an increase in employment, but the e®ects are just the opposite
if ° > 1: If we want to treat this as a simple model of the business
cycle, where °uctuations are driven by technology shocks (changes in
z); these results are troubling. In the data, aggregate output, aggregate
consumption, and aggregate employment are mutually positively correlated.
However, this model can deliver the result that employment and
output move in opposite directions. Note however, that the real wage
will be procyclical (it goes up when output goes up), as is the case in
the data.
1.1.6 Linear Technology - Comparative Statics
This section illustrates the use of comparative statics, and shows, in a
somewhat more general sense than the above example, why a productivity
shock might give a decrease or an increase in employment. To
make things clearer, we consider a simpli¯ed technology,
y = zn;
i.e. we eliminate capital, but still consider a constant returns to scale
technology with labor being the only input. The social planner's problem
for this economy is then
max
`
u[z(1 ¡ `); `];
and the ¯rst-order condition for a maximum is
¡zu 1[z(1 ¡ `); `] + 2[z(1 ¡ `); `] = 0: (1.24)
Here, in contrast to the example, we cannot solve explicitly for `; but note that the equilibrium real wage is
w =
@y
@n
= z;
so that an increase in productivity, z, corresponds to an increase in the
real wage faced by the consumer. To determine the e®ect of an increase
1.1. A STATIC MODEL 13
in z on `; apply the implicit function theorem and totally di®erentiate
(1.24) to get
[¡u1¡ z(1 ¡ `)11+ u 21(1 ¡ `)]dz
+(z2u 11¡ 2zu12 + u22)d` = 0:
We then have
d`
dz
=
u1 + z(1 ¡ `)11 ¡ 21(1 ¡ `)
z2u11 ¡ 2zu12+ u 22
: (1.25)
Now, concavity of the utility function implies that the denominator
in (1.25) is negative, but we cannot sign the numerator. In fact, it
is easy to construct examples where d`
dz> 0; and where d`
dz< 0: The
ambiguity here arises from opposing income and substitution e®ects.
In Figure 1.3, AB denotes the resource constraint faced by the social
planner, c = z 1(1¡`); and BD is the resource constraint with a higher
level of productivity, 2 > z1: As shown, the social optimum (also the
competitive equilibrium) is at E initially, and at F after the increase in
productivity, with no change in ` but higher c: E®ectively, the representative
consumer faces a higher real wage, and his/her response can
be decomposed into a substitution e®ect (E to G) and an income e®ect
(G to F).
Algebraically, we can determine the substitution e®ect on leisure
by changing prices and compensating the consumer to hold utility constant,
i.e.
u(c; `) = h; (1.26)
where h is a constant, and
¡zu1(c; `) + 2(c; `) = 0 (1.27)
Totally di®erentiating (1.26) and (1.27) with respect to c and `; and using
(1.27) to simplify, we can solve for the substitution e®ect d`
dz(subst:)
as follows.
d`
dz
(subst:) =
u1
z2u11 ¡ 2zu12+ u 22
< 0:
From (1.25) then, the income e®ect d` dz(inc:) is just the remainder,
d`
dz
(inc:) =
z(1 ¡ `)11¡ u21(1 ¡ `)
z2u11 ¡ 2zu12+ u22
> 0;
14 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
Figure 1.3:
1.2. GOVERNMENT 15
provided ` is a normal good. Therefore, in order for a model like this
one to be consistent with observation, we require a substitution e®ect
that is large relative to the income e®ect. That is, a productivity shock,
which increases the real wage and output, must result in a decrease in
leisure in order for employment to be procyclical, as it is in the data.
In general, preferences and substitution e®ects are very important in
equilibrium theories of the business cycle, as we will see later.
1.2 Government
So that we can analyze some simple ¯scal policy issues, we introduce a
government sector into our simple static model in the following manner.
The government makes purchases of consumption goods, and ¯-
nances these purchases through lump-sum taxes on the representative
consumer. Let g be the quantity of government purchases, which is
treated as being exogenous, and let ¿ be total taxes. The government
budget must balance, i.e.
g = ¿: (1.28)
We assume here that the government destroys the goods it purchases.
This is clearly unrealistic (in most cases), but it simpli¯es matters,
and does not make much di®erence for the analysis, unless we wish
to consider the optimal determination of government purchases. For
example, we could allow government spending to enter the consumer's
utility function in the following way.
w(c; `; g) = u(c; `) + v(g)
Given that utility is separable in this fashion, and g is exogenous, this
would make no di®erence for the analysis. Given this, we can assume
v(g) = 0:
As in the previous section, labor is the only factor of production,
i.e. assume a technology of the form
y = zn:
Here, the consumer's optimization problem is
c;`
u(c; `)
16 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
subject to
c = w(1 ¡ `) ¡ ¿;
and the ¯rst-order condition for an optimum is
¡wu 1+ u2 = 0:
The representative ¯rm's pro¯t maximization problem is
max n
(z ¡ w)n:
Therefore, the ¯rm's demand for labor is in¯nitely elastic at w = z:
A competitive equilibrium consists of quantities, c; `; n; and ¿; and
a price, w; which satisfy the following conditions:
1. The representative consumer chooses c and ` to maximize utility,
given w and ¿:
2. The representative ¯rm chooses n to maximize pro¯ts, given w:
3. Markets for consumption goods and labor clear.
4. The government budget constraint, (1.28), is satis¯ed.
The competitive equilibrium and the Pareto optimum are equivalent
here, as in the version of the model without government. The social
planner's problem is
max
c;`
u(c; `)
subject to
c + g = z(1 ¡ `)
Substituting for c in the objective function, and maximizing with respect
to `; the ¯rst-order condition for this problem yields an equation
which solves for ` :
¡zu1[z(1 ¡ `) ¡ g; `] 2[z(1 ¡ `) ¡ g; `] = 0: (1.29)
In Figure 1.4, the economy's resource constraint is AB, and the Pareto
optimum (competitive equilibrium) is D. Note that the slope of the
resource constraint is ¡z = ¡w:
1.2. GOVERNMENT 17
Figure 1.4:
18 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
We can now ask what the e®ect of a change in government expenditures
would be on consumption and employment. In Figure 1.5, g
increases from g 1to g2; shifting in the resource constraint. Given the
government budget constraint, there is an increase in taxes, which represents
a pure income e®ect for the consumer. Given that leisure and
consumption are normal goods, quantities of both goods will decrease.
Thus, there is crowding out of private consumption, but note that the
decrease in consumption is smaller than the increase in government
purchases, so that output increases. Algebraically, totally di®erentiate
(1.29) and the equation c = z(1 ¡ `) ¡ g and solve to obtain
d`
dg
= ¡zu11 + u12
z2u11 ¡ 2zu12+ u 22
< 0
dc
dg
=
zu 12¡ u22
z2u11 ¡ 2zu12+ u 22
< 0 (1.30)
Here, the inequalities hold provided that ¡zu 11 + u12> 0 and zu 12 ¡
u22 > 0; i.e. if leisure and consumption are, respectively, normal goods. Note that (1.30) also implies that dy
dg < 1; i.e. the \balanced budget
multiplier" is less than 1.
1.3 A \Dynamic" Economy
We will introduce some simple dynamics to our model in this section.
The dynamics are restricted to the government's ¯nancing decisions;
there are really no dynamic elements in terms of real resource allocation,
i.e. the social planner's problem will break down into a series of
static optimization problems. This model will be useful for studying
the e®ects of changes in the timing of taxes.
Here, we deal with an in¯nite horizon economy, where the representative
consumer maximizes time-separable utility,
1X
t=0
¯ u(c t; t);
where ¯ is the discount factor, 0 < ¯ < 1: Letting ± denote the discount
rate, we have ¯ = 1
1+± ; where ± > 0: Each period, the consumer
is endowed with one unit of time. There is a representative ¯rm
1.3. A \DYNAMIC" ECONOMY 19
Figure 1.5:
20 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
which produces output according to the production function y t= z n :
The government purchases g tunits of consumption goods in period t;
t = 0; 1; 2; :::; and these purchases are destroyed. Government purchases
are ¯nanced through lump-sum taxation and by issuing one-period government
bonds. The government budget constraint is
g t+ (1 + r )b t= ¿ t+ b t+1; (1.31)
t = 0; 1; 2; :::; where b tis the number of one-period bonds issued by
the government in period t¡1: A bond issued in period t is a claim to
1+r t+1units of consumption in period t+1; where r t+1is the one-period
interest rate. Equation (1.31) states that government purchases plus
principal and interest on the government debt is equal to tax revenues
plus new bond issues. Here, b 0= 0:
The optimization problem solved by the representative consumer is
max
ft+;c;tg1t
=0;
1X
t=0
¯ u(c t; t)
subject to
ct= w (1 ¡ `t) ¡ ¿t¡ st+1+ (1 + r )st; (1.32)
t = 0; 1; 2; :::; s0= 0; where s t+1is the quantity of bonds purchased
by the consumer in period t, which come due in period t + 1: Here, we
permit the representative consumer to issue private bonds which are
perfect substitutes for government bonds.
We will assume that
lim
n!1
sn Q n¡1
i=1(1 + r ) = 0; (1.33)
which states that the quantity of debt, discounted to t = 0; must equal
zero in the limit. This condition rules out in¯nite borrowing or \Ponzi
schemes," and implies that we can write the sequence of budget constraints,
(1.32) as a single intertemporal budget constraint. Repeated
substitution using (1.32) gives
c0 +
1X
t=1
ctQ ti
=1(1 + r )
= w 0 (1 ¡ 0 ) ¡ ¿0+
1X
t=1
w t(1 ¡ t) ¡ ¿tQ ti
=1(1 + r )
: (1.34)
1.3. A \DYNAMIC" ECONOMY 21
Now, maximizing utility subject to the above intertemporal budget
constraint, we obtain the following ¯rst-order conditions.
¯ u 1(ct; t) ¡
¸
Q ti
=1(1 + r )
= 0; t = 1; 2; 3; :::
¯ u 2(ct; t) ¡
¸w tQ ti
=1(1 + r )
= 0; t = 1; 2; 3; :::
u 1(c0; `0) ¡ ¸ = 0
u 2(c0; `0) ¡ ¸w0 = 0
Here, ¸ is the Lagrange multiplier associated with the consumer's intertemporal
budget constraint. We then obtain
u 2(ct; t)
u 1(ct; t)
= w t; (1.35)
i.e. the marginal rate of substitution of leisure for consumption in any
period equals the wage rate, and
¯u 1(c t+; `t+)
u 1(ct; t)
=
1
1 + r t+1
; (1.36)
i.e. the intertemporal marginal rate of substitution of consumption
equals the inverse of one plus the interest rate.
The representative ¯rm simply maximizes pro¯ts in each period, i.e.
it solves
max
nt
(zt¡ w t)nt;
and labor demand, n ; is perfectly elastic at w t= z t: A competitive equilibrium consists of quantities, fc ; `; nt; st+; bt+1; ¿tg1t=0;
and prices fw t; t+1g1t=0 satisfying the following conditions.
1. Consumers choose fc ; `; st+1; g1t=0optimally given f¿ g and fw t; t+1g 1t=:
2. Firms choose fn g 1t=0optimally given fw g 1t=:
3. Given fg tg1t=0; fbt+1; ¿g 1t=0satis¯es the sequence of government
budget constraints (1.31).
22 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
4. Markets for consumption goods, labor, and bonds clear. Walras'
law permits us to drop the consumption goods market from
consideration, giving us two market-clearing conditions:
st+1 = bt+1; t = 0; 1; 2; :::; (1.37)
and
1 ¡ `t= n ; t = 0; 1; 2; :::
Now, (1.33) and (1.37) imply that we can write the sequence of
government budget constraints as a single intertemporal government
budget constraint (through repeated substitution):
g 0+
1X
t=1
g tQti
=1(1 + r )
= ¿ 0 +
1X
t=1
¿ tQ ti
=1(1 + r )
; (1.38)
i.e. the present discounted value of government purchases equals the
present discounted value of tax revenues. Now, since the government
budget constraint must hold in equilibrium, we can use (1.38) to substitute
in (1.34) to obtain
c0 +
1X
t=1
ctQ ti
=1(1 + r )
= w 0(1 ¡ `0) ¡ g0 +
1X
t=1
w (1 ¡ `t) ¡ gtQ ti
=1(1 + r )
: (1.39)
Now, suppose that fw t; t+1g 1t=0are competitive equilibrium prices.
Then, (1.39) implies that the optimizing choices given those prices remain
optimal given any sequence f¿ tg 1t=0satisfying (1.38). Also, the
representative ¯rm's choices are invariant. That is, all that is relevant
for the determination of consumption, leisure, and prices, is the present
discounted value of government purchases, and the timing of taxes is
irrelevant. This is a version of the Ricardian Equivalence Theorem. For
example, holding the path of government purchases constant, if the
representative consumer receives a tax cut today, he/she knows that
the government will have to make this up with higher future taxes. The government issues more debt today to ¯nance an increase in the
government de¯cit, and private saving increases by an equal amount,
since the representative consumer saves more to pay the higher taxes
in the future.
1.3. A \DYNAMIC" ECONOMY 23
Another way to show the Ricardian equivalence result here comes
from computing the competitive equilibrium as the solution to a social
planner's problem, i.e.
max
fg1t
=0
1X
t=0
¯tu[z (1 ¡ `) ¡ gt; t]
This breaks down into a series of static problems, and the ¯rst-order
conditions for an optimum are
¡ztu1[zt(1 ¡ `) ¡ g; `] + u 2[z(1 ¡ `) ¡ gt; t] = 0; (1.40)
t = 0; 1; 2; ::: . Here, (1.40) solves for ` ; t = 0; 1; 2; :::; and we can solve
for ctfrom c t= zt(1 ¡ `): Then, (1.35) and (1.36) determine prices.
Here, it is clear that the timing of taxes is irrelevant to determining the
competitive equilibrium, though Ricardian equivalence holds in much
more general settings where competitive equilibria are not Pareto optimal,
and where the dynamics are more complicated.
Some assumptions which are critical to the Ricardian equivalence
result are:
1. Taxes are lump sum
2. Consumers are in¯nite-lived.
3. Capital markets are perfect, i.e. the interest rate at which private
agents can borrow and lend is the same as the interest rate at
which the government borrows and lends.
4. There are no distributional e®ects of taxation. That is, the present
discounted value of each individual's tax burden is una®ected by
changes in the timing of aggregate taxation.
24 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS
Chapter 2
Growth With Overlapping
Generations
This chapter will serve as an introduction to neoclassical growth theory
and to the overlapping generations model. The particular model introduced
in this chapter was developed by Diamond (1965), building on
the overlapping generations construct introduced by Samuelson (1956).
Samuelson's paper was a semi-serious (meaning that Samuelson did not
take it too seriously) attempt to model money, but it has also proved
to be a useful vehicle for studying public ¯nance issues such as government
debt policy and the e®ects of social security systems. There
was a resurgence in interest in the overlapping generations model as a
monetary paradigm in the late seventies and early eighties, particularly
at the University of Minnesota (see for example Kareken and Wallace 1980).
A key feature of the overlapping generations model is that markets
are incomplete, in a sense, in that economic agents are ¯nite-lived,
and agents currently alive cannot trade with the unborn. As a result,
competitive equilibria need not be Pareto optimal, and Ricardian
equivalence does not hold. Thus, the timing of taxes and the size of the
government debt matters. Without government intervention, resources
may not be allocated optimally among generations, and capital accumulation
may be suboptimal. However, government debt policy can
be used as a vehicle for redistributing wealth among generations and
inducing optimal savings behavior.
25
26CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS
2.1 The Model
This is an in¯nite horizon model where time is indexed by t = 0; 1; 2; :::;1.
Each period, L ttwo-period-lived consumers are born, and each is endowed
with one unit of labor in the ¯rst period of life, and zero units
in the second period. The population evolves according to
Lt= L 0(1 + n) ; (2.1)
where L 0 is given and n > 0 is the population growth rate. In period 0
there are some old consumers alive who live for one period and are collectively
endowed with K 0 units of capital. Preferences for a consumer
born in period t; t = 0; 1; 2; :::; are given by
u(c y
t; co t+);
where c y
tdenotes the consumption of a young consumer in period t and
cot
is the consumption of an old consumer. Assume that u(¢; ¢) is strictly
increasing in both arguments, strictly concave, and de¯ning
v(cy; co) ´
@u
@c y
@u
@c o
;
assume that lim cy!ov(cy; co) = 1 for co > 0 and lim co!ov(cy; co) = 0
for cy> 0: These last two conditions on the marginal rate of substitution
will imply that each consumer will always wish to consume positive
amounts when young and when old. The initial old seek to maximize
consumption in period 0:
The investment technology works as follows. Consumption goods
can be converted one-for-one into capital, and vice-versa. Capital constructed
in period t does not become productive until period t+1; and
there is no depreciation.
Young agents sell their labor to ¯rms and save in the form of capital
accumulation, and old agents rent capital to ¯rms and then convert
the capital into consumption goods which they consume. The representative
¯rm maximizes pro¯ts by producing consumption goods, and
renting capital and hiring labor as inputs. The technology is given by
Y t= F(K t; L); 2.2. OPTIMAL ALLOCATIONS 27
where Y tis output and K tand L tare the capital and labor inputs,
respectively. Assume that the production function F(¢; ¢) is strictly increasing,
strictly quasi-concave, twice di®erentiable, and homogeneous
of degree one.
2.2 Optimal Allocations
As a benchmark, we will ¯rst consider the allocations that can be
achieved by a social planner who has control over production, capital
accumulation, and the distribution of consumption goods between
the young and the old. We will con¯ne attention to allocations where
all young agents in a given period are treated identically, and all old
agents in a given period receive the same consumption.
The resource constraint faced by the social planner in period t is
F(K ; Lt) + K t= K t+1+ c y
tLt+ c ot
L t¡; (2.2)
where the left hand side of (2.2) is the quantity of goods available in
period t; i.e. consumption goods produced plus the capital that is left
after production takes place. The right hand side is the capital which
will become productive in period t + 1 plus the consumption of the
young, plus consumption of the old.
In the long run, this model will have the property that per-capita
quantities converge to constants. Thus, it proves to be convenient to
express everything here in per-capita terms using lower case letters.
De¯ne k t´K t
Lt
(the capital/labor ratio or per-capita capital stock) and
f(kt) ´ F(kt; 1): We can then use (2.1) to rewrite (2.2) as
f(kt) + kt= (1 + n)k t+1 + cy
t+
cot
1 + n
(2.3)
De¯nition 1 A Pareto optimal allocation is a sequence fc y
t; cot
; kt+1g1t=0
satisfying (2.3) and the property that there exists no other allocation
f^cy
t; ^cot
; ^kt+1g 1t=0which satis¯es (2.3) and
^c o
1 ¸ co
1
u(^c y
t; ^co t+1) ¸ u(cy
t; co t+)
for all t = 0; 1; 2; 3; :::; with strict inequality in at least one instance.
28CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS
That is, a Pareto optimal allocation is a feasible allocation such
that there is no other feasible allocation for which all consumers are at least as well o® and some consumer is better o®. While Pareto
optimality is the appropriate notion of social optimality for this model,
it is somewhat complicated (for our purposes) to derive Pareto optimal
allocations here. We will take a shortcut by focusing attention on steady
states, where k t= k; cy
t= cy; and c ot
= co; where k; c y; and c oare
constants. We need to be aware of two potential problems here. First,
there may not be a feasible path which leads from k 0to a particular
steady state. Second, one steady state may dominate another in terms
of the welfare of consumers once the steady state is achieved, but the
two allocations may be Pareto non-comparable along the path to the
steady state.
The problem for the social planner is to maximize the utility of each
consumer in the steady state, given the feasibility condition, (2.2). That
is, the planner chooses c y; o; and k to solve
max u(c y; o)
subject to
f(k) ¡ nk = cy+
co
1 + n
: (2.4)
Substituting for c oin the objective function using (2.4), we then solve
the following
max
c;k
u(c ; [1 + n][f(k) ¡ nk ¡ ])
The ¯rst-order conditions for an optimum are then
u1 ¡ (1 + n)u2 = 0;
or
u1
u2
= 1 + n (2.5)
(intertemporal marginal rate of substitution equal to 1 + n) and
f0(k) = n (2.6)
(marginal product of capital equal to n): Note that the planner's problem
splits into two separate components. First, the planner ¯nds the
2.3. COMPETITIVE EQUILIBRIUM 29
capital-labor ratio which maximizes the steady state quantity of resources,
from (2.6), and then allocates consumption between the young
and the old according to (2.5). In Figure 2.1, k is chosen to maximize
the size of the budget set for the consumer in the steady state, and then
consumption is allocated between the young and the old to achieve the
tangency between the aggregate resource constraint and an indi®erence
curve at point A.
2.3 Competitive Equilibrium
In this section, we wish to determine the properties of a competitive
equilibrium, and to ask whether a competitive equilibrium achieves the
steady state social optimum characterized in the previous section.
2.3.1 Young Consumer's Problem A consumer born in period t solves the following problem.
max
t;ot
+1;t
u(c y
t; co t+)
subject to
cy
t= w t¡ st(2.7)
co t+1= s t(1 + r t+) (2.8)
Here, w tis the wage rate, r tis the capital rental rate, and s t is saving
when young. Note that the capital rental rate plays the role of an interest
rate here. The consumer chooses savings and consumption when
young and old treating prices, w tand r t+1; as being ¯xed. At time t
the consumer is assumed to know r t+1: Equivalently, we can think of
this as a rational expectations or perfect foresight equilibrium, where
each consumer forecasts future prices, and optimizes based on those
forecasts. In equilibrium, forecasts are correct, i.e. no one makes systematic
forecasting errors. Since there is no uncertainty here, forecasts
cannot be incorrect in equilibrium if agents have rational expectations.
30CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS
Figure 2.1:
2.3. COMPETITIVE EQUILIBRIUM 31
Substituting for c y
tand c o t+1in the above objective function using
(2.7) and (2.8) to obtain a maximization problem with one choice variable,
st; the ¯rst-order condition for an optimum is then
¡u 1(w ¡s ; st(1+r t+1))+u 2(w t¡s; st(1+r t+1))(1+r t+1) = 0 (2.9)
which determines s ; i.e. we can determine optimal savings as a function
of prices
st = s(w t; rt+): (2.10)
Note that (2.9) can also be rewritten as u1
u2
= 1 + r t+1; i.e. the intertemporal
marginal rate of substitution equals one plus the interest
rate. Given that consumption when young and consumption when old
are both normal goods, we have @s
@w t
> 0; however the sign of @s
@r t+1
is
indeterminate due to opposing income and substitution e®ects.
2.3.2 Representative Firm's Problem
The ¯rm solves a static pro¯t maximization problem
max
Kt;t
[F(K t; Lt) ¡ wtLt ¡ tK ]:
The ¯rst-order conditions for a maximum are the usual marginal conditions
F 1(K ;L ) ¡ rt= 0;
F 2(K ;L ) ¡ w t= 0:
Since F(¢; ¢) is homogeneous of degree 1, we can rewrite these marginal conditions as
f0(kt) ¡ t= 0; (2.11)
f(kt) ¡ tf0(k) ¡ w t= 0: (2.12)
2.3.3 Competitive Equilibrium
De¯nition 2 A competitive equilibrium is a sequence of quantities,
fkt+1; sg 1t=0and a sequence of prices fw ; rg 1t=0; which satisfy (i) consumer
optimization; (ii) ¯rm optimization; (iii) market clearing; in
each period t = 0; 1; 2; :::; given the initial capital-labor ratio k 0:
32CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS
Here, we have three markets, for labor, capital rental, and consumption
goods, and Walras' law tells us that we can drop one marketclearing
condition. It will be convenient here to drop the consumption
goods market from consideration. Consumer optimization is summarized
by equation (2.10), which essentially determines the supply of
capital, as period t savings is equal to the capital that will be rented in
period t+1: The supply of labor by consumers is inelastic. The demands
for capital and labor are determined implicitly by equations (2.11) and
(2.12). The equilibrium condition for the capital rental market is then
kt+1(1 + n) = s(w t; t+1); (2.13)
and we can substitute in (2.13) for w tand r t+1from (2.11) and (2.12)
to get
kt+1(1 + n) = s(f(k t) ¡ k0(k ); 0(k t+)): (2.14)
Here, (2.14) is a nonlinear ¯rst-order di®erence equation which, given
k0; solves for fk g 1t=. Once we have the equilibrium sequence of capitallabor
ratios, we can solve for prices from (2.11) and (2.12). We can then
solve for fs tg1t=0from (2.10), and in turn for consumption allocations.
2.4 An Example
Let u(c y; co) = ln c y+ ¯ ln c o; and F(K; L) = °K ®L 1¡®; where ¯ > 0;
° > 0; and 0 < ® < 1: Here, a young agent solves
max
st
[ln(w t¡ s) + ¯ ln[(1 + r t+1)s)];
and solving this problem we obtain the optimal savings function
st=
¯
1 + ¯
w : (2.15)
Given the Cobb-Douglass production function, we have f(k) = °k ®
and f 0(k) = °®k ®¡1: Therefore, from (2.11) and (2.12), the ¯rst-order
conditions from the ¯rm's optimization problem give
rt= °®k ®¡1
t; (2.16)
2.4. AN EXAMPLE 33
w t= °(1 ¡ ®)k ®
t: (2.17)
Then, using (2.14), (2.15), and (2.17), we get
kt+1(1 + n) =
¯
(1 + ¯)
°(1 ¡ ®)k ® t: (2.18)
Now, equation (2.18) determines a unique sequence fk tg1t=1given k0
(see Figure 2m) which converges in the limit to k ¤; the unique steady
state capital-labor ratio, which we can determine from (2.18) by setting
kt+1= kt= k ¤and solving to get
k¤ =
"
¯°(1 ¡ ®)
(1 + n)(1 + ¯)
# 1
1¡®
: (2.19)
Now, given the steady state capital-labor ratio from (2.19), we can solve
for steady state prices from (2.16) and (2.17), that is
r¤=
®(1 + n)(1 + ¯)
¯(1 ¡ ®)
;
w ¤= °(1 ¡ ®)
"
¯°(1 ¡ ®)
(1 + n)(1 + ¯)
# ®
1¡®
:
We can then solve for steady state consumption allocations,
cy= w ¤ ¡
¯
1 + ¯
w ¤=
w ¤
1 + ¯
;
co =
¯
1 + ¯
w ¤(1 + ¤):
In the long run, this economy converges to a steady state where the
capital-labor ratio, consumption allocations, the wage rate, and the
rental rate on capital are constant. Since the capital-labor ratio is
constant in the steady state and the labor input is growing at the rate
n; the growth rate of the aggregate capital stock is also n in the steady
state. In turn, aggregate output also grows at the rate n:
Now, note that the socially optimal steady state capital stock, ^k; is
determined by (2.6), that is
°®^k ®¡1= n;
34CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS
or
^k
=
μ
®° n
1¡®
: (2.20)
Note that, in general, from (2.19) and (2.20), k ¤ 6= ^k; i.e. the competitive
equilibrium steady state is in general not socially optimal, so this
economy su®ers from a dynamic ine±ciency. There may be too little or
too much capital in the steady state, depending on parameter values.
That is, suppose ¯ = 1 and n = :3: Then, if ® < :103; k ¤> ^k; and if
® > :103; then k ¤< ^k:
2.5 Discussion
The above example illustrates the dynamic ine±ciency that can result
in this economy in a competitive equilibrium.. There are essentially two
problems here. The ¯rst is that there is either too little or too much
capital in the steady state, so that the quantity of resources available
to allocate between the young and the old is not optimal. Second,
the steady state interest rate is not equal to n; i.e. consumers face
the \wrong" interest rate and therefore misallocate consumption goods
over time; there is either too much or too little saving in a competitive
equilibrium.
The root of the dynamic ine±ciency is a form of market incompleteness,
in that agents currently alive cannot trade with the unborn. To
correct this ine±ciency, it is necessary to have some mechanism which
permits transfers between the old and the young.
2.6 Government Debt
One means to introduce intergenerational transfers into this economy
is through government debt. Here, the government acts as a kind of
¯nancial intermediary which issues debt to young agents, transfers the
proceeds to young agents, and then taxes the young of the next generation
in order to pay the interest and principal on the debt.
Let B t+1denote the quantity of one-period bonds issued by the
government in period t: Each of these bonds is a promise to pay 1+r t+1
2.6. GOVERNMENT DEBT 35
units of consumption goods in period t +1: Note that the interest rate
on government bonds is the same as the rental rate on capital, as must
be the case in equilibrium for agents to be willing to hold both capital
and government bonds. We will assume that
B t+1= bL ; (2.21)
where b is a constant. That is, the quantity of government debt is ¯xed
in per-capita terms. The government's budget constraint is
B t+1+ Tt= (1 + rt)Bt; (2.22)
i.e. the revenues from new bond issues and taxes in period t, T t; equals
the payments of interest and principal on government bonds issued in
period t ¡ 1:
Taxes are levied lump-sum on young agents, and we will let ¿ tdenote
the tax per young agent. We then have
T = ¿ tLt: (2.23)
A young agent solves
max
st u(w t¡ st¡ ¿t; (1 + rt+1)st);
where s tis savings, taking the form of acquisitions of capital and government
bonds, which are perfect substitutes as assets. Optimal savings
for a young agent is now given by
st= s(w t¡ ¿ ; rt+): (2.24)
As before, pro¯t maximization by the ¯rm implies (2.11) and (2.12).
A competitive equilibrium is de¯ned as above, adding to the de¯nition
that there be a sequence of taxes f¿ tg1t=0 satisfying the government
budget constraint. From (2.21), (2.22), and (2.23), we get
¿ t=
μ
rt¡ n
1 + n
¶
b (2.25)
The asset market equilibrium condition is now
kt+1(1 + n) + b = s(w t¡ ¿t; t+1); (2.26)
36CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS
that is, per capita asset supplies equals savings per capita. Substituting
in (2.26) for w t; ¿t; and r t+; from (2.11), we get
kt+1(1+n)+b = s
Ã
f(kt) ¡ tf0(kt) ¡
Ã
f0(kt) ¡ n
1 + n
!
b; f0(kt+1)
!
(2.27)
We can then determine the steady state capital-labor ratio k ¤(b) by
setting k ¤(b) = k t= k t+1in (2.27), to get
k¤(b)(1+n)+b = s
Ã
f(k¤(b)) ¡ k¤ (b)f0(k¤(b)) ¡
Ã
f0(k¤(b)) ¡ n
1 + n
!
b; f0(k¤(b))
!
(2.28)
Now, suppose that we wish to ¯nd the debt policy, determined by b;
which yields a competitive equilibrium steady state which is socially
optimal, i.e. we want to ¯nd ^b such that k ¤(^b) = ^k: Now, given that
f0(^k) = n; from (2.28) we can solve for ^b as follows:
^b
= ¡^k(1 + n) + s
³
f(^k) ¡ ^kn; n
´
(2.29) In (2.29), note that ^b may be positive or negative. If ^b < 0; then debt
is negative, i.e. the government makes loans to young agents which
are ¯nanced by taxation. Note that, from (2.25), ¿ t= 0 in the steady
state with b = ^ b; so that the size of the government debt increases at
a rate just su±cient to pay the interest and principal on previouslyissued
debt. That is, the debt increases at the rate n, which is equal to
the interest rate. Here, at the optimum government debt policy simply
transfers wealth from the young to the old (if the debt is positive), or
from the old to the young (if the debt is negative).
2.6.1 Example
Consider the same example as above, but adding government debt.
That is, u(cy; co) = ln cy+ ¯ ln co; and F(K; L) = °K ®L1¡®; where
¯ > 0; ° > 0; and 0 < ® < 1: Optimal savings for a young agent is
st=
Ã
¯
1 + ¯
!
(w t¡ ¿): (2.30)
2.7. REFERENCES 37
Then, from (2.16), (2.17), (2.27) and (2.30), the equilibrium sequence
fkg 1t=0is determined by
kt+1(1 + n) + b =
Ã
¯
1 + ¯
!"
(1 ¡ ®)°k®
t¡
(®°k ®¡1
t¡ n)b
1 + n
#
;
and the steady state capital-labor ratio, k ¤(b); is the solution to
k¤(b)(1+n)+b =
Ã
¯
1 + ¯
!"
(1 ¡ ®)° (k¤(b))® ¡
(®° (k¤(b))®¡1 ¡ n)b
1 + n
#
Then, from (2.29), the optimal quantity of per-capita debt is
^b
=
Ã
¯
1 + ¯
!
(1 ¡ ®)° μ
®°
n
¶®
1¡®
¡
μ
®°
n
1¡®
(1 + n)
= °
μ
®°
n
¶®
1¡®
"
¯(1 ¡ ®)
1 + ¯ ¡
®
n
#
:
Here note that, given °; n; and ¯; ^b < 0 for ® su±ciently large, and
^b
> 0 for ® su±ciently small.
2.6.2 Discussion
The competitive equilibrium here is in general suboptimal for reasons
discussed above. But for those same reasons, government debt matters.
That is, Ricardian equivalence does not hold here, in general,
because the taxes required to pay o® the currently-issued debt are not
levied on the agents who receive the current tax bene¯ts from a higher
level of debt today. Government debt policy is a means for executing
the intergenerational transfers that are required to achieve optimality.
However, note that there are other intergenerational transfer mechanisms,
like social security, which can accomplish the same thing in this
model.
2.7 References
Diamond, P. 1965. \National Debt in a Neoclassical Growth Model,"
American Economic Review 55, 1126-1150.
38CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS
Blanchard, O. and Fischer, S. 1989. Lectures on Macroeconomics,
Chapter 3.
Kareken, J. and Wallace, N. 1980. Models of Monetary Economies,
Federal Reserve Bank of Minneapolis, Minneapolis, MN.
Chapter 3
Neoclassical Growth and Dynamic Programming
Early work on growth theory, particularly that of Solow (1956), was
carried out using models with essentially no intertemporal optimizing
behavior. That is, these were theories of growth and capital accumulation
in which consumers were assumed to simply save a constant
fraction of their income. Later, Cass (1965) and Koopmans (1965) developed
the ¯rst optimizing models of economic growth, often called
\optimal growth" models, as they are usually solved as an optimal
growth path chosen by a social planner. Optimal growth models have
much the same long run implications as Solow's growth model, with the
added bene¯t that optimizing behavior permits us to use these models
to draw normative conclusions (i.e. make statements about welfare).
This class of optimal growth models led to the development of stochastic
growth models (Brock and Mirman 1972) which in turn were the
basis for real business cycle models.
Here, we will present a simple growth model which illustrates some
of the important characteristics of this class of models. \Growth model"
will be something of a misnomer in this case, as the model will
not exhibit long-run growth. One objective of this chapter will be to
introduce and illustrate the use of discrete-time dynamic programming
methods, which are useful in solving many dynamic models.
39
40CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING
3.1 Preferences, Endowments, and Technology
There is a representative in¯nitely-lived consumer with preferences given
by
1X
t=0
¯tu(ct);
where 0 < ¯ < 1; and c tis consumption. The period utility function
u(¢) is continuously di®erentiable, strictly increasing, strictly concave,
and bounded. Assume that lim c!0u0(c) = 1: Each period, the consumer
is endowed with one unit of time, which can be supplied as labor.
The production technology is given by
yt= F(k t; t); (3.1)
where y tis output, ktis the capital input, and n tis the labor input.
The production function F(¢; ¢) is continuously di®erentiable, strictly
increasing in both arguments, homogeneous of degree one, and strictly
quasiconcave. Assume that F(0; n) = 0; lim k!0F (k; 1) = 1; and
limk!1F (k; 1) = 0:
The capital stock obeys the law of motion
kt+1= (1 ¡ ±)kt+ it; (3.2)
where i tis investment and ± is the depreciation rate, with 0 · ± · 1 and
k0 is the initial capital stock, which is given. The resource constraints
for the economy are
ct+ it· y; (3.3)
and
nt· 1: (3.4)
3.2 Social Planner's Problem There are several ways to specify the organization of markets and production
in this economy, all of which will give the same competitive
equilibrium allocation. One speci¯cation is to endow consumers with
3.2. SOCIAL PLANNER'S PROBLEM 41
the initial capital stock, and have them accumulate capital and rent it
to ¯rms each period. Firms then purchase capital inputs (labor and
capital services) from consumers in competitive markets each period
and maximize per-period pro¯ts. Given this, it is a standard result
that the competitive equilibrium is unique and that the ¯rst and second
welfare theorems hold here. That is, the competitive equilibrium
allocation is the Pareto optimum. We can then solve for the competitive
equilibrium quantities by solving the social planner's problem, which is
max
=0;n;;kt+g1t
1X
t=0
¯ u(c t)
subject to
ct + it· F(k ; n ); (3.5)
kt+1 = (1 ¡ ±)k t + i; (3.6)
n t· 1; (3.7)
t = 0; 1; 2; :::; and k 0 given. Here, we have used (3.1) and (3.2) to
substitute for y tto get (3.5). Now, since u(c) is strictly increasing in
c; (3.5) will be satis¯ed with equality. As there is no disutility from
labor, if (3.7) does not hold with equality, then n t and c tcould be
increased, holding constant the path of the capital stock, and increasing
utility. Therefore, (3.7) will hold with equality at the optimum. Now,
substitute for i tin (3.5) using (3.6), and de¯ne f(k) ´ F(k; 1); as n t= 1
for all t: Then, the problem can be reformulated as
max
fc;t+1g1t
=0
1X
t=0
¯ u(c t)
subject to
ct + k t+1= f(k ) + (1 ¡ ±)k ;
t = 0; 1; 2; :::; k0 given. This problem appears formidable, particularly
as the choice set is in¯nite-dimensional. However, suppose that we solve
the optimization problem sequentially, as follows. At the beginning
of any period t; the utility that the social planner can deliver to the
consumer depends only on k t; the quantity of capital available at the
beginning of the period. Therefore, it is natural to think of k tas a \state
42CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING
variable" for the problem. Within the period, the choice variables,
or \control" variables, are c tand k t+1: In period 0, if we know the
maximum utility that the social planner can deliver to the consumer as
a function of k 1; beginning in period 1, say v(k 1); it is straightforward
to solve the problem for the ¯rst period. That is, in period 0 the social
planner solves
max
c0;1 [u(c 0) + ¯v(k 1)]
subject to
c0 + k 1= f(k 0) + (1 ¡ ±)k 0:
This is a simple constrained optimization problem which in principle
can be solved for decision rules k 1 = g(k 0); where g(¢) is some function,
and c 0= f(k 0) + (1 ¡ ±)k 0¡ g(k 0): Since the maximization problem is
identical for the social planner in every period, we can write
v(k 0) = max
c0;1
[u(c 0) + ¯v(k 1)]
subject to
c0 + k 1= f(k 0) + (1 ¡ ±)k 0;
or more generally
v(k ) = max
c;kt+1
[u(c ) + ¯v(k t+1)] (3.8)
subject to
ct+ k t+1= f(k ) + (1 ¡ ±)k t: (3.9)
Equation (3.8) is a functional equation or Bellman equation. Our primary
aim here is to solve for, or at least to characterize, the optimal
decision rules k t+1= g(k ) and c t= f(kt)+(1¡±)k t¡g(k ): Of course,
we cannot solve the above problem unless we know the value function
v(¢). In general, v(¢) is unknown, but the Bellman equation can be used
to ¯nd it. In most of the cases we will deal with, the Bellman equation
satis¯es a contraction mapping theorem, which implies that
1. There is a unique function v(¢) which satis¯es the Bellman equation.
3.2. SOCIAL PLANNER'S PROBLEM 43
2. If we begin with any initial function v 0(k) and de¯ne v i+(k) by
vi+1(k) = max
c;0
[u(c) + ¯v (k0)]
subject to
c + k 0 = f(k) + (1 ¡ ±)k;
for i = 0; 1; 2; :::; then, limi!1vi+1(k) = v(k):
The above two implications give us two alternative means of uncovering
the value function. First, given implication 1 above, if we are
fortunate enough to correctly guess the value function v(¢); then we can
simply plug v(k t+1) into the right side of (3.8), and then verify that v(k t)
solves the Bellman equation. This procedure only works in a few cases,
in particular those which are amenable to judicious guessing. Second,
implication 2 above is useful for doing numerical work. One approach is
to ¯nd an approximation to the value function in the following manner.
First, allow the capital stock to take on only a ¯nite number of values,
i.e. form a grid for the capital stock, k 2 fk 1; k2; :::m g = S; where
m is ¯nite and k i< k i+1: Next, guess an initial value function, that
is m values v i0
= v 0(ki); i = 1; 2; :::;m: Then, iterate on these values,
determining the value function at the j thiteration from the Bellman
equation, that is
vij
= max `;c
[u(c) + ¯v `
j¡]
subject to
c + k `= f(k ) + (1 ¡ ±)k i:
Iteration occurs until the value function converges. Here, the accuracy
of the approximation depends on how ¯ne the grid is. That is,
if ki¡ ki¡= °; i = 2; :::m; then the approximation gets better the
smaller is ° and the larger is m: This procedure is not too computationally
burdensome in this case, where we have only one state variable.
However, the computational burden increases exponentially as we add
state variables. For example, if we choose a grid with m values for
each state variable, then if there are n state variables, the search for
a maximum on the right side of the Bellman equation occurs over m n
grid points. This problem of computational burden as n gets large is
sometimes referred to as the curse of dimensionality.
44CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING
3.2.1 Example of \Guess and Verify"
Suppose that F(k ; nt) = k®
tn 1¡®
t; 0 < ® < 1; u(c ) = ln c t; and ± = 1
(i.e. 100% depreciation). Then, substituting for the constraint, (3.9),
in the objective function on the right side of (3.8), we can write the
Bellman equation as
v(k ) = max
kt+1
[ln(k®
t¡ kt+1) + ¯v(k t+1)] (3.10)
Now, guess that the value function takes the form
v(k ) = A + B ln k ; (3.11)
where A and B are undetermined constants. Next, substitute using
(3.11) on the left and right sides of (3.10) to get
A + B ln k t= max
kt+1
[ln(k®
t¡ kt+1) + ¯(A + B ln k t+1)]: (3.12)
Now, solve the optimization problem on the right side of (3.12), which
gives
kt+1 =
¯Bk ®
t
1 + ¯B
; (3.13)
and substituting for the optimal k t+1in (3.12) using (3.13), and collecting
terms yields
A + B ln k t= ¯B ln ¯B ¡ (1 + ¯B) ln(1 + ¯B) + ¯A
+(1 + ¯B)® ln k t:
(3.14)
We can now equate coe±cients on either side of (3.14) to get two equations
determining A and B:
A = ¯B ln ¯B ¡ (1 + ¯B) ln(1 + ¯B) + ¯A (3.15) B = (1 + ¯B)® (3.16)
Here, we can solve (3.16) for B to get
B =
®
1 ¡ ®¯
: (3.17)
3.2. SOCIAL PLANNER'S PROBLEM 45
Then, we can use (3.15) to solve for A; though we only need B to
determine the optimal decision rules. At this point, we have veri¯ed
that our guess concerning the form of the value function is correct.
Next, substitute for B in (3.13) using (3.17) to get the optimal decision
rule for kt+1;
kt+1= ®¯k ®
t: (3.18)
Since c t= k®
t¡ kt+; we have
ct= (1 ¡ ®¯)k ®
t:
That is, consumption and investment (which is equal to k t+1given 100%
depreciation) are each constant fractions of output. Equation (3.18)
gives a law of motion for the capital stock, i.e. a ¯rst-order nonlinear
di®erence equation in k t; shown in Figure 3.1. The steady state for the
capital stock, k ¤; is determined by substituting k t= kt+1= k ¤in (3.18)
and solving for k ¤ to get
k¤ = (®¯)
1
1¡®:
Given (3.18), we can show algebraically and in Figure 1, that k tconverges
monotonically to k ¤; with k tincreasing if k 0< k ¤; and k tdecreasing
if 0 > k : Figure 3.1 shows a dynamic path for k twhere the initial
capital stock is lower than the steady state. This economy does not
exhibit long-run growth, but settles down to a steady state where the
capital stock, consumption, and output are constant. Steady state consumption
is c¤= (1 ¡ ®¯)(k ¤)®; and steady state output is y ¤ = (k¤)®:
3.2.2 Characterization of SolutionsWhen the Value
Function is Di®erentiable
Benveniste and Scheinkman (1979) establish conditions under which
the value function is di®erentiable in dynamic programming problems.
Supposing that the value function is di®erentiable and concave in (3.8),
we can characterize the solution to the social planner's problem using
¯rst-order conditions. Substituting in the objective function for c tusing
in the constraint, we have
v(k ) = max
kt+fu[f(kt) + (1 ¡ ±)kt¡ kt+] + ¯v(k t+1)g (3.19)
46CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING
Figure 3.1:
3.2. SOCIAL PLANNER'S PROBLEM 47
Then, the ¯rst-order condition for the optimization problem on the right
side of (3.8), after substituting using the constraint in the objective
function, is ¡u0 [f(t) + (1 ¡ ±)k t¡ kt+1] + ¯v 0(kt+1) = 0: (3.20)
The problem here is that, without knowing v(¢); we do not know v 0(¢):
However, from (3.19) we can di®erentiate on both sides of the Bellman
equation with respect to k tand apply the envelope theorem to obtain
v0(k ) = u 0[f(kt) + (1 ¡ ±)k t¡ kt+][f0(kt) + 1 ¡ ±];
or, updating one period,
v0(k t+) = u 0[f(kt+1) + (1 ¡ ±)k t+1¡ kt+2][0(k t+) + 1 ¡ ±]: (3.21)
Now, substitute in (3.20) for v 0(kt+1) using (3.21) to get
¡u0 [f(t) + (1 ¡ ±)k t¡ kt+1]
+¯u 0[f(kt+) + (1 ¡ ±)k t+1 ¡ kt+][f0(kt+1) + 1 ¡ ±] = 0;
(3.22)
or
¡u0 (c) + ¯u 0(ct+1)[0(k t+) + 1 ¡ ±] = 0;
The ¯rst term is the bene¯t, at the margin, to the consumer of consuming
one unit less of the consumption good in period t; and the second
term is the bene¯t obtained in period t + 1; discounted to period t;
from investing the foregone consumption in capital. At the optimum,
the net bene¯t must be zero.
We can use (3.22) to solve for the steady state capital stock by
setting k t= kt+1 = kt+2 = k¤ to get
f0(k¤) =
1
¯ ¡ 1 + ±; (3.23)
i.e. one plus the net marginal product of capital is equal to the inverse
of the discount factor. Therefore, the steady state capital stock depends
only on the discount factor and the depreciation rate.
48CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING
3.2.3 Competitive Equilibrium
Here, I will simply assert that the there is a unique Pareto optimum
that is also the competitive equilibrium in this model. While the most
straightforward way to determine competitive equilibrium quantities in
this dynamic model is to solve the social planner's problem to ¯nd the
Pareto optimum, to determine equilibrium prices we need some information
from the solutions to the consumer's and ¯rm's optimization
problems.
Consumer's Problem
Consumers store capital and invest (i.e. their wealth takes the form
of capital), and each period they rent capital to ¯rms and sell labor.
Labor supply will be 1 no matter what the wage rate, as consumers
receive no disutility from labor. The consumer then solves the following
intertemporal optimization problem.
max
=0;t+g1t
1X
t=0
¯ u(c t)
subject to
ct+ k t+1 = w t+ rtkt+ (1 ¡ ±)k ; (3.24)
t = 0; 1; 2; :::; k0given, where w tis the wage rate and r tis the rental
rate on capital. If we simply substitute in the objective function using (3.24), then we can reformulate the consumer's problem as
max
ft+g 1t
=0
1X
t=0
¯ u(w t+ rtkt+ (1 ¡ ±)k t¡ k t+)
subject to k t¸ 0 for all t and k 0given. Ignoring the nonnegativity
constraints on capital (in equilibrium, prices will be such that the consumer
will always choose k t+1 > 0), the ¯rst-order conditions for an
optimum are
¡¯tu 0(w t+ rtkt+ (1 ¡ ±)k t¡ k t+)
+¯ t+1u0(w t+1 + rt+1kt+1+ (1 ¡ ±)k t+1¡ kt+2)(t+1 + 1 ¡ ±) = 0
(3.25)
3.2. SOCIAL PLANNER'S PROBLEM 49
Using (3.24) to substitute in (3.25), and simplifying, we get
¯u 0(c t+)
u 0(ct)
=
1
1 + r t+1¡ ±
; (3.26)
that is, the intertemporal marginal rate of substitution is equal to the
inverse of one plus the net rate of return on capital (i.e. one plus the
interest rate).
Firm's Problem
The ¯rm simply maximizes pro¯ts each period, i.e. it solves
max
k;nt
[F(k ; n ) ¡ w n t¡ tkt];
and the ¯rst-order conditions for a maximum are
F 1(kt; n) = rt; (3.27)
F 2(kt; n) = w t: (3.28)
Competitive Equilibrium Prices
The optimal decision rule, k t+1 = g(k ); which is determined from the
dynamic programming problem (3.8) allows a solution for the competitive
equilibrium sequence of capital stocks fk g 1t=1given k 0: We can
then solve for fc g 1t=0using (3.9). Now, it is straightforward to solve
for competitive equilibrium prices from the ¯rst-order conditions for
the ¯rm's and consumer's optimization problems. The prices we need
to solve for are fw t; tg1t=0; the sequence of factor prices. To solve for
the real wage, plug equilibrium quantities into (3.28) to get
F 2(kt; 1) = w :
To obtain the capital rental rate, either (3.26) or (3.27) can be used.
Note that r t¡ ± = f 0(k) ¡ ± is the real interest rate and that, in the
steady state [from (3.26) or (3.23)], we have 1+r ¡± = 1
¯ ; or, if we let
¯ = 1
1+´; where ´ is the rate of time preference, then r¡± = ´; i.e. the
real interest rate is equal to the rate of time preference.
50CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING
Note that, when the consumer solves her optimization problem, she knows the whole sequence of prices fw t; g1t=: That is, this a \rational
expectations" or \perfect foresight" equilibrium where each period the
consumer makes forecasts of future prices and optimizes based on those
forecasts, and in equilibrium the forecasts are correct. In an economy
with uncertainty, a rational expectations equilibrium has the property
that consumers and ¯rms may make errors, but those errors are not
systematic.
3.3 References
Benveniste, L. and Scheinkman, J. 1979. \On the Di®erentiability of
the Value Function in Dynamic Models of Economics," Econometrica
47, 727-732.
Brock, W. and Mirman, L. 1972. \Optimal Economic Growth and
Uncertainty: The Discounted Case," Journal of Economic Theory
4, 479-513.
Cass, D. 1965. \Optimum Growth in an Aggregative Model of Capital
Accumulation," Review of Economic Studies 32, 233-240.
Koopmans, T. 1965. \On the Concept of Optimal Growth," in The
Econometric Approach to Development Planning, Chicago, Rand-
McNally.
Chapter 4
Endogenous Growth
This chapter considers a class of endogenous growth models closely
related to the ones in Lucas (1988). Here, we use discrete-time models,
so that the dynamic programming methods introduced in Chapter 2
can be applied (Lucas's models are in continuous time).
Macroeconomists are ultimately interested in economic growth because
the welfare consequences of government policies a®ecting growth
rates of GDP are potentially very large. In fact, one might argue, as
in Lucas (1987), that the welfare gains from government policies which
smooth out business cycle °uctuations are very small compared to the
gains from growth-enhancing policies.
Before we can hope to evaluate the e±cacy of government policy in a
growth context, we need to have growth models which can successfully
confront the data. Some basic facts of economic growth (as much as
we can tell from the short history in available data) are the following:
1. There exist persistent di®erences in per capita income across
countries.
2. There are persistent di®erences in growth rates of per capita income
across countries.
3. The correlation between the growth rate of income and the level
of income across countries is low.
4. Among rich countries, there is stability over time in growth rates
51
52 CHAPTER 4. ENDOGENOUS GROWTH
of per capita income, and there is little diversity across countries
in growth rates.
5. Among poor countries, growth is unstable, and there is a wide diversity in growth experience.
Here, we ¯rst construct a version of the optimal growth model in
Chapter 2 with exogenous growth in population and in technology, and
we ask whether this model can successfully explain the above growth
facts. This neoclassical growth model can successfully account for
growth experience in the United States, and it o®ers some insights with
regard to the growth process, but it does very poorly in accounting for
the pattern of growth among countries. Next, we consider a class of
endogenous growth models, and show that these models can potentially
do a better job of explaining the facts of economic growth.
4.1 A Neoclassical GrowthModel (Exogenous
Growth)
The representative household has preferences given by
1X
t=0
¯tN t
c°
t
°
; (4.1)
where 0 < ¯ < 1; ° < 1; c tis per capita consumption, and N tis
population, where
N t= (1 + n) N 0; (4.2)
n constant and N 0given. That is, there is a dynastic household which
gives equal weight to the discounted utility of each member of the
household at each date. Each household member has one unit of time
in each period when they are alive, which is supplied inelastically as
labor. The production technology is given by
Y t= K ®
t(N A t1¡®; (4.3)
where Y tis aggregate output, K tis the aggregate capital stock, and A t
is a labor-augmenting technology factor, where
A t= (1 + a) A 0; (4.4)
4.1. A NEOCLASSICAL GROWTHMODEL (EXOGENOUS GROWTH)53
with a constant and A 0 given. We have 0 < ® < 1; and the initial
capital stock, K 0; is given. The resource constraint for this economy is
N c t+ K t+1= Y : (4.5)
Note here that there is 100% depreciation of the capital stock each
period, for simplicity.
To determine a competitive equilibrium for this economy, we can
solve the social planner's problem, as the competitive equilibrium and
the Pareto optimum are identical. The social planner's problem is to
maximize (4.1) subject to (4.2)-(4.5). So that we can use dynamic
programming methods, and so that we can easily characterize longrun
growth paths, it is convenient to set up this optimization problem
with a change of variables. That is, use lower case variables to de¯ne
quantities normalized by e±ciency units of labor, for example y t´Y t
AtNt
:
Also, let xt´ ct
At : With substitution in (4.1) and (4.5) using (4.2)-(4.4),
the social planner's problem is then
max
fx;t+1g1t
=0
1X
t=0
[¯(1 + n)(1 + a) ]t
Ã
x°
t
°
!
subject to
xt + (1 + n)(1 + a)k t+1= k ®
t; t = 0; 1; 2; ::: (4.6)
This optimization problem can then be formulated as a dynamic program
with state variable k tand choice variables x tand k t+1: That is,
given the value function v(k ); the Bellman equation is
v(k ) = max
x;kt+1
"
x°
t
°
+ ¯(1 + n)(1 + a) °v(k t+1)
#
subject to (4.6). Note here that we require the discount factor for the
problem to be less than one, that is ¯(1 +n)(1 +a) °< 1: Substituting
in the objective function for x tusing (4.6), we have
v(k ) = max
kt+1
"
[k®
t¡ kt+1(1 + n)(1 + a)] °
°
+ ¯(1 + n)(1 + a) °v(k t+1)
#
(4.7)
54 CHAPTER 4. ENDOGENOUS GROWTH
The ¯rst-order condition for the optimization problem on the right side
of (4.7) is
¡(1 + n)(1 + a)x °¡1
t+ ¯(1 + n)(1 + a) °v 0(kt+1) = 0; (4.8)
and we have the following envelope condition
v0 (kt) = ®k ®¡1
tx °¡1
t: (4.9)
Using (4.9) in (4.8) and simplifying, we get
¡(1 + a) 1¡x °¡1
t+ ¯®k ®¡1
t+1x °¡1
t+1= 0: (4.10)
Now, we will characterize \balanced growth paths," that is steady states where x t= x¤ and k t= k ¤; where x ¤ and k ¤are constants. Since (4.10)
must hold on a balanced growth path, we can use this to solve for k ¤;
that is
k¤ =
"
¯®
(1 + a) 1¡°
# 1
1¡®
(4.11)
Then, (4.6) can be used to solve for x ¤to get
x¤ =
"
¯®
(1 + a) 1¡°
1¡®
"
(1 + a) 1¡°
¯® ¡ (1 + n)(1 + a)
#
: (4.12)
Also, since y t= k ®
t; then on the balanced growth path the level of
output per e±ciency unit of labor is
y¤ = (k¤)® =
"
¯®
(1 + a) 1¡°
# ®
1¡®
: (4.13)
In addition, the savings rate is
st=
K t+1
Y t
=
kt+1(1 + n)(1 + a)
k®
t
;
so that, on the balanced growth path, the savings rate is
s¤ = (k¤)1¡®(1 + n)(1 + a):
4.1. A NEOCLASSICAL GROWTHMODEL (EXOGENOUS GROWTH)55
Therefore, using (4.11) we get
s¤ = ¯®(1 + n)(1 + a) °: (4.14)
Here, we focus on the balanced growth path since it is known that
this economy will converge to this path given any initial capital stock
K 0> 0: Since k ¤; ¤; and y ¤ are all constant on the balanced growth
path, it then follows that the aggregate capital stock, K , aggregate
consumption, N c ; and aggregate output, Y t; all grow (approximately)
at the common rate a + n; and that per capita consumption and output
grow at the rate a: Thus, long-run growth rates in aggregate variables are determined entirely by exogenous growth in the labor force
and exogenous technological change, and growth in per capita income
and consumption is determined solely by the rate of technical change.
Changes in any of the parameters ¯; ®; or ° have no e®ect on long-run
growth. Note in particular that an increase in any one of ®; ¯; or °
results in an increase in the long-run savings rate, from (4.14). But
even though the savings rate is higher in each of these cases, growth
rates remain una®ected. This is a counterintuitive result, as one might
anticipate that a country with a high savings rate would tend to grow
faster.
Changes in any of ®; ¯; or ° do, however, produce level e®ects. For
example, an increase in ¯; which causes the representative household to
discount the future at a lower rate, results in an increase in the savings
rate [from (4.14)], and increases in k ¤and y ¤; from (4.11) and (4.13).
We can also show that ¯(1+n)(1+a) ° < 1 implies that an increase in
steady state k ¤will result in an increase in steady state x¤: Therefore,
an increase in ¯ leads to an increase in x ¤: Therefore, the increase in ¯
yields increases in the level of output, consumption, and capital in the
long run.
Suppose that we consider a number of closed economies, which all
look like the one modelled here. Then, the model tells us that, given
the same technology (and it is hard to argue that, in terms of the logic
of the model, all countries would not have access to A t); all countries
will converge to a balanced growth path where per capita output and
consumption grow at the same rate. From (4.13), the di®erences in
the level of per capita income across countries would have to be explained
by di®erences in ®; ¯; or °: But if all countries have access
56 CHAPTER 4. ENDOGENOUS GROWTH
to the same technology, then ® cannot vary across countries, and this
leaves an explanation of di®erences in income levels due to di®erences
in preferences. This seems like no explanation at all.
While neoclassical growth models were used successfully to account
for long run growth patterns in the United States, the above analysis
indicates that they are not useful for accounting for growth experience
across countries. The evidence we have seems to indicate that growth
rates and levels of output across countries are not converging, in contrast
to what the model predicts.
4.2 A Simple Endogenous Growth Model
In attempting to build a model which can account for the principal facts
concerning growth experience across countries, it would seem necessary
to incorporate an endogenous growth mechanism, to permit economic
factors to determine long-run growth rates. One way to do this is
to introduce human capital accumulation. We will construct a model
which abstracts from physical capital accumulation, to focus on the
essential mechanism at work, and introduce physical capital in the next
section.
Here, preferences are as in (4.1), and each agent has one unit of
time which can be allocated between time in producing consumption
goods and time spent in human capital accumulation. The production
technology is given by Y t= ®h tutN ;
where ® > 0; Y tis output, h tis the human capital possessed by each
agent at time t, and u tis time devoted by each agent to production.
That is, the production function is linear in quality-adjusted labor input.
Human capital is produced using the technology
h t+1= ±h t(1 ¡ u ); (4.15)
where ± > 0, 1¡u tis the time devoted by each agent to human capital
accumulation (i.e. education and acquisition of skills), and h 0 is given.
Here, we will use lower case letters to denote variables in per capita
terms, for example y t´ Yt
Nt
: The social planner's problem can then
4.2. A SIMPLE ENDOGENOUS GROWTH MODEL 57
be formulated as a dynamic programming problem, where the state
variable is h tand the choice variables are c t, ht+1; and u : That is, the
Bellman equation for the social planner's problem is
v(h ) = max
c;u;ht+1
"
t°
°
+ ¯(1 + n)v(h t+1)
#
subject to
ct = ®h u t(4.16)
and (4.15). Then, the Lagrangian for the optimization problem on the
right side of the Bellman equation is
L =
c°
t
°
+ ¯(1 + n)v(h t+1) + ¸t(®h tu t¡ c) + ¹t[±h t(1 ¡ ut) ¡ ht+1];
where ¸ tand ¹ tare Lagrange multipliers. Two ¯rst-order conditions
for an optimum are then
@L
@c t
= c °¡1
t¡ ¸t= 0; (4.17)
@L
@h t+1
= ¯(1 + n)v 0(ht+1) ¡ ¹t= 0; (4.18)
(4.15) and (4.16). In addition, the ¯rst derivative of the Lagrangian
with respect to u tis
@L
@u t
= ¸t®h t¡ t±h t
Now, if @L @u t
> 0; then u t= 1: But then, from (4.15) and (4.16), we have
h s= c s = 0 for s = t + 1; t + 2; ::: . But, since the marginal utility of
consumption goes to in¯nity as consumption goes to zero, this could
not be an optimal path. Therefore @L @u · 0: If @L @u t < 0; then u t= 0; and
ct = 0 from (4.16). Again, this could not be optimal, so we must have
@L
@u t
= ¸t®h t¡ t±h t= 0 (4.19)
at the optimum.
58 CHAPTER 4. ENDOGENOUS GROWTH
We have the following envelope condition:
v0(h t) = ®u t¸t+ ¸t®(1 ¡ u );
or, using (4.17),
v0(h t) = ®c °¡1
t(4.20)
From (4.17)-(4.20), we then get
¯(1 + n)±c °¡1
t+1¡ c°¡1
t= 0: (4.21)
Therefore, we can rewrite (4.21) as an equation determining the equilibrium
growth rate of consumption:
ct+1
ct
= [¯(1 + n)±]
1
1¡: (4.22)
Then, using (4.15), (4.16), and (4.22), we obtain:
1¯(1 + n)±]
1¡=
±(1 ¡ u t)ut+1
u t
;
or
u t+1=
[¯(1 + n)± °]
1
1¡u t
1 ¡ u t
(4.23)
Now, (4.23) is a ¯rst-order di®erence equation in u tdepicted in Figure
4.1 for the case where [¯(1+n)] 1¡± ¡°< 1, a condition we will assume
holds. Any path fu g 1t=0satisfying (4.23) which is not stationary (a
stationary path is u t= u; a constant, for all t) has the property that
lim t!u t= 0; which cannot be an optimum, as the representative
consumer would be spending all available time accumulating human
capital which is never used to produce in the future. Thus the only
solution, from (4.23), is
1 t= u = 1 ¡ [¯(1 + n)± °]
1¡°
for all t: Therefore, substituting in (4.15), we get
h t+1
h t
= [¯(1 + n)±]
1 1¡;
4.3. ENDOGENOUS GROWTHWITH PHYSICAL CAPITAL AND HUMAN
CAPITAL59
and human capital grows at the same rate as consumption per capita.
If [¯(1 + n)±]
1
1¡> 1 (which will hold for ± su±ciently large), then
growth rates are positive. There are two important results here. The
¯rst is that equilibrium growth rates depend on more than the growth
rates of exogenous factors. Here, even if there is no growth in population
(n = 0) and given no technological change, this economy can
exhibit unbounded growth. Growth rates depend in particular on the
discount factor (growth increases if the future is discounted at a lower
rate) and ±; which is a technology parameter in the human capital accumulation
function (if more human capital is produced for given inputs,
the economy grows at a higher rate). Second, the level of per capita
income (equal to per capita consumption here) is dependent on initial
conditions. That is, since growth rates are constant from for all t; the
level of income is determined by h 0; the initial stock of human capital.
Therefore, countries which are initially relatively rich (poor) will tend
to stay relatively rich (poor).
The lack of convergence of levels of income across countries which
this model predicts is consistent with the data. The fact that other
factors besides exogenous technological change can a®ect growth rates
in this type of model opens up the possibility that di®erences in growth
across countries could be explained (in more complicated models) by
factors including tax policy, educational policy, and savings behavior.
4.3 Endogenous GrowthWith Physical Capital
and Human Capital
The approach here follows closely the model in Lucas (1988), except
that we omit his treatment of human capital externalities. The model is
identical to the one in the previous section, except that the production
technology is given by
Y t= K ®
t(N h u )1¡®;
where K tis physical capital and 0 < ® < 1; and the economy's resource
constraint is
N c t+ K t+1= K ®
t(N h u )1¡®
60 CHAPTER 4. ENDOGENOUS GROWTH
Figure 4.1:
4.3. ENDOGENOUS GROWTHWITH PHYSICAL CAPITAL AND HUMAN
CAPITAL61
As previously, we use lower case letters to denote per capita quantities.
In the dynamic program associated with the social planner's optimization
problem, there are two state variables, k tand h t; and four choice
variables, u ; ct; ht+; and k t+1: The Bellman equation for this dynamic
program is
v(k ; ht) = max
c;u;kt;ht+1 "
c°
t
°
+ ¯(1 + n)v(k t+1; ht+)
#
subject to
ct+ (1 + n)k t+1= k ®
t(h u )1¡® (4.24)
h t+1= ±h t(1 ¡ ut) (4.25)
The Lagrangian for the constrained optimization problem on the right
side of the Bellman equation is then
L =
c°
t
°
+¯(1+n)v(k t+; ht+1)+¸ [k®
t(h u )1¡®¡c ¡(1+n)k t+1]+¹t[±h (1¡u t)¡ht+1]
The ¯rst-order conditions for an optimum are then
@L
@c t
= c °¡1
t¡ ¸t= 0; (4.26)
@L
@u t
= ¸t(1 ¡ ®)k ®
th 1¡®
tu ¡®
t¡ ¹±h t= 0; (4.27)
@L
@h t+1
= ¯(1 + n)v 2(kt+1; ht+1) ¡ t= 0; (4.28)
@L
@k t+1
= ¡¸t(1 + n) + ¯(1 + n)v 1 (kt+1; ht+) = 0; (4.29)
(4.24) and (4.25). We also have the following envelope conditions:
v1(k ; ht) = ¸t®k ®¡1
t(h u )1¡® (4.30)
v2(k ; ht) = ¸t(1 ¡ ®)k ®
th ¡®
tu 1¡®
t+ ¹t±(1 ¡ u t) (4.31)
Next, use (4.30) and (4.31) to substitute in (4.29) and (4.28) respectively,
then use (4.26) and (4.27) to substitute for ¸ t and ¹ tin (4.28)
and (4.29). After simplifying, we obtain the following two equations:
¡c°¡1
t+ ¯c °¡1
t+1®k ®¡1
t+1(h t+u t+1)1¡®= 0; (4.32)
62 CHAPTER 4. ENDOGENOUS GROWTH
¡c°¡1 tk®
th ¡®
tu ¡®
t+ ±¯(1 + n)c °¡1
t+1k ®
t+1h¡®
t+1u¡®
t+1= 0: (4.33)
Now, we wish to use (4.24), (4.25), (4.32), and (4.33) to characterize
a balanced growth path, along which physical capital, human capital,
and consumption grow at constant rates. Let ¹ k; ¹h; and ¹ cdenote
the growth rates of physical capital, human capital, and consumption,
respectively, on the balanced growth path. From (4.25), we then have
1 + ¹ h= ±(1 ¡ u );
which implies that
u t= 1 ¡
1 + ¹ h
±
;
a constant, along the balanced growth path. Therefore, substituting
for u , ut+1; and growth rates in (4.33), and simplifying, we get
(1 + ¹ c)1¡(1 + ¹ k)¡®(1 + ¹h)® = ±¯(1 + n): (4.34)
Next, dividing (4.24) through by k t, we have
ct
kt
+ (1 + n)
kt+1
kt
= k ®¡1
t(h u )1¡®: (4.35)
Then, rearranging (4.32) and backdating by one period, we get
(1 + ¹ c)1¡°
¯®
= k ®¡1
t(h u )1¡® (4.36)
Equations (4.35) and (4.36) then imply that
ct
kt
+ (1 + n)(1 + ¹ k) =
(1 + ¹ c)1¡°
¯®
:
But then ct
kt
is a constant on the balanced growth path, which implies
that ¹c = ¹k: Also, from (4.36), since u tis a constant, it must be the
case that ¹ k = ¹h: Thus per capita physical capital, human capital, and
per capita consumption all grow at the same rate along the balanced
growth path, and we can determine this common rate from (4.34), i.e.
1 + ¹ c= 1 + ¹ k = 1 + ¹ h = 1 + ¹ = [±¯(1 + n)] 1¡: (4.37)
4.4. REFERENCES 63
Note that the growth rate on the balanced growth path in this model
is identical to what it was in the model of the previous section. The
savings rate in this model is
st=
K t+1
Y t
=
kt+1(1 + n)
ktk®¡1
t(htu)1¡®
Using (4.36) and (4.37), on the balanced growth path we then get
1t= ® [± ¯(1 + n)]
1¡(4.38)
In general then, from (4.37) and (4.38), factors which cause the savings
rate to increase (increases in ¯; n; or ±) also cause the growth rate of
per capita consumption and income to increase.
4.4 References
Lucas, R.E. 1987. Models of Business Cycles, Basil Blackwell, New
York.
Lucas, R.E. 1988. \On the Mechanics of Economic Development,"
Journal of Monetary Economics 22, 3-42.
64 CHAPTER 4. ENDOGENOUS GROWTH
Chapter 5
Choice Under Uncertainty
In this chapter we will introduce the most commonly used approach to
the study of choice under uncertainty, expected utility theory. Expected
utility maximization by economic agents permits the use of stochastic
dynamic programming methods in solving for competitive equilibria.
We will ¯rst provide an outline of expected utility theory, and then
illustrate the use of stochastic dynamic programming in a neoclassical
growth model with random disturbances to technology. This stochastic
growth model is the basis for real business cycle theory.
5.1 Expected Utility Theory
In a deterministic world, we describe consumer preferences in terms of
the ranking of consumption bundles. However, if there is uncertainty,
then preferences are de¯ned in terms of how consumers rank lotteries
over consumption bundles. The axioms of expected utility theory imply
a ranking of lotteries in terms of the expected value of utility they
yield for the consumer. For example, suppose a world with a single consumption
good, where a consumer's preferences over certain quantities
of consumption goods are described by the function u(c); where c is
consumption. Now suppose two lotteries over consumption. Lottery i
gives the consumer c 1i
units of consumption with probability p , and c 2i units of consumption with probability 1¡p ; where 0 < p i< 1; i = 1; 2:
65
66 CHAPTER 5. CHOICE UNDER UNCERTAINTY
Then, the expected utility the consumer receives from lottery i is
piu(c 1i
) + (1 ¡ p)u(c 2i
);
and the consumer would strictly prefer lottery 1 to lottery 2 if
p1u(c 11
) + (1 ¡ p1)u(c21
) > p2u(c 12
) + (1 ¡ p2)u(c22
);
would strictly prefer lottery 2 to lottery 1 if
p1u(c 11
) + (1 ¡ p1)u(c21
) < p2u(c 12
) + (1 ¡ p2)u(c22
);
and would be indi®erent if
p1u(c 11
) + (1 ¡ p1)u(c21
) = p2u(c 12
) + (1 ¡ p2)u(c22):
Many aspects of observed behavior toward risk (for example, the observation
that consumers buy insurance) is consistent with risk aversion.
An expected utility maximizing consumer will be risk averse with respect
to all consumption lotteries if the utility function is strictly concave.
If u(c) is strictly concave, this implies Jensen's inequality, that
is
E[u(c)] · u (E[c]) ; (5.1)
where E is the expectation operator. This states that the consumer
prefers the expected value of the lottery with certainty to the lottery
itself. That is, a risk averse consumer would pay to avoid risk.
If the consumer receives constant consumption, ¹c; with certainty,
then clearly (5.1) holds with equality. In the case where consumption
is random, we can show that (5.1) holds as a strict inequality. That
is, take a tangent to the function u(c) at the point (E[c]; u(E[c])) (see
Figure 1). This tangent is described by the function
g(c) = ® + ¯c; (5.2)
where ® and ¯ are constants, and we have
® + ¯E[c] = u(E[c]): (5.3)
Now, since u(c) is strictly concave, we have, as in Figure 1,
® + ¯c ¸ u(c); (5.4)
5.1. EXPECTED UTILITY THEORY 67
for c ¸ 0; with strict inequality if c 6= E[c]: Since the expectation
operator is a linear operator, we can take expectations through (5.4),
and given that c is random we have
® + ¯E[c] > E[u(c)];
or, using (5.3), u(E[c]) > E[u(c)]:
As an example, consider a consumption lottery which yields c 1units
of consumption with probability p and c 2units with probability 1 ¡ p;
where 0 < p < 1 and c 2> c1: In this case, (5.1) takes the form
pu(c 1) + (1 ¡ p)u(c2) < u (pc1 + (1 ¡ p)c2) :
In Figure 2, the di®erence
u (pc 1+ (1 ¡ p)c2) ¡ [pu(c1) + (1 ¡ p)u(c2)]
is given by DE. The line AB is given by the function
f(c) =
c2u(c 1) ¡ 1u(c2)
c2 ¡ 1
+
"
u(c ) ¡ u(c1)
c2 ¡ 1
#
c:
A point on the line AB denotes the expected utility the agent receives
for a particular value of p; for example p = 0 yields expected utility
u(c ) or point A, and B implies p = 1: Jensen's inequality is re°ected
in the fact that AB lies below the function u(c): Note that the distance
DE is the disutility associated with risk, and that this distance will
increase as we introduce more curvature in the utility function, i.e. as
the consumer becomes more risk averse.
5.1.1 Anomalies in Observed Behavior Towards Risk
While expected utility maximization and a strictly concave utility function
are consistent with the observation that people buy insurance,
some observed behavior is clearly inconsistent with this. For example,
many individuals engage in lotteries with small stakes where the
expected payo® is negative.
68 CHAPTER 5. CHOICE UNDER UNCERTAINTY
Another anomaly is the \Allais Paradox." Here, suppose that there
are four lotteries, which a person can enter at zero cost. Lottery 1
involves a payo® of $1 million with certainty; lottery 2 yields a payo®
of $5 million with probability .1, $1 million with probability .89, and
0 with probability .01; lottery 3 yields $1 million with probability .11
and 0 with probability .89; lottery 4 yields $5 million with probability
.1 and 0 with probability .9. Experiments show that most people prefer
lottery 1 to lottery 2, and lottery 4 to lottery 3. But this is inconsistent
with expected utility theory (whether the person is risk averse or not
is irrelevant). That is, if u(¢) is an agent's utility function, and they
maximize expected utility, then a preference for lottery 1 over lottery
2 gives
u(1) > :1u(5) + :89u(1) + :01u(0);
or
:11u(1) > :1u(5) + :01u(0): (5.5)
Similarly, a preference for lottery 4 over lottery 3 gives
:11u(1) + :89u(0) < :1u(5) + :9u(0);
or
:11u(1) < :1u(5) + :9u(0); (5.6) and clearly (5.5) is inconsistent with (5.6).
Though there appear to be some obvious violations of expected utility
theory, this is still the standard approach used in most economic
problems which involve choice under uncertainty. Expected utility theory
has proved extremely useful in the study of insurance markets, the
pricing of risky assets, and in modern macroeconomics, as we will show.
5.1.2 Measures of Risk Aversion
With expected utility maximization, choices made under uncertainty
are invariant with respect to a±ne transformations of the utility function.
That is, suppose a utility function
v(c) = ® + ¯u(c);
5.1. EXPECTED UTILITY THEORY 69
where ® and ¯ are constants with ¯ > 0: Then, we have
E[v(c)] = ® + ¯E[u(c)];
since the expectation operator is a linear operator. Thus, lotteries are
ranked in the same manner with v(c) or u(c) as the utility function. Any
measure of risk aversion should clearly involve u 00(c); since risk aversion
increases as curvature in the utility function increases. However, note
that for the function v(c); that we have v 00(c) = ¯u00(c); i.e. the second
derivative is not invariant to a±ne transformations, which have no e®ect
on behavior. A measure of risk aversion which is invariant to a±ne
transformations is the measure of absolute risk aversion,
ARA(c) = ¡
u00(c)
u0(c)
:
A utility function which has the property that ARA(c) is constant for
all c is u(c) = ¡¡®; ® > 0: For this function, we have
ARA(c) = ¡¡® 2e¡®c
®e ¡®c= ®:
It can be shown, through Taylor series expansion arguments, that the
measure of absolute risk aversion is twice the maximum amount that
the consumer would be willing to pay to avoid one unit of variance for
small risks.
An alternative is the relative risk aversion measure,
RRA(c) = ¡c
u00(c)
u0(c)
:
A utility function for which RRA(c) is constant for all c is
u(c) =
c1¡°¡ 1
1 ¡ °
;
where ° ¸ 0: Here,
RRA(c) = ¡c¡°c ¡(1+°)
c¡°= °
70 CHAPTER 5. CHOICE UNDER UNCERTAINTY
Note that the utility function u(c) = ln(c) has RRA(c) = 1:
The measure of relative risk aversion can be shown to be twice the maximum amount per unit of variance that the consumer would be
willing to pay to avoid a lottery if both this maximum amount and
the lottery are expressed as proportions of an initial certain level of
consumption.
A consumer is risk neutral if they have a utility function which is
linear in consumption, that is u(c) = ¯c; where ¯ > 0: We then have
E[u(c)] = ¯E[c];
so that the consumer cares only about the expected value of consumption.
Since u 00(c) = 0 and u 0(c) = ¯; we have ARA(c) = RRA(c) = 0:
5.2 Stochastic Dynamic Programming
We will introduce stochastic dynamic programming here by way of
an example, which is essentially the stochastic optimal growth model
studied by Brock and Mirman (1972). The representative consumer
has preferences given by
E 0
1X
t=0
¯tu(c );
where 0 < ¯ < 1; c tis consumption, u(¢) is strictly increasing, strictly
concave, and twice di®erentiable, and E 0 is the expectation operator
conditional on information at t = 0: Note here that, in general, c twill
be random. The representative consumer has 1 unit of labor available in
each period, which is supplied inelastically. The production technology
is given by
yt= z tF(kt; n);
where F(¢; ¢) is strictly quasiconcave, homogeneous of degree one, and
increasing in both argument. Here, k tis the capital input, n tis the labor
input, and z tis a random technology disturbance. That is, fz g 1t=0is
a sequence of independent and identically distributed (i.i.d.) random
variables (each period z tis an independent draw from a ¯xed probability
distribution G(z)): In each period, the current realization, z , is learned
5.2. STOCHASTIC DYNAMIC PROGRAMMING 71
at the beginning of the period, before decisions are made. The law of
motion for the capital stock is
kt+1= it+ (1 ¡ ±)k t;
where i tis investment and ± is the depreciation rate, with 0 < ± < 1:
The resource constraint for this economy is
ct+ it= y :
5.2.1 Competitive Equilibrium
In this stochastic economy, there are two very di®erent ways in which
markets could be organized, both of which yield the same unique Pareto
optimal allocation. The ¯rst is to follow the approach of Arrow and
Debreu (see Arrow 1983 or Debreu 1983). The representative consumer
accumulates capital over time by saving, and in each period he/she
rents capital and sells labor to the representative ¯rm. However, the
contracts which specify how much labor and capital services are to
be delivered at each date are written at date t = 0: At t = 0; the
representative ¯rm and the representative consumer get together and
trade contingent claims on competitive markets. A contingent claim is
a promise to deliver a speci¯ed number of units of a particular object (in this case labor or capital services) at a particular date (say, date
T) conditional on a particular realization of the sequence of technology
shocks, fz 0; z1; z2; :::; Tg: In a competitive equilibrium, all contingent
claims markets (and there are potentially very many of these) clear at
t = 0; and as information is revealed over time, contracts are executed
according to the promises made at t = 0: Showing that the competitive
equilibrium is Pareto optimal here is a straightforward extension of
general equilibrium theory, with many state-contingent commodities.
The second approach is to have spot market trading with rational
expectations. That is, in period t labor is sold at the wage rate w t
and capital is rented at the rate r : At each date, the consumer rents
capital and sells labor at market prices, and makes an optimal savings
decision given his/her beliefs about the probability distribution of
future prices. In equilibrium, markets clear at every date t for every
72 CHAPTER 5. CHOICE UNDER UNCERTAINTY
possible realization of the random shocks fz 0; z1; z2; :::; g: In equilibrium
expectations are rational, in the sense that agents' beliefs about
the probability distributions of future prices are the same as the actual
probability distributions. In equilibrium, agents can be surprised
in that realizations of z tmay occur which may have seemed, ex ante,
to be small probability events. However, agents are not systematically
fooled, since they make e±cient use of available information.
In this representative agent environment, a rational expectations
equilibrium is equivalent to the Arrow Debreu equilibrium, but this
will not be true in models with heterogeneous agents. In those models,
complete markets in contingent claims are necessary to support Pareto
optima as competitive equilibria, as complete markets are required for
e±cient risk sharing.
5.2.2 Social Planner's Problem
Since the unique competitive equilibrium is the Pareto optimum for this
economy, we can simply solve the social planner's problem to determine
competitive equilibrium quantities. The social planner's problem is
max
ft;t+g1t=0
E 0
1X
t=0
¯ u(c t)
subject to
ct+ k t+1 = ztf(kt) + (1 ¡ ±)k ;
where f(k) ´ F(k; 1): Setting up the above problem as a dynamic
program is a fairly straightforward generalization of discrete dynamic
programming with certainty. In the problem, given the nature of uncertainty,
the relevant state variables are k tand z ; where k tis determined
by past decisions, and z tis given by nature and known when decisions
are made concerning the choice variables c tand k t+1: The Bellman
equation is written as
v(k ; zt) = max
c;kt+1
[u(c ) + ¯E v(k t+1; zt+)]
subject to ct + k t+1= z f(k ) + (1 ¡ ±)k t:
5.2. STOCHASTIC DYNAMIC PROGRAMMING 73
Here, v (¢; ¢) is the value function and E tis the expectation operator
conditional on information in period t: Note that, in period t; c t is
known but c t+ii = 1; 2; 3; :::; is unknown. That is, the value of the
problem at the beginning of period t + 1 (the expected utility of the
representative agent at the beginning of period t + 1) is uncertain as
of the beginning of period t: What we wish to determine in the above
problem are the value function, v(¢; ¢); and optimal decision rules for
the choice variables, i.e. k t+1 = g(k t; z) and c t= z tf(kt) + (1 ¡ ±)k t¡ g(k ; zt):
5.2.3 Example
Let F(k t; n) = k ®
tn 1¡®
t; with 0 < ® < 1; u(c t) = ln c ; ± = 1; and
E[ln z t] = ¹: Guess that the value function takes the form
v(k ; zt) = A + B ln k t+ Dln z t
The Bellman equation for the social planner's problem, after substituting
for the resource constraint and given that n t= 1 for all t, is
then
A+B ln k +Dln z t= max
kt+fln[z k ®
t¡ kt+1] + ¯E [A + B ln k t+1+ Dln z t+1]g ;
or
A+B ln k +Dln z t= max
kt+fln[z k ®
t¡ kt+1] + ¯A + ¯B ln k t+1+ ¯D¹g :
(5.7)
Solving the optimization problem on the right-hand side of the above
equation gives
kt+1 =
¯B
1 + ¯B
ztk®
t: (5.8)
Then, substituting for the optimal k t+1 in (5.7), we get
A + B ln k t+ Dln z t= ln
Ã
ztk®
t
1 + ¯B
!
+ ¯A + ¯B ln
Ã
¯Bz tk®
t
1 + ¯B
!
+ ¯D¹
(5.9)
74 CHAPTER 5. CHOICE UNDER UNCERTAINTY
Our guess concerning the value function is veri¯ed if there exists a solution for A;B; and D: Equating coe±cients on either side of equation
(5.9) gives
A = ln
Ã
1
1 + ¯B
!
+ ¯A + ¯B ln
Ã
¯B
1 + ¯B
!
+ ¯D¹ (5.10)
B = ® + ®¯B (5.11)
D = 1 + ¯B (5.12)
Then, solving (5.10)-(5.12) for A; B; and D gives
B =
®
1 ¡ ®¯
D =
1
1 ¡ ®¯
A =
1
1 ¡ ¯
"
ln(1 ¡ ®¯) +
®¯
1 ¡ ®¯
ln(®¯) +
¯¹
1 ¡ ®¯
#
We have now shown that our conjecture concerning the value function
is correct. Substituting for B in (5.8) gives the optimal decision rule
kt+1= ®¯z k®
t; (5.13)
and since c t= zk ®
t¡ kt+; the optimal decision rule for c tis
ct= (1 ¡ ®¯)z tk®
t: (5.14)
Here, (5.13) and (5.14) determine the behavior of time series for c tand
kt(where k t+1is investment in period t): Note that the economy will not
converge to a steady state here, as technology disturbances will cause
persistent °uctuations in output, consumption, and investment. However,
there will be convergence to a stochastic steady state, i.e. some
joint probability distribution for output, consumption, and investment.
This model is easy to simulate on the computer. To do this, simply
assume some initial k 0, determine a sequence fz gT
t=0using a random
number generator and ¯xing T; and then use (5.13) and (5.14) to determine time series for consumption and investment. These time series
5.3. REFERENCES 75
will have properties that look something like the properties of post-war
de

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