Representative Agent Models in Macroeconomics: A Comprehensive Analysis, Lecture notes of Compensation and Reward Management

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Notes on Macroeconomic Theory

Steve Williamson

Dept. of Economics

University of Iowa

Iowa City, IA 52242

August 1999

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 represen- tative ¯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 competi- tive 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

1.1. A STATIC MODEL 3

subject to c · w(1 ¡ `) + rks (1.2)

0 · ks ·

k 0 N

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 pre- ferred to less), we must have ks = k N^0 ; 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 equi- librium we will never have ` = 1; as then nothing would be produced, so we can safely ignore this case. The optimization problem for the con- sumer is therefore much simpli¯ed, and we can write down the following Lagrangian for the problem.

L = u(c; `) + ¹(w + r

k 0 N

¡ w` ¡ c);

where ¹ is a Lagrange multiplier. Our restrictions on the utility func- tion assure that there is a unique optimum which is characterized by the following ¯rst-order conditions.

@L @c

= u 1 ¡ ¹ = 0

@L @`

= u 2 ¡ ¹w = 0

@L @¹

= w + r

k 0 N

¡ w` ¡ c = 0

Here, ui is 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

k 0 N

¡ w;) ¡ u 2 (w + r

k 0 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 k 0 N :^ Equation (1.6) can be rewritten as u 2 u 1

= 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 curve, where an indi®erence curve has slope ¡u u^21 :

Firm's Problem

Each ¯rm chooses inputs of labor and capital to maximize pro¯ts, treat- ing 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 zf 1 = r; (1.7)

zf 2 = w; (1.8)

where fi denotes the partial derivative of f (¢; ¢) with respect to argu- ment 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) = zf 1 k + zf 2 n: (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)

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) k 0 = k (1.13)

That is, supply equals demand in each market given prices. Now, the total value of excess demand across markets is

N c ¡ y + w[n ¡ N (1 ¡ `)] + r(k ¡ k 0 );

but from the consumer's budget constraint, and the fact that pro¯t maximization implies zero pro¯ts, we have

N c ¡ y + w[n ¡ N (1 ¡ `)] + r(k ¡ k 0 ) = 0: (1.14)

Note that (1.14) would hold even if pro¯ts were not zero, and were dis- tributed 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 con- dition 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 mar- ket 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 repre- sentative 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:

zf 2 (k 0 ; 1 ¡ )u 1 (zf (k 0 ; 1 ¡); ) ¡ u 2 (zf (k 0 ; 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:

zf 1 (k 0 ; 1 ¡ `) = r (1.16)

zf 2 (k 0 ; 1 ¡ `) = w (1.17)

n = 1 ¡ `

k = k 0

c = zf(k 0 ; 1 ¡ `) (1.18)

It is not immediately apparent that the competitive equilibrium exists and is unique, but we will show this later.

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 ¡u u^21 :

In more general settings, it is true under some restrictions that the following hold.

1. A competitive equilibrium is Pareto optimal (First Welfare The- orem).
2. Any Pareto optimum can be supported as a competitive equilib- rium 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 ab- sence 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 com- petitive 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 market- clearing 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 mar- ginal 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:

12 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS

on the quantity of leisure (and therefore on employment) are ambigu- ous. 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 corre- lated. 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 produc- tivity 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 prob- lem for this economy is then

max u[z(1 ¡); `];

and the ¯rst-order condition for a maximum is

¡zu 1 [z(1 ¡ );] + u 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 [¡u 1 ¡ z(1 ¡)u 11 + u 21 (1 ¡ )]dz +(z^2 u 11 ¡ 2 zu 12 + u 22 )d = 0:

We then have

d` dz

u 1 + z(1 ¡ )u 11 ¡ u 21 (1 ¡) z^2 u 11 ¡ 2 zu 12 + u 22

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 ddz > 0 ; and where ddz < 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, z 2 > z 1 : 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 repre- sentative 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 con- stant, i.e. u(c; `) = h; (1.26)

where h is a constant, and

¡zu 1 (c; ) + u 2 (c;) = 0 (1.27)

Totally di®erentiating (1.26) and (1.27) with respect to c and ; and us- ing (1.27) to simplify, we can solve for the substitution e®ect ddz (subst:) as follows. d` dz

(subst:) =

u 1 z^2 u 11 ¡ 2 zu 12 + u 22

From (1.25) then, the income e®ect (^) dzd` (inc:) is just the remainder,

d` dz

(inc:) =

z(1 ¡ )u 11 ¡ u 21 (1 ¡) z^2 u 11 ¡ 2 zu 12 + u 22

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 man- ner. 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

max 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 + u 2 = 0:

The representative ¯rm's pro¯t maximization problem is

maxn (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 re- spect to ; the ¯rst-order condition for this problem yields an equation which solves for :

¡zu 1 [z(1 ¡ ) ¡ g;] + u 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:

18 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS

We can now ask what the e®ect of a change in government expen- ditures would be on consumption and employment. In Figure 1.5, g increases from g 1 to g 2 ; shifting in the resource constraint. Given the government budget constraint, there is an increase in taxes, which rep- resents 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

¡zu 11 + u 12 z^2 u 11 ¡ 2 zu 12 + u 22

dc dg

zu 12 ¡ u 22 z^2 u 11 ¡ 2 zu 12 + u 22

Here, the inequalities hold provided that ¡zu 11 + u 12 > 0 and zu 12 ¡ u 22 > 0 ; i.e. if leisure and consumption are, respectively, normal goods. Note that (1.30) also implies that dydg < 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 alloca- tion, 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 represen- tative consumer maximizes time-separable utility,

X^1 t=

¯tu(ct; `t);

where ¯ is the discount factor, 0 < ¯ < 1 : Letting ± denote the dis- count rate, we have ¯ = (^) 1+^1 ± ; where ± > 0 : Each period, the con- sumer is endowed with one unit of time. There is a representative ¯rm

1.3. A \DYNAMIC" ECONOMY 19

Figure 1.5: