LA COMPETENCIA INTERCULTURAL

Real business cycle models are straightforward extensions of equilibrium models of the kind that we use throughout this course. In most cases, the models feature infinitely lived consumers, and business cycles are generated by random disturbances to production pos-sibilities. Unfortunately, solving that kind of model is difficult. Often no explicit solution is available, so numerical approximations have to be used. To keep the presentation tractable, in this chapter we will use a simpler framework in which people live for two periods only.

The model does not fit the facts as well as a full-scale real business cycle model, but it serves its purpose as a simple illustration of the main ideas of real business cycle theory.

In the model world there is a sequence of overlapping generations. Each period a new gen-eration of consumers is born, and each consumer lives for two periods. We will sometimes refer to the periods as years, and for simplicity we assume that exactly one consumer is born each year. People work in the first period when they are young. In the second period they are retired and live on savings. Throughout the model, superscripts refer to the year when a person was born, while subscripts refer to the actual year. For example,^ctt is the period-^tconsumption of a consumer who was born in year^t, so such a consumer would be young in period^t. Similarly,^ctt+1is the consumption of the same consumer in period^t+ 1, when he is old. The consumers do not care about leisure. A consumer born in year^thas the following utility function:

u(^ctt

;c t

t+1) = ln(^ctt) + ln(^ctt+1)^:

We could introduce a discount factor, but for simplicity we assume that the consumers value both periods equally. Note that at each point of time there are exactly two people around: one who was just born and is young, and another who was born the year before and is now retired. In each period the young person supplies one unit of labor and receives wage income^wt. The labor supply is fixed, since consumers do not care about leisure. The wage income can be used as savings^ktand as consumption^ct. The budget constraint of a

9.2 A Real Business Cycle Model 73

young worker is:

c t

t+^kt=^wt;

i.e., consumption plus savings equals income from labor. In period^t+ 1 the consumer born in^tis old and retired. The old consumer lends his savings^ktto the firm. The firm uses the savings as capital and pays return^rt₊₁to the old consumer. A fraction^Æof the capital wears out while being used for production and is not returned to the consumer. ^Æis a number between zero and one, and is referred to as the depreciation rate. The budget constraint for the retirement period is:

c t

t+1= (1 ^Æ+^rt₊₁)^kt; i.e., consumption equals the return from savings.

The household born in period^tmaximizes utility subject to the budget constraints, and takes prices as given:

We can use the constraints to eliminate consumption and write this as:

max

kt

fln(^{wt kt}) + ln((1 ^Æ+^rt+1)^kt)^g:

This is similar to the problem of the consumer in the two-period credit market economy that we discussed in Section 3.2. From here on we will drop the practice of denoting opti-mal choices by superscripted stars, since the notation is already complicated as it is. The first-order condition with respect to^ktis:

0 = 1 Solving this for^ktyields:

k

t= ^wt 2 ^: (9.1)

Thus, regardless of the future return on capital, the young consumer will save half of his labor income. Again, this derives from the fact that wealth and substitution effects cancel under logarithmic preferences. This feature is is helpful in our setup. Since there will be productivity shocks in our economy and^rt+1 depends on such shocks, the consumer might not know^rt+1in advance. Normally we would have to account for this uncertainty explicitly, which is relatively hard to do. In the case of logarithmic utility, the consumer does not care about^rt₊₁anyway, so we do not have to account for uncertainty.

Apart from the consumers, the economy contains a single competitive firm that produces output using capital^kt 1and labor^lt. Labor is supplied by the young consumer, while the supply of capital derives from the savings of the old consumer.The rental rate for capital is

r

t, and the real wage is denoted^wt. The production function has constant returns to scale and is of the Cobb-Douglas form:

f(^lt;kt ₁) =^Atlt k

1 

t 1^:

Hereis a constant between zero and one, while^Atis a productivity parameter.^Atis the source of shocks in this economy. We will assume that^Atis subject to random variations and trace out how the economy reacts to changes in^At. The profit-maximization problem of the firm in year^tis:

The first-order conditions with respect to^ltand^kt ₁are:

A

Using the fact that the young worker supplies exactly one unit of labor,^lt = 1, we can use these first-order conditions to solve for the wage and return on capital as a function of capital^kt 1:

Since the production function has constant returns, the firm does not make any profits in equilibrium. We could verify that by plugging our results for^wtand^rtback into the firm’s problem. Note that the wage is proportional to the productivity parameter^At. Since^At is the source of shocks, we can conclude that wages are procyclical: when^Atreceives a positive shock, wages go up. Empirical evidence suggests that wages in the real world are procyclical as well.

To close the model, we have to specify the market-clearing constraints for goods, labor, and capital. At time^tthe constraint for clearing the goods market is:

c

On the left hand side are goods that are used: consumption ^ctt of the currently young consumer, consumption^ctt ¹of the retired consumer who was born in^t 1, and savings

k

tof the young consumer. On the right hand side are all goods that are available: current production and what is left of the capital stock after depreciation.

The constraint for clearing the labor market is^lt= 1, since young consumers always supply one unit of labor. To clear the capital market clearing we require that capital supplied by

9.2 A Real Business Cycle Model 75

the old consumer be equal to the capital demanded by the firm. To save on notation, we use the same symbol^kt ₁both for capital supplied and demanded. Therefore the market-clearing for the capital market is already incorporated into the model and does not need to be written down explicitly.

In summary, the economy is described by: the consumer’s problem, the firm’s problem, market-clearing conditions, and a random sequence of productivity parameters^fAtg1t=1. We assume that in the very first period there is already an old person around, who some-how fell from the sky and is endowed with some capital^k0.

Given a sequence of productivity parameters^fAtg1t=1, an equilibrium for this economy is an allocation^fctt

;c t 1

t

;k

t 1^;ltg1t=1and a set of prices^frt;wtg1t=1such that:

 Given prices, the allocation ^fctt

;c t 1

t

;k

t 1^;ltg1t=1 gives the optimal choices by con-sumers and firms; and

 All markets clear.

We now have all pieces together that are needed to analyze business cycles in this economy.

When we combine the optimal choice of savings of the young consumer (9.1) with the expression for the wage rate in equation (9.2), we get:

k

t= 1 2^Atk

1 

t 1^:

(9.4)

This equation shows how a shock is propagated through time in this economy. Shocks to^At have a direct influence on ^kt, the capital that is going to be used for production in the next period. This implies that a shock that hits today will lead to lower output in the future as well. The cause of this is that the young consumer divides his income equally between consumption and savings. By lowering savings in response to a shock, the consumer smoothes consumption. It is optimal for the consumer to distribute the effect of a shock among both periods of his life. Therefore a single shock can cause a cycle that extends over a number of periods.

Next, we want to look at how aggregate consumption and investment react to a shock. In the real world, aggregate investment is much more volatile than aggregate consumption (see Barro’s Figure 1.10). We want to check whether this is also true in our model. First, we need to define what is meant by aggregate consumption and investment. We can rearrange the market-clearing constraint for the goods market to get:

c t

t+^ctt ¹+^kt (1 ^Æ)^kt 1=^Atlt k

1 

t 1^:

On the right-hand side is output in year^t, which we are going to call^Yt. Output is the sum of aggregate consumption and investment. Aggregate consumption^Ctis the sum of the consumption of the old and the young person, while aggregate investment^It is the

difference between the capital stock in the next period and the undepreciated capital in

Consumption can be computed as the difference between output and investment. Using equation (9.4) for^ktyields:

C

Aggregate investment can be computed as output minus aggregate consumption. Using equation (9.5) for aggregate consumption yields:

I

We are interested in how^Ctand^Itreact to changes in the technology parameter^At. We will look at relative changes first. The elasticity of a variable^x with respect to another variable^yis defined the percentage change in^xin response to a one percent increase in^y. Mathematically, elasticities can be computed as^@x

@y y

x

. Using this formula, the elasticity of consumption with respect to^Atis:

@C and for investment we get:

@I

It turns out that the relative change in investment is larger. A one-percent increase in^At leads to an increase of more than one percent in investment and less than one percent in consumption. Investment is more volatile in response to technology shocks, just as real-world investment is. Of course, to compare the exact size of the effects we would have to specify the parameters, likeand^Æ, and to measure the other variables, like^kt.

3More precisely,^Itin the model is gross investment, which includes replacement of depreciated capital. The net difference between capital tomorrow and today^{kt k}t 1is referred to as net investment.