# 6.8 The Structure of a Real Business Cycle Model

In the typical real business cycle model, aggregate output of a single good,

which can be used for both consumption or investment purposes, is produced

according to a constant returns to scale neoclassical production function

shown by equation (6.3):

Yt AtF(Kt , Lt ) (6.3)

where Kt is the capital stock, Lt is the labour input, and At represents a

stochastic productivity shift factor (shocks to technology or total factor productivity

= TFP). The evolution of the technology parameter, At, is random

and takes the form shown in equation (6.4):

At1 At t1, where 0 < <1, (6.4)

Here is large but less than 1, and is a random disturbance to technology.

Equation (6.4) tells us that the level of technology in any given period

depends on the level prevailing in the previous period plus a random disturbance

(Kydland and Prescott, 1996). In real business cycle models it is

usually assumed that the typical economy is populated by identical individuals.

This allows group behaviour to be explained by the behaviour of a

representative agent (Plosser, 1989; Hartley, 1997). The representative agent’s

utility function takes the general form given by (6.5):

Ut f (Ct , Let ), where f ′(Ct ) 0, and f ′(Let ) 0 (6.5)

Here Ct is units of consumption and Let hours of leisure for our representative

agent. It is assumed that the objective function of the representative agent

(Robinson Crusoe) is to maximize the expected discounted sum of their

current and future utility over an infinite time horizon. This maximization

problem is given by equation (6.6):

Ut Et u C L

t j

j

t j −t j t

∞

max , −| ,

0

1 1 0 (6.6)

where Ct is the representative agent’s level of consumption, Lt is the number

of hours of work, 1–Lt is the hours of leisure consumed, Et {·} is the mathematical

expectations operator, t is the information set on which expectations

are based, and is the representative agent’s discount factor. Equation (6.6)

provides a specification of a representative agent’s willingness to substitute

consumption for leisure. Thus the choice problem for the representative agent

is how to maximize their lifetime (infinite) utility subject to resource constraints

shown in equations (6.7) and (6.8):

Ct It ,≤AtF(Kt , Lt ) (6.7)

Lt Let ≤1 (6.8)

Equation (6.7) indicates that the total amount of consumption (Ct) plus investment

(It) cannot exceed production (Yt), and equation (6.8) limits the total

number of hours available to a maximum of 1. The evolution of the capital

stock depends on current investment (= saving) and the rate of depreciation,

, as given in equation (6.9):

Kt1 (1−)Kt It (6.9)

In this setting a disturbance to the productivity shift factor At (technological

shock) will result in a dynamic response from the utility-maximizing representative

agent such that we will observe variations in output, hours worked,

consumption and investment over many periods.

To illustrate how a ‘business cycle’ can occur in a world without money

or financial institutions, let us take the extreme case of Robinson Crusoe on

a desert island. Suppose an exogenous shock occurs (a change in At in

equation 6.3), raising Robinson Crusoe’s productivity. In this particular

example we can think in terms of an unusual improvement in the weather

compared to what Crusoe has been used to over the previous years. With

the same number of hours worked Crusoe can now produce much more

output given the more favourable weather. Because Crusoe is concerned

about consumption in the future as well as the present (see equation 6.6), it

is likely that he will choose to reduce current leisure and work more hours

in the current period; that is, Crusoe will engage in intertemporal labour

substitution.

The incentive to save and work longer hours will be especially strong if

Crusoe believes the shock (better-than-normal weather) is likely to be shortlived.

Because some of the increase in output is saved and invested, according

to equation (6.9), the capital stock will be higher in the next period, and all

future periods. This means that the impact of even a temporary shock on

output is carried forward into the future. Moreover, the response of the

representative agent to the economic shock is optimal, so that Crusoe’s

economy exhibits dynamic Pareto efficiency. When the weather returns to

normal the following year Crusoe reverts to his normal working pattern and

output declines, although it is now higher than was the case before the shock.

Remember, Crusoe now has a higher capital stock due to the accumulation

that took place during the previous year. As Plosser (1989) argues, the outcomes

we observe in response to a shock are ones chosen by the representative

agent. Therefore the social planner should in no way attempt to enforce a

different outcome via interventionist policies. Note that throughout this hypothetical

example we have just witnessed a fluctuation of output (a business

cycle) on Crusoe’s island induced entirely by a supply-side shock and Crusoe’s

optimal response to that shock. At no time did money or financial variables

play any part.

In the Crusoe story we noted how our representative agent engaged in

intertemporal labour substitution when the price of leisure increased (in

terms of lost potential current output) due to more favourable weather. According

to real business cycle theorists, the large response of the labour

supply to small changes in the real wage, resulting from the intertemporal

substitution of labour, acts as a powerful propagation mechanism. According

to this hypothesis, first introduced by Lucas and Rapping (1969), households

shift their labour supply over time, being more willing to work when real

wages are temporarily high and working fewer hours when real wages are

temporarily low. Why should this be the case?

Since the aggregate supply of labour depends on the labour supply decisions

of individuals, we need to consider the various factors which influence

the amount of labour individuals choose to supply. The benefits of current

employment relate primarily (but obviously not entirely) to the income earned

which allows the individual worker to consume goods and services. In order

to earn income, workers will need to allocate less of their time to leisure, a

term used to encapsulate all non-income-producing activities. The utility

function for the representative worker indicates that consumption and leisure

both yield utility. But in making their labour supply decisions workers will

consider future as well as current consumption and leisure. In taking into

account the future when deciding how much labour to supply in the current

period, workers will need to consider how much the current real wage offers

are above or below the norm. The substitution effect of a higher real wage

offer will tend to increase the quantity of labour supplied. However, since

higher real wages also make workers feel wealthier, this will tend to suppress

the supply of labour. This wealth or income effect works in the opposite

direction to the substitution effect. The impact of an increase in the current

real wage on the amount of labour supplied will clearly depend on which of

the above effects predominates. Real business cycle theorists distinguish

between permanent and temporary changes in the real wage in order to

analyse how rational maximizing individuals respond over time to changes in

their economic circumstances that are brought about by technological shocks.

The intertemporal labour substitution hypothesis suggests two things. First, if

a technological shock is transitory, so that the current above-normal real

wage offers are temporary, workers will ‘make hay while the sun shines’ and

substitute work for current leisure. Less work will be offered in the future

when the real wage is expected to be lower and hence the decision to supply

more labour now is also a decision to consume more leisure in the future and

less leisure now. Therefore real business cycle theory predicts a large supply

response from temporary changes in the real wage. Permanent technological

shocks, by raising the future real wage, induce wealth effects which will tend

to lower the current labour supply.

Second, some theorists have stressed the importance of real interest rates

on labour supply in flexible price models (see Barro, 1981, 1993). An increase

in the real interest rate encourages households to supply more labour

in the current period, since the value of income earned from working today

relative to tomorrow has risen. This effect would show up as a shift of the

labour supply curve to the right.

We can therefore express the general form of the labour supply function in

the real business cycle model as equation (6.10), where r = real interest rate:

SL SL (W/P, r) (6.10)

The appropriate intertemporal relative price (IRP) is given by (6.11):

IRP (1r)(W/P)1/(W/P)2 (6.11)

According to (6.11) any shocks to the economy that cause either the real

interest rate to rise or the current real wage (W/P)1 to be temporarily high

relative to the future real wage (W/P)2, will increase labour supply and hence

employment.

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