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Chapter 5
Real business cycles
5.1 Real business cycles
The most well known paper in the Real Business Cycles (RBC) literature is Kydland and Prescott (1982). That paper introduces both a specific theory of business cycles, and a methodology for testing competing theories of business cycles.
The RBC theory of business cycles has two principles:
- Money is of little importance in business cycles.
- Business cycles are created by rational agents responding optimally to real (not nominal) shocks - mostly fluctuations in productivity growth, but also fluctuations in government purchases, import prices, or pref- erences.
The āRBCā methodology also comes down to two principles:
- The economy should always be modeled using dynamic general equi- librium models (with rational expectations).
- The quantitative implications of a proposed model should be taken se- riously. In particular, a modelās suitability for describing reality should be evaluated using a quantitative technique known as ācalibrationā. If the model āfitsā the data, its quantitative policy implications should be taken seriously.
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Romer, writing just a few years ago, treats these two meanings of the term as essentially the same. But since then, the RBC methodology has taken hold in a lot of places - researchers have analyzed āRBCā models with money, with all sorts of market imperfections, etc. In order to reflect this change, researchers are trying to change the name of the methodology to the more accurate (if awkward) āstochastic dynamic general equilibrium macroeconomics.ā
5.1.1 Some facts about business cycles
Business cycles are the reason why macroeconomics exists as a field of study, and theyāre the primary consideration of many macroeconomists. What char- acterizes business cycles? The two obvious characteristics are fluctuations in unemployment and output. A few definitions:
- Procyclical: a variable that usually increases in booms, decreases in recessions. For example, productivity is procyclical.
- Countercyclical: a variable that usually decreases in booms, increases in recessions. For example, unemployment is countercyclical.
- Acyclical: a variable that shows no systematic relationship to the busi- ness cycle.
- Fiscal policy: the governmentās policy for taxes and spending.
- Monetary policy: the governmentās policy for how much money to put into the economy.
Before we go into the details of an RBC model, letās establish some stylized facts about business cycles. Most of these are outlined in the beginning of Chapter 4 in Romer.
- Labor input varies considerably and procyclically (goes up in booms, down in recessions). Most of this variation is variation in employment rates, though some is in average weekly hours.
- The capital stock varies little at business cycle frequencies (1-3 years).
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technology level is below average, output is low, so investment is low, so the next periodās capital stock is also below average. So even if the technology level returns to normal next period, output will be below normal.
A fourth observation (not so much a puzzle) is why investment spending is more variable than consumption spending. This is not so hard to explain in the baseline RBC model (or any other model): an agent with the preference to smooth consumption over time will invest in productive periods and eat capital in unproductive periods.
5.1.2 The model
Now letās put this into a model. Weāll take the equilibrium growth model that weāve already looked at, and add a labor-leisure choice, and a stochastic technology. This produces whatās commonly known as the ābaseline real business cycle modelā. The version of the model Iām outlining here is from Chapter 1 in Thomas Cooleyās Frontiers of Business Cycle Research. The chapter is written by Cooley and Prescott. The model described in Romer is basically the same.
Consumers
The consumerās problem is the same as before, except that he values leisure. He supplies labor ht ā [0, 1] in period t, just like before, but now he receives utility from leisure. The consumer also has uncertainty over future prices, so he maximizes expected utility:
E 0
ā^ ā
t=
Ī²tu(ct, 1 ā ht) (5.1)
subject to the constraints:
xt + ct + bt+1 ā¤ wtht + rtkt + Rtbt + Ļt kt+1 ā¤ (1 ā Ī“)kt + xt kt ā„ 0 k 0 given N P G
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We assume that the consumer is making all time-t choices (xt, ct, kt+1, bt+1, ht) conditional on time t information (all variables subscripted t and below, plus the interest rate on bonds Rt+1).
Notice that we will be modeling fluctuations in employment as a representa- tive consumer varying his hours. Of course, employment fluctuations are a lot ālumpierā than that - this may affect the outcomes we look at.
The firm
Weāve talked about the consumer dealing with stochastic prices. We havenāt discussed the source of uncertainty. That will be on the firm side. The firm is the same as before. However, the production function is subject to random productivity shocks: The firmās problem is:
max Kt,Ht
ezt^ F (Kt, Ht) ā wtHt ā rtKt (5.2)
where F is a neoclassical production function and zt follows an AR(1) process:
zt = Ļztā 1 + t (5.3)
where t is white noise.
Equilibrium
When variables are stochastic, equilibrium is defined slightly differently.
An equilibrium in this economy is a joint distribution of prices and allocations such that, etc.
5.1.3 Solving the model
The consumerās Lagrangian is:
E 0
ā^ ā
t=
Ī²tu(ct, 1 ā ht) + Ī»t((1 ā Ī“)kt + wtht + rtkt + Ļt ā ct ā kt+1) (5.4)
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5.1.4 Calibration
Kydland and Prescott suggest a way to identify if this model can explain business cycles. Their method is known as calibration. The procedure is:
- Use microeconomic studies or theory to find values for all of the pa- rameters.
- Solve the model numerically, and simulate the economy.
- Compare the moments (standard deviations, correlations, etc) of the simulated economy with those in the actual economy.
- If the moments are matched, success!
- If not, the moments which donāt match up suggest areas of potential model improvement.
Weāre going to do this with the model. First, we need to put parametric functional forms. First, utility:
u(c, 1 ā h) =
( c^1 t āĪ±(1 ā ht)Ī±
) 1 āĻ
1 ā Ļ
This is a variation on the CRRA form weāve been using. Note that 1/Ļ is the intertemporal elasticity of substitution. When Ļ = 0, utility is linear, when Ļ = ā, utility is Leontief, and when Ļ = 1, utility is Cobb-Douglas (logarithmic).
Production is assumed to be Cobb-Douglas:
F (K, H) = KĪøH^1 āĪø^ (5.17)
Now, we need plausible parameter values for this model. Iāll use the values from Cooley and Prescott (1995). Their model is identical to the one here, except that they allow for population growth at rate Ī½ and productivity growth at a rate Ī³.
First, Ī². At the non-stochastic steady state, we have R = (^) Ī²^1. The average real interest rate in the U.S. is usually around 4% annually, which is about
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1% quarterly. So R ā 1 = (^) Ī²^1 ā 1 = 0.01, or Ī² ā 0 .99. Cooley and Prescott follow a slightly more complex process and calibrate Ī² as 0.987.
Next, Īø. If you remember, 1 ā Īø will be labourās share of output, a quan- tity that can be estimated from the national income accounts. Cooley and Prescott set Īø to 0.40.
Ļ is set to one. Estimates from micro studies of the typical workerās intertem- poral elasticity of substitution are in this range, and Ļ = 1 gives us a simple Cobb-Douglas form of the utility function
u(c, h) = (1 ā Ī±) ln c + Ī± ln(1 ā h) (5.18)
Now, solving for the steady state relationship between consumption, labour supply, and output, we get the following:
Ī± 1 ā h
1 ā Ī± c
w (5.19)
1 ā Ī± c
(1 ā Īø)
y h
Ī± is set so that, in the non-stochastic steady state, 31% of available time is spent working (h = 0.31)and the steady state output to consumption ratio is about 1.33 (c/y = 1.33). Substituting into the equations above and solving for Ī±, we get Ī± = 0.64.
Cooley and Prescott estimate that depreciation is 4.8% annually, so 1.2% quarterly (Ī“ = 0.012).
The parameters of the stochastic process for technology is easy to estimate. This model has perfect competition and constant returns to scale. So zt āztā 1 is the Solow residual. The average value of the Solow residual gives us our estimate for Ī³. Cooley and Prescott set Ī³ = 0.0156, giving about 1.6% annual TFP growth. Once we subtract out this average, we can estimate an AR(1) model and get Ļ = 0.95 and Ļ = 0.007.
Finally the quarterly population growth rate Ī½ is set at 0.012.
Numerical solution of the model
Once we have set up the model, and calibrated parameters, we next need to find a numerical solution to the model.
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As you can see, the variability in output is a little smaller, but the same order of magnitude. As in reality, consumption is much less variable than output and investment is much more variable. Just like in real life, the modelās variables are closely correlated with output. Theyāre actually a little too closely correlated, which Cooley and Prescott note is due to the fact that thereās only one source of uncertainty in this model.
To an RBC theorist, these numbers represent success. Weāve managed to write down a very simple model that duplicates many of the properties (mo- ments) of the actual data. There are a few failures though. This model seems to understate the variability of both consumption and hours. The RBC ap- proach to this failing is to investigate why the model doesnāt match, and adjust the model so that it does match.
The consumption variability is simple. Even with careful measurement, a lot of āconsumptionā is actually purchase of consumer durables, which really belongs in investment.
5.1.5 Other issues
Variation in labor
For an example of how the dynamic GE approach is supposed to work, letās consider the low variation in hours. Gary Hansen, in a 1985 paper, developed an explanation and a fix. All of the variation in hours worked in this model take the form of changes in the hours worked by each worker. We know that most of the variation in hours over the business cycle take the form of variations in employment. Recall that the functional form of utility is:
u(c, h) =
( c^1 t āĪ±(1 ā ht)Ī±
) 1 āĻ ā 1 1 ā Ļ
In order to generate higher variation in hours worked for each individual worker, we need to make them more willing to substitute intertemporally - work less when wages are low and more when they are high. For example, if Ļ = 0, then their IES is infinity - they will work all day or not at all, depending on wages. But micro studies show a low IES, so we canāt justify simply lowiering Ļ.
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Hansenās solution is to note that much of the variation in hours takes the form of variation in employment itself. He argues that it is not profitable to vary hours, given the fixed costs of employment. Instead he sets up the following model of the labor market. At the beginning of each period, a worker signs a binding contract to work hā (say 8) hours with probability Ļt and zero hours with probability (1Ļt). Whether the worker works or not is decided by lottery. The worker gets paid wage wtĻt whether or not he works. He doesnāt get to choose the value of hā, but does choose Ļt.
Expected utility is:
E[1 ā Ī± ln c + Ī± ln(1 ā h)] = (1 ā Ī±) ln c + ĻĪ± ln(1 ā hā) (5.22)
Now notice that this is linear in Ļt. Per capita hours in this economy are Ļthā, so the aggregate IES is quite high. It turns out that once you make this change, hours are much more variable, and the problem is āfixedā.
Internal propagation mechanism
One of the most difficult problems for RBC adherents to deal comes from the results of Cogley and Nason (AER 1995). They analyze the baseline RBC model from a different perspective. One of the selling points of the RBC model is that fluctuations in the model are persistent. However, their persistence really isnāt much more than that of the Solow residual, which is the exogenous source of shocks. Remember that the source of persistence in the baseline RBC model is that investment is higher in booms, so capital is higher in the near future. The problem is that new investment is very small relative to the capital stock, so the capital stock itself varies little.
Many papers since Cogley and Nason aim to find a better internal persistence mechanism. These include:
- Financial markets - for example suppose that an unexpected nega- tive shock causes solvent but illiquid companies to become cashflow- constrained or even go bankrupt. They could then become less efficient as a result.
- Labor market search - suppose that it takes time for workers to find new jobs that they match well with.
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One approach that some proponents of the RBC theory have suggested is that the Solow residual is poorly measured. Capital (and labor) utilization rates tend to vary significantly and procyclically. If you use the capital stock to measure the flow of capital services, you will overstate the extent of fluctuations in technical progress. Thereās a paper by King and Rebelo in the Handbook of Macroeconomics that adds variable utilization to an RBC model and gets plausible fluctuations without negative Solow residuals.
