# VI. Real Business Cycles Models

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1 VI. Real Business Cycles Models Introduction Business cycle research studies the causes and consequences of the recurrent expansions and contractions in aggregate economic activity that occur in most industrialized countries. Kydland and Prescott (1982) and Long and Plosser (1983) strikingly illustrated the promise of the application of rational expectations and general equilibrium analysis in business cycle research. Three revolutionary ideas in Kydland and Prescott (1982). Business cycles can be studied using Dynamic General Equilibrium models. It is possible to unify business cycle and growth theory by insisting that business cycle models should be consistent with the empirical regularities of long-run growth. We can quantitatively compare features of a model economy with stylized facts of real economy. Simple equilibrium models, when driven by shifts in total factor productivity, could generate time series with the same complex patterns of volatility, persistence and comovement as those of actual economies. Real business cycle analysis now occupies a major position in the core curriculum of nearly every graduate program. The methods of the RBC research program are now commonly applied, being used in work in monetary economics, international economics, public finance, labor economics, asset pricing and so on. 1

2 Data Detrending and Measuring Business Cycles Many real macroeconomic quantities grow overtime. (King and Rebelo (1999) Figure 1) Business Cycles: Those fluctuations in economic time series that have periodicity of eight years or less. For example, there are alternating periods of high and low output, but these episodes are of unequal duration and amplitude. To identify the business cycle properties of a series with a trend we must decompose the series in trend and cyclical components Detrending via log-linear regressions y t = y c t + y g t ln Y t = β 1 + β 2 t + e t y g t = ˆβ 1 + ˆβ 2 t An alternative log-linear regression y c t = ê t ln Y t = β 1 + β 2 t + β 3 t 2 + e t y g t = ˆβ 1 + ˆβ 2 t + ˆβ 3 t 2 Hodrick-Prescott Filter (HP filter) y c t = ê t the HP filter identifies a trend that fluctuates slightly overtime. specifically, the trend component (y g t ) is chosen to minimize a loss function: min {y g t } t=0 {(y t y g t ) 2 + λ[(y g t+1 y g t ) (y g t yt 1)] g 2 } t=1 as λ, the trend component approaches a linear trend; as λ 0, the trend coincides with the original series. 2

3 Business cycle statistics for the U.S. economy (King and Rebelo (1999) Table 1) Volatility consumption, wages, labour productivity, capital stock and TFP are less volatile than output; investment and capital utilization are more volatile than output; total hours worked and employment are about as volatile as output; hours per worker are much less volatile than output. Persistence First-order autocorrelations are large and positive. Cyclicality strongly procyclical: consumption, investment, total hours worked, labour productivity and TFP. Imports are more procyclical than exports. acyclical: wages, capital stock and government expenditures. countercyclical: real interest rate. Stylized Facts of Economic Growth Balanced Growth output grows at a (more or less) constant rate; capital grows faster than labour input; growth rates of output and capital stock are about the same; the rate of profit on capital has no trend. Other Facts the ratio of labour income to output has no trend; the ratio of investment to output has no trend; the ratio of consumption to output has a small (not statistically significant) positive trend. 3

4 Standard Real Business Cycles Model Keep the structure and assumptions of the neo-classical growth model. However, the parameter A in the production function is now a random variable (productivity shock). Y t = A t Kt α (N t γ t ) 1 α γ t : deterministic component of productivity A t : stochastic component of productivity Productivity shocks are stationary and follow an AR(1) process in logs ln A t+1 = ρ ln A t + ε t+1 0 < ρ < 1, ε t+1 iid(0, σ 2 ε) The competitive equilibrium is optimal in the standard RBC model. Therefore, we focus on a central planner s maximizing (under constraints) the expected lifetime utility of a representative household. max E 0 β t [ln C t + θ ln(1 N t )] t=0 subject to Ỹ t = A t Kα t N 1 α t C t + Ĩt = Ỹt γ K t+1 = (1 δ) K t + Ĩt ln A t+1 = ρ ln A t + ε t+1 0 < ρ < 1, ε t+1 iid(0, σ 2 ε) K 0 > 0 and A 0 > 0 given Unless δ = 1 and utility is logarithmic, this model cannot be solved analytically (except in steady state). 4

5 With 0 < δ < 1, numerical methods must be used to solve the model (e.g. linearization method of King, Plosser and Rebelo). The use of numerical methods requires numerical values for the parameters of the model calibration or estimation. Calibration Select parameter values based on microeconomic empirical studies and on long-run properties of the economy. There are broadly two parts of calibration. One must begin by choosing functional forms which imply that certain parameters are important and then one must assign parameter values. Labour income share in the model is 1 α. Labour income share in U.S. data is roughly 2/3. So α = 1/3. The mean of A affects only the scale of the economy, we normalize its unconditional mean to unity, A = 1. In the model, γ is the growth rate of Y (and X, C, I, K). We set γ = 1.004, the quarterly gross growth rate of per capita output in post-war U.S. data. The crucial assumption in RBC analysis is that the stochastic component of productivity can be extracted from the empirical Solow Residual (SR) measured as ln SR t = ln Y t (1 α) ln N t α ln K t If the theoretical model (the production function) is correct, ln SR t = ln A t + (1 α) ln(γ) t. Therefore, given values for α and γ, the productivity shocks are calculated as ln A t = ln SR t (1 α) ln(γ) t. 5

6 Given values for α and γ, we calculate ln A t = ln SR t (1 α) ln(γ) t and then estimate ln A t+1 = ρ ln A t + ε t+1 to obtain ˆρ = 0.979, ˆσ ε = With logarithmic periodic utility u( C t, N t ) = ln C t + θ ln(1 N t ) θ = 3.48 is selected to insure that steady-state hours worked equal The discount factor β is chosen so that the steady-state real interest rate equals the average return to capital in the post-war U.S. data. In steady state, the capital Euler equation implies β = γ 1 + r k δ = / Capital depreciation rate δ = is chosen so that the capital-output ratio is approximately 10 in the model s steady state: K Y = 10. Predictions of the Standard RBC Model King and Rebelo (1999) Figure 7 compares historical and simulated paths for the U.S. economy. Looking at Figure 7, we can see that the basic RBC model gives quite a good account of the quarter-to-quarter variation in the output time series. Turning to the individual components of output, the performance of the RBC model is also surprisingly good for such a simple model. One way of evaluating the predictions of the basic RBC model is to compare moments that summarize the actual experience of an economy with similar moments from the model. Therefore, we compare Tables 1 and 3 in King and Rebelo (1999). 6

7 From the comparison, we can see that the RBC model produces a surprisingly good account of US economic activity. However, there are also evident discrepancies. Internal propagation of shocks is weak in the standard RBC model. Autocorrelation Model Data Prod. Shocks (A) Output (Y ) Impulse response functions (IRFs): King and Rebelo (1999) Figures 9 and 10. Main Criticisms of the RBC Model I. Labour Market A few statistics 1. corr(w, Y ) is too high; 2. corr(w, N) is too high; 3. var(n)/var(w) is too small; 4. In the model, the labour supply elasticity is much larger than in the data. The labour market in the standard RBC model labour demand comes from firm s FONCs w t = (1 α) A t K α t N α t N d t = [ (1 α) A t K α t w t ] 1 α labour supply comes from the representative household s FONCs w t = (graphical representation) θ C t 1 N t N s t = 1 θ C t w t 7

8 II. Productivity Shocks 1. It is hard to identify the macro shocks that produce the productivity variations suggested by the Solow residuals. 2. Solow residuals often decline suggesting that recessions are caused by technological regress. 3. Measurement problems: Solow residual based measures of shocks that do not account for unmeasured variations in labor and capital will tend to be more volatile and procyclical than true shocks to technology. For example, if we take into account variable labor effort (e t ) and variable capital utilization rates (z t ), Y t = A t (z t K t ) α (e t N t X t ) 1 α ln Y t α ln K t (1 α) ln N t = ln A t + (1 α) ln X t + α ln z t + (1 α) ln e t ln SR t = ln NSR t + α ln z t + (1 α) ln e t. Thus, Solow residual (SR t ) is overestimated. 8

9 Table 1 of King and Rebelo (1999) Business Cycle Statistics for the U.S. Economy SD SD/SD(Y ) Autocor. Cor. with Y Y C I N Y/N w r A SD: Standard Deviation; SD/SD(Y ): Standard Deviation relative to Output; Autocor.: First-Order Autocorrelation; Cor. with Y : Contemporaneous Correlation with Output. 2. Y : output per capita; C: consumption per capita; I: investment per capita; N hours worked per capita; w: real wage (compensation per hour); r: real interest rate; A: total factor productivity. 3. All variables are in logarithms (with the exception of the real interest rate) and have been detrended with the HP filter. Data sources are described in Stock and Watson (1998), who created the real rate using VAR inflation expectations. Table 3 of King and Rebelo (1999) Business Cycle Statistics for Basic RBC Model SD SD/SD(Y ) Autocor. Cor. with Y Y C I N Y/N w r A Note: All variables have been logged (with the exception of the real interest rate) and detrended with the HP filter. 9

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