2.2 A Standard One-Good Economy
2.2.8 Standard RBC Model: An Example
To end this section, I use a standard RBC model as an example to illustrate the business cycle properties of the one-good economy under different types of shocks.
Suppose that prices are perfectly flexible (φp = 0) and all markets are perfectly com-petitive. Shocks to the price elasticity of goods demand and the wage elasticity of labor demand are no longer considered as both converge to infinity: εp → ∞ and εw → ∞. The production function for the standard RBC model is assumed to be Cobb-Douglas (CD):
yt= (zl,tlt)αk1−αt , where α ∈ (0, 1) is the CD share of labor. The required effective labor is given by L (y, k) = yα1k1−α1 and the convexity of the variable cost function in steady state is ζ = 1−αα > 0.
Calibration
There are seven parameters remained to be calibrated. The depreciation rate δ is calibrated to match the average ratio of gross private domestic investment to private fixed assets from 1947 to 2016 in the National Income and Product Accounts (NIPA) published by the U.S. Bureau of Economic Analysis (BEA). The curvature of the consumption utility
Table 2.1: Parameters and Calibration Targets – Standard RBC
Parameter Value Target
δ 0.0210 Quarterly depreciation rate 0.021 γ 1.0000 Elasticity of inter-temporal substitution 1 η 0.0000 Frisch elasticity of labor supply ∞ α 0.6200 Labor share of income 0.62 β 0.9747 Investment to output ratio 0.17
γ is calibrated to give an elasticity of inter-temporal substitution of 1, which corresponds to a log utility of consumption. The curvature of the dis-utility of labor η is chosen to imply an infinite Frisch elasticity of labor supply as in the indivisible labor setup (e.g., Hansen, 1985 and Rogerson, 1988). This choice of η helps the RBC model to generate a sizable labor volatility under productivity shocks (e.g., King and Rebelo, 1999). The CD share of labor α is calibrated to match the average labor share of income estimated by the Bureau of Labor Statistics (BLS) from 1946 to 2016. The subjective discount factor in steady state β is calibrated to be such that the investment to output ratio in steady state is equal to the average of the gross private domestic investment to the gross domestic product (GDP) ratio from 1947 to 2016 in the NIPA.
Table 2.1 summarizes the calibrated parameter values and their mostly associated targets.
Quantitative Results and Discussions
Tables 2.2 compares the business cycle statistics in the calibrated RBC model under dif-ferent shocks with those in the data. There are five stylized business-cycle facts that we aim to match simultaneously using a single type of shock. First, there is a co-movement between consumption, investment, and hours. Second, the volatility of investment is much larger than the volatility of consumption. Third, the real wage rate is not too counter-cyclical.5 Fourth, the labor productivity measured as the output to labor ratio is not too countercyclical. Fifth, the Solow residual measured based on the average labor share of income is strongly pro-cyclical.
The investment specific technology shock zi,t and the patience shock βtare both shocks to
5The cyclicality of the real wage rate shown in Table 2.2 is for the average real wage rate of the business sector deflated with the GDP deflator. For robustness, Appendix 4.6 shows the cyclicality of the average real wage rates of different sectors including the business sector, the no-farm sector, and the non-financial corporations. The average nominal wage rates are deflated using different deflators, e.g., the GDP deflator, the consumer price index (CPI), and the own sector deflator. The results show that the real wage rates on average are roughly acyclical. It is also well known in the literature that cyclicality of the average real wage rate tends to underestimate the pro-cyclicality of the real wage rate at the individual level because of a composition bias (e.g., Bils, 1985).
investment demand. Both are subject to the Barro-King curse and thus fail to generate the co-movement between consumption, investment, and hours. A positive investment demand shock, though induces more hours, inevitably crowds out consumption.
The consumption demand shock zc,t is not subject to the Barro-King curse because the marginal rate of substitution between consumption and leisure is directly affected. Hence, consumption and hours can move in the same direction. If the shock is persistent, the desire to invest could be strong enough so that there is a co-movement among consumption, investment, and hours. However, the relative volatility of investment to consumption is too small. Intuitively, capital resources are tight in the RBC model: a short run increase in output would cause an increase in the real marginal cost of production and a decrease in the marginal product of labor. Thus, it is not wise to concentrate investment in a single period. The tendency to smooth investment under consumption demand shocks causes a small volatility of investment relative to that of consumption.
The labor dis-utility shock ¯ωtis able to generate a co-movement between consumption, in-vestment, and hours and a large relative volatility of investment to consumption. However, like all other shocks that do not affect productivity, e.g., zc,t, zi,t, βt, the labor dis-utility shock ¯ωt generates a strongly countercyclical real wage rate, a strongly countercyclical labor productivity, and an acyclical Solow residual.
The labor productivity shock zl,tturns out to be the only type of shock that can generate all the stylized business-cycle facts. Therefore, the business cycle literature tends to consider productivity shocks as the primary driving forces of business cycles since RBC theory has been introduced by Kydland and Prescott (1982). This view, however, contradicts the Keynes tradition which emphasizes on real demand shocks as the main driving forces of business cycles. The new Keynesian (NK) literature developed based on RBC theory tries to re-introduce the importance of demand by including nominal rigidities. However, the standard NK literature emphasizes more on the effectiveness of monetary demand than on the importance of real demand. For shocks that directly affect the real economy, supply shocks, including productivity shocks and markup shocks, remain to be the most important (e.g., Smets and Wouters, 2007).
Table 2.2: Business Cycle Statistics – Standard RBC
The U.S. Data zc,t zi,t βt ω¯t zl,t
Persistence ρ 0.00 0.99 0.00 0.99 0.00 0.99 0.00 0.99 0.00 0.99
ρ (c, h) 0.58 [0.42, 0.72] 0.88 0.96 -0.99 -0.91 -0.99 -0.93 0.42 0.90 0.39 0.83 volatility of investment to consumption. Cov(w,y)/σ2y stands for the covariance between x and output relative to the variance of output. The brackets are the 95% confidence intervals calculated using a parametric bootstrapping method (see Appendix 4.5 for details). The U.S. data is from the BEA and the BLS. A path of 5,000 quarters is simulated to calculate the statistics for each calibrated model. All variables are Hodrick-Prescott (HP) filtered logarithms of the original series.