With Realistic Parameters the Basic Real Business Cycle Model Acts Like the Solow Growth Model

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1 Preliminary Draft With Realistic Parameters the Basic Real Business ycle Model Acts Like the Solow Growth Model Miles Kimball Shanthi P. Ramnath University of Michigan ABSTRAT The business cycle aspect of the real business cycle model was driven by unrealisitic values chosen for parameters. When empirical measures of key exogenous parameters are incorporated into the real business cycle model, the model responds to a technology shock with strikingly similarity to the Solow growth model. The similarity results from labor hours and the investment to output ratio remaining fixed. We argue that these key parameters include the technology process, the elasticity of intertemporal substitution, and the utility function. Based on prior empirical studies, technology shocks are found to be permanent while the elasticity of intertemporal substitution is estimated to be low. Also, King-Plosser-Rebello preferences are constructed to match the data which shows no long-run trend in labor hours. After imposing these three facts through a recalibration of parameters, the RB model behaves identically to the Solow model with respect to the adjustment of capital in response to a technology shock. What s more, these conclusions do not depend on the size of the labor supply elasticity. 1. Introduction By endogenizing labor supply and introducing mean-reverting technology shocks, the real business cycle model provided a mechanism that successfully replicated various features of a business cycle. Though puzzles remained, there was a marked change in the way growth theory and business

2 2 cycle theory were viewed. But soon the classic Prescott model became the target of criticism for what seemed to be an arbitrary methodology. Watson (1993) describes the RB methodology and aruges since the economic model does not provide a complete probability structure, inference procedures lack statistical foundations and are necessarily ad hoc. Various parameter values came under attack, including the largely disputed labor supply elasticity, which was said to be much larger than empirical results suggested. In the sections to follow, we will discuss three key parameters that drive the results and what their proper estimates should be. We will also discuss the irrelevance of the labor supply elasticity once the other parameters are chosen to have empirically sound values. 2. Realistic alibration of RB Parameters 2.1. Permanent versus Temporary Technology Shocks The RB model, as Prescott formulated it, assumes that technology shocks follow a persistent autoregressive process. Intuitively, this assumption is unrealistic. A technological improvement should not be temporary for the same reason good ideas should not be forgotten. And this intuition indeed appears in the data. It s true empirically that the Solow residual often mimics a business cycle and thus it seems plausible that technology shocks are autoregressive. However, the cyclicality of the Solow residual comes from changes in utilization and hence is the result of mismeasurement. Basu, Fernald, Kimball (2006) account for this mismeasurement by cleaning up the Solow residual. After estimating a purified technology series they find that after controlling for non-technological factors in measured aggregate total factor producitivity, technology shocks are in fact permanent. But permanent technology shocks do not produce business cycles. ogley and Nason (1995) find that standard RB models must rely heavily on exogenous sources of dynamics and output dynamics are essentially the same as impulse dynamics. This suggests that business cycles produced by the RB model are largely driven by the autoregressive technology process, which is not substantiated in the data. Permanent technology shocks are also supported by the existance of a unit root in output data.

3 3 If US gross domestic product contains a unit root, then shocks to GDP are permanent and the shocks likely to affect GDP are technological improvements. There has been some debate in the past as to whether output data is can be described by cyclical fluctuations around a deterministic trend, or whether output data is a non-stationary process. The first view is consistent with business cycle theory in that once data is detrended, the residual component is interpreted as a business cycle. ampbell and Mankiw (1987) question this traditional view and find that shocks to GDP in postwar US data are largely permanent, suggesting output data is non-stationary. Nelson and Plosser (1982) find that the existence of a unit root cannot be rejected in various US aggregate times series data. hristiano and Eichenbaum (1990) dismiss ampbell and Mankiw s (1987) methodology, however they conclude that whether or not output data has a unit root is indeed uncertain. The use of permanent technology shocks is now more accepted in the literature. Rotemberg and Woodford (1996) argue that technology shocks must be permanent in order for the model, rather than an exogenous factor, to explain growth. Gali (1999) also assumes permament technolgy shocks when identifying the effect of technology shocks on labor hours. The identifying assumption is that only a permanent techology shock can affect labor productivity in the long run King-Plosser-Rebello Preferences Much research has found that the long-run labor supply elasticity is zero. In other words, in the long run, income and substitution effects cancel. In light of this evidence, King-Plosser-Rebello preferences are appropriate to model consumer utility. KPR preferences are constructed such that income and substitution effects cancel and they allow for nonseparability of consumption and labor. Kimball and Shapiro (2004) argue for the use of KPR preferences to assure key features of the data are maintained in preferences.

4 Low Elasticity of Intertemporal Substitution The RB model uses the elasticity of intertemporal substitution as its internal propogation mechanism. A low elasticity of intertemporal substitution implies that consumption responds modestly to changes in the interest rate. Thus in order to incite dynamics, higher values are more favorable in RB models. However, empirical estimates of the elasticity of intertemporal substitution tend to be low. Hall (1988) estimates the elasticity by looking at changes in consumption with respect to changes in the real interest rate. The elasticity of substitution with respect to Treasury bills and savings accounts are inconclusive. Stock market estimates reveal an elasticty close to zero. Basu and Kimball (2002) find similar results that the elasticity of intertemporal substitution is small. They measure an elasticity of 0.50 that is significantly different than zero The Debate over Labor Supply Elasticity The labor supply elasticity parameter used by Prescott has been often criticized for being too high. In macroeconimics, a high labor supply elasticity is preferable since it implies a large labor response to technology shocks thereby amplifying the business cycle produced through simulation. But labor economists have done numerous studies to estimate labor supply elasticities and tend to estimate values much lower than the macroeconomist hopes for. In Pencavel s handbook chapter on male labor supply, he reports that using US data, Ashenfelter and Heckman (1973) estimate a male income-compensated wage elasticity of Hausman and Ruud (1984) estimate a response of 0.55, while Wales and Woodland (1979) estimate the highest elasticity of These estimates are much lower than the calibration that typical RB models use for labor supply elasticity, which can range anywhere from a value of 3 to infinite. Once empirical estimates for the key parameters we discuss above are used to calibrate the basic RB model, the magnitude of the labor supply elasticity becomes irrelevant. In fact, when empirical parameters are used, labor is constant after a technology shock.

5 5 3. Analyzing Basic Real Business ycle Models The basic real business cycle model begins with a representative agent maximizing the stream of present-value discounted utility. There is a representative firm seeking to maximize profits subject to a material balance condition and an equation of motion for capital. We will consider the centralized problem, where the representative agent owns the capital and labor. We will impose KPR preferences and for simplicity, assume a obb-douglas production function. The problem is then to maximize subject to the constraint that U = 0 e ρt 1 1 s 1 1 e [( 1 s 1)v(N)] s = ZK α N 1 α and the materials balance constraint, = + I + G. apital evolves according to following the equation of motion, K = I δk. For now, we will assume that there is no government spending, giving us the current-value Hamiltonian, H = 1 1 s 1 1 e ( 1 s 1)v(N) + Λ[ZK α N 1 α δk]. s Taking first order conditions we get equilibrium conditions giving us the H : 1 s e ( 1 s 1)v(N) = Λ H N : 1 1 s e [( 1 s 1)v(N)] v (N) = ΛW Λ = ρλ R + δ. King-Plosser-Rebello utility is constructed such that the real wage is is equivalent the disutility of labor scaled by consumption. In other words, v (N) = w where w is the real wage, is

6 6 consumption and v (N) is the marginal disutility from working. Using KPR utility and obb Douglas technology we show that labor is fully characterized by the consumption to output ratio. Proposition 1. With obb Douglas technology and King-Plosser-Rebello utility N monotonically relates to /. Proof. v (N) = w Nv (N) = wn Nv (N) = wn Nv (N) = Nv (N) = wn wn obb-douglas production implies that wn = 1 α where α is capital s share in production and 1 α labor s share. This implies that, Nv (N) = 1 α. (1) Note that v (N) is the marginal disutility from working and increasing in N. If / is constant, then N will also remain constant. However, if / rises (income effect dominates), then N must fall and if / falls (substitution effect dominates), then N must rise. This is will prove to be a useful result for the rest of our analysis. 4. Steady-State and Deviations from We will now derive the steady-state equations for all contemporanous variables in terms of capital, labor, and the marginal value of capital. We start by deriving the steady-state values for

7 7 / and I/ in terms of parameters. I = δk = δα R = δα ρ + δ = I W N + RK I = W N + RK I = (1 α) + RK I = (1 α) + (ρ + δ ρ + δ )α = (1 α) + ρδ ρ + δ δα ρ + δ To begin we can derive the log-linarized equilibrium equations. onsumption: Substituting in Output: 1 α (/ ) 1 s e ( 1 s 1)v(N) = Λ 1 s + ( 1 1)ṽ(N) = Λ s 1 s c + (1 s 1)v (N) = λ for Nv (N) we finally get an expression for the log deviation of consumption, c = (1 α) (1 α) + ( ρ s)n λs (2) ρ+δ )α(1 y = αk + (1 α)n + z Investment: i = y( I ) c( I ) i = αk + (1 α)n + z( I ) ( (1 α) (1 α) + ( ρ ρ+δ )α(1 s)n λs)( I ) Rental rate: marginal product of capital r = (α 1)k + (1 α)n + z

8 8 Labor demand: and the price measure of the solow residual, n + z k = σ KN (r + z w) (1 α)z = α r+(1 α)wassuming obb-douglas and normalizing prices to 1, y = αk + (1 α)(n + z) and, and so we get, 0 = αr + (1 α)(w z) n + z + k = σ KN (r + z w). If we assume technology is not changing and therefore = 0, then demand and supply for capital becomes, r = 1 α (w z). α To get labor demand eliminate r from the equation to get, n + z + k = σ KN [ 1 α (w z) + z w] α n = k + ( σ KN α 1)z σ KN α w. (3) Labor supply: H N : 1 1 s e [( 1 s 1)v(N)] v (N) = ΛW n = η(w + sλ) (4) factors] [Here is where I would set up the table to summarize variables in terms of n, λ, and exogenous

9 9 By setting labor supply equal to labor demand, we can solve for the optimal amount of labor, n and the wage rate, w. Multiplying equation 1 by σ α and equation 2 by η and adding them together we get: ( σ α + η)n = σ α sλ + +η( σ α 1)z n = s σ α λ + +η( σ α 1)z σ α + η Subtracting second equation from the first gives, n = η(w + sλ) n = k + ( σ α 1)z σ α w 0 = sηλ k ( σ α 1)z + σ α w + wη k + ( σ α 1)z sηλ = ( σ α + η)w w = k + ( σ α 1)z sηλ σ α + η. With our equation for n we can now substitute back in, to solve for our contemporaneous variables in terms of k and λ. In doing so we will then solve for our two log-linearized equations of motion. 5. Empirical Parameter Estimates and Simulations 5.1. Results 6. onclusion