Production, growth and business cycles: I. The basic neoclassical model

The paper’s stated aim is to provide “an introduction to the real business cycle research program by considering the basic concepts, analytical methods and open questions on the frontier of research,” with this first essay focused specifically on “the dynamic aspects of the basic neoclassical model of capital accumulation” (Sec. 1, p. 195-196). The authors are explicit that they bracket the role of money: “the role of money in an equilibrium theory of economic growth and fluctuations remains an open area for research,” and that understanding real fluctuations is a prerequisite for assigning money an important causal role (p. 196). The paper is organized around four questions: what role growth plays in restricting the model, what analytical methods can be used to study its time-series implications, what its dynamics are in response to technology shocks, and whether a technology-shock-driven neoclassical model replicates important features of macroeconomic time series (p. 195-196).

The economy is populated by identical infinitely-lived individuals with preferences U = sum of beta^t u(C_t, L_t), beta < 1 (eq. 2.1), producing a single good via a constant-returns neoclassical technology Y_t = A_t F(K_t, N_t X_t) (eq. 2.2), with capital accumulating as K_{t+1} = (1-delta_K)K_t + I_t (eq. 2.3) and resource constraints L_t + N_t <= 1 and C_t + I_t <= Y_t (eqs. 2.4-2.5). A_t represents temporary total-factor-productivity shocks, while X_t represents permanent, labor-augmenting technological progress, a distinction the paper says is required by Swan (1963) and Phelps (1966) for a feasible steady state (Sec. 2.3). Because the setup satisfies the conditions for the second welfare theorem, the optimal (social-planner) allocation coincides with the competitive equilibrium allocation (Sec. 2.2, citing Debreu 1954).

Any feasible steady state requires the growth rates of output, consumption, capital and investment to be equal (gamma_Y = gamma_C = gamma_K = gamma_I = gamma_X, eq. 2.7a) and the growth rate of work effort to be exactly zero (gamma_N = 1, eq. 2.7b), since hours are bounded by the fixed time endowment. Reconciling zero hours growth with a marginal product of labor that rises through labor-augmenting technical progress requires that the income and substitution effects of productivity growth on labor supply exactly offset (Sec. 2.3), which the authors show restricts momentary utility to u(C,L) = [1/(1-sigma)]C^(1-sigma)v(1-N) for sigma != 1 (eq. 2.8a) or u(C,L) = log(C) + v(1-N) for sigma = 1 (eq. 2.8b), where 1/sigma is the (constant, scale-invariant) intertemporal elasticity of substitution in consumption. “In all of the parameterizations of the neoclassical model that we consider, variations in work effort are associated with intertemporal substitution made possible in equilibrium by capital accumulation” (Sec. 2.3, p. 202) – without capital (a production function of the form A_t(N_tX_t)^alpha alone), hours would not respond to productivity shocks at all in general equilibrium.

After transforming the growing economy into a stationary one (dividing by X_t) and forming the Lagrangian (eq. 2.10), the optimum is characterized by five conditions (eqs. 2.11-2.15): two intratemporal first-order conditions, an intertemporal Euler-type condition, the resource constraint, and a transversality condition, which together pin down sequences of consumption, effort, capital and shadow prices given the initial capital stock k_0. The paper notes that, under strict concavity of the production function in capital, “the optimal capital stock converges monotonically to a stationary point” (Sec. 3, p. 204, citing Boyd 1986 and Becker, Boyd and Foias 1986 for the fixed-labor case). The competitive-equilibrium prices supporting this allocation are the real wage w_t = A_t D_2F(k_t,N_t) and real interest rate (1+r_t) = gamma_X*lambda_t/(lambda_{t+1}beta) (Sec. 2.4).

Linearizing the efficiency conditions around the steady state yields a first-order system k_hat_{t+1} = mu_1*k_hat_t + … (eq. 3.7) with a single stable root mu_1 governing the speed of adjustment, and Table 1 shows that with the baseline calibration (labor share alpha=0.58, depreciation 10%/year, real return 6.5%/year, technological growth 1.6%/year) and log utility (sigma=1), mu_1 = 0.966, implying that one-half of any initial deviation of capital from its steady state is worked off in 20 quarters (5 years). Larger sigma (lower intertemporal substitution, hence smoother consumption) slows adjustment (mu_1 closer to 1; e.g., mu_1=0.992 at sigma=10, half-life 86 quarters), while smaller sigma speeds it up (mu_1=0.886 at sigma=0.1, half-life 6 quarters) (Sec. 3.2.1, Table 1). Two pi-coefficients are invariant to sigma in the fixed-labor case: output and real wages move directly with capital with elasticity (1-alpha)=0.42, and the real interest rate moves inversely with capital with elasticity -0.024 (Sec. 3.2.1).

Reinstating labor as a choice variable (with u(c,L)=log(c)+theta_l*log(L) calibrated so steady-state hours N=0.20, the 1948-1986 U.S. average) gives mu_1=0.953 – a half-life of just under 14 quarters, slightly faster than the comparable fixed-labor case – because work effort provides an additional margin for intertemporal substitution (Sec. 3.2.2). Starting from an initially low capital stock, Fig. 1 shows “a sustained period in which output and consumption are low, but rising, while work effort and investment are high, but declining” (p. 209): low capital means low non-human wealth and a high shadow real interest rate, both of which induce more work, while the low marginal product of labor (low shadow wage) works in the opposite direction; with the calibrated parameters the wealth and interest-rate effects dominate, so hours are high (not low) during the transition from a capital shortfall, even though steady-state hours themselves are invariant to growth.

Rather than solving the full stochastic dynamic program (the Brock-Mirman 1972 approach, which the authors call computationally burdensome), the paper invokes certainty equivalence: it linearizes the efficiency conditions around the steady state (as in Sec. 3) and posits that productivity A_t follows a first-order autoregressive process with parameter rho, replacing future A’s with their conditional expectations (Sec. 4.1, eq. 4.1-4.2). This differs from Kydland and Prescott’s (1982) approximation (which takes a quadratic approximation to a return function built by substituting constraints into utility) mainly in centering the linearization on the steady state rather than a likely range of variation; the authors note this choice “allows us to replicate exactly the Long and Plosser (1983) closed-form solution” (footnote 25) and facilitates computing both impulse-response functions and population moments of the joint (z_t, s_t) process (Sec. 4.1).

The paper examines (1) Long-Plosser’s original parameterization with 100% depreciation (which has an exact closed-form solution, serving as a benchmark for the approximation method), (2) the same preferences but with realistic 10%-per-year depreciation (the “baseline” model, structurally identical to Hansen’s 1985 divisible-labor economy), (3) a “panel data” calibration with a labor-supply elasticity ten times smaller, based on Pencavel’s (1986) survey, and (4) an “infinite labor supply elasticity” calibration motivated by Rogerson’s (1988) result that individual indivisibility can make a finite-elasticity economy behave as if elasticity were infinite (Sec. 4.2, Table 2). Complete depreciation implies output, consumption and investment share an identical stochastic structure and hours are totally unresponsive to productivity (mu_1=(1-alpha)=0.42, eqs. 4.3-4.5) because income and substitution effects of a productivity shock on labor supply exactly cancel under log utility and 100% depreciation; moving to realistic depreciation raises mu_1 to 0.953 and makes the capital-accumulation response to productivity (pi_kA) sensitive to shock persistence, falling from 0.166 (rho=0) to 0.137 (rho=0.9) because capital accumulation responds less to a shock that agents know is more permanent (and hence mostly a wealth effect) (Sec. 4.3).

With purely temporary shocks (rho=0) in the baseline (realistic-depreciation) model, consumption is only about two-tenths as variable as output, investment is more than three times as variable, and labor input is “much more variable than consumption and about three fourths as variable as output” (Sec. 5.1, p. 223); with highly persistent shocks (rho=0.9), consumption becomes six-tenths as variable as output (consistent with the permanent-income hypothesis) while labor becomes less than half as variable as output, reflecting “diminished desirability of intertemporal substitution of effort with more persistent shocks.” Reducing the labor-supply elasticity to the panel-data value lowers labor’s relative volatility to 0.27 (roughly half the 1948-1986 U.S. relative standard deviation of hours), while the infinite-elasticity (indivisible-labor) parameterization makes labor “too volatile relative to output,” mirroring Hansen’s (1985) finding (Sec. 5.1, footnote 32).

Table 5 shows that with purely temporary shocks (rho=0.0), output, investment and hours exhibit essentially no serial correlation (first-order autocorrelations of 0.03, -0.01 and -0.02 respectively), while consumption, wages and the real interest rate are smoother and more persistent (autocorrelations of 0.95, 0.44 and 0.95) purely through consumption-smoothing and capital accumulation. When shocks are instead highly persistent (rho=0.9), output, investment and hours inherit high autocorrelation (0.93, 0.88, 0.86 at lag 1) because the shock process itself is persistent. The paper states plainly that “the principle serial correlation in output – one notable feature of economic fluctuations – derives mainly from the persistence of technology shocks” rather than from the model’s internal propagation mechanism (Sec. 6, p. 231), echoing the earlier statement (Sec. 1, p. 198) that the basic model “does not generate serial correlation in output and employment close to that exhibited by macroeconomic data” when shocks are purely temporary.

Using quarterly 1948.1-1986.4 U.S. data (CITIBASE: GNP, consumption of nondurables and services, gross fixed investment, and average weekly hours per capita, each detrended by a common estimated linear trend of 0.4% per quarter), output deviations have a standard deviation of 5.6% (range -13.0% to 10.0%); consumption is less volatile (3.9% SD, correlation 0.85 with output) and investment more volatile (7.6% SD, correlation 0.60) than output (Sec. 5.2, p. 226). Hours have a 3.0% SD but a correlation with output of “essentially zero” in the full sample, even though splitting the sample into roughly five-year subperiods raises the average correlation to 0.77 (never below 0.30) – a sensitivity the authors attribute to a low-frequency component not removed by the common deterministic trend, consistent with output’s own highly persistent autocorrelation structure (Sec. 5.2, p. 230). The authors caution, citing their own related work (King and Rebelo 1987, in progress), that low-frequency (e.g., Hodrick-Prescott) filtering can materially distort the economic picture – for instance raising the model’s own output-labor correlation while sharply lowering its output-capital correlation – which is why they report unfiltered population moments as their primary results (Sec. 5.2, p. 230, footnote 31 and 37).

The paper concludes that steady-state growth restricts preferences to a class in which labor effort is constant only in the steady state itself, not along transition paths or in response to shocks, where intertemporal substitution via capital accumulation applies to both consumption and effort; that highly persistent technology shocks let the basic model replicate procyclical employment, consumption and investment and the observed ranking of relative volatilities; but that the model is “less satisfactory” along other dimensions, since its serial correlation in output derives mainly from the assumed persistence of shocks rather than genuine internal propagation, and the model is silent on government and the heterogeneity of economic agents (a point attributed to McCallum 1987) (Sec. 6, p. 231). The authors flag as “perhaps the most intriguing possibility raised by the basic model” that economic fluctuations might be “just a manifestation of the process of stochastic growth,” which – along with government and heterogeneity – is the subject of the companion essay, Part II (“New Directions”) (Sec. 6, p. 231).