Indivisible Labor and the Business Cycle

Transcription

1 Indivisible Labor and the Business Cycle By Gary Hansen Zhe Li SUFE Fall 2010 Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

2 Motivation Kydland and Prescott (1982) Equilibrium theory of the business cycle It fails to account for some important labor market phenomena existence of unemployment uctuations in the rate of unemployment uctuations in hours worked are large relative to productivity uctuations It depends too heavily on the willingness of individuals to substitute leisure across time in response to wage or interest rate changes when accounting for the uctuations in hours worked Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

3 Extensive margin vs. intensive margin Heckman and McCurdy (1980) var(log H t ) = var(log h t ) + var(log N t ) + 2cov(log h t, log N t ) 20% 55% rest intensive extensive hours per worker employment Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

4 Rationale of non-convexity individual preferences for leisure utility function exibites decreasing marginal utility of leisure at low levels of leisure and increasing marginal utility at higher levels in the technology marginal productivity of an individual s work e ort is increasing during the rst part of the workday or workweek (warm up), and then decreasing later on Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

5 Model with divisible labor a continuum of identical in nitely lived households [0, 1] a single rm with access to a technology f (λ t, k t, h t ) = λ t k θ t h 1 t θ constraint c t + i t f (λ t, k t, h t ) law of motion for the capital stock k t+1 = (1 δ) k t + i t 0 δ 0 Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

6 Technology First-order Markov process λ t+1 = γλ t + ε t+1 ε t+1 i.i.d with distribution function F (mean 1 γ) Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

7 Households solve the following problem max E fc t,h t g β t u (c t, 1 h t ), given k 0 and λ 0 t=0 c t + i t w t h t + r t k t k t+1 = (1 δ) k t + i t where w t = f h (λ t, k t, h t ) according to labor demand Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

8 Model with indivisible labor α t contract on probability of working: ex anti identital households, ex post employed and unemployed U(c t, α t ) = α t (log c t + A log (1 h 0 )) + (1 α t ) (log c t + A log 1) U(c t, α t ) = log c t + Aα t log (1 h 0 ) h 0 hours worked if employed, 0 hours workerd if unemployed risk of unemployed is insured (note that c t is the same) h t = α t h 0 per capita hours worked Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

9 Households problem max E fc t,α t g β t U(c t, α t ), given k 0 and λ 0 t=0 c t + i t w t α t h 0 + r t k t k t+1 = (1 δ) k t + i t where w t = f h (λ t, k t, h t ) according to labor demand Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

10 A key property The elasticity of substitution between leisure in di erent periods for the "representative agent" is in nite 1 l t = h t = α t h 0 α t = 1 l t h 0 U(c t, α t ) = log c t + Aα t log (1 h 0 ) U(c t, α t ) = log c t + A 1 l t h 0 log (1 h 0 ) Linear in leisure: either consume leisure as much as possible, or not consume it at all (marginal utility of leisure is constant) risk aversion parameter: 1 U(c t, α t ) = log c t + Bl t B = A log(1 h 0 )/h 0 Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

11 Solution method Economies with aggregate uncertainty: decision rules are solved approximately Dynamic optimization problems with decision variables i t and h t (divisible labor) / α t (indivisible labor), and state variables λ t and k t Bellman s equation: V (k, λ) = max u (c, 1 h) + βe V k 0, λ 0 jλ V (k, λ) = max u (c, α) + βe V k 0, λ 0 jλ Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

12 Calibration Choose functional forms, distribution F parameter γ and σ ε : 0.95 and Parameter values for θ, δ, β, A, h 0 From divisible labor model (steady state) θ capital share 0.36 δ annual rate of depreciation 10% β interest rate 4% 0.99 A 1/3 of time on market activities 0.2 From indivisible labor model h 0 = 0.53 Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

13 Results Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14

14 Conclusion non-convexity: 0 or 1: the elasticity of substitution between leisure in di erent periods for the aggregate economy is in nite independent of the elasticity of substitution implied by the individuals utility function large uctuations in hours worked relative to uctuations in productivity, even when individuals are relatively unwilling to substitute leisure across time Zhe Li (SUFE) Advanced Macroeconomics III Fall / 14