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Employment and Capital Utilization Mr. Vaughan Income and Employment Theory (402) Updated: 4/6/09 1 Lecture Outline • Allowing Factor Inputs to Vary in Equilibrium Model – Labor Supply – Capital Services Supply (Capacity Utilization) • Empirical Predictions vs. Stylized Facts Total Slides: 22 2 Variations in Labor Input Labor Supply: Households have a fixed amount of time to allocate to labor and leisure. – More labor supplied means less leisure time. Household preferences about leisure/labor are stable. – More leisure is always better (as well as more consumption goods). – Leisure is a normal good. Real wage (w/P) is “price” of leisure. – A rise the (w/P) generates substitution and income effects. Total Slides: 22 3 Variation in Labor Input Substitution Effect: If real wage rate (w/P) rises, household can buy more consumption goods for each extra hour worked. – Put another way, opportunity cost of leisure is higher, so households demand smaller quantity. Household respond to higher real wage (w/P↑) by working more, i.e., increasing quantity of labor supplied (Ls↑). Income Effect: Higher w/P means higher real wage income for any given labor supplied, (w/P)·Ls. Household spends extra income on consumption and leisure. So higher (w/P↑) leads to smaller quantity of labor supplied (Ls↓) In Sum: Substitution effect of increase in w/P increases quantity of labor supplied and income effect reduces quantity of labor supplied; total effect is ambiguous (Ls?). Total Slides: 22 4 Variations in Labor Input Total Effect of (w/P) Change on Labor Supply: Resolve ambiguity by assessing strength of income effect (i.e., is shock temporary or permanent?) Recall, multi-year budget constraint: Uses of funds: C1 + C2/(1+i1) + C3/[( 1+i1)·(1+i2) ] + · · · Sources of funds: (1+i0)·(B0/P+K0) + (w/P)1·Ls1+(w/P)2·Ls2/(1+i1) + (w/P)3·Ls3/[(1+i1)·(1+i2)] + · · · Consider positive technology shock that increases demand for labor (MPL↑) and real wage. Permanent shock [w/P↑ permanently] produces large income effect. – Large income effect (work less) could swamp substitution effect (work more), so total effect of real wage hike on quantity of labor supplied remains ambiguous. Less-than-permanent shock [w/P↑ long-lasting] produces smaller income effect. – Smaller income effect (work less) allows substitution effect (work more) to dominate, so total effect of real wage hike is rise in quantity of labor supplied (i.e., positively sloped laborsupply curve, Ls↑). Total Slides: 22 5 Variations in Labor Input Model Predictions A long-lasting technology shock raises demand for labor (MPL), thereby raising the real wage. Given positively sloped labor-demand curve, labor input rises as well. Note: Technology shock shifts labor-demand curve (“A” exogenous) but induces movement along labor supply curve (affects Ls through real wage). Total Slides: 22 6 Matching Theory with Facts Evidence on Fluctuations in Labor Input: Measures of labor input are pro-cyclical (i.e., they move in same direction as real GDP during booms and recessions). Employment Total hours worked Total Slides: 22 7 Matching Theory with Facts Labor Input Total Slides: 22 8 Matching Theory with Facts Labor Input Total Slides: 22 9 One last empirical test on labor… Can model explain cyclical behavior of labor productivity? – Prediction: Positive technology shock raises average as well as marginal product of labor. Common measures of (average) labor productivity – Real GDP per worker – Real GDP per worker-hour Both measures of labor productivity are procyclical. Total Slides: 22 10 Capital Input Capital Utilization Rate: Fraction of capital stock used in production. κ (Greek letter kappa) represents utilization rate of capital stock, K. – Production function now given by: Y= A·F(κ K, L) κK, rises with utilization rate, κ. – Therefore, increase in “κ” raises real GDP (Y) for given technology level, A, capital stock, K, and labor input, L. Equilibrium capital input (κK) determined by intersection of demand for/supply of capital services. Total Slides: 22 11 Cyclical Behavior of Capacity Utilization Total Slides: 22 12 Capital Input Demand for Capital Services: As before, derived from profit maximization w.r.t capital services – now equal to (κK). Firms maximize real profit: π/P = A·F[(κK)d, Ld)] - (w/P)·Ld - (R/P)·(κK)d ∂(π/P)/∂(κK)d = A·F(κK)d - R/P = 0 A·F(κK)d = R/P In words: Use additional capital services up to point where marginal revenue earned by selling additional output (MPK = A·F(κK)d ) equals marginal cost of hiring capital services to produce that output (R/P). Total Slides: 22 13 Capital Input Total Slides: 22 14 Capital Input Demand for Capital Services Technology Shocks and Demand for Capital Services Assume technology level rises from A to A’. MPK rises for every quantity of capital services. Demand for capital services shift to rightward (increases). Total Slides: 22 15 Capital Input Supply of Capital Services: For given stock of capital, K, owners can supply more or less capital services per year by varying κ. One reason to set utilization rate, κ, below its maximum: Increases in κ raise depreciation rate, δ. – Hence, depreciation rate is function of capacity utilization, δ = δ(κ) Total Slides: 22 16 Capital Input Supply of Capital Services: (Like demand for capital services, determined by optimization) Net real income from supplying capital services: Real Rental Payments − Depreciation = (R/P)·κK − δ(κ)·K Factoring “K” out yields: K·[(R/P)·κ − δ(κ)] (where bracketed term is real rate of return from owning capital). Owners of capital (households) select utilization rate, κ, that maximizes net real income: (R/P)·κ - δ(κ) Total Slides: 22 17 Capital Input Supply of Capital Services Assumptions 1. Idle machines depreciate (i.e., δ(κ)>0, when κ = 0). 2. Depreciation rate increases with capacity utilization (κ) at increasing rate. Note: Vertical distance between curves equals real income from supplying capital. Real income is maximized when vertical distance is at a max (i.e., marginal revenue from working machines more intensely (R/P) equals marginal depreciation cost [δ’(κ)]. Total Slides: 22 18 Capital Input Supply of Capital Services Analysis Increase in real rental price raises capital utilization rate. Higher real rental price makes it worthwhile to raise κ, despite resulting increase in depreciation rate, δ(κ). Supply curve of capital services slopes upward because increase in real rental price, R/P, motivates higher capital utilization rate, κ. Total Slides: 22 Note: Technology level, A, does not affect choice of the capital utilization rate, κ 19 Capital Input Putting It All Together Market Clearing and Capital Utilization When technology level rises demand curve for capital services shifts rightward (“A” exogenous). Supply curve does not shift (“A” affects κ* by raising R/P). Increase in demand for capital services produces increase in quantity of services supplied. Market clears at higher real rental price, [(R/P)∗], and larger quantity of capital services, [(κK)∗]. Model Predictions: R/P, κK, and i are pro-cyclical. WHY? – Rate of return on bonds must equal rate of return on capital, i = (R/ P)·κ − δ(κ) – Increase in technology, A, raises rate of return on capital, so interest rate, i, must increase. Total Slides: 22 20 Matching Theory with Facts Total Slides: 22 21 Questions over Employment and Capital Utilization? Mr. Vaughan Income and Employment Theory (402) Total Slides: 22 22