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Employment and Capital Utilization
Mr. Vaughan
Income and Employment Theory (402)
Updated: 4/6/09
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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
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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
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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
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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
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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
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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
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Matching Theory with Facts
Labor Input
Total Slides: 22
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Matching Theory with Facts
Labor Input
Total Slides: 22
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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
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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
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Cyclical Behavior of Capacity Utilization
Total Slides: 22
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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
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Capital Input
Total Slides: 22
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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
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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
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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
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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
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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
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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
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Matching Theory with Facts
Total Slides: 22
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Questions
over
Employment and Capital Utilization?
Mr. Vaughan
Income and Employment Theory (402)
Total Slides: 22
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