Factor Prices, the Labor-Leisure Tradeoff, and
General Equilibrium
Brian C. Jenkins
University of California, Irvine
May 1, 2017
Brian C. Jenkins University of California, Irvine
Factor Prices & Labor-Leisure
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Figure 1: GDP, total factor productivity, the capital stock, and
total labor hours for the US from 1948 to 2015. Annual. Source: FRED.
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Model Overview
Static general equilibrium model
Components:
1
2
Representative firm
Representative household
No government consumption, taxes, or international trade
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Model Overview
The household:
Owns an exogenously determined amount of physical capital
Has 1 unit of time to allocate between leisure and labor
Earns income by selling labor and capital services to the firm
Receives utility from consumption and leisure
The firm:
Rents capital from the household and hires labor from the
household
Produces output and sells it to the household
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Figure 2: The demand for capital and labor. The quantity of capital
demanded by firms increases as the real rental rate falls (left panel) and
the quantity of labor demanded by firms increases as the real wage falls
(right panel).
Capital Demand
Labor Demand
W
P
R
P
MPL
MPK
L
K
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Figure 3: An increase in the quantity of capital used by the firm from K̄0
to K̄1 while holding labor constant at L̄. The real rental rate falls (left
panel) while the real wage rises (right panel).
Capital Market
R
P
K0S
Labor Market
W
P
K1S
LS
W
P 1
R
P 0
MPL0
W
P 0
R
P 1
MPL
MPK
K̄0
K̄1
Brian C. Jenkins University of California, Irvine
K
L
L̄
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Figure 4: An increase in the quantity of labor used by the firm from L̄0
to L̄1 while holding capital constant at K̄ . The real wage falls (right
panel) while the real rental rate rises (left panel).
Capital Market
R
P
K
Labor Market
W
P
S
R
P 1
LS0
LS1
W
P 0
MPK 0
R
P 0
W
P 1
MPK
K̄
Brian C. Jenkins University of California, Irvine
K
MPL
L̄0
L
L̄1
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Figure 5: An increase in total factor productivity while holding capital
and labor constant at K̄ and L̄. The rental rate rises (left panel) and the
real wage rises (right panel).
Capital Market
R
P
K
Labor Market
W
P
S
LS
W
P 1
R
P 1
MPL0
MPK 0
W
P 0
R
P 0
MPL
MPK
K̄0
Brian C. Jenkins University of California, Irvine
K
L
L̄
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Figure 6: Typical indifference curves. Household utility increases with
both consumption C and leisure `.
Indifference curves
C
+
u3
u2
u1
u0
`
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Factor Prices & Labor-Leisure
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Figure 7: The supply of labor. The quantity of labor supplied increases
as the real wage rises. The supply of labor is bounded below by 0 and
above by 1.
Labor Supply
W
P
LS
1
Brian C. Jenkins University of California, Irvine
L
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Figure 8: The supply of labor following an increase in either the real
rental rate R/P or capital K̄ . An increase in capital income reduces the
household’s willingness to supply labor and shifts the labor supply curve
up.
Labor Supply
W
P
LS1
LS0
1
Brian C. Jenkins University of California, Irvine
L
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Equilibrium
Y = AK̄ α L1−α
R
= αAK̄ α−1 L1−α
P
W
= (1 − α)AK̄ α L−α
P
1
R/P
L=
1−ϕ
K̄
1+ϕ
W /P
Y =C
Brian C. Jenkins University of California, Irvine
Factor Prices & Labor-Leisure
(1)
(2)
(3)
(4)
(5)
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Figure 9: Equilibrium in the factor markets.
Capital Market
R
P
Labor Market
W
P
KS
LS
W
P 0
R
P 0
MPL
MPK
K̄
Brian C. Jenkins University of California, Irvine
K
L
L̄
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Figure 10: Equilibrium in the factor markets following an increase in
capital K̄ . The real rental rate falls (left panel). The demand for labor
increases and the supply of labor decreases by an offsetting amount (right
panel). Equilibrium labor is unchanged and the real wage rises.
Capital Market
R
P
K0S
Labor Market
W
P
K1S
LS1
W
P 1
R
P 0
LS0
R
P 1
W
P 0
MPL1
MPK
K̄0
K̄1
Brian C. Jenkins University of California, Irvine
K
MPL0
L
L̄
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Figure 11: Equilibrium in the factor markets following an increase in
total factor productivity A. The demand for capital increases and the real
rental rate rises (left panel). The demand for labor increases and the
supply of labor decreases by an offsetting amount (right panel).
Equilibrium labor is unchanged and the real wage rises.
Capital Market
R
P
Labor Market
W
P
KS
LS1
W
P 1
R
P 1
MPK1
LS0
MPL1
W
R
P 0
P
MPK0
K̄
Brian C. Jenkins University of California, Irvine
K
0
MPL0
L
L̄
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Figure 12: Equilibrium in the factor markets following an increase in the
household’s relative preference for leisure ϕ. The supply of labor falls
reducing the equilibrium quantity of labor and raising the real wage (right
panel). The marginal product of capital falls, reducing the real rental rate
(left panel).
Capital Market
R
P
Labor Market
W
P
KS
LS1
W
P 1
R
P 0
LS0
W
P 0
MPK0
R
P 1
MPL
MPK1
K̄
Brian C. Jenkins University of California, Irvine
K
L̄1
L
L̄0
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