New Keynesian Unemployment: Practice Problem Key
Intermediate Macroeconomics
John T. Dalton
Question 1
a) The firm chooses its efficiency wage by minimizing total costs.
w D
min
L +c
w
p
p
w
LD
p
w
⇒ L = LD − ln
p
w
LD
p
∂L
LD
D
=
L
−
w = 0
∂ wp
p
LD =
LD
w
p
w
=1
⇒
p eff
Given the efficiency wage, the firm then maximizes its profit, which determines the labor demand function.
w
w
D
max Y −
L −c
LD
S
D
p eff
p eff
Y ,L
S
1
⇒ L = 2(LD ) 2 − LD −
s.t.
ln(1)LD
| {z }
=0 because ln(1)=0
1
∂L
= (LD )− 2 − 1 = 0
D
∂L
1
1
Y S = 2(LD ) 2
LD = 1
b) Given the
labor supply function and the efficiency wage from part
= 2(1) = 2. Therefore, this economy’s natural rate of
a), LS = 2 wp
eff
unemployment = LS − LD = 2 − 1 = 1.
Question 2
a) From our discussion of the firm’s cost minimization problem in lecture,
we know the marginal benefit of increasing the real wage is equal to the
negative of the derivative of the per worker turnover cost function w.r.t. the
real wage, i.e. marginal benefit = −c0 ( wp ). In this case, c0 ( wp ) = −4( wp )−2 , so
the marginal benefit = 4( wp )−2 .
b) The firm chooses its efficiency wage by minimizing total costs.
w D
L +c
min
w
p
p
w
LD
p
w
⇒ L = LD + 4
p
∂L
D
w = L −4
∂p
−1
w
LD
p
−2
w
LD = 0
p
−2
w
4
=1
p
w
⇒
=2
p eff
2
c) To find the market-clearing real wage, which is the same as ( wp )∗ from the
classical model, set the labor demand curve and labor supply curve equal to
one another and solve for the real wage:
−1
w
w
2
=
p
p
2
w
2=
p
∗
w
= 1.41,
⇒
p
which, as we would expect, is lower than the efficiency wage.
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