Classical Model: Practice Problems Key
Intermediate Macroeconomics
John T. Dalton
Question 1
a)
∂U (Y D ,LS )
∂Y D
=
b)
∂U (Y D ,LS )
∂LS
1
= − 1−L
S
c)
∂Y S
∂LD
1
YD
=4
d) A competitive equilibrium in the classical model is prices (p∗ , w ∗) and
allocations (LD∗ , LS∗ , Y D∗ , Y S∗ ) such that the following conditions hold:
1) Given prices (p∗ , w ∗ ), the representative household solves
max U(Y D , LS )
Y D ,LS
s.t.
p∗ Y D = w ∗ LS
2) Given prices (p∗ , w ∗ ), the representative firm solves
max p∗ Y S − w ∗ LD
s.t.
Y S ,LD
Y S = f (LD )
3) Markets clear
Y S∗ = Y D∗
LS∗ = LD∗
e) Labor demand is derived from the firm’s problem.
max pY S − wLD
s.t.
Y S ,LD
1
Y S = 4LD
⇒ max 4pLD − wLD
LD
L = 4pLD − wLD
∂L
= 4p − w = 0
∂LD
w
=4
p
This is an elastic labor demand curve, so
w∗
p∗
= 4.
Labor supply is derived from the household’s problem.
max ln(Y D ) + ln(1 − LS )
Y D ,LS
s.t.
pY D = wLS
wLS ⇒ max ln
+ ln(1 − LS )
p
LS
wLS + ln(1 − LS )
L = ln
p
p w
∂L
1
=
−
=0
S
S
∂L
wL p
1 − LS
Solving for LS ,
1
2
This is an inelastic labor supply curve, so LS∗ = 12 . Using the market
LS =
clearing condition for the labor market, LD∗ = LS∗ = 21 . Using the firm’s
production function, Y S∗ = 4(LD∗ ) = 4( 12 ) = 2. Finally, using the market
clearing condition for the goods market, Y D∗ = Y S∗ = 2.
2
The competitive equilibrium for this example is the price
allocations LS∗ =
1
, LD∗
2
=
1
, Y S∗
2
w∗
p∗
= 4 and
= 2, and Y D∗ = 2.
f) π ∗ = p∗ Y S∗ − w ∗ LD∗ = (1)(2) − (4)( 12 ) = 0
g) See Figure 1 below.
h) p =
MS
kY D
=
2
YD
i) Using the aggregate supply curve, Y S∗ = Y D∗ = 2, which means p∗ =
1.
Question 2
a)
∂U (Y D ,LS )
∂Y D
=1
b)
∂U (Y D ,LS )
∂LS
= −LS
c)
∂Y S
∂LD
= 5 − LD
d) pY D = wLS (1 − τ )
e)
1
max Y D − (LS )2
D
S
2
Y ,L
⇒ max
LS
L=
s.t.
pY D = wLS (1 − τ )
w S
1
L (1 − τ ) − (LS )2
p
2
1
w S
L (1 − τ ) − (LS )2
p
2
∂L
w
= (1 − τ ) − LS = 0
S
∂L
p
3
2
2
=
LS =
w
(1 − τ )
p
f)
max pY S − wLD
Y
S ,LD
s.t.
1
Y S = 5LD − (LD )2
2
1
⇒ max p 5LD − (LD )2 − wLD
2
LD
1
L = p 5LD − (LD )2 − wLD
2
∂L
= p5 − pLD − w = 0
∂LD
LD = 5 −
w
p
g) Using the market clearing condition in the labor market,
LS∗ = LD∗
w∗
w∗
(1
−
τ
)
=
5
−
p∗
p∗
w∗
5
=
p∗
2−τ
h) Using the labor supply curve, L∗ = LS∗ =
w∗
(1
p∗
1−τ
− τ ) = 5 2−τ
OR
Using the labor demand curve, L∗ = LD∗ = 5 −
i) Real Tax Revenue =
w∗ ∗
Lτ
p∗
4
w∗
p∗
2−τ
= 5 2−τ
−
5
2−τ
1−τ
= 5 2−τ
j) Plugging in the equilibrium values for L∗ and
Revenue =
5
5 1−τ τ
2−τ 2−τ
=
w∗
p∗
derived above, Real Tax
(1−τ )
25 (2−τ
τ.
)2
k) “Supply-siders” argued the government should cut taxes to increase revenue. They believed U.S. tax rates were such that we were on the “wrong
side” of the Laffer Curve, that is, to the right of the tax rate which generates
the maximum amount of revenue.
Question 3
a) The balanced government budget shows total tax revenue equals total
expenditure: τ pY S = T .
b)
1
max Y D − (LS )2
2
Y D ,LS
⇒ max
LS
L=
s.t.
pY D = wLS + T
1
w S T
L + − (LS )2
p
p
2
w S T
1
L + − (LS )2
p
p
2
∂L
w
=
− LS = 0
∂LS
p
LS =
w
p
c)
max (1 − τ )pY S − wLD
Y S ,LD
s.t.
Y S = 3LD
⇒ max(1 − τ )p3LD − wLD
LD
5
L = (1 − τ )p3LD − wLD
∂L
= (1 − τ )p3 − w = 0
∂LD
w
= 3(1 − τ )
p
d) Using the labor demand curve,
w∗
p∗
= 3(1 − τ ). Plugging the equilibrium
real wage into the labor supply curve gives LS∗ = 3(1 − τ ). Market clearing
in the labor market implies LD∗ = LS∗ = 3(1 − τ ). Plugging the equilibrium
quantity of labor demanded into the firm’s production function gives Y S∗ =
9(1 − τ ). Market clearing in the goods market then implies Y D∗ = Y S∗ =
9(1 − τ ). Notice, the goods market equilibrium can also be solved through
the household’s budget constraint:
pY D = wLS + T,
which we know also holds with equality in equilibrium, so we have the rewritten budget constraint
Y D∗ =
w ∗ S∗ T
L + ∗.
p∗
p
Using the government budget constraint and plugging in the equilibrium
values for the real wage and labor supplied gives the following:
Y D∗ = 3(1 − τ )3(1 − τ ) + τ Y S∗ ,
which simplifies to
Y D∗ = 9(1 − τ ).
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e) See Figure 2 below. Household consumption in equilibrium, Y D∗ , increases
after the government eliminates the tax on total revenue. Eliminating the
tax shifts the labor demand curve of the firm outward, which increases the
equilibrium quantity of labor demanded, LD∗ . Using the firm’s production
function, output supplied in equilibrium, Y S∗ , also increases. Through the
market clearing condition in the goods market, Y D∗ must also increase.
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Figure 1: Aggregate Supply Curve
P
AS
S*
Y =2
8
Y
Figure 2: Eliminating the Tax on Firm Revenue
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