FIN2005 Problem Set 3 - Solution
JOONKYO HONG
November 22, 2024
1
How Does the Cost Function Look When Production
Technology is Leontief?
(1) The optimal input allocation should satisfy
2l ∗ = 3k∗ = y.
(2) Since k is fixed at 10 in the short run, the firm either produces 30 = 3 × 10 by
employing l ∗ = 15 or 0 by choosing l ∗ = 0. If the firm decides to produce, it must pay
2 × 15 + 6 × 10 = 90. The decision not to produce surely incurs no costs. Thus, the
short-run cost function is
cs ( y) =

0
if the firm doesn’t produce.
90
if the firm produces.
(3) The short-run cost is fixed at 90, conditional upon the decision to produce. Hence,
the fraction of the short-run cost accruing to variable parts is zero.
(4) In the long run, the firm can flexibly choose the optimal input allocation (l ∗ , k∗ )
given the production target y.
2l ∗ = 3k∗ = y,
implying that the conditional factor demand functions for labor and capital are given
1
by
1
y,
2
1
k∗ = y,
3
l∗ =
implying that the long-run cost function is c( y) = 2 × 12 y + 6 × 13 y = y + 2 y = 3 y. Thus,
LAC( y) = LM C( y) = 3.
(5) We have already discussed the Returns to Scale of a Leontief production function
in the class: Constant Returns to Scale. Thus, we can guess LAC( y) is flat without any
calculation.
2
Perfectly Competitive Market
(1) The short-run supply curve of each individual firm is the inverse marginal cost function.
P = M C( y ∗ ) = 40 + 12 y ⇒ y ∗ (p) =
p − 40
12
(2) We can get the market supply curve by adding up all the individual firms’ supply
curves.
Y = S(p) = 6 · y ∗ (p) = 6 ·
(3) As the inverse demand function is Y =
p − 40
p − 40
=
12
2
400−p
2 , the competitive equilibrium price and
quantity are obtained by equating the demand to the supply:
400 − p c
p c − 40
=
⇒ (p c , Y c ) = (220, 90)
2
2
2
(4) We will get the followings from a graphical illustration of the market equilibrium:
1
· 180 · 90 = 8100
2
1
PS = · 180 · 90 = 8100
2
SW = C S + PS = 16200
CS =
(5)
Y c = 90 ⇒ y c =
90
= 15
6
(6) If Mercedes-Benz enters the market, the new market supply becomes
Snew (p) = 7 · y ∗ (p) =
7(p − 40)
12
Therefore, the new competitive equilibrium is
c
400 − pnew
2
=
c
7(pnew
− 40)
12
c
c
⇒ (pnew
, Ynew
)=

2680 1260
,
13
13
‹
1
180
Each firm sells 1260
13 · 7 = 3 units. The profit for Mercedes-Benz is

‹
180 2
2680 180
180
·
− 40 ·
−6
= 1150.2959 < 1200
13
13
13
13
This shows the firm will not choose to enter the market.
3
Profit-Maximizing Output and Demand Elasticity
Proof. The optimal level of output y m is determined by the following optimality condition:
M R( y m ) = M C( y m ).
(1)
m
Since the cost function is non-decreasing, we
€ observeŠ that M C( y ) ≥ 0, implying that
M R( y m ) ≥ 0. Given that M R( y m ) = p( y m ) 1 + ϵ( y1m ) , the following holds at the opti3
mal:

p( y ) 1 +
m
‹
1
≥ 0.
ϵ( y m )
(2)
In other words,

1+
‹
1
≥ 0,
ϵ( y m )
(3)
which implies that ϵ( y m ) ≤ −1.
In addition, since we have chosen p( y) arbitrary, ϵ( y m ) ≤ −1 for any optimal output
y m.
4
Monopoly
(1) The cost function of Tesla is an anti-derivative of the marginal cost function, so we
will get
y = S(p) =
⇒ c( y) =
p − 40
⇒ M C( y) = S −1 ( y) = 2 y + 40
2
Z
M C( y)d y = y 2 + 40 y
(2)
π( y) = p( y) · y − c( y) = (400 − 2 y) y − ( y 2 + 40 y) = 360 y − 3 y 2
(3) By the first-order condition, we have
π′ ( y m ) = 360 − 6 y m = 0 ⇒ y m = 60, p m = 400 − 2 y m = 280
⇒ (p m , y m ) = (280, 60)
(4) At the equilibrium,
M C( y m ) = 2 · 60 + 40 = 160,
p m = 280 ⇒ markup =
4
pm
7
=
m
M C( y ) 4
(5) By drawing the graph, we can get
1
· 120 · 60 = 3600
2
1
PS = · ((280 − 160) + (280 − 40)) · 60 = 10800
2
1
DW L = · (280 − 160) · 30 = 1800
2
CS =
(6) The payment to the other 5 firms is the profit they earned in a perfectly competitive
market. By our calculation in Problem 3, p c = 220, y = 15, and each firm earns
π( y) = p c · y − (40 y + 6 y 2 ) = 1350
Moreover, by (5), the profit for Tesla is PS = 10800. Thus,
10800 − 1350 · 5 = 4050 > 0,
implying that we would advise Tesla to proceed with this M&A plan.
5
How Does Labor Market Condition Influences Market
Power?
(1) As the long-run cost minimizing input choice (l ∗ , k∗ ) should obey
y = l ∗ = k∗ ,
we obtain
c(w, r, y) = wl ∗ + r k∗ = [4 + 6] y = 10 y.
5
(2) The profit function is given by
π( y) = (100 − y) × y − 10 y
= 90 y − y 2 .
(3) The first-order condition implies
dπ( y)
= 90 − 2 y m = 0
dy
⇒ y m = 45.
(4) The equilibrium market price is p m = 100 − 45 = 55. Hence, the markup charged
by the monopolist is 55
10 = 5.5.
(5) The long-run cost function now becomes c(w, r, y) = [6 + 6] y = 12 y. Thus, when
we repeat the same procedures from (2) to (3), we get
y m = 44
6
and
p m = 56.
Sometimes, Price Discrimination is Welfare-Improving
(1)
π( yk , y t ) = (400 − 2 yk ) yk + (200 − 6 y t ) y t − 60( yk + y t )
(2)
∂
π( yk , y t ) = 400 − 4 yk − 60 = 0 ⇒ yk = 85
∂ yk
∂
35
π( yk , y t ) = 200 − 12 y t − 60 = 0 ⇒ y t =
∂ yt
3
35
⇒ ( yk∗ , y t∗ ) = (85, )
3
6
(3) Inserting the results in (2) into the corresponding demand curves, we obtain
(pk∗ , p∗t ) = (230, 130)
(4) Drawing graphs for the two markets respectively, one can get
CS =
+
85 · (230 − 60)
|
{z
}
35
· (130 − 60)
3
|
{z
}
CS of EV drivers on highways to Kaohsiung
PS =
1 35
·
· (200 − 130)
2
3
|
{z
}
1
· 85 · (400 − 230)
2
|
{z
}
+
profits from highways to Kaohsiung
= 7633.3333
CS of EV drivers on highways to Taichung
= 15266.6667
profits from highways to Taichung
SW = C S + PS = 22900
(5)
y(p) =

 y (p) + y (p) if 0 ≤ p ≤ 200
k
t
 yk (p)

 700−2p
if 0 ≤ p ≤ 200
 400−p
2
if 200 < p ≤ 400

400 − 2 y
if 0 ≤ y ≤ 100
350 − 3 y
2
if 100 < y ≤ 700
3
⇒ y(p) =
⇒ p( y) =
if 200 < p ≤ 400
3
(6) Notice that the optimal point will occur in 0 ≤ y ≤ 100. (One can check this by
maximizing on the other interval, and then find a contradiction). Hence, Tesla would
not serve EV drivers on highways to Taichung. By the result in (2) and (3), we have
(p m , y m ) = (230, 85).
7
(7) We can now narrowly focus on the welfare of the Kaohsiung market under monopoly:
1
· 85 · (400 − 230) = 7225
2
PS = 85 · (230 − 60) = 14450
CS =
SW = C S + PS = 21675
(8) The consumer surplus calculated in (7) is less than the one in (4), so the citizens
are worse off. Take note of that the decreased amount of CS is exactly the same as the
CS of EV drivers on highways to Taichung.
8