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Feb 20, 2024
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1 FIN 501: Financial Economics Professor Deryugina Problem Set 5 Solutions TA Sarah Kim Question 1 a.
Suppose that coal is produced in a competitive world market. The world price of coal is a constant $40
per ton (similar to our trade example). Assume that US firms are price takers
in this world market. The marginal private cost of coal production
in the US, aggregated across all firms, is 𝑀𝐶
=0.8
?
, where ?
is tons of coals per day. Fixed costs are zero. How many tons of coal will US firms produce per day? To maximize profits, firms will choose to produce the quantity such that MC=MR. In a competitive market, price is constant, so total revenue TR=P*Q and marginal revenue MR=P. 𝑀𝐶 = ? → 0.8? = 40 ?
∗
=
40
0.8
= 50
b.
Maintain the same general set up as before (competitive world market, price-taking firms). The world price of coal is a constant $40
per ton. The marginal private cost of coal production
in the US is 𝑀𝐶
=0.8
?
. Now let’s take into account the fact that coal produces sulfur dioxide when burned. The US government has estimated that the external social cost of coal
to the US is ??𝐶
=1.4
?
2
. This does not
include the marginal private cost of production. What is the socially optimal level of coal production by US firms, assuming all the coal that is produced in the US gets burned in the US? ????? ???𝑔𝑖??? ???𝑖?? ???? (𝑀?𝐶)
= ???𝑔𝑖??? ??𝑖𝑣??? ???? (𝑀?𝐶) + ???𝑔𝑖??? ?𝑥?????? ???𝑖?? ???? (𝑀𝐸?𝐶)
MPC=0.8Q and 𝑀𝐸?𝐶 =
𝜕??𝐶
𝜕?
= 2.8?
, so MSC=3.6Q. Thus, the socially optimal level of coal production satisfies the condition MSC=P 3.6? = 40 → ?
?𝑜𝑐
∗
= 11.11
c.
Maintain the same general set up as before (competitive world market, price-taking firms). The world price of coal is a constant $40
per ton. The marginal private cost of coal production
in the US is 𝑀𝐶
=0.8
?
. The external social cost of coal
to the US is ??𝐶
=1.4
?
2
. This does not
include the marginal private cost of production. The US government wants to impose a tax on each ton of coal that the US produces. What size tax will induce US firms to produce the socially optimal level of output? As found in (b), ?
?𝑜𝑐
∗
= 11.11
, so MPC will be 0.8*11.11=8.888 at the socially optimal Q. This MPC is much lower than the world price, $40. If taxes are collected, firms will internalize the tax as additional costs, and they will still be incentivized to produce Q=11.11. The optimal tax amount satisfies the condition, 8.888+tax=40. Therefore, tax=40-8.888
≈
$31.11
2 d.
Maintain the same general set up as before (competitive world market, price-taking firms). The world price of coal is a constant $40
per ton. The marginal private cost of coal production
in the US is 𝑀𝐶
=0.8
?
. The external social cost of coal
to the US is ??𝐶
=1.4
?
2
. This does not
include the marginal private cost of production. If the government decides to solve the externality problem with tradable coal-mining permits, how many daily permits should it issue (it’s ok to have fractional permits)?
The government would want the quantity of production to be at the socially optimal level. As determined in (b), this level is ?
?𝑜𝑐
∗
= 11.11.
This corresponds to the exact number of daily permits the government would issue.
3 Question 2 a.
Apple is a monopolist selling iPhones. The demand for iPhones in the United States is ?
?
= 300−5
?
. Apple’s cost function is 𝐶
(
?
)=3.5
?
2
. How much should Apple charge per iPhone to maximize profits? Inverse demand function is ? = 60 −
1
5
?
, so 𝑀? =
𝜕{?(?)∗?}
𝜕?
= 60 −
2
5
?
. Additionally, MC=7Q. Setting MC=MR, 7? = 60 −
2
5
?
37
5
? = 60 ?
𝑀
∗
= 8.1081 Plugging ?
𝑀
∗
into the inverse demand function yields P=58.38 b.
Apple is a monopolist selling iPhones. The demand for iPhones in the United States is ?
?
= 5
00−
10
?
. Apple’s cost function is 𝐶
(
?
)=3
?
2
. What is the deadweight loss from the monopoly? Under perfect competition, assuming that C(Q) represents total cost in the US market, MC will be the equivalent to a market inverse supply. Hence, the optimal market level of production will be at the point of intersection between the inverse demand and supply curves: 50 −
1
10
? = 6?
Therefore, 61
10
? = 50
or ?
𝑐
∗
= 8.1967
. If Apple is a monopoly, then it produces where MC=MR, where MR = 𝜕{?(?)∗?}
𝜕?
= 50 −
2
10
? = 50 −
1
5
?
. 𝑀𝐶 = 𝑀? → 50 −
1
5
? = 6?
Hence, ?
𝑀
∗
= 8.065
and ?
𝑀
∗
= 50 −
8.065
10
= 49.1935
. Also at this ?
𝑀
∗
, 𝑀𝐶 = 6(8.065) = 48.39
Therefore, the area of the triangle representing DWL is 1
2
∗ ? ∗ ℎ =
1
2
(49.1935 − 48.39)(8.1967 − 8.065) = 0.05
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