It's a homework for master degree

1

Winter Term 20/21

Chair for European Economic Policy International Economic Policy Prof. Dr. Carsten Hefeker Konstantinos Theocharopoulos, M.Sc.

Problem Set 1

1. Consider a small economy with 𝐿𝐿 households, indexed by 𝑖𝑖. The utility function of the household 𝑖𝑖 is

𝜔𝜔𝑖𝑖 = 𝑎𝑎0 + ln(𝑎𝑎) (1)

where good 0 is a numeraire and 𝑎𝑎 is the consumption of the traded labor intensive

good. Each household owns one unit of labor and capital stock 𝛫𝛫𝑖𝑖 ≥ 0. The

consumption of the traded good is

𝑎𝑎 = 𝜑𝜑(𝜋𝜋) = 1 𝜋𝜋

(2)

with 𝜑𝜑 denoting the demand for good 𝑎𝑎 and 𝜋𝜋 the relative domestic price of it. The

household’s remaining income is spent on the numeraire.

a. Show that the utility function of household 𝑖𝑖 is quasi-linear.

The domestic supply of the traded good is 𝑦𝑦(𝜋𝜋) with 𝑦𝑦′(𝜋𝜋) > 0 and the relationship

between its relative domestic price and the exogenous world market price 𝜋𝜋∗ is given

by 𝜋𝜋 = 𝜋𝜋∗ + 𝑡𝑡 with 𝑡𝑡 the tariff rate.

b. Find an expression for the total imports 𝑚𝑚 and the tariff revenue 𝜏𝜏. What is the impact of a change in the traded good’s domestic price on total imports?

The tariff revenue is distributed lump sum to households while the total GDP 𝑦𝑦𝑜𝑜(𝜋𝜋) +

𝜋𝜋𝑦𝑦(𝜋𝜋) = 𝑤𝑤𝐿𝐿 + 𝑟𝑟𝑟𝑟 is distributed to factor owners via wages 𝑤𝑤 and interest payments

𝑟𝑟.

2

c. The household income is 𝜀𝜀. Specify the household’s indirect utility function 𝜈𝜈𝑖𝑖(𝜋𝜋, 𝜀𝜀).

Given that 𝜁𝜁𝑖𝑖 = 𝐾𝐾 𝑖𝑖

𝐾𝐾 𝐿𝐿

is the individual capital endowment, normalized by the average

capital to labor ratio in the economy, compute 𝜀𝜀.

d. Use the Envelope-Theorem (𝑦𝑦0′ (𝜋𝜋) + 𝜋𝜋𝑦𝑦′(𝜋𝜋) = 0) to calculate the median voter’s preferred tariff rate 𝑡𝑡𝜇𝜇. Does the fact that the households differ in their factor

endowments matter? How does the type of the import competing good (labor/capital

intensive) influence the median’s preferred tariff rate? Explain.

Assume now that the median voter has less than the average capital stock. Thus, 𝜁𝜁𝜇𝜇 = 𝐾𝐾𝜇𝜇 𝐾𝐾 𝐿𝐿

< 1.

e. Find a way to measure inequality in the given framework. Consider two countries, one industrialized 𝐴𝐴 and one developing 𝐵𝐵. Country 𝐴𝐴 and 𝐵𝐵 import a labor and a

capital intensive good, respectively. Given that in both countries inequality increases,

what does this model predict for the respective trade policies?

2. Consider a political system consisting of two parties. One of them is predisposed to a liberal (𝜏𝜏0) and the other to a protectionist (𝜏𝜏1) trade policy with 𝜏𝜏0 < 𝜏𝜏1. The parties

are ideologically committed to their positions. There is an import competing firm

producing a good that is imperfectly substitutable in domestic consumption of

imports. The domestic firm makes a campaign contribution 𝐿𝐿 to influence the outcome

of the electoral contest between the above-mentioned parties. There is also a foreign

firm. The foreign firm seeks to influence the domestic trade policy as well with a

contribution 𝐿𝐿∗ and has the same per unit cost of production 𝛿𝛿 as the domestic firm.

The inverse demand functions are

𝑃𝑃 = 𝛼𝛼2 − 1 𝛽𝛽 𝑥𝑥 + 𝛾𝛾𝑃𝑃∗ (3)

and

𝑃𝑃∗ = 𝛼𝛼2 − 1 𝛽𝛽 𝑥𝑥∗ + 𝛾𝛾𝑃𝑃 (4)

3

where 𝑥𝑥 and 𝑥𝑥∗ denote quantities of domestically produced and foreign goods, 𝑃𝑃 and

𝑃𝑃∗ are the respective prices of the goods and 𝑎𝑎, 𝛽𝛽, 𝛾𝛾 > 0.

a. Given that τ is the tariff rate, specify the profit functions 𝛺𝛺 and 𝛺𝛺∗ for domestic and foreign firm, respectively. Find the optimal output for both firms.

Assume that 𝜕𝜕𝛺𝛺 𝜕𝜕𝜏𝜏

> 0 and 𝜕𝜕𝛺𝛺 ∗

𝜕𝜕𝜏𝜏 < 0. The election probabilities of the political parties

are given as a function of relative contribution by the firms. Therefore, 1 − 𝜑𝜑 = 𝐿𝐿 𝐿𝐿+𝐿𝐿∗

is the probability that the protectionist party has to be elected.

b. Show that the expected profits of the firms, before elections take place, depend on the expected trade policy. Find the optimal lobbying effort that maximizes the expected

profits of both firms. How do the optimal 𝐿𝐿 and 𝐿𝐿∗ depend on the ratio 𝛥𝛥𝛺𝛺 𝛥𝛥𝛺𝛺∗

=

𝛺𝛺(𝜏𝜏1)−𝛺𝛺(𝜏𝜏0) 𝛺𝛺(𝜏𝜏0)−𝛺𝛺(𝜏𝜏1)

?

c. Assume that the stakes of both firms are equal. Thus, 𝛥𝛥𝛺𝛺 = 𝛥𝛥𝛺𝛺∗ = 𝜅𝜅 > 0. Plot the functions 𝐿𝐿(𝐿𝐿∗) and 𝐿𝐿∗(𝐿𝐿) on the same graph. What part of its stakes each firm is

willing to contribute in this case?

d. Consider the case in which lobbying resembles a sequential game. The domestic firm moves first and the foreign observes 𝐿𝐿 before deciding on its lobbying effort.

Calculate again the optimal lobbying efforts of both firms.

Hint: The game here is sequential. The domestic firm moves first and the foreign

follows. Start with the foreign firm’s maximization problem and find 𝐿𝐿∗(𝐿𝐿). Plug it in

the domestic firm’s objective function and solve its maximization problem to find 𝐿𝐿.

Finally, use L to find 𝐿𝐿∗.

Deadline for submission: Tuesday 24.11.2020, 12:00 p.m. (noon)

Upload your solutions via MOODLE at the requested time or earlier in one PDF

file.

Please, stick to the rules and deadlines. There will be no exceptions.