Econ 441: International Trade After derivations that need several steps, always mark your final answer clearly: put a box around it, underline it or highlight it. You should aim for brief, concise explanations. 1 US/Canada Free Trade Agreement This question asks you to work throw the quantitative implications of integration between two different-sized countries in the monopolistic competition model. The market sizes are S US = 100 and S CA = 10. Firms in each country can enter the market by paying a fixed cost 1 and produce with a variable cost 1 for every unit q produced, so T C(q) = 1 + Q. As in lecture, firms in each country face the demand curve Q = S · 1/N − (P − P ̄) where S is the market size, N is the number of firms, P is the price charged by the firm and P ̄ is the average price charged by all firms. All firms are identical. Note: don’t worry about “fractions” of firms in your answer

ANSWER

Econ 441: International Trade

Fall 2017
Problem Set 3

Please, write your full name and umich id on your problem set.
Please, pay attention to the organization of your answers. Always keep the order of the questions,
write legibly, and clearly indicate what you are doing. Please staple your answer sheets together.
Don’t forget: the grader can only give you credit if they can read and comprehend what you have
written.
After derivations that need several steps, always mark your final answer clearly: put a box
around it, underline it or highlight it. You should aim for brief, concise explanations.

1 US/Canada Free Trade Agreement
This question asks you to work throw the quantitative implications of integration between two
different-sized countries in the monopolistic competition model. The market sizes are S

US = 100 and

S
CA = 10. Firms in each country can enter the market by paying a fixed cost 1 and produce with
a variable cost 1 for every unit q produced, so T C(q) = 1 + Q. As in lecture, firms in each country
face the demand curve Q = S ·

1/N − (P − P ̄)

where S is the market size, N is the number of
firms, P is the price charged by the firm and P ̄ is the average price charged by all firms. All firms
are identical. Note: don’t worry about “fractions” of firms in your answer. We will interpret these
quantities as millions (i.e. 2.5 firms equals 2.5 million firms), but just to keep the notation reasonable
we’ll use the lower numbers.
1. Find the marginal revenue curve for each country under autarky.
2. Assume that, under autarky, free entry drives profits to zero. Derive the long run equilibrium
number of firms in each market, and compute the ratio of U.S. firms to Canadian firms.
3. Derive the scale of firms in each country, i.e. the quantity Q produced by each firm. Which
country has larger firms? How much larger are they (i.e. 2 times? 3 times?).
4. Derive the markup of price over marginal cost μ = P/c − 1 in each market. Which country has
larger markups? How much larger are they (i.e. 2 times? 3 times?).

1

Department of Economics

Economics 441

5. Now lets assume that the two countries sign a free trade agreement. Effectively, firms in each
country now share the same market. In the short run, i.e. no firms exits, compute the new
scale of each firm (now common across countries) and the new markups (also common across
countries). Do they go up or down compared to their autarky values for each country?
6. Describe the two sources of gains from trade for consumers in each economy. Which country
do you think gains more from trade in the short run?
7. Compute the total profits per firm under free trade, in the short run. Explain the logic behind
your answer.
8. Compute the long-run equilibrium number of firms using the zero profit condition under free
trade. Is it larger or smaller than the short-run number of firms?
9. Compute the long-run scale Q and markup under free trade. Compare them to their short-run
values and provide intuition for the difference.

2 US/Canada Free Trade, Heterogeneous Firms (Chal-
lenge)

Let’s stick with the same example, except now there are two types of firms. One type has marginal
cost equal to 1 and one type has marginal cost equal to 1/2. Assume that initially, under autarky,
half the firms in each country are of one type and half of the other. All other parameters are the
same.
1. For each country, find the price and quantity for each type of firm, given the average price P ̄
and the number of firms N. Hint: first find the prices charged by low and high cost firms as a
function of the quantities sold by each type of firm using the condition MR = c. Then solve
for the quantities that each firm sells (given the average price) by plugging your solution to
MR = c into the demand curve.
2. For each country, find the average firm price P ̄ charged in each country for a given number of
firms N under autarky. Hint: use P ̄ =
P
high+P
low
2
.

3. For each country, find the total profits of each type of firm (i.e. including fixed costs) for a
given number of firms N under autarky. It’s fine not to simplify your answer too much.
4. BONUS: Find the number of firms in each country under autarky, under the assumption that
the high-cost firms earn zero profits (and that they make up half of all firms). Compute the
ratio NUS/NCA. Hint: remember the quadratic formula, and pick the lower of the two solutions
if they are both positive. Explain why you should pick the lower solution.
5. BONUS: Find the scale and markup of each type of firm in each country under autarky.
6. BONUS: Now assume the countries sign a free trade agreement. In the short run, i.e. no
firms exits, compute the new scale and markups for each type of firm. Do they go up or down
compared to their autarky values for each country? Does anyone have incentive to exit the
market?
7. BONUS Demonstrate whether or not the long run equilibrium (in which firms are allowed to
exit if the earn negative total profits but no new firms enter) will feature positive numbers of
high cost firms or only low cost firms.

ANSWER