Assignment 3

March 26, 2020

Explain the economic intuition behind your results.

Problem 1

Consider a Krugman-style model in which preferences are given by:

U =

X

N

i=1

c

σ−1

σ

i

, (1)

where N is the number of firms and we assume σ > 1.

The labor requirement to produce yi units for firm i is:

Li = α +

1

β + 1

y

β+1

i

, (2)

where β > 0.

The number of workers in the economy is given by

L (3)

Normalize the wage to w = 1 throughout.

Question 1.

Part A: Find the optimal pricing strategy of a firm, and show that the profit

maximizing price is then given by:

p =

σ

σ − 1

MC =

σ

σ − 1

y

β

i

. (4)

Part B: Comment on your finding in Part A. In particular, explain how p

depends on yi for β > 0 and why.

Question 2.

Find average costs for an individual firm.

Question 3.

What are profits per firm?

1

Question 4.

Impose a zero profit condition and find the equilibrium production per firm, y,

the number of firms, N, and utility, and show that the latter is given by

U =

L

1

σ

α

1 + σβ

σ(1 + β)

α(σ − 1)(1 + β)

1 + σβ 1

1+β

! σ−1

σ

(5)

Question 5.

How does the number of firms depend on β? Why? Interpret.

2

Problem 2

We have previously assumed constant returns to scale production, i.e. y1 =

a1L1, where a1 is labor productivity in sector 1 and L1 is use of labor in sector

1 and a1 is considered a constant. This is not always a fitting description for

the productive structure of an industry. In some industries, the productivity of

a single firm in a given industry is increasing in how large the overall production

of the industry as a whole is. This might be the case because

• Producers share the same suppliers, so many producers means many suppliers and it’s cheaper to get inputs for production

• The labor market is deeper. There is always turnover in labor markets. If

a company is located near similar other companies it will be easier for it

to find workers

• Technological spillovers. You might learn something about how to conduct your business from eavesdropping in the local restaurant because the

people at the next table are in the same line of work and they are talking

about the exact same problem you’re having.

We could model those things explicitly. Instead we will just say that productivity for a single representative firm is given by:

a1(Y¯

1) = Y

α

1

(6)

where a1(Y¯

1) is labor productivity, Y¯

1 is aggregate production (of good 1) in

the economy and 1 > α > 0. Labor productivity in sector 2 is given by a2.

Normalize a2 to 1 for simplicity throughout, a2 = 1. In other words, we have

that for a representative firm in either sector, the production levels are

Y1 = Y¯ α

1 L1 (7)

Y2 = L2. (8)

Question 1.

Why have we assumed that α > 0?

Question 2:

The single representative firm takes output prices as given (p1 = p in sector 1

and p2 = 1 in sector 2), input prices of labor w and aggregate production Y¯

1

as given. Show that an equilibrium with positive and finite production in each

sector requires:

a1(Y¯

1)p = w, (9)

1 = w. (10)

Question 3.

First, consider autarky and let aggregate preferences be given by:

U = C

2/3

1 C

1/3

2

. (11)

3

The country has labor L = 1. Find Marshallian demand for each good.

Question 4.

Impose the autarky condition of Y1 = C1, Y2 = C2, i.e. we consume what we

produce, and show that the equilibrium production levels will be:

Y1 = (2/3)

1

1−α (12)

Y2 = 1/3. (13)

For the remainder of the Assignment, open up to trade and let foreign be

identical to home: same number of workers and same technology. There are no

transportation costs between the two countries.

Question 5.

Show that there is a Nash equilibrium in which home specializes in producing

good 1 and foreign specializes in producing good 2. State the production levels

in both countries, and show that the wages in this equilibrium are given by

w = 2 ∧ w

∗ = 1. (14)

Show explicitly that no firm wishes to deviate, and hence, that this is in fact an

equilibrium.

Hint: A Nash equilibrium is a situation in which no agent (e.g. firm) has an

incentive to deviate, e.g. start production in the other sector. They do not

wish to deviate when marginal profits are lower in the sector that they consider

deviating to.

Question 6:

Explain that there are actually 3 possible equilibria in total: The one described

in question 5, one where home produces good 2 and foreign produces good 1

and finally one where both countries produce both.

Show that wages are given by

w = 1 ∧ w

∗ = 2 (15)

and

w = 1 ∧ w

∗ = 1 (16)

for these latter two equilibria, respectively.

Show that both of these are actually equilibria, i.e. that no firm wishes to

deviate.

Question 7.

Show that there exist conditions under which it is not welfare improving for

a country to open up to trade. Specifically, show that when α → 1 it holds

that welfare for the representative agent is higher without trade than it is under

trade in the country that does not specialize in industry 1. Explain the economic

intuition behind this result.