Selected solutions to homework 2

(c) Jan Hagemejer

Question 2 Person A has a utility function of

uA(x1A, x2A) = x1A + 2x2A and ωA = (2, 0)

Person B has a utility function ofuB(x1B , x2B) = min(2x1B , x2B) and ωB = (0, 4).

(b) What is the equilibrium relationship between p1 and p2.

Since consumer B will always want to consume both goods, we will have an interior solution. Therefore the budget line forconsumr A has to coincide with his indifference curve. Therefore we know that p1 = 1 and p2 = 2 or any combination such that2p1 = p2.

(c) What is the competitive equilibrium allocation?

If we take (b) into account and also the fact that B has Leontief utility function, therefore we can find the demand for bothgoods for B:

x1B =4 ∗ p2

p1 + 2p2=

4 ∗ 2

1 ∗ 1 + 2 ∗ 2=

8

5

x2B =2 ∗ 4 ∗ p2p1 + 2p2

=2 ∗ 4 ∗ 2

1 ∗ 1 + 2 ∗ 2=

16

5

The other consumer consumes what is left from the endowment of both goods:

x1A = 2− 8

5=

2

5

x2A = 4− 16

5=

4

5.

Question 3 Consider the Varian’s revealed preference version of the second welfare theorem in the pure exchange economy.What assumption on preferences do you need to have x′i = x∗i for all i. (In other words what assumptions do you need for theWalrasian equilibrium to guarantee that the competitive equlibrium will be exactly in the allocated Pareto efficient point).

The only thing that need you need to do here is that for x′i = x∗i preferences need to be strictly convex. If only convex, forexample with perfect subsitututes, you can have multiple equilibria (a small picture would do).

Question 4 Explain why if preferences are locally non-satiated, then the condition “if xi �i x∗i then p · xi ≥ p · x∗i ” is equivalent

to saying “x∗i is expenditure minimizing for the price vector p in the set {xi ∈ Xi : xi �i x∗i }.” You can use a diagram and explain

why they are not equivalent when preferences are not non-satiated.

If preferences are not non-satiated, then a bundle that satisfies xi �i x∗i then p · xi ≥ p · x∗i may not be expenditure minimizing.

If you consider a not nonsatiated preferences (eg. with indifference curves that are thick and not infinitely thin), then a bundleon the upper boundary of the indifference curve satisfies the first condition while it does not satisfy the second.

Question 5 Consider a production economy with one consumer with a Cobb-Douglas utility function for leisure x1 and con-sumption x2 : u(x1, x2) = 0.5 lnx1 + 0.5 lnx2. Endowment of leisure is one. Consumer can supply its leisure to the firm withtechnology q = L−a, where L is the supply of labour to the firm and a ≥ 0 is a constant and q is the output of good 2. Consumerowns the firm.

(a) Does the competitive equilibrium exist? Why? What is the condition for a for the competitive equilibrium to exist?

If a > 0 no equilibrium exisits, because we have IRS. If a = 0, we have CRS and we can find an equilibrium. The social plannersolution exists and it coincides with competitive equlibrium only when we have CRS.

Question 8 (Small open economy with a Leontief function) Assume that we have two countries, each producing two goods:crackers and beer. Technology of production is the same in the two countries:

Qc = min{1

2K, 1L} andQb = min{1

3K,

1

4L}.

Endowments are: at home: ωL = ωK = 100, abroad: ω∗L = 120, ω∗K = 80.

1

a) Which of the goods is more labour intensive? What would be your expectations towards the production pattern i.e. who willproduce more of what?

The unit factor requirements with our functions are:aKC = 2, aLC = 1, aKB = 3, aLB = 4, so for sure aKC

aLC= KC

LC> KB

LB= aKB

aLB

so C is more capital intensive. The expectations toward the production pattern in a trading situation would be that the countrythat has relatively more capital will produce more of C.

b) Compute equilibrium production level assuming that there is full employment, i.e. all factors of production are used. Do itfor every country separately.

Factors are immobile internationally, so we do it for every country separately. We need to have full employment, so:

KC + KB = ωK

and

LC + LB = ωL

And similar thing for abroad. For the home country it is easy as the system of equation becomes:

2 ∗ C + 3 ∗B = 100

And full employment in the capital market:

1 ∗ C + 4 ∗B = 100

Solve the system:

C = 20, B = 20

However for the abroad we have a problem, why?

We have full employment in the labour market:

2 ∗ C∗ + 3 ∗B∗ = 80

And full employment in the capital market:

1 ∗ C∗ + 4 ∗B∗ = 120

This equation produces a system that has one of the quantities negative

C∗ = −8, B∗ = 32.

Therefore abroad we cannot have an equilibrium with full employment1. What happens here is that we have relatively to littlecapital to produce both goods with full employment. Therefore the abroad will only be producing B in the trading situation.What will be the production level?

Check the capital constraint:

ω∗K/aBK = 80/3 = 262

3= B∗

What is the labour employment level?

z∗LB = 262

3∗ 4 = 106

2

3

so in fact some labour will remain unemployed - that is actually important for the determination of wages.

c) What would be the equilibrium wages given the world price level at pC = 5, pB = 10 (Hint: what is the unit cost for a Leontieffunction)? Is there factor price equalization?

In order to find the equilibrium wages for home, we will be using the zero profit conditions (the unit cost = price):

1(check if we can have it when alternatively ωK = 120 and ωL = 80)

2

2wK + 1wL = 5

3wK + 4wL = 10

And the solution to this is:

wK = 2, wL = 1

What happens abroad?

Remember that in equlibrium we have excess supply of labour. Therefore labour is free (there is no disutility of labour, sothe household is indifferent between supplying labour at a price equal to zero and not supplying it). If there was disutility ofworking then the supply of labour would go down with the price assuring equilibrium. Therefore we only use the zero profitcondition for good B:

3wK + 4 ∗ 0 = 10, wK =10

3

If there was full employment in both countries, the wages would be equal, so there would be factor price equal-ization. When we do not have full employment in both countries, there would not be factor price equalizationbetween countries.

3