1. Suppose the production possibility frontier for guns (x) and butter (y) is given by

x²+2y² = 900.

a. Graph this frontier.

b. If individuals always prefer consumption bundles in which y = 2x, how much x and y will be produced?

c. At the point described in part (b), what will be the RAT and hence what price ratio will cause production to take that point? (This slope should be approximated by considering small changes in x and y around the optimal point.)

d. Show your solution on the figure from part (a).

2. Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (x) or chicken soup (y). Smith's utility function is given by

U_s = x^0.3y^0.7,

Whereas Jones' is given by

 U_l = x^0.5y^0.5.

The individuals do not care whether they produce x or y, and the production function for each good is given by

x = 2l and y= 3l,

where l is the total labor devoted to production of each good.

a. What must the price ratio, p_x/p_y, be?

b. Given this price ratio, how much x and y will Smith and Jones demand? Hint: Set the wage equal to 1 here.

c. How should labor be allocated between x and y to satisfy the demand calculated in? (b)?

3. Smith and Jones are stranded on a desert island. Each has in his possession some slices of ham (H) and cheese (C). Smith is a choosy eater and will eat ham and cheese only in the fixed proportions of 2 slices of cheese to 1 slice of ham. His utility function is given by U_s = min(H, C/2).

Jones is more flexible in his dietary tastes and has a utility function given by U_I = 4H + 3C. Total endowments are 100 slices of ham and 200 slices of cheese.

a. Draw the Edgeworth box diagram that represents the possibilities for exchange in this situation. What is the only exchange ratio that can prevail in any equilibrium?

b. Suppose Smith initially had 40H and 80C. What would the equilibrium position be?

c. Suppose Smith initially had 60H and 80C. What would the equilibrium position be?

d. Suppose Smith (much the stronger of the two) decides not to play by the rules of the game. Then what could the final equilibrium position be?

4. In the country of Ruritania there are two regions, A and B. Two goods (x and y) are produced in both regions. Production functions for region A are given by         

x_A = √l_x,

y_A = √l_y;

here l_x and l_y are the quantities of labor devoted to x and y production, respectively. Total labor available in region A is 100 units; that is,

l_x + l_y = 100.

Using a similar notation for region B, production functions are given by

x_B = ½√l_x,

y_B = ½√l_y.

There are also 100 units of labor available in region B:

l_x + l_y = 100.

a. Calculate the production possibility curves for regions A and B.

b. What condition must hold if production in Ruritania is to be allocated efficiently between regions A and B (assuming labor cannot move from one region to the other)?

c. Calculate the production possibility curve for Ruritania (again assuming labor is immobile between regions). How much total y can Ruritania produce if total x output is 12? Hint: .A graphical analysis may be of some help here.

 5. An example of Walras' law

Suppose there only three goods (x₁, x₂, x₃) in an economy and that the excess demand functions for x₂ and x₃ are given by

ED₂ = -(3P₂/P₁) + (2P₃/P₁) - 1,

ED₃ = -(4P₂/P₁) - (2P₃/P₁) - 2.

a. Show that these functions are homogeneous of degree 0 in P₁, P₂, and P₃.

b. Use Walra's law to show that, if ED₂ = ED₃ = 0, then ED₁ must also be 0. Can you also use Walras' law to calculate ED₁?

c. Solve this system of equations for the equilibrium relative prices P₂/P₁ and P₃/p₁. What is the equilibrium value for P₃/P₂?