Question

3. Consider a pure exchange economy with two goods, wine (x) and cheese (y) and two con- sumers, A and B. Let cheese be the numeraire good with price of $1. Consumer A’s utility function is UA(x,y) = xy and B’s utility function is UB(x,y) = min[x,y]. A has an initial allocation of 10 x and no y, and B has an initial allocation of 10 units of y and no x.

(a) Put wine x on the horizontal axis and cheese y on the vertical axis. Measure goods for consumer A from the lower left and goods for consumer B from the upper right. Solve for the contract curve of Pareto efficient allocations in this economy and show this on your graph.

(b) Mark the initial allocation with the letter W. Draw the indifference curves for each person through this point. Calculate utility at this allocation for both consumers. Is the initial resource allocation consistent with Pareto efficiency? Explain.

(c) Find the competitive equilibrium prices and consumption for each type of consumer. Derive A’s and B’s demand functions (Marshallian). Calculate the equilibrium price of wine assuming price of cheese is $1. Using demand for wine, show that Walras’ Law holds. Show the budget constraint and indifference curves at the equilibrium. Label the equilibrium E. Show that E is on the contract curve.

(d) Suppose instead, you wished to obtain the equilibrium E1, where utility of A is equal to UB = 7. Find corresponding price of wine p1 that must hold in this instance, and show the set of possible new endowments (which will be a line) that satisfies the Second Theorem of Welfare Economics, given p1.

Answer 1