Question

Anne and Bill are left stranded on a desert island with nothing else but some water x and bread y. There are 100 units available of each good. Suppose that initially Anne has all the water and Bill has all the bread. Anne and Bill have different preferences over the consumption of water and bread. Anne’s utility function is ??(?,?)=? raised to 2/5, y raised to 3/5, and Bill’s utility function is ??(?,?)=? raised to 1/4 y raised to3/4 [30 marks] a) [3 marks] Is the initial consumption efficient in terms of consumption egalitarianism or welfare egalitarianism? (We assume that utilities are interpersonally comparable.) b) [10 marks] Derive Anne’s and Bill’s marginal rate of substitution of water for bread. Define the contract curve and draw it on the Edgeworth box. Is the initial allocation Pareto efficient? Justify your answer. c) [7 marks] Characterize the set of allocations of water and bread which Pareto improve

Answer 1

a. The initial allocation is not consumption egalitarian, as Anne and Bob have different amounts of the two goods. It is welfare egalitarian, as their utilities are the same (both are equal to zero).

b. MRS_A = $\frac{2y}{3x}$
MRS_B = $\frac{y}{3x}$

The contract curve is given by y = $\frac{200x}{100+x}$ , where x and y are the amounts held by Anne. The initial endowment is clearly not on this curve.

c. Pareto-efficient allocations are those which are on the contract curve, so they must satisfy y = $\frac{200x}{100+x}$ .
They must also give higher utility to Anne, so x^2/5 y^3/5 > 0, and to Bob, so (100 − x)^1/4(100 − y)^3/4 > 0.