Consider an exchange economy with only 2 agents, Arnold and Brigitte, and 2 goods, Xand Y. Let Arnold have utility function Ua(Xa,Y1/ 3a)= XaY1/ 3a and Brigitte UB (XB ,YB ) = XB YB . Here (X A , Y A) and (X B, Y B) represent the good bundles of Arnold and Brigitte, respectively. Let Arnold have an initial endowment of (X A , Y A ) = (1,5) and Brigitte (X B , Y B) = (2,0)
(a) Draw the initial endowment, as well as a few indifference curves of Arnold and Brigitte in an Edgeworth box. Prove using the Edgeworth box whether or not the initial endowment is Pareto efficient.
(b) To avoid confusion, draw up a new Edgeworth box and indicate clearly the contract curve, the set of individually rational trades between Arnold and Brigitte, and the core of this economy.
(c) (1) Give a formula for Arnold's budget restriction if the price of good X is 3 and the price of good Y is 1. (2) Draw up a fresh Edgeworth box again and show what is Arnold's optimum bundle given these prices.
(d) Define what is meant by a competitive equilibrium allocation.(e) Prove mathematically that the prices in (c) constitute a competitive equilibrium and find the competitive equilibrium allocation?