Consider a 2-agent, 2-good pure exchange economy, where agents are denoted by superscripts and goods by subscripts. The endowments for the economy are ?1 = ?1 1 + ?1 2 and ?2 = ?2 1 + ?2 2 for goods 1 and 2, respectively, where ?? ? is the endowment of good ? ? {1,2} for agent ? ? {1,2}. Consumption of goods 1 and 2 are denoted by ?1 and ?2, respectively, where the Edgeworth box for the economy is drawn with ?2 on the vertical axis. Let ??1 and ??2 denote the allocations of goods 1 and 2, respectively, for agent 1, implying the corresponding allocations for agent 2 are ?1 ? ??1 and ?2 ? ??2. The utility functions are ? 1 = ?1 ??2 1?? for agent 1 and ? 2 = ?1 ? ?2 1?? for agent 2, where ?, ? ? (0,1).

2. [16 marks] Consider a 2-agent, Z-good pure exchange economy, where agents are denoted by superscripts and goods by subscripts. The endowments for the economy are E1 = e: + of and 2 = e21 + e for goods 1 and 2, respectively, where e3, is the endowment of good 111 E {1,2} for agent n E {1,2}. Consumption of goods 1 and 2 are denoted by :71 and x2, respectively, where the Edgeworth box for the economy is drawn with x: on the vertical axis. Let 1 and 1?; denote the allocations of goods 1 and 2, respectively, for agent 1, implying the corresponding allocations for agent 2 are E1 f1 and E2 2. The utility functions are u1 = xi'x'\" for agent 1 and u2 = x13 sci-'3 for agent 2, where or, ,3 E (0,1). a. Show that the contract curve is given by the function: (1 - lngi \"22"?\" = a0 ms, + (a an, Verify that the contract curve connects the lower left and upper right corners of the Edgeworth box. I]. Let :3, (p) denote the demand function of agent n for good m, where p is the price of good 2 and the price ofgood 1 is set equal to 1 without loss of generality. Derive the four demand functions. c. Derive the competitive equilibrium allocation 1:}, xi, xf and x3. :1. Show that the First Welfare Theorem holds for the economy