3. (25 Minutes - 30 Points) Consider an economy with three goods. Suppose that a consumer has a continuous utility function satisfying local nonsatiation. Suppose also that the consumer's Walrasian demands for goods 1 and 2 when p3 = 1 satisfy 21 (P1, P2, 1, W) = a1 + bipi + cp2 +d1P1P2 22(P1, P2,1,W) = 22 + b2p1 + c2p2 + d2p1p2 (a) State Walras" law and use it to find the Walrasian demand for good 3. (It's fine to just give the demand when p3 = 1.) (b) State a result about the homogeneity of Walrasian demands and use it to find the consumer's Walrasian demands at other values of p3. (c) Note that the Walrasian demands for goods 1 and 2 are independent of wealth. Show that this makes it very easy to find the Hicksian demands for goods 1 and 2. State the Compensated Law of Demand. Show that this law puts some restrictions on the possible values for (a,b,c,d1, 22, B2, C2, d2). (d) Define the Slutsky substitution matrix. What properties must it have if demands are derived from maximizing a continuous, locally nonsatiated, and strictly quasiconcave utility function? Give at least one additional restriction on (a1, 61,C1,21,22,b2, C2, d) that this implies. 3. (25 Minutes - 30 Points) Consider an economy with three goods. Suppose that a consumer has a continuous utility function satisfying local nonsatiation. Suppose also that the consumer's Walrasian demands for goods 1 and 2 when p3 = 1 satisfy 21 (P1, P2, 1, W) = a1 + bipi + cp2 +d1P1P2 22(P1, P2,1,W) = 22 + b2p1 + c2p2 + d2p1p2 (a) State Walras" law and use it to find the Walrasian demand for good 3. (It's fine to just give the demand when p3 = 1.) (b) State a result about the homogeneity of Walrasian demands and use it to find the consumer's Walrasian demands at other values of p3. (c) Note that the Walrasian demands for goods 1 and 2 are independent of wealth. Show that this makes it very easy to find the Hicksian demands for goods 1 and 2. State the Compensated Law of Demand. Show that this law puts some restrictions on the possible values for (a,b,c,d1, 22, B2, C2, d2). (d) Define the Slutsky substitution matrix. What properties must it have if demands are derived from maximizing a continuous, locally nonsatiated, and strictly quasiconcave utility function? Give at least one additional restriction on (a1, 61,C1,21,22,b2, C2, d) that this implies