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 p 3 1 satisfy x 1 (p 1 , p 2 , 1, W) a 1 b 1 p 1 c 1 p 2 d 1 p 1 p 2 x 2 (p 1 , p 2 , 1, W) a 2 b 2 p 1 c 2 p 2 d 2 p 1 p 2 (a) State Walras' law and use it to find the Walrasian demand for good 3 (It's fine to just give the demand when p 3 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 p 3 (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 1 , b 1 , c 1 , d 1 , a 2 , b 2 , c 2 , d 2 ) (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 (a 1 , b 1 , c 1 , d 1 , a 2 , b 2 , c 2 , d 2 ) that this implies

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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

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

x₁(p₁, p₂, 1, W) = a₁ + b₁p₁ + c₁p₂ + d₁p₁p₂

x₂(p₁, p₂, 1, W) = a₂ + 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 (a₁, b₁, c₁, d₁, a₂, 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, b₁, c₁, d₁, a₂, b₂, c₂, d₂) that this implies.