Let there be two goods with quantities = (1, 2 ) + 2 1 0, 2 0. Let prices = (1, 2 ) ++ 2 1 > 0, 2 > 0. Let the consumer's preferences be represented by the CES utility function (1, 2 ) = (1 + 2 ) 1 where < 1, 0. Note: parameter in the utility function is the Greek letter rho not the price vector p. Derive the expenditure function (1, 2,). (14)

2. Let there be two factors (inputs), L and K, (,) + 2 , with factor prices w and r, respectively, (, ) ++ 2 . Let the firm's production function be = where , > 0.

a. Find the long run total cost function (, , ). (10)

b. Let capital be fixed at = in the short run. Find the short run total cost function (, , ; = ). (6)

c. Let p be the price of y. If this is a competitive firm, find how much would it produce in the short run. (6)

3. Let the firm's output be +. Let the output price be ++. Let the quantities of the m inputs be = (1, ... , ) + . Let input prices be = (1, ... , ) ++ . The firm takes input prices as given ie they are parameters. Prove that the profit function (, ) is nondecreasing in p. (6)