Question 3: Cost minimization [35 points]

Consider a firm that has a Cobb-Douglas technology. The firm wishes to minimize the cost of producing y units of output and has access to perfectly competitive factor markets. The firm's cost minimization problem is given by:

s.t. = y

Let ? denote the Lagrange multiplier on the output constraint.

1.What are the parameters of the problem? [1 point]

2.Find the conditional factor demand functions. Label them l?(w, r, y) and k?(w, r, y). (Hint: You can check your algebra with Varian pp 54-55.) [15 points]

3.Find the cost function: c(w, r, y). What is its interpretation? [8 points] ?

4.Find ?^* . What is its interpretation? [7 points]

5.Find and show that it is equal to ?^* . [4 points]

6.How does depend on (? + ?)? [graded for completion]

7.Show that the cost function is homogeneous of degree in output. [graded for completion]

How do the returns to scale (degree of homogeneity of the production function) relate to the properties of the cost function? Does this make

Question 3: Cost minimization [35 points] Consider a rm that has a Cobb-Douglas technology. The rm wishes to minimize the cost of producing y units of output and has access to perfectly competitive factor markets. The firm's cost minimization problem is given by: min W! + rk {kl} sekal= y Let n denote the Lagrange multiplier on the output constraint. NH POT-19541:\"? What are the parameters of the problem? [1 point] Find the conditional factor demand functions. Label them 11w, r, y) and k*(w, r, y). (Hint: You can check your algebra with Varian pp 54-55.) [15 points] Find the cost function: QLW, r, y]. What is its interpretation? [8 points] a: Find uLIWhat is its interpretation? [7 points] Find 3: and show that it is equal to [[1,3[4 points] I 626 How does F depend on (a + [3)? [graded for completion] Show that the cost function is homogeneous of degree \"13 in output. [graded for completion] How do the returns to scale [degree of homogeneity of the production function) relate to the properties of the cost function? Does this make sense to you? Explain clearly why. [graded for completion]