Problem Set 4 Economics 511 1. Suppose q is produced using three factors of production, z1,z2 and z3. In the short run, the firm takes z1 as fixed and then chooses z2 and z3 to minimize the cost of producing some given level of output. Under what circumstances will an increase in z1 (1) raise short run costs, (2) lower short run costs, or (3) leave short run costs unchanged? That is, if C(w1,w2,w3,z1, q)is a firm's short run cost function, what determines the sign of C/z1 ? Give an intuitive explanation of your results(HINT: Think about the relationship between the price ratio and the MRTS). 2. For each of the following functions, (1) derive the marginal rate of technical substitution, (2) say whether it exhibits diminishing marginal product of labor, and (3) say whether it exhibits increasing, decreasing or constant returns to scale. a. q=K+L10KL b. q=K2L c. q=min(3K,4L) 3. For the functions in #3, (1) Find the function for (or the value of) the elasticity of substitution for each production function and (2) find the cost minimizing input combination to produce 10 units of output when w=$1 and r=$4. 4. For the function in part 2a above, derive the conditional factor demand functions. Problem Set 4 Economics 511 1. Suppose q is produced using three factors of production, z1,z2 and z3. In the short run, the firm takes z1 as fixed and then chooses z2 and z3 to minimize the cost of producing some given level of output. Under what circumstances will an increase in z1 (1) raise short run costs, (2) lower short run costs, or (3) leave short run costs unchanged? That is, if C(w1,w2,w3,z1, q)is a firm's short run cost function, what determines the sign of C/z1 ? Give an intuitive explanation of your results(HINT: Think about the relationship between the price ratio and the MRTS). 2. For each of the following functions, (1) derive the marginal rate of technical substitution, (2) say whether it exhibits diminishing marginal product of labor, and (3) say whether it exhibits increasing, decreasing or constant returns to scale. a. q=K+L10KL b. q=K2L c. q=min(3K,4L) 3. For the functions in #3, (1) Find the function for (or the value of) the elasticity of substitution for each production function and (2) find the cost minimizing input combination to produce 10 units of output when w=$1 and r=$4. 4. For the function in part 2a above, derive the conditional factor demand functions