5. (40%) Suppose a competitive firm may manufacture electronics using machinery and labor. Let z; and z; denote the quantity of machinery and labor respectively. The firm may use one of the two production technology. Under Technology A. the machinery will be manually operated by workers, so that both factors are needed for production. The production function is given L by FA4(z1, 2z2) = 4{z122)7 s Technology B leads to full automation. The production function is given by 3 \\ FB(z1,22) = %{zl}?. The firm uses no workers. Suppose the factor prices are given by wy = 1 and we = 4. We are interested in how the profit-maximizing input bundle changes with good price p. We break down the analysis into two steps. We first find out the profit-maximizing bundle under each technology. Then we work out the firm's choice of technology, and hence the corresponding input bundle. {a) First suppose that the firm is using technology B given wy = 1 and ws = 4. i. Under technology B, find out the firm's quantity demanded for machinery Zf in term of the good price p. ii. Show that the maximal profit under technology B is given by 1 foodag =37 o~ i {b) Now suppose that the firm is using technology A given wy = 1 and ws = 4. i. Show that under technology A, the profit-maximizing input bundle is given by - .'!_ 2 H: E 1 ii. Show that the maximal profit under technology A is given by ! ~d i i | ! &= = A = p? () We now study the firm's choice of technology. Explain why the firm will use technology B if the good price above p and technology A if the good price is below p. Find out the cut-off value of good price p