Question

4. (35 points) Hannah and Sam run Moretown Makeovers, a home remodeling business. The number of square feet they can remodel in a week is described by the Cobb- Douglas production function Q = F(L, K) = 10L0.5Kos where L is their number of workers and K is units of capital. The wage rate is $1000, and a unit of capital costs Hannah and Sam $1000 per week. (a) (10 points) Graph the isoquants F(L, K) = 10 and F(L, K) = 20. 9 6 (b) (10 points) What is their least-cost input combination for remodeling 100 square feet a week?

Answer 1

The given production function is, The wage rate (w) is $1,000 and cost of capital (r) is S1,000 per week. (a) To graph the isoquant, substitute the value in production function and simplify as follows When F (L, K) = 10. F (K,L)=1025K's 10-10r5x05 I-k1-1 When F (L, K) = 20. 20-10rsK°5 Now, substitute the above simplified functions as follows

4.5 4 3.5 2.5 1.5 F (L, K)20 0.5 0 0 234 5 67 89 (b) To determine the least-cost input combination, the following condition must satisfy: ratio of marginal products ratio of input prices MP w Using the optimum condition, compute the tangency condition as follows I W 05 (L) 1,000 1,000 Thus, the tangency condition is K L. In addition to this tangency condition, the combination of input must result in 100 quare feet a week. Therefore, substitute the output in production function and simply in terms of L as follows

0 10105 100 1005xa5 10 100 The above steps result in two equations of K. Equate the above two equations and solve for input combinations as follows 100 1 100 Hence, the cost-minimizing input combination of labor and capital is!L = 101 and K = 10