Question

suppose a firm has a cobb-douglas weekly production function q=f(l,k)=25l^.5k^.5, where l is the number of workers and k is units of capital.mrtslk is k/l. the wage rate is $900 per week, and a unit of capital costs $400 per week. assuming no fixed cost, what is the firm's total cost of production if it uses least-cost input combination to produce 675 units of output?

Answer 1

q = 25l^0.5k^0.5

Cost is minimized when MPl/MPk = w/r = 900/400 = 9/4

MPl = $\dpi{80} \partial$ q/$\dpi{80} \partial$l = 25 x 0.5 x (k/l)^0.5

MPk = $\dpi{80} \partial$ q/$\dpi{80} \partial$k = 25 x 0.5 x (l/k)^0.5

MPl/MPk = k/l = 9/4

k = (9/4) x l

Substituting in production function,

25l^0.5k^0.5 = q = 675

l^0.5k^0.5 = 27

l^0.5[(9/4) x l]^0.5 = 27

l^0.5 x l^0.5 x (3/2) = 27

l = 18

k = (9/4) x 18 = 40.5

Total cost = wl + rk = 900 x 18 + 400 x 40.5 = 16,200 + 16,200 = 32,400