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. what is the least cost input combination for producing 675 units of output?

Answer 1

Least cost combination is determined where MRTS = wage rate/capital cost = 900/400 = 2.25

MRTS = MPL/MPK = $\frac{\frac{\partial q}{\partial L}}{\frac{\partial q}{\partial K}} = \frac{25(0.5)l^{0.5-1}k^{0.5}}{25(0.5)l^{0.5}k^{0.5-1}} = \frac{l^{-0.5}k^{0.5}}{l^{0.5}k^{-0.5}} = \frac{k^{0.5+0.5}}{l^{0.5+0.5}} = \frac{k}{l}$

So, MRTS = 2.25 gives,
k/l = 2.25
So, k = 2.25l

q = 675 = 25l^.5k^.5 = 25l^.5(2.25l)^.5 = 25l^.5(1.5)(l)^.5 = 37.5l^.5+.5 = 37.5l
So, l = 675/37.5 = 18
k = 2.25l = 2.25*(18) = 40.5

So, l = 18; k = 40.5