Question

Only two goods are made in a city: mike and cookies. milk is made using the tchonology Qm=Min(1/2L,K) and cookies is made using the techonology Qc=Min(L,1/2K), each good sells for 10 pennies a unit. What wage and capital rate is consistent with zero profits.

Answer 1

From production function we see that L = 2Qm and K = Qm. Similarly L= Qc and K = 2Qc

Cost function is C = wL + rK

Cost function for milk is Cm = 2wQm + rQm and revenue Rm = 10Qm. Profits are zero so Rm = Cm

10Qm = 2wQm + rQm

This gives the first equation 2w + r = 10

Similarly for cookies, we have Cc = wQc + 2rQc and Rc = 10Qc. This gives the second equation 10 = w + 2r

Solve them to see 2w - w + r - 2r = 10 - 10 or w = r

Now use this in first equation to get 2w + w = 10 or w* = 10/3. Since w = r, we have r* = 10/3

The required wage and capital rate are both 10/3 pennies.