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. There are 30 units of capital and 36 units of labor. Find the level of milk and cookies outputs that are consistent will full employment.

Answer 1

Let the labor and capital be L1 + L2 = 36 and K1 + K2 = 30. We have

Qm = L1/2 or L1 = 2Qm and Qm = K1 or K1 = Qm.

Qc = L2 or L2 = Qc and K2 = 2Qc

Use the labor and capital constraint

L1 + L2 = 36 K1 + K2 = 30

2Qm + Qc = 36 Qm + 2Qc = 30

Solve them (first subtract the two equations)

2Qm - Qm + Qc - 2Qc = 6

Qm - Qc = 6 and similarly (now add the two equations)

2Qm + Qm + Qc + 2Qc = 66

Qm + Qc = 22

Now solve Qm - Qc = 6 and Qm + Qc = 22 to get Qc = 8 and Qm = 14.

The level of milk and cookies outputs that are consistent will full employment are milk = 14 and cookies = 8.