Question

Consider a firm using the production technology given by q = f(K, L) = ln(L^K)

If capital is fixed at K = 2 units in the short run, then what is the profit maximizing allocation of output if the price of output and respective input prices of labor and capital are given by (p, w, r) = (2, 1, 5)?

Answer #1

**q =
lnLk **

**K = 2**

**q =
lnL2 **

**q =
2lnL **

**Profit
= pq - wL - rK **

**= 2(2lnL) - L -
5(2) **

**= 4lnL - L -
10 **

**d/dL
= 4/L - 1 **

**put d/dL
= 0 **

**4/L - 1 =
0 **

**4/L = 1**

**L =
4 **

**d2/dL2 =
- 4/L2 < 0 **

**Thus L = 4 is profit
maximising labour**

**q =
2lnL **

**=
2ln(4) **

**= 2.772**

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