Question

2 TSF manufacturing is a producer of dinning table. It may currently sell all the table it can produce at $4 each. Its production is described by the function Q=8K"L4. It may buy all the capital and labor it wants at the constant input prices of $16 per unit and $8 per unit, respeetively.
i. Which type returns to scale is present in TSF's roadrunner trap production?
ii. If TSF wishes to maximize its eurrent profit, what are the optimal quantities of K and L that TSE should employ?

Answer 1

Part i

$Q(L,K)=8LK^4\\ Let \space t >0\\ Q(tL,tK)=8(tL)(tK)^4\\ Q(tL,tK)=t^58LK^4\\ Q(tL,tK)=t^5Q\\$

Increasing returns to sales is exhibited

Part ii

Maximizing profit is equal to minimizing cost , so

$Min_{L,K} \space \space 16K+8L \space s.t. \space Q= 8KL^4\\$

Thus

$Z= 16K +8L + \lambda[Q-8KL^4]\\ min Z_{L,K; \lambda}$

$Z_L = 8- \lambda 32 KL^3 = 0----(i)\\ Z_K = 16- \lambda 8 KL^4 = 0----(ii)\\ Z_{\lambda} = Q- 8 KL^4 = 0-----(ii)$

From (i) and (ii)

$\frac{1}{2}=4 \frac{K}{L}\\ L= 8K-----(iv)$

Using (iv) and (iii)

$Q= 8^5 K^5\\ K^*= \frac{Q^{\frac{1}{5}}}{8}\\ L^*=8 K^*\\ L^*=Q^{\frac{1}{5}}$

Which are the optimal demands.