Question

a.) EASTEK Co. operates in a competitive market. Its production function is q=L (to the power alpha) * K (to the power beta). The exponents alpha and beta are both less than 1. If the firm's capital is fixed, and it takes the wage and price as given, what is the firm's short-run demand for labour ? [13 marks]

b.) Pipi Co Ltd. operates in a competitive market. Its marginal product of labour is 1 over L (1÷L), and it takes the wage and prices as given. Derive the firm's short-run demand for labour as a function of "w" and "p".

How much labour will the firm hire if w = 2 and p = 10? [10 marks]

Answer 1

(a)

$f(L,K)=Q=AL^\alpha K^\beta$

The partial derivative of the CD function w.r.t. Labor(L) is:

$\frac{\delta Q}{\delta L}=A\alpha L^{\alpha-1}K^\beta$

$\frac{\delta Q}{\delta L}=\alpha AL^\alpha L^{-1}K^\beta$

$\frac{\delta Q}{\delta L}=\frac{\alpha AL^\alpha K^\beta}{L}$

Quantity produced is based on labor and capital, therefore:

$Q=AL^\alpha K^\beta$ ,

$\frac{\delta Q}{\delta L}=\alpha \frac{Q}{L}$

$\alpha =\frac{\delta Q}{\delta L}\frac{L}{Q}$ ..This will yield $MP_L$

$\beta=\frac{\delta Q}{\delta K}\frac{K}{Q}$ ..This will yield $MP_K$

The linear model of the function:

$f(L,K)=Q=AL^\alpha K^\beta$

$log(Q)=log(A)+\alpha log(L)+\beta log(K)$

We come up with constraint minimization problem:

$C(w,r.Y,A)=min wL+rK$

s.t. $f(L,K)=Q=AL^\alpha K^\beta$

To find optimal amount of inputs (L and K), solve the minimization constraint using Lagrange multiplier method:

$l(w,r,L,K,Q)=wL+rK+\lambda (Q-AL^\alpha K^\beta$

$\frac{\delta l}{\delta L}=w=\lambda \alpha AL^{\alpha -1}K^\beta$

$\frac{\delta l}{\delta K}=r=\lambda \beta AL^\alpha K^{\beta -1}$

$\frac{\delta l}{\delta \lambda }=Q-AL^\alpha K^\beta=0.$

Solve for L.

$\frac{\frac{\delta l}{\delta K}}{\frac{\delta l}{\delta L}}=\frac{r}{w}=\frac{\lambda \beta AL^\alpha K^{\beta-1}}{\lambda \alpha AL^{\alpha-1}K^\beta}$

$\frac{r}{w}=\frac{\beta}{\alpha} \frac{L}{K}$

$L=\frac{r}{w}\frac{\alpha}{\beta}K$

L(as a function of Q,A,w and r):

$L=(\frac{Q}{A})^\frac{1}{\alpha +\beta}(\frac{r}{w}\frac{\alpha}{\beta})^\frac{\beta}{\alpha-\beta}$