Question

You manage a firm that has the production function q = 12L1/2K1/3. You can hire labor (L) at a wage of $6 per unit of labor and rent capital (K) at a rate of $15 per unit of capital. Your boss gives you a budget of $1200. Write your budget constraint in terms of inputs: 1200=6L+15K What is your optimal amount of L and K? L = K =

Answer 1

The production is the process which can be transformed the factors of the production in to the final product. There are four factors of the production. These four factors of the production are land , labour, capital and entrepreneur. The production function shows the relationship between employed inputs and produced output. The functional relationship is given as Q= f( L,K) where Q is the total output produced and L is the amount of labour employed and K is the amount of capital employed.

An iso-cost line is a line that represents all combinations of the inputs of the firm which can be measured the same total cost. The iso cost line can be given as C= Lw+Kr where C is the total cost of the production , L is the amount of labour employed and w is the price of the labour that is wage , K is the amount of capital employed and r is the price of the capital that is rent.

The production function of firm is given as

q= 12 L^{​​​​​1/2} K^{​​​​​​1/3} where q is the total amount of output and L is the amount of labour and K is amount of capital.

Labour (L) can be hired at the wage rate w= $6

Capital (K) can be hired at the rent r= $15

The total budget cost is given as C= $1200.

The iso -cost line is given as 1200= 6L+15K.

To calculate the optimal amount of labour (L) and capital (K) , the Lagrangian expression can be used. To solve the problem, the total output must be maximised subject to the budget constraint given by boss. This problem can be solved following way

To solve the optimisation problem , here the Lagrangian expression be used. By using the Lagrangian method the optimum value of labour and capital be calculated .

The optimum amount of labour be employed by firm is L= 120 unit.

The optimum amount of labour be employed by the firm K = 32 unit