Answer to Question #190522 in Microeconomics for Alicia

Question

Suppose that a firm’s production function is Q =10 K^(3/4)L^(1/4). The cost of a unit of labor is $1 and the cost of a unit of capital is $3. The manager of this firm is interested in finding the following information:

a)    Type of returns to scale and the marginal products of labor and capital.

b)    The marginal rate of technical substitution of labor for capital. Graph the isoquant map.

c)    The firm is currently producing 100 units of output. Find the optimal cost-minimizing quantities of labor and capital. Graphically illustrate this optimal solution using isoquants and isocost lines. What is the minimum total cost?

d)    The manager now wants to know the K/L ratio to produce anyoutput level at the minimum total cost. Represent graphically the expansion path in the long run.

Answer 1

Given,

$Q=10K^{\frac{3}{4}}L^{\frac{1}{4}}$

unit cost of labor (P_L)=$1

unit cost of capital (P_k)=$3

(a). The calculation of returns to scale is given below:

Let us multiply labor and capital by a fraction λ

$Q(\lambda L,\lambda K)=10(\lambda K)^{\frac{3}{4}}(\lambda L)^{\frac{1}{4}}$

$Q(\lambda L, \lambda K)=10\lambda ^{\frac{3}{4}}K^{\frac{3}{4}}\lambda ^{\frac{1}{4}}L^{\frac{1}{4}}$

$Q(\lambda L, \lambda K)=10\lambda ^{\frac{3}{4}+\frac{1}{4}}K^{\frac{3}{4}}L^{\frac{1}{4}}$

$Q(\lambda L, \lambda K)=\lambda 10K ^{\frac{3}{4}}L^{\frac{1}{4}}$

substitute $10K ^{\frac{3}{4}}L^{\frac{1}{4}}=Q$

$Q(\lambda L, \lambda K)=\lambda Q$

Since the power of λ

is one there is constant returns to scale. It means if we multiply both factors by λ

then the total production also increases by the same fraction λ

so there is constant returns to scale.

Calculation of marginal product of labor:

$MP_L(marginal\space product \space of \space labor)=\frac{\delta Q}{\delta L}$

$MP_L=10\times\frac{1}{4}\times K^{\frac{3}{4}}\times L^{\frac{-3}{4}}$

$MP_L=\frac{10}{4}K^{\frac{3}{4}}L^{\frac{-3}{4}}$

Calculation of marginal product of capital:

$MP_K(marginal \space product\space of\space labor)=\frac{\delta Q}{\delta K}$

$MP_K=10\times\frac{3}{4}\times K^{\frac{-1}{4}}\times L^{\frac{1}{4}}$

MP_K=10\times\frac{3}{4}\times K^{\frac{-1}{4}}\times L^{\frac{1}{4}}

$MP_K=\frac{30}{4}K^{\frac{-1}{4}}L^{\frac{1}{4}} ​$

(b)

The marginal rate of technical substitution (MRTS) is given below:

$MRTS=\frac{\frac{10}{4}K^{\frac{3}{4}}L^{\frac{-3}{4}}}{\frac{30}{4}K^{\frac{-1}{4}}L^{\frac{1}{4}}}$

$MRTS=\frac{K^{\frac{3}{4}+\frac{1}{4}}}{3L^{\frac{1}{4}+\frac{3}{4}}}$

$MRTS=\frac{K}{3L}$

The graphical presentation of the isoquant map is given below:

According to the above figure, the x-axis measures the units of labor and the y-axis measures the units of capital. Q1 shows the iso-quant when the production level is 100, Q2 shows the iso-quant when the production level is 200, and the Q3 shows the iso-quant when the production level is 300.

(c)

At Q=100 the production function will become as given below:

$100=10K^{\frac{3}{4}}L^{\frac{1}{4}}$

$L^{\frac{1}{4}}=\frac{100}{10K^\frac{3}{4}}=\frac{10}{K^{\frac{3}{4}}}$

$L=\frac{10^4}{K^3}........................Equation\space 1$

The equation of the iso-cost line is given below:

$C(Total\space cost\space )=LP_L+KP_K$

$C=L+3K$

At the cost-minimizing level the following condition must satisfy:

slope of the iso-quant line

$MRTS=\frac{P_L}{P_K}$

$\frac{K}{3L}=\frac{1}{3}$

$K=L$

now, substituting the value of K=L in equation 1

$L=\frac{10^4}{L^3}$

$L^4=10^4$

$L=10$

and

$K=10$

Graphical presentation:

According to the above figure, the firm minimizes its total cost at point E where the iso-cost line (C=40) and iso-quant line (Q1=100) are tangent to each other. The firm employs 10 units of labor and 10 units of capital to minimize the cost.

(d)

$\frac{K}{L}\space ratio=\frac{\frac{10}{4}K^{\frac{3}{4}}L^{\frac{-3}{4}}}{\frac{30}{4}K^{\frac{-1}{4}}L^{\frac{1}{4}}}$

$\frac{K}{L}=\frac{1}{3}$

Since this is an example where the MRS will go on decreasing/diminishing, the expansion path will be, 

the IC(indifference curve) is curved in a convex way, which shows that the MRTS will go on diminishing.