Question

The production function for Superlite Sailboats, Inc., is

Q = 20K0.5L0.5

with marginal product functions

MPK=10L0.5K-0.5 and MPL=10K0.5L-0.5

a. If the price of capital is $5 per unit and the price of labor is $4 per unit, determine the expansion path for the firm.

b. The firm currently is producing 200 units of output per period using input rates of L=4 and K=25. Is this an efficient input combination? Why or why not? If not, determine the efficient input combination for producing an output rate of 200. What is the capital-labor ratio?

c. If the price of labor increases from $4 to $8 per unit, determine the efficient input combination for an output rate of 200. What is the capital-labor ratio now? What input substitution has the firm made?

Answer 1

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Answer on Question #73899-Economics - Microeconomics

The production function for Superlite Sailboats, Inc., is

$$Q = 20K^{0.5}L^{0.5}$$

with marginal product functions

$$MPK = 10L^{0.5}K^{-0.5} \quad \text{and} \quad MPL = 10K^{0.5}L^{-0.5}$$

a. If the price of capital is $5 per unit and the price of labor is $4 per unit, determine the expansion path for the firm.

b. The firm currently is producing 200 units of output per period using input rates of $L=4$ and $K=25$. Is this an efficient input combination? Why or why not? If not, determine the efficient input combination for producing an output rate of 200. What is the capital-labor ratio?

c. If the price of labor increases from $4 to $8 per unit, determine the efficient input combination for an output rate of 200. What is the capital-labor ratio now? What input substitution has the firm made?

Answer.

a) Expansion path is determined by the condition

$$\frac{MP_L}{MP_K} = \frac{w}{r}$$

So,

$$\frac{10L^{-0.5}K^{0.5}}{10L^{0.5}K^{-0.5}} = \frac{4}{5}$$ $$\frac{K}{L} = \frac{4}{5}$$ $$4L = 5K$$ $$L = 1.25K$$

b) It is not efficient input combination, as does not response with expansion path.

To find the efficient one let's solve the system of equations

$$\left. \begin{array}{l} 200 = 20K^{0.5}L^{0.5} \\ L = 1.25K \end{array} \right\}$$ $$200 = 20K^{0.5}(1.25K)^{0.5}$$ $$200 = 20K \times 1.12 \Rightarrow 200 = 22.4K \Rightarrow K = 8.93, L = 11.16$$

Capital- labor ratio is

$$\frac{8.93 \times 5}{11.16 \times 4} = 1$$

c)

$$\frac{K}{L} = \frac{8}{5}$$ $$\left. \begin{array}{l} 200 = 20K^{0.5}L^{0.5} \\ L = 0.625K \end{array} \right\}$$ $$200 = 20K^{0.5}(0.625K)^{0.5}$$ $$200 = 20K \times 0.79 \Rightarrow 200 = 15.8K \Rightarrow K = 12.65, L = 7.91$$

Capital- labor ratio is

$$\frac{12.65 \times 5}{7.91 \times 8} = 1$$

The firm increased the capital by 42% and reduced the labor by 29%.

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