Question

Microeconomics

Posted 5 months ago

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Hufflepuff Industries produces badger sweaters and currently uses 20 units of labor and 30 units of capital. The marginal product of the $20 t h$ unit of labor is 100 sweaters, and the marginal product of the $30 t h$ unit of capital is 200 sweaters.
If the cost of each unit of labor $(w)$ is $\$ 20$ and this firm minimizes costs, what is the rental rate of capital $(r)$ ?
Choose 1 answer:
(A) $\$ 30$
(B) $\$ 50$
(C) $\$ 20$
(D) $\$ 40$
(E) $\$ 10$
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Answer from Sia

Posted 5 months ago

Solution

a

Marginal Product of Labor (MPL): The marginal product of the 20th unit of labor is 100 sweaters

b

Marginal Product of Capital (MPK): The marginal product of the 30th unit of capital is 200 sweaters

c

Cost of Labor (w): The cost of each unit of labor is $20

d

Cost Minimization Condition: For cost minimization, the ratio of the marginal product to the cost should be equal for both labor and capital. This can be expressed as $\frac{MPL}{w} = \frac{MPK}{r}$

e

Substitution: Substitute the given values into the cost minimization condition: $\frac{100}{20} = \frac{200}{r}$

f

Solving for r: Simplify the equation to find the rental rate of capital: $\frac{100}{20} = \frac{200}{r} \implies 5 = \frac{200}{r} \implies r = \frac{200}{5} = 40$

Answer

$40$

Key Concept

Cost Minimization in Production

Explanation

To minimize costs, a firm must equalize the ratio of the marginal product to the cost for both labor and capital. By using the given marginal products and costs, we can solve for the unknown rental rate of capital.

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