A firm's production function is given by $Q = K L$. The marginal products of labor and capital are, respectively, $M P _ { L } = K$ and $M P _ { K } = L$. The wage rate of labor is $w = \$ 10$ and the rental rate of capital is $r = \$ 20$. The firm wants to produce 1,800 units of output in the most efficient way possible. How much does the firm spend?
A)$2,000
B)$1,300
C)$1,200
D)$1,100

Question

Quizplus · Accepted Answer

To produce 1,800 units in the most efficient way, the firm needs to find the optimal combination of capital and labor that minimizes cost. We can use the formula for cost minimization, which is:

$MP_L/w = MP_K/r$

Substituting the given marginal products and prices, we get:

$K/10 = L/20$

Solving for K in terms of L, we get:

$K = 2L$

Substituting into the production function, we get:

$Q = K L = 2L^2$

To find the value of L that produces 1,800 units, we solve for L:

$2L^2 = 1,800$

$L^2 = 900$

$L = 30$

Substituting back into the equation for K, we get:

$K = 2L = 60$

So the firm spends:

$30 \times 10 + 60 \times 20 = \$1,200$ The most efficient way to produce 1,800 units of output is to choose the combination of capital and labor that minimizes the cost of production while still producing 1,800 units.

One way to approach this is to use the formula for the cost-minimizing combination of inputs:

$$\frac{M P _ { L } }{w} = \frac{M P _ { K } }{r}$$

Substituting in the given marginal products and prices, we get:

$$\frac{K}{10} = \frac{L}{20}$$

Simplifying, we get:

$$2K = L$$

We also know that:

$$Q = K L$$

Substituting in the cost-minimizing combination of inputs, we get:

$$Q = \frac{1}{2} L^2$$

To produce 1,800 units of output, we need:

$$1,800 = \frac{1}{2} L^2$$

Solving for L, we get:

$$L = 60$$

Using the equation we found earlier, we have:

$$K = \frac{1}{2} L = 30$$

So the cost of production is:

$$(30 \times \$20) + (60 \times \$10) = \$1,200$$

Therefore, the answer is C.

A Firm's Production Function Is Given By Q=KLQ = K LQ=KL

A Firm's Production Function Is Given By $Q = K L$