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Say a firm is facing a budget line as stated below:($K$ stands for capital and $L$ stands for labor)

$$200K + 100L = 1000$$

Under this circumstance, a rational firm will look for ways to maximize its total production so as to lower its average total cost.

The current rental rate for capital is 200 dollars, and the going wage rate is at 100 dollars.

Now, assume that the marginal product of 1 unit of capital is $MPK_1$. And as the firm moves from ($K=0$, $L=10$) to ($K=1, L=8$), the number of outputs increases by $MPK_1$, and the number of outputs it forgoes should equal to the marginal product at the output level of 9 labors PLUS the marginal product at the output level of 10 labors, which could be denoted as ($MPL_9+MPL_{10}$).

My textbook says that if $\frac{MPK_1}{200} > \frac{MPL_8}{100}$, the firm will choose to purchase more capital. But the problem is, instead of $MPL_8$, the $MPK_1$ should be compared against ($MPL_9+MPL_{10}$), since the forgone product is ($MPL_9+MPL_{10}$) as the firm shifts from ($K=0$, $L=10$) to ($K=1$, $L=8$), right?

There's one more question I'd like to ask. My textbook also has this equality: $\frac{100}{MPL} = \frac{200}{MPK} = MC$ when the firm has maximized its profit. But I don't understand. The total amount of money spent should remain constant under the constraint of the budget line.

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