Consider a competitive firm that produces bots. Labor (L) and capital (K) are the only two inputs of production; each unit of labor is paid the market wage (w), and each unit of capital is rented at the rental price of capital rr). Output (Y) is therefore a function of labor and capital, orY=f(K,L)Y and is sold at the market price (P).

The goal of this firm is to maximize profit given the price of bots, the wage rate, the rental rate of capital, and production technology by choosing its labor and capital inputs.

Fill in the following equations with the firm's profit maximization problem:

Profit =?

Revenue - Labor Cost or Labor Cost + Capital Cost - Revenue or Revenue - Capital Cost or Revenue - Labor Cost - Capital Cost.

Profit-maximizing firms must determine how renting an additional unit of capital will affect profit. In order to complete this, the firm can compare the change in revenue from renting an additional unit of capital against the cost of that extra unit of capital. The additional output produced by an extra unit of capital is equal to the marginal product of capital (MPK). Because each bot sells for the market price (P), the change in revenue from renting an additional unit of capital isY/P or P*MPK or MPK/P or P*Y/MPK or P/MPK, and the change in cost is r or r*MPK or P or r*Y.

Firms will continue to hire workers until the change in profit is equal to zero or positive or negative. Given this condition, which of the following equations correctly describes the profit-maximizing capital choice of the firm?

1. r=MPK/P
2. MPK=r/P
3. MPK=r*P
4. MPK=P/r

Demonstrate your understanding of the previous concepts in the following example. Suppose the price of a bot is equal to $2, and the market rental price is $18. Moreover, suppose the marginal product of capital for this firm is equal to the following:

MPK = 14?K

On the graph, use the blue line (circle symbol) to show the firm's demand for capital. Then, place the black point (plus symbol) on the graph to indicate the real rental price and the profit-maximizing quantity of capital.

Note: Dashed drop lines will automatically extend to both axes.

Graph below:

20 UNITS OF OUTPUT (Bots) PER UNIT OF CAPITAL 18 O 16 Capital 14 + 12 Real r, capital 10 8 B 10 12 14 16 18 20 UNITS OF CAPITAL P W