For more information on these topics, see Varian Chapter 19: Profit Maximization.
Profit maximization is a fundamental goal for firms. This chapter examines how firms make production decisions to maximize their profits, given market prices and their own cost structures.
When discussing profit maximization, it’s crucial to understand the difference between economic profit and accounting profit.
Accounting Profit is the profit typically reported on a company’s financial statements. It’s calculated as total revenue minus explicit costs (the direct, out-of-pocket expenses a firm incurs, such as wages, rent, and materials).
Economists are interested in decision-making, so we need a broader definition of profit which also includes implicit costs, which often come in the form of opportunity costs.
An opportunity cost is the value of the next-best alternative use for a resource. For example, the opportunity cost of studying for an exam might be the wages you could have earned working a part-time job during those hours. And if you use your savings to start a business, the opportunity cost is the interest you could have earned by investing that money instead. If a farmer decides to plant wheat, the opportunity cost is the profit they could have made by planting corn or another crop on that same land.
A firm’s profit is their total revenue minus their total costs:
\[\pi = TR - TC\]
Suppose a firm is a price-taker, meaning it is small relative to the entire market and it therefore cannot influence the market price. Let’s say that market price is $10. The firm’s total revenue will therefore be \(TR = 10 Q\), where \(Q\) is the quantity the firm decides to produce and sell.
Suppose the firm’s total cost is given by: \(TC = 100 + 5Q + 0.01Q^2\).
It wouldn’t be smart for the firm to set \(Q\) to be too small: the firm incurs a fixed cost of $100 to begin production (the fixed part of the total cost function), and if \(Q\) is too small, revenue doesn’t make up for those high initial costs.
It also wouldn’t be smart for the firm to set \(Q\) to be too large: the \(Q^2\) term in the total cost function means that total costs start to explode when the firm is over its capacity.
The best choice the firm can make for \(Q\) is its profit-maximizing quantity. One way to solve for that value is to find the maximum of the profit function:
\[\begin{align} \pi &= TR - TC\\ &= 10 Q - (100 + 5Q + 0.01Q^2)\\ &= -100 + (10 - 5) Q - 0.01 Q^2\\ &= -100 + 5 Q - 0.01 Q^2\\ \end{align}\]
This is a parabola facing down, so we know profit \(\pi\) is maximized when its derivative (slope of the tangent line) is equal to 0.
\[\begin{align} \frac{d \pi}{d Q} &= 5 - 0.02 Q\\ 0 &= 5 - 0.02 Q\\ 0.02 Q &= 5\\ Q &= 250\\ \end{align}\]
We found the profit-maximizing quantity for the firm, but how much profit can the firm expect to gain if they set Q to be 250?
\[\begin{align} \pi &= TR - TC\\ &= 10 (250) - (100 + 5(250) + 0.01(250)^2)\\ &= 2500 - 100 - 1250 - .01 (62500)\\ &= 525 \end{align}\]
The firm can make $525 in profit.
Let’s verify our answer by plotting the profit function.
library(tidyverse)
ggplot() +
stat_function(
fun = function(Q) 10 * Q - (100 + 5 * Q + 0.01 * Q^2),
color = "lightcoral"
) +
ylab("Profit") +
xlab("Q") +
xlim(0, 500) +
annotate(
geom = "point", x = 250, y = 525,
size = 4, color = "lightseagreen"
)
Finding the maximum of the profit function was pretty easy, but there’s a rule to make finding the profit maximizing Q even easier. It will also help you understand the crucial economic concept of marginal thinking, which connects to many other types of decision-making problems.
The rule is that, in order to profit maximize, a firm should produce at the level of output where marginal revenue (\(MR\)) equals marginal cost (\(MC\)). Marginal revenue is the amount that producing the last unit increased a firm’s total revenue. Marginal cost is the amount that producing the last unit increased a firm’s total cost.
Consider what happens if \(MR > MC\). In this case, producing one more unit will add more to revenue than to costs, increasing overall profit. The firm should keep expanding production as long as \(MR > MC\). Similarly, if \(MR < MC\), reducing production by one unit will decrease costs more than revenue, again increasing profit.
Therefore, profit is maximized only when \(MR = MC\). At this point, producing one more unit or one fewer unit would reduce profit.
Consider again \(TR = 10 Q\) and \(TC = 100 + 5Q + 0.01Q^2\). We’ll use the \(MR = MC\) principle to find the firm’s profit-maximizing level for \(Q\). The first step is to take the derivatives of the total revenue and total cost functions to find marginal revenue and marginal cost:
\[\begin{align} TR &= 10 Q\\ MR &= \frac{d TR}{dQ} = 10 \end{align}\]
The marginal revenue is a constant $10 per unit. This makes sense: each extra unit you sell earns you $P more in revenue, because you sell it at a price of $P. Now we’ll find the marginal cost function:
\[\begin{align} TC &= 100 + 5Q + 0.01Q^2\\ MC &= \frac{d TC}{dQ} = 5 + 0.02 Q \end{align}\]
Find the \(Q\) where marginal revenue is equal to marginal cost:
\[\begin{align} MR &= MC\\ 10 &= 5 + 0.02 Q\\ 5 &= 0.02 Q\\ Q &= 250 \end{align}\]
And we end up with the same answer as we did previously.
Here are some multiple-choice questions on the topic of marginal thinking. In each one, you’ll see the concept of setting \(MR = MC\) at work.
A firm has the following total revenue and total cost functions: \(TR = 100Q - Q^2\) and \(TC = 20 + 20Q + 2Q^2\) Where \(Q\) is the quantity produced. Note that here, TR does not have the form \(P \times Q\). This means that the firm is not a price-taker, and the quantity they supply the market helps to determine the price that the good is sold.
Find the profit-maximizing level of output by maximizing the profit function.
Verify your answer using the \(MR = MC\) rule.
Calculate the maximum profit and verify your answers using a plot of the profit function.
Consider a firm with a production function given by \(Q = 10L^{0.5}\), where \(Q\) is output and \(L\) is units of labor input. The price of output is $20 and the wage rate is \(w\).
Express the firm’s total revenue as a function of \(L\).
The marginal revenue product of labor is the amount that total revenue increases when labor increases by a little: it is \(\frac{d TR}{dL}\). Find an expression for the marginal revenue product of labor.
We can derive the firm’s demand for labor by using the \(MR = MC\) rule. That is, when \(MRPL > MC\), the firm should keep hiring workers, because they are adding to revenue more than they are adding to costs. The firm should stop when \(MRPL = MC\): at that point, hiring more workers will only start subtracting from the firm’s total profit. Set MRPL equal to the marginal cost of workers (that is, the wage rate \(w\)), and solve for \(L\). The result is the firm’s demand for labor.
If the wage rate is $10, how many units of labor will the firm demand?
Plot the firm’s demand for labor. As the wage rate increases, the firm profit maximizes by using (more/less) labor.
Note that for a price-taking firm, the marginal revenue product of labor is: \(\frac{dTR}{dL} = \frac{d PQ}{dL} = P \frac{dQ}{dL}\). Therefore, the MRPL is equal to the output price times the marginal product of labor, and the profit maximizing firm hires until \(P \times MP = w\). An interpretation is that if the price of the output increases, this firm will hire (more/less) workers. And if workers become more productive, this firm will hire (more/less) workers.