Section5.1Profit maximization in perfect competition
Firms in perfect competition have no market power. They are unable to influence the price they sell for, and as a result, take the price in the market as given. For these firms, total revenue is expressed as
where the price is a constant value. In general, this value is expressed as the parameter \(P\text{,}\) though if the price in a particular market were, say, $5, it would take the value of an actual number, giving \(P = 5\) and \(TR = 5q\text{.}\)
How does this lack of market power influence a perfectly competitive firm’s profit maximizing choice of output? Our study of marginal analysis concluded that any firm - regardless of market power - will find it optimal to produce the level of output \(q^*\) where the firm’s marginal revenue equals its marginal cost. Any lower and the firm could increase its profit by producing more; any higher and the firm is overproducing.
This choice rule applies to any firm, but since no firm has market power, perfect competition is a special case! Competitive firms do not face a tradeoff between price and quantity: producing an additional unit of output will not change the price. This means the firm’s marginal revenue, the added revenue generated from another unit of output, is just equal to the price! If, for example, the price in the market is $10, then the first unit produced will sell for $10, the second will sell for $10, and the \(60^{th}\) unit will sell for $10, and so on. The constant price faced by a perfectly competitive firm renders the marginal revenue constant as well.
This can be observed on the graph of a competitive firm’s residual demand curve. Recall that the curve is horizontal because of the lack of market power. Since total revenue at a point is the shaded area under the curve at that point, the marginal revenue can be interpreted as the added area generated from increasing output by one unit. On the graph, you can see that when output increases by one, there is a vertical strip added to total revenue. This vertical strip is one unit of output wide, and \(P\) dollars tall, giving it an area of \(1\times P = P\text{.}\) Therefore, \(MR = P\) for every unit of output: total revenue keeps growing by \(P\) dollars for each unit added!
Figure5.1.1.Graph of residual demand in perfect competition. This shows that for every value of \(q\text{,}\)\(MR = P\text{,}\) since the shaded area under the curve \((TR)\) grows by \(P\) dollars for each additional unit. Here, \(P = 10\text{.}\)
This can also be observed on the graph of total revenue. In perfect competition, total revenue is linear and increasing. Since marginal revenue in general is defined as
it can be interpreted here as the slope of the total revenue function. The total revenue function is linear, and therefore has a constant slope. Moreover, this slope is exactly equal to \(P\text{!}\) As discussed, each additional unit of \(q\) will increase total revenue by \(P\) dollars. In this way, it must be true that for every value of \(q\text{,}\) marginal revenue is constant and \(MR = P\). The lack of market power shows itself in the constant slope of \(TR\text{,}\)\(MR = P\text{.}\) 1
Figure5.1.2.Graph of total revenue in perfect competition. The slope of total revenue is marginal revenue (\(MR = \frac{\Delta TR}{\Delta q}\)) and is constant. For every value of \(q\text{,}\) total revenue grows by \(P\text{,}\) showing that \(MR = P\) everywhere along the curve. Here, \(P = 10\text{.}\)
Now, we return to the profit maximization condition for the perfectly competitive firm. If the optimal level of output \(q^*\) can be founded wherever
\begin{equation*}
MR = MC
\end{equation*}
but for a perfectly competitive firm, it is true that for every value of \(q\)
\begin{equation*}
MR = P
\end{equation*}
then a perfectly competitive firm will maximize its profit when it chooses the \(q^*\) at which
\begin{equation*}
P = MC
\end{equation*}
The firm should produce up until the point where the added cost of another unit (\(MC\)) is equal to the added benefit of another unit (\(P\)). If the price in the market is $20, and producing another unit will cost a firm $12, this unit will be rational to produce. It will add more to the firm’s total revenue than it will to its cost, increasing its total profit. If the firm were to make $20 for selling a unit it cost them $35 to produce, this unit would not be rational to produce!
This can be seen in a graph of marginal revenue and marginal cost for the perfectly competitive firm. We now know that marginal revenue is constant and equal to the price, which gives a flat marginal revenue curve. We also know that under diminishing marginal product of labor, marginal cost is an increasing curve. Combining the two, we can see that if production is too low, then price exceeds marginal cost, and the firm is incentivized to produce more to maximize its profit; if production is too high, then marginal cost exceeds price and it should cut back. Finally, the optimal quantity \(q^*\) is found at the intersection of the two curves: where \(P = MC\text{.}\)
Figure5.1.3.Profit maximization in perfect competition. MC is increasing, while P = MR is flat and constant. At \(q^*\text{,}\) MC = P. If \(q\) is too low, \(P > MC\) and \(q\) should increase; if \(P \lt MC\text{,}\) then it is optimal to lower \(q\text{.}\)
The perfectly competitive firm’s profit maximization rule can be adapted from the general profit maximization rule in the following way:
If, for some level of output \(q\text{,}\)\(P > MC\text{,}\) then it is optimal to produce more output than \(q\).
If, for some level of output \(q\text{,}\)\(MC > P\text{,}\) then it is optimal to produce fewer than \(q\) output.
If, for some level of output \(q\text{,}\)\(P = MC\text{,}\) then \(q^*\) is the exact optimal number of units of output to produce!
We can see this rule in action with an example of cost data for a firm, reminiscent of Chapter 3:
\(q\)
\(TC\)
\(MC = \frac{\Delta TC}{\Delta q}\)
0
40
-
10
50
\(1\)
18
60
\(1.25\)
24
70
\(1.67\)
26
80
\(5\)
Marginal cost and total cost for a perfectly competitive firm.
When initially introduced, we didn’t specify (or care) whether the firm had market power or not. But now, let’s suppose the firm above competes in a perfectly competitive market. It has no market power, but faces a market price of \(P = 4\text{.}\) Since every unit it sells will generate $4 in revenue, the firm’s marginal revenue is equal to its price of $4. We can then apply the decision rule to determine how many units the firm should produce to maximize its profit.
If the firm produces the first 10 units, each unit will cost it an additional dollar to make, while each unit will bring in 4 dollars. This is rational: \(P > MC\text{!}\) The same is true for up to 18 and even 24 units. But if the firm considers upping its production to 26 units, each unit will now costs 5 dollars to make, while only generated 4 dollars per unit. This tradeoff isn’t worth it, so it would not maximize the firm’s profit to produce 26 units. Therefore, the firm should stop at 24: \(q^* = 24\text{.}\)
\(q\)
\(TC\)
\(MC\)
\(P = MR\)
0
40
-
4
10
50
\(1\)
4
\textcolor{blue}{\checkmark}
18
60
\(1.25\)
4
\textcolor{blue}{\checkmark}
\textcolor{blue}{\(q^* = 24\)}
\textcolor{blue}{70}
\textcolor{blue}{\(1.67\)}
\textcolor{blue}{4}
\textcolor{blue}{\checkmark}
26
80
\(5\)
4
\textcolor{red}X
Marginal cost and price for a perfectly competitive firm. The firm should produce as long as \(P > MC\text{,}\) and stop once \(P \lt MC\text{.}\) The optimal quantity is the highest quantity at which \(P > MC\text{.}\) Here, that occurs at \(q^* = 24\text{.}\)
We can confirm the profit maximization decision rule works by calculating profit directly. Recall that we define a firm’s profit (\(\pi\)) as the difference between its total revenue and its total cost:
\begin{equation*}
\pi = TR - TC
\end{equation*}
If \(\pi \lt 0\text{,}\) we say the firm makes a loss, because the firm incurs more cost than it takes in in revenue. If \(\pi = 0\text{,}\) we say the firm breaks even: the firm brings enough revenue to exactly cover its cost. The firm makes a positive profit anytime \(\pi > 0\text{.}\)
With knowledge of the price, we can compute the total revenue earned, and therefore the firm’s profit, at each level of \(q\text{.}\) For example, if the firm produces \(q = 0\text{,}\) the firm does not generate any revenue, but still has a total cost of 40. 2 If the firm produces \(q = 10\text{,}\) it generates a total revenue of 40, but incurs a total cost of 50. Its profit improves to \(\pi = -10\text{,}\) which, while still a loss, is a smaller loss than if it had produced 0. If the objective of the firm is to maximize its profit, the choice which does this is the same one we identified just above: at \(q^* = 24\text{,}\) the firm generates a revenue of 96, only has total cost of 70, and makes the most profit possible. 3
\(q\)
\(TC\)
\(MC\)
\(TR = 4Q\)
\(P = MR\)
\(\pi = TR - TC\)
0
40
-
0
4
-40
10
50
\(1\)
40
4
-10
18
60
\(1.25\)
72
4
12
\textcolor{blue}{\(q^* = 24\)}
\textcolor{blue}{70}
\textcolor{blue}{\(1.67\)}
\textcolor{blue}{96}
\textcolor{blue}{4}
\textcolor{blue}{26}
26
80
\(5\)
104
4
24
The profit maximizing quantity of output, \(q^* = 24\text{,}\) generates total revenue of 96, total cost of 70, and the optimal level of profit, 96 - 70 = 26.
We can take the same approach if output is a continuous variable, and can take any numerical value 0 and higher. This mirrors what we saw with the general model of marginal analysis. Our analysis is restricted when we use the table above, because it does not allow us to calculate profit if the firm produces 8 units of output, or \(15\frac{1}{2}\) units, or 400 units.
So, as we have done in other models, we can use a mathematical function to expand the model by allowing an infinite number of possible \(q\)’s. In a perfectly competitive model of profit maximization, the key mathematical function we need is the firm’s marginal cost function. That is, we need to know how costly additional units of output are at every stage in the production process. If, for example, a competitive firm had a marginal cost function of
\begin{equation*}
MC = 4q
\end{equation*}
then when the firm is producing 2 units of output, its marginal cost will be \(MC = (4)(q) = (4)(2) = 8\text{.}\) This means when the firm is already producing 2 units, additional units will cost 8 dollars each. If the firm is instead producing 10 units already, its marginal cost will be 40. As the firm produces more and more output, added units get more and more costly. (Why does this happen again? 4 )
If the firm’s marginal cost function is given by \(4q\text{,}\) and the firm faces a price of 80, how many units should it produce to maximize its profit? We can now use the condition that \(P = MC\) at the optimal choice, and solve the equation for \(q^*\text{.}\) At \(q^*\text{,}\)
\begin{equation*}
P = MC
\end{equation*}
\begin{equation*}
80 = 4q
\end{equation*}
\begin{equation*}
q^* = 20
\end{equation*}
The profit maximization condition, which is derived from our model of marginal analysis, guarantees this choice will generate the most profit for the firm. If the price in the market increased to 100, we can solve the new problem to find the new optimal quantity:
\begin{equation*}
P = MC
\end{equation*}
\begin{equation*}
100 = 4q
\end{equation*}
\begin{equation*}
q^{**} = 25
\end{equation*}
Figure5.1.4.We can find \(q^*\) where P = MC. MC is the upward sloping curve \(MC = 4q\text{,}\) and \(MR = P = 80\) gives the price. Here, \(q^* = 20\text{.}\)
Having only the marginal cost function does not give enough data to calculate profit. Above, if the price is 80, then the optimal quantity of output is 20 and the firm would generate total revenue of \((80)(20) = 1600\text{.}\) We know the firm will have a marginal cost of 80, but we do not know what the firm’s total cost is. We would need a total cost function to determine the profit. If \(TC = 2q^2 + 200\text{,}\) for example 5 , then when it maximizes its profit, the firm incurs a cost of \((2)(20)^2 + 200 = 1000\text{.}\) With this data, we can calculate that when maximizing its profit, the firm earns a profit of
We can visualize the profit maximization solution from this angle as well. With a price of 80, marginal cost of \(4q\text{,}\) and total cost of \(2q^2 + 200\text{,}\) we can plot all three relevant functions of output.
Total revenue: \(TR = 80q\) is linear and increasing.
Total cost: \(TC = 2q^2 + 200\) is increasing and convex, with a y-intercept at 200 (its fixed cost).
Figure5.1.5.TR and TC are both increasing functions. At \(q^*\text{,}\) the slope of total revenue (\(MR = P\)) is equal to the slope of total cost (\(MC\text{,}\) shown by the line tangent to \(TC\)).
Total profit is the difference between total revenue and total cost, and is the vertical distance between the two curves. At the two points where the curves intersect, \(TR\) and \(TC\) are the same: these are break even points 6 . To the far left and the far right are areas where total cost exceeds total revenue: these are the quantities of output where the firm makes a loss. This should make sense. If the firm produces too much, its total cost shoots up too sharply (from diminishing marginal product of labor, remember?); if the firm produces too little, it isn’t generating enough revenue to compensate for its fixed cost.
The optimal choice for the firm is the level of output, \(q^* = 20\text{,}\) where \(P = MC\text{.}\) But we know that marginal cost is the slope of total cost, and that price (which is equal to marginal revenue) is the slope of total revenue. Therefore, \(q^*\) occurs where the \(TR\) and \(TC\) curves have the same slope - where their tangent lines have the slope. Geometrically, this maximizes the distance between total revenue and total cost, maximizing the firm’s profit!
The profit function can be graphed directly as well. Since it can be expressed directly as
we can see it on the same graph. Notice that the function is a negative quadratic (negative sign in front of the \(2q^2\)) shifted down. Profit takes a value of zero at the two break-even points, dips negative to the far left and far right, and takes positive values in the middle \(q\) range - where total revenue exceeds total cost. Not coincidentally, the peak of the profit function is reached at the optimal quantity \(q^* = 20\text{,}\) where price equals marginal cost for the firm.
Figure5.1.6.The profit function (\(\pi\)) is shown above in below as the difference between \(TR\) and \(TC\text{.}\) Graphically, \(\pi\) is the vertical distance between the two curves.
In Depth5.1.7.The return of eventually diminishing marginal product of labor.
In Chapter 3, we explore the difference between diminishing marginal product of labor (Assumption P2) and eventually diminishing marginal product of labor (Assumption P2B). P2B includes both a period of specialization for the first several units of output and a period of eventual crowding out beyond a certain point. This leads to a production function which is convex then concave - and, importantly, to a total cost function that is concave then convex.
So which assumption should we make? In one sense, assumption P2B seems more realistic. It is reasonable to consider that specialization occurs in many real-world production contexts. However, assumption P2 is more straightforward to model. In most models, economists are willing to forgo assumption P2B in favor of assumption P2. But what do economists lose when they ignore specialization?
In the context of the firm’s profit maximization problem, now, we can better contextualize the distinction between these two assumptions. Under assumption P2, the firm’s total cost is convex everywhere, which leads to a concave profit function. This concave function (pictured below) guarantees that the spot where marginal revenue (slope of TR) equals marginal cost (slope of TC) is the peak of the firm’s profit function.
The only point where MR = MC, \(q^* = 20\text{,}\) corresponds to the peak of the firm’s profit function.
However, under assumption P2B, the total cost function is concave then convex. The S-shaped total cost curve cause the profit function to be S-shaped as well, as shown below. In general, this does not prevent us from finding the firm’s profit-maximizing level of output, \(q^*\text{.}\) But it does complicate things a bit. Now there are two levels of output where marginal revenue equals marginal cost. One such point is \(q^*\text{,}\) the firm’s optimal choice of output, which corresponds to the peak of the profit function. The second is \(q_1\text{,}\) and this corresponds to a valley in the profit function! If we are not careful in seeking out a point where MR = MC, we may accidentally find \(q_1\) and not \(q^*\text{.}\)
There are now two points where MR = MC: (1) \(q^* = 20\) corresponds to the peak of the firm’s profit function; (2) \(q_1\) corresponds to a valley of the firm’s profit function.
Ultimately, this profit function analysis suggests that ignoring specialization as an assumption - and losing the S-shape of the profit function - doesn’t cost much. It only eliminates potentially-confusing output choices like \(q_1\) which are some of the worst possible choices the firm has as its disposal.
