Barry Haworth
University of Louisville
Department of Economics
Economics 201

### Profit maximization and the Perfectly Competitive firm

Our first look at firm behavior comes within the context of perfect competition. What comes below is a step by step explanation of how perfectly competitive firms maximize their profits, both algebraically and graphically, and a discussion of our result.

Remember that, in perfectly competitive markets, no individual firm has any influence over the market price (since there are many firms and each is a small player in the overall market). Since each firm's product is identical to that of other firms (i.e. products are homogeneous), all firms face the same price.

While firms cannot individually influence the market price through their actions, they can collectively. Therefore, our starting point will be the market demand and supply curves. These are the same demand and supply curves from the earlier material on Consumer Theory (i.e. they do all the same tricks, like demand shifting when there's a change in income, that those other demand and supply curves did).

 (Market Demand) P = 100 - .078Qd (Market Supply) P = .02Qs + 2

Solving for equilibrium price and quantity, we get: P*= \$22 and Q*= 1000 units. These values represent the price that each firm will charge and the total number of units that will be produced overall.

A typical firm within this market has the following costs:

 (Total Costs) TC = q2 + 2q + 100 (Average Costs) AC = q + 2 + (100/q) (Marginal Costs) MC = 2q + 2

Let's note a few things about the first two equations before proceeding. In the TC equation, q2 + 2q represents the firm's variable costs and 100 represents the fixed costs. The AC equation is obtained by dividing the TC equation by q. This means that, in the AC equation, q + 2 are the average variable costs and 100/q are the average fixed costs.

1. Given these costs, how much should the firm produce?

The firm will always produce where the MC of a certain level of output equals the market price. That is, the firm will adjust its output level until P = MC. To find this output level, we set the MC equation equal to the equilibrium price:

P* = MC

22 = 2q + 2

q = 10

The firm will maximize its profits by producing 10 units. It is possible to characterize this firm and market level information with the following pair of demand and supply graphs. The graph on the right represents the market, while the graph on the left represents the firm.

The equilibrium price corresponds with where the market demand (DM) intersects the market supply (S). The firm accepts this price and decides how much to produce. This occurs where the firm's marginal cost curve (MC) crosses the firm's demand curve (Df). Note that the firm's demand curve is a horizontal line at the equilibrium price of \$22.

Another way to see whether the firm is maximizing profits is to assume that our P = MC rule isn't true. Suppose that the firm decides to test this rule by varying its output. If profits decline as we move away from where q = 10 (e.g. as we move between 8 and 12 units), then profits must be maximized in the row where P = MC.

 P q MC AC Profits 22 8 18 22.5 - 4.0 22 9 20 22.1 - 0.9 22 10 22 22 0 22 11 24 22.1 - 1.1 22 12 26 22.3 - 3.6

As the table makes clear, profits reach their highest level when the firm produces 10 units. Although it is true that the price equals both marginal and average cost in this row, this is only coincidence right now (in the short run). Profit maximization only necessitates that P = MC.

2. How do we calculate the firm's profits?

To find the firm's profits, we take one of two approaches (where TR = total revenue, which is (P x q)):

 TR - TC approach P - AC approach Profit = TR - TC Profit = (P - AC)q Profit = (22 x 10) - [(10)2 + 2(10) + 100] Profit = (22 - [(10) + 2 + (100/(10)]) x (10) Profit = 220 - 220 Profit = (0) x 10 Profit = 0 Profit = 0

The result is that this firm produces 10 units and makes zero economic profit. Graphically, we find this result by comparing P and AC. Recall that P comes from the action of the market (as a whole), and it is represented by the horizontal demand curve Df. AC is found by: (a) locating the firm's output level, (b) tracing a dotted line from this output level to the AC curve, and (c) from the point where the dotted line hits AC - go left, over to the vertical axis.

In the graph above, both P and AC are the same. We find TR by multiplying P and q, and TC by multiplying AC and q. By this method, the firm's TR and TC are represented by the same shaded area on the graph.

3. Why would the firm produce if it makes zero profit?

One way to answer this question is by seeing what happens if the firm shuts down. Then we'll compare the profits (or losses) under the two situations: producing vs. shut down. Recall that the firm has fixed costs of \$100. Assume that these fixed costs are all sunk (i.e. non-recoverable). If so, shutting down will cost the firm its \$100 in sunk costs. This is worse than making zero profits, so the firm will produce.

Supposing that the fixed costs are all recoverable, then the firm would be indifferent between producing and shutting down since both situations would involve making zero profit. In a lot of introductory economic analysis, however, fixed costs are implicitly assumed to be 100% sunk.

The important thing to remember here is that these profits are economic profits, not accounting profits. To see why this is important, consider how economic profits and accounting profits are calculated:

Economic profit = Actual revenue - (Actual costs + Opportunity costs)

Accounting profit = Actual revenue - Actual costs

While zero accounting profit would be undesirable, zero economic profit is not. A person could work all day to make \$1 in accounting profits and be very unhappy since that person could probably do better in some other money-making activity (i.e. the next best alternative occupation). By including opportunity cost, economic profit accounts for things like the value of one's time in producing a good or service.