A Worked Long-Run Entry-and-Exit Adjustment

From an Initial Price to Long-Run Equilibrium

A competitive market can begin with a price that gives firms positive economic profit or persistent losses. Those outcomes create incentives for firms to enter or leave. To calculate where adjustment leads, combine each firm's cost information with market demand: the firm's cost data determine its output at a given price, while market demand determines how much buyers want to purchase at that price.

The calculations below use a simple model to make the adjustment explicit. Firms are identical and price-taking, entry and exit are unrestricted, and there are no changes in demand or production costs during adjustment. In the short run, the number of firms is fixed; over time, firms can enter or exit. Each market outcome is treated as a competitive equilibrium, so total quantity supplied equals quantity demanded. A firm chooses output to maximize profit, and it remains in the market in the long run only if it can cover all costs.

Definition: A firm's economic profit is total revenue minus total economic cost, including the opportunity cost of resources and any fixed cost. In a long-run competitive equilibrium with free entry and exit, active firms earn zero economic profit: price equals the minimum of average total cost. Zero economic profit does not mean that owners receive no return; it means their resources earn their best available alternative return.

The core calculation has three parts. First, use the firm's marginal cost to find its profit-maximizing output at the initial market price. Second, compare price with average total cost at that output to identify profit or loss. Third, use entry or exit and market demand to find the long-run outcome.

Why the Long-Run Price Covers Minimum Average Cost

For an interior, positive output choice, a competitive firm's profit is $\pi(q)=Pq-C(q)$, where $P$ is the market price and $C(q)$ is total cost. If cost is differentiable, the profit-maximizing condition is $P=MC(q)$. Zero economic profit at positive output also requires $P=ATC(q)$, because revenue $Pq$ must equal cost $C(q)$. These conditions together imply that price equals marginal cost and average total cost at the firm's selected output.

At an interior minimum of average total cost, marginal cost equals average total cost. To see why, differentiate $ATC(q)=C(q)/q$:

\frac{d\,ATC(q)}{dq} =\frac{qMC(q)-C(q)}{q^2} =\frac{MC(q)-ATC(q)}{q}.

At a differentiable interior minimum, the derivative is zero. Since $q>0$, this gives $MC(q)=ATC(q)$. Therefore, in the standard case in which the minimum is attained at positive output, the long-run zero-profit conditions meet at the minimum of average total cost. This conclusion relies on the assumptions stated above; it is not a rule for a firm with market power or for a market where entry is blocked.

Key result: In a long-run competitive equilibrium with active firms and free entry and exit, each firm's output is at the minimum of its average total cost, the price equals that minimum, and economic profit is zero. Market demand then determines total sales and, for identical firms, how many firms serve the market.

Worked Market Adjustments

Worked Example: Entry After Positive Economic Profit

Each firm has total cost $C(q)=q^2+16$, where $q$ is output per period and costs are dollars per period. Market demand is $Q_D=200-10P$. Initially, 20 identical firms operate. Find the initial equilibrium price and each firm's output and profit, then calculate the long-run equilibrium after entry.

Method and initial price. Marginal cost is $MC(q)=2q$. At a positive price, the firm's profit-maximizing output satisfies $P=MC$, so $q=P/2$. With 20 firms, market supply is $Q_S=20(P/2)=10P$. Equating market supply and demand gives $10P=200-10P$, so $20P=200$ and the initial price is $P=10$.

Each firm produces $q=10/2=5$, and total market output is $20(5)=100$. Demand at the price is $200-10(10)=100$, confirming market clearing.

Initial profit. A firm's total revenue is $Pq=10(5)=50$. Its total cost is $C(5)=5^2+16=25+16=41$. Thus its economic profit is $\pi=50-41=9$ dollars per period. Since firms earn positive economic profit, entry is encouraged.

Long-run calculation. Average total cost is $ATC(q)=q+16/q$. Its derivative is $1-16/q^2$, which is zero at $q=4$ for positive output. Average total cost there is $4+16/4=8$, its minimum. Hence the long-run price is $P=8$, and each active firm produces $q=P/2=4$. Demand at this price is $Q_D=200-10(8)=120$. The number of firms needed to supply 120 units is $120/4=30$.

Check. At $q=4$, revenue per firm is $8(4)=32$, and cost is $4^2+16=32$, so economic profit is $32-32=0$. The 30 firms supply $30(4)=120$, equal to demand. Entry takes the number of firms from 20 to 30; it does not change the assumed cost or demand schedule.

Common mistake avoided: The initial revenue of 50 minus variable cost of 25 is 25, but that is not economic profit. It is the amount remaining to cover fixed cost. Subtracting the fixed cost of 16 gives the correct profit, $25-16=9$.

Worked Example: Exit After Persistent Losses

Now use the same firm cost $C(q)=q^2+16$, but let market demand be $Q_D=240-10P$, with 50 firms initially operating. Calculate the initial competitive equilibrium and the long-run outcome if losses lead to exit.

Initial price and output. As before, $MC=2q$, so each firm supplies $q=P/2$. With 50 firms, $Q_S=25P$. Equilibrium requires $25P=240-10P$, so $35P=240$, giving $P=48/7$, approximately $\$6.8571$. Each firm produces $q=(48/7)/2=24/7$, approximately $3.4286$ units. Total output is $50(24/7)=1200/7$, approximately $171.4286$; demand is $240-10(48/7)=1200/7$, so the market clears.

Initial profit. Revenue per firm is $(48/7)(24/7)=1152/49$. Total cost is $(24/7)^2+16=576/49+784/49=1360/49$. Profit is therefore $1152/49-1360/49=-208/49$, approximately $-\$4.2449$ per firm per period. Price is below the minimum average total cost of $\$8$, so the firms incur economic losses. With persistent losses, some firms exit.

Long-run calculation. The minimum average total cost remains $\$8$, attained at $q=4$, so the long-run price is $P=8$. Demand at that price is $240-10(8)=160$. Each active firm supplies $q=8/2=4$, so the number of firms is $160/4=40$. This is a reduction from 50 firms.

Check. At the long-run output, each firm receives revenue $8(4)=32$ and incurs cost $4^2+16=32$, giving zero economic profit. The 40 firms produce $40(4)=160$, equal to market demand. The initial losses are consistent with exit: the lower initial price rises to the break-even price as the number of firms falls.

Common mistake avoided: Do not infer that a loss-making firm must immediately produce zero. A short-run output decision and a long-run exit decision are different. Here the firm produces positive output at the initial price under the specified supply condition, but persistent losses eventually prompt exit.

Worked Example: A Different Cost Structure and an Exit Adjustment

Suppose each firm has cost $C(q)=2q^2+32$, and market demand is $Q_D=280-8P$. There are initially 48 firms. Calculate the initial equilibrium, the loss per firm, and the long-run number of firms.

Initial equilibrium. Marginal cost is $MC=4q$, so a firm produces $q=P/4$. With 48 firms, market supply is $48(P/4)=12P$. Set this equal to demand: $12P=280-8P$, so $20P=280$ and $P=14$. Each firm produces $q=14/4=3.5$, and total output is $48(3.5)=168$. Demand is $280-8(14)=168$, confirming the initial equilibrium.

Loss at the initial price. Revenue per firm is $14(3.5)=49$. Cost is $2(3.5)^2+32=2(12.25)+32=24.5+32=56.5$. Thus profit is $49-56.5=-\$7.50$ per firm per period.

Long-run price and output. Average total cost is $ATC(q)=2q+32/q$. Its derivative is $2-32/q^2$, which is zero at $q=4$ for positive output. At that output, $ATC=2(4)+32/4=8+8=16$, the minimum. The long-run price is therefore $\$16$, and marginal-cost output at that price is $q=16/4=4$.

Demand at the long-run price is $280-8(16)=152$. The required number of firms is $152/4=38$. Thus exit reduces the number of active firms from 48 to 38 while the price rises from $\$14$ to $\$16$.

Check. At $P=16$ and $q=4$, revenue is $16(4)=64$, and cost is $2(4)^2+32=32+32=64$, so profit is zero. Total supply is $38(4)=152$, equal to demand. The calculation also confirms that the long-run output satisfies both $P=MC=16$ and $P=ATC=16$.

Common mistake avoided: Using the first example's marginal-cost rule $q=P/2$ would be incorrect here. The cost function changed, so its marginal cost is $4q$, and the correct output rule is $q=P/4$.

A Reliable Calculation Sequence

Keeping firm-level and market-level calculations separate helps prevent errors. The following sequence works for the scenarios in this tutorial when the stated competitive assumptions hold.

Find the firm's supply rule.
Differentiate total cost to obtain marginal cost. For a producing, price-taking firm with an interior optimum, set price equal to marginal cost. Check any stated operating or shutdown condition before using that rule.

Find the initial market-clearing price.
Multiply the output of one firm by the fixed number of firms to obtain market supply, then equate market supply and demand.

Calculate economic profit.
For each firm, calculate $Pq-C(q)$, including fixed cost. Positive economic profit invites entry; persistent economic loss invites exit.

Find the long-run break-even output and price.
Minimize average total cost. In the standard interior case, set price equal to minimum average total cost and use marginal cost to identify the firm's output there.

Use demand to find the number of firms.
Evaluate market demand at the long-run price, then divide total market quantity by output per firm. Check that total supply equals demand and that each firm's economic profit is zero.

These steps are a way to solve a specified market adjustment, not a claim that firms enter or exit in a particular numerical sequence. The final number of firms must satisfy both the market-clearing condition and the zero-economic-profit condition. With discrete firms, an exact market-clearing count may not always be possible; the examples here use values that produce whole numbers.

Common Traps and Interpretation

Leaving fixed cost out of profit. Revenue minus variable cost is not economic profit when fixed cost is positive. Use total revenue minus total cost.
Using market demand for one firm's output. A firm chooses its own output using the market price and its cost. Market demand gives total industry sales, not the quantity demanded from one firm.
Confusing entry with a change in firm costs. Under the assumptions used here, entry changes the number of suppliers, not the cost curve of each existing firm.
Stopping after finding the break-even price. A long-run market solution also needs market quantity at that price and the implied number of active firms.
Calling zero economic profit a zero return. It means revenue covers economic costs, including the opportunity cost of the firm's resources.

The main practical lesson is to keep the three quantities distinct: output per firm, total market quantity, and the number of firms. Use cost data for the first, demand for the second, and their ratio for the third. Then check the result from both sides: the market must clear, and active firms must have no incentive to enter or exit.

Check Your Understanding

For each problem, use the competitive assumptions stated in the question and check both the market-clearing condition and the firm's profit condition.

Firms have cost $C(q)=q^2+9$, market demand is $Q_D=180-10P$, and 18 firms initially operate. Find the initial equilibrium price and determine whether firms have an incentive to enter or exit. Then find the long-run price and number of firms. (Answer: Initial $P=180/19\approx \$9.4737$, $q=90/19\approx 4.7368$, with positive profit of approximately $\$9.2438$ per firm; long run $P=\$6$, $q=3$, and 40 firms.)
Firms have cost $C(q)=q^2+25$, demand is $Q_D=300-10P$, and 50 firms initially operate. Find the initial price and profit per firm, then the long-run number of firms. (Answer: Initial $P=60/7\approx \$8.5714$, $q=30/7$, profit $-325/49\approx-\$6.6327$ per firm; long run $P=\$10$, $q=5$, and 40 firms.)
Firms have cost $C(q)=q^2+4$, demand is $Q_D=120-5P$, and 20 firms initially operate. Find the initial price, output per firm, and profit, then calculate the long-run price and firm count. (Answer: Initial $P=\$8$, $q=4$, profit $\$12$ per firm; long run $P=\$4$, $q=2$, and 50 firms.)
Return to $C(q)=q^2+16$ and $Q_D=200-10P$. If 25 firms are operating, find the market-clearing price, output per firm, and profit per firm. Is entry or exit encouraged? (Answer: $P=80/9\approx \$8.8889$, $q=40/9\approx 4.4444$, profit $304/81\approx \$3.7531$ per firm; entry is encouraged.)
For the same cost and demand as in problem 4, suppose the long-run price is $\$8$. Calculate the quantity demanded, the output of each firm, and the number of firms needed for long-run equilibrium. (Answer: Quantity demanded $=120$, output per firm $=4$, and 30 firms; each firm earns zero economic profit.)

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