Why the Same Demand Shift Has Two Effects
A rise in demand for a product can increase its market price quickly, while the number of firms takes time to adjust. The immediate price rise changes the output and profit of firms already in the market. If that profit attracts entry, the market then moves toward a different long-run outcome. A demand decrease follows the reverse pattern: existing firms first face a lower price, and persistent losses can lead to exit.
The distinction matters because a short-run equilibrium and a long-run equilibrium answer different questions. In the short run, firms already in the industry respond to the new price, but the number of firms is fixed. In the long run, entry or exit is possible. This tutorial traces both stages in a constant-cost industry, where changes in industry size do not change firms' input prices or cost curves.
A Benchmark Firm and Its Industry
To make the adjustment concrete, suppose identical competitive firms have the cost function \(C(q)=100+q^2\), where \(q\) is a firm's output per period and money is measured in dollars. The fixed cost is 100, and variable cost is \(q^2\). The firm chooses output to maximize economic profit, taking the market price \(P\) as given. For a positive price, its profit-maximizing output satisfies \(P=MC\), where \(MC=2q\); therefore \(q=P/2\). This supply rule applies when the firm operates, and it is consistent with the shutdown condition because average variable cost is \(q\), whose minimum is zero.
The firm's average total cost is \(AC=100/q+q\). Its minimum occurs at \(q=10\): differentiating gives \(dAC/dq=-100/q^2+1=0\), and the second derivative \(200/q^3\) is positive. At that output, \(AC=20\) and \(MC=20\). Thus the long-run competitive price is 20, each firm produces 10, and each firm earns zero economic profit.
The model also gives a useful short-run industry supply rule. With \(N\) identical firms, each producing \(P/2\), market supply is \(Q^S=N(P/2)\) for positive prices. This is the market supply of the existing firms, not the long-run supply after entry or exit has occurred.
Worked Example: A Demand Increase, Then Entry
An industry has 100 firms, each with cost function \(C(q)=100+q^2\). Initial market demand is \(Q^D=2000-50P\). Demand then rises to \(Q^D=2600-50P\). Find the initial equilibrium, the short-run changes in price, output, and profit, and the long-run number of firms.
Method. First hold the number of firms at 100 and equate demand with their combined supply. Then allow entry and use the zero-economic-profit long-run price. This avoids assuming that entry occurs immediately.
Initial equilibrium. Each firm's supply is \(q=P/2\), so the 100 firms together supply \(Q^S=50P\). Equating initial demand and supply gives \(50P=2000-50P\), hence \(100P=2000\) and \(P=20\). Market output is \(Q=50(20)=1000\), or \(q=1000/100=10\) per firm. Each firm earns \(20(10)-(100+10^2)=200-200=0\).
Short run after the increase. The number of firms remains 100, so supply is still \(50P\). Equilibrium requires \(50P=2600-50P\), giving \(P=26\). Market output is \(Q=50(26)=1300\), and each firm produces \(q=1300/100=13\). Its profit is \(26(13)-(100+13^2)=338-269=69\) dollars per period. The higher price raises each firm's output and creates positive economic profit.
Long run. Positive profit encourages entry. In this constant-cost industry, entry does not change the firms' cost curves, so the price returns to the minimum average cost, \(P=20\). At that price the shifted demand is \(Q^D=2600-50(20)=1600\). Each firm produces 10, so the long-run number of firms is \(N=1600/10=160\). Check: each firm earns \(20(10)-(100+10^2)=0\). The demand increase ultimately raises industry output and the number of firms, but not the price or output of each firm.
Why the Long-Run Price Returns to Minimum Average Cost
The result follows from profit maximization and the opportunity for entry and exit. For an operating competitive firm, profit is \(\pi=Pq-C(q)\). At an interior profit-maximizing output, price equals marginal cost: \(P=MC\). In a long-run equilibrium with free entry and exit, economic profit is zero, so \(P=AC\) at the firm's chosen output. Combining these conditions gives \(MC=AC\). For the usual U-shaped average-cost curve, marginal cost crosses average cost at its minimum. Therefore the long-run price equals minimum average cost.
Here is the same reasoning as a sequence of conditions:
For an operating price-taking firm with an interior optimum, \(P=MC\).
With free entry and exit, a long-run equilibrium cannot sustain positive economic profit or persistent economic loss; thus \(P=AC\) for the representative firm.
At the minimum of a differentiable average-cost curve, \(dAC/dq=0\). Since \(dAC/dq=(MC-AC)/q\), this means \(MC=AC\), so the long-run price is minimum average cost.
The derivation assumes competitive firms, free entry and exit, and a stable cost environment. The constant-cost assumption is especially important: it ensures that expansion or contraction of the industry does not change the input prices or cost curves that determine minimum average cost. If industry expansion raises input prices, the long-run price need not return to its original level.
Worked Example: A Demand Decrease, Then Exit
An industry has 100 firms with \(C(q)=100+q^2\). After a demand decrease, market demand is \(Q^D=1200-20P\). Find the short-run price, each firm's output and profit, and the long-run number of firms.
Method. Keep the existing 100 firms in the short-run supply calculation. Then, if the resulting loss persists, use exit and the zero-profit price to find the long-run firm count.
Short-run price and output. Industry supply with 100 firms is \(Q^S=50P\). Equilibrium satisfies \(50P=1200-20P\), so \(70P=1200\) and \(P=120/7\), or approximately 17.1429 dollars per unit. Each firm's output is \(q=P/2=60/7\), or approximately 8.5714 units.
Short-run profit. Using the exact output, revenue is \(Pq=(120/7)(60/7)=7200/49\). Cost is \(100+(60/7)^2=100+3600/49=8500/49\). Therefore profit is \(7200/49-8500/49=-1300/49\), approximately \(-26.53\) dollars per firm per period, rounded to the nearest cent. These losses create pressure for exit.
Long run. The zero-profit price remains \(P=20\), and each remaining firm produces 10. At that price, demand is \(1200-20(20)=800\). The long-run number of firms is \(800/10=80\). Check: market output is 800, and each of 80 firms produces 10; each earns \(20(10)-(100+10^2)=0\). The price and firm output recover, while the number of firms falls below its initial level of 100.
Tracing the Stages in a Constant-Cost Industry
The sequence below summarizes how to solve this class of problem. The first calculation uses the existing firms' supply; the later calculation uses the long-run zero-profit condition. Keeping those stages separate prevents a common error: using the long-run result to describe the immediate response.
Identify the number of firms, their cost conditions, and initial market demand. If the initial market is in long-run equilibrium, verify that price equals minimum average cost.
Aggregate the existing firms' supply and solve for the new short-run market price and quantity after the demand shift.
Find its output from its supply rule, then calculate economic profit as revenue minus total cost. Positive profit invites entry; a persistent loss invites exit.
Find the long-run price from minimum average cost, evaluate the shifted demand at that price, and divide market output by each firm's long-run output to find the number of firms.
Worked Example: A Different Cost Curve and a Larger Industry
Suppose each firm instead has cost function \(C(q)=64+q^2\), and 50 firms are initially in long-run equilibrium. Market demand shifts to \(Q^D=1600-50P\). Determine the short-run price, output and profit per firm, then the long-run number of firms.
Method. Find the firm's minimum average-cost point to establish the long-run benchmark, then calculate the immediate equilibrium with 50 firms fixed.
Long-run benchmark. Average cost is \(AC=64/q+q\). Setting its derivative \(-64/q^2+1\) equal to zero gives \(q=8\). At that output, \(AC=64/8+8=16\), so the initial long-run price is 16. Initial market output is \(50(8)=400\).
Short-run adjustment. Marginal cost is \(2q\), so each firm supplies \(q=P/2\), and 50 firms supply \(Q^S=25P\). The new equilibrium satisfies \(25P=1600-50P\), giving \(P=64/3\), or approximately 21.3333 dollars. Each firm produces \(q=32/3\), approximately 10.6667 units. Its revenue is \((64/3)(32/3)=2048/9\). Its cost is \(64+(32/3)^2=64+1024/9=1600/9\). Profit is therefore \(2048/9-1600/9=448/9\), or approximately 49.78 dollars per firm per period, rounded to the nearest cent.
Long-run adjustment. Entry brings price back to 16, and each firm again produces 8. Shifted demand at that price is \(1600-50(16)=800\), so the long-run number of firms is \(800/8=100\). Check: \(100\) firms producing \(8\) each supply \(800\), and each earns \(16(8)-(64+8^2)=128-128=0\).
Common Mistake: Mixing Short-Run and Long-Run Supply
A tempting but incorrect shortcut is to say, “The long-run price is 20, so the price immediately after a demand increase stays at 20.” That uses the long-run entry-and-exit result before entry has changed the number of firms. In the first example, this shortcut would put the post-shock price at 20 and output at 1,600 immediately. But with only the original 100 firms, short-run supply at price 20 is \(50(20)=1000\), not 1,600. The short-run equilibrium is instead \(P=26\), with market output 1,300. Only after entry does price return to 20 and output rise to 1,600.
For a decrease in demand, avoid the reverse error of assuming firms exit before calculating their short-run losses. First solve with the existing number of firms. The resulting price and profit indicate the direction of the later adjustment; they do not describe the final number of firms.
Check Your Understanding
For each problem, keep short-run and long-run calculations separate. Money is in dollars per period, and quantities are units per period.
- A firm has \(C(q)=36+q^2\). At the long-run competitive output, what are its output and price? (Answer: \(q=6\), \(P=12\).)
- An industry with 60 firms has \(C(q)=100+q^2\) per firm. After a demand increase, \(Q^D=2400-40P\). Find the short-run price, output and profit per firm, then the long-run number of firms. (Answer: Short run \(P=240/7\), \(q=120/7\), profit \(9500/49\) dollars; long run \(P=20\), \(q=10\), \(N=160\).)
- After a demand decrease, an industry has 80 firms with \(C(q)=100+q^2\) per firm and demand \(Q^D=600-10P\). Find the short-run price and profit per firm, then the long-run number of firms. (Answer: Short run \(P=12\), \(q=6\), profit \(-64\) dollars; long run \(P=20\), \(q=10\), \(N=40\).)
- An industry has 50 firms with \(C(q)=25+q^2\) per firm. After a demand increase, \(Q^D=900-30P\). Find short-run price and profit per firm, and the long-run number of firms. (Answer: Short run \(P=180/11\), \(q=90/11\), profit \(5075/121\) dollars; long run \(P=10\), \(q=5\), \(N=120\).)