Examples of long run competitive equilibrium

Procedure

To find a long run competitive equilibrium in a constant cost industry we need to

find the minimizer of the LAC, which is the output of each firm in a long run competitive equilibrium
find the minimum of the LAC, which is the long run equilibrium price
add together the consumers' demand functions to get the aggregate demand
divide the the aggregate demand at the equilibrium price by the output of each firm to get the number of firms.

Example

Each firm in an industry has the long run cost function

TC(y) = 0 if y = 0

1000 + 10y² if y > 0.

The aggregate demand function for the output of the firms is Q = 5000

2p. Find the long run equilibrium price, number of firms, and output of each firm.

We have LAC(y) = 1000/y + 10y. This is minimized where the derivative is zero: 1000/y² + 10 = 0, or y = 10. Thus output per firm is 10. [I'm assuming that this is in fact a minimum, not a maximum; it is, since LAC is U-shaped.] [You can alternatively find the minimizer of LAC by equating LAC and LMC: 1000/y + 10y = 20y.]
The minimum of LAC is LAC(10) = 200. Thus the long run equilibrium price is 200.
Aggregate demand at the price 200 is 5000400 = 4600, so the number of firms is 4600/10 = 46.

Example

Each firm in an industry has the cost function

0 if y = 0

TC(y) = 12+8y if 0 < y 6

if y > 6.

(It has a capacity of 6: it simply cannot produce more than 6 units.) Aggregate demand is given by Q_d(p) = 72

6p.

Find the long run equilibrium price, output of each firm, and number of firms.

We have

LAC(y) = 12/y+8 if 0 < y 6

if y > 6.

This function is shown in the following figure. It is decreasing from 0 to 6, and hence attains a minimum at y = 6. Hence the long run equilibrium output of each firm is 6.
The minimum of LAC is LAC(6) = 10. Hence the long run equilibrium price is 10.
The aggregate demand at the price 10 is Q_d(6) = 36, so the long run equilibrium number of firms is 36/6 = 6.

Example

The total cost function is TC(y) = 2y for each firm. (Note that this implies that the technology has CRTS.) Aggregate demand is Q_d(p) = 100

4p.

LAC(y) = 2, so every output minimizes LAC. Thus the output of each firm is indeterminate. (All outputs are efficient for the firm: the LAC does not have a unique minimum.)
The minimum of LAC is 2, so the long run equilibrium price is 2.
The aggregate demand at the price 2 is Q_d(2) = 92. Since the equilibrium output of each firm is indeterminate, the number of firms is also indeterminate; we know only that the aggregate output of the firms is 92.

Example: the effect of an excise tax

Each firm in an industry has the same U-shaped LAC. The minimum of LAC is $20, which is attained at the output of 50. Aggregate demand is Q = 1000

10p. What is the effect of an excise tax of $10 per unit on the long run equilibrium? How much money does the tax raise?

Before the tax is imposed the equilibrium price is $20 and the equilibrium output of each firm is 50. Total demand at the price $20 is 800, so that there are 16 firms.

The tax raises LAC(y) by 10 for every value of y. Thus the equilibrium price increases to $30 and the output of each firm remains the same. Total demand at the price $30 is 700, so the number of firms decreases to 14.

The amount of money the tax raises is ($10)(700) = $7000.

Example: the effect of an excise tax

Each firm in an industry has LAC(y) = y²

200y + 10,100. Aggregate demand is Q_d(p) = 4000

10p. Find the long run equilibrium.

LAC is minimized where 2y 200 = 0, or y = 100. Thus the long run equilibrium output of each firm is 100.
The minimum of LAC is LAC(100) = (100)² 20,000 + 10,100 = 100. Thus the long run equilibrium price is 100.
The aggregate demand at the price 100 is Q_d(100) = 3000, so there are 3000/100 = 30 firms.

Now suppose a $10 excise tax is imposed on each unit the firm sells. Then, as in the previous example, the firm's LAC rises by $10 at each output, so that the price rises by $10, to $110, the output of each firm stays the same, and demand falls to 2900, so there are 29 firms.

The excise tax therefore raises (29)($10)(100) = $29,000. Each firm pays $1000.

Example: the effect of a lump sum tax

A lump sum tax adds a fixed amount to TC, independent of output. Thus it raises LAC everywhere, but less for larger outputs. Thus the output at which LAC is smallest increases, and the minimal value of LAC increases too. Hence the equilibrium price increases and the output of each firm increases. The change in LAC that the tax causes is shown in the following figure, in which p* and y* are the original equilibrium price and output per firm and p' and y' are the post-tax equilibrium price and output per firm.

Now compare the effect of this lump sum tax with that of an excise tax that raises the equilibrium price by the same amount. Now, an excise tax does not affect output per firm: each firm produces y* after the tax, as it did before the tax. Thus on each unit sold the excise tax yields p' p*. On the other hand, on each unit sold the lump sum tax raises p' LAC(y'), which is less than p' p*. Given that the post-tax price is the same in both cases, so too is the aggregate demand, so that the excise tax raises more revenue than the lump sum tax.

Looking at this result differently, if an excise tax and a lump sum tax raise the same revenue then the lump sum tax must increase price by more than does the excise tax.

LAC(y) =	12/y+8	if 0 < y 6
		if y > 6.