## Long run competitive equilibrium in an economy with production

### Basic theory

In the long run firms can enter and exit the industry.

Theory: A situation is a long run equilibrium if

• no firm in the industry wants to leave
• no potential firm wants to enter.

Implications: Given the definition of economic profit, the theory implies that in a long run equilibrium

• no existing firm makes a loss
• any potential firm that entered would make a loss

Assuming that the technology (and hence cost functions) of every firm are the same, and ignoring the discrete change that may occur in the firms' maximal profits when a firm enters, the theory thus implies that

in a long run equilibrium every firm's maximal profit is zero
or, equivalently,
price is equal to minimum average cost.
Terminology: the output at which LAC is minimal is the efficient scale of production. LAC at the efficient scale of production is thus the minimum average cost. In the following figure, y* is the efficient scale of production and p* is the minimum average cost.

Given this terminology, another implicaton of the theory is:

every firm produces at the efficient scale of production.
What determines the equilibrium number of firms? Given the equilibrium price (minimum average cost), the aggregate demand function Qd gives us the total amount Y* = Qd(p*) that must be produced in equilibrium. We know how much each firm produces in equilibrium (y*, the output equal to its efficient scale of production), so if we divide Y* by this amount we obtain the equilibrium number of firms.

### Changes in industry output may affect the cost function

In order to define a long run competitive equilibrium more precisely, we need to take account of the fact that changes in industry output may affect the firms' cost functions. In the long run, as all firms expand or contract, the change in the industry's demand for inputs may lead to input prices to change. (This is likely to be the case for any input for which the industry uses a significant fraction of the total amount of that input that is available in the economy.) Thus we should index the firms' cost function by industry output: TCY(y) denote the total cost of producing y units of output when the output of the industry is Y. If an increase in industry output leads to an increase in the prices of the inputs used then TC increases in Y: TCY'(y) > TCY(y) for all y whenever Y' > Y. A logical possibility is that TC is decreasing in Y, though this possibility seems unlikely to occur.

Some terminology:

### Definition of a long run competitive equilibrium

Denote by LACY(y) the long run average cost corresponding to TCY(y) (i.e. LACY(y) = TCY(y)/y) and by Qd the aggregate demand function. Then we can define a long run competitive equilibrium precisely as follows.

The long run competitive equilibrium when every firm's long run average cost curve is the same, given by LACY, is characterized by a price p*, an output y* for each firm, and a number n* of firms such that

p* is the minimum of LACn*y*
y* is the minimizer of LACn*y*
Qd(p*) = n*y*.
These conditions are interrelated: the variables p*, y*, and n* appear in each of them. Thus to solve for a long run equilibrium in general we need to solve three simultaneous equations in the three variables p*, y*, and n*.

In the case of a constant cost industry, in which LAC is independent of industry output, the three conditions reduce to

p* is the minimum of LAC
y* is the minimizer of LAC
Qd(p*) = n*y*.
These conditions have a very simple structure: the first one determines p*, the second determines y*, and the last determines n*, given p* and y*. Thus it is straightforward to find the long run equilibrium in a constant cost industry.

Given how the short run and long run cost curves are related, note that in a long run equilibrium we have:

p* = LACY(y*) = SACY,y*(y*) = LMCY(y*) = SMCY,y*(y*),
where SACY,y* and SMCY,y* are the short run cost curves when the aggregate demand is Y = n*y* and the firm's plant is optimal for producing y* units of output.

### The long run industry supply function

If the aggregate demand curve shifts (consumers' tastes change, the prices of other goods change, the population increases or decreases, ...) then the long run equilibrium changes. The path of the pairs (Y,p) (where Y is aggregate demand and p is price) traced out as demand changes is called the long run supply function.

#### Constant cost industry

In a constant cost industry, LAC is independent of industry output, so the long run supply function is horizontal, as in the following figure, which shows the effect of a shift in the aggregate demand curve from D1 to D2. In each case the long run equilibrium price is p* and the output of each firm is y*. When the demand is D1 the number of firms is n1*, and when demand is D2 the number of firms is n2*.

We can tell a dynamic story when the demand shifts. If it shifts to the right, for example,

• in the short run each firm produces more, and makes profit
• then more firms enter
• the short run supply (given the number of firms) therefore moves out
• the price falls, and each firm reduces its output again.

#### Increasing cost industry

In an increasing cost industry the cost curve shifts up as industry output increases, so the industry supply curve is upward-sloping. The following figure is drawn under the assumption that as the cost curve shifts up, the efficient scale for each firm remains the same, equal to y*. When the aggregate demand curve is D1 the equilibrium price is p1* and the number of firms is n1*; when the aggregate demand curve is D2 the equilibrium price is p2* and the number of firms is n2*.

Again we can tell a dynamic story: Initially each of n1* firms produces y* and the price is p1*. Suppose demand increases from D1 to D2.

• in the short run price rises, as each firm expands and moves up its short run supply function
• the profit induces more firms to enter
• input prices rise as the demand for inputs increases, so LAC rises
• in the new long run equilibrium there are n2* firms, each producing y* as before.
Examples and exercises on long run competitive equilibrium