Long Run Competitive Equilibrium

General setup

There are \(N\) identical consumers and \(M\) identical firms. The number of consumers is fixed, but \(M\) can change over time, as firms can freely enter and exit the market.

A commodity is traded in the market at price \(p\). Consumers and firms are both price takers.

The consumers each have income \(Y\). Their utility function over the numeraire good \(c\) and commodity \(q\) is:

\[u(c,q) = c + v(q)\]

The cost for a firm to produce \(q\) units of the commodity is \(c(q)\). Firms maximize their profit:

\[\Pi(q) = pq - c(q)\]

The market is said to be in long run equilibrium if the total quantity consumed by consumers equals the total quantity produced by firms, AND the profit of each firm is \(0\), so that there is no more entry and exit by firms.

Economic Insight

Firms make zero economic profit (profit in excess of opportunity cost) in the long run equilibrium of a competitive market with free entry and free exit.

We want to find:

  1. The long-run equilibrium price of the commodity \(p\)
  2. The quantity consumed by each consumer in the long-run equilibrium, \(q_d\)
  3. The quantity produced by each firm in the long-run equilibrium, \(q_s\)
  4. The number of firms in the long-run equilibrium, \(M\)

Equilibrium conditions

In the long run equilibrium of the market, three equations have to hold:

  1. The consumer’s first order condition
  2. The firm’s first order condition
  3. The market equilibrium condition (quantity demanded = quantity supplied)
  4. The zero profit condition (firms don’t want to enter or exit)

The consumer’s first order condition

The consumer solves:

\[\max_{q} ~ Y - pq + v(q)\]

The first order condition is:

\[\text{(Eq.1)} ~ ~ ~ ~ p = v^\prime(q_d)\]

The firm’s first order condition

The producer solves:

\[\max_{q} ~ pq - c(q)\]

The first order condition is:

\[\text{(Eq.2)} ~ ~ ~ ~ p = c^\prime(q_s)\]

The equilibrium condition

Total quantity demanded has to equal total quantity supplied:

\[\text{(Eq.3)} ~ ~ ~ ~ Nq_d = Mq_s\]

The zero profit condition

Firm profit has to be zero, so that no more firms want to enter or exit:

\[\text{(Eq.4)} ~ ~ ~ ~ pq_s - c(q_s) = 0\]

Solving the system of equations

The above conditions contain 4 equations: (Eq.1, Eq.2, Eq.3, and Eq.4), and 4 unknowns: (\(p\), \(q_d\), \(q_s\), and \(M\)). The long run equilibrium is the solution to this system of equations.

Example 1

A commodity is traded in a competitive market at price \(p\). Consumers and firms are price takers.

Consumers. There are 3,000 identical consumers each with income \(Y=100\). Each consumer has a utility function over numeraire consumption \(c\) and commodity \(q\) that is equal to:

\[u(c,q) = c + 10q - 0.5q^2\]

Firms. There are \(M\) identical firms each with cost function equal to:

\[c(q) = 32 + 0.5q^2\]

Firms can freely enter and exit the market, so the number of firms is flexible in the long-run.

  1. Find the price and total quantity in the long-run equilibrium of the market.
  2. How much does each consumer consume and how much does each firm produce?
  3. How many firms are there in the long-run equilibrium?

Answer.

Step 1: Write down the consumer’s optimization problem:

\[\begin{align} \max_{q} ~ 100 - pq + 10q - 0.5q^2 \end{align}\]

Step 2: Solve the consumer’s first order condition:

\[\begin{align} -p + 10 - q_d &= 0 \\ q_d &= 10 - p \end{align}\]

Step 3: Write down the firm’s optimization problem:

\[\begin{align} \max_{q} ~ pq - 32 - 0.5q^2 \end{align}\]

Step 4: Solve the firm’s first order condition:

\[\begin{align} p - q_s &= 0 \\ q_s &= p \end{align}\]

Step 5: Solve the zero profit condition for \(q_s\):

\[\begin{aligned} pq_s - 32 - 0.5q_s^2 &= 0 & \\ (q_s) q_s - 32 - 0.5q_s^2 &= 0 & \text{(plug in } p=q_s \text{ from firm FOC)} \\ q_s^2 - 32 - 0.5q_s^2 &= 0 & \\ 0.5q_s^2 &= 32 & \\ q_s^2 &= 64 & \\ q_s &= 8 & \end{aligned}\]

Step 6: Use the firm’s first order condition to solve for \(p\):

\[\begin{align} p &= q_s \\ &= 8 \end{align}\]

Step 7: Use the consumer’s first order condition to solve for \(q_d\):

\[\begin{align} q_d &= 10 - p \\ &= 10 - 8 \\ &= 2 \end{align}\]

Step 8: Use \(Q = Nq_d\) to solve for the total quantity:

\[\begin{align} Q &= N q_d \\ &= 3,000 \times 2 \\ &= 6,000 \end{align}\]

Step 9: Use \(Q = Mq_s\) to solve for \(M\):

\[\begin{align} Q &= M q_s \\ q_s &= \frac{Q}{q_s} \\ &= \frac{6,000}{8} \\ &= 750 \end{align}\]

Summary

To find the long run equilibrium of a competitive market with free entry and free exit, follow these steps:

  1. Write down the consumer’s optimization problem and solve the first order conditions to get \(q_d\) as a function of \(p\).
  2. Write down the firm’s optimization problem and solve the first order conditions to get \(q_s\) as a function of \(p\). At this point, it would also be helpful to write \(p\) as a function of \(q_s\).
  3. Write down the zero profit condition, plug in \(p\) as a function of \(q_s\), and solve for \(q_s\).
  4. Plug \(q_s\) into the firm’s first order condition to get \(p\).
  5. Plug \(p\) into the consumer’s first order condition to get \(q_d\).
  6. Use \(Q = Nq_d\) to get the total quantity.
  7. Use \(Q = Mq_s\) to get \(M\).
  8. Use the above to calculate any other relevant variable, like consumer utility or firm profit.