5.6 Industry Supply

Let \(q\) be a firm’s supply and let \(Q\) be the industry supply. We’ll assume that all firms are identical, so \(Q = nq\) where \(n\) refers to the number of firms operating in the market.

Also suppose:

• Market demand is given by: \(P = 100 - Q\)
• Each firm’s costs given by: \(TC = q^2 + 25\)
• There is free entry/exit (firms can freely enter or exit the market)

1. Derive the Firm’s Supply Curve

   Show that each firm’s supply curve is \(P = 2q\). Hint: recall that a firm will supply output up to the point where its marginal cost (MC) equals the marginal benefit (MB), which is the market price \(P\).

Answer:

2. Determine the Number of Firms in Long-Run Equilibrium

   Hints: In a long-run equilibrium with free entry and exit, two conditions must be satisfied:

   • Market Clearing: Total supply equals total demand.
   • Zero Economic Profit: Each firm earns zero economic profit, meaning total revenue (TR) equals total cost (TC). This occurs because if firms were earning positive economic profits, new firms would enter the market, increasing industry supply and driving down the market price until profits are eliminated. Conversely, if firms were incurring losses, some would exit the market, reducing supply and increasing the price until losses are eliminated. In equilibrium, firms have no incentive to enter or exit.

Answer:

\[\begin{align} &\text{Firm Supply:}\\ &q = \frac{P}{2}\\ &\text{Industry Supply:}\\ &Q = nq = \frac{nP}{2}\\\\ &\text{Demand:}\\ &P = 100 - Q\\ &\text{So market clearing gives us:}\\ &\text{Zero profit gives us:}\\ &\text{Solve for n:}\\ \end{align}\]

3. Report the market price and total quantity produced in equilibrium.

Answer: