Download Calculus Worked-Out Problem 14.2: The Problem Consider again

Calculus Worked-Out Problem 14.2:
The Problem Consider again the pizza market in Chicago. Assume the daily demand for pizza
is Qd = 32,900- 600P, where P is the price of a pizza. The daily costs for a pizza company are the
same as in Calculus Worked-Out Problem 14.1. They include $845 in fixed costs and variable
costs equal to VC = 5Q + Q2/80, where Q is the number of pizzas produced in a day. Suppose
that in the long run, there is free entry into the market and the fixed cost is avoidable. What are
the long-run market equilibrium price and quantity? How many firms are active, and how much
does each produce?
Now suppose that demand doubles to Qd =65,800 - 1,200P. If, in the short run, the number of
firms is fixed (so that neither entry nor exit is possible) and fixed costs are sunk, what is the new
short-run market equilibrium? What is the new market equilibrium in the long run?
The Solution Recall from Calculus Worked-Out Problem 14.1that the efficient scale of
production is 260 pizzas per day, and that ACmin equals $11.50, so that the long-run market
supply curve is horizontal at a price of $11.50.
Step 1: Finding the initial long-run competitive equilibrium. In a long-run equilibrium, the
market price equals ACmin, so the price in the initial long-run equilibrium must be $11.50. We
next determine how many pizzas are produced and sold each day in this equilibrium using the
demand function. When the price is $11.50, the total quantity demanded is Qd=32,900 600(11.50) = 26,000 pizzas per day. Finally, in a long-run equilibrium, each active firm produces
260 pizzas per day, its efficient scale. That means it takes 100 active firms to produce the 26,000
pizzas that are bought and sold in the initial long-run equilibrium.
Step 2: Finding the new short-run competitive equilibrium. Now suppose that demand doubles.
In the short run, the number of active firms is fixed at 100. To find the short-run market supply,
we need to first find the supply function for each of these individual firms. In the short-run, their
fixed costs are sunk. Each of the firms therefore has the same supply function that we solved for
in Calculus Worked-Out Problem 9.3, in the case with no avoidable fixed cost:
( )
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}
Since there are 100 such firms, the short-run market supply function is:
( )
{
}
To find the new short-run competitive equilibrium, we equate supply and demand and solve for
the equilibrium price:
The solution is P=$16.50. The total number of pizzas produced and sold each day in this shortrun equilibrium is 46,000 and each active firm sells 460 pizzas. Since the price is above ACmin =
$11.50, the active firms each make a positive profit.
Step 3: Finding the new long-run competitive equilibrium. In the long run, the price falls back to
ACmin = $11.50. Since demand has doubled, the total numbers of pizzas produced and sold must
double to 52,000 a day. And since each active firm is again producing at its efficient scale of 260
pizzas a day, the number of active firms doubles to 200.