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Exercise: Price control

Your city council voted to set a price ceiling on the market for good X. The price ceiling is set to be equal to 75% of the current equilibrium price. 

The current market demand can be modeled by the equation:
 
The market supply curve corresponds to the following equation:
 
Using the given information, find:
  • the new price and the new amount of quantity sold 
  • the shortage
  • a tax per unit of good sold that would result in the same quantity that is available at the price ceiling
    • Derive an equation for the new market supply curve with the tax.
First, let's derive the equilibrium price and quantity for this market. Set the demand curve equal to the supply curve.

Now that we know the original price (i.e Pe), we can derive the price ceiling and the quantity sold at that price:
From our notes, we know that the quantity sold at that price must be equal to the amount that can be supplied by the producers. Therefore, all we need to do is to plug the new price into the supply curve equation and solve for Q.

The shortage is the difference between the quantity demanded and the quantity produced at a given price:

So the shortage is equal to 31.25 units. Or in other words, if producers were able to produce 31.25 more units at the given price, no price ceiling would be needed.

In order to find a per unit tax that would result in the same amount of quantity consumed as the price ceiling, one needs to understand that the tax would shift the supply curve to the left. Intersecting the demand curve where the quantity is equal to 12.5 (why?). Once we know the price that consumers are willing to pay, the tax is equal to the distance between the price ceiling and the new price for the demand curve at the given quantity.

Since we found the per unit tax, the new supply curve can be modeled by the following equation:


Let's visualize the problem:



The new supply curve (S2) has the same slope as S1. The only difference is that the constant factor in S2 includes the per unit tax, which causes this leftward shift. Furthermore, the tax is equal to difference between PT and PC on the graph. 









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