Finding the market equilibrium price and quantity

Finding the equilibrium price and quantity is perhaps one of the most common problems in an introduction to Microeconomics course, which is exactly what we are going to do in this post!

Consider the following problem, you are tasked with finding the equilibrium price and quantity for a particular good X.

You are told that the producers are willing and able to sell 2 units for the price of 16 U.S dollars a unit, and if the price increases, they are able to make 10 units for a price of 20 U.S dollars per unit. Assuming that the supply curve is linear, then it can be modeled by the following equation:

$Q_s=2P-30$

The same process can be used to determine the market demand (assuming linearity and that we are given information about the market demand schedule). At a price of 30 U.S dollars consumers are willing to buy 10 units, if the price decreases to 12 U.S dollars, consumers want to buy 20 units. Thus the demand curve corresponds to this equation:

$Q_d=-\frac{P}{5}+16$

Finally, all we need to do is to set both equations equal to one another and solve for P. Remember, the equilibrium price and quantity are determined by the intersection of the demand and supply curves (note that in the equation below P=P_e).

$\\(1.)-\frac{P}{5}+16=2*(P)-30\\(2.)\frac{11}{5}P=46\\ (3.)P_e\approx 20.91$

Now that we have the equilibrium price, we can plug this result in any of our two previous equations representing the market demand and supply curves and solve for Q. They should both give the quantity demanded when the market is in equilibrium. This is also a nice way to double check your work (i.e if you get two totally different results you probably made a mistake somewhere).

$-\frac{P_e}{5}+16\approx 11.82\\2*(P_e)-30=11.82$

Let's visualize that:

In a more realistic fashion, we would say that the market clearing price is 20.91 U.S dollars and the quantity of goods provided is 12 units. It would not make much sense to buy 11 units plus 82% of one good, so we can round that number to 12.

Source: example inspired from Zeder,Raphael.How to Calculate the Equilibrium Price. Quickonomics (2018).

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