How to find equilibrium price and quantity mathematically -, Learning Economics... Solved!


How to find equilibrium price and quantity mathematically

Edit: Updated August 2018 with more examples and links to relevant topics.

Summary:  To solve for equilibrium price and quantity you should perform the following steps:

1) Solve for the demand function and the supply function in terms of Q (quantity).

2) Set Qs (quantity supplied) equal to Qd (quantity demanded). The equations will be in terms of price (P)

3) Solve for P, this is going to be your equilibrium Price for the problem.

4) Plug your equilibrium price into either your demand or supply function (or both--but most times it will be easier to plug into supply) and solve for Q, which will give you equilibrium quantity.

When solving for equilibrium price and quantity, you need to have a demand function, and a supply function.  Sometimes you will be given an inverse demand function (ie. P = 5 –Q) in this case you need to solve for Q as a function of P.  Once you have both your supply and demand function, you simply need to set quantity demanded equal to quantity supplied, and solve.

This is best explained by using an example...

Suppose your monthly quantity demand function for a product is Qd = 10,000-80P, and your monthly quantity supply function for a product is Qs=20P, then we need to follow the first step outlined above and set Qd=Qs and solve for price.

1)    Qd=Qs   which is also equal to--
2)    10,000 – 80P = 20P

We then must add 80P to both sides, then divide by 100 to get:

100 = P

(we got the above by adding 80P to both sides which gave us 10,000=100P, then divided both sides by 100 to get 100 = P)

Which is our equilibrium price.  Now to find equilibrium quantity we can plug our equilibrium price (100) into either our demand or supply function.  If we plug it into our demand function we get:

Qd= 10,000 – 80*100 = 2,000

We get 2,000 because 80*100 is 8,000... and 10,000 minus 8,000 is 2,000.

If we plug it into our supply function we get:

Qs = 20*100 = 2,000

Luckily, our quantity supplied is equal to our quantity demanded so we know that we did it right. It can also help to look at the graphs associated with market equilibrium if you are having problems developing the intuition for the math.

So the steps are:

1) Get functions solved for Qs (quantity supplied) and Qd (quantity demanded).
2) Set Qs equal to Qd
3) Solve for P (equilibrium price)
4) Plug your P back into your Qs and Qd functions to get equilibrium quantity

What’s going on behind the scenes?

The reason we set Qs equal to Qd is because we know that in equilibrium they must be equal.  Since supply and demand will only cross at one point, we know that when Qs = Qd that we are at equilibrium.  We can use this information to solve for equilibrium price even though we don’t know what Qd and Qs are!  Once we do have equilibrium price, we can use this information to back out what Qs and Qd are.

Another example:

Suppose that demand is given by the equation QD=500 – 50P, where QD is quantity demanded, and P is the price of the good. Supply is described by the equation QS= 50 + 25P where QS is quantity supplied. What is the equilibrium price and quantity?

So here we get:
or (subtract 50 from both sides, and add 50P to both sides to get)

divide both sides by 75 to get P = 6.

Plug P = 6 into both quantity functions:
500-50(6) = 200
50+25(6) = 200

So we know that equilibrium price is 6, and equilibrium quantity is 200.

Related lessons: Sometimes you will what to solve for equilibrium after a shift in either supply or demand. Other times you will want to calculate a change in equilibrium after an income change. Sometimes people will refer to the equilibrium price and quantity formula, but that is a bit of a misnomer. The formula that you use to calculate equilibrium price and quantity is Qd=Qs and then following the steps that are outlined above.