9/26/11

How to find equilibrium price and quantity mathematically


Summary:  To get equilibrium price and quantity,
1) Solve for the demand and supply function in terms of Q (quantity).
2) Set Qs (quantity supplied) equal to Qd (quantity demanded).
3) Solve for P, this is your equilibrium Price.
4) Plug your equilibrium price into either your demand or supply function (or both) 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.

For example, if 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 set Qd=Qs and solve.

Qd=Qs   -or-
10,000 – 80P = 20P
Add 80P to both sides, then divide 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

If we plug it into our supply function we get:
Qs = 20*100 = 2,000

Luckily, our quantity supplied equaled our quantity demanded so we know that we did it right.

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:
Qd=Qs=500-50P=50+25P
or (subtract 50 from both sides, and add 50P to both sides to get)
450=75P

divide both sides by 75 to get P = 6.

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

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


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