## 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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#### 18 comments:

Anonymous said...

This helped me so well! Thanks for using such a clear way to explain things

Anonymous said...

Thank u so much :)

zakari yussif on September 8, 2015 at 3:15 PM said...

Thanks

Unknown on September 8, 2015 at 5:10 PM said...

Excellent!! Thank you for making this easy to follow example at a pace I could follow. Thank you!!!

Megan Smith on October 14, 2015 at 9:12 PM said...

Yes, this was an awesome lesson. Straight forward explanations and good examples. Thank you!

Traci Gonzalez on October 16, 2015 at 12:17 AM said...

This was an awesome way for me to finally get the formula and how to use it and it's purpose. Now I have a question, how would I explain it in a graph?

Basel Alshehri on December 22, 2015 at 3:25 AM said...

Thanks a lot!

Zuzeey on February 18, 2016 at 9:56 AM said...

nice lecture

Teboho Mokati on March 24, 2016 at 11:09 AM said...

Thanks a lot. This is so clearly illustrated. But now my two questions are : 1. What would the quantities demanded and supplied be (Diagrammatically or otherwise)? [Traci Gonzalez above is on point]. 2. What would happen in the markets?

Ram Katariya on May 30, 2016 at 5:38 AM said...

This article solved my problem, i appreciate for sharing the solution. Commodity tips

Elizabeth FelizG on June 10, 2016 at 7:35 AM said...

help!!
consider a competitive market for a good where the demand is determined by the demand function: O=6-Qd and supply curve is determined by the supply function: P=2+Qs.

what is the quantity demanded of the good when the price level is: P=\$2?

what is the quantity supplied of the good when the price level is: P=\$2?

what is the equilibrium quantity level for the good in the competitive market?

what is the equilibrium price level for the good in the competitive market?

what is tge consumer and producers surplus in the competitive narket?

what is the level surplus in the competitive market?

Anonymous said...

Pls help i got supply equation Qs=-100➕10p calculate price of supply given price change from 20 to 35

Abbey Edward on July 15, 2017 at 4:52 AM said...

That was a great mathematics! I am already noted this in my hand note. You know I am a student of mathematics. By the way, the post is really impressive. Thank you so much and I appreciate that kind of work. mathematics typing

Nomty Mkhize on August 9, 2017 at 2:17 PM said...

Please help how to solve this Qs=28000+35p and Qd=80000-25p ,Qd=60000-10p , Qd=40000-6p. How do I determine the equilibrium market price and quantity

Kris said...

I have P= 100-.01QD
P= 10 + .01QS
Calculate equilibrium price and quantity

Sophie Gooch on September 8, 2017 at 3:49 AM said...

I am so much excited after reading equilibrium price and quantity. Your math is very much innovative and much helpful for any industry as well as for person.Harrison Ford Jacket

Lalnunpuii Jinhlawng said...

Thankyou so much this help in my preparation for the exam❤

Robert Welain on November 23, 2017 at 7:35 AM said...

Wow, this site is awesome guys!

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