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

8/7/18

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:
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.

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.

30 comments:

  1. Anonymous08 October

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

    ReplyDelete
  2. Anonymous06 April

    Thank u so much :)

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

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

    ReplyDelete
  5. 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?

    ReplyDelete
    Replies
    1. Anonymous11 March

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

      Delete
  6. 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?

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

    ReplyDelete
  8. 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?

    ReplyDelete
  9. 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

    ReplyDelete
  10. 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

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

    ReplyDelete
  12. 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

    ReplyDelete
  13. Lalnunpuii Jinhlawng23 November

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

    ReplyDelete
  14. Anonymous07 February

    Hey this helped me a lot. Thanks admin

    ReplyDelete
  15. Awesome...nice lecture

    ReplyDelete
  16. Find the equilibrium price P_e and quantity Q_efor each of the following marketS using (1) equations and (2) graphs:
    Supply: P=1/4 Q+2 Demand: P=-3/4 Q+22

    ReplyDelete
  17. I think this is a really good article. You make this information interesting and engaging. I guess I am not only one having all the enjoyment right here! Lara Croft Tomb Raider Jacket

    ReplyDelete
  18. I have Pd=450-2Qd;Ps=100+5Qs
    (i)Calculate the equilibrium quantity and price
    (ii)If the government provides a subsidy of $70 per unit sold, find the new equilibrium quantity and price

    ReplyDelete
  19. Anonymous07 May

    It has been helpful thank you

    and im also stuck with this one
    Qd=15-p
    Qs=3-P

    the requirement is to solve the equilibrium qty and Equilibrium pricw

    ReplyDelete
  20. P=1600-80q



    S there missing i cant solve this

    ReplyDelete
  21. Anonymous17 May

    Demand Q= 15 P
    Supply P=3

    ReplyDelete
  22. Anonymous17 May

    Assist to solve for equilibrium quantity and equilibrium price

    ReplyDelete
  23. Given marginal cost is $5, 20 trips per day and demand function is 1200-peP. Calculate the equilibrium price and equilibrium quantity

    ReplyDelete
  24. Keep posting the good work. Some really helpful information in there. Bookmarked. Nice to see your site. Thanks! learning videos

    ReplyDelete
  25. Sultan Faris17 August

    Thanks really help me to build quality backlinks.Tutuapp

    ReplyDelete