Solving for equilibrium price and quantity after a shift in supply and demand

 This economics post goes over the tricky problem of determining the change in equilibrium price and quantity after a shift occurs.  This changes both the supply and demand function.  The trick is to know how to enter the shift into the supply and demand equations.  Generally you need to solve the functions for quantity (Q) and change the intercept.

The question in this post is:

Assuming Pd = 250 - 0.5Q and Ps = 100 + 0.25Q, then
What if quantity demanded at every price level increases by 10 and quantity supplied also rises by 5 at every price level?

First we can confirm equilibrium price and quantity by setting Pd equal to Ps and solving for Q.

Pd = Ps = 250 – 0.5Q = 100 + 0.25Q

If we add 0.5Q to both sides and subtract 100 from both sides we get:
0.75Q = 150 now we divide both sides by .75 to get:
Q = 200 and we can plug this into either our Ps or Pd equation to confirm that equilibrium price is indeed 150.

The next part of this question shifts the demand and supply curves by the given amounts.  Note that this is a change in the intercept and therefore a SHIFT in the curves.  Nothing has changed the slopes of the curves.  To add this change to our original equations we change the intercept values, but first we should change the equations to be true demand and supply functions.

To do this we need to solve both functions for Q instead of P.  To do this for the demand function we add 0.5Q to both sides, subtract P from both sides, and multiply both sides by 2 to get:

Q = 500 – 2P

For the supply function, we need to subtract 50 from both sides and multiply both sides by 4 to get:
Q = -400 + 4P

Now we can just make the associated changes to the intercepts, which is adding 10 to the intercept of the demand function to get 510 (500 + 10) and 5 to the supply function to get -395 (-400+5).

We now set these new supply and demand functions equal to each other and solve for P to find our new equilibrium price:

Qd = Qs = 510 – 2P = -395 + 4P

So we must add 2P and 395 to both sides to get:
905 = 6P  Now we divide both sides by 6 to get:
P = 150.833

We can plug this new equilibrium price into our demand and supply functions, and we will get an equilibrium quantity of 208.333 (which is the same whether we plug it into our demand or supply function so we know we did it right).

This means that both equilibrium price and equilibrium quantity have gone up as a result of the shifts.  It makes sense to see the equilibrium quantity go up because both consumers and suppliers had a positive shift (an increase in both supply and demand).  However, an increase in both supply and demand leaves the change in price ambiguous, but by solving for the new equilibrium price we see that it has gone up.  This should make sense because the relative magnitude of the shift in the demand curve was greater than the shift in supply.  Since the shift in demand has a positive impact on price and the shift in supply a negative, we would expect the positive force to dominate and we have confirmed this by seeing the increase in equilibrium price.

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Anonymous said...

Thanks so much for helping with this question!!!

freeeconhelp on March 28, 2012 at 7:41 AM said...

no problem, double check my work and let me know if there is anything else that would help.

Anonymous said...

Just one more question relating to this, how large would the supply shift need to be assuming the demand shift remains at +10? Thanks!

Anonymous said...

How large would the supply shift need to be assuming the demand shift remains at +10 for the price to decrease instead of increasing*. I have fiddled with the equations a bit and anything >10 seems to do the trick. Just wondering if my working is correct?

Qd = 410 - 2p
Qs = -200 +4p + X
410 - 2p = -200 + 4p + X
610 = 6p + x
Price decrease is anything less than 100. Subbing into P = 99.9
610 = 6(99.9) +x
610 - 599.4 = x
x = 10.6

So basically anything over 10 decreases the price?

freeeconhelp said...

Looks right! Another way to do it is to plug in 100 for P and X will end up being 10 which shows you the same thing. The reason they are equal is because each curve essentially shifts right at a quantity of 10 (regardless of the slope).

If you have a graph and mess around with it a little I think you will see what I mean.

Rakesh Mahatha on May 7, 2016 at 4:57 AM said...

if demand function D=10-5p2 and supply function S=-5+5p then what is price and quntity of eqilibrium

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