# Thread: Supply and Demand equations are P = -5Q + 80 and P = 2Q + 10

1. ## Supply and Demand equations are P = -5Q + 80 and P = 2Q + 10

Hi all,
Trying to solve the following problem from Mathematics for Economics and Business by Ian Jacques - Supply and Demand Analysis. The demand and supply functions of a good are given by

P = -5Q + 80
P = 2Q + 10

For this first part I've already worked out the equilibrium price and quantity. P = 30; Q = 10

Next, if there's 15% government tax on the market price, determine the new equilibrium price and quantity. I had no issue when the question provide a nominal amount i.e. £5, but I'm struggling with the %age. I'm know this only affects the supply equation.

So I thought it would start as follows:

P x 1.15 = 2Q + 10 (divide both sides by 1.15)
P = (2Q + 10) / 1.15
Then
-5Q + 80 = (2Q + 10) / 1.15 (multiple both sides by 1.15)
-5Q + 80 x 1.15 = 2Q + 10 (add 5Q both sides)
80 x 1.15 = 7Q + 10 (minus 10 both sides)
70 x 1.15 = 7Q
80.5 = 7Q (divide both sides by 7)
11.5 = Q

0.85P = 2Q + 10
P = 33.6
Q = 9.28

I thought the tax was added on, hence 1.15, but it's seems it's deducted, so 0.85 - why is that?

Anyway, I used 0.85 and still got the wrong answer.

0.85 x P = 2Q + 10 (divide both sides by 0.85)
P = (2Q + 10) / 0.85
Then
-5Q + 80 = (2Q + 10) / 0.85 (multiple both sides by 0.85)
-5Q + 80 x 0.85 = 2Q + 10 (add 5Q both sides)
80 x 0.85 = 7Q + 10 (minus 10 both sides)
70 x 0.85= 7Q
59.5 = 7Q (divide both sides by 7)
8.5 = Q

Where am I going wrong?

Thank you for any help.

Hi all,
Trying to solve the following problem from Mathematics for Economics and Business by Ian Jacques - Supply and Demand Analysis. The demand and supply functions of a good are given by

P = -5Q + 80
P = 2Q + 10

For this first part I've already worked out the equilibrium price and quantity. P = 30; Q = 10

Next, if there's 15% government tax on the market price, determine the new equilibrium price and quantity. I had no issue when the question provide a nominal amount i.e. £5, but I'm struggling with the %age. I'm know this only affects the supply equation.

So I thought it would start as follows:

P x 1.15 = 2Q + 10 (divide both sides by 1.15)
P = (2Q + 10) / 1.15
Then
-5Q + 80 = (2Q + 10) / 1.15 (multiple both sides by 1.15)
-5Q + 80 x 1.15 = 2Q + 10 (add 5Q both sides)
80 x 1.15 = 7Q + 10 (minus 10 both sides)
70 x 1.15 = 7Q
80.5 = 7Q (divide both sides by 7)
11.5 = Q

0.85P = 2Q + 10
P = 33.6
Q = 9.28

I thought the tax was added on, hence 1.15, but it's seems it's deducted, so 0.85 - why is that?

Anyway, I used 0.85 and still got the wrong answer.

0.85 x P = 2Q + 10 (divide both sides by 0.85)
P = (2Q + 10) / 0.85
Then
-5Q + 80 = (2Q + 10) / 0.85 (multiple both sides by 0.85)
-5Q + 80 x 0.85 = 2Q + 10 (add 5Q both sides)
80 x 0.85 = 7Q + 10 (minus 10 both sides)
70 x 0.85= 7Q
59.5 = 7Q (divide both sides by 7)
8.5 = Q

Where am I going wrong?

Thank you for any help.
You do not say so, but I am guessing that the problem says that the tax is imposed on the seller. That is, the seller must turn over 15% of the price to the government. So the seller gets only 85% of the price, and the buyer is not DIRECTLY affected by the tax so only the supply function is changed.

Let's say that price a elicits quantity b without the tax and price c elicits quantity b with the tax. But the sellers only get 85% of c, which must be the same as a in order to elicit quantity b.

$0.85c = a = 2b + 10 \implies c = \dfrac{2b + 10}{0.85}.$

So our new supply curve is $p = \dfrac{2q + 10}{0.85}.$

At equilibrium, price and quantity must be the same for both functions.

$\dfrac{2q + 10}{0.85} = 80 - 5q \implies 2q + 10 = 0.85(80 - 5q) = 68 - 4.25q \implies$

$6.25q = 58 \implies q = 9.28 \implies p = 80 - 5 * 9.28 = 33.6.$

3. Originally Posted by JeffM
You do not say so, but I am guessing that the problem says that the tax is imposed on the seller. That is, the seller must turn over 15% of the price to the government. So the seller gets only 85% of the price, and the buyer is not DIRECTLY affected by the tax so only the supply function is changed.

Let's say that price a elicits quantity b without the tax and price c elicits quantity b with the tax. But the sellers only get 85% of c, which must be the same as a in order to elicit quantity b.

$0.85c = a = 2b + 10 \implies c = \dfrac{2b + 10}{0.85}.$

So our new supply curve is $p = \dfrac{2q + 10}{0.85}.$

At equilibrium, price and quantity must be the same for both functions.

$\dfrac{2q + 10}{0.85} = 80 - 5q \implies 2q + 10 = 0.85(80 - 5q) = 68 - 4.25q \implies$

$6.25q = 58 \implies q = 9.28 \implies p = 80 - 5 * 9.28 = 33.6.$

I see where I went wrong now. When I multiplied both sides by 0.85, I didn't multiply the whole function, i.e. -5Q + 80 x 0.85, whereas it should have been 0.85(-5Q + 80).

Thank you JeffM. Much appreciated.

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