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

  1. #1

    Question 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

    But the answer key starts the answer as
    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.

  2. #2
    Senior Member
    Join Date
    Sep 2012
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    2,396
    Quote Originally Posted by radnorgardens View Post
    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

    But the answer key starts the answer as
    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.

    [tex]0.85c = a = 2b + 10 \implies c = \dfrac{2b + 10}{0.85}.[/tex]

    So our new supply curve is [tex]p = \dfrac{2q + 10}{0.85}.[/tex]

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

    [tex]\dfrac{2q + 10}{0.85} = 80 - 5q \implies 2q + 10 = 0.85(80 - 5q) = 68 - 4.25q \implies[/tex]

    [tex] 6.25q = 58 \implies q = 9.28 \implies p = 80 - 5 * 9.28 = 33.6.[/tex]

  3. #3
    Quote Originally Posted by JeffM View Post
    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.

    [tex]0.85c = a = 2b + 10 \implies c = \dfrac{2b + 10}{0.85}.[/tex]

    So our new supply curve is [tex]p = \dfrac{2q + 10}{0.85}.[/tex]

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

    [tex]\dfrac{2q + 10}{0.85} = 80 - 5q \implies 2q + 10 = 0.85(80 - 5q) = 68 - 4.25q \implies[/tex]

    [tex] 6.25q = 58 \implies q = 9.28 \implies p = 80 - 5 * 9.28 = 33.6.[/tex]



    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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