Answers to elasticity problems

  1. Given the following demand schedule:

    Demand schedule
    PRICE QUANTITY REVENUE
    15 10 __ 150 __
    10 55 __ 550 __
    5 100 __ 500 __

    1. Fill in the revenue column; without doing any further computations, is the demand curve elastic or inelastic? Why? Answer: since revenue increased when price declines, demand will be elastic .

    2. Compute the coefficient of elasticity between a price of $5 and of $15 using the midpoint formula. If you have forgotten the midpoint formula, review the hypertext link here

      Answer: The percentage change in price, using the midpoint formula is:
      ( 5 - 15) / 10 = -10/10 = - 1.0 or - 100 percent.

      The percentage change in quantity, using the midpoint formula is:
      (100 - 10) / 55 = 90/55 = 1.636.

      The coefficient of elasticity is the percentage change in quantity divided by the percentage change in price, or - 90/55 = - 1.636. Since this is greater than 1 demand is elastic over this range, as we expected from the increase in revenue.

    3. Answer the above questions for the following demand schedule:

      Demand schedule
      PRICE QUANTITY REVENUE
      100 100 - 10,000 -
      300 90 - 27,000 -
      500 80 - 40,000 -

      Note that revenue is increasing as the price is increasing. We expect that demand will prove to be inelastic in this case.

      The percentage change in price, using the midpoint formula, is:
      (100 - 500)/300 = - 400/300 = - 1.33 or - 133 percent.

      The percentage change in quantity, using the midpoint formula, is:
      (100 - 80)/90 = 20/90 = .22 or 22 percent.

      The coefficient of elasticity is the percentage change in quantity divided by the percentage change in price:
      2/9 divided by 4/3 = (2/9)*(3/4) = 1/6 or .167

      Demand, as expected on the basis of the change in revenue, is very inelastic in this case.

    4. Given the demand curve Q = 200 - 4P
      1. Graph the demand curve, showing exactly where it cuts the axes.
      2. How much is demanded at a price of 10 dollars? 11 dollars? 9 dollars?
      3. Use the above information to find the elasticity of the curve at a price of $10, that is between prices of $9 and $11. (Note: given a demand curve in algebraic form, we can find the elasticity at a point by raising and lowering the price by a dollar. We then construct a table similiar to the tables in the first two probems and compute the elasticity between the points defined by the given price plus or minus one dollar.)

        Note in constructing the table that the slope coefficient -4 shows that the quantity demanded goes down by 4 for every dollar the price is increased. You do not need to use the full equation more than once; after finding that
        Q = 200 - 4(10) = 160
        simply adjust the quantity by 4.

        Demand schedule
        PRICE QUANTITY
        9 164
        10 160
        11 156

        Percent change in price = (11 - 9)/10 = 2/10 or 20 percent.
        Percent change in quantity = (156 - 164)/160 = -8/160 or - 5 percent
        Coefficient of elasticity =
        (percent change in quantity)/(percent change in price) = -5/20 = - .25

        Note that the revenue goes from $ 9 x 164 = $ 1476 at a price of $ 9 to
        a revenue of $ 11 x 156 = $ 1716 at a price of $ 11. Since revenue goes up with price in this case, we would expect that demand is inelastic, and the computation of the coefficient of elasticity confirms this.

    5. Using the same demand curve, Q = 200 - 4P
      1. How much is demanded at a price of 40 dollars? at a price of 39 dollars?at a price of 41 dollars?

        Demand schedule
        PRICE QUANTITY
        39 44
        40 40
        41 36

      2. What is the coefficient of elasticity at a price of 40 dollars?

        The percentage change in price is (41 - 39)/40 = 2/40 = 5 percent.
        Note that the same two dollar price increase was a 20 percent price increase in the last problem. The difference is that in this case the average price is much higher, 40 dollars rather than 10 dollars.

        The percentage change in quantity is (36 - 44)/40 = -8/40 = 20 percent.
        Note that the percentage change in quantity was only 5 percent in the previous case; the average quantity is much lower in this case

        The coefficient of elasticity is
        (Pct. Change in Q)/(Pct. change in P) = -20/5 = 4.

      3. How does this compare with the coefficient of elasticity found in the previous problem? Is elasticity the same anywhere along a straight line demand curve? How does it vary with price?

        The coefficient of elasticity was .25 in the previous case; it is 4.0 in this case. The higher absolute value means that

        The elasticity of demand increases as the price increases if the demand curve is a straight line.

        (Note: an increase in the elasticity of demand with price may not take place if the demand curve is curved; constant or declining elasticity as price increases is possible in that case.)

    6. Given the demand curve Q = 100 - 1/2 P
      1. Can we say it is less elastic than the previous demand curve?

        No, since elasticity varies as the price increases along both curves. We must specify the price at which we are making the comparison.

      2. Is it less elastic than the previous demand curve at a price of 30 dollars?

        Demand schedules
        PRICE Q = 200 - 4P Q = 100 - 1/2 P
        29 84 84.5
        30 80 85
        31 76 85.5

        The coefficient of elasticity for the demand curve Q = 200 - 4 P
        is the percentage quantity change of 8/80 divided by 2/30 = 240/160 = 1.5 at a price of $30.

        The coefficient of elasticity at a price of $30 for the demand curve Q = 100 - 1/2 P
        is 1/85 divided by a percentage price change of 2/30 = 30/170 = .1765.

        Accordingly, we can say that the curve introduced in this problem is less elastic than that in the previous problem if both curves are evaluated at the price of $30.

      3. Is it less elastic than the previous demand curve at a price of one dollar?

        Demand schedules
        PRICE Q = 200 - 4 P Q = 100 - 1/2 P
        0 200 100
        1 196 99.5
        2 192 99

        The elasticity of demand for Q = 200 - 2P is the quantity percentage change of (200 - 192)/196 = 8/196 = 4 percent divided by the price percentage change of 2/1 or 200 percent -- that is, the COE is .02.

        The elasticity of demand for Q = 100 - 1/2 P is the percent change in quantity of 1/99.5 or about 1 percent divided by the price percentage change of 200 percent; that is, the COE is .005. Again, the curve introduced in this problem is less elastic at a common price.

      4. Does elasticity vary along this demand curve? Explain how -- does elasticity increase or decrease with price? Is this the same as the previous curve?

        Again, as in the previous case, demand is more elastic as price increases.


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