Gwartney et al. listed the following data relating the price of a monthly cellul

Question

Gwartney et al. listed the following data relating the price of a monthly cellular bill (in dollars) and the demand (in millions of subscribers). *For a TED video on this phenomenon, see http://www, ted.com/talks/sean_gourley_on_the_mathematics_of_war.html. The authors point out that these are actual prices and quantities annually for 1988 to 1994. If they could assume that other demand determinants, such as income, had remained constant during that period, this would give an accurate measurement of the demand function. a. Using a graphing calculator, plot the natural logarithm of the price against the natural logarithm of the quantity. Does the relationship appear to be linear? b. Find the best-fitting line to the natural logarithm of the data, as plotted in part a. Plot this line on the same axes as the data. c. Plot the price against the quantity. What is different about the trend in these data from the trend in Figures 59(a) and 61(a)? What does this tell you about the exponent of the best-fitting power function for these data? What conclusions can you make about how demand varies with the price? d. Find the best-fitting power function for the data plotted in part c. Verify that this function is equivalent to the least squares line through the logarithm of the data found in part b.

Explanation / Answer

a)

yes, the relationship between ln(price) and ln(qty) is linear

b) plotted above the ln(price) and ln(qty) and the best fitting line for them is

ln(price) = -0.3289 * ln(qty) +5.0649

c) figures for 59a and 61a not available for comparison

As the demand increases, the price decreases. In fact, there is a logarithmic reduction in the price for the logarithmic increase in demand.

d) the best fitting power function is price = 158.378 * qty -0.329 (eqn 1)

ln(price) = 158.37 *ln(qty)-0.329 (eqn 2 from part b)

taking exponent function on both sides

price = exp158.37 exp ln(qty)^-0.329

price = 158.37 * qty-0.329

therefore eqn 1 is same as eqn 2

qty price ln qty ln price 2.1 123           0.74           4.81 3.5 107           1.25           4.67 5.3 92           1.67           4.52 7.6 79           2.03           4.37 11 73           2.40           4.29 16 63           2.77           4.14 24.1 56           3.18           4.03

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