Problem #1

The city of Philadelphia wants to study the production of bus service. The city officials hire a local group of academics to conduct the research study. The academics decide to use the natural log (ln) form of the Cobb-Douglas production function to estimate. The functional form of the Cobb-Douglas production function looks as the following:

Q = A*KL G

Since the Cobb-Douglas production function is non-linear the academics decide to take the natural log (ln) linear form which looks like the following:

ln Q = ln A + *ln K + *ln L + *ln G

This equation is now in a linear form which allows linear regression to be utilized. The following is the estimated equation performed by the research team:

Ln Q = 2.303 + 0.40 ln K + 0.60*ln L + 0.20*ln G (3.40) (4.15) (3.05) R2 = 0.89 and DW = 2.15

Where Q = number of miles driven; K = number of buses utilized; L = number of bus drivers employed each day and G = gallons of gasoline used. The corresponding estimated t-statistics are in parentheses below the estimated regression coefficients. The sample size, n, is 48.

Note the following learning points:

i. if you take the ln Q = %Q (that is also true of K, L & G). Thus the change in the ln Q is the percentage change in Q. Thus, in the estimated equation above the estimated coefficient on ln G is 0.02 which means if you change ln G (ln G) holding K and L fixed, we have ln Q = 0.20*ln G => ln Q/ln G = 0.20 => %Q/%G = 0.20. Note this is similar to an elasticity coefficient or the elasticity of miles driven concerning gas usage is +0.20.

ii. To take the anti-natural log in an Excel cell type = Exp(the numerical value). Exp is shorthand for exponential which is the base of the natural logarithms. As an example if ln Q = 5.75 then in a cell in Excel type in a cell =exp(5.75 or reference the appropriate cell) the result will equal 314.19 to 2 decimal points. Output Q will equal 324.19.

a. Are the estimated coefficients on K, L and G statistically significant at = 0.025? Explain.

b. Interpret the R2 .

c. Using equation "3" What is the number of miles driven, Q, if K = 250, L = 425 and G = 3750?

d. Using the estimated coefficients in "equation 3" on A, , , & find Q using "equation 1".

e. Using the marginal product equations calculate the MPK, MPL and the MPG.

f. Using the same footnote calculate the output elasticity of K, L and G.

g. For each output elasticity in part f if K increases by 10% how much does Q increase by? If L increases by 10% how much does Q increase by? If G increases by 10% how much does Q increase by?

h. Do the results of this study show that economies of scale are increasing, decreasing or constant? Explain.