There are at least two different possible approaches to the problem of building a model of the costs of production of electric power.
I: Model I hypothesizes that per-unit costs (C) as a function of the number of kilowatt-hours produced (Q) continually and smoothly falls as production is increased, but it falls at a decreasing rate.
II: Model II hypothesizes that per-unit costs (C) decrease fairly steadily as production (Q) increases, but costs decrease at a much faster rate for hydroelectric plants than for other
kinds of facilities.
Given this information,
(a) What functional form would you recommend for estimating Model I? Be sure to write out a specific equation.
(b) What functional form would you recommend for estimating Model II? Be sure to write out a specific equation.
(c) Would $\overline { \mathrm { R } } ^ { 2 }$ be a reasonable way to compare the overall fits of the two equations? Why or why not?
Note: In the following question, you may want to change the absolute size of the coefficients, depending on the size of your school, but remember to change the size of the standard errors proportionally.

Question

Quizplus · Accepted Answer

The Answer is (a) A number of forms are possible, but a reciprocal form would be perhaps the most appropriate:

\mathrm { C } _ { \mathrm { t } } = \rho _ { 0 } + \rho _ { 1 } / \mathrm { Q } _ { \mathrm { t } } + \varepsilon _ { \mathrm { t } }

(b) Such a hypothesis calls for the use of a slope dummy defined (for instance) as D= 1 if the plant is hydroelectric and 0 otherwise. The resulting equation would be:

\mathrm { C } _ { \mathrm { t } } = \beta _ { 0 } + \beta _ { 1 } \mathrm { Q } _ { \mathrm { t } } + \beta _ { 2 } \mathrm { D } _ { \mathrm { t } } + \beta _ { 3 } \mathrm { D } _ { \mathrm { t } } \mathrm { Q } _ { \mathrm { t } } + \varepsilon _ { \mathrm { t } }

(c)

\overline { \mathrm { R } } ^ { 2 }

is perfectly appropriate for comparing the overall fits of the two equations because the number of independent variables changes but the functional form of the dependent variable does not.

There Are at Least Two Different Possible Approaches to the Problem