Question

See the multivariate demand function below. Please show work so I can learn how this is done.

Setting: U.S. Auto manufacturers are trying to develop a multivariate function with which to estimate the demand for their gas-electric hybrid compact cars. Here is one that Motors General developed for its Jolt: Qj = 65000 – 20Pj + 20Pf + 35Pt – 5Pb + 0.2Tc + 0.05Y + 10Mg + 0.04A

Where:

Qj = the number of Jolts demanded per week.Pj = the price of each new Jolt (in $). Pf = the price of each new Ford gas-electric hybrid (in $). Pt = the price of each new Toyota gas-electric hybrid (in $). Pb = the price of replacement batteries for the Jolt (in $). Tc = the amount of tax credit incentive offered with the purchase of a new hybrid (in $). Y = average weekly disposable income of a typical Jolt purchaser (in $). Mg = the miles per gallon of gas rating of the Jolt (in miles per gallon). A = average weekly Jolt advertising expenditure (in $).

1. What is the partial derivative of the demand for Jolts with respect to the gas mileage rating (Mg) of Jolts?

Answer 1

Qj = 65000 – 20Pj + 20Pf + 35Pt – 5Pb + 0.2Tc + 0.05Y + 10Mg + 0.04A

Qj = the number of Jolts demanded per week.

Pj = the price of each new Jolt (in $).

Pf = the price of each new Ford gas-electric hybrid (in $).

Pt = the price of each new Toyota gas-electric hybrid (in $).

Pb = the price of replacement batteries for the Jolt (in $).

Tc = the amount of tax credit incentive offered with the purchase of a new hybrid (in $).

Y = average weekly disposable income of a typical Jolt purchaser (in $).

Mg = the miles per gallon of gas rating of the Jolt (in miles per gallon).

A = average weekly Jolt advertising expenditure (in $).

(dQj)/dM_g= d(65000 – 20Pj + 20Pf + 35Pt – 5Pb + 0.2Tc + 0.05Y + 10Mg + 0.04 A)/dM_g

= > (dQj)/dM_g = 10

Note :- 65000 is a constant

              dP_j /dM_g = 0   ;   Since, P_j is independent of M_g i.e., exogenoeus hence any change in M_g doesn’t   affect or cause any change in P_j.

Similarly all other variables are independent of M_g.

Hence the Partial derivative of Q_j with respect to M_g yields the number 10. Therefore for a unit change in miles per gallon of gas rating of the jolt increases the demand of jolts per week by 10 units.