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.

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