A mathematical demand function for new Toyotas sold per year for a dealer is as

Question

A mathematical demand function for new Toyotas sold per year for a dealer is as follows                                        Qr = 200 - 0.001Pt + 0.005Pm - 10Pg + 0.01I +0.003A          

where  

                Qt = quantity purchased,

                Pt = the average price of Toyotas,

                Pm = the average price of mazdas,

                Pg = the price of gasoline,

                I = per capita income, and

                A = dollars spent annually on advertising

1. Characterize this function by circling all in the following list that are applicable:

univariate, bivariate, multivariate, linear, exponential, logarithmic, curvilinear, 1st degree, 3rd             degree, additive, multiplicative, linearly homogeneous

2. What is the numerical value of the partial derivative of the function with respect to advertising spending (be sure to also include the + or – sign, but I do not want the symbol for this)?

3. Write the mathematical symbol representing the coefficient of the price of gas (the numerical value of this coefficient is -10, but the answer you give is to be the symbol for this partial derivative).

4. Assuming there is a $1 increase in the price of gasoline, what change in Toyota demand will result (give the numerical value, too)?

5. Are Mazadas a substitute for or complementary to Toyotas? What feature of the function tells you?

6. Explain in words what the intercept (which is 200) includes (or does not include).

7. The sign of the variable I is +. Suppose it were negative (that is -.01). What would the negative sign tell you about the type of good Toyotas are?

8. Glory be! Enough pipe lines have finally been constructed to carry significantly more oil from the Cushing, OK, petroleum exchange to refineries in Houston. As a result, the price of gasoline decreases by $2.00 per gallon. What change in Toyoto demand will result (give the numerical value, too)?

9. The amount budgeted for advertising spending is cut by $100,000. What change in Toyoto demand will result (give the numerical value, too)?

10. Which will cause a greater influence on Toyota demand: a $1,000 decrease in the price of a Toyota or a $1000 increase in the price of a Mazada? Explain what items in the given function give you this information.

Explanation / Answer

Qr = 200 - 0.001Pt + 0.005Pm - 10Pg + 0.01I +0.003A

(1) The function is Multivariate (More than 1 explanatory variables), Linear, 1st-Degree & Additive.

(2) Partial derivative with respect to advertising (A) = + 0.003

(3) Coefficient of price of gas (Pg) = - 10

(4) If price of gas increases by $1, demand will decrease by 10.

(5) Coefficient of price of Mazda, Pm, is the partial derivative of demand with respect to Pm.

Cross-price elasticity of demand between Toyota & Mazda is = (dQr / dPm) x (Pm / Qr)

Here, dQr / dPm = +0.005, Pm > 0, Qr > 0

So, croos-price elasticity > 0

Or, if price of Maza increases (decreases), demand for Toyota increases (decreases).

So they are substitutes.

(6) The intercept is 200, which is the autonomous demand component which is independent of the explanatory variables.

(7) If the Income-coefficient was negative, it means that as income increased, demand for Toyota would decrease. This would imply that Toyota is an inferior good (whose consumption decreases with increase in income).

(8) If gasoline price increases by $1, demand decreases by 10 [See answer (3) & (4)].

So, if gasoline price increases by $2, demand decreases by 20.

(9) Coefficient of Advertising (A) = 0.003.

So, if A decreases by $1, demand decreases by 0.003.

If A decreases by $100,000 then demand decreases by $(100,000) x 0.003 = 300

(10) Coefficient of own price = - 0.001

Coefficient of Mazda price = + 0.005

So, increase in Mazda price will have a 5 time bigger effect on Toyota demand, compared to decrease in Toyota's own price.

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