Assignment: DEMAND INTERPRETATON EXERCISE
Analyze the demand function for Toyotas in problem C4, page 82. Please also read "What is a Symbol" located in the folder with this assignment.
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 the price of Mazdas (be sure to also include the + or - sign. Note: I do not want the symbol for this partial derivative)?
3. Write the mathematical symbol representing the coefficient of advertising (the numerical value of this coefficient is +.003, but the answer you give is to be the symbol representing this partial derivative).
4. Assuming there is a $1 increase in advertising expenditures, what change in Toyota demand will result (give the numerical value, too)?
5. Is gasoline a substitute for or complementary to Toyotas? What feature of the function tells you?
6. Are Mazadas substitutes or complements to Toyotas? What feature of the function tells you?
7. Is a Toyota a normal or an inferior good? What feature of the function tells you?
8. Assuming there is a $1000 Toyota price decrease, what change in quantity demanded of Toyotas will result (give the numerical value, too)?
9. Assuming there is a $1000 increase in average family income, what change in Toyota demand will result (give the numerical value of it, too)?
10. Assuming there is a $1000 decrease in the price of Mazdas, what change in Toyota demand will result (give the numerical value of it, too)
WHAT IS A SYMBOL (as it relates to Assignment 7)?
A symbol is a shorthand way of representing something else (OK -- this is not Webster's definition). I can write, "find the price of an item and square it" or I can simply write Px2 (assuming you let me get away with saying that X is the item).
Suppose I have a function with one variable, say Z, whose dependent variable is Y. If I want the first derivative of this function, I can ask you to find the first derivative, I can also ask you to find dy/dz or perhaps I get really lazy, I can ask you to find Y'.
Want the derivative of a derivative, that is, a second derivative? Find d2y/dz2 or being lazy, I could simply say, find Y'' (I claim NO originality for how these are written --- they are very standard).
NOW in Lesson 15, Interpreting Multivariate Demand (multivariate because the function has several independent variables), you get introduced to the concept of a partial derivative. Why partial?
The concept of a total derivative means that you want the incremental (marginal) influence on demand of a simultaneous incremental change in all of the independent variables. This is a mess, and does not allow you to do analysis of the influence of a change in only ONE variable.
It is useful to find the incremental (marginal) influence on demand of a change in one variable of interest to you. Suppose Q = [ a function of several variables, one of which is H]. If you are interested in the marginal influence of a change in H on dependent variable Q, how do you state what you want with a symbol ?
You cannot use dQ/dH because then H would have to be the only independent variable in the equation. Mathematicians have a very special symbol for a partial derivative. They would ask you to find ∂Q/∂H. This symbol sort of looks like a backward 6. See footnote 23 on page 59 of your text for an example of its use in a point elasticity. This symbol means find the marginal influence on the demand for Q of an incremental change in H, assuming there is no change in any of the other independent variables at the same instance. By the way, if a multiple linear regression equation is estimated of this multivariate function, the estimated coefficients of the variables ARE estimates of the partial derivatives of the respective variables.
Text Book: Managerial Economics: Analysis, Problems, Cases, 8th edition (revised, 2006), by Truett and Truett published by John Wiley & Sons. ISBN 13 is #9780470009932. The ISBN 10 is #0470009934.