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Identifying necessities and luxuries
48 Part I: The Nature of Managerial Economics
You obtain a partial derivative by applying the rules for finding a derivative, while treating all independent variables, except the one of interest, as constants. Thus, in the example, you hold constant both price and income. And the great thing about constants is their derivative equals zero!
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Assume the following equation describes the relationship between the quantity sold of a good and its price, consumer incomes, and the amount spent on advertising
where q is the number of units sold per month, p is the price per unit in dollars, Y is average consumer income in dollars, and A is advertising expenditures in dollars.
In order to determine the partial derivative of quantity with respect to advertising, you should take the following steps:
1. First, remember that both p and Y are treated as constants. Therefore, you treat them exactly as you would a number when taking the derivative.
2. To take the partial derivative of q with respect to A, start with the first term “1,000” and its derivative equals zero in the partial derivative. 3. The second term “–10p” has a partial derivative equal to zero because you treat the p like a constant or number. 4. The next term “+0.01Y” also has a partial derivative equal to zero because you treat the Y like a constant. 5. The derivative of the term “0.2A” equals 0.2, because you treat
the A like a variable in this partial derivative. You’re interested in determining how changes in A’s value affect q. 6. The derivative of the term “–0.01A×p” equals –0.01p. Remember, you treat p the same as any number, while A is the variable.
7. Finally, derivative of the term “–0.0001A2” equals –0.0002A.
Putting each of these steps together yields a partial derivative of q with respect to A of
Similarly, the partial derivative of quantity with respect to price, δq/δp, and the partial derivative of q with respect to Y, δq/δY, can be determined by treating any variables other than those specified in the partial derivative as constants. Those partial derivatives would be
