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A.6 MULTIVARIABLE FUNCTIONS 743
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we vary Q 2 , but holds Q 1 constant, with Q 1 4 . Point B in the top of the profit hill, point A in Figure A.8. Let’s also
this figure is therefore the same as point B in Figure A.8. consider point A in Figure A.9, where Q 1 4 and Q 2 3 ,
We have also drawn RS, the line tangent to the profit hill at the outputs that maximize profits. Point A in this figure is
point B. (The tangent line RS is the same in Figures A.8 and therefore the same as point A in Figure A.8. Since we have
reached the top of the profit curve, the slope of the profit
A.9.) The partial derivative of profit with respect to Q 2
(denoted by 0p/0Q 2 ) measures the slope of this tangent hill at A in Figure A.9 is zero; this means that the partial
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line. At point B the slope is 4. derivative 0p/0Q 2 , is zero.
Similarly, we could ask how an increase in Q 1 affects p, To practice taking partial derivatives, you might try the
holding constant the other independent variable Q 2 . To following exercises.
find this information, we find the partial derivative of p
. We take the deriva- LEARNING-BY-DOING EXERCISE A.7
with respect to Q 1 , denoted by 0p/0Q 1
tive of equation (A.10), but treat the level of Q 2 as a constant.
Marginal Utility with Two Independent Variables
When we do this, the last two terms in equation (A.10) will
be a constant because they depend only on Q 2 ; therefore the In Chapter 3 (Learning-By-Doing Exercise 3.1), we intro-
partial derivative of these terms with respect to Q 1 is zero. duced the utility function U 1xy . Here U is the de-
The partial derivative of the third term (Q 1 Q 2 ) with respect pendent variable and x and y are the independent variables.
to Q 1 is just Q 2 . The partial derivative of the first two terms The corresponding marginal utilities function are
with respect to Q 1 will be 13 4Q 1 . When we put all of MU x 1y/(21x ) , and MU y 1x/(21y ) .
this information together, we learn that
Problem Use partial derivatives to verify that these ex-
0p pressions for marginal utilities are correct.
13 4Q 1 Q 2 (A.12)
0Q 1 Solution It may help to rewrite the utility function as
U x 1/2 1/2 . The marginal utility of x is just the partial
y
Equation (A.12) measures the marginal profit of Q 1 , derivative of U with respect to x, that is, 0U/0x. To find this
that is, the rate of change of profit as we vary Q 1 , but hold derivative, we treat y as a constant. Therefore, we only need
Q 2 constant. Let’s evaluate this partial derivative at point B to find the derivative of the term in brackets: U [x 1/2 ] y 1/2 .
in Figure A.8. When Q 1 4 , and Q 2 2 , we find that (The y 1/2 is just a multiplicative constant.) We observe that
13 4(4) 2 1 . Let’s draw the line tan-
0p/0Q 1 x 1/2 is a power function, with the derivative (1/2)x 1/2 ,
gent to the profit hill at point B, holding Q 2 constant which can be rewritten as 1/(21x ) . The marginal utility is
(Q 2 2 ), and label this line MN. The tangent line will have then MU x 1y/(21x) .
a slope of 1.
Similarly, the marginal utility of y is just the partial
derivative of U with respect to y, that is, 0U/0y. To find this
Finding a Maximum or a Minimum derivative, we treat x as a constant. Therefore, we only need
How can we find the top of the profit hill in Figure A.8? At to find the derivative of the term in brackets: U x 1/2 [ y 1/2 ].
a maximum, the slope of the profit hill will be zero in all di- We observe that y 1/2 is a power function, with the deriva-
rections. This means that at a maximum the partial derivatives tive (1/2)y 1/2 , which can be rewritten as 1/(21y ) . The
must both be zero. Thus, in the example, marginal utility is then MU y 1x/(21y ) .
0p/0Q 1 and 0p/0Q 2
0p
13 4Q 1 Q 2 0 LEARNING-BY-DOING EXERCISE A.8
0Q 1
0p Marginal Cost with Two Independent Variables
Q 1 8 4Q 2 0
0Q 2
Problem Suppose the total cost C of producing two prod-
When we solve these two equations, we find that ucts is C Q 1 1Q 1 Q 2 Q 2 , where Q 1 measures the
number of units of the first product and Q 2 the number of
Q 1 4 and Q 2 3 . These are the quantities that lead us to
units of the second. When Q 1 16 and Q 2 1, find the
marginal cost of the first product, MC 1 .

3 Another way to see the meaning of the partial derivative illus- 4 To ensure that we have a maximum, or to distinguish a maximum
trated in Figure A.9 is to substitute Q 1 4 into the profit function from a minimum, we would also have to examine the second-order
2
p 13Q 1 2(Q 1 ) Q 1 Q 2 8Q 2 2(Q 2 ) 2 . Profits then become conditions for an optimum. In this appendix we do not present these
p 20 12Q 2(Q 2 ) 2 . This is the equation of the profit hill conditions for a function with more than one independent variable
in Figure A.9, because we have assumed Q 1 is held constant at 4. and refer you to any standard calculus text. Also, the techniques we
The slope of the profit hill in Figure A.9 is therefore dp/dQ 2 have discussed in this appendix may show you where a local maxi-
12 4Q 2 . At point B, where Q 2 2 , we find that dp/dQ 2 4 , mum or minimum exists, but you may need to check further to see if
which is the slope of the tangent line RS. the local maximum or minimum is a global optimum (see footnote 2).