Chapter 6 NAME
Demand
Introduction. In the previous chapter, you found the commodity bundle
that a consumer with a given utility function would choose in a specific
price-income situation. In this chapter, we take this idea a step further.
We find demand functions, which tell us for any prices and income you
might want to name, how much of each good a consumer would want. In
general, the amount of each good demanded may depend not only on its
own price, but also on the price of other goods and on income. Where
there are two goods, we write demand functions for Goods 1 and 2 as
x1(p1,p
2,m)andx2(p1,p
2,m).∗
When the consumer is choosing positive amounts of all commodities
and indifference curves have no kinks, the consumer chooses a point of
tangency between her budget line and the highest indifference curve that
it touches.
Example: Consider a consumer with utility function U(x1,x
2)=(x1+
2)(x2+ 10). To find x1(p1,p
2,m)andx2(p1,p
2,m), we need to find a
commodity bundle (x1,x
2) on her budget line at which her indifference
curve is tangent to her budget line. The budget line will be tangent to
the indifference curve at (x1,x
2) if the price ratio equals the marginal
rate of substitution. For this utility function, MU1(x1,x
2)=x2+10 and
MU2(x1,x
2)=x1+ 2. Therefore the “tangency equation” is p1/p2=
(x2+ 10)/(x1+ 2). Cross-multiplying the tangency equation, one finds
p1x1+2p1=p2x2+10p2.
The bundle chosen must also satisfy the budget equation, p1x1+
p2x2=m. This gives us two linear equations in the two unknowns, x1
and x2. You can solve these equations yourself, using high school algebra.
You will find that the solution for the two “demand functions” is
x1=m−2p1+10p2
2p1
x2=m+2p1−10p2
2p2
.
There is one thing left to worry about with the “demand functions” we
just found. Notice that these expressions will be positive only if m−2p1+
10p2>0andm+2p1−10p2>0. If either of these expressions is negative,
then it doesn’t make sense as a demand function. What happens in this
∗For some utility functions, demand for a good may not be affected by
all of these variables. For example, with Cobb-Douglas utility, demand
for a good depends on the good’s own price and on income but not on the
other good’s price. Still, there is no harm in writing demand for Good
1 as a function of p1,p2,andm. It just happens that the derivative of
x1(p1,p
2,m) with respect to p2is zero.
68 DEMAND (Ch. 6)
case is that the consumer will choose a “boundary solution” where she
consumes only one good. At this point, her indifference curve will not be
tangent to her budget line.
When a consumer has kinks in her indifference curves, she may choose
a bundle that is located at a kink. In the problems with kinks, you
will be able to solve for the demand functions quite easily by looking
at diagrams and doing a little algebra. Typically, instead of finding a
tangency equation, you will find an equation that tells you “where the
kinks are.” With this equation and the budget equation, you can then
solve for demand.
You might wonder why we pay so much attention to kinky indiffer-
ence curves, straight line indifference curves, and other “funny cases.”
Our reason is this. In the funny cases, computations are usually pretty
easy. But often you may have to draw a graph and think about what
you are doing. That is what we want you to do. Think and fiddle with
graphs. Don’t just memorize formulas. Formulas you will forget, but the
habit of thinking will stick with you.
When you have finished this workout, we hope that you will be able
to do the following:
•Find demand functions for consumers with Cobb-Douglas and other
similar utility functions.
•Find demand functions for consumers with kinked indifference curves
and for consumers with straight-line indifference curves.
•Recognize normal goods, inferior goods, luxuries, and necessities from
looking at information about demand.
6.1 (0) Charlie is back—still consuming apples and bananas. His util-
ity function is U(xA,x
B)=xAxB. We want to find his demand func-
tion for apples, xA(pA,p
B,m), and his demand function for bananas,
xB(pA,p
B,m).
(a) When the prices are pAand pBand Charlie’s income is m, the equation
for Charlie’s budget line is pAxA+pBxB=m. The slope of Charlie’s indif-
ference curve at the bundle (xA,x
B)is−MU1(xA,x
B)/M U2(xA,x
B)=
NAME 69
(b) You now have two equations, the budget equation and the tan-
gency equation, that must be satisfied by the bundle demanded. Solve
these two equations for xAand xB. Charlie’s demand function for ap-
ples is xA(pA,p
B,m)= m
(c) In general, the demand for both commodities will depend on the price
of both commodities and on income. But for Charlie’s utility function,
the demand function for apples depends only on income and the price
of apples. Similarly, the demand for bananas depends only on income
and the price of bananas. Charlie always spends the same fraction of his
6.2 (0) Douglas Cornfield’s preferences are represented by the utility
function u(x1,x
2)=x2
1x3
2. The prices of x1and x2are p1and p2.
(b) If Cornfield’s budget line is tangent to his indifference curve at (x1,x
2),
then p1x1
(c) Other members of Doug’s family have similar utility functions, but
the exponents may be different, or their utilities may be multiplied by a
positive constant. If a family member has a utility function U(x, y)=
cxa
1xb
2where a,b,andcare positive numbers, what fraction of his or her
6.3 (0) Our thoughts return to Ambrose and his nuts and berries. Am-
brose’s utility function is U(x1,x
2)=4
√x1+x2,wherex1is his con-
sumption of nuts and x2is his consumption of berries.
(a) Let us find his demand function for nuts. The slope of Ambrose’s
indifference curve at (x1,x
2)is −2
√x1.Setting this slope equal to
the slope of the budget line, you can solve for x1without even using the
70 DEMAND (Ch. 6)
(b) Let us find his demand for berries. Now we need the budget equation.
In Part (a), you solved for the amount of x1that he will demand. The
budget equation tells us that p1x1+p2x2=M. Plug the solution that
you found for x1into the budget equation and solve for x2as a function
of income and prices. The answer is x2=M
(c) When we visited Ambrose in Chapter 5, we looked at a “boundary
solution,” where Ambrose consumed only nuts and no berries. In that
example, p1=1,p2=2,andM= 9. If you plug these numbers into the
the budget line x1+2x2= 9 is not tangent to an indifference curve when
x2≥0. The best that Ambrose can do with this budget is to spend all
of his income on nuts. Looking at the formulas, we see that at the prices
p1=1andp2= 2, Ambrose will demand a positive amount of both goods
6.4 (0) Donald Fribble is a stamp collector. The only things other
than stamps that Fribble consumes are Hostess Twinkies. It turns out
that Fribble’s preferences are represented by the utility function u(s, t)=
s+lntwhere sis the number of stamps he collects and tis the number
of Twinkies he consumes. The price of stamps is psand the price of
Twinkies is pt. Donald’s income is m.
(a) Write an expression that says that the ratio of Fribble’s marginal
utility for Twinkies to his marginal utility for stamps is equal to the ratio
(b) You can use the equation you found in the last part to show that if he
buys both goods, Donald’s demand function for Twinkies depends only
on the price ratio and not on his income. Donald’s demand function for
(c) Notice that for this special utility function, if Fribble buys both goods,
then the total amount of money that he spends on Twinkies has the
peculiar property that it depends on only one of the three variables m,
pt,andps, namely the variable ps.(Hint: The amount of money that
he spends on Twinkies is ptt(ps,p
t,m).)
NAME 71
(d) Since there are only two goods, any money that is not spent on
Twinkies must be spent on stamps. Use the budget equation and Don-
ald’s demand function for Twinkies to find an expression for the number
of stamps he will buy if his income is m, the price of stamps is psand the
price of Twinkies is pt.s=m
(e) The expression you just wrote down is negative if m<p
s. Surely
it makes no sense for him to be demanding negative amounts of postage
stamps. If m<p
s, what would Fribble’s demand for postage stamps be?
(f) Donald’s wife complains that whenever Donald gets an extra dollar,
he always spends it all on stamps. Is she right? (Assume that m>p
s.)
Yes.
(g) Suppose that the price of Twinkies is $2 and the price of stamps is $1.
On the graph below, draw Fribble’s Engel curve for Twinkies in red ink
and his Engel curve for stamps in blue ink. (Hint: First draw the Engel
curves for incomes greater than $1, then draw them for incomes less than
$1.)
0246
8
2
4
6
Quantities
Income
8
Blue
line
Red line
1
0.5
6.5 (0) Shirley Sixpack, as you will recall, thinks that two 8-ounce cans
of beer are exactly as good as one 16-ounce can of beer. Suppose that
these are the only sizes of beer available to her and that she has $30 to
spend on beer. Suppose that an 8-ounce beer costs $.75 and a 16-ounce
beer costs $1. On the graph below, draw Shirley’s budget line in blue ink,
and draw some of her indifference curves in red.
72 DEMAND (Ch. 6)
0102030
40
10
20
30
16-ounce cans
8-ounce cans
40
Blue
budget
line
Red
curves
Red curve
(a) At these prices, which size can will she buy, or will she buy some of
(b) Suppose that the price of 16-ounce beers remains $1 and the price of
(c) What if the price of 8-ounce beers falls to $.40? How many 8-ounce
(d) If the price of 16-ounce beers is $1 each and if Shirley chooses some
8-ounce beers and some 16-ounce beers, what must be the price of 8-ounce
(e) Now let us try to describe Shirley’s demand function for 16-ounce beers
as a function of general prices and income. Let the prices of 8-ounce and
16-ounce beers be p8and p16, and let her income be m.Ifp16 <2p8, then
6.6 (0) Miss Muffet always likes to have things “just so.” In fact the
only way she will consume her curds and whey is in the ratio of 2 units of
whey per unit of curds. She has an income of $20. Whey costs $.75 per
unit. Curds cost $1 per unit. On the graph below, draw Miss Muffet’s
budget line, and plot some of her indifference curves. (Hint: Have you
noticed something kinky about Miss Muffet?)
NAME 73
(a) How many units of curds will Miss Muffet demand in this situation?
0 8 16 24 32
8
16
24
Curds
Whey
32 w = 2c
Budget
line
Indifference
curves
(b) Write down Miss Muffet’s demand function for whey as a function
of the prices of curds and whey and of her income, where pcis the price
of curds, pwis the price of whey, and mis her income. D(pc,p
w,m)=
m
pw+pc/2
.(Hint: You can solve for her demands by solving two equa-
tions in two unknowns. One equation tells you that she consumes twice
as much whey as curds. The second equation is her budget equation.)
6.7 (1) Mary’s utility function is U(b, c)=b+ 100c−c2,wherebis the
number of silver bells in her garden and cis the number of cockle shells.
She has 500 square feet in her garden to allocate between silver bells and
cockle shells. Silver bells each take up 1 square foot and cockle shells each
take up 4 square feet. She gets both kinds of seeds for free.
(a) To maximize her utility, given the size of her garden, Mary should
(b) If she suddenly acquires an extra 100 square feet for her garden, how
74 DEMAND (Ch. 6)
(c) If Mary had only 144 square feet in her garden, how many cockle
(d) If Mary grows both silver bells and cockle shells, then we know that
6.8 (0) Casper consumes cocoa and cheese. He has an income of $16.
Cocoa is sold in an unusual way. There is only one supplier and the more
cocoa one buys from him, the higher the price one has to pay per unit.
In fact, xunits of cocoa will cost Casper a total of x2dollars. Cheese is
sold in the usual way at a price of $2 per unit. Casper’s budget equation,
therefore, is x2+2y=16wherexis his consumption of cocoa and yis
his consumption of cheese. Casper’s utility function is U(x, y)=3x+y.
(a) On the graph below, draw the boundary of Casper’s budget set in
blue ink. Use red ink to sketch two or three of his indifference curves.
04812
16
4
8
12
Cheese
16
Cocoa
Red
indifference
curves
Blue budget line
(b) Write an equation that says that at the point (x, y), the slope
of Casper’s budget “line” equals the slope of his indifference “curve.”
6.9 (0) Perhaps after all of the problems with imaginary people and
places, you would like to try a problem based on actual fact. The U.S.
government’s Bureau of Labor Statistics periodically makes studies of
family budgets and uses the results to compile the consumer price index.
These budget studies and a wealth of other interesting economic data can
be found in the annually published Handbook of Labor Statistics. The