Intermediate Microeconomic Theory, 10/9/2001
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PS#2 Answers
1. a) If Coke and Pepsi are perfect substitutes their indifference curves are a straight line
with a slope of –1, as in the graph below. The thick lines represent the indifference
curves. Letting the price of Pepsi vary, holding the price of Coke constant, is
represented by pivoting the budget line, represented by the thin lines on the graph.
The intercept remains constant, at Point A, since the price of Coke and income are
fixed, but the change in the price of Pepsi causes the slope to change. The movement
of the budget lines from AB, to AC, to AD, etc. represent a fall in the price of Pepsi.
Since the slope of the indifference curves is –1, the absolute value of the slopes of AB
and AC are greater that one. Since the slope of the budget line is –Ppepsi/Pcoke , this
means that for AB and AC, the price of Pepsi is greater than the price of Coke. At any
such price ratio, steeper than the indifference curve, the consumer will buy only Coke
and no Pepsi. Why? Buying any Pepsi at all will put her on a lower indifference
curve. When the budget line becomes AD, its slope is –1, which overlays one of the
indifference curves. This means that the price of Pepsi and Coke are the same. When
they are the same, the consumer could purchase any combination of the two and still
be on the highest indifference curve possible. When the price of Pepsi falls below the
price of Coke, the consumer will choose to purchase only Pepsi, because any Coke
purchase would put her on a lower indifference curve. Therefore, the priceconsumption line follows the thick blue line in the diagram going from A to D, then,
it turns and follows the X-axis from E to F, continuing outward infinitely.
To indicate the price of Pepsi corresponding to each budget line, let’s use the notation
PAB, PAC, PAD, etc., to represent the price of Pepsi on budget lines AB, AC, AD, etc.,
respectively. Then the demand curve for Pepsi is as in the second graph below. Note
that it has two parts, a flat part, corresponding with the section of the priceconsumption path where the price of Pepsi and Coke are the same, such that there is
an entire range of points, ranging from zero consumption of Pepsi to zeroconsumption of Coke, all of which are possible optimal choices for that price ratio.
For all price ratios below that, there is only one optimal choice, always involving only
Pepsi consumption, but the amount consumed increases as the price of Pepsi falls.
Coke
slope = -1
A
B
C D
E
F
Pepsi
Intermediate Microeconomic Theory, 10/9/2001
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PAB
PAC
PAD = 1
PAE
PAF
D
F
Pepsi
If the price is above PAD = $1, there is no demand for Pepsi. If the price is equal to $1,
the demand could be zero or it could be as high as the amount of Pepsi indicated by
the Point D in the first and second graphs, or anything in between. If the price of
Pepsi less than $1, consumers buy only Pepsi (since Coke is $1 and they don’t care
which they buy). But as the price falls further and further, consumers have more
purchasing power, so they will buy more and more Pepsi. As for the downward
sloping section, the curve is not necessarily linear. In fact, it may be shaped like the
dotted line. However, for our purposes, what I want you to recognize is that it is
downward sloping, hence if you drew a linear downwardly sloping segment for prices
below $1, that is adequate.
b) If the price of Pepsi is $1 a can, consumers pay the same price for Coke or Pepsi.
Since they are always indifferent to a one-for-one exchange between the two, any
combination of the two is possible. But if the price of Pepsi falls just 1 cent, the
consumer is placed on the highest possible indifference curve by buying only Pepsi.
She will buy no Coke at that price.
c) If she has the preferences below, the demand curve is found using the same
exercise as before of varying the price of Pepsi, holding the price of Coke constant,
which causes the budget line to pivot. The main difference is that the consumer will
exclusively consume only one of the two only when the prices are extremely
different, and there is no price which will cause the budget line to coincide with )or
overlay) an indifference curve. That means there is no horizontal section to the
demand curve, where more than one optimal level of Pepsi consumption corresponds
with a single price. Overall, as the price of Pepsi falls, and the price of Coke remains
constant, the demand for Pepsi increases, which the demand curve below illustrates.
Note that what I most care about is that you recognize that the curve is downward
sloping. In fact, it could have a different shape, such as the dotted curve, but you
really don’t have enough information to determine its precise shape. Therefore, any
downward sloping line is sufficient.
Intermediate Microeconomic Theory, 10/9/2001
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Coke
A
Price consumption path
Pepsi
B
C D
E
F
Price
of
Pepsi
Pepsi
d) The cross-price effect is a substitute effect. As the price of Pepsi falls relative to
the price of Coke, the demand for Coke falls.
2. The income consumption path for McDonald’s hamburgers, as estimated by these
market analysts, as the shape as below of the thicker curve. Notice the dotted line—a
ray—which is tangent to the income-consumption path. If the demand for McDonald’s
hamburger were to increase in the same proportion as the increase in income, it would
follow a straight line from the origin, like this one. The income-consumption path drawn
shows the demand for hamburgers increasing in greater proportion than the increase in
income in the zone below the tangent and in smaller proportion to the increase in income
above the tangent. Below the tangent, in lower income neighborhoods, McDonald’s
hamburgers are luxuries, but in higher income neighborhoods, above the tangent, they are
non-luxuries.
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McDonald‘s
Hamburgers
Income-consumption path
other goods
3. The MRS = -2(Y/X). Assuming diminishing MRS, this expression must indicate the
slope of the indifference curve when X is on the horizontal axis. If X is on the horizontal
axis, the MRS falls as X increases, but Y is on the horizontal axis, it increases, which is
contrary to our standard assumptions about preferences.
The budget line is: I = p X X + pY Y or Y =
The tangency condition is:
p
Y
−2 =− X
or
X
pY
Y =
p
I
− x X.
pY
py
1 pX
X
2 pY
Substituting into the budget equation:
 1 pX  3
I = p X X + pY Y = p X X + pY 
X  = p X X
 2 pY  2
The optimal bundles, satisfying the tangency condition and budget constraint are:
2 I
3 pX
1 pX 2 I
1 I
Y=
=
2 pY 3 p X 3 pY
X=
Intermediate Microeconomic Theory, 10/9/2001
a) X =
2 120
⋅
= 40
3 2
1 120
Y= ⋅
= 40
3 1
b) X =
2 120
⋅
= 20
3 4
1 120
Y= ⋅
= 40
3 1
c) X =
2 120
⋅
= 10
3 8
1 120
Y= ⋅
= 40
3 1
d) X =
2 240
⋅
= 20
3 8
1 240
Y= ⋅
= 80
3 1
5
The own-price effect on X is inverse, as one would expect. The cross-price effect is zero.
That is, as the price of X increases, there is no effect on the demand for Y. The income
effect indicates that both X and Y are normal goods.
4. a) Judy’s indifference map is as shown in the graph below as the three indifference
curves at right angles falling along the ray with a slope of 1/k.
M
slope = 1/k
480
320
16
F
b) Her time constraint is:
M = 20(16 – F).
On the graph, the downwardly sloping line crossing the Y-axis at 320 and the X-axis at
16 represents the time constraint. The slope of the line is -20. If she enjoys no free time,
she will make $320. If she does not work, she can enjoy 16 hours of free time (excluding
sleep), but she earns no income.
c) If her hourly wage were to increase to $30, the slope of the time constraint will rise, in
absolute value, from -20 to -30. The total hours she can enjoy of free time, if she does not
work remains 16, but the amount she can earn if she enjoys no free time is now $480.
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The point at which she will choose to work is the point where the new time constraint
line and the ray with the slope of 1/k intersect. The ray is the income-consumption path.
At the higher wage, she will choose to work less or to enjoy more free time.
d) The indifference curves that are right angles indicate that M and F are perfect
complements – always consumed in the same proportion. M and F are both normal
goods.
5. a) Without food stamps, the budget constraint is a line with a slope of -$1/$1 = -1
which intercepts the vertical axis at I/pN = $1000/$1 = 1000. It intercepts the horizontal
axis as I/pF = $1000/$1 = 1000. With food stamps, the family can use the $1000 just as
before, but in addition, they have means to purchase $100 worth of food. That means (i)
it is now possible, if the family wishes, to purchase $100 of food and $1000 of non-food
– that is 100 units of F and 1000 units of N, a point on the budget constraint with food
stamps, but not without. (ii) If the family hypothetically desires to purchase no food,
setting F = 0, the family may still purchase no more than 1000 units of N. (iii) If the
family wishes to use its resources entirely for food, with food stamps, it can obtain 1100
of units of food. F= 0, N=1000 and F=100, N=1000 are on the constraiting, but at no
point can N exceed 1000. That implies the flat portion of the constraint between F = 0
and F = 100. If F exceeds 100, the rate of exchange between F and N is a constant –1 per
unit, since the prices are the same. That implies the linear portion of the constraint
running from F = 100, N = 1000 to F = 1100, N = 0 (indicated by the heavy line on the
graph.) An alternative cash subsidy would be linear and parallel to the non-food-stamp
budget line but higher, with intercepts of 1100 and 1100 on both axes.
Non-Food (N)
1100
Tangency showing optimum
with a cash-subsidy budget
1050
Budget constraint
without food stamps
1000
Budget constraint with
food stamps
Food (F)
100
1000
1100
Intermediate Microeconomic Theory, 10/9/2001
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b) The three indifference curves drawn illustrate this. If the family has food stamps that
cannot be used for N, it will choose to consume 1000 N and 100 F, at point on the corner
of the two segments of the food-stamp budget constraint. If the same family were given a
cash subsidy instead, It could reach a higher level of satisfaction by substituting away
from F toward N, i.e consuming less F and more N, at the tangency of the cash-subsidy
budget line with the higher indifference curve drawn.
c) This discussion may vary, but the issue should center on whether society is better (or
worse) off when food-stamp recipients are permitted to use the subsidy for non-food
items.
d) If the family can exchange food stamps for money (which can then be used to
purchase non-food) at a rate of 50¢ in money for each $1 worth of food stamps, the foodstamp budget line would be the same as the former food-stamp budget line for amounts
of F exceeded 100, but for all amounts of F between 0 and 100, the line will be a
downwardly sloped line with a slope of –1/2, intercepting the vertical axis at 1050. That
would be the amount of N the family may purchase if it exchanges all its food-stamps for
money to buy non-food items with.
6. a) (NB: Your answers may differ marginally from mine because of round-off error.
Different calculators or computer programs cut-off decimals at different lengths and
introduce different round-off error.)
Assuming OPEC supplies all the world’s crude:
Point elasticity estimate:
∆y / yo
(23.2 − 24.2) / 24.2
−0.041
=
=
= 0.50
∆p / po
(26.84 − 24.78) / 24.78
0.083
Arc elasticity estimate:
∆Y /[(Yo + Y1 ) / 2]
(23.2 − 24.2) /[(24.2 + 23.2) / 2]
=
∆p /[( po + p1 ) / 2] (26.84 − 24.78) /[(24.78 + 26.84) / 2]
∆Y /[(Yo + Y1 ) / 2]
−0.042
=
= 0.51
∆p /[( po + p1 ) / 2]
0.080
b)
Adding the additional information that OPEC supplies about 40% of the world’s
crude oil will give different estimates. Let Yt represent world production in period t and yt
Intermediate Microeconomic Theory, 10/9/2001
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represent OPEC’s production, for the periods t = 0, 1. Then in each period yt = 0.4Yt , or
Yt =
yt
0.4
. We need to find the price elasticity of demand , which equals
∆Y / Yo
. We
∆p / po
can find Yo easily. Since yo = 24.2 , Yt = 24.2 / 0.4 = 60.5 . We assume non-OPEC
producers do not change their production. That means the entire change in the market is
represented by the change in OPEC’s production, that is: 23.2 – 24.2 = – 1. That
produces a fall in world production of 60.5 – 1 = 59.5 million barrels per day.
The point elasticity is
∆Y / Yo
(−1) / 60.5
−0.017
=
=
= 0.20
∆p / po
(26.84 − 24.78) / 24.78
0.083
The arc elasticity estimate is:
∆Y /[(Yo + Y1 ) / 2]
−1/[(60.5 + 59.5) / 2]
=
∆p /[( po + p1 ) / 2] (26.84 − 24.78) /[(24.78 + 26.84) / 2]
∆Y /[(Yo + Y1 ) / 2]
−0.017
=
= 0.21
∆p /[( po + p1 ) / 2]
0.080
c) The arc estimate is probably more accurate. The point estimate is more
convenient and easier to calculate. They both give us a small elasticity and lead us
to the same conclusions.
d) The oil market is price inelastic because the price elasticity of demand is less than
one.
e) The action of reducing output with the objective of raising the market price will
be more effective if the market is price inelastic.