Law 678: Introduction to Microeconomics
STANFORD LAW SCHOOL
Autumn Quarter 2012
Instructor: Alex Gould
Problem Set #2: Solutions
Answers to Chapter 3 Questions for Review
1. You will be as well off as a year ago; your budget line will remain the same.
2. False. The slope of the budget constraint tells us only the ratio of the prices of the two
goods.
3. False. Diminishing MRS explains the convexity of the indifference curve, but not the
downward slope.
6. One bundle may be within the individual's opportunity set while the other is not.
8. True. The corner solution (a) is on a higher indifference curve than the corresponding
tangency (b). Which corner becomes the solution depends on the slope of the budget
constraint. There can be a solution in either corner, as shown in the graphs below. Quantity
discounts will not change this outcome scenario.
Y
a
Y
B
b
a
X
X
9. Suppose that Ralph's current consumption bundle is given by the point A in the diagram.
The information given tells us that on the budget with M+10 units of income, Ralph would
consume at the point B, and that B is equally preferred to C. This can happen only if the
indifference curve passing through B and C does not have the usual convex shape. His
indifference curve through B and C could, for example, be a straight line, indicating that tuna
fish and Marshallian money are perfect substitutes in this region. (If the indifference curve
through B and C were convex, then Ralph's optimal bundle would lie between B and C,
which means that he would spend some of the extra $10 on tuna fish.)
Y
M + 10
M
B
A
C
Answers to Chapter 3 Problems
3. a) Pecans are equally preferred to macadamias, which are preferred to almonds, which are
preferred to walnuts, so by transitivity it follows that pecans are preferred to walnuts.
b) Macadamias are preferred to almonds and cashews are preferred to almonds. Transitivity
tells us nothing here about the preference ranking of macadamias and cashews.
4. True. Each price increases by 15%, so that – Px/Py is unchanged.
Y
M/80
M/92
slope = 120/80 = 3/2
Slope = 138/92 =3/2
M/138 M/120
X
5. a)
Y
150
60
Milk Balls
b) The opportunity cost of an additional unit of the composite good is 1/2.5 = 0.4 bags of milk
balls.
6. a)
Y
150
100
60
Milk Ball
b) The opportunity cost of a unit of the composite good is now 0.6 bags of milk balls
7.a)
Y
150
90
Milk Balls
7. b) The opportunity cost of a unit of the composite good is again 0.6 bags of milk balls.
8. a) To get any enjoyment from them, Picabo must consume skis and bindings in exactly the
right proportion. This means that the satisfaction Picabo gets from the bundle consisting of 4
pairs of skis per year and 5 pairs of bindings will be no greater than the satisfaction provided
by the bundle (4, 4). Thus the bundle consisting of 4 pairs of skis per year and 5 pairs of
bindings lies on exactly the same indifference curve as the original bundle. By similar
reasoning, the bundle consisting of 5 pairs of skis per year and 4 pairs of bindings lies on this
indifference curve as well. Proceeding in like fashion, we can trace out the entire
indifference curve passing through the bundle (4, 4).
b) Skis (pairs/yr)
20
16
I4
12
I3
8
I2
4
I1
0
4
8
12
16
18
20
Bindings (pairs/yr)
9. Picabo's budget cnstraint is B = 15 - 2S. Initially, she needs the same number of pairs of skis
and bindings S = B. Inserting this consumption equation into her budget constraint yields B =
15 - 2B, or 3B = 15, which solves for B = 5 pairs of bindings (and thus S = 5 pairs of skis).
As an aggressive skier, she needs twice as many skis as bindings S = 2B. Inserting this
consumption equation into her budget constraint yields B = 15 - 4B, or 5B = 15, which solves
for B = 3 pairs of bidings (and thus S = 6 pairs of skis). She consumes more skis and fewer
bindings as an aggressive skier than as a recreational skier. See graph below.
Pairs of Bindings per Year (B)
15
B = 15 – 2S
B+S
5
B = S/2
3
0
5
6
Pairs of Skis per year (S)
7.5
10. Alexi's budget constraint is T = 75 - (3/4)C. Her perfect substitute preferences yield linear
indifference curves with slope equal to negative one, such as T = 75 - C and T = 100 - C. By
consuming 90/0.90 = 100 cups of coffee each month, she reaches a higher indifference curve than
consuming 90/1.20 = 75 cups of tea (or any affordable mixture of coffee and tea). Thus Alexi
buys 100 cups of coffee and no tea. Any increase in the price of coffee would force Alexi to a
lower indifference curve, and thus lower her standard of living.
Cups of Tea/month
(T)
100
T = 100 – C
75
T = 75 – (3/4)C
0
100
Cups of Coffee per month (C)
11. In the diagram, suppose we start at bundle A and then take away ΔP units of pears. How
many more units of apples would we have to give Eve to make her just as happy as at A?
The answer is none, because she didn't care about pears in the first place, and therefore
suffered no loss in satisfaction when we took ΔP units of pears away. Bundle B is thus on the
same indifference curve as bundle A, as are all other bundles on the horizontal line through
A. All of Eve's indifference curves are in fact horizontal lines, as shown.
Apples (lbs/wk)
Increasing satisfaction
⇑
B
ΔP
A
Pears (lbs/wk)
12. Again start at a given bundle, such as A in the left panel of the diagram below. Then take
away a small amount of food, ΔF, and ask what change in smoke, ΔS, would be required to
restore Koop's original satisfaction level. In the standard case, when we take one good away
we need to add more of the other. This time, however, we compensate by taking away some
of the other good. Thus, when we take ΔF units of food away from Koop, we must reduce
the smoke level by ΔS in order to restore his original satisfaction level. This tells us that the
indifference curve through A slopes upward, not downward. Koop would be just as happy
with a smaller meal served in a restaurant with a no-smoking section as he would with a
larger meal served in a restaurant without one. It is usually possible to translate the
consumer's indifference curves into ones with the conventional downward slope by simply
redefining the undesirable good. Thus, if we might focus not on smoke, an undesirable good,
but on cleanliness (the absence of smoke), which is clearly desirable. So doing would recast
the indifference map in the left panel of the diagram as the much more conventional-looking
one in the right panel.
Food (lbs/wk)
Increasing Satisfaction
⇑
I3
I2
I1
Food
Increasing Satisfaction
⇑
B
A
ΔF
I3
I2
ΔS
Smoke (micrograms/wk)
I1
Cleanliness
13. You prefer to maximize profit, which is the same under the two rate structures, making you
indifferent between them.
14. a)
b) If plays cost $12 and movies cost $4, the budget line is Bo, which has exactly the same
slope as Paula's indifference curves. She will be indifferent between all the bundles on B0.
c) On B1, she will consume 10 plays.
15.
Y
Increasing
satisfaction
Y
Increasing
Satisfaction
Garbage
Garbage
16. Let C = coffee (ounces/week) and M = milk (ounces/week). Because of Boris's preferences,
C = 4 M. At the original prices we have:
4M(l) + M(0.5) = 9
4.5M = 9
So M=2 and C=8
Let M' and C' be the new values of milk and coffee. Again, we know that C'=4M'. With the
new prices we have:
(4M')(3.25) + M'(.5) = 9
13M' + 0.5M' = 9, 13.5M' = 9, M' = 2/3
C = 8/3
17. An unrestricted cash grant would correspond to the budget B1 in the diagram. On B1 the
university would want to spend more than 2M on non-secular activities anyway, so the
restriction will have no effect. This result is analogous to the result in the text concerning the
restriction that food stamps not be spent on cigarettes. Provided the recipient would have
spent more on food than he received in stamps, such a restriction has no effect.
Non-secular Activities
14
12
10
B0
B1
8
6
4
2
0
18.
2
4
6
8
10
12
14
16 Secular Activities
23.
Quantity of Soft Drinks
Note that the budget constraint is not a line but rather the set of points that are shown in the
diagram and the ones that are below them. To construct this, for each level of composite
good, from 0 to 12, determine the maximum number of bottles you can buy with the leftover
money. For example, for Y=4, you have $8 left. The best you can do is 1 large and 1 small,
which gives 11 tickets. Remember that you can't buy a fraction of a set. Notice that point
(0,12) is also on the budget constraint.
24. Assume that the quality of the food is the same in both restaurants, so that price is the only
difference that matters to consumers. In the first restaurant, the $15 flat tip is a fixed cost: it
does not affect the cost of additional items ordered from the menu. In the second restaurant,
by contrast, the price will be 15 percent higher for each extra item you order. The marginal
cost is higher. The average meal is $100 in the first restaurant, which with tip comes to $115.
The same amount of food would cost the same in the second restaurant. But because the cost
of each additional item is higher there, we expect that less food will be consumed in the
second restaurant. Note the similarity of this problem to the pizza experiment described in
Chapter 1.
Answers to Chapter 4 Questions for Review
6. Vertical summation would mean that each good could be jointly consumed. Horizontal
summation means each person consumes their commodity and excludes others from it.
7. An elastic demand leads to revenue increases if price falls. An inelastic demand leads to
revenue increases if the price increases. A unitary demand curve results in constant revenue
no matter what price does. If price goes the opposite direction from that listed above, the
revenue moves in the opposite direction also.
8. The slope of the demand will give only an absolute change number. It does not give a
proportionate change. Since price sensitivity has little meaning apart from the proportion of
change, elasticity is far better than slope at showing a useful responsiveness of demand to
price.
9. Unitary
13. False. In the diagram below, an increase in the price of X leads to a reduction in the
amount of X consumed, but an increase in the quantity of Y.
Y
Positive income effect for both X
and Y because the quantity of
both X and Y increase when
income is increased .
Chang
e in Y
X
Change in X
14. The demand for tennis balls is elastic. When its price goes up, the total expenditure on the
balls goes down. Thus, the share of income available for tickets increases. Since their price is
constant, he consumes more tickets.
15. False. Look at Figure 4-13 in the text. Both individuals have linear demand curves, but
the aggregate demand curve is kinked, not straight.
16. No. If bread is an inferior good, then as income increases, quantity demanded of bread
decreases. If butter were an inferior good also, then likewise, quantity demanded of butter
would decline as income grows. However, spending on both goods cannot decline, because
there would be no way of spending the added income. Thus, not all goods can be inferior.
Answers to Chapter 4 Problems
1. Sam’s budget constraint is 2OJ + AJ = 6 or OJ = 3 – (1/2)AJ. Sam’s indifference curves
are straight lines with constant MRS = 1/3. Sam’s optimal bundle is to consume no apple
juice and three cups of orange juice. When the price of apple juice doubles, Sam would
not need any additional income to afford his original consumption bundle, since he does
not consume any apple juice.
Orange Juice in
Cups
3
Bs’ = B1
B0
0
3
6
Apple Juice in cups/week
ICs
9
4. First solve the demand curve for Q and multiply the result by 10. Then solve back in terms
of P to get P = 101 – Q for the market demand. At price $1/cup the individual consumes 10
cups and the market consumes 100 cups.
Price
101
10.1
101 Cups
P
2
1
elastic
unit-elastic
.
inelastic
50
100
Q
6.a) (see diagram above)
b) At (1, 50), total revenue is maximized since this is the unit-elastic point. At higher prices,
revenue decreases since it is the elastic region. At lower prices, revenue again decreases since
it is the inelastic region.
7. a) P=$3, Q=8000, Revenue=$21,000
b) Ep = (P/Q)(1/slope) = (3/7000)(-1000) = - 3/7
c) A price increase will increase revenue since current price is in the inelastic region.
d) Since substitution chances are increased, demand for the bridge will become more elastic.
8. We can’t know. We are only given that income elasticity of demand for safety (Ei) is positive.
For necessities, we have 0 < Ei< 1, and for luxury goods we have Ei> 1.
We need more information to determine whether Ei> 1 or not.
13. a) 300 = 1800 - 15P, so P = 100, which gives TR = 100(300) = 30000 cents/day.
b) Expressing the demand curve in terms of price, we have P = 120 - Q/15. Price elasticity =
(P/Q) (1/slope) = (1/3)(-15) = -5 .
c) Since demand is elastic with respect to price, a reduction in price will increase total revenue.
d) Maximum total revenue occurs where price elasticity = -1.
(P/Q)(1/slope) = (P/Q)(-15) = -1, so maximum TR will occur when P = Q/15.
Substituting P = Q/15 back into the demand curve we get Q/15 = 120 - Q/15, or
2Q/15 = 120, which solves for Q = 900. At Q = 900, we have P = 60.
14. In absolute value terms, where price elasticity = Ep
Ep A = Q2A/AP2 = 2
Ep B = Q2B/P2B = 1
Ep C = Q1C/P1C = 1
Ep D = Q1D/P1D = 3
Ep E = Q1E/P2E = 1
So Ep D > Ep A > Ep B = Ep C = Ep E
18. Wheat and rice are perfect substitutes for Smith, and her indifference curves are shown as the
heavy downward-sloping 45° lines in the diagram. The lighter downward-sloping straight
lines, B1_B4, are the budget constraints that correspond to four arbitrarily chosen prices of
wheat, namely, $12/lb, $4/lb, $2/lb, and $1.50/lb, respectively. The first two of these prices
exceed the price of rice, so Smith ends up spending all of her food budget on rice. Bundle A
denotes the optimum purchase of wheat when the price of wheat is $12/lb (budget constraint
B1); and bundles C, D, and F are the corresponding bundles for the remaining prices (budget
constraints B2, B3, and B4, respectively). As noted, the amount of wheat in both A and C is
zero. Once the price of wheat falls below the price of rice, Smith does best to spend all of her
food budget on wheat. When wheat costs $2/lb, for example, she will buy
($24/wk)/($2/lb)=12 lbs/wk (bundle D on B3); and at $1.50/lb, she will buy 16 lbs/wk
(bundle F on B4). The heavy line labeled PCC is Smith's price-consumption curve.
Rice (lbs/wk)
18
16
14
12
PCC
10
8
AC
6
4
B1
2
0
B4
B3
B2
2
4
6
8
F
D
10
12
14
16
18
20
Wheat (lbs/wk)
To construct Smith's demand curve for wheat, we can retrieve the price-quantity pairs
from her PCC and plot them in a separate diagram, just as before. But an even easier way
is available in this particular case. It is to note that her behavior may be summarized by
the following purchase rule: when the price of wheat, PW, is below the price of rice, she
will buy $24/PW pounds of wheat, and when PW is above the price of rice, she will buy
no wheat at all. The demand curve that corresponds to this purchase rule is plotted as the
heavy line in the diagram below
22
P ($/lb)
W
Demand curve for wheat
6
5
4
Price of rice =
3
2
1.5
1
0
19.
D
Wheat (lbs/wk)
4
8
12
16
20
24
.
Rice (lbs/wk)
12
PCC
10
8
6
4
2
0
24/9
2
24/5
24/3
Wheat (lbs/wk)
24/2
3 4 5
Price of Wheat ($/lb)
D
9
8
7
6
5
4
3
2
D
1
Wheat (lbs/wk)
0
1
2
3
4
5
6
7