ECON 300 Homework 2
Due Wednesday, October 20th
1. Jack’s preferences for jeans (J) and shirts (S) are given by the utility function:
a. Derive a formula for the marginal utility of jeans
b. Derive a formula for the marginal utility of shirts
c. Derive a formula for the marginal rate of substitution of shirts for jeans
d. Draw 1 indifference curve that shows Jack’s preferences (Hint: start with some point, then figure out what other
points on the same indifference curve would be)
I’ll start with the point (3,3). At this point, U(3,3)=27
All other points on this indifference curve must also have U(J,S)=27
So when S=1,
When S=2,
When S=3,
Alternatively, you could solve this equation for S and graph that equation:
e. Jack has $150 to spend on clothes every month. The price of jeans is $50 and the price of shirts is $20. What is
Jack’s marginal rate of transformation between jeans and shirts?
f.
Set up the Lagrangian and used constrained optimization to determine how many jeans and shirts Jack will buy.
Re-write (1) as
Re-write (2) as
Plug (4) into (3)
Jack will buy 1 shirt and 5 pairs of jeans.
2. Many water utilities charge customers in block pricing, where the price charged increases as you consume more
water. For instance, say Sarah’s utility charged her the following rates for water: $3/hundred cubic feet (HCF) for the
first 15 HCF consumed, $4/HCF for the second HCF consumed and $5/HCF for any water consumed over 30 HCF. Sarah
has $200 a month to spend on water and food (we’ll measure in meals). Let’s say meals cost $5 each. Draw Sarah’s
budget curve for food vs. water.
If Sarah spent all her money on food, she could buy $200/$5 = 40 meals, so the intercept on the food axis if 40.
If Sarah spent all her money on water, it would cost her $45 for the first 15 HCF (15*$3/HCF) and $60 for the
second 15 HCF (15*$4/HCF). So she would spend $105 on the first 30 HCF. That would leave $95 to buy water at
$5/HCF, so she could buy 19 more HCF. Thus if she spent all her money on water, she could buy 15 + 15 + 19 =
49 HCF.
We know the slope of the budget constraint will change at 15 and 30 HCF, since the price of water gets more
expensive. If water is on the x-axis, the slope will get steeper since the slope is
. At w=15, the
remaining income of 155 buys 31 meals. At w=30, the remaining income of $95 would buy 19 meals. So we can
plot the points where the slope changes at (15,31) and (30, 19).
3. We’ve talked about 2 special cases of goods, perfect substitutes and perfect complements. For both of these types,
a. Give a personal example of goods that are perfect substitutes and perfect complements
For me, I’m indifferent to Heintz vs. Hunts brand ketchups, so those would be perfect substitutes.
My bike lock and the key to it are perfect complements, since they are only useful if I have both.
b. Draw an indifference curve for each that goes through the point (3,3)
c. Draw an example of a budget constraint and show where the utility-maximizing bundle is
d. Make up a utility function for each type
Perfect substitutes:
Perfect complements:
Chapter 4 problems: 8, 14, 16, 26, 33, 39
8.
14. The 10,000 gallon limit won’t affect Max’s optimal bundle if that bundle includes water consumption of less than
10,000 gallons, since the limit would not be binding (as seen on left-hand graph). If Max was consuming more than
10,000 gallons, then the limit will be binding and Max will have to consume less water. This will put him on a lower
indifference curve, since if consuming less water brought him greater utility, he would have already been consuming less
water. The highest indifference curve he can hit is when he’s consuming 10,000 gallons, since he would have greater
utility from more water consumption, this is as high as his water consumption (and utility) can be.
16. Since consumers in both cities maximize their utility, we know that the marginal rate of substitution must be equal
to the marginal rate of transformation for both groups of customers. In Boston consumers pay twice as much for
avocados as tangerines, so
. In San Diego, the price of avocados is half the price of tangerines, so
The marginal rate of substitution of avocados for tangerines is
utility, the MRS equals the MRT,
. When consumers are maximizing
, so the MRS of Boston consumers is -1/2 while the MRS of San Diego
consumers is -2. So consumers in San Diego have the higher MRS, in that they will give up more avocados to get an
additional tangerine.
26. The slope of the budget line if you don’t join the pool is
You will consume where
your indifference curve is tangent to this budget line, shown by A on the graph. The slope of the budget line if you do
join the pool is
. Thus to be indifferent between joining and not joining the pool, you would
have to be on the same indifference curve as before, but with it tangent to a budget line with a slope of -5, as shown at
point B. Since the price of other goods is 1, you can buy Y goods if you don’t join the pool and Y-F goods if you do, the
fee to join F will be the size required to make a budget line with slope of -5 be tangent to the same IC.
33. Yes, those two utility functions do give the same amount of utility. We are really only interested in the rank of utility
values, not the actual values. Squaring and adding a constant to each possible value of U(Z,B) would not change the rank
of the various utilities.
39.
Chapter 5 problems: 5, 9, 10, 19, 30, 32, 34
5. a. The substitution effect causes her to buy more clothing when the price of clothing decreases. Clothes are now
relatively less expensive in terms of foregone food than they used to be, Michelle will want to shift her consumption to
include more clothes and less food.
b. When the price of clothes decreases, it increases Michelle’s real income since her buying power has increased. If
clothes are a normal good, the income effect will cause Michelle’s consumption of clothes to increase. If clothes are an
inferior good, then the income effect will cause her consumption of clothes to decrease. The income effect is
ambiguous in this case since we only know the total affect. On the one hand, it could be that both the substitution and
income effects increase consumption (i.e. clothes are a normal good). On the other hand, it could be that clothes are
inferior but the substitution effect is stronger than the income effect, so that the total effect would be an increase in
consumption.
9. During his first year, Ximing spends $400 on textbooks. For the second year, Ximing’s dad’s offer of $80 would
increase the amount Ximing has to spend on books by 20% (b/c 80/400=0.20). While that would exactly cover the
increase in textbook prices if all book prices rose 20% (leaving Ximing the same before and after the price increase), in
this case used textbooks only increase by 10%. That means Ximing will substitute away from new books towards used
books, and he therefore would not need a full 20% increase in income to buy all the books he would need. He will be
better off since his dad didn’t consider the substitution effect and thus overestimated the amount it would require cover
textbook price increases.
10. Jean is equally well off as she was before the cost of living adjustment. That is because with perfect complements,
there is no substitution effect. If the price of cream increases by more than the price of coffee (as shown below), Jean
will not be better off by consuming more coffee and less cream since she uses these goods together only. The COLA is
designed to maintain Jean’s current consumption, so she maintains her current utility.
19. The constraint on Bessie only working only 8 hours decreases her utility, since she would have chosen to work more
(the constraint is binding). Working 8 hours puts her on IC2,which brings less utility than IC1.
30.
Ice cream
IC1
IC2
BL2
BL1
Pie
Price
P1
P2
Demand
Pie
Q1
Q2
If ice cream (I) and pies (P) are perfect complements for Olivia, then her L-shaped indifference curves will be tangent to
budget lines where
. This can be substituted into the budget constraint
,.
32.
34.
Before the tax, Steve’s optimal utility is when he consumes 60 packs of cigarettes and 30 hamburgers.
With the tax,
After the tax, Steve’s optimal utility is when he consumes 60 packs of cigarettes and 20 hamburgers.
4. Grant earns $20/hour and gets an allowance of $100/week. He has 168 hours per week to spend on work and leisure
(L). The price of consumption goods (C) is $1.
a. Graph Grant’s budget constraint.
C
C = 3460
Slope = -20
L=0
H = 168
Kink @ H = 0
L = 168
C = 100
L
b. Suppose that Grant’s utility function is U(L,C)=LC. Solve for Grant’s optimal choice of labor supply, leisure and
consumption. Illustrate your answer in a diagram including the budget constraint and the indifference curve
corresponding to his optimal choice.
(1)Budget Line: C = 3460 – 20L
(2)MRS = PL/PC => C/L = 20 => C = 20L
(1) and (2) => 20L = 3460 – 20L => 40L = 3460
 L* = 3460/40 – 86.5
 H* = 168 – 86.5 = 81.5
 C* = 86.5*20 = 1730
C
L* = 86.5
C = 3460
L=0
H* = 81.5
C* = 1730
H = 168
Kink @ H = 0
L = 168
C = 100
L
b. Suppose Grant can earn time and a half for all work beyond 40 hours. Graph his budget constraint now.
Kink is @ H = 40, L = 128, C = 100+20*40 = 900
Cmax = 900 + 128*30 = 4740
C
Slope = -30
C = 4740
L=0
Kink @ L = 40, H = 128, C =
800 Slope = -20
Slope = -20
Slope = -20
H = 168
Kink @ H = 0, L = 168
C = 100
L
d. Suppose instead that if Grant earns more than $300 he is subject to a tax of 10% on all labor earnings beyond
that $300. Graph his budget constraint (drop the assumption from part c).
Kink @ WH = 300, 20H = 300,
H = 15, L = 168 – 15 = 153, C = 300 + 100 = 400 (Point “A”)
Cmax = 400 + 18*153 = 3154
W when H < 15 = 20
W’ when H≥15 = 20*.9 = 18
This is a progressive tax – like the US income tax.
C
Kink @ “A”
Slope = -18
C = 3154
L=0
H = 168
Slope = -20
Kink @ H = 0
L = 168
C = 100
L