Intermediate Micro Midterm
Econ 101
July 23, 2014
You will have 2 hours to complete this exam. There are a total of 120 points. Point values
correspond with the suggested time to spend on each question. Point values are given after
the question number. Please record your work and answers in the blue book. You may use
your exam sheet for scratch paper, but marks on the exam sheet will not be scored or
considered for partial credit. When you have finished, please turn in your exam sheet
along with your blue book.
15
1. For each of the following descriptions, select a utility function, and give a value for the constant k. The
goods in this economy are newspapers (n) and cups of coffee (c).
• u(n, c) =
√
2n2 + kc2
• u(n, c) = (2n + kc)2
• u(n, c) = min{kn, c}
• u(n, c) = kln(n) + c
• u(n, c) = nk c3
√
• u(n, c) = kln(n) + 1.5 c
5
(a) Betty finds herself spending half her income on newspapers, and half on coffee, no matter what the
prices.
This is a description of demand functions for Cobb-Douglas preferences, with parameters a = b.
u(n, c) = nk c3 , k = 3
5
(b) No matter what he currently has, Donald would be indifferent between trading 2 newspapers for a
cup of coffee, or not.
This is a description of indifference curves that are straight lines, with slope − 2newspapers
. u(n, c) =
1cup
(2n + kc)2 is a monotonic transformation of the more standard perfect substitutes utility function.
1 cup of coffee is worth 2 newspapers, so k = 2 ∗ 2 = 4.
5
(c) Lois will only read a newspaper while she drinks a cup of coffee, and will only drink a cup of coffee
while she reads a newspaper. If she has an excess of either, she doesn’t care.
Since these two goods will only be consumed in specific proportions, they are perfect complements
for Lois. u(n, c) = min{kn, c}, and since both goods are consumed in equal proportion, k = 1.
1
25
2. According to the Post, a typical millenial lives in their parents’ basement, and so pays no rent (sigh).
Suppose Max, a typical millenial, has the following utility function and budget for food (f ) and entertainment (n).
1
3
ln(f − 4) + ln(n)
4
4
100 = pf + 5n
u(f, n) =
10
(a) Find the demand functions for f and n. Take as given that there is an interior solution. (This is
the case as long as p ≤ 25).
This utility function looks similar to Cobb-Douglas but is not the same. So, we need to solve it
with MRS and the price ratio (or one of the other calc-based methods).
1 1
4f −4
31
M Un =
4n
1 n
M RS =
3f −4
1 n
p
=
5
3f −4
3p
(f − 4)
n=
5
3p
100 = pf + 5 ∗ (f − 4)
5
100 + 12p = 4pf
25
f (p) = 3 +
p
3
n(p) = 15 − p
5
M Uf =
10
(
pf
= M RS)
pn
(Budget)
(b) Find a formula for the own price elasticity of demand for f . What is f,p when p = 2?
Taking the derivative of (1),
df (p)
25
=− 2
dp
p
(1)
(2)
(3)
The formula for elasticity is
df (p) p
dp f (p)
25 p
= 2
p 3 + 25
p
f,p = −
f,p =
When p = 2, f,p (p = 2) =
5
25
3p + 25
(4)
25
31 .
(c) Are goods n and f complements, substitutes, or neither, at p = 2? How do you know? **For full
credit, answer without taking the derivative of the demand function for n.**
Complements. Demand for good f is inelastic when p = 2, or at any price other than 0. This
means that an increase in the price of good f leads to an increase in expenditure on good f . There
is less money left to spend on good n, so n decreases. By definition, if ↑ pf causes ↓ n, they are
complements.
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25
3. Ann has an endowment of (ωx , ωy ) = (5, 5). She has utility function u(x, y) = 3ln(x) + ln(y). Fix the
price of good y at py = 1, and let px = p.
10
(a) Find Ann’s gross demand functions for goods x and y.
If Ann sells her endowment, she’ll have m = 5p + 5 to spend. With Cobb-Douglas utility, her
demand functions are
15 15
3 5p + 5
=
+
4 p
4
4p
5p + 5
y(p) =
4
x(p) =
(5)
(6)
5
(b) At p = 5, is Ann a net demander or supplier of good x? What is her net demand or net supply of
good x at this price?
15
1
1
1
Ann’s gross demand is x(p = 5) = 15
4 + 4∗5 = 4 2 . Her net demand is 4 2 − 5 = 2 , so she is a net
1
supplier of good x, by 2 of a unit.
5
(c) If p rises to from 5 to 5.41924, is Ann better off or worse off?
Ann is a net supplier of x at p = 5. If the price of x goes up, she’ll continue to be a net supplier,
but be able to afford more. She is better off.
5
(d) At what price is Ann neither a net demander, nor a net supplier of good x?
Solve
5 = x(p)
15 15
5=
+
4
4p
15
5
=
4
4p
p=3
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35
4. The way he sees it, Jared splits his budget between two things: laser tag, and everything else. Let ` be
the number of games of laser tag he plays in a month, and e be dollars spent on everything else.
His utility function is u(`, e) = 200ln(`) + e. He has $2000 of income every month. Laser tag games are
$8 each.
5
(a) Fill in the blank: With price $1, good e is a numeraire good.
Composite, normal, elastic, and luxury were also acceptable answers.
5
(b) How many games of laser tag will Jared play this month? Call this `∗ .
200
=8
`
`∗ = 25
5
(c) The laser tag arena will have a sale in September: $5 per game. What will Jared’s demand for laser
tag games be in September? Call this `∗∗
200
=5
`
`∗∗ = 40
5
(d) What income would just allow Jared to afford this month’s optimal (`∗ , e∗ ) bundle in September?
m0 = 5`∗ + 8(m − `∗ )
m0 = 1925
5
(e) What would be Jared’s Slutsky compensated demand from this price drop?
200
=5
`
`c = 40
5
(f) What are the income effect, and the substitution effect of the price drop, for Jared?
∆`s = `c − `∗ = 15
∆`n = `∗∗ − `c = 0
5
(g) Are laser tag games a normal good, an inferior good, or neither, for Jared?
Neither. The income effect was zero.
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20
5. Wendy the worker gets a wage of $10 per hour worked, and has 3000 hours to divide between work and
leisure.
The government has decided that everyone who makes less than (or equal to) $20, 000 should get assistance. It gives a $2500 payment to everyone who makes no more than $20, 000 (in wages).
On the other hand, the government has decided that everyone who makes more than $20, 000 in wage
income can afford to pay taxes. There is a 50% tax rate on wage income above $20, 000. For every dollar
in wage income above $20, 000, it also reduces the low-income payment by $0.50. Once wage income
reaches $25, 000, the low-income supports have been completely phased out, so there is no more subsidy
reduction, but there is still the 50% tax rate.
Define Wendy’s income, m, as the total of her wage income plus any government benefits minus any
taxes paid. Define leisure as any time not spent working ` = 3000 − work.
15
(a) Make a large graph of all possible bundles of (`, m) for Wendy, with ` on the horizontal (x-)axis, and m on the vertical (y-)axis. Label the endpoints, and any kink points, and all slopes.
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5
(b) Wendy has monotone preferences. Are there any (`, m) bundles on the budget line that she will
definitely not choose? Circle them on your graph, and explain your choice in no more than 3
sentences.
If Wendy works between 2000 and 2500 hours (500-1000 hours of leisure), her income is the same
because of the payment phase-out. Given her monotone preferences, she would rather work less
and earn the same.
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