Econ 100A Course Pack - Answers to Practice Problems Spring 2022.
Econ 100A: Microeconomics Course Pack
Answers to problems
Jim Campbell
UC Berkeley
Spring 2022
This document has answers to the problems from the course pack. Some are answers only, and
some have a little more explanation; I’ve tried my best to make sure that the question numbering
matches the course pack itself, but since the course pack changes slightly every semester there
might be the odd discrepancy. Note that this pack doesn’t cover discussion prompts, despite the
fact that I think they are very important to test your understanding. The reason is that the
discussion prompts are typically subjective, with no single correct approach or answer, and so I
prefer you to think through them independently or with classmates rather than me leading you in
any particular direction.
Finally, I strongly encourage you to try the problems yourself or watch the demonstrations of
them in class or section before you turn to the answers here. If you look at the answers too soon, it
can be too easy to convince yourself that you could have gotten the answer! The places where you
are less confident will reveal what you need to study a little more, and working on the problems
yourself will make you remember the techniques better.
1
,Econ 100A, Spring 2022 Course pack suggested answers
Contents
1 Preferences and utility 3
2 The rational choice model 7
3 Demand 14
4 An application of consumer theory to labor supply 22
5 Choice under uncertainty and risk aversion 28
6 General equilibrium in an exchange economy 32
7 Producer theory 40
8 Perfect competition and partial equilibrium 45
9 General equilibrium with production 52
10 Monopoly and market power 56
11 Externalities 62
12 Public goods 68
13 Oligopoly theory 74
14 Markets with asymmetric information 79
15 Answers to example midterms from the course pack 82
16 Answers to math refresher problems 92
2
,Econ 100A, Spring 2022 Course pack suggested answers
1 Preferences and utility
1.
2. Jim likes both watching baseball (x1, measured in ‘hours per day’) and teaching microeco-
nomics (x2, measured in ‘hours per day’). He has decided that his utility function for these
two goods is u(x1, x2) = x3x1 2.
a) MU1 = δu = 3x21 x2
δx1
b) MU2 = δu = x31
δx2
c) MRS = MU1 3x x2 2
3x2
MU2 = x131 = x1
d) At this point MRS = 6 = 3 . This means the slope of his indifference curve at that point
8 4
is − 43 ; at this point he would be willing to swap hours of teaching for hours of baseball
at the rate of 34 of an hour of teaching to 1 hour of baseball.
6
2 = x2 , so MRS = 12 = x1∗ 2
6 2 x 6 x2
e) We could calculate this: MU 1 = x1 and MU = 3xx21. It’s
x2
the same! This is because the new utility function is a monotonic transformation of the
old one: if we took the natural log of the original function and multiplied by 2, we’d
get the second. Both of those transformations are order-preserving: whenever the first
function gives a higher utility number to bundle A than B, the second utility function
definitely does as well. This shows the ordinal property of utility functions. Note that
despite all that, the MRS, which is a real, behavioral thing for this decision maker, is
everywhere identical for both utility functions!
4
3. a) MRS = MU1 = x1
= 2x2
MU2 2 x1
x2
1
√
b) MRS = MU1
MU 2
= 2 x1
2 = √1
4 x1
3
, Econ 100A, Spring 2022 Course pack suggested answers
f ′(x1)
c) MRS = MU 1
MU 2
= 1 = f J(x1)
d) MRS = MU
MU
1
= 1
1
=1
2
4. Using two or three non-technical sentences:
a) Nonconvexity could be because the consumer prefers bundles with lots of either one good
or the other rather than bundles that have a balance of different types of good. For
example, maybe I’d prefer to spend all my time on playing soccer or all my time on
playing basketball so that I can practice and get as good as possible at one sport rather
than splitting my time between the two sports.
b) Upward-sloping indifference curves represent a case in which the consumer gains utility
from more of one good but loses utility from more of the other good—colloquially, to
them, one good is good and one is bad. u = x1−x2 would be a simple example that
generates this. Sketch it and see! This is one example of preferences that do not satisfy
monotonicity.
c) The picture in part a) shows indifference curves for preferences that have increasing MRS.
Increasing MRS would mean that the consumer would be willing to give up more units
of good 2 to get an extra unit of good 1 when they have a little of good 2 and a lot of
good 1. That is: they are more willing to give up something when they have a little of
it than when they have a lot of it. Equivalently, they prefer bundles with an extreme
amount of one of the other type of good over bundles that have a weighted average of
the two extreme amounts. For example: they would prefer a bundle (10,0) or (0,10)
over a bundle(5,5). Extremes are preferred to averages. MRS is reflected in the slope of
the indifference curve for a consumer, and in the picture we see that as we move from
the upper left to the lower right the slope of each of the indifference curves gets steeper.
5. Explain using one or two non-technical sentences:
a) Informally: monotonicity means ‘more is better’. Formally: there is at least one good
such that the consumer would prefer a bundle with a higher amount of that good, all
else equal. (We can, by the way, draw a distinction between strict monotonicity and
weak monotonicity: ‘more is better’ vs. ‘more is at least as good’.)
4
Econ 100A: Microeconomics Course Pack
Answers to problems
Jim Campbell
UC Berkeley
Spring 2022
This document has answers to the problems from the course pack. Some are answers only, and
some have a little more explanation; I’ve tried my best to make sure that the question numbering
matches the course pack itself, but since the course pack changes slightly every semester there
might be the odd discrepancy. Note that this pack doesn’t cover discussion prompts, despite the
fact that I think they are very important to test your understanding. The reason is that the
discussion prompts are typically subjective, with no single correct approach or answer, and so I
prefer you to think through them independently or with classmates rather than me leading you in
any particular direction.
Finally, I strongly encourage you to try the problems yourself or watch the demonstrations of
them in class or section before you turn to the answers here. If you look at the answers too soon, it
can be too easy to convince yourself that you could have gotten the answer! The places where you
are less confident will reveal what you need to study a little more, and working on the problems
yourself will make you remember the techniques better.
1
,Econ 100A, Spring 2022 Course pack suggested answers
Contents
1 Preferences and utility 3
2 The rational choice model 7
3 Demand 14
4 An application of consumer theory to labor supply 22
5 Choice under uncertainty and risk aversion 28
6 General equilibrium in an exchange economy 32
7 Producer theory 40
8 Perfect competition and partial equilibrium 45
9 General equilibrium with production 52
10 Monopoly and market power 56
11 Externalities 62
12 Public goods 68
13 Oligopoly theory 74
14 Markets with asymmetric information 79
15 Answers to example midterms from the course pack 82
16 Answers to math refresher problems 92
2
,Econ 100A, Spring 2022 Course pack suggested answers
1 Preferences and utility
1.
2. Jim likes both watching baseball (x1, measured in ‘hours per day’) and teaching microeco-
nomics (x2, measured in ‘hours per day’). He has decided that his utility function for these
two goods is u(x1, x2) = x3x1 2.
a) MU1 = δu = 3x21 x2
δx1
b) MU2 = δu = x31
δx2
c) MRS = MU1 3x x2 2
3x2
MU2 = x131 = x1
d) At this point MRS = 6 = 3 . This means the slope of his indifference curve at that point
8 4
is − 43 ; at this point he would be willing to swap hours of teaching for hours of baseball
at the rate of 34 of an hour of teaching to 1 hour of baseball.
6
2 = x2 , so MRS = 12 = x1∗ 2
6 2 x 6 x2
e) We could calculate this: MU 1 = x1 and MU = 3xx21. It’s
x2
the same! This is because the new utility function is a monotonic transformation of the
old one: if we took the natural log of the original function and multiplied by 2, we’d
get the second. Both of those transformations are order-preserving: whenever the first
function gives a higher utility number to bundle A than B, the second utility function
definitely does as well. This shows the ordinal property of utility functions. Note that
despite all that, the MRS, which is a real, behavioral thing for this decision maker, is
everywhere identical for both utility functions!
4
3. a) MRS = MU1 = x1
= 2x2
MU2 2 x1
x2
1
√
b) MRS = MU1
MU 2
= 2 x1
2 = √1
4 x1
3
, Econ 100A, Spring 2022 Course pack suggested answers
f ′(x1)
c) MRS = MU 1
MU 2
= 1 = f J(x1)
d) MRS = MU
MU
1
= 1
1
=1
2
4. Using two or three non-technical sentences:
a) Nonconvexity could be because the consumer prefers bundles with lots of either one good
or the other rather than bundles that have a balance of different types of good. For
example, maybe I’d prefer to spend all my time on playing soccer or all my time on
playing basketball so that I can practice and get as good as possible at one sport rather
than splitting my time between the two sports.
b) Upward-sloping indifference curves represent a case in which the consumer gains utility
from more of one good but loses utility from more of the other good—colloquially, to
them, one good is good and one is bad. u = x1−x2 would be a simple example that
generates this. Sketch it and see! This is one example of preferences that do not satisfy
monotonicity.
c) The picture in part a) shows indifference curves for preferences that have increasing MRS.
Increasing MRS would mean that the consumer would be willing to give up more units
of good 2 to get an extra unit of good 1 when they have a little of good 2 and a lot of
good 1. That is: they are more willing to give up something when they have a little of
it than when they have a lot of it. Equivalently, they prefer bundles with an extreme
amount of one of the other type of good over bundles that have a weighted average of
the two extreme amounts. For example: they would prefer a bundle (10,0) or (0,10)
over a bundle(5,5). Extremes are preferred to averages. MRS is reflected in the slope of
the indifference curve for a consumer, and in the picture we see that as we move from
the upper left to the lower right the slope of each of the indifference curves gets steeper.
5. Explain using one or two non-technical sentences:
a) Informally: monotonicity means ‘more is better’. Formally: there is at least one good
such that the consumer would prefer a bundle with a higher amount of that good, all
else equal. (We can, by the way, draw a distinction between strict monotonicity and
weak monotonicity: ‘more is better’ vs. ‘more is at least as good’.)
4
