Econ 100A: Microeconomics Course Pack
Answers to problems
Jim Campbell
UC Berkeley
Fall 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.
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Contents
1 Preferences and utility
3
2 The rational choice model
7
3 Demand
15
4 An application of consumer theory to labor supply
24
5 Choice under uncertainty and risk aversion
31
6 General equilibrium in an exchange economy
36
7 Producer theory
45
8 Perfect competition and partial equilibrium
51
9 General equilibrium with production
60
10 Monopoly and market power
64
11 Externalities
72
12 Game theory
78
13 Oligopoly theory
84
14 Markets with asymmetric information
90
15 Answers to example midterms from the course pack
93
16 Answers to math refresher problems
104
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Course pack suggested answers
Preferences and utility
1.
2. Jim likes both watching baseball (x1 , measured in ‘hours per day’) and teaching microeconomics (x2 , measured in ‘hours per day’). He has decided that his utility function for these
two goods is u(x1 , x2 ) = x31 x2 .
a) M U1 =
δu
δx1
= 3x21 x2
b) M U2 =
δu
δx2
= x31
c) M RS =
M U1
M U2
=
3x21 x2
x31
=
3x2
x1
d) At this point M RS = 68 = 43 . This means the slope of his indifference curve at that point
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.
e) We could calculate this: M U1 =
6
x1
and M U2 =
2
,
x2
so M RS =
6
x1
2
x2
=
6
x1
∗
x2
2
=
3x2
.
x1
It’s
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!
3. a) M RS =
M U1
M U2
=
b) M RS =
M U1
M U2
=
4
x1
2
x2
2
=
√1
x1
2
2x2
x1
=
√1
4 x1
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c) M RS =
M U1
M U2
=
f 0 (x1 )
1
d) M RS =
M U1
M U2
=
1
1
= f 0 (x1 )
=1
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’.)
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b) Informally: convexity means ‘averages are preferred to extremes’. Formally: strictly
convex indifference curves imply that for any two bundles x and y the weighted average
bundle αx + (1 − α)y for any α ∈ (0, 1) is strictly preferred by the consumer over either
x or y. (Again, we can draw the distinction between strict and weak convexity.)
c) If indifference curves are convex, it means that the consumer has a lower MRS when they
have more of good 1 and less of good 2 (in a two good model). Intuitively: it means
that the consumer is more willing to give up some of good A to get more of good B
when they have a lot of good A and/or not much of good B, all else equal.
6. Utility functions u2 and u5 represent exactly the same preferences. It’s always the case that
a bundle that has a higher utility number than another bundle under one of those functions
also has a higher utility number under the other function. This is because the two utility
functions are order-preserving transformations of each other—for example, if we take the
natural log of u2 and multiply it by 3, we get exactly u5 . Both of those operations (the
natural log and the multiplication) are order-preserving, and so nothing changes about how
different bundles are ranked relative to each other.
7. Part of a consumer’s indifference map:
a) The consumer’s MRS is the slope of the indifference curve at some point; here MRS is
constant at 3, since the slope of the indifference curve is everywhere -3. (If you said
the MRS is -3 that’s fine too—remember in class I mentioned that some textbooks take
MRS to be -1 times the slope of the indifference curve, while some call it just the slope
itself.)
What does this mean? The consumer’s utility would be unchanged if they gave up three
units of good 2 in exchange for 1 unit of good 1. Or: they would be willing to give up
three units of good 2 in exchange for at least one more unit of good 1 (assuming their
preferences are monotonic). Or: they are always willing to give up good 2 to get more
good 1 at the rate three to one.
b) I’ll pick u1 = 3x1 + x2 , u2 = 6x1 + 2x2 , and u3 = −3x1 − x2 . All three of these have
M RS = 3 always. The first two have monotonic preferences: the consumer likes more
stuff to less, so higher up indifference curves are further from the origin. The third one
has non-monotonic preferences: the consumer likes less stuff over more, so higher up
indifference curves are the ones closer to the origin.
(Something like u = x1 + x2 or anything with a different MRS represents a different
preference ordering, but could not have generated this indifference map. u = x1 + 3x2
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is a common mistake: that would give you a situation where (3, 0) was the same utility
as (0, 1) rather than the other way around like it is in the diagram.)
8. a)
This would violate transitivity of the preference ordering (or, since together with completeness and reflexivity this gives us a rational preference ordering, we could say that
this violates rationality of the preference ordering). In the diagram we can see that for
this consumer a ∼ b and b ∼ c but a c: that is, the consumer is indifferent between
bundles a and b and between bundles b and c but they strictly prefer a to c. This preference ordering cannot be rationalized by a utility function that puts higher numbers
on more preferred bundles: we can’t give a and b the same utility number, b and c the
same utility number, and a a higher utility number than c.
b) Two functions that would work would be e.g. u1 = x2 or u2 = −x2 . These represent
different preference orderings despite both having M RS = 0. (Something like u3 = 2x2
has the same M RS = 0 but represents the same preference ordering as u1 .) M RS = 0
means that the consumer is never willing to give up any positive amount of x2 to get
any amount more of x1 (in the case where they like x2 ); equivalently, if you add or take
away any amount of x1 without changing x2 they remain on the same indifference curve
as where they started.
9. a) All we can say here is that the consumer prefers bundle B to bundle C. Just like in part
a), since utility is an ordinal concept (higher utility means more preferred) rather than
cardinal (the magnitude of the utility number has no significance) we cannot say they
like bundle B twice as much as bundle C. Equivalently: if they are offered a choice
between bundle B and bundle C they will pick B (that’s the revealed preference idea);
equivalently, they rank bundle B higher on their preference ordering than C. If you said
‘they like B better but we don’t know by how much’, I find that an acceptable answer,
but be a little careful since ‘prefer’ is a more general, abstract, and ambiguous concept
than just ‘like’ !
b) This is not correct. The MRS for this utility function is M RS =
M U1
M U2
=
5
x1
1
x2
=
5x2
.
x1
Now,
at the point (1, 1) this is equal to 5, and so the statement is one way to approximately
convey their MRS at that point. However, this MRS is not constant. At any point in
bundle space where x1 6= x2 , their MRS takes on different values. Equivalently: the
slope of an indifference curve for this consumer is not 5 for any bundle where x1 6= x2 .
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10. a)
Person B has convex preferences. Marginal rate of substitution is equal to the negative
slope of an indifference curve. From the formula, we can see that the slope of A’s
indifference curves is low for bundles with lots of good 2 and a little good 1, and high
for bundles with lots of good 1 and a little good 2. This looks like the left hand panel
of the diagram: this shape is non-convex.
On the other hand, from A’s MRS we see that the slope of A’s indifference curves is
low for bundles with lots of good 1 and a little good 2, and high for bundles with lots
of good 2 and a little good 1. This looks like the right hand panel of the diagram: this
shape is convex. ‘Convex’ here means that each level set of the utility function is a
convex set; for monotonic preferences, this gives us the familiar shape on the right hand
side.
b) The Cobb-Douglas utility function of the form either u = x21 x2 , u = 2 ln x1 + ln x2 , or
any other monotonic transformation of these would generate this MRS. We can confirm
2
for a, b > 0 is associated
this by calculating the MRS. In general, MRS of the form ax
bx1
with a Cobb-Douglas form.
2
The rational choice model
1. The equation for a) is 4x1 + 2x2 ≤ 60 and for b) x1 + 5x2 ≤ 30.
2. a) The consumer’s MRS: M RS =
M U1
M U2
1
x1
4
x2
=
=
x2
.
4x1
The slope of the budget line:
p1
p2
= 32 .
x2
At the point of tangency, these are the same: 4x
= 23 ⇒ 2x2 = 12x1 ⇒ x2 = 6x1 .
1
Finally, we can use the budget line equation 3x1 + 2x2 = 30 together with the tangency
relationship: 3x1 + 12x1 = 30 ⇒ x∗1 = 2, x∗2 = 12.
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√
= 2 x2 . The slope of the budget line:
√
√
p1
= 28 = 4. At the point of tangency, these are the same: 2 x2 = 4 ⇒ x2 =
p2
2 ⇒ x∗2 = 4. Finally, we can use the budget line equation 8x1 + 2x2 = 12 to get x1 :
8x1 + (2 ∗ 4) = 12 ⇒ x∗1 = 21 .
b) The consumer’s MRS: M RS =
c) The consumer’s MRS: M RS =
M U1
M U2
M U1
M U2
=
1
x1
2
x2
=
1
√1
2 x2
=
x2
.
2x1
The slope of the budget line:
p1
p2
= 2.
x2
= 2 ⇒ x2 = 4x1 . Finally, we can
At the point of tangency, these are the same: 2x
1
use the budget line equation 2x1 + x2 = 30 together with the tangency relationship:
2x1 + 4x1 = 30 ⇒ x∗1 = 5, x∗2 = 20.
3.
4. a)
U1
b) M RS = M
= 3. This is smaller than the price ratio he faces: he is willing to give up
M U2
3 Pawtucket tickets for 1 Red Sox ticket, but at these prices he would have to forgo 5
Pawtucket tickets per Red Sox ticket.
c)
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d) The logic is from b). All the better bundles than x∗ are above the illustrated indifference
curve, but none of them are feasible (in the budget set). Jim is not willing to give up
as many Pawtucket tickets as it takes to get any Boston tickets.
5. a) If M RS 6= pp12 then the rate at which the consumer is willing to swap one good for the
other is different than the rate at which they can actually swap one good for the other.
For example, they may be willing to give up three units of good 1 for one unit of good 2,
but they would actually only have to give up two units; they can reach a more preferred
bundle by making that change.
b) The rational choice model: when economists use this model they are assuming that the
decision maker makes choices by selecting their most preferred bundle from those that
are available. In the ‘standard’ model we typically assume that the decision maker has
a single, stable preference relation that captures things like their needs, wants, purpose,
or motivation, and that their choices reflect those preferences.
c) There are many possibilities here. Three that would work: u = x1 (so the DM prefers
bundles with more good 1 and is indifferent to any amount of good 2 in the bundle),
u = x1 − x2 (so the DM prefers bundles with more of good 1 and/or less of good 2),
or u = x1 + x2 where the price of good 1 is less than the price of good 2 (so the DM
prefers bundles with a larger total amount of stuff). In all three cases the DM would
choose to spend all of their money on good 1.
6.
Here the consumer’s preferences are such that they find the two goods to be perfect substitutes. Their utility function shows that they prefer bundles with a higher total number of
goods. So, they will spend all of their money on the cheaper good—in this case, good 1—to
get the largest total number of goods in their bundle within their budget.
7. a),b)
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The optimal choice is at (15, 5). There are a couple of ways to find this; here’s what I
did. I looked for a point of tangency with a standard budget set with income 20 and
prices of 1 each. With this utility function, the point of tangency with that budget
would have been at (16, 4). But that point isn’t in the actual budget set. Since the
indifference curves are convex, I know that the highest available point is (15, 5).
Equivalently, the consumer’s MRS evaluated at (15, 5) is 43 . This is steeper than 1, the
slope of the budget line. So I know that the utility at points to the left of (15, 5) is
lower than at (15, 5) itself.
c) Without the purchase limit, Jim would have gotten more utility. He could have reached
(16, 4), which we just argued is on a higher indifference curve than (15, 5).
8. In this situation the consumer has M RS = 21 . Their indifference curves therefore everywhere
have a downward slope of 12 .
The question, then, is going to be whether the budget line is steeper or flatter than that. For
sufficiently low p2 , the consumer’s optimal choice will be to spend all of their money on good
2; geometrically, they can reach a higher indifference curve this way as long as the most x2
they can afford is at least 5 units. On the other hand, for sufficiently high p2 , the consumer’s
optimal choice will be to spend all of their money on good 1; geometrically, they can reach a
higher indifference curve whenever the most x2 they can afford is less than 5. In the picture
I’ve illustrated with two examples:
This threshold is crossed when p2 = 4. For p2 > 4, we are in the second case. Intuitively:
the consumer’s marginal utility from good 2 is always twice as much as for good 1, and so
they spend all of their budget on good 2 unless it is at least twice as expensive as good 1. So
when p2 > 4, they can achieve higher utility by spending all of their money on good 1 than
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with any other choice. Still another way to think about this: the consumer is always willing
to swap two units of good 1 for one unit of good 2. When p2 > 4, the opportunity cost of a
unit of good 2 is more than two units of good 1, so they will not make that change; the best
they can do is all good 1.
(By the way, you definitely didn’t need a super long explanation like this, I just wanted to
give a few examples of ways you could have answered this question! As long as you made a
good, clear explanation that makes sense, that’s fine.)
9. Consider a model with two goods, Subway sandwiches (x1 ) and tickets to Drake shows (x2 ).
Each good must be consumed in a non-negative amount. Jim’s preferences over these two
goods can be represented by the utility function u = 10 − x1 − x2 .
a) I chose u = 5 and u = 8 to illustrate. This indifference map has higher levels of utility
closer to the origin: monotonicity is violated.
b) His optimal choice is x∗ = (0, 0). He gets more utility from this bundle than any bundle
in the budget set in which he has positive amounts of either good. This is because his
marginal utility from both goods is always negative.
c) Any order-preserving transformation of the utility function would work. For example,
adding or subtracting any constant (e.g. u = −x1 − x2 or u = 500 − x1 − x2 ) or
multiplying by a positive constant (e.g. u = 20 − 2x1 − 2x2 or u = 5 − 12 x1 − 12 x2 ). These
represent exactly the same preferences as the original because whenever the original
function gives a higher utility number to one bundle over another, any of these orderpreserving alternatives would also give it a higher utility number. This is related to
the fact that utility functions are ordinal: all that matters is the relative magnitude of
utility, not its numerical value.
Note that something like u = x1 + x2 does not work here. It does not preserve the
ordering of the original function. For example, the original utility function ranks (0, 0)
above (1, 1) but u = x1 + x2 ranks (1, 1) above (0, 0).
10. a) M U1 = x61 and M U2 = x32 . For each of these goods, the consumer has diminishing
marginal utility: each subsequent increase in the amount of the good they consume
brings less and less increase in their utility. It is not the case though that they start
to ‘dislike’ the good or lose utility from it: marginal utility is always positive for both
goods. (Other things you may have mentioned: the MU for one good does not in this
example depend on how much of the other good the consumer has; for a given unit of
each good the consumer derives more MU from the first than the second (i.e. the 2nd
unit of good 1 gives a higher increase in utility than the second unit of good 2).
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b) We know that since the consumer’s preferences are Cobb-Douglas and since the budget
set is triangular that their optimal choice is a point of tangency between the budget line
U1
2
= 2x
,
and an indifference curve. So: M RS = pp21 at their optimal choice. M RS = M
M U2
x1
2x2
4
and so we know that x1 = p2 . Since we’re told that the optimal bundle is (5, 5) this
means that
2(5)
5
=
4
,
p2
which we can solve to find p2 = 2.
How big is m? We know their choice is on the budget line (i.e. they spend all of their
money). To afford (5, 5) at these prices, the consumer would need (4 ∗ 5) + (2 ∗ 5) = 30.
Their income is $30.
11. a)
I have two suggestions. On the left is a situation where preferences are non-convex.
There’s a tangency point, but the indifference curves are bent out from the origin, so
the tangency point is worse than available points on a higher indifference curve. (By
the way, one way to generate this is with a utility function u = x21 + x22 —it’s a really
popular micro trick question to ask for the utility maximizing choice for that utility
function, since tangency gets it wrong!)
On the right is a situation where preferences are non-monotonic. There’s a tangency
point, but the indifference curves are getting higher up back towards the origin, so the
tangency point is worse than available points on a higher indifference curve. (Something
like u = −x1 x2 gets this done.)
b)
We’ve seen a few of these cases in our course so far—I’ve shown just two of many
possibilities here. On the left we have a consumer that only cares about good 1; their
indifference curves are straight, vertical lines, and their optimal choice is on the budget
line but the it’s not a tangency point since the slope of the indifference curve is not
equal to the slope of the budget line.
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On the right, perfect complements. The optimal choice is at the corner of an L, where
the slope of the indifference curve is undefined; this is not a tangency point. Other
possibilities could be perfect substitutes when M RS 6= pp21 , or a situation where the
consumer gets positive marginal utility from one good and negative marginal utility
from the other.
12. a)
This budget set is unbounded—it’s goes on forever in the horizontal direction! The
person can afford up to 5 units of good 2, but they can afford any number of good 1 to
go along with that because good 1 costs nothing. The slope of the budget line is zero.
b)
The utility function u = x2 − x1 would work here (as would any function where x1
entered negatively and x2 positively). This person prefers bundles with less x1 rather
than more x1 and bundles with more x2 rather than less x2 . The bundle that minimizes
x1 and maximizes x2 within the available bundles is (0, 5). Their indifference curves
are straight, upward-sloping lines that increase in utility level as we go up and to the
left in the diagram—this means that the highest indifference curve available within the
budget is the one that passes through (0, 5). The optimal choice is unique because no
other available bundle is on this same indifference curve.
13. a) Jim is willing to give up more video games to get an extra book than Stephanie is.
Equivalently, he is willing to accept fewer books in exchange for giving up a video game
than Stephanie is.
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Examples of utility functions that would work: uJ = 3x1 +x2 , uS = x1 +2x2 . (Anything
of the following form works: uJ = ax1 + bx2 , uS = cx1 + dx2 , ab > dc . This inequality is
equivalent to saying ‘Jim’s MRS is higher than Stephanie’s’.)
b) Jim’s optimal bundle also has x1 = 10 and x2 = 0. Both have monotonic preferences so
will pick a point on the budget line. The exact nature of each person’s optimal choice
in this scenario with constant MRS depends on whether MRS is bigger or smaller than
the price ratio. MRS bigger than the price ratio will lead the consumer to choose a
bundle that’s all x1 and no x2 . MRS smaller than the price ratio, a bundle that’s all x2
and no x1 . Here, since Stephanie’s MRS is lower, if her MRS is bigger than the price
ratio, then Jim’s certainly is too.
In the diagram, we see this graphically. Jim’s ICs are steeper than Stephanie’s, but
all are straight, downward-sloping lines. We can see that if x∗ is the highest available
bundle for Stephanie, it must be for Jim as well.
14. There are two goods, x1 and x2 . Jim’s well-behaved preferences over bundles of goods can
√
be represented by the utility function u = 3x1 + 4 x2 . The price of good 1 is p1 = 3, the
price of good 2 is p2 = 2, and Jim has m = 14.
a) Use the tangency method (or another valid method if you prefer) to find Jim’s optimal
choice of consumption bundle.
δu
δu
First, let’s find Jim’s MRS. M U1 = δx
= 3 and M U2 = δx
= √2x2 . So: M RS =
1
2
M U1
M U2
=
3
√2
x2
=
√
3 x2
.
2
Next, we know that at a point of tangency between
√
3 x
indifference curve and budget line we have M RS = pp12 ⇒ 2 2 = 32 ⇒ x∗2 =
1. Finally, we can use the budget line equation 3x1 + 2x2 = 14 to find x∗1 :
3x∗1 + 2(1) = 14 ⇒ x∗1 = 4. Jim’s optimal choice is the bundle x∗ = (4, 1).
b) Consider the bundle (2, 4). Sketch Jim’s budget set, and sketch the indifference curve for
Jim on which that bundle lies (just the right general shape is OK, no need to precisely
plot it out). Explain why your sketch shows that this bundle cannot be Jim’s optimal
choice.
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√
At the bundle (2, 4), Jim’s MRS is 3 2 4 = 3. This means his indifference curve
(which we know is well-behaved) has a steeper downward slope at that point
than the downward slope of the budget line, pp21 = 32 . This shows us that there
are bundles ‘above’ the indifference curve through (2, 4) which are available
to Jim in the budget set: (2, 4) cannot be his optimal choice.
3
Demand
x2
U1
x2
x2
1. M RS = M
= 2x12x2 = 2x
. At tangency, 2x
= pp12 , or p2 x2 = 2p1 x1 . By combining this with
M U2
1
1
the budget line p1 x1 + p2 x2 = m we can solve to find x∗1 = 31 pm1 and x∗2 = 23 pm2 . Both goods are
normal (since demand increases when m increases); both goods are ordinary (since demand
increases when own price decreases); and the goods are neither substitutes nor complements
(since the other good’s price does not appear in the demand function for either one).
1
M U1
= x21 = 2x11 . At tangency, this means 2x11 = 1 since the price ratio is 1:1 here;
2. M RS = M
U2
rearranging we see that x∗1 = 12 . Using the budget line x1 + x2 = m we see that x∗2 = m − 12 .
The income expansion path is therefore a straight vertical line as shown in the diagram: more
income means more spending on good 2 but the same amount of good 1 for this consumer.
The example of course depends on the consumer’s preferences, but maybe something like
toothpaste would work for good 1: a person may plausibly buy the same amount of toothpaste
regardless of how much money they have.
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3. a)
b) The Engel curve for good 1 slopes up, and the Engel curve for good 2 slopes down. The
Engel curve plots income against the consumer’s optimal choice of a good; for good 1,
as income increases they choose more of the good, but for good 2, as income increases
they choose less of the good.
4. The income expansion path here is a straight line out from the origin. The slope of this line
depends on the parameters of the Cobb-Douglas function and on the prices of each good:
all else equal, for a higher weight on good 1 relative to good 2 in the utility function the
line is flatter; all else equal, for a lower price of good 1 the line is flatter. As an example,
with p1 = p2 and equal weights on the two goods in the utility function, the Cobb-Douglas
demand functions tell us that the consumer always spends exactly half of their income on
each good and so the income expansion path follows a 45 degree line from the origin in that
case. The diagram below has a generic example.
U1
2
5. M RS = M
= x21 . At tangency, M RS = pp12 and so x∗1 = 2p
. From the budget constraint
M U2
p1
∗
we know p1 x1 + p2 x2 = m, and substituting in for x1 we can solve to get x∗2 = pm2 − 2. Price
2
2
1 p1
elasticity is given by − dx
= −(− 2p
)( xp11 ) = p2p
. At the optimal choice (substituting in
dp1 x1
p21
1 x1
∗
x1 ) this is exactly 1.
6. No matter what decomposition we do here, the substitution effect will be zero. The whole
change in demand is due to the income effect. The consumer’s optimal choice on the hypothetical budget line will be exactly the same as in the original budget line: the bundle at
which the amount of each good is the same.
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7. a) The income expansion path will just follow along the axis of whatever good is cheaper—
the consumer always spends all of her income on the cheaper good.
b) Let’s fix p2 and think of varying p1 . When p1 is larger, the consumer spends all her
income on good 2, so the offer curve in that range is a point on the good 2 axis. When
p1 = p2 , any choice on the budget line is utility-maximizing for this consumer, so the
offer curve is the budget line at that point. When p1 is less than p2 , the consumer
spends all of her money on good 1, so the offer curve follows the good 1 axis.
8. First we will calculate the consumer’s original optimal choice x by the tangency method. At
the optimal choice,
p1
p2
M U1
1
=
M U2
1
M RS =
√1
2 x1
1
=1
1
x1 = .
4
(1)
(2)
(3)
(4)
We can find x2 by using the budget constraint:
p 1 x1 + p 2 x2 = m
1
(1 ∗ ) + (1 ∗ x2 ) = 2
4
7
x2 =
4
So we have bundle x =
1 7
,
4 4
(5)
(6)
(7)
.
Next let’s find the consumer’s final optimal choice y. At the optimal choice,
M RS =
p01
p2
1
M U1
= 2
M U2
1
√1
2 x1
1
1
2
x1 = 1.
=
(8)
(9)
(10)
(11)
Again we can find x2 by using the budget constraint:
p01 x1 + p2 x2 = m
1
( ∗ 1) + (1 ∗ x2 ) = 2
2
3
x2 =
2
17
(12)
(13)
(14)
Econ 100A, Fall 2022
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So we have bundle y = 1, 32 .
i. Hicks The Hicks decomposition asks the following question: what if we changed the price
ratio, but made it so that the consumer’s optimal choice was on the same indifference
curve as before (at x)? This hypothetical optimal choice, point z, is therefore characterized by two things:
(a) u(x) = u(z): the consumer’s utility at z is the same as at x, and
p0
(b) M RS = p21 : z is a point at which the consumer’s indifference curve is tangent to
the hypothetical budget line.
So to find z, let’s first invoke tangency:
M RS =
p01
p2
1
M U1
= 2
M U2
1
√1
2 x1
1
1
2
x1 = 1.
=
(15)
(16)
(17)
(18)
To get x2 , we need to make use of fact 1:
r
u(x) =
1 7
+
4 4
9
4
= u(z)
√
= x1 + x2
√
= 1 + x2
5
⇒ x2 =
4
=
(19)
(20)
(21)
(22)
(23)
(24)
So we have z = 1, 54 . Now we can get the size of the income and substitution effects
on good 1. The income effect is the x1 value at y minus the x1 value at z, which in this
case is 0. The substitution effect is the x1 value at z minus the x1 value at x, which is
3
.
4
Typically making use of the two facts we know about z will give us two relationships
between x1 and x2 that we solve together to find the numerical answer. In this special
case, we got x1 directly from fact 2.
ii. Slutsky The Slutsky decomposition asks the following question: what would the consumer choose if we changed the price ratio, but changed the consumer’s income so that
she could still just afford x? So first we find what it would cost to buy x at the new
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prices:
1 1
7
p01 x1 + p2 x2 = ( ∗ ) + (1 ∗ )
2 4
4
15
= .
8
(25)
(26)
If we gave the consumer this income, what would she choose? Her hypothetical optimal
choice would again be characterized by tangency with the hypothetical budget:
M RS =
p01
p2
1
M U1
= 2
M U2
1
√1
2 x1
1
1
2
x1 = 1.
=
(27)
(28)
(29)
(30)
To get x2 , we think about the hypothetical budget:
15
8
1
15
( ∗ 1) + (1 ∗ x2 ) =
2
8
11
⇒ x2 = .
8
p01 x1 + p2 x2 =
(31)
(32)
(33)
So we have z = (1, 11
). Similarly to the Hicks case, the income effect is the x1 value at
8
y minus the x1 value at z, which in this case is 0. The substitution effect is the x1 value
at z minus the x1 value at x, which is 43 .
iii. A note on the special case The consumer’s preferences in this example, represented
√
by u = x1 + x2 , are called quasilinear preferences. Her utility is linear in good 2 plus
a function of good 1. For preferences like these, the income effect is always zero, as we
found here. Why is this? Notice that the M RS depends only on the level of good 1
and not at all on the level of good 2: this means that indifference curves are horizontal
translates of each other. So a given slope—a given price ratio—pins down the tangency
point in the x1 direction, and for any income level that x1 must be part of the optimal
choice.
In general if M RS depends on both x1 and x2 , both the Hicks and Slutsky decompositions will be a little trickier since the x1 value won’t be pinned down so immediately by
the tangency condition. You’ll typically have two equations in two unknowns (x1 and
x2 ) to solve together to get the answer for the z bundle, just as you had on the example
in your problem set. But the principles are the same as in this example.
9. a) On the diagram, x = (10, 0) is Jim’s original optimal choice and y = (5, 0) is his new
optimal choice after the price change.
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b)
The substitution effect is zero and the income effect is −5 units of x1 . We added a
hypothetical Slutsky budget line (with the new prices but compensated to make x still
just affordable). From the diagram we can see that Jim’s optimal choice from the
Slutsky budget set is z = (10, 0), just the same as his original optimal choice. So the
relative price change doesn’t change his demand at all (so no substitution effect) and
the whole change in demand overall is therefore due to the income effect.
c) What is the equation of the budget line that you added to your diagram in part b)?
Explain why that’s the equation of the budget line you added, with reference to both
the specific numbers in your equation and to the point of the Slutsky decomposition.
This budget line has an equation 6x1 + 6x2 = 60. The prices associated with each good
are the new prices ($6 each), and the income on the right hand side is the amount of
money that Jim would need to have to be able to afford his original optimal choice of
(10, 0) at those new prices. This is because the Slutsky decomposition asks what would
happen if the relative prices had changed but the consumer’s purchasing power (in the
sense of being able to still afford their original bundle) was unchanged. In this case,
though, the change in relative prices alone has no effect on Jim’s choice since he always
spends all of his money on good 1 given his preferences here.
(Note that for full credit your explanation of the hypothetical budget must be the
Slutsky method, not Hicks: it’s not about constructing a budget that reaches the same
indifference curve or utility level as before; that’s a Hicks explanation.)
10. a) These Engel curves show that for both goods Jim’s demand does not increase or decrease
when his income changes (i.e. both are neither normal nor inferior). This pattern
of choices is inconsistent with monotonicity of preferences. When income increases,
bundles with more of either or both goods are available, and so a utility-maximizing
consumer would choose more of at least one of the goods if they had higher income.
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b)
The diagram shows that Mo demands (i) more lobster rolls when the price of lobster
rolls is lower and (ii) fewer bowls of ramen when the price of lobster rolls is lower. (i)
means that the demand curve slopes down, since the demand curve plots price against
demand, and (ii) means that Mo finds these goods to be substitutes, since that’s the
term for the situation where a lower price of good j leads to lower demand for good i.
11. a) The price of good 1 has increased. This is a Hicks decomposition, because the hypothetical
budget line shows the same price ratio as the new budget line but leaves the consumer
just able to achieve the same utility as they obtained at their original optimal choice.
b) These effects are about the effect of the change in the price of good 1 on the consumer’s
demand for good 1. The substitution effect is 5−7 = −2, which is the difference between
the consumer’s original optimal choice of good 1, 7 units, and their optimal choice at
the new relative prices but controlling for purchasing power, 5 units. The income effect
is -1.5, the difference between their hypothetical optimal choice, 5 units, and their true
final optimal choice, 3.5 units. The substitution effect isolates the effect of the change
in the relative price of the two goods, controlling for the fact that purchasing power
is lower after an increase in the price of good 1. The income effect is the effect of the
change in purchasing power, controlling for the change in relative prices.
Goods 1 and 2 are both normal. We can see this by comparing the consumer’s optimal
choice from the new budget line and the hypothetical budget line. The consumer chooses
more of both goods from the hypothetical budget line, which is a parallel shift out of
the new budget line, just like an increase in income. Good 1 is ordinary, because
the consumer chooses more of good 1 from the original budget line than from the new
budget line—higher p1 , lower demand for good 1. Whether good 1 is ordinary or Giffen:
there are two different answers we’ll consider correct here. One: just looking directly at
the diagram, we can’t say whether good 2 is ordinary or Giffen, because that requires
knowing how the consumer’s demand for good 2 changes when the price of good 2
changes, which we don’t observe here. Two: it’s true that with monotonic preferences,
a normal good is definitely ordinary (this is the same logic that tells us that a Giffen
good is ‘very inferior’. So if you said ‘ordinary’ with this particular justification, that’s
good too!
12. a) The original prices are p1 = 2, p2 = 2, and the new prices are p1 = 4, p2 = 2 (which we can
get using the intercepts of the two budget lines). The Slutsky budget line will feature
the new price ratio but give the consumer (hypothetically) enough money to afford
their original optimal bundle at the new prices. Since their original optimal bundle was
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x∗O = (9, 1), they will need m = 38 to afford it at the new prices. The Slutsky budget
line (in the usual p1 x1 + p2 x2 = m form) is 4x1 + 2x2 = 38.
√
U1
= √11 = x2 . The tangency condition is M RS = pp12 ⇒
b) The consumer’s MRS is M
M U2
x2
√
x2 = 42 ⇒ x∗2 = 4. The budget line equation is 4x1 + 2x2 = 38 and so at the
optimal choice (4x1 + 2(4) = 38 ⇒ x∗1 = 7.5. The consumer’s optimal bundle from the
hypothetical budget set is x∗ = (7.5, 4).
The substitution effect is 7.5−9 = −1.5, which is the difference between the hypothetical
optimal choice (isolating the effect of the relative price change, controlling for purchasing
power) and the original optimal choice. The income effect is 3 − 7.5 = −4.5, which is
the difference between the final optimal choice and the hypothetical optimal choice.
13. Consider a standard consumer choice model with two goods. Say that a person has demand
functions given by x∗1 = 0, x∗2 = pm2 .
a) Suggest a utility function that might represent this person’s preference ordering. Describe
that preference ordering in simple terms.
This person’s demand functions tell us that they will always choose to spend
all of their money on good 2, regardless of the prices of the goods and
regardless of how much money they have. They will always choose the
highest point on the x2 axis from any triangular budget set.
There are many possible answers here. u = x2 , u = x2 − x1 are (to me)
the simplest examples. Any monotonic transformation of those or similar
examples like u = x2 − 0.5x1 would also work. Notice that something like
u = x1 + x2 is not quite right: the optimal choice for this utility function is
not always to spend everything on x2 (it depends on prices).
I’ll pick u = x2 . The preference ordering here says: this person prefers
bundles with more of good 2 over bundles with less of good 2, regardless of
how much good 1 is in either bundle. They are indifferent between bundles
with the same amount of x2 —more x1 or less x1 does not change their ranking
of the bundle if it has the same amount of good 2. One might also more
loosely say something like ‘this person likes good 2 and doesn’t care about
good 1’; that’s less strictly accurate (and doesn’t really keep the focus on
bundles) but I think it gets the gist of the preference ordering well enough.
b) Let’s sketch the offer curve for this person for changes in the price of good 2. On a
diagram, sketch (i) three budget lines, (ii) the person’s optimal choice from each budget
line, (iii) the indifference curve on which each of the three optimal choices lies (following
the utility function you picked in part a), and (iv) the offer curve.
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14. The diagram shows a situation before and after an increase in the price of good 1 in a
two good model. The original (O) and new (N ) optimal choices for the decision maker are
illustrated, as are the associated indifference curves.
a) On the diagram, add a line representing the Hicks decomposition’s ‘hypothetical’ budget
line. Mark the decision maker’s optimal choice (both the bundle H and the amount of
good 1 in the bundle x1,H ) from that budget line.
b) Explain what the point of the Hicks decomposition is and how that relates to where you
drew the budget line in part a).
The point of the Hicks decomposition is to break down the effect of a price
change on a person’s demand for a good into two components: a part that
is due to the fact that the relative price of the good compared to the price
of other goods has changed, and a part that is due to the fact that the
consumer’s purchasing power changed. Here, good 1 got more expensive
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relative to good 2, and the consumer’s purchasing power decreased since
they can afford less stuff overall after the change.
The Hicks decomposition says: ‘what if the relative price of the goods
changed, but the consumer still had enough purchasing power to be able
to reach their original utility level?’ We draw the ‘hypothetical’ budget line
such that it has the new price ratio but the consumer can still achieve their
original utility level (can still reach the indifference curve on which their
original optimal choice lies). Once we’ve done this, the change x1,H − x1,O
gives us the so-called ‘substitution effect’ (the effect of the relative price
change only) and the change x1,N − x1,H gives is the ‘income effect’ (the effect
of the change in purchasing power only).
4
An application of consumer theory to labor supply
1.
UL
2. a) M RS = M
= 20. This person would always be willing to give up 20 units of consumpM Uc
tion for at least one extra hour of leisure, or, equivalently, willing to give up one hour
of leisure for at least 20 units of consumption.
UL
b) M RS = M
= Lc . This consumer has convex preferences: loosely speaking, they prefer
M Uc
a balance of leisure and consumption. Equivalently, they are more willing to give up
consumption for leisure if they have a lot of money and little free time; they are more
willing to give up leisure for consumption if they have a lot of free time and little money.
(This is the idea of diminishing marginal rate of substitution that’s captured by convex
preferences.)
3. a) M RS = pp21 gives Lc = wp = 32 . So at the optimum 2c = 3L. The budget is pc + wL = wT
which is 2c + 3L = 36, and so we can solve for L = 6, c = 9. Since he takes 6 hours of
leisure, he works for 12 − 6 = 6 hours.
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Econ 100A, Fall 2022
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b) His utility from his optimal choice in a) is u = 6 ∗ 9 = 54, so he must get at least this
much utility from consuming L = 12 and c = mp = m2 . Solving u = 12 m2 ≥ 54, we see
that m ≥ 9 for him to accept the deal.
c) Utility if he doesn’t work at all is u = 12 ∗ 6, since he can take L = 12 and c = m2 = 6.
His new budget constraint is pc + wL = wT + m, which for the given values is 2c + 3L =
36 + 12. We know from a) that 2c = 3L at the tangency point, so we can solve to find
L = 8, c = 12. This is new optimal choice (we could also check to make sure that he
is happier with this than at the corner point where he does not work to confirm that it
beats the corner point). He works less than in a): only 4 hours now.
d) If he doesn’t work at all his utility is u = 12 m2 . His budget constraint pc + wL = wT + m
is now 2c + 3L = 36 + m. There are a couple of ways to solve this question, but the
approach I took is to figure out what’s the smallest m such that the tangency point
‘would’ be at L > 12 (which of course we can’t actually get since the budget set is
chopped off at L = 12). So: since 2c = 3L at tangency, substitute to the budget line
equation to get 6L = 36 + m, or L = 6 + m6 . If m ≥ 36 then L ≥ 12. So m of at least
36 will induce Jim to work zero hours: his preferred point in the budget set is at the
corner, L = 12.
4. a) Optimal choice is characterized by tangency:
p1
p2
w
M UL
=
M Uc
1
2Lc
=w
L2
wL
c=
.
2
M RS =
(34)
(35)
(36)
(37)
We can substitute this back into the budget constraint:
pc + wL = wT
c + wL = 12w
wL
+ wL = 12w
2
L = 8.
L = 8 and so c =
wL
2
(38)
(39)
(40)
(41)
= 4w. Jim works for 12 − L = 4 hours.
b) w = 14 and so at Jim’s optimal choice in a) he consumes L = 8, c = 1 for a total utility
of u = L2 c = 64.
If he accepts the deal, he must work for 8 hours and so will only get L = 12 − 8 = 4
hours of leisure. His utility will then be u = L2 c = 16c. For this to be bigger than his
utility before we need 16c > 64, or c > 4.
How much consumption can he get? His income will now be 8w + b, which means he
can afford 8w + b units of consumption at the price p = 1. The wage is w = 14 , so we
need (8 ∗ 41 ) + b > 4, or b > 2.
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5. a)
2
UL
c
c
= 2cL
= 2L
. The fact that it’s not constant means that the wage he’d need
b) M RS = M
M Uc
to be paid to induce him to give up an extra hour of his time depends on how much
time and money he currently has. His preferences here are convex—he prefers a balance
of time and money.
c
c) M RS = pp12 at tangency gives us 2L
= 5 or c = 10L. The budget line is pc + wL = wT or
c + 5L = 60. Solving those together: 15L = 60 ⇒ L∗ = 4 and so c∗ = 40. He works 8
hours (12 free hours minus 4 hours of leisure).
d) If he took this job, he’d get utility u = (502 ) ∗ 2 = 5000. If he took the wage job, his
utility would be (from part c) u = (402 ) ∗ 4 = 6400. He prefers the wage job.
6. a)
The budget set is shaded in the above picture. Note that you have to indicate the whole
set—the budget line alone is not the same thing as the budget set. If you made this
mistake it was worth a small deduction.
b) First, Ivan’s MRS is
M UL
M Uc
=
√2 .
L
At tangency, this is equal to the price ratio:
4p2
√2
L
=
⇒ L∗ = w2 , which gives us the demand function for leisure. Since preferences are
monotonic, the optimal choice will be on the budget line, which has equation wL +
pc = wT . By substituting in what we found from the tangency relationship, we get
2
w 4p
+ pc = wT ⇒ c∗ = wT
− 4p
, which gives us the demand function for consumption.
w2
p
w
w
p
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c)
The ‘time expansion path’ would be a vertical line. (Increases in T would shift the
budget set out in parallel, just like increases in income in the standard model.) The
reason is that Ivan’s demand for leisure doesn’t depend on the amount of free time
T that he has available, but his demand for consumption increases as time increases.
When Ivan has more free time, he uses it to work (and consume the proceeds).
d) The most he’d be willing to pay is w. The value of the extra hour of free time is w; if he
had to pay more than w to get it, it wouldn’t be worth it! There are a couple of ways
to see this.
First, let’s think about the budget line equation: wL + pc = wT . If T is increased by
1, it means that the budget line is now wL + pc = w(T + 1) = wT + w. Since Ivan’s
preferences are monotonic, he would be able to reach a higher level of utility on this
new, higher budget line. If he had to pay to get that extra hour of free time (maybe
he buys a car, for example, to avoid a longer commute on the train), then his budget
would be wL + pc = wT + w − K, where K is the amount he pays. If K < w, he comes
out ahead: his budget line after paying for that extra hour is higher up than his old
one, so he can reach a more preferred bundle. But if K > w, he comes out behind: his
budget line after paying for the extra hour is lower than his old one, so he can reach
only worse bundles than he started with!
A second way to see this is to think about how much consumption he could buy with
the extra hour. He could work that hour at the wage w and afford wp extra units of
consumption. If he paid w for that extra hour, though, this would mean he’d have to
forgo wp units of consumption and he’d be right back where he started!
Your explanation doesn’t have to be this precise, as long as you got the right idea!
7. a)
b) The answer is that for a wage between $10 and $30, only Ritesh will work. (Below $10
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Econ 100A, Fall 2022
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neither will work, and above $30 both will work.)
2c
UL
UL
= 3L
= 3c
. For Ritesh, M RS = M
= Lc . Let’s evaluate
For Saleem, M RS = M
M Uc
L3
L
M Uc
their MRS at the corner of the budget set: L = 10, c = 100. Saleem’s MRS at that
point is 30, so his indifference curve at that point has a slope of -30; for Ritesh the MRS
is 10, so his indifference curve at that point has a slope of -10. The slope of the budget
set at that point is −w.
If the slope of the indifference curve is steeper than the budget set, then we know that
all of the points that are better than (10, 100) are outside of the budget set. If the slope
of the indifference curve is flatter than the budget, then we know that there are points
inside the budget set that are better than (10, 100).
So: if the wage is below $10, both people have an indifference curve that’s steeper at
(10, 100) than the budget line, and above $30, both have an indifference curve that’s
flatter. In between, Ritesh has a flatter IC and Saleem steeper: Saleem cannot do better
within the budget set than the corner, but Ritesh can do better by working some hours.
(A picture is below if you’re interested!)
8. a) The consumer’s MRS is
M UL
M Uc
=
2
1
√
2 c
√
= 4 c. At the point of tangency, we know two
things: the slope of their indifference curve is equal to the slope of the budget line√(the
price ratio) and the point lies somewhere on the budget line. So: M RS = wp ⇒ 4 c =
20 ⇒ c∗ = 25. The budget line equation is wL + pc = wT and so at the optimal choice
(20 × L) + (1 × 25) = (20 × 10) ⇒ 20L = 175 ⇒ L∗ = 8.75.
If they’d had more available hours this person would have worked the same amount.
They work 1.25 hours, which is enough to afford the 25 units of consumption we found
from the tangency condition. Since this 25 units of consumption they desire is independent of T (it depends only on the price ratio), they will still work exactly 1.25 hours if
they had more time.
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b)
9. a)
b)
This person prefers bundles with more consumption than less, all else being equal, and
prefers bundles with less leisure than more, all else being equal. Loosely speaking, they
like consumption but dislike leisure (they like work). Their optimal choice from any
budget set will be the bundle with c − L as big as can be. Here that means L = 0,
c = 300.
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10. Stephanie got a raise! The diagram shows her budget set before and after a change in her
wage rate. She has 10 hours of available time. The price per unit of consumption is $2.
a) How much non-labor income
does Stephanie have? What
was her wage initially, and
what was it after the raise?
What is the slope of the new
budget line, and what does
that slope represent?
Stephanie has $30 in non-labor income. We know this because she can afford
15 units of consumption even if she works zero hours—consumption is $2
per unit and so she must have $30.
Before the raise, her wage is $12 per hour. After, her wage is $18 per hour.
We know this because before the raise she can afford 60 additional units
of consumption with her labor income if she works all 10 hours (over and
above her 15 from non-labor income). This costs $120, which is 10 hours at
$12 per hour. Similarly for the post-raise case: she can afford 90 additional
units of consumption, which costs $180, meaning 10 hours at $18 per hour.
(Note for the first two parts I chose not to specifically ask for explanations!
If you gave them, great! If not, that was fine in this case!)
The slope of the new budget line is −9. This is the relative price of leisure
. The reason is that after the raise
and consumption; the price ratio wp = 18
2
for each hour that Stephanie takes as leisure, she must sacrifice the chance
to earn $18 and so sacrifice the chance to buy 9 units of consumption. Put
differently, this is the opportunity cost of one hour of her time, in the context
of this simple model.
b) Say that Stephanie’s preferences can be represented by the utility function u = αL + c,
where α > 0. Say that Stephanie chose to work zero hours before the raise, but now
chooses to work 10 hours. What do we know about α? Explain carefully how you know.
It must be that α is between 6 and 9. Stephanie’s MRS here is equal to
M U1
= α1 = α; this is the downward slope of each one of her straight, parallel
M U2
indifference curves. If the point (10, 15) was optimal from the original budget
and (0, 105) was optimal from the new budget (as specified in the question)
then it must be the case that her indifference curves are steeper than the
original budget line but flatter than the new budget line. The diagram
illustrates:
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Econ 100A, Fall 2022
Course pack suggested answers
Since the new budget line has slope -9 and the old has slope -6, we need α
between 6 and 9 to yield an indifference map and optimal choices consistent
with the information given!
5
Choice under uncertainty and risk aversion
1. a) EU = 21 u(4 + 12) + 21 u(4 + 0) = 3
b) If he sold the ticket for
√ an amount x, he’d have 4 + x dollars for sure. He’d sell the
ticket if EU (sell) = 4 + x > EU (keep) = 3. Solving gets us that he’d sell for a price
of at least $5. Another way to see it: his EU in a) is the same as if he’d had $9 for
sure—that’s $5 more than his initial $4.
2. (Risk aversion)
00
1
2x
00
1
x
a) − uu0 =
b) − uu0 =
c) If they start with the same initial wealth, ARA for individual A is definitely smaller than
ARA for individual B. That means that A is less risk averse than B for gambles of some
fixed cash amount. Therefore, if A rejects gamble g because it’s too risky for them then
it’s definitely too risky for B also: B will reject it.
d) Here you could illustrate the curvature of each utility function. More curvature means
higher risk aversion, so B has a more curved utility function here at some common
starting point—see slide 33 of the posted notes!
e) This is the constant ARA utility function:
u0 (x) = αe−αx
u00 (x) = −α2 e−αx
u00 (x)
rA (x) = − 0
=α
u (x)
31
(42)
(43)
(44)
Econ 100A, Fall 2022
Course pack suggested answers
f ) This is the constant RRA utility function:
u0 (x) = x−ρ
u00 (x) = −ρx−ρ−1
xu00 (x)
rA (x) = − 0
=ρ
u (x)
(45)
(46)
(47)
Note that ρ = 1 gives us a special case of CRRA: u(x) = ln x
3. a) EU = 12 u(1) + 21 u(0) =
1
2
b) We need the EU after selling the ticket to be bigger than the EU of keeping the ticket
as found in a). So we need the amount y to be big enough that u(y) > 12 . That comes
out to ≈ $0.71.
c) CARA = − x1 . Risk-lover since CARA negative. Less risk-loving as wealth increases.
d) CRRA = −1. Risk-lover, CRRA negative. Constant CRRA—same risk tolerance for a
gamble of a given fraction of wealth no matter initial wealth.
√
√
4. a) EU = 0.95 10, 000 + 0.05 0 = 95
√
b) EU = 10, 000 − C
c) We need the expression in b) to be at least as big as 95; this is true if C is no bigger than
$975.
5. a) For example: Stephanie would prefer some amount for sure over a gamble whose expected
value was that amount.
1 √
y
b) EU = [pr(win) ∗ u(win)] + [pr(lose) ∗ u(lose)] = 100
√
c) Stephanie’s expected utility if she doesn’t buy is equal to 1 = 1. Stephanie will buy if
EU (buy) > EU (don0 t)
1 √
y>1
100
y > 10, 000
d) Stephanie’s expected utility if she doesn’t buy is equal to
if
EU (buy) > EU (don0 t)
1 √
y > 10
100
y > 1, 000, 000
(48)
(49)
(50)
√
100 = 10. Stephanie will buy
(51)
(52)
(53)
e) In d) Stephanie must risk a larger amount of utility, and so the prize must be larger to
induce her to accept the bet.
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Econ 100A, Fall 2022
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f)
EU (noinsurance) = [pr(noloss) ∗ u(noloss)] + [pr(loss) ∗ u(loss)]
9 √
1 √
= ( ∗ 100) + ( ∗ 0)
10
10
=9
√
EU (insurance) = 100 − c
(54)
(55)
(56)
(57)
g) She will buy if
EU (insurance) > EU (noinsurance)
√
100 − c > 9
c < 19
(58)
(59)
(60)
h) Jim will buy if
EU (buy) > EU (don0 t)
1 2
y > 12
100
y > 10
(61)
(62)
(63)
i) By inspecting the utility functions we see that Jim is a risk-lover and Stephanie is riskaverse (you can check by calculation their coefficients of risk aversion). Therefore if Jim
rejects some gamble, Stephanie also certainly rejects it.
6. a) The inequality is u(2000) > 0.9u(2500) + 0.1u(0). (You may have normalized u(0) = 0;
that’s OK but remember to say so if you do.)
b) To get something equivalent to C, we need a mix of 12 chance of lottery A and a 12 chance
of getting nothing.
c) To get something equivalent to D, we need a mix of 12 chance of lottery B and a 12 chance
of getting nothing.
d) By the consequentialism assumption, the EUT maximizer cares about the ultimate
probability distribution over final consequences. If a EUT maximizer prefers C to
D, then 21 u(0) + 12 u(2000) < 12 u(0) + 21 (0.9u(2500) + 0.1u(0)), which simplifies to
u(2000) < 0.9u(2500) + 0.1u(0), a contradiction with what we had in part a).
This is an example of the independence axiom in action. In parts b) and c) we showed
that you can mix in the same probability of a third lottery (here the degenerate lottery
paying nothing) to A and B to construct C and D. The independence axiom says that
an EUT maximizer’s preference can’t flip when we do that.
7. See the lecture notes for these!
8. a) For Z ≤ 48 no risk-averse person will buy the ticket. If Z ≤ 48 then the expected dollar
value of the lottery ticket is no bigger than $36, and no risk-averse person would prefer
a risky bet with an expected dollar value of $36 over getting $36 for sure. For them to
prefer the risky bet it would have to have a higher expected dollar value than $36 (how
much higher depends on their level of risk aversion).
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Econ 100A, Fall 2022
Course pack suggested answers
√
√
b) Their expected utility from buying the ticket would be 14 u(0) + 34 u(Z) = 14 0 + 43 Z =
√
√
3
Z. Their expected utility if they don’t buy the ticket is 1u(36) = 36 = 6. For
4
√
them to want to buy the ticket it must be that 43 Z > 6 ⇒ Z > 64.
If the decision maker had started with more wealth, this threshold Z would have been
lower. The decision maker is less risk averse with respect to gambles of a given dollar
amount when they are wealthier, since their coefficient of absolute risk aversion is decreasing in x. The extra expected dollar value they need from the risky gamble in order
to prefer it over keeping the $36 is therefore smaller at higher levels of wealth.
9. a)
+ 12 u(0) > u(600). This decision maker is risk loving: they prefer a gamble
with an expected dollar value of $500 over a certain $600. A risk averse or risk neutral
would, by contrast, have a certainty equivalent less than $500 for this risky gamble,
meaning they would definitely prefer the sure $500.
1
u(1, 000)
2
b) For X = 2 we know that the decision maker definitely prefers C to D. If we take the
1
and then add 0.96u(0) to both
inequality from a), we can multiply both sides by 25
sides while preserving the inequality (since both of those are monotonic operations):
1
1
u(1, 000) + u(0) > u(600)
2
2
2
2
4
u(1, 000) +
u(0) >
u(600)
100
100
100
2
98
4
96
u(1, 000) +
u(0) >
u(600) +
u(0)
100
100
100
100
(64)
(65)
(66)
The last line is: the expected utility of C must be greater than the expected utility
of D. (You may also have talked about the independence axiom from Expected Utility
Theory here, which is the key assumption that is driving this aspect of that model.)
10. a) The first derivative of this function is u0 =
1
2
3x 3
. The second derivative is u00 =
applying the formula for the coefficient of absolute risk aversion: rA (x) =
−
−2
5
9x 3
1
2
3x 3
=
2
.
3x
−2
5
. So,
9x 3
00 (x)
− uu0 (x)
=
00
(x)
The formula for relative risk aversion: rR (x) = − xu
= 23 .
u0 (x)
These are measures related to the curvature of the Bernoulli function (since they rely
on the second derivative). For Bernoulli functions like this one that have diminishing
marginal utility of money (characteristic of a risk averse decision maker), the bigger the
coefficients, the more downward curvature the Bernoulli function has, which means the
more risk averse the decision maker is.
b) Since this decision maker is risk averse, they will accept some gambles with positive
expected cash value but reject others. The fact that they are risk averse means that
there are definitely some such gambles they reject. For example, if they faced a choice
between “50% chance of -$100, 50% chance of +$101” or “stay with what you have”,
this has (small) positive expected value, but even a little risk aversion would be enough
to make a decision maker say no to that, since the expected cash value is only $0.50,
small compared to the risk. However, if they faced a choice between “50% chance of
-$1, 50% chance of +$1,000,000” or “stay with what you have”, then even a person who
was quite risk averse is likely to accept that extremely valuable (risky) gamble!
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Econ 100A, Fall 2022
Course pack suggested answers
The person with Bernoulli function u = x3 is a risk lover. We know that because their
Bernoulli function displays increasing marginal utility of money (their coefficients of
risk aversion would be negative). They will accept all gambles with positive expected
cash value. They not only don’t dislike risk, they actively like it—a risky gamble with
positive expected value is certainly one they would accept since they like both the
positive expected value and the fact that the gamble is risky.
√
11. a) The √
expected utility of option B is given by EU (B) = 0.5u(0)√
+ 0.5u(400) = 0.5 0 +
0.5 400 = 10. The expected utility
√ of option A is EU (A) = 1 50 + X. Jim prefers A
over B if EU (A) > EU (B) ⇒ 50 + X > 10 ⇒ X > 50. The expected cash value of
B is (0.5 × −50) + (0.5 × 350) = 150; Jim is risk averse—his Bernoulli function shows
diminishing marginal utility of money—and so he is willing to take a sure thing that is
worth less than the expected cash value of the risky gamble.
b) Saeromi is a risk lover. There are lots of ways to express what that means! She will
prefer a risky option with some given expected value over a sure thing with that value
for sure; her risk premium is negative; certainty equivalents for her are bigger than the
cash value of risky gambles. You could also talk about the Bernoulli function, which
must have an increasing slope as her wealth increases; the following diagram shows a
Bernoulli function with increasing marginal utility of money, as would be consistent
with her risk-loving preferences over lotteries:
Since the risky bet offers an expected gain of $150, if X ≤ 150 then Saeromi will
definitely prefer the risky gamble over the sure thing. Since she is risk-loving, she will
definitely prefer a risky gamble with expected cash value of $150 over a certain amount
of cash that’s less than $150.
12. Jim currently has wealth of $50 but he faces risk. With probability 15 , he will end up with
4
double his current wealth; with probability
√ 5 , he will end up with half his current wealth.
His Bernoulli utility function is u(w) = w, where w is his final wealth. His preferences over
lotteries can be represented by the Expected Utility form.
a) Say that Jim could pay some price p to eliminate the risk he faces. What is the most he
would be willing to pay? Is this more or less than the dollar amount of the expected
loss he currently faces, and why?
We need to check whether Jim would prefer to stay with his current situation
or pay p to eliminate the risk. His expected utility in the current scenario is
35
Econ 100A, Fall 2022
Course pack suggested answers
√
√
EU = 15 u(100) + 45 u(25) = 51 100 + 45 25 √
= 6. His expected utility if he pays to
eliminate the risk is EU = u(50 − p) = 50 − p. So: he will be prefer to pay if
√
50 − p > 6 ⇒ 50 − p > 36 ⇒ p < 14. He’s willing to pay up to $14.
This is more than the dollar amount of the expected loss. His expected
change in wealth in dollar terms is 51 (50) + 45 (−25) = −10. He is willing to
pay more than the expected loss of $10 to avoid the risk because he is risk
averse. He prefers a smaller sure thing over a risky situation with a higher
expected value—equivalently, he is willing to pay a risk premium to avoid
risk.
b) Make a rough sketch of Jim’s Bernoulli function. Explain in simple terms why we could
say that Jim has ‘diminishing marginal utility of money’.
Diminishing marginal utility of money means that the rate at which Jim’s
utility is increasing as his wealth increases is smaller when his wealth is
higher. That is: as we increase his wealth, each subsequent dollar of wealth
increases his utility by less than the previous dollar did. This corresponds
to the slope of the Bernoulli function getting flatter as we move from left to
right on the diagram; the second derivative of the Bernoulli function with
respect to w is negative.
6
General equilibrium in an exchange economy
1.
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Econ 100A, Fall 2022
Course pack suggested answers
2. a), b) For a) and b):
c), d), e) For c), d) and e):
f ) There is a competitive equilibrium at the Pareto efficient allocation marked on the diagram: Jim gets bundle (4, 0) and Stephanie gets bundle (0, 3). If we have a price
ratio ppnc = 23 , then (4, 0) is a utility maximizing choice for Jim, and (0, 3) is a utility
maximizing choice for Stephanie (check that you can see why). The market clears since
the total amount of each good allocated is no more than the total endowment. So both
conditions for a competitive equilibrium are satisfied. This allocation and price ratio
are a competitive equilibrium.
3. Edgeworth box for this example:
Jim’s income: mJ = pc ωc + pd ωd = 8. His demand is the highest bundle he can afford such
that cJ = dJ ; since mJ = 8 this is cJ = dJ = 4.
TJ’s income: mT = pc ωc + pd ωd = 4. His demand is to spend all of his income on good c;
thus cT = 4, dT = 0.
Allocation cJ = 4, dJ = 4, cT = 4, dT = 0 and a price ratio
37
pc
pd
= 1 is a competitive equilib-
Econ 100A, Fall 2022
Course pack suggested answers
rium. Each consumer’s allocation is utility-maximizing at these prices given the endowment,
and markets clear since the amount of each good allocated is equal to the amount available.
Jim sells two units of coffee and buys two units of donuts; TJ sells two units of donuts and
buys 2 units of coffee.
4. mA = 2p1 + 2p2 = 2 + 2p2 and mB = 3p1 = 3.
4+4p2
,
3
x1∗
A =
2 mA
3 p1
=
x1∗
B =
1 mB
3 p1
= 1, x2∗
A =
x2∗
A =
2 mB
3 p2
1 mA
3 p2
=
=
2
3p2
+ 32 .
2
.
p2
1∗
1
1
2∗
2∗
2
2
Market clearing: x1∗
A + xB = ωA + ωB = 5 and xA + xB = ωA + ωB = 2 when p2 = 2.
So we have a competitive equilibrium where xA = (4, 1), xB = (1, 1), and pp21 = 12 . (The
consumers’ bundles are utility maximizing given the prices, and markets clear.)
5. We know that in competitive equilibrium (i) both consumers are making a utility-maximizing
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Econ 100A, Fall 2022
Course pack suggested answers
choice and (ii) markets must clear. First we will find the demand functions for each consumer
to deal with (i). To find each consumer’s income, we calculate the value of the endowment
that the consumer owns as a function of prices p1 and p2 :
mA = p1 ωA1 + p2 ωA2 = (p1 ∗ 1) + (p2 ∗ 0) = p1 ,
mB = p1 ωB1 + p2 ωB2 = (p1 ∗ 0) + (p2 ∗ 1) = p2 .
(67)
(68)
Next, we will solve for each consumer’s optimal choice of good 1. In this example we can use
the Cobb-Douglas demand functions that we learned earlier in the course. In general though
you might need to use tangency or graphical methods here.
x1∗
A
c mA
=
=
c + d p1
x1∗
B =
c mB
=
c + d p1
1
2
1
2
1
p1
=
2
+ p1
1
2
1
3
1
3
p2
p2
=
3p1
+ p1
2
3
(69)
(70)
Now we can exploit our fact (ii) that markets must clear. We know that both consumers like
good 1, and so we know that the market clearing condition will hold with equality:
x1A + x1B = ωA1 + ωB1 .
(71)
We can now combine equations 69, 70 and 71 to find a price ratio that will clear the market
for good 1 at each consumer’s optimal choice:
1
1∗
1
x1∗
A + xB = ωA + ωB
1
p2
+
=1
2 3p1
p1
2
= .
p2
3
(72)
(73)
(74)
At this price ratio, consumer 1’s optimal choice of good 1 is 21 and consumer 2’s optimal
choice of good 1 is 12 , which adds up to precisely the total endowment of 1. By Walras’ Law,
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Econ 100A, Fall 2022
Course pack suggested answers
we know that the market for the remaining good will also clear at this price ratio; let’s make
sure this is the case. By our Cobb-Douglas demand functions, we have:
x2∗
A =
d mA
=
c + d p2
x2∗
B =
d mB
=
c + d p2
1
2
1
2
p1
1 2
1
= ∗ = ,
2 3
3
+ p2
(75)
1
2
2
3
1
3
p2
2
= .
3
+ p2
(76)
2
3
And we’re done! The competitive equilibrium is at xA =
1 1
,
2 3
, xB =
1 2
,
2 3
,
p1
p2
= 23 .
Remember that a competitive equilibrium is an allocation and a price ratio that implements
it, so you must write down both things. To check your understanding, what if the price
ratio that the auctioneer had called out had instead been pp12 = 1? Why would this not have
resulted in a competitive equilibrium? Can you relate the answer to that question to the
consumers’ preferences?
6. Please see my posted lecture notes for discussion and diagrams for these questions!
7. a) (x1A )∗ =
1 mA
4 p1
=
1 4p1
4 p1
= 1, and (x2A )∗ =
3 mA
4 p2
=
3 4p1
4 p2
=
3p1
.
p2
b) At pp21 = 1, x1A = 1, x2A = 3 is the utility maximizing choice for A. If we give B the
remaining goods so that x1B = 3, x2B = 1, the market clears. This is trivially also
utility maximizing for B since all bundles for B give the same utility. So we have all the
requirements for a competitive equilibrium.
c) The allocation x1A = 4, x2A = 4, x1B = 0, x2B = 0 is a Pareto improvement over the
allocation in c), since it yields higher utility for A and the same utility for B.
d) Consumer B has preferences which do not satisfy local nonsatiation. From the utilitymaximizing bundle in c), goods could be taken from B and given to A without lowering
B’s utility.
8. a) (x1A )∗ =
1 4p1 +4p2
4
p1
= 1 + p2 , (x2A )∗ =
3 4p1 +4p2
4
p2
=
3
p2
+ 3, (x1B )∗ = 0, (x2B )∗ = 0
b) At pp21 = 13 , the utility maximizing bundles are xA = (4, 4) and xB = (0, 0). These satisfy
the market clearing conditions since x1A + x1B = 4 and x2A + x2B = 4, so we have a
competitive equilibrium with this price ratio and these bundles.
c) Since both consumers have well-behaved and continuously differentiable preferences, the
contract curve is characterized by tangency between their indifference curves: M RSA =
x2
x2
M RSB , which gives 3xA1 = xB1 .
A
B
d) At xA = (2, 3), xB = (2, 1), M RSA = M RSB and so the allocation is Pareto efficient.
By STWE it can be supported as a competitive equilibrium. To implement this, the
auctioneer must move the endowment and call a price ratio such that the proposed
allocation is utility maximizing for each consumer. (Up to here is enough for a correct
answer.)
What exactly can the auctioneer do? The price ratio must be equal to the M RS for
each consumer at this allocation: pp12 = 12 . If the endowment is moved to any point on
a budget line through the proposed allocation with this slope, we’re good to go. One
40
Econ 100A, Fall 2022
Course pack suggested answers
easy way to do this is to move the endowment to precisely the proposed allocation,
ωA = (2, 3), ωB = (2, 1), and call prices pp12 = 12 . You can check that in this case
optimal choice for both consumers is to stay at the endowment point and have zero
excess demand for each good.
9. a)
In the diagram we can see that the markets do not clear. Given the prices that were
called, the bundles xA and xB are utility maximizing for each consumer, but more of
good 2 has been chosen by both consumers than the total amount in the economy,
which violates the market clearing condition. For a competitive equilibrium, we need
that each consumer’s bundle is utility maximizing and the market for each good clears,
so this is not a competitive equilibrium.
(Your diagram must clearly show that both consumers are at a utility maximizing
bundle, which means we need to see budget lines and indifference curves to illustrate
that. I shaded in each consumers’ budget set above to make things nice and clear, but
that’s not necessary for full credit. Your diagram must show a failure of market clearing,
and you must have explained what that means to get full credit.)
b)
The diagram shows an allocation x in the exact center of the Edgeworth box. This is
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Econ 100A, Fall 2022
Course pack suggested answers
because if each consumer has an identical bundle, they have the same amount of each of
the goods and so the allocation is precisely half way across the box in both dimensions.
It also shows the indifference curve that each consumer reaches at x. The gray shaded
area shows allocations that would be on a higher indifference curve for both A and B.
Therefore x is not Pareto efficient, since there are alternatives that would make both
consumers better off than they are at x.
10. a) Yes. At a Pareto efficient allocation we cannot find an alternative allocation that is
better for one person without it being worse for anyone else. We can certainly find an
alternative allocation that’s better for one person. In the diagram, x is Pareto efficient
but consumer A prefers y over x.
b) No. If we could find an alternative allocation that was better for both consumers, the
original allocation was by definition not Pareto efficient. In the diagram, x is Pareto
efficient, and as we can see there is no alternative that is better for both consumers: if
the alternative is better for one, it’s definitely worse for the other.
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Econ 100A, Fall 2022
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11. a)
b) Since the second theorem of welfare economics holds in this economy, we know that
any allocation that is Pareto efficient can be supported as a competitive equilibrium
in this economy. Since the first theorem of welfare economics holds, we know that if a
competitive equilibrium exists, it must be Pareto efficient. So, let’s figure out whether
any of these allocations is Pareto efficient.
Since both consumers have Cobb-Douglas preferences, we know that the Pareto efficient
allocations will be characterized by tangency between an indifference curve for Tatiana
and an indifference curve for Martin. That is, where M RST = M RSM . Tatiana’s
4i3 c
c2
M Ui
M Ui
M RS = M
= iT4 T = 4ciTT . Martin’s M RS = M
= 2iMMcM = 2icMM . Therefore the
Uc
Uc
T
contract curve (Pareto efficient allocations) are those that satisfy 4ciTT = 2icMM .
Of the three allocations mentioned in the question, only allocation 1 satisfies this equation: both people have M RS = 2 at that point. This allocation can be supported as a
competitive equilibrium. The others are not Pareto efficient, and cannot be supported
as competitive equilibria.
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Econ 100A, Fall 2022
Course pack suggested answers
12. a)
A Pareto improvement would mean that we’ve found a feasible allocation that May
would prefer over her starting allocation (6, 2) and that Joel would prefer over his
starting allocation (2, 6)—we’ve found a way to re-allocate the resources in a way that
makes both people better off than they started; a win-win re-allocation. (More precisely:
it’s a different allocation that someone ‘likes better’ (in the sense of preference) than the
starting allocation, and the other person does not like less than the starting allocation.)
b) This allocation can be Pareto efficient. For example, say that uM = x1,M and uJ = x1,J .
That is: say that each consumer derives utility only from the amount of good 1 in their
bundle. Then there is no other allocation that gives someone higher utility without it
giving the other person lower utility, which is the definition of Pareto efficiency. To
increase one person’s utility, we’d have to give them more good 1, but that would mean
we’d have to take some good 1 away from the other, lowering their utility. The amount
of good 2 that each person has is irrelevant, since both are indifferent over bundles with
different amounts of good 2 but the same amount of good 1. There is no re-allocation
of good 2 that will improve either person’s utility.
(This is not a unique answer here, by the way. Another trivial example is e.g. uM =
1, uJ = 1... we need your example to both work and be justified with your brief
explanation. If you wrongly answered that this could never be Pareto efficient, you’ll
get some credit for any knowledge displayed along the way!)
13. An exchange economy has two people, A and B, and two goods, j and k. Person A has an
endowment of 8 units of good j and none of good k; their preferences can be represented
by the Cobb-Douglas utility function uA = jA kA (where jA is the amount of good j that A
has, and so on). B has an endowment of no units of good j and 6 units of good k; their
2
preferences can be represented by the Cobb-Douglas utility function uB = jB kB
. Let the
price of good j be 1. Denote the price of good k with p.
a) Write demand functions for each consumer for each good as a function of p (for this
question, it’s not required to derive the demand functions). Explain what plays the role
of ‘income’ in the demand functions.
Since both people have Cobb-Douglas preferences, we can directly use the
Cobb-Douglas demand function form. Each person’s ‘income’ is the value
of their endowment at the prices called by the auctioneer. This is because
we have an exchange economy: the consumers start with stuff that they can
sell, not money that they have to spend. So at the prices pj = 1 and pk = k,
the value of each person’s endowment is mA = 8 and mB = 6p.
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Econ 100A, Fall 2022
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Then, we can apply the Cobb-Douglas demand form. For consumer A:
jA∗ = 21 mpjA = 4 and kA∗ = 12 mpkA = p4 . For consumer B: jB∗ = 13 mpjB = 2p and
∗
kB
= 23 mpkB = 4.
b) Find a competitive equilibrium in this economy. Explain why your answer is a competitive
equilibrium.
A competitive equilibrium in this exchange economy is (i) an allocation for
each person and (ii) a price for each good (or price ratio) such that the
allocation for each person is utility-maximizing given the prices and their
endowment, and the market for each good clears, meaning that the amount
of each good in the allocations does not exceed the amount we have in the
endowments.
Let’s see if we can have the markets clear with equality at some price ratio
at each consumer’s utility-maximizing demands. First, for good j: jA∗ + jB∗ =
∗
8 ⇒ 4 + 2p = 8 ⇒ p = 2. For good k: kA∗ + kB
= p4 + 4 = 6 ⇒ p = 2. So, if we have
a price of 2 for good k, the utility maximizing choices for each consumer at
these prices do indeed clear the markets for each good.
The competitive equilibrium is therefore: a bundle (4, 2) for person A, a
p
bundle (4, 4) for person B, and a price ratio of pkj = 12 . It’s important that you
expressed all of that: an amount of each good for each consumer and a price
ratio (or a price for each good, equivalently). A competitive equilibrium
is made up of all of these things! To summarize: these allocations are the
most preferred bundles for each consumer given their endowments and these
prices, and the market for each good clears.
7
Producer theory
δy
δy
P1
1. In each case M P1 = δx
and M P2 = δx
; T RS = M
. Returns to scale is found by
M P2
1
2
computing how much would be produced if we used k > 0 times the amount of each and
every input. If we get more than k times the original output we have increasing returns to
scale, if we get exactly k times the original output we have constant returns to scale, and if
we get less than k times the original output we have decreasing returns to scale.
1
1
a) M P1 =
x23
2 , M P2 =
3x13
1
b) M P1 =
x13
2
3x23
, T RS =
x2
,
x1
decreasing returns to scale
, T RS =
x2
,
x1
constant returns to scale
1
x22
1 , M P2 =
2x12
x12
1
2x22
c) M P1 = x2 , M P2 = x1 , T RS =
x2
,
x1
increasing returns to scale
d) M P1 = 1, M P2 = 4, T RS = 14 , constant returns to scale
Each of the three has isoquants that are the same shape (Cobb-Douglas shape) but they are
spaced differently: constant returns they are spaced evenly, but for increasing returns they
get closer together as we move away from the origin, and for decreasing returns they get
further apart.
2. a) Average product is
y
x
=
√4 .
x
Marginal product is
45
dy
dx
=
√2 .
x
Econ 100A, Fall 2022
Course pack suggested answers
√
b) π = py − wx = 4 xp − wx. The first order condition for a maximum is
which we can solve for x =
4p2
w2
dπ
dx
=
2p
√
x
− w = 0,
.
√
c) Back to the production function to find y ∗ = 4 x∗ =
8p
.
w
d) With w = 1, y ∗ = 8p. Sketch this on the usual supply curve axes, output on the x axis
and price on the y axis. Notice that if we take different w values the supply curve moves
in the intuitive direction.
√
3. π = 10(2 x) − 5x. The first order condition for a maximum is dπ
= 0 ⇒ √10x − 5 = 0 which
dx
we can solve for x∗ = 4 and therefore y ∗ = 4 from the production function. π(x∗ ) = 20.
4. The production
returns to scale. Scale up both inputs by k > 1:
√
√ function√ has decreasing
√ √
10 min{ kx1 , kx2 } = k10 min{ x1 , x2 }. This is less than k times the original amount.
We know that at the profit-maximizing choice, Jim Corp. will choose x1 = x2 (since otherwise
it could reduce whichever is bigger and save on costs without losing any output). Call
x1 = x2 = x. Therefore we can be sneaky and write the profit function as a function of just
one variable:
π = py − c(y)
√
= p10 x − (w1 + w2 )x
√
= 120 x − 12x
(77)
(78)
(79)
We can maximize this in the usual way. The first order condition, dπ
= 0, is √60x − 12 = 0, so
dx
we solve for x∗1 = x∗2 = 25, and the production function tells us that output is then y ∗ = 50.
5. a) Fixed 10, variable 0.5y 2 . M C = y, AC =
10
y
+ 0.5y.
b) Fixed 0, variable 3y. M C = 3, AC = 3.
c) Fixed 100, variable 2y + 0.1y 3 . M C = 2 + 0.3y 2 , AC =
100
y
+ 2 + 0.1y 2 .
6. i. y = x1 + x2 . Each unit of labor has marginal product of 1, and scaling up both inputs by
a constant factor increases output by that same factor.
1
1
ii. y = x12 x22
1
1
iii. y = x14 x24
P1
= 51 = 5. This tells us the rate at which the producer could swap robots for
7. a) T RS = M
M P2
humans while maintaining the same level of output. In this case it’s constant at 5 (the
isoquants here would be straight, parallel lines with slope −5). This means that if the
producer could replace 5 humans with one robot and reach the same level of production
as before; equivalently, if they reduced the number of robots by 1 they would need to
add 5 humans to reach the same level of production as before.
b) If we scale up each factor of production by a constant proportion k, we’d get 5(kr)+(kh) =
k(5r + h) = ky—that is, we’d get the same proportionate increase in output as the
proportion by which we scaled up all of the inputs. This means we have constant
returns to scale. For example, if this producer used twice as many robots and twice as
many humans, they’d produce exactly twice as many drinks.
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Econ 100A, Fall 2022
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c)
8. Kristal Corp. is organizing a one-day massage pop-up where they will produce massages y
2
using a single input, massage therapists x, according to the production function y = 6x 3 .
The price of massages is p = 50 and the wage rate for a massage therapist for the day is
w = 100.
−1
dy
a) M Px = dx
= 4x 3 = 41 . So as x increases, the marginal product of a massage therapist
x3
gets smaller and smaller. This production function has decreasing returns to massage
therapists.
b) We’re going to write the profit function and then look for the x that maximizes it.
π =R−C
= py − wx
2
= 50(6x 3 ) − 100x
2
3
= 300x − 100x
−1
dπ
= 200x 3 − 100 = 0
dx
⇒ x∗ = 8
y ∗ = 6x
∗ 23
= 24
(80)
(81)
(82)
(83)
(84)
(85)
(86)
This point satisfies the first order condition for a maximum; since the second derivative
−4
d2 π
= −1
400x 3 is always negative, this point is a maximizer. Kristal Corp. hires 8
dx2
3
massage therapists and 24 massages are provided.
c) The average product is 24
= 3. This lets us see a slightly different way to calculate profit:
8
on average each massage therapist produces 3 massages (price 50 each) and their wage is
100—so each generates 150 − 100 = 50 in profit, and all 8 of them generate 8 ∗ 50 = 400
in profit. You can verify this if you like using the usual profit function!
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Econ 100A, Fall 2022
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9. a)
= 0 ⇒ 25−5−4y = 0 ⇒ y ∗ = 5. Profit is positive—one
b) π = 25y −(8+5y +2y 2 ) and so dπ
dy
quick way to see that is that average cost y8 + 5 + 2y at y = 5 is less than 25, so the firm
makes a positive profit margin.
c) Average variable cost is 5 + 2y. As long as price is bigger than AVC then the firm will
produce something in the short run. The lowest AVC can get is 5, so as long as p > 5
they will produce in the short run.
d) Average cost is y8 + 5 + 2y. Price has to be at least as big as average cost or the firm
will prefer to shut down in the long run. The lowest point of AC occurs at y = 2 when
average cost is 13 (we can get this either by using calculus to find the minimum of
average cost, or by looking for where MC=AC since that occurs at the lowest point of
average cost). For p < 13 the firm would shut down in the long run.
2
10. a) The profit function here is π = R − C = py − wx = 30x 3 − wx − F . The first order
condition for a maximum is therefore dπ
= 201 − 20 = 0 ⇒ x∗ = 1, y ∗ = 3. This does
dx
x3
not depend at all on F . The profit-maximizing point is found where M R = M C, but
F does not appear in marginal cost since it doesn’t vary with x or y. Marginal cost is
the rate at which cost changes as we increase output; F does not change.
b) The marginal revenue curve would be flat at 10. For the producer in a perfectly competitive industry, marginal revenue is simply the (constant, unchanging) price—their
choice of output has no bearing on the price at all. If they produce more output, it
can simply be sold at whatever the market price is. The marginal cost curve would
be upward-sloping. Since returns to the input x are diminishing, it takes more and
more extra input to produce each subsequent unit of output, and since the price of x is
constant, this means that it costs more and more to produce each subsequent unit of
output.
11.
a) M Pt =
δy
δt
=
2
√
t
and M PL =
δy
δL
= 0.5. T RS =
M Pt
M PL
=
2
√
t
1
2
=
4
√
.
t
At Jim’s current production plan, the marginal product of tea is √24 = 1. This means
that, holding fixed the amount of labor, an increase in cups of tea increases the number of
quiz questions at the rate 1:1. (If you made the linear approximation, you could loosely
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Econ 100A, Fall 2022
Course pack suggested answers
speaking say ‘1 more cup of tea yields 1 extra quiz question, if labor is unchanged’.)
The marginal product of labor is 0.5, which similarly means that, holding fixed cups of
tea at 4, an increase in hours of labor increases quiz questions produced at the rate of
one hour of labor to half a quiz question.
Technical rate of substitution is √44 = 2. This means that we can swap labor for tea
at the rate of 2 hours of labor to 1 cup of tea and leave output unchanged. (At the
linear approximation: if you reduced labor by 2 and increased tea by 1, output would
be unchanged.)
√
b) If√we√scaled up both tea and labor by √
k > 1, we’d get output equal to 4 kt + 0.5(kL) =
4 k t+0.5kL, which is less than k(4 t+0.5L): the production function has decreasing
returns to scale. If we increase the amount of both tea and labor in the same proportion,
we will get less than that proportion of an increase in quiz questions produced.
This means that the isoquants are getting ‘further apart’ as we move in a straight line
away from the origin. It takes a bigger scaling up of inputs to go from y = 20 to y = 25,
for example, than to go from y = 15 to y = 20.
12.
a) Since we know that the M RT for this production function is always exactly 13 , we can
tell that the production function must be Y = L + 3K. This has the right M RT , and
the point given on the Y = 100 isoquant.
Since returns to scale are decreasing, we know that the isoquants are getting further
and further apart—it takes a bigger scaling up of inputs to go from y = 200 to y = 300
than it does to go from y = 100 to y = 200.
b) We know that the profit-motivated producer’s optimal choice will be to choose the
cheapest way to produce the their optimal output. Since they’ve picked K = 0 they
must be using only labor to produce the output. Since labor is always three times
more productive than capital, it can be up to three times more expensive and it will
be the cheapest way for the producer to go. So we need 3wL < wK . (For example, to
produce 100 output would take either 100 labor or 100
capital; the labor way is cheaper
3
100
if 100wL < 3 wK ⇒ 3wL < wK .
13. Two questions about producer theory:
49
Econ 100A, Fall 2022
Course pack suggested answers
a) The diagram shows a very rough
sketch of an isoquant map. Consider three production functions:
0.75
ˆ y = x0.75
1 x2
ˆ y = 0.2x1 + 0.2x2
0.3
ˆ y = x0.3
1 x2
Which of the three production
functions does the sketch most
closely resemble and why?
0.3
The third production function y = x0.3
1 x2 most resembles this isoquant map.
The second production function is not correct since these isoquants are not
straight lines; the second production function has a technical rate of substitution equal to 1, meaning that its isoquants would be straight, downwardsloping lines with slope −1. The first production function is not correct since
the isoquant map pictured is consistent with decreasing returns to scale—
the gap between each isoquant is getting bigger and bigger for the same 10
unit increase in output. The first production function has increasing returns
to scale since its Cobb-Douglas coefficients sum to more than 1 (we could
check this with the ‘scaling up by k’ method we used in class).
The second production function fits the map best. It has decreasing returns
0.3
to scale (to check: for k > 1, (kx1 )0.3 (kx2 )0.3 = k 0.6 x0.3
< ky) and it has
1 x2
x2
a technical rate of substitution of x1 , generating the convex level sets we
see in the diagram. For a successful answer here, we needed you to give
an explanation that shows why only the second production function can
be consistent with the diagram—the slope and the spacing are what will
let us identify that. However! Since production functions 1 and 3 are both
constant returns to scale production functions, we can accept an explanation
that is only about the decreasing returns (spacing) part as long as it was
clear and accurate!
b) Say that a profit-motivated producer in a perfectly competitive industry had a cost
function c(y) = F + 0.2y + 0.1y 2 . How would the size of F impact the producer’s
decision of how much to produce in (i) the short run and (ii) the long run? Why?
The size of F will have no effect at all on the producer’s optimal choice in the
short run. The fixed cost is irrelevant to the short run decision. There are
a few different ways to see this. Mathematically, when we maximize profit
with respect to quantity of output, the fixed cost plays no role since it does
not change with quantity. Intuitively, the fixed cost is sunk and must be paid
in the short run no matter what quantity of output the producer chooses. In
the model of perfect competition, the usual M R = M C condition for profit
maximization (produce up until the point when extra revenue from more
output no longer outweighs the extra cost of producing more output) boils
down to p = M C since marginal revenue is constant at price under perfect
50
Econ 100A, Fall 2022
Course pack suggested answers
competition. Marginal cost does not have F in it since F does not change
with output. Further, comparing price to average variable cost will tell us
whether the producer prefers to produce at that p = M C point rather than
nothing at all—can they cover their variable costs?
In the long run, F tells us whether this producer will want to stay in the
industry or not. This is because F certainly does impact the economic
profit that the firm makes at their optimal choice. If that economic profit
is negative, the producer will exit the industry when they can (in the long
run) and so produce nothing going forward. If economic profit is positive,
the producer will stay in the industry and produce their profit-maximizing
quantity of output. So: at some threshold a sufficiently large F would lead
the producer to want to shut down and produce zero in the long run; a
smaller F would lead them to stick around.
8
Perfect competition and partial equilibrium
1. a) The long run is the length of time after which all costs are variable. For example, if your
fixed cost is rent, then the long run is when your lease is up. This means that the long
run isn’t a set length of time; it could be different in different contexts. The perfectly
competitive model predicts entry by new firms when there is positive profit to be made,
and exit by firms who are making losses. These waves of entry or exit push the price
in the industry up or down respectively. So that means that (taking the case where
firms are identical for simplicity) in the long run the model predicts that firms make
zero profit. At that point there is no entry and exit. This point is characterized where
applicable by p = M C = min AC due to the fact that M C intersects AC at AC’s lowest
point and profit-maximizing firms produce where p = M C.
b) In a perfectly competitive industry, price is fixed and taken as given by the producer
(this, by the way, can be motivated by the idea that price elasticity of demand for a
given producer is infinite when there are arbitrarily many producers...). This means
that marginal revenue for a producer in this industry is p, the price. Since a profit
maximizing firm will produce at a point where M R = M C (from our usual calculus of
profit maximization) we get that p = M C.
2. a) qS = qD when p − 5 = 25 − p. So p = 15 and q = 10. CS and P S we can compute as the
areas of the triangles in the first of the three diagrams. CS = P S = 50.
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Econ 100A, Fall 2022
Course pack suggested answers
b) With a 2 per unit tax we know that pS = pD − 2. So we need pS − 5 = 25 − pD which
we can rewrite (pD − 2) − 5 = 25 − pD . This solves for pD = 16 and so pS = 14. Now
CS = 40.5, P S = 40.5, T = 18, and DW L = 1.
c) Since equilibrium price is above the ceiling the price that will prevail will be 10 (if the
equilibrium had been below the price ceiling then the ceiling would have no effect).
Quantity traded is 5 since that is the quantity supplied at p = 10 and it is smaller than
the quantity demanded. From the diagram we get CS = 12.5 + 50 = 62.5, P S = 12.5,
and DW L = 25.
dqD
× pqDD . Why? This is the slope of demand curve normalized
3. Price elasticity of demand is dp
D
for units—it gives us the percentage change in demand divided by the percentage change in
dqD
× pqDD = 1 ∗ pqDD .
price. In this example: dp
D
For example, say that price is 10. Then qD = 25 − 10 = 15 and so price elasticity of demand
is 1 ∗ pqDD = 10
= 32 .
15
4. The equilibrium price here would be where QS = QD , which is P − 20 = 40 − 12 P or P = 40
with an associated quantity of Q = 20. So what we have here is a price above the equilibrium
level. That means we’ll get excess supply.
From the diagram: CS = 0.5(30 ∗ 15) = 225, P S = (15 ∗ 15) + 0.5(15 ∗ 15) = 337.5, and
DW L = 0.5(5 ∗ 15) = 37.5.
5. a) 85 − 12 q = p = 10 + q at q = 50 and p = 60.
b) 85 − 12 q = p = 25 + q at q = 40, p = 65.
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Econ 100A, Fall 2022
Course pack suggested answers
c)
d) A perfectly inelastic demand curve is vertical. So the whole response to the shift in
supply is through price. In this case when the (upward-sloping) supply curve shifts,
the point of intersection is at the same quantity as before, and there is a bigger price
response than with a downward-sloping demand curve.
e) More elastic demand means a smaller price response in the market equilibrium following
a shift in supply.
6. a) 50 − p = 21 p − 10 at q = 10 and p = 40.
b)
c) From the graph we can use areas: CS =
1
2
∗ 10 ∗ 10 = 50, P S =
1
2
∗ 10 ∗ 20 = 100.
d) pD = 50 − qD and pS = 20 + 2qS . We know that with this tax, pD = pS + 3, so we need
50 − q = 20 + 2q + 3. This occurs at q = 9, pD = 41, pS = 38.
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Econ 100A, Fall 2022
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e)
f ) Again using areas: CS = 12 ∗ 9 ∗ 9 = 40.5, P S = 21 ∗ 18 ∗ 9 = 81, Revenue = 3 ∗ 9 = 27.
Two ways to calculate deadweight loss. First, before consumers’ and producers’ surplus
was 150, now the sum of surpluses plus revenue is 148.5, for a difference of DW L = 1.5.
We could also calculate the area of the DW L triangle on the diagram for DW L =
1
∗ 3 ∗ 1 = 1.5.
2
7.
a) QD = QS where 65 − P = 2P − 10 ⇒ 3P = 75 ⇒ P = 25. The associated quantity is
65 − 25 = 40 units.
b) Quantity traded is 30. The price ceiling at $20 binds; at that price quantity demanded
is 45 and quantity supplied is 30.
c) A tax of $15 per unit would result in 30 units traded. The reason is the price paid by
buyers exceeds the price received by sellers by exactly $15 at 30 units traded. This is
therefore the point that we’d arrive at if there was a $15 per unit tax.
Producers are just as well off as under the price ceiling. They would still receive $20 per
unit for 30 units sold. Buyers, however, would be worse off under the tax. They would
pay $35 per unit, not $20 as before, for the 30 units and so would get lower consumer
surplus. (Tax revenue accounts for the difference between the old CS+PS and the new
one; deadweight loss would be the same in both cases. I didn’t ask for a diagram in the
question but here it is for your information:)
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Econ 100A, Fall 2022
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d) 40 units are traded. The reason is that the price ceiling of $30 is above the equilibrium
price of $25 and so it has no effect on the equilibrium situation in the market. We
remain at the original price and quantity as in part a).
8. a) Fixed costs are 25, variable 5y + y 2 .
b) M C =
dc
dy
= 5 + 2y and AC =
c
y
=
25
y
+ 5 + y.
c) π = py − c(y) = 35y − (25 + 5y + y 2 ). At the optimal choice
dπ
=0
dy
35 − 5 − 2y = 0
y ∗ = 15.
(87)
(88)
(89)
At this output, π = (35 ∗ 15) − (25 + (5 ∗ 15) + 152 ) = 200.
d) In the long run, since this industry is perfectly competitive, the supply curve is where
price equals minimum average cost:
dAC
=0
dy
25
− 2 +1=0
y
y = 5.
(90)
(91)
(92)
Putting this back into average cost, we get p = 15 at the minimum.
e) We already got y = 5 in the previous part as the output that minimizes average cost. So
π = (15 ∗ 5) − (25 + (5 ∗ 5) + 52 ) = 0, which is as we expected in a long run perfectly
competitive industry.
9.
a) AC =
c(y)
y
=
450
y
+ 10 + 12 y and M C =
55
dc(y)
dy
= 10 + y.
Econ 100A, Fall 2022
Course pack suggested answers
(Notice that the intersection of MC and AC is certainly at the point where AC is at its
lowest.)
b) Price right now must be above 40. At the firm’s optimal choice (where p = M C, since
they’re profit maximizers in a competitive industry) it must be that p > AC since we’re
told that they make a positive profit.
In the long run, price will be exactly 40. Free entry and exit of producers in response
to positive or negative profit drives the price towards the level where economic profit is
precisely zero. That occurs where, at the producer’s optimal choice, p = AC. (You may
also have noted that we have used the homogenous products/firms assumption, since
that means that each firm makes zero economic profit, not just the marginal entrant.)
10.
a) π = Revenue − Cost = ps − c(s) = ps − 500 − 0.5s − 0.05s2 . The first order condition
= 0 ⇒ p − 0.5 − 0.1s = 0 ⇒ s = 10p − 5. The supply curve, drawn
for a maximum is dπ
ds
as usual with quantity on the horizontal axis and price on the vertical axis, is:
b) Tsunoda Corp. may make positive, negative, or zero profit in the short run—we don’t
know which because we don’t know the price of sushi. (If price is above the average
cost of production at their optimal choice, they’ll make positive profit; if it’s below,
they’ll make negative profit.) In the long run Tsunoda Corp. will make zero profit.
In the perfectly competitive model, positive profit induces entry by new producers,
driving price down and reducing profit for each producer; negative profit induces exit
by producers, driving prices up and increasing profit for each producer. This process
leads, in the model, to zero economic profit in the long run for all producers in the
perfectly competitive industry.
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11. a) No. None of the three plans is less-preferred than another plan by all three groups.
b) Plan 1 is welfare-maximizing under a utilitarian SWF.
c) The utilitarian SWF considers the sum of wellbeing to be paramount. This captures
ethics that, for example, accept some person being hurt by some amount as long as
some other person benefits by a greater amount.
d) Plan 2 is welfare-maximizing under a minimax SWF.
e) The minimax SWF considers the wellbeing of the least well off to be paramount. This
captures ethics that, for example, say that some change makes society better off only if
it makes the person with the least utility better off.
12. a) It is not. At the allocation xT = 6, xJ = 0, UT = 6 and UJ = −36. At an equal division,
UT = UJ = 3 − 92 = −6. Since TJ is better off in the original allocation, the equal
division is not a Pareto improvement.
b) Using the information above, for the original allocation W = 6 + (−36) = −30 and
W = (−6) + (−6) = −12. The equal division is better under a utilitarian SWF.
c) Using the information above, for the original allocation W = min{6, −36} = −36 and
W = min{−6, −6} = −6. The equal division is better under a minimax SWF.
d) One way to quickly see the first part is that at an allocation (0, 0) then the utilitarian
SWF takes a value of W = 0; this is better than any allocation in which 6 beers are
allocated among the two people, since the square of any number bigger than one is
bigger than the number itself. The least number of beers that should be thrown away
is 4. If there are two beers or fewer (and remembering there must be a whole number of
beers for each person) then there is definitely a way to achieve W = 0. No better level
of W can be achieved, so at least 4 bottles thrown away gets us to the highest possible
W.
13. a) First, note that the utilitarian SWF is W = uJ + uS : it looks for the biggest sum of
the utilities. The minimax SWF is W = min{uJ , uS }, it looks to make the smallest
utility as big as possible. We gave credit for both a) and b) if you demonstrated this
knowledge.
i. Utilitarian SWF is maximized by either J = 200, S = 0 or J = 0, S = 200.
This is because squares increase exponentially: to make the sum as big as possible,
we should make one of the amounts J or S as big as possible. Minimax SWF is
maximized by J = S = 100, since J 2 = S 2 for a maximum with that SWF.
ii. With these utility functions, both the utilitarian SWF and minimax SWF are maximized by J = √
S = 100.
√ For minimax SWF the logic is the same as in part i.
For utilitarian, 100 + 100 is bigger than any other allocation with J 6= S since
square roots decrease exponentially.
iii. Utilitarian: J = 200, S = 0. The sum of utilities is maximized when Jim gets all
the money. Minimax: J = S + J gives us S = 0 and so J = 200. This gives both
people U = 200.
b) Utilitarian: J = 200, S = 0. Each dollar gives Jim 3 utility and Saeromi only 1. Minimax:
3J = S gives us J = 50, S = 150.
Cobb-Douglas: W = 3J(200 − J) and so dW
= 600 − 6J = 0: the maximum is J = 100,
dJ
S = 100.
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c) Any split of the $200 is Pareto efficient for the utility functions in a)i. since in that case to
increase the utility of any person requires taking a dollar away from the other, lowering
their utility.
For a)iii. only J = $200 and S = $0 is Pareto efficient. If Saeromi gets any money in
the allocation, we can always make a Pareto improvement by taking a dollar from her
and giving it to Jim—her utility is unchanged but Jim’s would increase.
14. a) An example of a reallocation would be for Jim to get (4, 2) and Martin to get (1, 2). This
is because this reallocation gives both Jim and Martin a higher utility than the original
allocation X, which satisfies the definition of a Pareto improvement. At the original
allocation X, Jim gets utility uJ = 32 × 3 = 27 and Martin gets uM = 2 × 12 = 2. At
the proposed reallocation Jim gets uJ = 42 × 2 = 32 and Martin gets uM = 1 × 22 = 4.
Intuitively: Jim ‘favors’ good 1 in his utility function and Martin ‘favors’ good 2: if we
swap one of Jim’s good 2 for one of Martin’s good 1, it’s a win-win reallocation.
b) For part (i): we need something such that uJ + uM is higher than in allocation X, but
min{xJ , xM } is smaller than in X. One easy example: give everything to Jim. Then
uJ = 52 × 4 = 100 and uM = 0 × 02 = 0. The sum of the two utilities is higher than
in X, but the lowest utility is smaller. The utilitarian SWF finds that this has higher
welfare than X since the sum of utilities is higher but the minimax SWF finds this to
have lower welfare than X because the worst-off person is worse off.
For part (ii): we need uJ + uM to be lower than in X, but min{xJ , xM } to be higher
than in X. Let’s take something from Jim and give it to Martin, to try to raise the
lower of the two utilities! An example: Jim gets (3, 2) and Martin gets (2, 2). Then
uJ = 32 × 2 = 18 and uM = 2 × 22 = 8. The sum of the utilities (utilitarian social
welfare) is now 26, less than 29 as it was in X, but the lowest utility (minimax social
welfare) is now 8, bigger than the 2 in X.
15. Consider an economy with two people, Bojack and Diane, and two goods, bags of chips c and
sodas s. Each person has Cobb-Douglas preferences: Bojack’s utility function is uB = cB sB
and Diane’s is uD = cD sD , where cB is the number of bags of chips Bojack has, and so on.
There are 10 total bags of chips and 10 total sodas in the economy.
a) Sketch an Edgeworth box to represent this economy. Mark an allocation Z in which
Bojack’s bundle is (8, 8) and Diane’s is (2, 2). Sketch the indifference curve for each
person on which their bundle lies (we need the right general shape and positioning for
the indifference curves, not a precise sketch). How do you know what the indifference
curves look like at that point?
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The allocation mentioned is Pareto efficient; the indifference curve for Bojack
at (8, 8) is tangent to the indifference curve for Diane at (2, 2). We know this
M Uc
because their M RS is the same. Bojack’s M RS = M
= scBB and Diane’s
Us
M Uc
M RS = M
= scDD , so each has M RS = 1 at the allocation Z. What we were
Us
looking for: was the Edgeworth box right, was Z in the right place, was the
shape of each indifference curve the right idea, and were the indifference
curves tangent to each other. (This means, by the way, that in this economy
the contract curve—the set of all Pareto efficient points—is a straight line
that connects the origins at either corner of the Edgeworth box. If we are
not on that line, the indifference curves would have crossed.)
b) Find a feasible allocation that is ‘better’ than Z under a utilitarian social welfare function.
Find a feasible allocation that is ‘better’ than Z under a minimax social welfare function.
Show your calculations and briefly explain why your allocations are ‘better’ in each case.
There are many, many possibilities here. (As an aside, I’ll note that there
are ‘better’ allocations that are also Pareto efficient; there are also ‘better’
allocations that are Pareto inefficient.) First, let’s observe that at Z Bojack’s
utility is uB = 8 × 8 = 64 and Diane’s is uD = 2 × 2 = 4. The utilitarian social
welfare function thus attaches W = 64+4 = 68 to this allocation; the minimax
social welfare function attaches W = min{64, 4} = 4 to this allocation.
An allocation with (10, 10) for Bojack and (0, 0) for Diane would be ‘better’
under the utilitarian social welfare function. Bojack’s utility is uB = 10×10 =
100 and Diane’s is uD = 0 × 0 = 0 and so the sum of their utilities—i.e.
utilitarian social welfare—is W = 100 + 0 = 100, which is bigger than at Z.
The utilitarian SWF considers higher total utilities to be ‘better’. (Notice
that the minimax social welfare function finds this allocation worse than Z.)
An allocation with (5, 5) for each person would be ‘better’ under the minimax
social welfare function. Each person would have a utility u = 5 × 5 = 25
and so minimax social welfare, given by the smaller of their utilities, is
W = min{25, 25} = 25, which is bigger than at Z. The minimax SWF considers
a higher smallest utility in society to be ‘better’. (Again we can note that
the utilitarian social welfare function finds this worse.)
(As one last aside, notice that (10, 9) for one person and (0, 1) for the other
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is also a utilitarian improvement on Z; (6, 4) and (4, 6) is also a minimax
improvement on Z. While all welfare-maximizing allocations are Pareto
efficient, welfare improvements can be found at Pareto inferior allocations!)
9
General equilibrium with production
√
= 0 when
1. a) π = py − wl = 4 l − wl. dπ
dl
∗
Substituting lD into π, we get π = w4 .
2
√
l
∗
− w = 0. Solving for l gives lD
=
4
.
w2
b) Value of profit plus labor income is w4 +wl and the price of pancakes is 1, so he could afford
4
+wl pancakes in total. Substituting this in to U = 2y−2l2 gives U = 2( w4 +wl)−2l2 =
w
8
+ 2wl − 2l2 . dU
= 0 when 2w − 4l = 0. Solving for l gives lS∗ = w2 .
w
dl
∗
c) The market for labor clears when lD
= lS∗ when w42 = w2 . Solving we get w = 2. At w = 2,
l = 1 and y = 4, which is the competitive equilibrium.
2. We can solve for the producer’s optimal demand for x and the consumer’s optimal supply
of x by using the same method as in question 1. When w = 4, we solve to find that the
producer’s optimal choice is to demand x∗D = 81 , and the consumer’s optimal choice is to
supply x∗S = 2. The market does not clear, and so we do not have a competitive equilibrium.
3. Consider an economy with one producer and one consumer. The producer is a profit1
maximizer who produces output y using labor l according to a production function y = 8l 2 .
The consumer is a utility-maximizer who consumes y and supplies l; their utility function is
U = y − 2l2 . Let the price of labor be w and normalize the price of y to be 1. The consumer
receives any profits earned by producer Jim and earns w per unit of labor supplied.
1
∗
a) π = py − wl = 8l 2 − wl. dπ
= 0 when √4l − w = 0. Solving for l gives lD
=
dl
32
16
∗
∗
means yS = w . Substituting lD into π, we get π = w .
60
16
.
w2
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Econ 100A, Fall 2022
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+ wl and the price of output is 1, so he could
b) Value of profit plus labor income is 16
w
16
afford w + wl units of output in total. Substituting this in to U = y − 2l2 gives
U = ( 16
+ wl) − 2l2 = 16
+ wl − 2l2 . dU
= 0 when w − 4l = 0. Solving for l gives lS∗ = w4 .
w
w
dl
∗
c) The market for labor clears when lD
= lS∗ when w162 = w4 . Solving we get w = 4. At w = 4,
l = 1 and y = 8, which is the competitive equilibrium.
d) At w = 1, the labor demand by the producer is 16 and the labor supply by the consumer
is 41 . The market for labor does not clear at the utility and profit maximizing choices
for each decision maker. (By Walras’ Law, the market for the output good will also fail
to clear.)
√
4. a) The profit function for the firm is π = py − wl = 12 l − wl. We can look for a candidate
for a maximizer like so:
dπ
6
= √ −w =0
dl
l
√
6
l=
w
36
∗
lD
= 2.
w
(93)
(94)
(95)
(Since the second derivative of profit with respect to l is always negative, this is indeed
p ∗ a
∗
∗
maximizer.) The associated yS we can get via the production function: yS = 12 LD =
72
. The amount of profit the firm makes (we’ll need this in the next part) is therefore:
w
∗
π = pyS∗ − wlD
72 36
−
=
w
w
36
=
w
(96)
(97)
(98)
b) Jim the consumer owns the firm, so receives 36
in profit. He can also work; if he works l
w
36
hours, then he’ll have a total of w + wl dollars. Since his preferences are increasing in
y, he’ll certainly spend all of this on coconuts. So his utility function can be written as
U = 4( 36
+ wl) − 32 l2 . The candidate for a maximizer can be found like this:
w
dU
= 4w − 3l = 0
dl
4
lS∗ = w.
3
(99)
(100)
(Again since the second derivative of utility with respect to l is negative, this is indeed
∗
a maximizer.) The associated demand for coconuts is then yD
= 36
+ 43 w2 .
w
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c) Markets clear when
∗
= lS∗
lD
36
4
= w
2
w
3
w3 = 27
w=3
(101)
(102)
(103)
(104)
∗
(We could also verify this by checking that yD
= y ∗ S; by Walras’ Law this will also be
true at w = 3 unless we made a math mistake!) At w = 3 the amount of labor supplied
and demanded is equal to e.g. w362 = 4 and therefore the amount of coconuts produced
√
and sold is y = 12 4 = 24. So the competitive equilibrium is: p = 1, w = 3, l = 4,
y = 24.
5. a)
b) With increasing returns, then for any finite price ratio wp , the firm’s optimal choice of
production plan is to employ infinite labor and produce infinite output. The consumer
with well-behaved preferences will certainly not choose to supply infinite labor, so the
market does not clear. The producer demands more labor than is supplied. (Note that
there’s a special case if the auctioneer calls out p = 0: then the firm produces nothing
at all, but the consumer certainly demands positive y in this case, so again the market
does not clear.)
6. a) The slope is the same because prices are the same for everyone: the slope of an isoprofit
line is the price ratio, and the slope of a budget line is the price ratio. The intercept is
the same because the firm’s profit level (divided by the output good price if we haven’t
normalized it to 1) is the intercept of the isoprofit line, and, since the consumer owns
the firm and receives its profits, this is also the value of the the non-labor income that
the consumer has (they can purchase πp units of the output good).
Another way to think of the slope is that the isoprofit line shows combinations of input
and output that would result in exactly the same level of profit. One more unit of
input and wp more units of output would leave profit unchanged, since the extra unit of
input costs w and the extra units of output bring in wp ∗ p = w. The budget line shows
combinations of input and output that would exhaust the consumer’s budget. One more
unit of input supplied brings in w in labor income, which if spent can purchase wp extra
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units of output.
b) ‘General’ means that the model is comprehensive and self-contained: it’s a model of all
markets and all people in the economy at once. Contrast with partial equilibrium: a
model of one market and its participants, holding ‘all else equal’. Only the price ratio
matters (and not the absolute value of prices) because the model is self-contained. If the
price of every good in the economy was to double, for example, nothing material would
change: it would double everything that’s measured in units of money, like profit and
income, but it wouldn’t change any of the physical quantities of stuff, like production
functions or budget constraints. By contrast, if the price of only one good was to double
in a partial equilibrium model, it would change, for example, the amount of that good
that a consumer could afford with their cash budget.
I’m open to the idea that there might have been different perspectives / answers to
these two questions a) and b) that show good understanding! In our grading we’ll be
asking whether you wrote something clear, accurate, and relevant to the question, not
whether your answer exactly matches mine. For example, in a) I think the important
pieces are the intercept being the same because the consumer owns the firm, and the
slope being the same because the tradeoff being captured on each line is happening at
the same rate; but your mileage may vary.
7. a)
This is not an equilibrium because the markets fail to clear at the producer’s profitmaximizing point and the consumer’s utility maximizing point. A competitive equilibrium allocation is one that is optimal for each producer and consumer, given prices, and
that satisfies market clearing. Here, at the optimal choices, we have more demand for
coconuts than supply, and less demand for labor than supply.
b) If its conditions hold, the first welfare theorem says that a competitive equilibrium, if
one exists, is Pareto efficient. In the Robinson Crusoe model such a point would occur
where the slope of the consumer’s indifference curve is exactly equal the slope of the
production function. That is: M RS = M RT . The rate at which Robinson is willing to
give up time (i.e. go to work) in exchange for coconuts is exactly the same, at such a
point, as the rate at which the economy can turn time into coconuts. At such a point,
there is no available alternative combination of leisure and coconuts that is better for
Robinson. (Notice that this is a little weird in a Robinson Crusoe model since there’s
only one person!)
8. a) First, Robinson will have
12
w
+ wl dollars to spend if they choose to work for l hours.
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This is because he owns the firm, receiving its profit, and earns wl labor income. We
+ wl units of y, since the price of y is 1 and since the
therefore know that he will buy 12
w
consumer’s preferences are monotonic in y and there are no other consumption goods
available.
We can therefore rewrite his utility function just as a function of one choice variable, l:
U = 3( 12
+ wl) − l2 . The first order condition for a maximum is dU
= 0 ⇒ 3w − 2l =
w
dl
3
∗
0 ⇒ l S = 2 w. (Since the second derivative of U w.r.t. l is always negative, this is
certainly a maximizer.)
b) In competitive equilibrium, the allocation must be profit-maximizing for the producer
and utility-maximizing for the consumer, given the prices, and the market for all goods
must clear. Here, we would need labor demand by the producer to be the same as labor
supply for the consumer: w122 = 32 w ⇒ w3 = 8 ⇒ w = 2.
The competitive equilibrium must have w = 2 since this clears the labor market. By
Walras’ Law, we know that this price ratio, wp = 2, will also clear the market for the
output good: in a two good model, if we find a price ratio that clears one market, it will
certainly clear the other. In general with n goods: if we find a price ratio that clears
n − 1 markets, it will certainly clear the last.
a) The consumer’s budget line would look exactly like the isoprofit line with intercept π2 .
This is the highest isoprofit line the producer can reach subject to the constraint of the
production function.
π = py − wx = y − wx ⇒ y = π + wx: the intercept of the isoprofit line is the profit
earned by the producer, and the slope is the price of input x (relative to the price of
output, here normalized to 1).
Since consumer Robinson owns the firm, π2 becomes their non-labor income, and so
this becomes the amount of output they can buy if they do not work. If they choose to
work more, each hour of work yields w extra dollars and so w extra units of output (at
the price 1). The intercept and slope of the consumer’s budget line are therefore both
identical to the intercept and slope of the isoprofit line that described the producer’s
optimal choice.
b) The consumer’s preferences must be such that their marginal rate of substitution between
output and working is not equal to the slope of the production function. In equilibrium
of a Robinson Crusoe model (in which the conditions of the first welfare theorem hold)
the consumer’s M RS is exactly equal to the M RT as given by the slope of the production function. (If this was not true, then at the prices called by the auctioneer at least
one of the two would not be maximizing their utility/profit.) Since we know there’s no
equilibrium at A, it must be that the consumer’s MRS is different than the slope of the
production function at that point.
10
Monopoly and market power
1. The profit function is π = p(y)y − c(y) = (60 − y)y − (100 + y 2 ). We can use the first order
= 0 is satisfied when 60 − 2y − 2y = 0, that is,
condition to search for a maximizer: dπ
dy
y ∗ = 15. The associated price is p = 60 − 15 = 45.
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By computing areas in the figure we can get CS = 112.5, P S = 450, and DW L = 37.5.
2. a) π(y) = p(y)y − c(y) = 20y − 12 y 2 − (40 + 4y), and at the maximum π 0 (y) = 20 − y − 4 =
0 ⇒ y ∗ = 16. The associated price is p = 20 − 12 16 = 12. The firm’s profit is
π = (16 ∗ 12) − 40 − (4 ∗ 16) = 88.
b) Check whether the price consumers are willing to pay for the next unit is bigger than
MC: M C = 4, W T P = p(17) = 20 − 21 17 = 11.5. Since W T P > M C it is socially
efficient to produce a 17th unit. To find the socially efficient output, we need p(y) =
M C ⇒ 20 − 12 y = 4 ⇒ ysoc = 32. p(32) = 20 − 12 32 = 4.
c) π(32) = (32 ∗ 4) − 40 − (4 ∗ 32) = −40. Profit is negative at y = 32: the firm prefers to
exit than stay in the market at p = 4.
3. a) R = 12y − y 2 , M R = 12 − 2y and so M R = M C ⇒ 12 − 2y = 6 ⇒ y = 3, p = 9. Profit
is π = (3 ∗ 9) − (3 ∗ 6) = 9.
b) p = M C = 6, lump sum payment is
π = 18 + (6 ∗ 6) − (6 ∗ 6) = 18.
65
1
2
∗ 6 ∗ 6 = 18. The monopolist does better:
Econ 100A, Fall 2022
Course pack suggested answers
4. A profit-maximizing monopolist faces a demand function P = 30−3Q and has a cost function
10 + 3Q.
dπ
= 0 when 30 − 6Q − 3 = 0, or Q∗ = 4.5 and
a) π = (30 − 3Q)Q − (10 + 3Q) and so dQ
P ∗ = 16.5. The socially efficient outcome would be where P = M C, which is given by
30 − 3Q = 3 or Q = 9, P = 3.
b) The optimal two-part tariff, assuming that this is a demand function for a representative
consumer, would be P = 3 and a lump sum of 121.5. The method is the same as the
previous question—the area for the lump sum is 12 ∗ 9 ∗ 27.
5. a) Inverse demand for each group is pA = 120 − yA , pB = 200 − yB . Revenue from each
group is therefore RA = 120yA − yA2 , RB = 200yB − yB2 and marginal revenue for each
group is M RA = 120 − 2yA and M RB = 200 − 2yB .
Optimal choice is given then by:
Group A: M RA = M C ⇒ 120 − 2yA = 10 ⇒ yA = 55, pA = 65.
Group B: M RB = M C ⇒ 200 − 2yB = 10 ⇒ yB = 95, pB = 105.
b) yA + yB = y = 320 − 2p, and so p = 160 − 21 y. R = 160y − 12 y 2 , M R = 160 − y.
M R = M C ⇒ 160 − y = 10 ⇒ y = 150, p = 85.
c) π(discrimination)
=
(55 ∗ 65) + (95 ∗ 105) − (150 ∗ 10)
=
12050.
π(no discrimination) = (150 ∗ 85) − (150 ∗ 10) = 11250
You could also just reason this out to avoid doing the calculation. The monopolist
could have charged the uniform price from b) in part a) if it had wanted. But it chose
something different. It must be that the restriction to uniform price means that the
monopolist does worse in b) than a).
6. This follows exactly slides 13 and 14 of the market power notes. Please check there for the
derivation. The intuition is raising price involves a tradeoff for the producer: less quantity
demanded but a higher price per sale. When demand is inelastic the gain from the higher
price outweighs the loss from less quantity demanded and so raising the price increases
revenue. Since we would imagine that producing less is less costly too it follows that profit
is higher as a result of raising the price as well.
dR
= 200 − Q.
7. a) In the first case: R = P (Q)Q = (200 − 21 Q)Q = 200Q − 21 Q2 and so M R = dQ
1
In the second case: M R = 200 − 2 Q. The answers are different because under first
degree price discrimination the monopolist can sell each unit of the good at a different
price without affecting the price for the other sales—it simply sells each marginal unit
for exactly the price that the next consumer is willing to pay. With a single constant
price, then selling more units means having to lower the price for all units as we move
down the demand curve. This means that M R is less than the price of the next unit,
since it includes the revenue-lowering effect of reducing price from where it would have
been to sell a smaller quantity.
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b)
Demand is price inelastic at prices below $100 (equivalently: at quantities above 200).
We know this because beyond that point marginal revenue is less than zero—MR is
less than zero when demand is price inelastic. (This is because the ‘downward drag on
prices’ effect on revenue outweighs the ‘extra sales’ effect on revenue at those points.
Conversely, at those points, raising prices (and so reducing quantity) increases revenue
for the monopolist because the increase in price increases revenue by more than is lost
in fewer sales.)
8. a)
b) R = (20 − y)y and so M R = 20 − 2y.
c) M R = M C at 20 − 2y = 0, so p∗ = 10, y ∗ = 10
d) π = (10 ∗ 10) − 50 = 50
e) CS =
1
2
∗ 10 ∗ 10 = 50
f ) p = M C when 20 − y = 0, so ysoc = 20, p = 0
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g) π = (20 ∗ 0) − 50 = −50, so Scott prefers not to develop; consumer surplus is therefore
zero and consumers are worse off
9. a) For consumer A: π = (8 − Q)Q and so the first order condition for a maximum is
dπ
= 8 − 2Q = 0 ⇒ Q∗ = 4, P ∗ = 4. For consumer B: π = (20 − 2Q)Q and so the
dQ
dπ
= 20 − 4Q = 0 ⇒ Q∗ = 5, P ∗ = 10. The
first order condition for a maximum is dQ
monopolist makes total profit of π = (4 ∗ 4) + (10 ∗ 5) = 66.
b) In part a) we found that M R = 8 − 2Q from consumer A and so marginal revenue is
-$2 at Q = 5. MR can be negative even with a positive price because while more sales
are made, they happen at a lower price. If the effect of reducing the price outweighs
the effect of the extra sales, MR is negative. (This happens where demand is price
inelastic.)
c) By sketching out the demand curves for both consumers, we can see the maximum that
they’re each willing to pay for some amount of the good. The monopolist can offer 8
units at a total price of $32 to consumer A, and 10 units at a total price of $100 to
consumer B. This extracts all consumer surplus from both consumers and so is the most
money that the monopolist can make from each one. They make a total profit of $132.
d) The situation in a) is not socially efficient but consumer surplus is positive; the situation
in c) is socially efficient but consumer surplus is zero. First degree price discrimination
results in a more efficient outcome but is worse for consumers than a single per-unit
price.
10. See the lecture notes for an example of this!
11. a) (For the answer key here I’ve added stuff to the diagram that was on the question paper
to illustrate the answers! We didn’t specifically need a diagram in your answers.) The
monopolist’s optimal choice is found where M R = M C, since this is the first order
condition of their profit-maximization problem (up to that point, increasing output
increases profit; after that point, it decreases profit). From the diagram, we can see
that at the optimal quantity Q∗ , price exceeds average cost and so the monopolist
makes positive profit.
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The socially efficient quantity is where p = M C, since that is the quantity that would
maximize the sum of producer and consumer surplus. At that point, Qe , we see that
price is less than average cost and so the monopolist makes negative profit.
b) The intersection of the demand curve with the average cost curve shows us this point:
it has quantity marked Q0 in the following diagram. That’s because at this pricequantity pair, the price is exactly the same as average cost and so the monopolist
makes zero profit. Despite the fact that profit is zero, we can see that producer surplus
is positive (the area between price and marginal cost of production). Producer surplus is
independent of fixed costs, which are of course part of average cost but do not contribute
to the marginal productivity of each unit of production which is what PS captures. (By
the way it’s just a total coincidence based on how I drew the diagram that the quantity
Q0 happens to be right where MR hits the quantity axis!)
12. a) The profit-maximizing two part tariff is a per-hour price of $0.50 and an entry fee of
$18. We know this because pricing at $0.50, the marginal cost per hour of pinball
provided, leads the consumer to stay for as many hours as their willingness and ability
to pay exceeds the marginal cost to Jim to provide each of those hours. It makes the
consumer’s surplus value from the per-unit price deal as big as possible, meaning that
Jim can then charge up to that amount as the entry fee.
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When charging $0.50 per hour, the customer chooses to stay for 12 hours, for which
they would be willing to pay the shaded area over and above the per-unit price of $0.50
each. That shaded area is 12 × 3 × 12 = 18. The customer is willing to pay an entry fee
of up to $18. $18 entry fee, $0.50 per-hour price is the optimal two-part tariff.
b) Jim is bored of keeping track of how many hours people stay. He decides to switch to a
new policy: pay to enter, then a price of zero per hour. What’s the biggest entry fee
he can charge now? How much less profit would he get compared to the plan in a),
assuming he charges the biggest entry fee possible in both cases?
At zero price per hour, the customer chooses to stay for 14 hours. Their total willingness
to pay for 14 hours is 12 × 3.50 × 14 = 24.50. The biggest entry fee he can charge is
now $24.50. If he charged the fee structure in a), his profit is π = R − C = 18 + (12 ×
0.5) − (10 + (12 × 0.5)) = 8. If instead he charged a $24.50 fee and $0 per hour, his
profit would be π = 24.50 − (10 + (14 × 0.5)) = 7.50. Jim would make $0.50 less profit
from this customer under the plan in b).
13. Consider a profit-motivated monopolist. The inverse demand curve for their product is
P = 21 − Q, and their cost of production is C(Q) = 3Q + 12 Q2 .
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a) Consider the case in which the monopolist
cannot price discriminate and must set a single, constant price for their product. Find an
equation for marginal revenue and add a line
representing it to this diagram. Find and
show on the diagram their optimal choice
of price and quantity. Indicate areas representing consumer surplus, producer surplus,
and/or deadweight loss, as applicable.
Revenue here will be R = P ×Q = (21−Q)Q = 21Q−Q2 and so marginal revenue
dR
= 21 − 2Q. The monopolist’s optimal choice can be found either
is M R = dQ
at M R = M C or, equivalently, at the first order condition for a maximizer of
their profit function. M R = M C ⇒ 21 − 2Q = 3 + Q ⇒ 18 = 3Q ⇒ Q∗ = 6. The
associated price, from the demand function, is P = 21 − 6 = 15. The diagram
illustrates. Consumer surplus (CS), producer surplus (PS), and deadweight
loss (DWL) are marked.
b) Consider the case in which the monopolist can
use first degree (a.k.a. ‘perfect’) price discrimination. Find and show on the diagram
their optimal choice of quantity. Indicate areas representing consumer surplus, producer
surplus, and/or deadweight loss, as applicable. What is going on with prices in this
case?
Here, the monopolist can sell each unit of its output for a different price,
personalized to the willingness to pay of each consumer. Each consumer is
charged precisely their willingness to pay (or if you find this bothersome
some price a tiny bit below their WTP). The monopolist will keep selling
until the next person is not willing to pay more than the marginal cost of
production: all the way up to where demand hits MC at 21 − Q = 3 + Q ⇒
Q = 9. (All surplus goes to the producer—all the way down the demand
curve, each unit of output is sold for precisely the height of the demand
curve. Consumers are left with no surplus.)
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Externalities
1. A pollutant is produced as a byproduct of production by a large number of profit-maximizing
polluters. The marginal private benefit of the pollutant to a polluter is M P B = 1000 − Q,
where Q is the total amount of the pollutant. There is no cost to the polluter to produce the
pollutant. However, the pollutant carries an externality effect on people in the area due to
its environmental impact, and the effect gets more severe the more pollution there is. The
marginal social cost of the pollutant is given by M SC = Q.
a) To find the efficient amount we look for M SB = M SC, the amount of pollution that
balances marginal social benefit and marginal social cost. Here since the benefit to the
polluter captures all benefits and the cost to the impacted people captures all the costs,
that equation is 1000 − Q = Q which we can solve for Q = 500.
The model instead predicts that polluters will produce pollution up until the point where
M P B = M P C, the amount of pollution that balances the marginal private benefit to
them with the marginal private cost to them. Since there is no cost to the polluter, this
equation is 1000 − Q = 0 or Q = 1000.
b) A Pigouvian tax of $500 per unit raises the marginal private cost of emissions to $500
for the polluter. The new point that balances MPB and MPC for the polluters is now
at 500 units of emissions, the socially efficient amount.
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c) The efficient number of permits is 500. If the market for permits was competitive (and
since the question specifies a large number of polluters, this is at least conceivable here)
then the supply and demand model might give us a useful look at what the permit price
would be.
The supply curve is vertical at the number of permits issued. Its intersection with the
demand curve is at a price of $500. The intuition is that there are 500 units of emissions
that are ‘profitable’ to make at the price $500, and so this is the price that clears the
market for permits.
(Note that a reasonable alternative way to view this question could be: what if the
permits were distributed randomly/equally to all current polluters? Then we would see
an upward sloping supply curve as polluters would presumably be willing to sell their
permits if the price was above the MPB of pollution to them.)
d) The polluters that end up emitting in both of these cases are the ones for whom the
permit is most valuable (i.e. those with a willingness to pay above $500. One argument
in favor of this is that we as a society may want emissions, if they must be made, to be
made by those industries/parties for whom it is most valuable. This depends a bit on
the extent to which price signals for those industries’ products reflect well their social
value, but aside from that it makes some sense. If you must burn fossil fuels, at least
let’s make it for the most valuable reasons.
A counterargument might be that the polluters who find it most valuable to emit may be
systematically different from those that are shut out. For example large producers may
be more willing and able to pay for permits than small producers. Random assignment
of permits may give a more superficially fair distribution, but it would leave gains from
trade on the table if they were not allowed to be traded.
There are surely many more arguments you could have used for this question—the
allocation of permits is an important issue in cap-and-trade systems.
2. a) π1 = px x − c1 (x, f ) = px x − c(x) + f (1 − f );
1 − 2f = 0 ⇒ f = 12
b) π2 = ps s − c2 (s, f ) = ps s − c(s) − f 2 ;
dc2 (s,f )
df
c) M SB = M SC ⇒ 1 − 2f = 2f ⇒ f =
1
4
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dπ1
df
= 2f
= 1 − 2f so first order condition is
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d) π1,2 = π1 + π2 = px x + ps s − c(x) − c(s) + f (1 − f ) − f 2 and via first order condition
dπ1,2
= 1 − 2f − 2f = 0 when f = 41
df
e) Since optimal x and s are unaffected, the only change is due to the cost difference when
2
f = 41 rather than 12 . When f = 12 the contribution of f to profits is 12 (1 − 12 ) − ( 12 ) = 0.
2
When f = 14 the contribution of f to profits is 14 (1 − 41 ) − ( 14 ) = 81 . Profits to the
merged firm are higher by 18 than profits to the two unmerged firms.
Figure 1: 1.1 a), e) and f)
3. a)
Un
b) M RS = − M
=
M Ux
1
√
2 n
M Uq
c) M RS = − M
=
Ux
√1
q
d) M RSA = M RSB when
1
√
2 n
=
√1
q
⇒ 4n = q. Since n + q = 10 this implies n = 2, q = 8.
e) By d) we know this is not Pareto efficient.
f ) By d) we know this is not Pareto efficient.
g) A Pareto improving trade would see consumer A trade some cash to B for the right to
make noise.
h) By d) we know that the distribution of noise will be the same in both cases since trade
will take us to such a point. But consumer A will end up with more cash than B after
trade from the endowment in e), and the opposite after trade from the endowment in
f).
4. a) Each considers how the average value per person (their benefit) compares to their cost
(zero). People show up until the average value has fallen to a point where it equals zero.
2
Average value is 10n−n
= 10 − n, which equals zero at n = 10.
n
b) It is socially efficient to add people until the marginal social value falls to equal marginal
social cost (zero). Marginal social value is the derivative of 10n − n2 , which is 10 − 2n.
This equals zero when n = 5.
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c) People will make their decision privately as in a), but the cost is now t, not zero. We
want to find how big t should be to make 10 − n = t for n = 5, our target for n. This
requires t = 5.
5. a) M SB = M SC ⇒
50
√
s
= 10 ⇒ s = 25
√ = 10 ⇒ s = 100. This is more than a) because each user doesn’t
b) AP = M P C ⇒ 100
s
have to account for the external cost to others of their use of the library—each extra
user reduces the value enjoyed by all other users.
100
√
25
= 10 + t ⇒ t = 10. This is a Pigouvian tax. It works by charging users the value
of marginal external harm evaluated at the efficient amount of the good, so raising the
marginal private cost at the efficient amount to match the true social cost, meaning
that the marginal user after n = 25 no longer finds it worthwhile to use the resource.
√
√
= √2j − 2 = 0 and
6. a) π = 4( j + m) − 2j − m. The two first order conditions are dπ
dj
dπ
= √2m − 1 = 0. Solving we get j = 1 and m = 4.
dm
√
b) π = 4( s − 18 m) − 2s. The first order condition is √2s − 2 = 0 or s = 1.
√
Stephanie’s profit in this scenario is π = 4( 1 − 18 4) − (2 ∗ 1) = 0. If instead m = 0 her
√
profit would be π = 4( 1 − 18 0) − (2 ∗ 1) = 2. She would be willing to pay up to 2 for
Jim to turn the music off.
√
√
c) Jim’s profit right now
is
π
=
4(
1
+
4) − (2 ∗ 1) − 4 = 6. If instead m = 0 his profit
√
√
would be π = 4( 1 + 0) − (2 ∗ 1) − 0 = 2. So Jim would not be willing to accept a
payment to turn off the music unless it was at least 4.
c)
d) The√easiest
√ way to do this is√to maximize the sum of the two profit functions, π =
4( j + m) − 2j − m + 4( r − 18 m) − 2r. Since this is separable in all arguments,
dπ
we can just look at the first order condition with respect to m: dm
= √2m − 1 − 12 = 0.
Solving we get m = 16
. This is smaller than the amount Jim originally chose. Jim
9
doesn’t account for the negative externality effect of his music on Stephanie, and so he
plays too much music. The socially efficient amount accounts for the effects on both
parties.
e) A pair of headphones would let Jim listen to music without disturbing Stephanie. So
he would
√ be√able to choose j = 1 and m = 4 as in part a) for a total payoff of
π = 4( 1 + 4) − (2 ∗ 1) − 4 = 6. In part b) we figured out that Stephanie would earn
a profit of 2 when m = 0. So their combined profit is 8.
The maximum combined payoff
is the one with the m
√ in√the case without headphones
√
we found in part d). π = 4( j + m) − 2j − m + 4( s − 18 m) − 2s with j = 1, s = 1,
m = 16
. At these values, π = 4(1 + 43 ) − 2 − 16
+ 4(1 − 18 16
) − 2 = 6 32 . So the social
9
9
9
value of a pair of headphones is the difference between these two amounts: 8 − 6 23 = 1 13 .
7. a) Total cost to Susan of n nights is 2n + n2 , so M P C = 2 + 2n. M P B = 10 since she gets
10 revenue per night. M P C = M P B when 2 + 2n = 10 ⇒ n∗ = 4. She will open the
club for 4 nights per week.
b) Total cost to everyone of the club being open n nights is 2n + n2 + n2 , so M SC = 2 + 4n.
M SB = 10 since there are no external benefits. So M SC = M SB when 2 + 4n = 10 ⇒
nS = 2.
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c) This would change Susan’s decision since it raises her M P C to 2+2n+f . M P C = M P B
is now 2 + 2n + f = 10, which at n = 2 means we need a fee f = 4. This internalizes
the externality for Susan—she now feels an additional cost to make her account for the
effect of her choice on Jim. (The amount, by the way, is equal to the marginal external
cost imposed by Susan on Jim at the socially efficient n = 2.)
d) First, what is the value to Jim of n = 2 compared to n = 4? At n = 4 the cost to Jim
is 42 = 16, while at n = 2 it’s only 22 = 4. So Jim would be willing to pay up to $12
to get Susan to only open 2 nights instead of 4. Second, what is the value to Susan of
opening 4 nights instead of 2? At n = 2 she gets a payoff (total benefit minus total
cost) of 2(10) − ((2 ∗ 2) + 22 ) = 12. At n = 4 she gets a payoff (total benefit minus total
cost) of 4(10) − ((2 ∗ 4) + 42 ) = 16. So Susan would be willing to accept anything bigger
than $4 to only open 2 nights instead of 4. Any b between $4 and $12 would leave both
better off after making the deal.
√
√
1
= √1e − 1 =
8. a) π1 = 2(2 x1 + e) − x1 − e. The first order condition for a maximum is dπ
de
0 ⇒ e∗ = 1. (Note that because x1 and e are additively separable in the profit function,
we only have to look at the first order condition with respect to e to find optimal e.)
√
b) π2 = x2 + 12 e − x2 and so when firm 1 produces 1 unit of e it raises firm 2’s profit by 12 .
c) One way to do this is to
e that maximizes the joint profits of both firms.
√ find the level of
√
√
π1 + π2 = 2(2 x1 + e) − x1 − e + x2 + 21 e − x2 . The first order condition for with
respect to e is √1e − 1 + 12 = 0, which we can solve for e = 4. This is more than in a)
because firm 1 does not take into account the positive external benefit of e to firm 2
when it makes its decision. The private benefit to them of increasing e was less than
the cost, but the social benefit to both firms of increasing e is bigger than the cost.
d) A Pigouvian subsidy of 21 per unit would induce firm 1 to choose the socially efficient
amount. This is because 12 is the size of the marginal effect of e on the other firm, and
so this subsidy brings the private benefit of e to firm 1 into line with the social benefit
of e, inducing them to choose the socially efficient amount.
9. a) La Noisette’s profit function: πN = R − C = 5d − (10 + 14 d2 . At the first order condition
for a maximum, dπ
= 5 − 21 d = 0 ⇒ d∗ = 10. (Note that since the second derivative is
dd
everywhere negative, this critical point is certainly a maximizer of the profit function.)
The socially efficient amount can be found by maximizing the joint payoff of both firms
(you might have alternatively looked to balance M SB and M SC; that’s equivalent and
1 2
t − d),
also fine if you did it correctly). So: πN + πJ = 5d − (10 + 14 d2 ) + 2t − (10 + 10
d(πN +πJ )
1
and at the first order condition
= 5 − 2 d + 1 = 0 ⇒ de = 12.
dd
b) First, let’s see how much worse off La Noisette is at the socially efficient amount compared
to their profit-maximizing amount. When d = 10, πN = (5 × 10) − (10 + 41 (102 )) = 15.
When d = 12, πN = (5 × 12) − (10 + 14 (122 )) = 14. La Noisette makes $1 more profit
at their optimal choice than at the socially efficient amount.
What about for Jim’s Tea? We could go ahead and calculate Jim’s optimal choice and
figure out how much profit he makes, but it’s quicker and easier just to notice that the
only way d affects πJ is that each unit of d reduces his total cost by 1. This means that
at d = 12 Jim’s total cost is $2 lower than at d = 10, and so his profit is $2 higher.
If Jim offers a payment between $1 and $2 to La Noisette in exchange for producing
d = 12 rather than d = 10, both parties are better off.
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10. a) The socially efficient amount could be zero if, for example, the marginal external effect
always exceeds $160. This would mean that the marginal social cost—the external
effect plus the constant $40 marginal private cost illustrated in the diagram—would
always exceed $200, so it would always be greater than marginal social benefit, whose
largest value is $200 as seen in the diagram. In simple terms, that means that the
harm to society from any incremental increase in the amount of this pollutant is big
enough to outweigh the benefits. In that case, producing any positive quantity of
the pollutant would reduce the net benefit to society. (Another valid answer is if the
marginal external effect is always sufficiently large to keep marginal social cost above
marginal social benefit. There are also more complicated potential cases where MSC
initially exceeds MSB but dips below it, to that total benefit exceeds total cost for some
quantity.)
b) A Pigouvian tax would increase the marginal private cost of emitting the pollutant.
Each extra unit of the pollutant emitted would now incur a tax, and so the cost to the
polluter at the margin would be increased by the amount of the tax. A famous realworld example is a carbon tax, which is a Pigouvian tax designed to raise the cost of
emitting CO2. This is a Pigouvian tax because it is designed to address an externality:
carbon emissions cause harm to third parties and so the Pigouvian tax is intended to
internalize the externality for the polluters. The tax forces them to take into account
the effect of their actions on the rest of society, bringing their private cost of polluting
in line with the true social cost.
11. Two roommates, A and B, live in an economy with three goods, cash (x), cooking TV shows
√
(c), and sitcom TV shows (s). A likes cooking shows: their utility function is UA = xA + 2 c
(where x√A is the amount of cash A has). B likes sitcoms: their utility function is UB =
3xB + 2 s. Their apartment has one TV so there is an externality problem: whenever one
type of TV show is on, the other one is not. Assume that each person has $200 in cash and
that they share the apartment and its TV for 20 hours per week.
a) Sketch a clearly and fully labeled Edgeworth box diagram (cash on the horizontal axes
please) to represent this economy. Do we know where the endowment point is in this
economy? If so, where is it? If not, why not?
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The Edgeworth box has width 400 (the total amount of cash) and height 20
(the total hours of either c or s on the apartment TV). I didn’t ask for that
specifically, but if you gave it, that’s great! I called the origin points OA and
OB , which is just my notation—what I was looking for was ‘did you label the
axes properly’ and ‘is it obvious which consumer’s being measured in which
direction’.
We don’t know where the endowment point is in the vertical direction. This
is because we don’t have any information on the initial allocation of property
rights. How is control of the TV initially allocated? Without knowing that,
we don’t know what bundle we’re starting at—we can’t trade if we don’t
know who owns what!
We do know that the endowment point must be half way across the Edgeworth box in the horizontal direction though, since we know that each person
initially has $200 in cash. But depending on property rights, it could be in
different spots in the vertical direction. For example, say that the presumption is that each person has the right, if they want, to control the TV for
half the time. Then the endowment point would be right in the center. Or
what if person B has the right to control the TV all the time. Then the
endowment point would be all the way at the bottom of the Edgeworth box.
b) The Coase Theorem could apply here under the assumptions that property rights are
well-defined and that there are no transaction costs of bargaining. If these assumptions
hold, how many hours of cooking shows and how many hours of sitcoms would the
theorem predict we’d see on this apartment’s TV? Show your work and explain your
answer.
The Coase Theorem predicts that bargaining to implement mutually beneficial trades will ultimately, in this scenario, take us to a Pareto efficient
point. Once we know who owns what and once we assume that bargaining
is ‘easy’, then the gains from trade can be realized to the mutual benefit of
both parties, until we reach efficiency (so the idea goes, at least).
So: here the socially efficient points
are where M RSA = M RSB√(the contract
√
1
Ux
Ux
=
=
c
and
M RSB = M
= √32 = 3 s. So at the
curve. M RSA = M
2
√
M Uc
M Us
2 c
2 s
√
√
efficient points, c = 3 s ⇒ c = 9s, and since c + s = 20, at the efficient points
c = 18, s = 2. We’d anticipate 18 hours of cooking shows and 2 hours of
sitcoms.
12
Game theory
1. a) Regardless of what row thinks column will do, their best option is B. If column was to
play A, then row gets 2 if they play A but 3 if they play B; if column was to play B,
then row gets 0 if they play A but 1 if they play B.
b) The unique Nash equilibrium is when both players choose B. In any other situation, at
least one person could have done better given what the other did, but when both play
B neither can do better than by choosing B given that the other chose B.
c) It is not. If both choose A then both players would get a higher utility than when both
chose B.
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d) Here’s one suggestion:
Column player
A
B
Row player A 1, 1 0, 3
B 3, 0 2, 2
e) $16. If everyone contributes $10 then $40 in total is contributed to the public good, and
so everyone receives 0.4 ∗ 40 = 16.
f ) In this scenario, there is $30 in the common pool, which means that everyone gets 0.4∗30 =
12 from the public good. The fourth player, having kept their $10, therefore gets $22
in total.
The unique Nash equilibrium in this game is for all four players to contribute nothing
to the public good, ending up with $10 each, if everyone cares only about maximizing
their own cash payoff.
g) Repeated games allow for patience, rewards, and punishment. It could be that players
contribute more if they expect to be shamed or sanctioned for free riding; even absent
explicit punishment opportunities they may fear retaliation (‘if you don’t contribute
today, I won’t contribute tomorrow’). Further, if players care about more than just
their own cash payoff, then they might choose not to free ride. As always, different
preferences may manifest as different choices in the same situation!
2. Game 1 has one pure strategy Nash equilibrium: player 1 plays D and player 2 plays R.
Game 2 has two pure strategy Nash equilibria: in the first, player 1 plays U and player 2
plays L; in the second, player 1 plays D and player 2 plays R.
Game 3 has two pure strategy Nash equilibria: in the first, player 1 plays U and player 2
plays L; in the second, player 1 plays D and player 2 plays R.
Game 4 has no pure strategy Nash equilibrium. (If you are interested, we can note that there
is a Nash equilibrium in ‘mixed strategies’ in this game, as there also would have been in
games 2 and 3. If you would like to know more, you can look this up in one of our reference
sources or ask!)
Game 5 has one pure strategy Nash equilibrium: player 1 plays M and player 2 plays C. (If
you are interested, we may notice that in this example the unique Nash equilibrium in pure
strategies is also the unique strategy profile that survives a process called ‘iterated deletion
of strictly dominated strategies’. If you would like to know more, you can look this up in
one of our reference sources or ask!)
3. a) One possible example:
Column player
A
B
Row player A 2, 1 0, 0
B 0, 0 1, 2
(A,A) and (B,B) are both pure strategy Nash equilibria here, but as we can see Row
prefers the first and B the second. (There would also be a mixed strategy Nash equilibrium somewhere, but that is beyond the scope of the course.)
b) One possible example:
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Column player
A
B
A
10,
10
0,
0
Row player
B 0, 0
1, 1
Again, (A,A) and (B,B) are both pure strategy Nash equilibria, but the outcome in
(A,A) Pareto dominates the outcome in (B,B). Notice that the concept of Nash equilibrium has no way to select among these—there is no sense in which we can say that
(A,A) is ‘more likely’ without introducing more theory. Economists have extensively
studied concepts like salience, risk dominance, and focal points to try to think through
these kind of situations. See Mehta, Starmer, and Sugden (1994), for example.
4. Two players, Row and Column, will simultaneously and independently decide whether to
‘share’ or ‘steal’. Cash outcomes have a ‘prisoner’s dilemma’ structure. However, Column
feels shame s > 0 if they choose ‘Steal’. Row’s payoff is their cash outcome; Column’s is their
cash outcome, minus their shame if they ‘Steal’. The following matrix summarizes (each cell
has Row’s payoff first, and Column’s second):
Column
Share
Steal
Share
5,
5
0,
8−s
Row
Steal
8, 0
2, 2 − s
This game has a unique Nash equilibrium, but it depends on s. What is the Nash equilibrium
of this game, for different values of s? Explain.
If s > 2 then the unique Nash equilibrium is when the Row plays Steal and
Column plays Share. If s < 2 then the unique Nash equilibrium is when Row
plays Steal and Column plays Steal.
First, we can see that Row’s best response is Steal, regardless of what strategy
Column uses. If Column shares, then Row gets 8 with Steal and only 5 with
Share. If Column steals, Row gets 2 with Steal and only 0 with Share. So there
can never be a Nash equilibrium in which Row plays Share, since that is never
a best response for Row.
Next, let’s figure out Column’s best responses. If Row shares, then Column’s
best response is Share if s > 3 and Steal if s < 3. If Row steals, then Column’s
best response is Share if s > 2 and Steal if s < 2.
So the mutual best response (in which each player is best responding, given what
the other did) is either at (Steal, Share) in the case with s > 2 or at (Steal, Steal)
if s < 2.
5. We will start at the bottom of the game tree and work our way back to the top.
The last decision notes are those in which player 3 chooses a strategy. If player 3 finds herself
at the node on the left side of the tree, she prefers playing X to playing Y. If she finds herself
at the node on the right side of the tree, she prefers playing Y to X. We can eliminate the
actions that result in the less preferred outcomes.
Next, we can go one step back up the tree, to the nodes at which player 2 chooses a strategy.
If player 2 finds himself at the left side node, he anticipates a payoff of 1 by playing A
(anticipating player 3’s rational action) and 4 by playing B. Thus he prefers to play B.
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Similarly at the right side node, he anticipates 1 by playing A and 2 by playing B, so again
prefers to play B. We can again eliminate the other actions.
Finally, we go one layer up again, now arriving at the initial decision node of player 1. Player
1 anticipates the previous logic for players 2 and 3 and so anticipates a payoff of 1 by playing
L and 2 by playing R. Thus R survives backward induction and L is eliminated.
Our subgame perfect Nash equilibrium (the one that survives backward induction) is this:
player 1 plays L; player 2 plays B if player 1 plays L and plays B if player 1 plays R; player
3 plays X if player 1 plays L and player 2 plays A and plays Y if player 1 plays R and player
2 plays A.
There are lots of different notation styles (more concise than the long elaboration I’ve just
written) that we may use to write down the strategy profile in equilibrium. Whichever you
choose, the crucial part is: you must write down a complete contingent plan of action for
each player. This means that the strategy you write for each player must specify an action
at each node in which they may be called on to make a decision, not just those nodes that
are on the equilibrium path.
6.
a) In pure strategies, NE1: both play Buy; NE2: both play Don’t Buy
b)
c) NE1: Stephanie plays buy, Jim plays buy if Stephanie buys, buy if Stephanie doesn’t.
NE2: Stephanie plays buy, Jim plays buy if Stephanie buys, don’t buy if Stephanie
doesn’t.
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NE3: Stephanie plays don’t buy, Jim plays don’t buy if Stephanie buys, don’t buy if
Stephanie doesn’t. (Try writing out the matrix form... this will be a 2x4 matrix!)
d) SPNE: Stephanie plays buy, Jim plays buy if Stephanie buys, don’t buy if Stephanie
doesn’t.
7. Jim Corp. and Ivan Corp. are racing to release new Carly Rae Jepsen remixes to the market.
Each of them must decide whether to remix song A (“Julien”) or song B (“Stay Away”).
Since Ivan is a much less lazy person than Jim, this will proceed as a sequential move game.
First Ivan will choose whether to work on song A or song B, and then Jim will observe his
choice and decide whether to work on song A or song B.
Two things determine each person’s payoff: which song they pick and whether they pick the
same as the other person. If they work on the same song, they each get 0 because their songs
will compete with other, but if they work on different songs, they each get 2. In addition to
that, since song A is more famous, anyone who chooses song A gets an extra 1.
a) (5 points) Sketch a game tree to represent the extensive form of this game.
b) (5 points) Sketch a matrix that represents this game. Find the Nash equilibria in pure
strategies. What outcomes are possible in the pure-strategy Nash equilibria of this
game?
For Jim’s strategies, I’ve written e.g. ‘A;B’ to represent the strategy where
he picks A at the left hand decision node and B at the right hand decision
node of the game tree as drawn above:
Jim
A; A A; B B; A B; B
A
1, 1
1, 1
3, 2
3, 2
Ivan
B 2, 3
0, 0
2, 3
0, 0
There are three pure strategy Nash equilibria: (B, A; A), (A, B; A), (A, B; B).
Two outcomes are possible: we either end up with Ivan remixing song A
and Jim remixing song B, or Ivan remixing song B and Jim remixing song
A.
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c) (5 points) Find the unique subgame perfect Nash equilibrium. Explain why this is different than your answer to b).
By backward induction, the unique SPNE in this game is (A, B; A). The
other two Nash equilibria in b) feature Jim playing an action at one of his
decision nodes that is not sequentially rational—it is not rational for him to
play A if Ivan plays A, and it is not rational for him to play B if Ivan plays
B. Anticipating this, Ivan’s best strategy is to play A.
(The standard way to think about SPNE versus NE, by the way, is using
the example of non-credible threats. That is: if you threaten to do something that’s not in your best interests, maybe to try to convince me to do
something, what if I call your bluff ? If something doesn’t survive that kind
of bluff-calling, it isn’t an SPNE. So here you could interpret one of the
non-SPNE Nash equilibria as e.g. ‘Jim is saying he’ll play A if you play A to
try to force Ivan to play B, but if Ivan actually called him on it and played
A, he’d cave and play B because it’s better for him’.)
8. Sale is made in stage 1 at a price p1 = (1 − δ)u.
9. Consider a 2 player game that infinitely repeats the following stage game:
Player 1 A
B
Player 2
A
B
2, 2 1, 5
5, 1 0, 0
Each player discounts the future at the rate δ according to the discounted utility model.
a) Player 1’s payoff (i) if they play B forever: in this case, player 1 would get a payoff of
5 today, and then 0 forever after. This is because today we end up in the cell (B,A),
but then player 2’s punishment strategy is triggered and player 2 plays B forever after,
meaning that we end up in the cell (B,B) in each subsequent period.
So, player 1’s payoff under the discounted utility model would be 5δ 0 + 0δ 1 + 0δ 2 + 0δ 3 +
... = 5.
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Player 1’s payoff if (ii) they use the grim punishment strategy: in this case, player 1
would get a payoff of 2 forever. This is because we will be in the cell (A,A) today
and neither player’s punishment strategy is ever triggered, meaning we stay in that cell
forever.
So, player 1’s payoff under the discounted utility model would be 2δ 0 + 2δ 1 + 2δ 2 + 2δ 3 +
2
. (We will derive this mathematical result in class; you may invoke without
... = 1−δ
proof that an infinite stream of X discounted at a rate δ between 0 and 1 is equal in its
X
.)
sum to 1−δ
b) If player i was using the grim strategy, then player 1 prefers also to use the grim strategy
2
> 5 ⇒ δ > 35 . Since the game is what we call
than to trigger the punishment if 1−δ
‘stationary’—that is, identical in its characteristics in each period—we know that this
logic will remain true for each period in the future so long as the punishment strategy
has never been triggered.
There is therefore a subgame perfect Nash equilibrium in which each player uses the
grim strategy if δ > 35 .
The intuition: if the players are sufficiently patient (that is, they care enough about
the future) then they will prefer to receive a lower payoff indefinitely rather than grab
a bigger payoff once and get nothing later. If they had been less patient, then grabbing
the bigger payoff once would have been preferable to a smaller stream indefinitely. (Try
thinking through the cases of δ = 0 and δ = 1 to get the intuition down!)
This kind of game has been used by economists to model things like temptation, reputation, and cooperative behavior in general. There is a formal result called the Folk
Theorem—far beyond the scope of our course—that more fully explores not just the existence of a ‘cooperative’ equilibrium enforced by the possibility of the grim punishment,
but the whole range of equilibria that are possible in infinitely repeated games!
10. First we can observe that the decision between grim and SPNE is identical from the point
of view of every period, since the game goes infinitely into the future.
For player 1 we compare two payoffs. First is the payoff to playing the grim strategy. If
both players play grim, then player 1 will receive 3 indefinitely. The discounted value of
3
(via the result that we will derive in
this stream is therefore 3δ10 + 3δ11 + 3δ12 + ... = 1−δ
1
class, and that which you may invoke without proof, for a geometric series of this type with
0 < δ < 1). Second is the payoff to playing SPNE. This gives 4 today and then 2 indefinitely
from tomorrow, since player 2’s punishment is triggered. The discounted value of this stream
2δ1
2δ1
3
is 4 + 1−δ
. The grim strategy is thus preferred if 1−δ
> 4 + 1−δ
; that is, δ1 > 12 .
1
1
1
A similar calculation for player 2 gives that she prefers grim if
δ2 > 52 .
13
5
1−δ2
> 7+
2δ2
;
1−δ2
that is, if
Oligopoly theory
1. a) Firm 1: π1 = p(y)y1 − c(y1 ) = (10 − y1 − y¯2 )y1 − 2y1 . Firm 2: π2 = (10 − y2 − y¯1 )y2 − 2y2 .
b) Firm 1 first order condition: π 0 (y1 ) = 10 − 2y1 − y¯2 − 2 = 0 ⇒ y1∗ = 4 − 21 y¯2 . Firm 2’s
reaction function, by the same method, is y2∗ = 4 − 21 y¯1 .
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c) Solve the two reaction functions together, for example by substituting one into the other.
y1 = 4 − 12 (4 − 21 y¯1 ), which we solve for y1 = 83 and y2 = 38 .
d) The single firm has a profit function π = (10 − y)y − 2y, which we can maximize to get
y = 4. As expected the Cournot duopolists produce a combined output that is bigger
than the monopoly output (and so price is lower under the duopoly).
e) Using the reaction function in b), if one firm produced 2 (half the monopolistic output)
then the best response by the other would be yi∗ = 4 − 12 2 = 3, bigger than half the
monopolistic output. The agreement is not a mutual best response (at least in this static
game! In the repeated game below we’ll see how collusion might be sustainable...)
2. a) Let’s derive this for firm 1; since the game is symmetric (the two firms face identical
decisions) the reaction function we get will also be true for firm 2. Firm 1’s profit is
π1 = R−C = py1 −c(y1 ) = (60−y1 −y2 )y1 −12y1 . Fix firm 2’s choice at y¯2 . The first order
1
= 60−2y1 −y¯2 −12 = 0 ⇒ 2y1 = 48−y¯2 ⇒ y1∗ = 24− 21 y¯2 .
condition for a maximizer is dπ
dy1
2
(Notice that since ddyπ21 = −2 < 0 the point that satisfies the first order condition will
1
indeed be a maximizer.) This is firm 1’s reaction function; firm 2’s, symmetrically, is
y2∗ = 24 − 21 y¯1 .
b) In a Nash equilibrium, each player’s choice is a best response to the other’s choice (a.k.a.
a mutual best response). So we need to find a pair y1∗ , y2∗ that satisfies both of the best
response functions. We can solve these two equations simultaneously by, for example,
substitution: y1∗ = 24 − 21 (24 − 21 y1∗ ) ⇒ 34 y1∗ = 12 ⇒ y1∗ = 16 and so, symmetrically,
y2∗ = 16. The Nash equilibrium occurs where both firms choose to produce 16 units.
The monopolist would have chosen a point to maximize π = (60 − y)y − 12y: dπ
=
dy
∗
60 − 2y − 12 = 0 ⇒ y = 24. The competitive output would have satisfied p = M C ⇒
60 − y = 12 ⇒ y = 48. The Cournot output of 32 units is therefore more than the
monopoly output but less than the competitive output.
c)
The intercepts correspond to the monopoly output and the competitive output. If
firm 2 produces 0 units, then the profit-maximizing choice for firm 1 is to produce the
monopoly quantity, so that point is on the blue best response function for firm 1. On the
other hand, if firm 2 produces the competitive quantity, then firm 1’s best response is
to produce nothing, since they will have marginal cost bigger than price for any output
they produce.
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3. a) π2 = py2 − c(y2 ) = (20 − y1 − y2 )y2 − 4y2
b)
dπ2
dy2
= 20 − y¯1 − 2y2 − 4 = 0 ⇒ y2∗ =
16−y¯1
2
c) π1 = py1 − c(y1 ) = (20 − y1 − y2 )y1 − 4y1 = (20 − y1 − ( 16−2 y¯1 ))y1 − 4y1 = (12 − 12 y1 )y1 − 4y1
d)
dπ1
dy1
= 12 − y1 − 4 = 0 ⇒ y1 = 8
e) y2 = 8 − 21 y1 = 4, and so p = 20 − 8 − 4 = 8
f ) π1 = (8 ∗ 8) − (4 ∗ 8) = 32 and π2 = (8 ∗ 4) − (4 ∗ 4) = 16; it is worth 16 to be the leader
instead of the follower.
4. a) π1 = (12 − y1 − y2 )y1 − 4y1 , π2 = (12 − y1 − y2 )y2 − 2y2
b)
c)
dπ1
dy1
dπ2
dy2
= 12 − 2y1 − y¯2 − 4 = 0 ⇒ y1∗ =
= 10 − y¯1 − 2y2 − 2 = 0 ⇒ y2∗ =
d) At mutual best response,
y2∗
=
10−y1∗
2
8−y¯2
2
10−y¯1
2
⇒
y2∗
=
10−
∗
8−y2
2
2
⇒ y1∗ = 2, y2∗ = 4.
e) p = 12 − y1∗ − y2∗ = 6
f ) Martin. π2 = 16, π1 = 4.
5. a) π2 = (12 − y1 − y2 )y2 − 2y2 and so
dπ
dy2
= 0 ⇒ 12 − y1 − 2y2 − 2 = 0 ⇒ y2 =
10−y1
2
= 5 − 12 y1 .
b) π1 = (12 − y1 − y2 )y1 − 2y1 , and substituting in the reaction function in a), π1 = (12 −
dπ
y1 − (5 − 12 y1 ))y1 − 2y1 = (7 − 12 y1 )y1 − 2y1 . dy
= 0 ⇒ 7 − y1 − 2 = 0 ⇒ y1∗ = 5 and
1
∗
therefore y2 = 2.5. Price is 12 − 5 − 2.5 = 4.5.
dπ
= 0, which gives y1∗ = 52 − 12 y¯2 .
dy1
dπ
π2 = (6 − y¯1 − y2 )y2 − 2y2 . First order condition is dy
= 0, which gives y2∗ = 2 − 12 y¯1 .
1
Solving the reaction functions together, we get y1∗ = 2, y2∗ = 1. For later reference I will
also note that π1∗ = 4, π2∗ = 1.
6. a) π1 = (6 − y1 − y¯2 )y1 − y1 . First order condition is
b)
c)
d) First let’s check the secret stealing. In that case, both firms would have a reaction
function like the one in a). y1∗ = 52 − 12 y¯2 and y2∗ = 52 − 12 y¯1 . Solving these together, we
get that y1∗ = y2∗ = 53 . Both firms make profit of 25
.
9
Second we have the sabotage case. In that case, both firms would have a reaction
function like the one in b). y1∗ = 2 − 21 y¯2 and y2∗ = 2 − 21 y¯1 . Solving these together, we
.
get that y1∗ = y2∗ = 34 . Both firms make profit of 16
9
So we see that the secret stealing proposal is better for firm 2. They would be willing
to pay up to 25
− 1 = 16
for this service.
9
9
. The
e) Firm 1 was making profit of 4 before, and if the plan happens, they make only 25
9
11
loss to them is 9 . They are not willing to pay enough to outbid firm 2: their potential
loss is not as much as firm 2’s potential gain.
7. a) If the competitor sets a price lower than 25 cents, Jim’s best response is to set any higher
price (earning zero rather than a negative payoff by matching or going lower). If the
competitor sets a price equal to 25 cents, Jim’s best response is to set any price equal
or higher (earning zero rather than a negative payoff by going lower). If the competitor
sets a price above 25 cents, Jim’s best response is to set a price just a tiny bit lower
(capturing all 200 customers at this price rather than half at the price equal to the
competitor’s or none with a higher price).
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b) A pair that represents a mutual best response is both to price at 25 cents.
c) Now if the competitor sets 25 cents, say Jim sets a price of $1. The competitor can serve
only 100 customers. There are 100 left over all willing to buy at any price up to $1 and
so Jim makes positive profit by serving them. This is better than zero profit by setting
25 cents.
d) If one firm prices at $1, the other can earn (1 − 0.25) ∗ 100 by also pricing at $1. A
higher price earns zero since no-one buys. A lower price earns strictly less, since the
capacity constraint means no more than 100 can be served. Intuitively, undercutting
cannot ‘steal’ customers from the competitor since the firm is at capacity.
8. a) Since this situation is symmetric (e.g. the firms are identical) we will just check firm 1’s
incentives, knowing that the same is true for firm 2 as well.
i. Say that firm 2 chooses p2 = 5. What is firm 1’s best response? If they had set a price
p1 = 5, they would have earned π = (50 ∗ 5) − (50 ∗ 1) = 200. If they’d set a price p1 > 5
they’d have earned π1 = 0 since their price would be greater than firm 2’s. If they’d set
a price of p1 = 4.99, they would have earned π = (100 ∗ 4.99) − (100 ∗ 1) = 399. This is
better than if they set p1 = 5; therefore, if the other firm had chosen $5, then it would
not be a best response to also set $5. The pair (5, 5) is not a mutual best response.
ii. Say that firm 2 chooses p2 = 1. What is firm 1’s best response? If they had set
a price p1 = 1, they would have earned π = (50 ∗ 1) − (50 ∗ 1) = 0. If they’d set a
price p1 > 1 they’d have earned π1 = 0 since their price would be greater than firm
2’s. If they’d set a price of p1 < 1, they would have earned a negative profit since they
would serve all 100 customers but at a price lower than the cost of producing each unit.
Therefore, they cannot do better than by setting p1 = 1 with any other price. The pair
(1, 1) is a mutual best response.
b) There is a unique mutual best response in this situation when firm 1 sets a price of $2.99
and firm 2 sets a price of $3. At this pair, each is best responding: the logic for firm
1 is the same as in a)i. where they do best by slightly undercutting p2 . Firm 2 earns
a profit of zero in this situation but given that firm 1 has set p1 = 2.99 they can’t do
better—if they’d matched or undercut firm 1, they’d have served all customers but at
a loss-making price. There are lots of other price pairs that are mutual best responses
in this case. For example: p1 = 2.04, p2 = 2.05. Can you show why?
9. Lots of people live equally spaced along a street. Two business owners are each deciding
where to locate a gas station on the street. The people who live on the street will go to
whatever gas station is closer. If the firms are located at the same point on the street they’ll
choose which one to go to at random. The business owners would each like to have as many
people come to their station as possible. Where could the two firms locate so that neither
one would rather have chosen a different location, given where the other one chose?
The proof of the Hotelling result is straight from the lecture notes. It is not a Nash equilibrium
for them to locate at different points, since then either could have gotten more people by
locating closer to where the other chose. It’s not Nash if they choose the same point that’s
not at the center, since then either could have gotten more people by locating closer to the
center. If both locate exactly at the center, they each get half the people and could not have
done better in another location: that is a Nash equilibrium since neither could have done
better given what the other did.
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10. a) Say that Pat locates at the midpoint. If Carlos also locates at the midpoint, everyone
between locations 41 and 34 will be willing to walk to the midpoint, and each will choose
at random whether to buy from Pat or Carlos. So: Carlos serves one quarter of the
block. However, he could have done better, given where Pat located. If Pat located at
the midpoint and Carlos located at (for example) 14 , then everyone between 0 and 38 is
closer to Carlos than to Pat. He will serve 83 of the block, more than if he’d located at
the midpoint.
Since a Nash equilibrium is a situation in which each player’s strategy is a best response
given what all other players are doing, it cannot be a Nash equilibrium for both to
locate at the midpoint because either one could have done better by choosing something
different, given what the other chose.
b) There is a Nash equilibrium in which one firm locates at 14 and the other locates at 34 .
Let’s fix the location of Pat at 34 and think about the options for Carlos (the same logic
will hold symmetrically for the Pat):
ˆ Carlos locates at 41 : Carlos gets 21 of the block, everyone between 0 and 12 .
ˆ Carlos locates closer to 0: Carlos gets less than 12 since some slightly below 12 will
now be further than 41 away from Carlos, so will not buy from him.
ˆ Carlos locates closer to 12 : Carlos gets less than 12 since some people close to 0 will
now be further than 41 away from Carlos, so will not buy from him. Some people
slightly above 21 will now be closer to Carlos than to Pat, but definitely fewer than
he will lose close to 0.
Carlos cannot do better than by locating at 14 given that Pat locates at 34 , and the same
is true symmetrically for Pat. Since this is a mutual best response pair, it is a Nash
equilibrium.
P
2
t
11. a) Π1 = ∞
t=0 δ ∗ 2 = 1−δ (this is a result from the sum of a geometric series since δ is
between 0 and 1).
P
t
b) Π1 = δ 0 4 + ∞
t=1 δ ∗ 0 = 4.
c)
2
1−δ
> 4 ⇒ δ > 12 . If the player doesn’t care enough about the future relative to the
present (that is, if δ is lower than this threshold) then they will prefer to grab the 4
today even though it costs them 2 in all future periods.
d) Bigger temptation would require more patience to resist. This is a seesaw: the more
temptation the player faces, the more patient they have to be to resist it, and, conversely,
the more patient the player is, the more temptation they can resist.
12. a) The unique Nash equilibrium is (Defect, Defect)—both firms play Defect. Note that it is
essential that you defined the Nash equilibrium in terms of actions, not payoffs: it is not
correct to say that the Nash equilibrium is (0,0), even if you had the correct reasoning.
A Nash equilibrium is a situation in which each player’s strategy is a best response given
all other players’ strategies (that is, a mutual best response, or, if you like, a situation
where no-one regrets their choice given what others did). Here, both playing Defect
is a Nash equilibrium because if one player plays Defect, the other gets -1 by playing
Collude but a larger payoff, 0, by playing Defect. There is no other Nash equilibrium,
since if one player plays Collude, the other player gets 3 if they’d played Collude, or
X > 3 if they’d played Defect. This means that there is no NE where anyone plays
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Collude. (Your logic may have been different to prove this; what we need is (i) why
is Defect, Defect a Nash equilibrium and (ii) why is no other pair of strategies a Nash
equilibrium.)
b) If firm 1 uses the same trigger strategy: the outcome in each period indefinitely will be
(Collude, Collude) for a payoff of 3 to firm 1, since neither player’s trigger strategy will
ever be activated. This means that firm 1’s total payoff is 3δ 0 + δ 1 + 3δ 2 + 3δ 3 + ...,
3
(as proved in class and on the
which since δ is between 0 and 1 is equivalent to 1−δ
lecture notes). This is bigger when δ is bigger: the relative weight that firm 1 puts on
payoffs further in the future is higher when they are more patient—higher δ—and so
the present value of the stream of indefinite 3s is bigger.
If firm 1 plays Defect in each period, they will get X in period 0 (as firm 2 is using
the trigger strategy so is playing Collude) but then 0 forever after (as firm 2’s trigger
strategy has been activated and firm 2 plays Defect forever). So the total payoff to firm
1 is Xδ 0 + 0δ 1 + 0δ 2 + 0δ 3 + ... = X. This does not depend on δ since the payoff X
arrives immediately and there is zero payoff in the future. Firm 1’s degree of patience
does not change the size of this total payoff since nothing is received in future periods.
c) Say that firm 1 has δ = 12 and firm 2 has δ = 34 . Is there an equilibrium in which both
firms use the trigger strategy? How and why does your answer depend on X?
We’re going to check whether using the trigger strategy is a best response for each firm
if the other firm is using the trigger strategy. First note that the best a firm can hope
for if they ever play Defect is X once then 0 forever, so we just need to check that the
payoff to using the trigger strategy beats that. So for firm 1, following b), we need that
3
> X ⇒ X < 6. For firm 2, analogously since the game is symmetric except for δ,
1−δ1
3
> X ⇒ X < 12.
we need 1−δ
2
We need X to be sufficiently small such that the least patient firm (here, firm 1) prefers
to get the indefinite stream of 3s rather than grabbing X one time. Intuitively, the
bigger is X, the more ‘temptation’ the firm faces, and so this is why firm 2’s threshold
X is higher: they are more patient, so value to the stream of 3s more, so can resist more
temptation!
13. Jim and TJ are Cournot duopolists. They each make an identical product. They will
simultaneously choose a quantity of output, yJ and yT respectively. The market for this
product has an inverse demand curve p = 15 − 21 y, where y = yJ + yT is the total quantity
of output produced by both Jim and TJ. Their production costs are identical: production
costs for each person i are c(yi ) = 3yi .
a) Say that both people are profit-motivated. Derive reaction functions for each person
(their optimal choice given the choice made by the other person). Thus find the quantity
produced by each person and the price in the Nash equilibrium of this game.
Person i’s profit function here will be πi = R − C = (15 − 12 yi − 12 yj )yi − 3yi .
For some choice yj made by the other person, their profit-maximizing choice
i
yi can be found with the first order condition for a maximizer: dπ
= 0 ⇒
dyi
1
1
∗
15 − yi − 2 yj − 3 = 0 ⇒ yi = 12 − 2 yj . (Notice, by the way, that the second
order condition is satisfied since the second derivative is −1, always less than
zero.) Symmetrically, for firm j, we’ll get yj∗ = 12 − 12 yi . This gives us the
best response of each player as a function of the other’s choice.
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So to find the Nash equilibrium, we will look for a pair yi∗ , yj∗ such that
each player is best responding given what the other did. To do so, let’s
solve the two reaction functions simultaneously; I’ll do it by substitution.
yi∗ = 12 − 12 (12 − 21 yi∗ ) ⇒ yi∗ = 12 − 6 + 41 yi∗ ⇒ yi∗ = 8. So in Nash equilibrium
each person chooses output 8: yJ = yT = 8. The price we can get from the
demand function: p = 15 − 12 (8 + 8) = 7.
b) Now say that Jim is not profit-motivated, but spite-motivated. He has one goal: to
ensure that TJ cannot make positive profit, no matter what yT he picks. At least how
much output yJ will Jim choose to produce? Explain the economic intuition behind
your answer.
If Jim produces at least 24 units of output, TJ could not possibly make
positive profit. There are a few different ways to see this (all equivalent).
First, notice that if Jim produces 24, then the price per unit of output can be
no greater than 3, even if TJ produces nothing: p = 15 − 12 yJ − 12 yT = 15 − 21 24 −
1
0 = 3. This is the (constant) marginal cost of production. Any quantity
2
that TJ produced would reduce the price even further, below marginal cost,
and TJ’s profit would certainly be negative: he’d be losing money on any
units he produced.
A second possible idea: 24 is the predicted amount produced in a hypothetical perfectly competitive market with this demand function and in which
all firms had this same cost function. This is the most that can be produced
in this market while still having non-negative profit for producers. This
is similar to the first explanation, and would be just fine if you made the
connection to why this stops TJ from making profit.
You may also have just checked the reaction function! yT = 12 − 21 yJ , and
so when yJ = 0 TJ’s best response is to produce zero output (thus making
precisely zero profit). Since zero profit is always an option for TJ, if his best
response is to produce more than zero it must be because he earns profit
by doing so. If you used this explanation, as long as you explained a little
beyond just ‘the reaction function said so’ I think this could be just fine as
well!
14
Markets with asymmetric information
1. a) The highest price any buyer is willing to pay for a car of a known quality exceeds the
seller’s lowest acceptable price for that quality.
b) Any seller with q < 12 is willing to sell. The expected quality in this pool is q̄ = 41 (we
know this because quality is assumed in this question to be uniformly distributed: the
mean of a uniform distribution from 0 to 12 is 14 ).
c) Highest WTP given expected quality q̄ =
1
4
is 54 q̄ =
d) At p = 21 , buyers’ willingness to pay is less than 12 .
e) q̄ = 21 p.
f ) WTP= 54 q̄ =
5
4
∗ 12 p = 85 p
g) Buyers’ WTP at any price p is less than p.
90
5
4
∗
1
4
=
5
.
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Course pack suggested answers
2. a) It is incentive compatible. In this scenario buyers expect a guarantee from good products
and not bad. So if the good seller does not offer a guarantee then buyers are willing
to pay at most $60. If they do offer a guarantee then buyers are willing to pay $100
but 5% of the time they have to pay the guarantee when the product fails. But that
$95 expected profit is bigger than the $60 from not offering a guarantee, so offering the
guarantee is indeed incentive compatible.
b) It is incentive compatible. The bad product earns $60 without a guarantee. With a
guarantee it sells for $100 but expected guarantee costs are $50 for the bad seller since
the product fails half the time. Then expected profit is E(π) = 50, less than the $60
earned by not offering a guarantee.
c) When buyers expect a guarantee only on good products and not on bad then neither
type of seller has an incentive to pretend to be something they’re not. The guarantee
therefore signals quality: there is a situation in which only the good sellers choose to
offer a guarantee and the bad sellers do not.
3. a) $1000 with a warranty, $0 without.
b) E(π) = 1000 − 100 = 900
c) E(π) = 0
d) Yes. If the good quality seller does not offer a warranty, it will not sell the unit and earns
0, while it earns 900 by offering the warranty.
e) No. The bad quality seller earns 0 by not offering a warranty, but by offering a warranty
sells at 1000 and pays 500 in expected repairs: E(π) = 500 by offering a warranty.
f ) Bad quality sellers have incentive to imitate the good quality seller and offer a warranty,
so a warranty cannot reveal to the buyer that the computer is of good quality.
4. a) This would be determined by what payment the candidates, and in particular the high
quality candidates, are willing to accept to do this job. The employer will be willing
to pay up to 5000p + 1000(1 − p) = 1000 + 4000p for a candidate that is good with
p probability. If this is an acceptable payment for the low quality candidate but not
for the high quality candidate, then only low quality candidates will accept, leaving
an adverse selection of candidates at this wage. (In turn, we can see that for a given
minimum dollar payment the good candidate will accept, the probability p plays a role,
since higher p will mean a higher willingness to pay by the employer, all else equal; you
may also have mentioned e.g. the candidate’s outside option mattering, as if the good
quality candidate has a better outside option they may be less willing to accept a given
payment.)
b) If the test is a credible signal in this case, it must be that only one type of candidate is
willing to take it. The employer can therefore infer a candidate’s quality from whether
or not they took the test. Since the employer is willing to pay more to high quality
candidates, it must be them that are taking the test (no-one would take the test in order
to get paid less, unless they were actively trying to avoid money). So: in this scenario,
test-takers would be inferred to be high quality and offered $5,000; people who did not
take the test would be inferred to be low quality and offered $1,000.
For this to be sustainable, we’d need that each type, high and low, prefers this scenario
over the alternative. The high quality type is taking the test and getting $5,000 rather
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than not taking the test and getting $1,000. The test must be less onerous than $4,000
for them—they would be willing to sacrifice less than $4,000 to avoid the test. The low
quality type is not taking the test, getting $1,000, rather than taking it and getting
$5,000. They must be willing to sacrifice more than $4,000 to avoid the test. If the cost
of the test to the high type is less than $4,000 and to the low type more than $4,000
then this can indeed be a sustainable equilibrium and the test can be a credible signal
of quality.
c) An incentive compatibility constraint in the context of a model with asymmetric information checks what choice a decision maker will prefer to make. Here, the incentive
compatibility constraint for a high type candidate says: does the candidate prefer taking the test and getting a higher wage, or not taking the test and getting a lower
wage? The incentive compatibility constraint for a low type candidate says: does the
candidate prefer not taking the test and getting a lower wage, or taking the test and
getting a higher wage? If both incentive compatibility constraints are satisfied, then
each type prefers to take a different action—neither prefers to imitate the other—and so
they can be distinguished by the less-informed party despite the presence of asymmetric
information.
5. a) u(high) ≥ 0 ⇒ E(w|high) − 50, 000 ≥ 0 ⇒ E(w|high) ≥ 50, 000
b) u(high) ≥ u(low) ⇒ E(w|high) − 50, 000 ≥ E(w|low) − 0 ⇒ E(w|high) − E(w|low) ≥
50, 000
c) u(high) = 100, 000 − 50, 000 = 50, 000 and u(low) = 0 − 0, and so u(high) > ū = 0 and
u(high) > u(low).
d) u(high) = 100, 000−50, 000 = 50, 000, which is less than u(low) = 100, 000−0 = 100, 000.
e) The employee puts in low effort, and so E(π) = ( 43 ∗ 0 + 14 ∗ 1m) − 100, 000 = 150, 000.
f ) u(high) = E(w|high) − 50, 000 = ( 43 ∗ 200, 000 + 14 ∗ 0) − 50, 000 = 100, 000
u(low) = E(w|low) − 0 = ( 14 ∗ 200, 000 + 43 ∗ 0) − 0 = 50, 000
High effort is incentive compatible since u(high) > u(low).
g) Yes. The employee earns u = 100, 000 by accepting the contract (and optimally choosing
high effort), which exceeds ū = 0.
h) E(π) = E(R) − E(w) = ( 43 ∗ 1m + 14 ∗ 0) − ( 34 ∗ 200, 000 + 41 ∗ 0) = 600, 000
6. a) UT (high) ≥ ū, and so we need E(w|high) − 4 ≥ 10.
b) UT (high) ≥ UT (low), and so we need E(w|high) − E(w|low) ≥ 4.
c) He will accept the contract and put in low effort. The participation constraint is satisfied
but the incentive compatibility constraint is not. Jim’s utility is -15: he pays the wage
of 15 always, and never gets any happiness.
d) Participation: need E(w|high) = 21 x + 21 0 ≥ 14, or x ≥ 28. Incentive: need E(w|high) −
E(w|low) = 21 x + 12 0 − 0 ≥ 4, or x ≥ 8. The payment for victory must be at least $28.
e) Jim’s utility is expected happiness minus expected wage, [ 12 100 + 12 0] − 12 28 + 12 0 = 36.
This is more than in c).
f ) If Toru loves to put in high effort (which he does. He never slacks off on the field) then
no victory bonus is needed. Jim just has to compensate Toru for his time. One way we
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could implement this in the model is to make the cost of high effort a negative number—
that is, a benefit from high effort. Or we could introduce a new positive variable to
capture his joy at playing hard.
If Toru loved high effort enough, Jim might not need to pay him at all—if his love was
bigger than the reservation wage, both the participation and incentive compatibility
constraints are satisfied with zero wage. Or maybe Jim still needs to give a positive
wage. But either way it doesn’t have to be a bigger wage in the case of victory.
7. a) The participation constraint: Ivan has to offer Jim a contract that Jim prefers over
whatever he would get by rejecting the contract (his outside option, reservation utility,
opportunity cost, etc). If not, Jim will not sign the contract. The incentive compatibility
constraint: to get Jim to work hard, Ivan has to offer a contract such that if Jim accepts
the contract he prefers to work hard rather than to slack off.
b) The lazier Jim is, the bigger wage premium Ivan will have to offer to Jim in the case
of thriving veggies to satisfy the incentive compatibility and participation constraints.
The reason is that when Jim is lazier slacking off is more appealing relative to working
hard, so Jim will require a bigger wage premium in order to prefer it. And the outside
option is more appealing relative to accepting a contract and working hard, so Jim will
require a bigger wage to want to sign the contract at all.
The more hard work matters relative to luck, the smaller wage premium Ivan will have
to offer to Jim to satisfy the incentive compatibility constraint. The more difference
Jim’s effort makes, then the more Jim boosts his chances at the thriving-veggies wage
instead of the dead-veggies wage by working hard. To reach the same expected wage
in the case of hard work now requires a smaller wage premium to success because the
probability of success is higher.
15
Answers to example midterms from the course pack
15.1
Answers to example midterm 1
1. There are two goods in the world, good 1 (x1 ), which has a price of $25 per unit, and good
2 (x2 ), which has a price of $10 per unit. Jim has $100 to spend, and his well-behaved
preferences can be represented by the utility function u = x31 x2 .
a) How many units of good 2 must Jim give up to get an extra unit of good 1? If we were
to sketch this problem, what would that correspond to on the diagram?
The opportunity cost of a unit of good 1 is 2.5 units of good 2. Since good
1 costs $25, Jim would need to forgo 2.5 units of good 2 at $10 each to be
able to get that extra unit of good 1. On a sketch, this corresponds to the
slope of the budget line, which in general is given by the price ratio pp12 .
b) Find Jim’s optimal choice of consumption bundle.
We can use the tangency method here since Jim has Cobb-Douglas preferences. First, let’s get his marginal rate of substitution: M U1 = 3x21 x2 and
3x2 x
M U1
2
M U2 = x31 and so M RS = M
= x13 2 = 3x
. Second, let’s use the tangency
U2
x1
1
p1
3x2
25
relationship M RS = p2 : x1 = 10 ⇒ 30x2 = 25x1 . Finally, let’s use the budget
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line equation: 25x1 + 10x2 = 100. Substituting what we know from the tangency condition, we get 30x2 + 10x2 = 100 ⇒ x∗2 = 2.5. Since 30x2 = 25x1 , we
know that at the optimum 30 × 2.5 = 25x1 ⇒ x∗1 = 3. Jim’s optimal choice of
consumption bundle is (3, 2.5).
2. Consider the following two-good world with a non-standard budget set. A person has m = 50
to spend, the price of good 1 (x1 ) is $0 per unit, and the price of good 2 (x2 ) is $10 per unit.
a) Sketch (a portion of) this person’s budget set, and explain what makes it unusual. In
terms of good 2, what is the opportunity cost of a unit of good 1?
This budget set is unbounded—it’s goes on forever in the horizontal direction! The person can afford up to 5 units of good 2, but they can afford any
number of good 1 to go along with that because good 1 costs nothing. The
slope of the budget line is zero.
b) Suggest a utility function such that the bundle (0, 5) would be the unique optimal choice
for this person from their budget set. On your diagram from a), add a couple of indifference curves for your chosen utility function, labeled with their utility level. Briefly
explain your answer.
The utility function u = x2 − x1 would work here (as would any function
where x1 entered negatively and x2 positively). This person prefers bundles
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with less x1 rather than more x1 and bundles with more x2 rather than less
x2 . The bundle that minimizes x1 and maximizes x2 within the available
bundles is (0, 5). Their indifference curves are straight, upward-sloping lines
that increase in utility level as we go up and to the left in the diagram—this
means that the highest indifference curve available within the budget is the
one that passes through (0, 5). The optimal choice is unique because no other
available bundle is on this same indifference curve.
3. Consider a labor supply model.
√ The decision maker has utility over leisure L and consumption
goods c given by u = 2L + c. They have 10 available hours and no non-labor income. The
wage rate is w = 20 and the price per unit of consumption is p = 1.
a) Use the tangency method to find the consumer’s optimal choice of L and c. Based on
your calculations, if this person had more available hours, would they have worked more,
less, or the same number of hours?
√
UL
= √21 = 4 c. At the point of tangency, we know
The consumer’s MRS is M
M Uc
2
c
two things: the slope of their indifference curve is equal to the slope of the
budget line (the price ratio)
√ and the ∗point lies somewhere on the budget
w
line. So: M RS = p ⇒ 4 c = 20 ⇒ c = 25. The budget line equation is
wL + pc = wT and so at the optimal choice (20 × L) + (1 × 25) = (20 × 10) ⇒
20L = 175 ⇒ L∗ = 8.75.
If they’d had more available hours this person would have worked the same
amount. They work 1.25 hours, which is enough to afford the 25 units of
consumption we found from the tangency condition. Since this 25 units of
consumption they desire is independent of T (it depends only on the price
ratio), they will still work exactly 1.25 hours if they had more time.
b) Say that the situation is the same as before except that after 7 hours of work the wage
rate increases from w = 20 to w = 30. Sketch the budget set for this situation on the
usual axes, with axes, intercepts, and slopes all labeled.
4. The following diagram shows the decomposition of the effect of a change in price for a
consumer. The line marked O is the original budget line, N is the new budget line, and H
the hypothetical budget line constructed during the decomposition. The lines marked IC are
indifference curves.
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a) Which price has changed and in what direction? Which type of decomposition
is this and how do you know?
b) What is the size of the substitution effect, and what is the size of the income
effect in this case? Explain briefly what
each of those things is. For each good,
1 and 2: is the good normal, inferior,
or can we not say? Is good 1 ordinary,
Giffen, or can we not say? Explain.
a) The price of good 1 has increased. This is a Hicks decomposition, because
the hypothetical budget line shows the same price ratio as the new budget
line but leaves the consumer just able to achieve the same utility as they
obtained at their original optimal choice.
b) These effects are about the effect of the change in the price of good 1 on
the consumer’s demand for good 1. The substitution effect is 5 − 7 = −2,
which is the difference between the consumer’s original optimal choice of
good 1, 7 units, and their optimal choice at the new relative prices but
controlling for purchasing power, 5 units. The income effect is -1.5, the
difference between their hypothetical optimal choice, 5 units, and their true
final optimal choice, 3.5 units. The substitution effect isolates the effect of
the change in the relative price of the two goods, controlling for the fact
that purchasing power is lower after an increase in the price of good 1. The
income effect is the effect of the change in purchasing power, controlling for
the change in relative prices.
Goods 1 and 2 are both normal. We can see this by comparing the consumer’s optimal choice from the new budget line and the hypothetical budget line. The consumer chooses more of both goods from the hypothetical
budget line, which is a parallel shift out of the new budget line, just like
an increase in income. Good 1 is ordinary, because the consumer chooses
more of good 1 from the original budget line than from the new budget
line—higher p1 , lower demand for good 1. Whether good 1 is ordinary or
Giffen: there are two different answers we’ll consider correct here. One: just
looking directly at the diagram, we can’t say whether good 2 is ordinary or
Giffen, because that requires knowing how the consumer’s demand for good
2 changes when the price of good 2 changes, which we don’t observe here.
Two: it’s true that with monotonic preferences, a normal good is definitely
ordinary (this is the same logic that tells us that a Giffen good is ‘very inferior’. So if you said ‘ordinary’ with this particular justification, that’s good
too!
15.2
Answers to example midterm 2
1. Two decision makers, Jim and Saeromi, each currently have $50 in initial wealth. They face
a choice between two options: (A) gain $X for sure, or (B) lose $50 with probability 12 and
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√
gain $350 with probability 21 . Jim has a Bernoulli utility function u = x, where x is ‘final
wealth’. We don’t know Saeromi’s Bernoulli function, but we do know that her coefficient of
absolute risk aversion is always less than zero. Both people’s preferences can be represented
with the expected utility form.
a) For what values of X will Jim choose option A? How does this compare to the expected
cash value of option B, and why?
The
=
√ utility of option B is given by EU (B) = 0.5u(0) + 0.5u(400)
√ expected
√
option A is EU (A) = 1 50 + X.
0.5 0 + 0.5 400 = 10. The expected utility of √
Jim prefers A over B if EU (A) > EU (B) ⇒ 50 + X > 10 ⇒ X > 50. The
expected cash value of B is (0.5 × −50) + (0.5 × 350) = 150; Jim is risk averse—
his Bernoulli function shows diminishing marginal utility of money—and so
he is willing to take a sure thing that is worth less than the expected cash
value of the risky gamble.
b) What does it mean that Saeromi’s coefficient of absolute risk aversion is less than zero?
For what values of X do we know that Saeromi will definitely choose option B over
option A? Explain how you know.
Saeromi is a risk lover. There are lots of ways to express what that means!
She will prefer a risky option with some given expected value over a sure
thing with that value for sure; her risk premium is negative; certainty equivalents for her are bigger than the cash value of risky gambles. You could
also talk about the Bernoulli function, which must have an increasing slope
as her wealth increases; the following diagram shows a Bernoulli function
with increasing marginal utility of money, as would be consistent with her
risk-loving preferences over lotteries:
Since the risky bet offers an expected gain of $150, if X ≤ 150 then Saeromi
will definitely prefer the risky gamble over the sure thing. Since she is riskloving, she will definitely prefer a risky gamble with expected cash value of
$150 over a certain amount of cash that’s less than $150.
2. Consider an exchange economy with two people, A and B, and two goods, x1 and x2 . A’s
preferences can be represented uA = x1,A x2,A and B’s uB = x41,B x2,B , where x1,A is the
amount of good 1 that A has, and so on. Their endowments are ωA = (3, 0) and ωB = (0, 5).
Normalize the price of good 1 to 1, and let the price of good 2 be p.
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a) Find each consumer’s demand for each good, as a function of p.
First, let’s get the ‘income’ for each consumer, which here is the value of
their endowment. mA = 3p1 + 0p2 = 3 and mB = 0p1 + 5p2 = 5p. Next, we
will find the demand functions for each person for each good. We could
use the tangency method, but since we know the functional form of demand
functions for case of Cobb-Douglas preferences, we know that x1,A = 12 mp1A = 23
3
and x2,A = 12 mp2A = 2p
for person A, and x1,B = 45 mp1B = 4p and x2,B = 15 mp2B = 1.
b) Thus find a competitive equilibrium in this economy.
At a competitive equilibrium, each person’s bundle must be utility maximizing for them, given the prices (i.e. it must satisfy their demand functions)
and the market for each good must clear. Since each consumer likes each
good, we know that the market clearing conditions will hold with equality
in equilibrium. So, let’s look for a value of p such that the market clearing
conditions are satisfied at each consumer’s utility maximizing choice. For
3
+ 1 = 5 ⇒ 3 = 8p ⇒ p = 38 . For
good 1, the market clearing condition is 2p
good 2, 32 + 4p = 3 ⇒ 4p = 23 ⇒ p = 38 . When p = 38 , the markets clear at the
consumer’s optimal choices; they will choose, respectively, xA = ( 32 , 4) and
xB = ( 23 , 1).
We therefore have a competitive equilibrium. p1 = 1, p2 = 38 , xA = ( 32 , 4), xB =
( 23 , 1). Note that all of these things are required: a competitive equilibrium
is a price for each good (or, if you like, a price ratio pp12 = 83 is also fine) and
an allocation for each consumer.
3. A profit-maximizing firm in a perfectly competitive industry has a cost function c(y) =
450 + 10y + 12 y 2 . The firm’s average cost reaches a minimum at y = 30. They currently make
positive profit.
a) Find expressions for the firm’s average cost and marginal cost. Sketch their average and
marginal cost curves, on the usual axes. (The right general shape is enough for average
cost, but make sure the curves are in the correct position relative to each other.)
AC =
c(y)
y
=
450
y
+ 10 + 12 y and M C =
dc(y)
dy
= 10 + y.
(Notice that the intersection of MC and AC is certainly at the point where
AC is at its lowest.)
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b) What do we know about the price in this industry at the moment? What will be the
price in this perfectly competitive industry in the long run? Explain your answers, with
reference to any assumptions of the perfectly competitive model that are important
here.
Price right now must be above 40. At the firm’s optimal choice (where
p = M C, since they’re profit maximizers in a competitive industry) it must
be that p > AC since we’re told that they make a positive profit.
In the long run, price will be exactly 40. Free entry and exit of producers
in response to positive or negative profit drives the price towards the level
where economic profit is precisely zero. That occurs where, at the producer’s
optimal choice, p = AC. (You may also have noted that we have used the
homogenous products/firms assumption, since that means that each firm
makes zero economic profit, not just the marginal entrant.)
4. Consider a standard Robinson Crusoe model with a single output good, coconuts. As a
consumer, Robinson has well-behaved preferences over leisure and coconuts. As a producer,
Robinson uses labor (the opposite of leisure) to produce coconuts, according to a production
function with diminishing returns to labor. Labor has price w and coconuts have price p.
a) Say that right now we are not in equilibrium because w is ‘too high’ relative to p. Sketch
a diagram that illustrates this situation. Why is it not an equilibrium?
This is not an equilibrium because the markets fail to clear at the producer’s
profit-maximizing point and the consumer’s utility maximizing point. A
competitive equilibrium allocation is one that is optimal for each producer
and consumer, given prices, and that satisfies market clearing. Here, at the
optimal choices, we have more demand for coconuts than supply, and less
demand for labor than supply.
b) If the conditions of the first theorem of welfare economics hold, what do we know about a
competitive equilibrium, if one exists? Specifically in this Robinson Crusoe model, how
are the consumer’s preferences and the production technology related at such a point?
If its conditions hold, the first welfare theorem says that a competitive equilibrium, if one exists, is Pareto efficient. In the Robinson Crusoe model such
a point would occur where the slope of the consumer’s indifference curve is
exactly equal the slope of the production function. That is: M RS = M RT .
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The rate at which Robinson is willing to give up time (i.e. go to work) in
exchange for coconuts is exactly the same, at such a point, as the rate at
which the economy can turn time into coconuts. At such a point, there is
no available alternative combination of leisure and coconuts that is better
for Robinson. (Notice that this is a little weird in a Robinson Crusoe model
since there’s only one person!)
15.3
Answers to example midterm 3
1. The diagram shows marginal cost, average cost, demand curve, and marginal revenue for a
profit-motivated monopolist which must set a single, constant price for its output.
a) At its optimal choice of quantity and
price, would this monopolist make positive, negative, or zero profit? At
the socially efficient quantity and price,
would they make positive, negative, or
zero profit? In both cases, explain how
you know from the diagram.
b) Say that a regulator gets to set the price
and wants to maximize quantity produced without forcing the monopolist
to make negative profit. What point on
the diagram tells us the price that the
regulator would choose? At that point,
is producer surplus positive, negative,
or zero? Explain.
a) (For the answer key here I’ve added stuff to the diagram that was on the question paper to illustrate the answers! We didn’t specifically need a diagram in
your answers.) The monopolist’s optimal choice is found where M R = M C,
since this is the first order condition of their profit-maximization problem
(up to that point, increasing output increases profit; after that point, it decreases profit). From the diagram, we can see that at the optimal quantity
Q∗ , price exceeds average cost and so the monopolist makes positive profit.
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The socially efficient quantity is where p = M C, since that is the quantity
that would maximize the sum of producer and consumer surplus. At that
point, Qe , we see that price is less than average cost and so the monopolist
makes negative profit.
b) The intersection of the demand curve with the average cost curve shows us
this point: it has quantity marked Q0 in the following diagram. That’s
because at this price-quantity pair, the price is exactly the same as average
cost and so the monopolist makes zero profit. Despite the fact that profit
is zero, we can see that producer surplus is positive (the area between price
and marginal cost of production). Producer surplus is independent of fixed
costs, which are of course part of average cost but do not contribute to the
marginal productivity of each unit of production which is what PS captures.
(By the way it’s just a total coincidence based on how I drew the diagram
that the quantity Q0 happens to be right where MR hits the quantity axis!)
2. La Noisette makes delicious desserts (d) according to a cost function c(d) = 10 + 41 d2 . Jim’s
1 2
t − d, which is
Tea, next door, produces tea (t) according to a cost function c(t, d) = 10 + 10
decreasing in d since the smell of delicious desserts makes Jim more productive. The price of
desserts is $5. The price of tea is $2. Both businesses are profit-motivated. No other party
is impacted by the amount of desserts, other than La Noisette and Jim’s Tea.
a) Find La Noisette’s optimal number of desserts and the socially efficient number of desserts.
La Noisette’s profit function: πN = R − C = 5d − (10 + 14 d2 . At the first order
condition for a maximum, dπ
= 5 − 12 d = 0 ⇒ d∗ = 10. (Note that since the
dd
second derivative is everywhere negative, this critical point is certainly a
maximizer of the profit function.)
The socially efficient amount can be found by maximizing the joint payoff
of both firms (you might have alternatively looked to balance M SB and
M SC; that’s equivalent and also fine if you did it correctly). So: πN + πJ =
1 2
5d − (10 + 41 d2 ) + 2t − (10 + 10
t − d), and at the first order condition d(πNdd+πJ ) =
5 − 21 d + 1 = 0 ⇒ de = 12.
b) Find a mutually beneficial Coasian bargain in which La Noisette agrees to produce the
socially efficient quantity of desserts in exchange for a cash payment from Jim’s Tea.
In what range must the cash payment fall?
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First, let’s see how much worse off La Noisette is at the socially efficient
amount compared to their profit-maximizing amount. When d = 10, πN =
(5 × 10) − (10 + 41 (102 )) = 15. When d = 12, πN = (5 × 12) − (10 + 41 (122 )) = 14. La
Noisette makes $1 more profit at their optimal choice than at the socially
efficient amount.
What about for Jim’s Tea? We could go ahead and calculate Jim’s optimal
choice and figure out how much profit he makes, but it’s quicker and easier
just to notice that the only way d affects πJ is that each unit of d reduces his
total cost by 1. This means that at d = 12 Jim’s total cost is $2 lower than
at d = 10, and so his profit is $2 higher.
If Jim offers a payment between $1 and $2 to La Noisette in exchange for
producing d = 12 rather than d = 10, both parties are better off.
3. Two competing profit-motivated firms, 1 and 2, will simultaneously choose a quantity of
output (y1 for firm 1, y2 for firm 2). The inverse demand function in their industry is given
by p = 60 − y, where y = y1 + y2 . The cost of production for each firm i is c(yi ) = 12yi .
a) For each firm, derive a reaction function that gives their optimal choice of output as a
function of the other firm’s choice of output.
Let’s derive this for firm 1; since the game is symmetric (the two firms face
identical decisions) the reaction function we get will also be true for firm
2. Firm 1’s profit is π1 = R − C = py1 − c(y1 ) = (60 − y1 − y2 )y1 − 12y1 . Fix
1
=
firm 2’s choice at y¯2 . The first order condition for a maximizer is dπ
dy1
1
∗
60 − 2y1 − y¯2 − 12 = 0 ⇒ 2y1 = 48 − y¯2 ⇒ y1 = 24 − 2 y¯2 . (Notice that since
d 2 π1
= −2 < 0 the point that satisfies the first order condition will indeed be
dy12
a maximizer.) This is firm 1’s reaction function; firm 2’s, symmetrically, is
y2∗ = 24 − 12 y¯1 .
b) Thus find the Nash equilibrium of this game. Explain what that means. Show whether
this is bigger, smaller, or the same as (i) the output that would have been produced
if this industry was a monopoly, and (ii) the output that would have been produced if
this industry had been perfectly competitive. (In both cases, assuming that the cost
function is always c(yi ) = 12yi for any producer.)
In a Nash equilibrium, each player’s choice is a best response to the other’s
choice (a.k.a. a mutual best response). So we need to find a pair y1∗ , y2∗
that satisfies both of the best response functions. We can solve these two
equations simultaneously by, for example, substitution: y1∗ = 24− 21 (24− 12 y1∗ ) ⇒
3 ∗
y = 12 ⇒ y1∗ = 16 and so, symmetrically, y2∗ = 16. The Nash equilibrium
4 1
occurs where both firms choose to produce 16 units.
The monopolist would have chosen a point to maximize π = (60 − y)y − 12y:
dπ
= 60 − 2y − 12 = 0 ⇒ y ∗ = 24. The competitive output would have satisfied
dy
p = M C ⇒ 60 − y = 12 ⇒ y = 48. The Cournot output of 32 units is therefore
more than the monopoly output but less than the competitive output.
c) Sketch the two reaction functions you found in a) on a diagram with y1 on the horizontal
axis and y2 on the vertical axis. Mark the Nash equilibrium on your diagram. Label
the intercepts with their appropriate values. What do those intercept values correspond
to? Why?
102
Econ 100A, Fall 2022
Course pack suggested answers
The intercepts correspond to the monopoly output and the competitive output. If firm 2 produces 0 units, then the profit-maximizing choice for firm
1 is to produce the monopoly quantity, so that point is on the blue best
response function for firm 1. On the other hand, if firm 2 produces the competitive quantity, then firm 1’s best response is to produce nothing, since
they will have marginal cost bigger than price for any output they produce.
4. Jim and Rahul are going to a party after work and each must decide whether to pick up
some beer. However, Rahul likes to work very hard and so will be leaving later than Jim.
Therefore the two will play a sequential move game. First, Jim will decide whether to “buy
beer” or “not”, and then Rahul will observe Jim’s choice and decide whether to “buy beer”
or “not”.
If one of the two buys beer, each drinks a good amount of beer, which gives each a payoff of
5. If neither brings beer, each drinks no beer and gets 0. However, if both buy beer, they
cannot help but drink too much and so get a payoff of −1 (and a hangover). In addition,
buying beer is costly and so subtracts 3 from the payoff of whoever buys beer.
a) Sketch a game tree to represent the extensive form of this game.
103
Econ 100A, Fall 2022
Course pack suggested answers
b) Find the unique subgame perfect Nash equilibrium. Find a Nash equilibrium that is not
subgame perfect and explain why it isn’t.
In the unique subgame perfect Nash equilibrium, Jim plays “don’t buy”;
Rahul plays “don’t buy if Jim buys, buy if Jim doesn’t buy”.
There are two other Nash equilibria, either of which would have been a valid
answer here.
First possibility. Jim: not buy; Rahul: buy if Jim buys, buy if Jim doesn’t
buy
Second possibility: Jim: buy; Rahul: not buy if Jim buys, not buy if Jim
doesn’t buy
These are both Nash equilibria, because they pass the test: in each case,
the strategy played by each player is a best response given what the other
has done—fixing the other player’s strategy, there is nothing else that would
have yielded a higher payoff. However, neither are SPNE because each has
a component that does not survive backward induction. In SPNE it cannot
be that Rahul’s strategy includes “don’t buy if Jim doesn’t buy”. This is
not sequentially rational (a.k.a. a non-credible threat): once Jim has not
bought, Rahul does better by buying. It also cannot be that Rahul’s strategy
in SPNE includes “buy if Jim buys”. Again this is not sequentially rational.
16
Answers to math refresher problems
1. a)
b)
c)
d)
e)
f 0 (x) = b
f 0 (x) = cbxc−1
f 0 (x) = x1
f 0 (x) = 3x2 − 16x + 30
f (x) = −5x−2
2. First-order condition is f 0 (x) = 1 − 2x = 0, solved by x = 21 . Second-order condition is
f 00 (x) = −2 < 0; everywhere negative, confirming that x = 21 is a maximum.
104
Econ 100A, Fall 2022
Course pack suggested answers
3. a) x5
b)
x2
x1
4. a)
δf
δx1
= 10 − 2x1
b)
δf
δx2
= 20 − 12 x2 2
5. a)
δf
δx1
= 10ax1a−1 xb2
b)
δf
δx2
= 10bxa1 xb−1
2
−1
6. a) x∗1 = 4, x∗2 = 2
b) x∗1 = 20, x∗2 = 40
105