Econ 303: Intermediate Macroeconomic Theory I
V1 ®
Instructor: Axel Watanabe*, Ph.D.
Fall 2026
This problem set summarizes the lecture and is intended
to help you consolidate the material by working through it
on your own. It is not to be turned in. You will benefit most
from working on it soon after the lecture, rather than waiting
until just before the exam.
This file is the intellectual property of the author and Concordia University. Any unauthorized use or distribution outside the Concordia server is strictly prohibited.
is a registered trademark of Sesame Workshop.
Difficulty levels. Each question is labeled as follows:
Core material that you should be able to solve directly.
A moderate extension of the core material.
P
Production-Possibility Frontier
P.1
Definition of Economics Pick whatever you have
at hand right now or the service you just received and
discuss the economic trade-off associated with it from
1) consumer’s perspective and 2) producer’s perspective.
P.2
Definition of Economics Explain why the air
you’re breathing in right now does not make an economic subject.
Substantially more challenging. These are optional unless you are aiming for an A. If your goal is to pass,
prioritize
and
questions.
Most questions are easier to solve while the material is
still fresh. You are encouraged to work regularly and compare answers with your group.
Vital questions. Questions marked with
are essential.
Many other questions build on them, so you should master
these first.
P.3
P.4
Optional questions. Questions marked with
are by
prescription only. Do not attempt them without prior approval. If you wish to go beyond the core material, check in
with me first.
questions may be slightly beyond the scope of the
course, repetitions for additional practice, or deliberately straightforward. They may appear in the exam pool, but your priority should be nonquestions.
→
→
P.5
PPF In the example we used today, suppose that
Zedd and I work 40 hours each rather than 20 hours.
Redraw the PPF.
→
P.6
PPF In the example we used today, suppose that
both Zedd and I got a steroid shot and now we can
produce (probably very moody but nonetheless) lecture and concerts twice as fast. In particular, I can produce one lecture in an hour or one concert in an hour,
and Zedd can produce one lecture in an hour or one
concert in 30 minutes. Redraw the PPF.
P.7
PPF In the example we used today, what would
happen to the PPF if Zedd killed me?1
P.8
PPF In class, starting from the top left end of the
PPF, we let me switch from concert to lecture production first. Sketch a PPF when you switch Zedd first
instead. Is the resulting line a PPF? Why or why not?
P.9
PPF Consider the technology specified in table 1.
What are the outputs and factor of production in table 1?
Take
only after completing the above, and only if prescribed.
Some questions are deceptively simple. When this is intentional, they are marked with
to signal that extra care
is required.
Some questions include partial answers
or hints
.
These are for self-checking only and will not appear on the
exam.
Occasionally, you may see:
Le maquereau
PPF In the economy we had today, are the following
part of PPF? Why or why not?
• 100 lectures and 100 concerts
• 10 lectures and 10 concerts
• 0 lectures and 0 concerts
Recommended order of work:
→
PPF Define PPF.
foo bar
Supplementary remarks for interest only. Not examinable.
1 He doesn’t even know that I exist. I’m probably safe.
* Department of Economics, Concordia University.
1
Chapter P
Québec
British Columbia
1 salmon / 1 worker
1 orange / 20 workers
1 salmon / 5 workers
1 orange/ 5 workers
1 salmon / 5 workers
1 orange/ 1 worker
P.17
Table 1.
PPF In table 1, if each province has 100 workers,
what is the maximum number of oranges and salmon
that can be produced?
P.11
PPF Which province is the most cost-effective in the
production of oranges in table 1?
P.12
PPF In table 1, which province is the least costeffective in salmon fishing?
P.13
PPF If all the provinces in table 1 are specialized in
salmon fishing and if they were to start producing oranges, which province should be the first one to switch
production from salmon to orange? On what ground?
And how many salmon do they have to give up to produce the first orange?
Le maquereau
50
40
30
20
10
Hello Seattle
0
0
Delta is cutting Minneapolis-Tokyo non-stop flights.
Is this an economic subject too?
It is. Wherever you have scarcity, there’s economics. Delta (actually Northwest, before it merged
with Delta) operates a hub-and-spoke network to
serve Asian cities. A dozen of flights leave major
US cities (including Minneapolis) and land in Tokyo
around three in the afternoon. Then a bunch of
flights take off from Tokyo around five in the afternoon to major destinations in Asia.
Delta is in the process of switching its hub from
Tokyo to Seattle. Take an itinerary from Milwaukee
to Shanghai for example. You would have to connect in Minneapolis and Tokyo (there is no non-stop
flight from Milwaukee to Tokyo). With the hub in
Seattle, you’ll have only one layover in Seattle (there
are non-stop flights between Milwaukee to Seattle).
What’s the scarcity here? – The fleet size. If
they spare an airplane to Minneapolis-Tokyo flight,
that same aircraft cannot be used to fly SeattleShanghai. Delta (or any airliner) does not have an
infinite number of airplanes. Canceling MinneapolisTokyo will only leave Minnesotans grumpy (they will
now have to have two layovers rather than one).
In turn, repositioning aircraft to Seattle-Shanghai
will benefit not only Milwaukee but any midsized Midwestern cities in the same situation. There you go,
the trade-off. Good luck finding something that
falls outside economics realm.
P.14
PPF Consider an economy with two producers, Olaf
and Baymax, who produce two commodities, snowballs and microbots. Olaf takes two minutes to make
one snowball and four minutes to make one microbot,
whereas Baymax takes one minute to make one snowball and one minute to make one microbot. Their PPF
is represented in figure 1. Who is switching in the red
60
Microbots
P.10
20
40
Snowballs
60
80
Figure 1.
segment (0 to 40 snowballs) and why?
P.18
PPF What is this economy’s MRT in figure 1?
It
will take two distinct values depending on the ongoing
number of snowballs.
P.19
Intertemporal Choice Consider the economy where
farmers produce corn. Its PPF is sketched in figure 2.
If the economy chooses point A, how much corn do
70
B
60
Corn Next Year (tons)
Salmon
Orange
Manitoba
50
40
30
A
20
PPF Define opportunity cost.
10
P.15
P.16
PPF In the example we used in class today, what
is Zedd’s opportunity cost of a lecture? And what is
Zedd’s opprotunity cost of a concert?
0
0
PPF In the example we used in class today, while
both Zedd and I can prepare one lecture in the same
two hours from an accounting point of view, I am a
better instructor than him from an economic point of
view. How so?
5
10
15
20
Corn This Year (tons)
25
30
Figure 2.
they consume this year and how much do they leave
for next year?
2
Chapter C1
P.20
Intertemporal Choice How much can they consume next year in item P.19?
P.21
Intertemporal Choice What is the opportunity cost
Don’t just
of corn this year at point A in figure 2?
give me a number but tell me in terms of what.
P.22
Intertemporal Choice Between A and B in figure 2,
which point will result in the larger PPF in the years to
follow?
Le maquereau
It’s an annual thing.
Each year, as the fall approaches, everything starts to get pumpkin spiced:
Pumpkin-spiced latte, pumpkin-spiced yogurt, pumpkinspiced beer, ... at this rate, I wouldn’t be surprised
if they start selling pumpkin-spiced two-bedroom
apartments. Now, a two-bedroom apartment is an
apartment as the name suggests. On the contrary,
pumpkin spice has a word pumpkin in it, but it doesn’t
have any actual pumpkin in it.
GDP falls into the latter category. Canadian GDP
is about $2.19 billion, but it does not have any money
in it. In fact, Canadian currency in circulation is
about $937 trillion, just as the amount of pumpkin and amount of pumpkin spice are different from
each other.
GDP is not money just as an orange is not $2 (it’s
a citrus fruit with succulent flesh, not two pieces
of banknotes). Likewise, instead of $, we opted to
measure everything in terms of a number of baskets. Real wage is 4 baskets per hour. But this does
not mean that labor is baskets. We just quote it
in terms of baskets. The choice of numéraire is arbitrary, as we just use it as a unit of measure. We
can equally say that Canadian GDP is about 1.1 billion
oranges (but then we are more familiar with Canadian dollars than oranges, and thus, GDP is quoted
in dollars).
C1 Budget Constraint
C1.1
Budget Constraint Figure 3 sketches
’s budget
constraint. What is the opportunity cost of fish in terms
of cheese?
10
9
8
Cheese (oz)
7
6
5
4
3
C1.3
Budget Constraint Suppose that the (nominal)
lump-sum tax is $1,000 and the (nominal) price of the
composite good is $2 per basket. How much is the real
lump-sum tax T ?
C1.4
Budget Constraint We took the composite good
as a numéraire in class today. What if we take the
leisure as a numéraire instead? What is the real price of
the composite good and leisure?
See that the choice
of a numéraire is arbitrary.
How many baskets of C
trade for one unit of l ?
C1.5
Budget Constraint For cheesecakes and tea, we can
observe how the budget constraint changes when the
price of tea changes. For the work-leisure choice, the
real price of C is always one. How, then, can we analyze the change in the budget constraint when the
(nominal) price of the basket changes?
Recall that
we defined the real wage to be W /2 (h/ ) in class for
example. How do we reflect the change in the nominal
price of C ?
C1.6
Budget Constraint Sketch the budget constraint when
real non-wage disposable income π − T = 0.
C1.7
Budget Constraint Suppose π − T ∈ (−4h, 0).2
π−T
If l = h +
, how much C does
have under his
4
budget?
C1.8
Budget Constraint Sketch
’s budget constraint
in item C1.7. In this case, is (C, l) = (0, h) within his
budget?
2
1
0
0
1
2
3
4
5
Fish (oz)
Figure 3. This is a culinary offense by the way.
C1.2
Pumpkin-Spiced GDP
Budget Constraint Fill in the budget constraint in
figure 3 when the price of cheese doubles.
PA from the cockpit: Suppose π − T ≥ 0 in what follows
unless otherwise specified.
2 Keep in mind, though, that we usually assume π − T ≥ 0.
3
Chapter C3
C2.1
Utility Suppose
’s utility function for punches
in the face x P and cheesecake x C is given by3
u(x P , x C ) = −x P + x C .
(1)
With the number of hits on the horizontal axis and the
number of slices on the vertical axis, sketch his indifference curve at u(x P , x C ) = −1, 0 and 1.
81
C2.2
Utility Are his preferences monotone in item C2.1?
Why or why not?
C2.3
Utility In item C2.1, how much is his marginal
rate of substitution of a punch in the face in terms of
The situation is
slices, and what does it represent?
a little different from cheesecake and tea. Deep down,
can you tell what MRS really represents?
C2.4
Utility Figure 5 represents
’s indifference curves
for left and right shoes. Are his preferences convex?
64
12
10
left shoes
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
C2 Utility
Budget Constraint Suppose that
has h = 12
units of time. His real non-wage disposable income is
π − T = 4 baskets. Sketch his budget constraints when
4 = 1 and 4 = 2 each in figure 4.
98
C
C1.9
98
81
64
10
8
8
6
6
4
4
2
2
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0
l
2
4
6
8
10
12
right shoes
Figure 4.
Figure 5.
C1.10
Budget Constraint The two budget lines in item C1.9
share one bundle. What is it and why do they share
this particular bundle?
C2.5
Utility Suppose that
’s preferences are represented in figure 6. Are his preferences monotone? Explain.
C1.11
Budget Constraint In item C1.9, sketch his budget
constraint when π − T = 6 while 4 = 1.
C2.6
Utility In item C2.5, are his preferences convex?
Explain.
Budget Constraint Suppose that the real wage rate
C2.7
Utility Suppose that
’s preferences are represented by
u(C, l) = C l.
(2)
C1.12
is 4 L for N s ≤ 34 h and rises to 4H (> 4 L ) past 43 h. Sketch
the budget constraint. What is (are) the opportunity
cost(s) of l in this case?
C1.13
Budget Constraint Sketch budget constraint but instead of l , measure N s on the x -axis.
You’ll have to
get comfortable going back and forth between l and N s
when we derive labor supply.
C1.14
Budget Constraint What do the slopes of PPF and
budget line have in common?
Fill his utility levels in table 2.4
C2.8
MRS Fill his marginal rate of substitution for
(2) in table 3.
See for yourself how M RSl, C (C, l)
changes depending on the bundle.
See figure 7.
C2.9
MRS Explain why MRS at (C, l) = (5, 1) is higher
than the one at (1, 5) in item C2.8.
3 For exercise purposes only. Do not actually punch my sidekick.
4 The one for (C, l) = (4, 2) is already filled out for your reference.
4
Chapter C3
5
9
8
16
tea [cups]
7
composite goods C [baskets]
9
indifference curves
25
6
36
5
49
4
64
3
1
2
81
4
3
2
0.5
33
0.303.235
.2
0
1
0
0
0
1
2
3
4
5
6
7
8
9
1
0
4
1
MRSl, C (C, l)
2
1
5
4
3
10
1
2
3
4
5
leisure l [hr]
10
cheesecake [slices]
Figure 7.
Figure 6.
5
4
3
2
1
8
C
l 1
2
3
4
5
3
4
5
Table 2.
5
4
3
2
1
C
C
2
l 1
2
Table 3.
C3 Utility Maximization Problem
C3.1
UMP The curves in figure 4 on page 4 represent
’s indifference curves. Locate his optimal consumption bundles for each of the three budget constraints
you sketched last time.
C3.2
UMP Suppose that real non-wage disposable income doubled from 4 to 8 baskets in the example today.
Write down the utility maximization problem and update the optimal consumption bundle. Judging by your
answer, are C and l normal?
You can use figure 8
to check your answer if you like.
C3.3
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
16
2
12
8
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
l
Figure 8.
UMP Suppose, for some unknown reasons, left
and right shoes are sold separately, and a right shoe is
$2 and a left shoe is $1 each. Suppose
has $12 to
spend on his shoes. Fill in his budget line in figure 5
and locate his optimal consumption bundle.
Unfortunately,
5
we cannot use the tangency condition here because MRS
cannot be defined at the optimal bundle for this particular preference. That doesn’t stop
from finding his
optimal consumption bundle. In essence, what does
Chapter C3
UMP try to do?
C3.4
Le maquereau
UMP Suppose
’s utility function is given by
Geographers use a contour map (cf. figure 9(a) on
page 7) to represent 3-dimensional objects (terrain)
on a 2-dimensional surface (a map). If you go on
treks often, you probably have seen their works before. A bundle of weaving lines you see on figure 9(a) is called a contour. A contour is a set of
points that are at the same elevation. If two points
are on the same contour, then you can tell that
these two points have the same height above the
sea level. If this sounds familiar to you, right on.
A contour [indifference curve] is a set of all the
combinations of latitude [cheesecake] and longitude
[tea] that have the same elevation [©].
Thus, indifference curves are actually a 2D projection (contour map) of a 3D © (terrain). For
instance, figure 5 on page 4 is a 2D projection of
figure 10(b) on page 7. Why didn’t I introduce these
3D surfaces in class then? Actually, economists
barely use these full-fledged 3D graphs to represent
© for two reasons. For one, 3D objects (terrain
or ©) presented on a 2D surface (a piece of paper)
is harder to read than 2D objects (contour or indifference curve) presented on a 2D surface (... duh).
The other reason is that we only need MRS (the
slope of the indifference curve) to find the solution to UMP, not the height of the indifference
curve (note that tangency condition doesn’t mention anything about height). And if you just need
the slope,a you only need two axes (one for latitude/cheesecake and the other for longitude/tea).
The third axis (for elevation/©) is redundant for our
purposes. For these reasons, we usually use contour
maps rather than 3D maps when solving UMP.
In this light, we can give another spin on MRS.
is currently at an elevation of 100 m
Let’s say
u(C, l) = C + 4 log l
and 4 = 12 or 1, h = 10, and π − T = 3. Write down his
utility maximization problem and find his optimal con(C ∗ , l ∗ ) =
sumption bundle for each wage rate. Part of
(4, 8) when 4 = 12 .
C3.5
A Mountain of ©
UMP Explain in words why (C, l) = (6, 11) is
not a solution to utility maximization problem in C-4
in Graph Vault.
walks due east by u T (·) m (or have
(or 100 ©). If
u T (·) slices of cheesecake), he’ll climb by u T (·)uC (·) m.
wants to get back to an elevation of
Now if
100 m (or 100 ©) by moving due north (or by having
more tea) this time, he will walk −uC (·) m (i.e., uC (·) m
to the south). Then he will climb up by −uC (·)u T (·) m
climbed
(i.e., going down by uC (·)u T (·) m). In total,
up by u T (·)uC (·) m and then down by uC (·)u T (·) m, i.e., he
came back to 100 m. The ratio between the distance
he moved to the north and to the east, −uC (·)/u T (·)
has one more
is exactly what MRS is. That is, if
cheesecake, he will come back to the original elevation level if he gives up −uC (·)/u T (·) slices. And of
course MRS changes depending on where he stands
and the shape of the mountain (or his preferences).
Just for curious, geographically minded people out
there, I’ve left two more graphs for the combinations we learned in class in figure 10 on page 7.
Figure 10(a) is for perfect substitutes and at the
opposite extreme, we have figure 10(b) for perfect
complements.
If you find these 3D representation confusing, so
do I (and most economists). Sometimes, less (dimensions) is more (legible). Let’s just stick to 2D and
leave the actual mountains for climbing.
a The slope here means the slope of the contour, not the mountain.
6
Chapter C3
(:
60
54
48
42
36
30
24
18
12
6
0
30
24
18
12
5
4
6
3
2
Bottles
1
0
0
6−Packs
(a) The one I used today.
(:
(:
(a) From Google Maps.
100
90
80
70
60
50
40
30
20
10
0
10
9
8
7
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
Left Shoe
6
(b) This is what I drew figure 5 out of.
5
4
3
2
Tea (cups)
1
0
0
1
2
3
4
5
6
7
8
9
10
Cheesecake (slices)
(b)
Figure 9.
7
Figure 10.
0
1
2
3
4
5
6
7
8
9
10
Right Shoe
Chapter C3
Le maquereau
Can I Use a Lagrangian?
– But of course. What I explained in class is essentially the first-order conditions of Lagrangian
method (but I didn’t explicitly say so because some of
you haven’t learned it yet).
Let’s see how. First, in the
• ˜ interest of lighter
l
,
’s consumption
notations, let’s define x ··=
C
•
˜
h
, his endowment; z ··= x − ω, his
bundle; ω ··=
π−T
net demand, i.e., his consumption bundle net of his
endowment (He can self-supply h [hr] of time allotted and π − T of
• ˜ . The remainder needs to be
4
, the price vector.
purchased); p ··=
1
The two conditions I introduced were
tangency condition: M RS(x) = −4.
(3)
budget constraint: p · z = 0.
(4)
How are (3) and (4) the first-order conditions of
Lagrangian method? Let’s define L(x, λ) ··= u(x)−λp·z .
The first-order conditions are
• ˜
0
Dx L = ∇u − λp =
.
(5)
0
Dλ L = p · z = 0.
(6)
It is immediate that (4) and (6) are identical. They both
satisfies the budget constraint.
mean z ⊥ p when
Notice that p (in purple) and z (in blue) form a right
angle in figure 11 (top).
What about (3) and (5)? – They represent the same
condition as well. Let’s start with (5). It simply means
∇u ∥ p. Let 3 and q be a vector orthogonal to ∇u and
p respectively. Since 3 ⊥ ∇u and ∇u ∥ p, 3 ∥ q. The slope
of 3 is M RS(x) and the slope of q is −4. Then 3 ∥ q
implies (3). And vice versa, if (3) holds, ∇u ∥ p so that
(5) holds as well. Therefore, (3) and (5) represent the
same condition.
How do (5) and (6) translate to the graph that
we used in class? – Again, the same concept, just
expressed differently. To see this, imagine that
meets (6) (or (4)) but not (5) (or (3)). Now recall
that gradient ∇u (in red) denotes the direction in
which u(·) makes its steepest ascent.
can shift
3 in the direction of ∇u to find some other bundle
that meets (6) and gives him higher utility, meaning
his current selection is not a solution.
meets (5) but not
On the contrary, what if
(6)? I left you figure 11 (middle). See if you can tell
that this won’t fly either.
Now if
meets both (5) and (6), then he will no
longer find any affordable bundle in the direction of
∇u (figure 11 (bottom)). Recall that when 3 ⊥ ∇u, 3
marks the direction where u(·) makes neither ascent
nor descent: du ≈ ∇u · 3 = 0, (or 3 marks the slope
of the indifference curve, if you will). Thus, moving
along in the direction of 3 (or along the budget line
in this case) won’t help either. Therefore, this x is a
local extremum.
I used such terms as indifference curves (instead
of 3) and tangency (instead of ∇u = λp) in class to
explain the same concept above. However you name
them and whichever method you take, the idea is still
the same and you’ll reach the same solution in the
end. Thus, you are more than welcome to use a
Lagrangian in the exam.
Figure 11. Top: (6) is met but (5) is not. Middle: (5) is met but (6) is
not. Bottom: Both conditions are met.
π − T.
C3.7
Comparative Statics on Income Suppose that C
is normal but l is not necessarily so. Indicate the area
where
’s optimal bundle can be after tax breaks.
C3.8
Comparative Statics on Real Wage Explain in
words how income and substitution effects take place.
C3.9
UMP Suppose that
is endowed with h = 10
units of time and his preferences are represented by
u(C, l) = (2l + 4)C.
C3.6
Write down his utility maximization problem and find
his optimal consumption bundle for each case represented in table 4, and compute the maximum utility
level he achieves for the first two cases.
Comparative Statics on Income In class, we have
indicated the area where
’s optimal bundle can appear after an increase in π − T . Indicate the area where
’s optimal bundle can appear after a decrease in
8
Chapter F
π−T
4
8
-8
8
1
4
4
Table 4.
Comparative Statics on Real Wage Sketch
’s
three budget constraints in item C3.9 and mark his income and substitution effects when 4 goes from 1 to 4
in figure 12.
C3.12
Comparative Statics on Real Wage Sketch indifference curves where the substitution effect falls behind the income effect in magnitude.
We assume
the opposite in class, but this question should help you
see how substitution and income effects appear on the
graph.
C4
C
C3.11
Labor Supply
C4.1
Labor Supply With N s on the horizontal axis and
4 on the vertical axis, sketch
’s labor supply when
π − T = 8 in item C3.9.
We only have two points,
one for 4 = 1 and the other for 4 = 4. Locate them and
you can simply connect the dots.
C4.2
Labor Supply Sketch labor supply that corresponds
to item C3.12.
It should be downward sloping if
you sketched it right in item C3.12.
C4.3
Labor Supply Consider the utility maximization
problem we worked on in class last time. With N s
on the horizontal axis and 4 on the horizontal axis,
sketch the labor supply. Judging by the graph you
drew, does
’s substitution effect exceed his income
effect in magnitude?
It should be a hyperbola. For
this question, you’ll need to leave 4 as is rather than
setting it equal to 2 so that you can sketch the supply
curve rather than a point on the curve at 4 = 2.
392
Comparative Statics on Real Wage Compute
’s
income, substitution and overall effects in item C3.9
Substitution
when the wage rate changes from 1 to 4. Part of
effect is +10 on C and −5 on l .
Indifference Curves
200
C3.10
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
F
1
2
3
4
5
F.1
Production Function We sketched a production function of concerts in class. Sketch one for lectures.
F.2
Production Function In item F.1, suppose that
hired two workers in a wrong order, namely, starting production with the worker whose opportunity cost
of lecture is higher than the other. Plot the number of
lectures over the number of hours worked.
F.3
Marginal Product Sketch marginal product that
corresponds to item F.2 directly below it.
6
7
8
9
10 11 12
l
Firm
Figure 12.
9
F.4
Production Function Show that the line you plotted in item F.2 is not a production function.
For the
same reason item P.8 on page 1 is not a PPF.
F.5
Marginal Product From items F.3 and F.4, what
can you infer about production when
has increasing marginal product?
Again, he’s dumb for the
same reason item P.8 is dumb.
F.6
Production Function Write Zedd alone’s produc-
Chapter F
tion function of concert from the day #1.
Sensei.
F.7
Marginal Product What is his marginal product
of labor in item F.6?
F.8
Marginal Product Between Zedd and Sensei in item F.6,
who is more productive on the grounds of marginal
product?
F.9
R2S What is Zedd’s returns to scale in item F.6?
F.10
R2S Between Zedd and Sensei in item F.6, who is
more productive on the grounds of returns to scale?
F.11
Production Function Both marginal product and
returns to scale are a measurement of productivity. Explain, in words, the exact difference between them.
The
answer to item F.8 is different from the answer to item F.10.
We use both marginal product and returns to scale to
categorize different production technologies, but they
are distinct concepts. Be able to tell them apart and
know when to use which.
F.12
Marginal Product Suppose that
function is given by
d
z F (K, N ) = K
1/2
d 1/2
(N )
of labor of the economy made of Zedd, Avicii and Sensei. Does marginal product of labor decline with hours
You’ll add Avicii somewhere.
worked? If so, why?
Where would you place her? Before Zedd? After Sensei? Or between the two?
Ignore
M R N (K, N d ) and M CN .
graph?
(7)
Find his marginal product of labor when K = 4 and 9.
F.13
F.18
Marginal Condition Whereas marginal revenue can
change with N d , marginal cost does not in our model.
Why not?
F.19
Marginal Condition You should increase N d if
M R N (K, N d ) > M CN to seize the positive marginal profit.
Does this mean you should increase N d if revenue >
variable cost as well? Why or why not?
Do not mix
up a function with its slope.
F.20
Marginal Condition
’s profit structure is tabulated in table 5 (some cells are intentionally left blank).
Find his profit-maximizing level of N d ∗ .
’s production
.
Production Function Does production function
(7) satisfy the fourth assumption?
The fourth assumption implies that marginal product of labor (or
capital) increases with capital (or labor).
F.15
R2S Examine the returns to scale of the following
production functions:5
Nd
1
2
3
4
5
6
7
8
M R N (K, N d )
M CN
10
6
8
6
4
2
0
-2
-4
6
Table 5.
Marginal Product With N d on the horizontal axis
’s production funcand Y on the vertical axis, sketch
tion in item F.12 when K = 4 and 9. Then right beneath
it, sketch his marginal product of labor ( M PN ) for each
K.
F.14
F.21
Production Function Does the production function we used today satisfy the four assumptions?
F.22
Marginal Condition Discuss the similarities and
differences between tangency condition and marginal
condition.
F.23
PMP Suppose that
’s production function is
given by
3
1
zF K, N d ··= zK 4 N d 4 ,
with z = 2 and K = 16. Write his profit maximization
problem and find his profit maximizer N d ∗ (4).
(a) z F (K, N d ) = K + N d
p
(b) z F (K, N d ) = N d
F.24
(c) z F (K, N d ) = N d
(d) z F (K, N d ) = (N d )2
(e) z F (K, N d ) = K 2 + (N d )2
Labor Demand Compute N d ∗ (4) at 4 = 1 or 32 in
item F.23 and sketch his labor demand function with
N d ∗ on the horizontal axis and 4 on the vertical axis.
F.25
(f) z F (K, N d ) = K α (N d )β , with α > 0 and β > 0.
PMP Suppose that
’s production function is given
by
1
zF K, N d ··= z K + N d 2 ,
For the last one, you’ll have to make three cases
depending on what value α + β takes.
F.16
Marginal Condition Define marginal profit using
How do they appear on a
F.17
with z = 16 and K = 6.6 Write his profit maximization
problem and find his profit maximizer N d ∗ (4).
Production Function Suppose that Avicii joined
the production of concerts. He has 20 hours as well
but he can produce one concert in half an hour. Sketch
the concert production function and marginal product
F.26
PMP Compute N d ∗ (4) when 4 = 1 and 2 each in
item F.25.
6 This production function does not satisfy the first and fourth assumption by
the way (why?)
5 K doesn’t appear on some of them for pedagogical purposes. Imagine that
these are a production function of cheesecake and K is the number of space shuttles. For those who have never baked a cheesecake (why not?), you do not need a
space shuttle to bake one.
10
Chapter 5A
F.27
F.28
PMP We assumed that the consumer’s substitution
effect exceeds income effect in magnitude to have an
upward-sloping labor supply curve. What assumption
about the firm makes the labor demand curve downWhat condition did we use to derive
ward sloping?
N d (4)?
Labor Demand Consider the example from today.
Suppose that z = 3 instead of 2. Overlay the new labor
demand function with the one you have on your lecture
note. How does the total factor productivity change the
labor demand curve?
Le maquereau
Produce Bronze
5A
5A.1
Intertemporal Allocation Why isn’t T greater or
smaller than G if there is only one period in the economy?
5A.3
Competitive Equilibrium Suppose that the firms
are owned by the government rather than by the shareholders. In this case, π gets paid out to the government
rather than to the consumers. Redefine the competitive
equilibrium in this scenario.
5A.4
Competitive Equilibrium Does the income-expenditure
identity still hold in item 5A.3?
Know exactly what
constitutes the total income and expenditure.
5A.5
Competitive Equilibrium Walras’ law doesn’t
specify which markets have to be in equilibrium to
show that the remaining one is also in equilibrium. I
just picked the composite goods market as the remaining market in class. Suppose, instead, that the composite goods market is in equilibrium. Show that the
Be sure
labor market will be in equilibrium as well.
to show exactly which condition (other than N s = N d
of course) of the competitive equilibrium you used in
each step to establish that N s = N d .
5A.6
Competitive Equilibrium Suppose that there is excess supply in labor market, namely, N s > N d . In this
case, would the composite goods market be in equilibrium? If not, would there be excess supply or demand?
Note that 4 N s − N d > 0 in this case.
See that
the income-expenditure identity doesn’t hold unless
the economy is in equilibrium.
5A.7
Competitive Equilibrium In class, I employed condition (2), labor market clearance, to find the equilibrium. Use condition (1), goods market clearance instead and see if you can find the equilibrium without
condition (2).
See that Walras’ law just says one market is in equilibrium as long as the others are, but it
doesn’t specify which one, and hence the choice of the
said market is arbitrary.
It will be the same procedure up to the point where we employed condition (2):
N s = N d . Do not use this equality here but use condition (1) instead. After you derive the equilibrium, you
can check to see if that satisfies condition (2) (it does
thanks to Walras’ law but see it for yourself).
5A.8
Competitive Equilibrium Suppose that the production function is given by the following:

‹
1
zF K, N d = z K + N d ,
(8)
2
Well, That’s One Way to
You may have noticed in Econ 201 that production
functions are always defined over a positive domain.
Namely, be it raw materials or labor, inputs are always a positive amount. Can they be defined over a
negative domain? You can, but then the output has
to be negative in exchange. If a firm could produce
a positive output from a negative input, it means
they produce two outputs from nothing: a negative
input is in effect a positive output. Thus, scarcity is
gone and the firm is no longer a well-defined entity and falls outside economics’ purview. However,
so long as output is negative, scarcity maintains its
presence. How can you use an output to produce an
input though? Wouldn’t that mean that we take an
airplane (an output) apart to “produce" scrap metal
(an input)?
Germany used to import industrial bronze ovens
(seemingly an output) from the Soviet Union, to
melt it and harvest bronze (seemingly an input).
Check out NPR Planet Money episode . In this case,
an output is negative (Germany produces a negative
amount of ovens) as is an input (Germany takes in
a negative amount of bronze).
As can be seen, we deal with production functions
that appear in the first quadrant (the familiar one)
or in the third quadrant (as above). Soviet’s production function appears in the first quadrant (Poor
Soviet people. They had no idea they might as well
have simply sold bronze as is), and German production function appears in the third quadrant. We
assume that production does not take place in the
second quadrant, in which case, you would be able to
produce two outputs from nothing. No scarcities,
no economics.
As the bronze oven example shows, the labelling of
input and output is rather arbitrary. For this reason, in general equilibrium models (cf. Econ 302 and
401) we typically assume there are only outputs and
regard inputs as negative outputs. In our course,
however, we will retain inputs as they are not interchangeable with outputs in our economy. Unlike
cannot conceivably produce leisure out
bronze,
of consumption goods. Thus, our production function resides exclusively in the first quadrant.
5
5A.2
with z = 4 and K = 0. In addition G = 5 and h = 10
in this economy. Draw yourself three sets of grids. On
the first one, sketch the production function with N d
on the horizontal axis and Y on the vertical axis. On
the second one, sketch the production function with l
on the horizontal axis and Y on the vertical axis. On
the third one, sketch the production function with l on
the horizontal axis and C on the vertical axis.
From
the first to second, you’ll need to rewrite the production function by swapping N d with some function of
l . From the second to the third, you’ll need to swap Y
One-Period Model
Competitive Equilibrium
Intertemporal Allocation If there is only one period in the economy, points like A or B in figure 2 on
page 2 will never be selected. Why? Which point will
be selected then?
11
Chapter 5A
with some function of C .
I used a generic production function in class. The production function (8) is
simple enough that you can get your hands dirty and
sketch the exact function to observe the transition from
the N d − Y plane to the l − C plane by yourself.
5A.9
Competitive Equilibrium On the third one in item 5A.8,
shade in the set of (C, l) that are feasible in this economy.
5A.10
Competitive Equilibrium Compare the area you
shaded in in item 5A.9 to Zedd and Sensei’s PPF from
the first week. Discuss similarities and differences beAre they upward sloping or downtween them.
ward sloping? What can you say about their shape?
5A.11
Competitive Equilibrium The second graph I
sketched in class today is in fact Y s = z F (K, h − l). We
learned that the slope of this graph is −M PN K, N d (I
just flipped the graph about the vertical axis. “Rise over
run” simply becomes “fall the same amount over run”). Show algebraically that the slope is indeed −M PN K, N d .
Use the chain rule.
5A.12
MRT Explain in words why the marginal rate of
transformation of leisure coincides with the marginal
product of labor.
5A.13
Competitive Equilibrium Suppose that
duction function is given by
Le maquereau
For those who have taken physics in high school (if
you didn’t, just disregard this maquereau), you’ll find
that profit maximization is analogous to maximizing
the work applied to a point. To see the connection,
set up output vector 2 = (21 , 22 ). The negative output means an input. In this example, let’s say 21 ≤ 0
is labor input and 22 ≥ 0 is cookies. The price vector is p = (p1 , p2 ). In this example, p1 is a wage rate,
and p2 is the price of a cookie. The profit maximization is then max2 p · 2 subject to 22 ≤ f (21 ), where f (·)
is a production function.a In more familiar terms,
p · 2 = p1 21 + p2 22 , with p1 21 (≤ 0) being the total cost
and p2 22 the total revenue.
*
Now imagine that you are applying force 2 to a
*
particle moving in the direction of p as in figure 13.
*
*
*
*
is then p 2 cos θ , or the area size of the rectangle
that appears in figure 13.
Now recall that the law of cosines implies p · 2 =
*
*
*
*
p
p
2 cos θ .
Thus, profit p · 2 is the same as work
*
2 cos θ . You are effectively looking for 2 of the
*
form 22 ≤ f (21 ) that yields the largest work, given p .
*
To do so, you create a vector orthogonal to p (such
*
as a in the figure) and shift it so that it will barely
touch the production function. That will maximize
the work done, a.k.a. the profit generated. In fig-
’s pro-
*
ure 13, 2′ is not a profit-maximizer because we can
generate more work by tilting it clockwise. On the
*
5A.14
Competitive Equilibrium Write the PPF as a
function of l , i.e., write C = some function of l .
You
can start with item 5A.8 and then generalize.
5A.15
Competitive Equilibrium Suppose that
function is given by
other hand 2∗ is the maximizer because you cannot
make θ any smaller while keeping to the production
function given. And you can check by yourself to see
*
’s utility
l + 2.
*
*
that a being orthogonal to p at 2∗ is none other
than the marginal condition itself.
Is this analogy useful? – It is in the following two
senses: For one thing, your high school physics teachers (perhaps unknowingly) laid the groundwork for
Econ 303. Tell them I said thank you next time you
meet them. In addition, it makes a great tool to
visualize comparative statics. For instance, if both
p1 and p2 increased in proportion (i.e., extending the
*
length of p without changing its direction), we can
immediately see that that only increases the work /
profit (the box gets bigger) but it won’t change the
*
optimizer 2∗ itself. On the other hand, if the output
*
price goes up while the input price goes down, p will
(9)
Given 4 and π∗ found in item 5A.13, h = 16 and
G = 2, write his utility maximization problem, find his
optimal consumption bundle, and compute the maximized utility level.
*
7
5A.16
*
is an angle that 2 and p form. The resultant work
1
with z = 16 and K = 16
. Suppose also that 4 = 1.
Write his profit maximization problem, find N d ∗ and
compute the maximized profit.
p
*
*
The length of 2 projected on p is 2 cos θ , where θ
p
z F K, N d = z K N d ,
u(C, l) = C + 4
May the Force Be... GDP?
Competitive Equilibrium Would the labor market
clear with the labor demand and supply found in items 5A.13
and 5A.15? If not, is labor overpriced or underpriced?
In which direction does the labor market push 4?
spin counterclockwise, and a vector like 2′ will now
become a new optimizer. You can also easily visualize
gets more productive, i.e., when
what happens if
f (21 ) fluffs up towards the northeast.
Now you might be tempted to say “why don’t we
simply set θ = 0 to generate the maximum work?"
Yes in physics, but no in economics. (why?) This is
where we part our ways.
a The function will be a mirror image of the one we learned in class
because we flip the sign of 21 in this example. If
works for 5
hours, then 21 = −5, i.e., 5 hours of negative output (= 5 hours of
positive input).
7 This type of utility function is called a quasi-linear utility function.
If it is
linear, u(C, l) = C + l . In comparison, (9) is linear only in C (hence, quasi-linear).
This is a popular choice when you have composite goods. One of the notable
features of this function is that MRS does not increase with C for a given l . MRS
typically grows with C as C becomes relatively abundant in comparison to l . In
−1
contrast, MRS of (9) takes the same value M RSl, C (C, l) = 12 (l + 2) 2 whatever
C is. This implies that l does not increase with income. Therefore, quasi-linear
utility functions violate the assumption of normality. It is nevertheless an easy
function to work with, and I will often use this in what follows, except when we
conduct comparative statics in chapter 5 (this is the only time normality sinks in).
12
Chapter 5A
with K = 16. The government spending is G = 1 and
time allotted is h = 4. Find the equilibrium
in this econ
omy.
C, l, N s , N d , T, Y, Y s , Y d = (1, 3, 1, 1, 1, 2, 2, 2)
and 4 = 32 .
*
Figure 13. The right force 2∗ yields the maximum work (or profit if
*
you will) possible done to a particle moving in the direction of p .
5A.17
5A.18
Competitive Equilibrium Do the allocation and the
wage you found in items 5A.13, 5A.15 and 5A.16 constitute an equilibrium? If not, which part of the definition of the competitive equilibrium do they violate?
Competitive Equilibrium Repeat items 5A.13 and 5A.15
to 5A.17 with 4 = 32 and 4 = 12 .
You should be
able to solve items 5A.13 and 5A.15 to 5A.18 without
a graph but feel free to fill in the budget constraint in
figures 14 and 15 to check your answer.
5A.19
Competitive Equilibrium Between 4 = 12 and 32 , the
former earns
a higher profit. Why, then, does 4 = 23
constitute an equilibrium when it does not maximize
his profit?
5A.20
Competitive Equilibrium Among three wage rates
the
in items 5A.13 and 5A.15 to 5A.18, 4 = 32 gives
lowest utility level. Why, then, does 4 = 32 constitute
an equilibrium when it does not maximize his utility
level?
5A.21
Competitive Equilibrium In items 5A.13 and 5A.15
on page 12, I gave you three random wage rates, which
may or may not be the equilibrium price. This time, do
not fix 4 in advance and find the equilibrium.
5A.22
Competitive Equilibrium Suppose that
erences are represented by
u(C, l) =
’s pref-
4
log(C + 1) + l
3
and his production function is given by
3
1
z F K, N d = K 4 N d 4
13
5A.23
Competitive Equilibrium Suppose
’s preferences
are represented by u (C, l) = C + 32 log l . He is endowed with h = 24
punits of time. Production function
is zF K, N d = 8 K N d , where K = 16. The government expenditure is G = 2 [ ]. Find the equilibrium.
C, l, N s , N d , T, Y, Y s , Y d = (126, 8, 16, 16, 2, 128, 128, 128)
and 4 = 4.
5A.24
Competitive Equilibrium Fill the budget constraint
in equilibrium in figure 16.
14
34
Indifference Curves
Budget Constraint
C=zF(K, h-l)-G
22
Indifference Curves
C=zF(K, h-l)
28
13
12
11
20
26
10
16
18
9
20
8
24
23
C
7
22
6
5
4
3
2
18
16
1
14
12
10
14
12
16
8
10
8
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
24
C
Chapter 5A
0
-1
-2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
l
l
(a) For 4 = 1.
Figure 15. For 4 = 21 .
14
13
Indifference Curves
C=zF(K, h-l)
24
12
18
6
C [baskets]
C
7
5
4
3
2
14
10
16
8
1
12
0
18
-1
-2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
l
120
0
(b) For 4 = 23 .
Indifference Curves
C=zF(K, h-l)-G
21
16
20
8
240
0 .54
20 92
1
22
10
158
150
142
134
126
118
110
102
94
86
78
70
62
54
46
38
30
22
14
6
0
-2
160
0
12
80
040
11
9
18
4
8
12
l [hours]
Figure 14.
Figure 16.
14
16
20
24
Chapter 5A
Le maquereau
Why Is the Leading Digit
of Canadian GDP Two?
GDP (in log scale, base year 2009)
It is hard to predict GDP. It takes us three chapters
just to see how much our output is going to be in
equilibrium.
That said, it is actually quite easy to predict what
the leading digit of GDP is. I don’t know how useful
it is to know the leading digit, but well, we just do.
The probability that GDP starts with one is about
30.1%. Taking US for example, its GDP’s leading
digit is indeed one (about $1.8 trillion, cf. figures 17
and 18 on page 15).
This is due to an empirical observation called Benford’s law, which states that the probability that
the first digit of a naturally occurring collection of
numbers is d is proportionate to the ratio between
d +1 and d (cf. figure 18 on page 15). For instance, the
2
probability that d = 1 is proportionate to d+1
d = 1 = 2,
and d = 2 is 32 (Canadian GDP is here now) and d = 3
is 43 . As you can see, the probability declines with the
number and the probability that d = 9 is proportionate to only 10
9 .
But why? Shouldn’t any of nine numbers appear as
frequently as any other with probability 19 across the
board? I’ll show the proof in the next installment.
Long story short, GDP grows in such a way that
it takes more time to pass through leading digit of
one than of nine. That is, it trots in equal intervals
along the line in figure 19 (bottom) than in figure 19
(top) on page 15.
The law is applied to many areas. For instance,
attorneys check for deviation from Benford’s law if
embezzlement is suspected. If figures are artificially
altered, they won’t follow Benford’s law. Knowledge
is king. Even when you want to racketeer, you will
easily get caught if you are unaware of Benford’s
law (not that I’m condoning criminal acts, but be
knowledgeable at whatever you do, i.e., study hard,
people).
GDP
Benford's Law
Frequency of GDP Whose First Digit Is d
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1
2
3
4
5
6
7
8
9
First Digit d of GDP
Figure 18. First-digit distribution of US GDP
10 12
1
2
3
4
5
6
7
8
9
10
8
9 10
x (in linear scale)
10 11
1930
1940
1950
1960
1970
1980
1990
2000
2010
1
Year
2
3
4
5
6
7
x (in log scale)
Figure 17. US GDP. Red dots are the years whose GDP’s first digit
was one, orange when two, ... blue when nine.
Figure 19. Top: The difference between any two adjacent numbers is
the same. Bottom: The quotient between any two adjacent numbers
is the same.
15
Chapter 5B
5B
Optimality
to see if you get the gist of Pareto optimality.
Read
the definition of Pareto optimality very carefully. Also,
if you are tempted to quote the second fundamental
theorem of welfare economics, that’s a good instinct
but resist your temptation. Read the second theorem
No.
very carefully too. Short
Pareto Optimality Define a Pareto optimal allo-
5B.1
cation.
5B.2
5B.3
Pareto Optimality Suppose that there are 6 cookand Sensei. One of the Pareto
ies to be split between
optimal allocation is (x C , x S ) = (3, 3), where x C and
x S are the number of slices given to each one of us.
This allocation is Pareto improved upon by another allocation (3, 4). How, then, is (3, 3) a Pareto optimum?
Read the definition very carefully.
Pareto Optimality Suppose that there are 6 cookies to be split between
and Cookie Angel. Unlike
, Cookie Angel is gluten intolerant and his utility
function is decreasing in cookies. Find the Pareto opFor instance (3, 3) is not Pareto
timal allocation(s).
optimal because it will be Pareto improved upon by
(4, 2).
5B.4
Competitive Equilibrium Suppose that M RSl, C (C, l) >
M RTl, C (C, l) at some feasible allocation. Explain why
this allocation cannot be an equilibrium.
5B.5
Pareto Optimality In class today, we observed
why M RSl, C cannot be > M RTl, C at a Pareto optimal
allocation using figure 5-3 in Graph Vault® . Use figure 20 to show that M RSl, C cannot be < M RTl, C at a
Pareto optimal allocation either.
After proving this,
in conjunction with what we excluded in class, the only
possibility left is M RSl, C = M RTl, C at a Pareto optimal allocation, establishing the first theorem we introduced today.
(C, l) = (3, 11) is probably an easy
bundle to work with. Show that M RSl, C (C = 3, l =
11) < M RTl, C (C = 3, l = 11) on the graph to start with
and then find another feasible allocation that Pareto improves upon (3, 11).
5B.7
Pareto Optimality State and prove the first and
second fundamental theorem of welfare economics in
one-period model.
5B.8
Pareto Optimality Write UMP and social planner’s problem side by side, and discuss the similarities
For similarities, reand differences between them.
fer back to your lecture note from day 1. What is economics all about?
5B.9
Pareto Optimality What conditions does a social
planner use to find the solution to her problem?
5B.10
Pareto Optimality Does a social planner need to
determine 4 to solve her problem? Why or why not?
Short
No.
5B.11
Pareto Optimality Which one of the theorems we
learned today guarantees that the solution to item 5B.9
can be implemented as a competitive equilibrium?
5B.12
Pareto Optimality Solve item 5A.22 on page 13
again but as a social planner this time.
If done
right, you’ll have a lot less calculations to do than in
item 5A.22. This is the beauty of having a fictional social planner. Write the social planner’s problem first
and then use item 5B.9 to find a Pareto optimal allocation. After that, quote the appropriate theorem to show
that the Pareto optimal allocation that you’ve found is
actually the competitive equilibrium too.
5B.13
Pareto Optimality If you were to implement the
Pareto optimal allocation in item 5B.12 as a competitive equilibrium, how much should 4 be and why?
This is why the 2nd fundamental theorem does not
say that a Pareto optimal allocation is a competitive
equilibrium. There is no mention of price in Pareto optimality, whereas the price is a crucial component of
competitive equilibrium. Thus you have to supply the
right 4 to have it realized as a competitive equilibrium.
5B.14
Pareto Optimality Solve items 5A.21 and 5A.23 on
page 13 as a social planner.
5B.15
Pareto Optimality
’s preferences are represented
by u(C, l) = C + 2l . Sketch his indifference curves at
u(C, l) = 6, 8 and 10 in figure 21.
5B.16
Pareto Optimality In item 5B.15
is endowed
with h = 4 units of time. His production function is
p
Y s = zF K, N d = z K N d , with z = 2 and K = 4. The
government expenditure is G = 2. Find the equilibrium.
C, l, N s , N d , T, Y, Y s , Y d = (2, 3, 1, 1, 2, 4, 4, 4)
and 4 = 2.
5B.17
Pareto Optimality How much is profit in item 5B.16?
2.
15
Indifference Curves
C=zF(K, h-l)-G
64
14
13
12
44
10
11
15
0
10
9
C
8
7
6
5
56.5
4
685
3
2
1
10
0
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
l
Figure 20.
5B.6
Pareto Optimality Does 4 have to be equal to M RSl, C
at a Pareto optimal allocation? Why or why not?
Checking
16
Chapter 5C
8
convex and marginal product does not decline, and
consequently, supply and demand become a correspondence rather than a function, which would otherwise
. A social planner can byrender these questions
pass all these pesky businesses and keep to
.
C=zF(K, h-l)-G
6
5B.20
Pareto Optimality How much is profit in item 5B.19?
1.
2
5B.21
Pareto Optimality Suppose that
’s preferences
are represented by u(C, l)p= C +16 log l . The production
function is zF (K, N ) = z K N , with z = 4 and K = 16.
is endowed with 24 units of time. The government
expenditure is G = 12. Find the equilibrium and equilibrium profit.
C
4
0
-2
0
1
2
3
5C
4
l
5C.1
Figure 21.
Comparative Statics
Comparative Statics Suppose that
are given by
u(C, l) = C l,
’s preferences
with h = 15. Production function is given by
5B.18
p
zF K, N d = z K N d ,
is endowed with h = 5 units
of time. His production function is Y s = z F K, N d =
z K + N d , with z = 1 and K = 1. The government
expenditure is G = 2. Sketch the PPF in figure 22.
Pareto Optimality
1
with z = 2 and K = 16
. Suppose that G is currently 0
and the government will increase it to 1. Find equilibrium (CB , l B ) before G is increased.
See figure 23.
Feel free to solve this as a social
planner. That’ll get
p
you to
(CB , l B ) = 12 5, 10 much faster.
4
11
.1
80
1.5
2
3
1.1180
C [basket]
C [basket]
1
15
93
0.6
0.5
0
-0.5
3
Indifference Curves
1
-0.5 0
0
3
0.5
1.180
0.1180
0
Indifference Curves
Budget Constraint (w=(4 5.5 )-1 )
Budget Constraint (w=1/12)
C=zF(K, h-l)-0
C=zF(K, h-l)-1
-1
0
1
2
3
4
34
5
l [hr]
-1
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
l [hr]
Figure 22.
Figure 23.
5B.19
Pareto Optimality In item 5B.18,
’s preferences
are represented by u(C, l) = 12 log l + 12 log C . Find the
equilibrium.
C, l, N s , N d , T, Y, Y s , Y d = (2, 2, 3, 3, 2, 4, 4, 4)
and 4 = 1.
Items 5B.15 and 5B.19 are twins. In
item 5B.15, indifference curves are linear, whereas in
item 5B.19, the PPF is linear. Unlike items 5B.12 and 5B.14,
if you solve items 5B.15 and 5B.18 agent by agent (i.e.,
UMP and PMP separately), it does not only take longer
but also becomes harder: Preferences are not strictly
17
5C.2
Comparative Statics In item 5C.1, suppose that
simply
reduced his C by 1 to offset the tax increase. Would
(CB −1, l B ) constitute an equilibrium? In this case, how
much are ∆Y and ∆C ?
5C.3
Comparative Statics Compute equilibrium (CA, lA)
after the government spending increase in item 5C.1.
Compute ∆Y and ∆C in this case.
See figure 23 to
check your answer.
Chapter 5C
5C.4
Comparative Statics Compare your answers to
simply reduce C
items 5C.2 and 5C.3. Why doesn’t
by ∆G to offset tax increase?
5C.5
Comparative Statics In class, we have seen that
an increase in G triggers an increase in Y . This observation actually hinges on one of the three assumptions we
made about consumer preferences. Identify which one
it is and show why the violation of the said assumption
may leave Y unchanged (or even have it reduced) after
Cf. figure 24 (this
government spending increase.
is a case where Y d does not respond to a change in G )
and footnote 7.
8
6
C [basket]
9
8
4
7
6
2
5
4
3
0
Indifference Curves
C = zF(K, h-l) - varying G
-2
0
5
10
15
l [hr]
Figure 24.
5C.6
Comparative Statics In item 5C.5, would |∆C|
be larger than |∆G|? How would you interpret the reIt can be. Without the assumption in
sult? Short
item 5C.5, there is no guarantee that
will reduce l
in response to an increase in G .
18
Chapter 9A
10 4
Geometric Progression
Geometric progression
Arithmetic progression
3.5
GDP (in linear scale)
Le maquereau
and Benford’s Law
As I discussed in the previous installment, GDP is
known to follow Benford’s law. This is partly due
to multiplicative (instead of additive) nature of GDP
growth. Before I venture out to connect GDP
to Benford’s law, first, let me show a cheap proof
of Benford’s law.a Suppose that x ∈ [1, 10) and
log10 (x) follows the uniform distribution over its
domain (cf. figure 19 (bottom) on page 15). Then
the first digit of x is 1 when log10 x ∈ [log10 1, log10 2)
and 2 when log10 x ∈ [log10 2, log10 3) and so on. Then
P r(The first digit is d) = log10 (d + 1) − log10 d as the
probability is proportional to (and is in this case)
the length of each bin in figure 19 (bottom), leading to Benford’s law. Note that if x itself (rather
than the log thereof) is uniformly distributed, then
P r(The first digit is d) is just 19 for any d (Each bin
in figure 19 (top) has the same length).
Then the question becomes "Why does GDP behave as if its log is uniformly distributed (and thus
follows Benford’s law)?" To answer this, note that
GDP usually follows geometric progression rather
than arithmetic progression. An arithmetic progression is a sequence where any two adjacent numbers are apart from each other by the same distance, e.g., {1, 2, 3, . . .} (add 1 to reach the next) or
{103, 106, 109, . . .} (add 3 to reach the next). A geometric progression is a sequence where any two
adjacent numbers are apart from each other by the
same factor, e.g., {1, 2, 4, . . .} (multiply by 2 to reach
the next) or {10, 100, 1000, . . .} (multiply by 10 to reach
the next). GDP trots like the latter.
Now suppose that GDP starts off with $100 and
increases 2% annually as in figure 25 on page 19. This
creates a geometric progression where you multiply
the previous GDP by 1.02 to reach the next. In this
manner, GDP walks along the line in equal steps in
logarithmic scale figure 19 (bottom) rather than in
linear scale figure 19 (top). That is, GDP will spend
a lot of time to get out of the first bin (the red
segment) but after a while, since GDP will grow exponentially (at the rate of 2%), it doesn’t take much
to pass through later bins (like green or blue segments). In figure 25, notice that GDP will spend lots
of time getting out of the red segment than the
blue segment. Thus, GDP behaves as if it is uniformly
distributed along figure 19 (bottom) rather than figure 19 (top). Consequently, GDP follows Benford’s
law.
Canadian GDP is going through the orange bin in
figure 19 (bottom) right now. I expect it to move on
to the next bin quicker than we spent time getting
out of the red bin.
Still not convinced? Then consider the counterfactual: Suppose instead that GDP increases by $2
rather than 2%, i.e., an arithmetic progression. Then
Benford’s law won’t apply to GDP because GDP will
spend an equal amount of time to move across each
bin in linear scale in figure 19 (top). Consequently,
any number will appear as the first digit asymptotically
with the probability of 19 (cf. pale bars in figure 26).
3
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
300
Year
GDP (in log scale)
10 4
10 3
Geometric progression
Arithmetic progression
10 2
0
50
100
150
200
250
300
Year
Figure 25. Arithmetic progression (pale) and geometric progression
(dark) plotted in linear and log scale. Any geometric progression
appears as a straight line in log scale, whereas any arithmetic progression appears as a straight line in linear scale. Notice that US GDP
figure 17 on page 15 roughly lines up along a straight line.
Geometric progression
Arithmetic progression
Frequency of GDP Whose First Digit Is d
100
a Cheap because I will establish Benford’s law for one particular
90
80
70
60
50
40
30
20
10
0
1
class of numbers.
2
3
4
5
6
7
8
First Digit d of GDP
Figure 26. First-digit distribution for figure 25. Notice the close resemblance of geometric progression (dark) to figure 18 on page 15.
9 Two-Period Model
with Exogenous Output
9A
9A.1
19
Financial Instruments
Financial Instruments Figure 27 represents
’s
income and desired consumption level over time. His
9
Chapter 9B
income level fluctuates due to booms and busts of the
economy but he prefers to have a steady stream of
cookies throughout. His preferences are represented
9A.6
Financial Instruments Suppose that Zedd has
can build him
ceviche that he does not want and
a cookie-themed concert venue in three months. Zedd
wants the cookie-themed nightclub in three months and
needs to eat ceviche now. Who should write an IOU
to whom? Who should be a creditor and who should
be a debtor in this case?
9A.7
Financial instruments In item 9A.6, would the
economy with an IOU Pareto improve upon the one
Yes.
without it? Explain. Short
9A.8
Financial Instruments How is a hot slice of pizza
not a financial instrument but banknotes are, when you
can’t even eat the latter?
9A.9
Financial Instruments That $1 bill you have in
your wallet is a financial instrument, and as such, an
IOU. Who is an “I” and a “U” in this case? How about
a check you write?
6
y , c [baskets ]
income stream
desired consumption stream
5.5
5
4.5
4
22
20
3
2
20
24
20
5
2
20
26
20
7
2
20
2
20
8
30
20
29
20
9B
Figure 27.
9B.1
Utility Suppose that
’s preferences are strongly
monotonic, but not convex. Sketch his indifference
curves and show that in this case,
would hate to
smooth out his consumption over time.
Cf. figure 6
on page 5.
9B.2
Utility Does levelling sτ (rather than cτ ) help
improve the utility level over time? Namely, is it beneficial to set aside the same amount of baskets every
period to better prepare for retirement?
9B.3
Intertemporal Budget Constraint A bundle (c1 , c2 ) =
(21 − t 1 , 22 − t 2 ) is always affordable for the same reason
(C, l) = (π − T, h) is always affordable (cf. item C1.10
on page 4). Discuss their similarities and explain why
they are always available for consumption.
9B.4
Intertemporal Budget Constraint Suppose (21 , 22 ) =
(5, 6) and (t 1 , t 1 ) = (1, 2). Why isn’t the lifetime income 5 − 1 + 6 − 2 = 8 ?
As always, check the unit
of measure. Some of them are quoted in in period 1
and others are in period 2. Adding them as is wouldn’t
make any sense. Hence, the time machine.
9B.5
Intertemporal Budget Constraint In item 9B.4
if the lifetime income were actually 8 (in period 1),
how much should r be?
in figure 28. Use this to explain why the green line in
Indifference Curves
9
25
5
9
consumption in the following
period c +1 [baskets +1 ]
Utility Maximization Problem
25
9
1
0
0
1
5
9
current consumption c [baskets ]
Figure 28.
figure 27 gives him a higher utility level than the blue
line.
9A.2
Financial Instruments Fill in his consumption
stream if there were no financial instruments in figure 27.
9B.6
Intertemporal Budget Constraint In item 9B.4
if the lifetime income were actually 8 (in period 2),
how much should r be?
9A.3
Financial Instruments Assume that interest and
inflation rates are both zero. Sketch his consumption
stream when there are financial instruments in item 9A.1.
9B.7
Intertemporal Budget Constraint In item 9B.4
if the lifetime income is 6 (in period 1), how much
should r be?
r = 1.
9A.4
Financial Instruments In item 9A.3, is he a creditor or debtor? Short
a creditor.
9B.8
Intertemporal Budget Constraint In item 9B.4
if the lifetime income is 10 (in period 2), how much
should r be?
r = 12 .
9A.5
Financial Instruments Sketch his income stream
where he will instead be a debtor in item 9A.3.
20
Chapter 9C
9B.9
9B.10
9B.11
9B.12
9B.13
Intertemporal Budget Constraint In class, we
quoted period 1’s budget constraint in period 1’s ,
and period 2’s budget constraint in period 2’s . Write
period 1’s budget constraint in period 2’s , and write
When
period 2’s budget constraint in period 1’s .
we write the budget constraint period by period, the
most obvious choice of the unit of measure is the contemporary , and that’s what we did in class. But
who’s stopping us from writing them in different units?
There is probably not much practical value to this question, but it should help you get the hang of time machine operations.
9B.18
Intertemporal Budget Constraint In class, we
wrote the intertemporal budget constraint using s1 , the
buys. In fact, there is no need to
amount of IOU
define the intertemporal budget constraint in this way.
Write the intertemporal budget constraint with b1 instead of s1 , where b1 measures the amount of IOU
writes.
9B.19
Intertemporal Budget Constraint Suppose that
(a) neither save or borrow.
(b) save but not borrow.
(c) borrow but not save.
Intertemporal Budget Constraint In item 9B.9,
does
’s lifetime purchasing power change depending on which unit it is quoted in? Short
No.
Shade in the set of affordable bundles in each case
A rectangle, a trapezoid and another trapeabove.
zoid.
Intertemporal Budget Constraint While there
are only two periods in this chapter, let us add a third
period in this question. Suppose that the interest rate
is r baskets in period 2 per a basket in period 1, and
then again r baskets in period 3 per a basket in period
2. How much is the opportunity cost of a basket in period 1 in terms of baskets in period 3? Conversely, how
much is the opportunity cost of a basket in period 3 in
terms of baskets in period 1?
There will be not one
but two legs of a journey on the time machine.
Intertemporal Budget Constraint Suppose that
is
endowed with 21 baskets this year and 22 baskets next
year, whereas the government collects t 1 and t 2 baskets
in each period respectively. How much is his lifetime
endowment measured in terms of 1 , and in terms of
2?
9B.20
Intertemporal Budget Constraint We assumed that
the interest rate is r whether
is a debtor or creditor.
Suppose instead that the saving rate is rs and the borrowing rate is r b (> rs ). Sketch the budget constraint.
(c1 , c2 ) = (21 − t 1 , 22 − t 2 ) should still be on the
budget line because
doesn’t save or borrow at this
point, and thus the interest rates, whatever they are,
have no bearing on this bundle. Take this bundle as
your point of departure.
9B.21
Intertemporal Budget Constraint Technically, the
budget constraint in period 2 should be c2 + s2 = 22 −
t 2 + (1 + r)s1 . However, s2 = 0. Why?
For the same
reasons as item 5A.2 on page 11.
9B.22
UMP Suppose that
earns (21 , 22 ) = (16, 50)
baskets, out of which the government collects (t 1 , t 2 ) =
(1, 10). The interest rate is r = 3. His preferences are
given by
p
1p
u(c1 , c2 ) = c1 +
c2 .
4
Intertemporal Budget Constraint In class, s1 is
defined to be a number of IOU’s that
buys. Write
his budget constraint in period 1 and 2 respectively,
with a number of IOU’s that
issues instead.
You
can use whatever variable you prefer but let’s say a
number of IOU he issues is represented by b1 (b for
borrowing). Which side of the equation should b1 appear? Just for reference, s1 takes income away this year
and augments it next year.
Find his optimal consumption bundle. To maximize
his utility level, does he issue an IOU (a debtor) or buy
one (a creditor), and for how many baskets?
See
figure 29.
9B.23
9B.14
Intertemporal Budget Constraint What is the
relationship between s1 in class and b1 as in item 9B.13?
9B.15
Intertemporal Budget Constraint When constructing the intertemporal budget constraint, does it matter whether we build it via s1 as in class or via b1 in
item 9B.13? Short
It doesn’t.
9B.16
Intertemporal Budget Constraint Write consumer’s
intertemporal budget constraints in terms of baskets in
period 1, and in period 2. Sketch each on the graph
with c1 on the horizontal axis and c2 on the vertial axis.
9B.17
Intertemporal Budget Constraint The two budget constraints in item 9B.16 should be identical on a
graph. Why?
Suppose that cheesecake is 2 $/slice
and tea is 1 $/cup. Say you have $12. Does it make any
difference to say that your income is worth 6 slices or
12 cups?
can:
UMP In item 9B.22, assume that there are no financial instruments. Write his UMP, find his optimal
bundle and compare the realized utility level to that of
item 9B.22. Is your answer consistent with item 9A.7?
See figure 29.
9C
21
Comparative Statics on Income
9C.1
Comparative Statics on 21 In class, I raised 21 from
21L to 21H . Reduce 21 from 21H to 21L instead and indicate
∆c = (∆c1 , ∆c2 ) on your graph.
9C.2
Comparative Statics on 21 In class, I sketched a case
where
is a creditor. Reproduce a graph for when
he is a debtor instead to run comparative statics on 21 .
Indicate two s1 ’s on your graph before and after the
change in 21 . Is the effect on s1 the same to the case
of a creditor?
Draw the indifference curve so that
the optimal bundle will appear to the southeast of the
endowment.
Chapter 9C
100
9C.9
Indifference Curves
Budget Constraint
90
80
70
2
90 1
5.5.454
5
c2
60
50
40
30
25
Figure 29.
9C.3
Comparative Statics on 21 Explain why ∆s1 = 0
when c H is found due east of c L after replacing 21L with
21H (> 21L ).
9C.4
Comparative Statics on 21 and 22 Explain both in
words and with equations, why ∆s1 > 0 when ∆21 > 0,
and ∆s1 < 0 when ∆22 > 0.
Convexity.
c2
20
0
1
2
3
4
4
15
c1
9
10
43
5
74
0
426
4.2
0
Indifference Curves
4.7
10
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
4.9
20
Comparative Statics on 21 and 22 Fill in the three
budget constraints for items 9C.7 and 9C.8, find the
optimal bundle for each, and indicate two ∆c ’s in figure 30.
5
6
7
8
c1
9C.5
9C.6
Comparative Statics on 21 In class, I went over
the case where c H appears to the southeast of c L , resulting in ∆s1 < 0, and showed you why this cannot happen (in violation of what again?) Sketch a case where
c H appears to the northwest of c L and explain why this
cannot happen either.
Figure 30.
Comparative Statics on 21 Suppose that you raised
’s income from 21L to 21H . Show on a graph a set of
bundles where
• he uses up his intertemporal budget,
• his savings increase, but
• c1 is not normal.
This question is effectively the same as item 9C.5.
9C.7
Comparative Statics on 21 Suppose that
preferences are represented by
u(c1 , c2 ) =
p
c1 +
’s
1p
c2 .
2
His income is (21 , 22 ) = (6, 18) or (9, 18), out of which
the government collects (t 1 , t 2 ) = (2, 2). The interest
rate is r = 1 [ 2 / 1 ]. Write his UMP and find his optimal bundle for each case. Compute ∆c and ∆s1 when
21 increases by 3 [ 1 ]. Part of
∆c = (∆c1 , ∆c2 ) =
(+2, +2).
9C.8
Comparative Statics on 22 In item 9C.7, suppose
that the income changes further from (9, 18) to (9, 21).
Compute ∆c and ∆s1 when 22 increases by 3 [ 2 ]. Part
of
∆c = (+1, +1).
22
9
10 11 12 13 14 15 16
Chapter 9D
Le maquereau
Compound Interest and e
In class,
receives interest payments once because
there are only two periods. In reality, we receive
interest payments multiple times because we tend
to live longer than two periods. Let’s say that we
save $100 with the annual interest rate of 1%. By
the end of the first year, we receive $1. By the
end of the second year, do we receive another $1
again? – No. We receive $1.01 rather. The interest
payment of $1 at the end of the first year itself will
earn the interest in the second year (cf. item 9B.11).
This is called compound interest. Interest will accrue
not only to the principal but also to the preceding
interest payments all combined.
There is an unexpected connection between compound interest and some mathematical constant
that you are familiar with. To make the argument
saves $1 and instead of
simpler, let’s say that
expanding the periods, divide the payment intervals
within one year. Namely, instead of receiving 1%
over a year, receive .5% twice every half year, receive .25% four times every quater, receive 1/12%
every month, receive 1/365% every day, and so on.
In these cases, a dollar of investment will yield
• (1 + .01)1 = 1.01
• (1 + .01/2)2 = 1.010025
• (1 + .01/4)4 ≈ 1.01003756
• (1 + .01/12)12 ≈ 1.01004596
.
.
.
over a year. As we divide the interval finer and finer,
’s return seems to grow larger and larger. Can
he be a billionaire at the limit then?
Sadly no. There’s a cap: His compound interest is
at most e1% ≈ 1.010050167.
What is Euler’s number e doing
here? Recall
n
that e is equal to limn→0 1 + 1n . Let r be the
annual interest rate and h be the number of in’s compound interest
tervals over the year.
•€
Šh/r ˜ r
h
1
. Now let h → ∞, i.e.,
is 1 + hr
=
1 + h/r
9D.3
Comparative Statics on r Explain, in words, why
the income effect from increasing r on c1 has to be
is a debtor.
Quote the appropriate
negative if
assumption on his preferences.
9D.4
Comparative Statics on r The overall effect of increasing r on s1 is indeterminate when
is a creditor.
I sketched a case where ∆s1 happens to be > 0 in class.
I made the
Draw yourself a case where ∆s1 < 0.
substitution effect stronger than the income effect in
class. Reverse that here so that c1 will decline, leaving
∆s1 > 0.
9D.5
Comparative Statics on r The overall effect of increasing r on c2 is indeterminate when
is a debtor.
I sketched a case where ∆c2 happens to be > 0 in class.
I made the
Draw yourself a case where ∆c2 < 0.
substitution effect stronger than the income effect in
class. Reverse that here so that the overall effect on c2
will be positive.
9D.6
Comparative Statics on r We went over the cases
where s1 > 0 and < 0 in class. Sketch a case for s1 = 0.
Does increasing r affect
in the same way as when
he is a debtor or creditor?
9D.7
Comparative Statics on r In class, I sketched a
case where a creditor remains to be a creditor and a
debtor remains to be a debtor when r increases. Is it
possible for a creditor to turn into a debtor and vice
versa when r changes? Explain.
PA from the cockpit: For items 9D.8, 9D.10 and 9D.12, it’ll probably
be quicker to solve item 9D.8 without substituting 21 , 22 , t 1 , t 2
and r with given values first to find the solution c1∗ , c2∗ and s1∗ , and
then swap these in the solution with the given values. You don’t
have to but I’m just tossing in a time-saving idea.
make the interval infinitesimally small.
Then˜ his
•€
Šh/r r
1
compound interest becomes limh→∞ 1 + h/r
=
•
€
Šh/r ˜ r
1
1 n r
r
limh/r→∞ 1 + h/r
= limn→0 1 + n
= e . Thus,
9D.8
u (c1 , c2 ) = log c1 +
even when the interest rate is as high as a 100%
for example, no matter how thinly you slice the year,
a dollar of investment will yield at most e100% ≈ 2.718
dollars. That’s one handsome yield but still, not a
billion.
9C.10
9D
Comparative Statics on Interest Rate
Comparative Statics on r Suppose r went down.
Explain in words why the substitution effect on s1 will
be negative.
We covered the case where r went up
in class. Simply put the argument in reverse.
9D.2
Comparative Statics on r It is obvious that a
change in r alters the opportunity cost of c1 and thus
creates a substitution effect. How does it cause an income effect as well?
’s pref-
1
log c2 .
2
He earns (21 , 22 ) = (33, 13), out of which the government collects (t 1 , t 2 ) = (3, 1). The interest rate is r = 1
basket next year per a basket this year. Write his utility maximization problem and find his optimal bundle
and s1 . Compute his utility level at the optimal bundle.
Comparative Statics on 21 and 22 Compare ∆c1 in
item 9C.7 to ∆c1 in items 9C.7 and 9C.8 combined.
Do your results support permanent income hypothesis? Explain.
9D.1
Comparative Statics on r Suppose that
erences are represented by
9D.9
Comparative Statics on r Fill in
’s budget constraint in item 9D.8 and locate his optimal consumption
bundle in figure 31 on page 24.
9D.10
Comparative Statics on r In item 9D.8, suppose that
5
his income becomes (21 , 22 ) = (9 · 2 3 + 3, 1), and r = 3.
Write his utility maximization problem and find his optimal bundle. Compute his utility level at the optimal
€
Š
5
5
bundle.
(c1∗ , c2∗ ) = 6 · 2 3 , 12 · 2 3 , as represented
by the blue dot in figure 31. The utility level is the same
to the one you found in item 9D.8.
9D.11
23
Comparative Statics on r Compare your answers
to items 9D.8 and 9D.10 and indicate
’s substitution
effect in figure 31.
both substitution and income effects are absent in this case.
Indifference Curves
Budget Constraint
It’ll probably be easier to sketch the budget constraint and indifference curves to see what will or will
not happen. In particular, pay close attention to the
tangency condition.
9D.17
4.7671
132
128
124
120
116
112
108
104
100
96
92
88
84
80
76
72
68
64
60
56
52
48
44
40
36
32
28
24
20
16
12
8
4
0
1
4.983
c1
Chapter 9F
Comparative Statics on r Suppose that r goes
) tend to save or borrow
up. Do you yourself (not
more or less? Explain your behavior and determine
whether your income effect is larger or smaller than
your substitution effect.
It’s okay if you do not register one of the effects. You have your own preferences,
and
has his own.
9E
9E.1
Competitive Equilibrium
Government’s Budget Constraint Suppose that the
government needs to raise (G1 , G2 ) = (1, 1).
(a) If T1 = 0, how much should B1 and T2 be?
(b) If T2 = 0, how much should B1 and T1 be?
4.9
83
You can use 1 or 2 for a unit of measure. Both
work so long as you pick one and stick with it throughout.
1
4.76
71
9E.2
Government’s Budget Constraint Write the government’s intertemporal budget constraint.
9E.3
Government’s Budget Constraint Repeat items 9B.9
and 9B.10 but for the government.
Figure 31. I filled in the disposable endowment e (purple dot) for
your reference. Be sure to have both your budget lines pass through
this bundle.
9E.4
Comparative Statics on r In item 9D.10, revert his
earnings to item 9D.8 while maintaining r = 3. Write
his utility maximization problem and find his optimal
consumption bundle and s1 .
Consumer’s Budget Constraint Repeat item 9E.2
for consumers.
Watch out for the sign: B1 > 0 is a
debt but s1 > 0 is saving, and vice versa. The rest is the
same though.
9E.5
Government’s Budget Constraint Explain why B2
cannot be > 0 or < 0, while B1 can.
For the same
reasons as items 5A.2 and 9B.21 on pages 11 and 21.
9E.6
Comparative Statics on r Compare your answers to
’s income effect
items 9D.10 and 9D.12 and indicate
in figure 31.
Intertemporal Budget Constraint Does B1 or s1
increase or decrease the lifetime tax revenue or income?
If it does, then what should happen to the government or
?
9E.7
Comparative Statics on r Using your answers to
items 9D.11 and 9D.14, indicate
’s overall effect in
figure 31.
Intertemporal Budget Constraint If item 9E.6
is true, then what’s the point of leaving B1 or s1 , 0?
Cf. items 9A.1, 9A.7 and 9B.23.
9E.8
Competitive Equilibrium Define the competitive
equilibrium.
9E.9
Competitive Equilibrium In this chapter, we assume that the labor supply is inelastic, i.e., Nτs (4) = N̄τs
for any 4 > 0 and τ = 1, 2. In this case, there is one Yτ
that leaves the labor market in equilibrium (from which
Y
I derived 2τ as mτ so that labor market will be in equilibrium). What does Yτ have to be equal to?
Recall
the production function from the previous chapter.
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32 34 36
c0
9D.12
9D.13
9D.14
9D.15
9D.16
Comparative Statics on r Fill in
’s budget constraint in item 9D.12 and locate his optimal consumption bundle in figure 31.
Comparative Statics on r Suppose that
’s existing optimal consumption bundle is (c1∗ , c2∗ ) = (21 −
t 1 , 22 − t 2 ), that is, he achieves the highest utility level
without a help of an IOU. Validate or discredit the following statement:
Since he does not save or borrow at his optimal bundle, r does not affect his consumption choice. Accordingly, when r changes,
24
Chapter 9F
36
9F.1
Ricardian Equivalence State the Ricardian equivalence theorem.
32
9F.2
9F.3
9F.4
9F.5
9F.6
9F.7
9F.8
Ricardian Equivalence Suppose that the government needs to meet (G1 , G2 ) = (10, 12). Suppose also
that r = 1. Write the government’s budget constraint in
period 1 and 2.
Ricardian Equivalence In item 9F.2, write the
government’s intertemporal budget constraint, one measured in terms of baskets in period 1 and another in
terms of baskets in period 2.
16
12
8
0
0
4
6
8
10
12
14
16
Figure 32.
Ricardian Equivalence (T1 , T2 ) = (G1 , G2 ) is a
member of the set in item 9F.4. Why?
Once again,
. His disposable endowment (21 − t 1 , 22 − t 2 ) is always on his intertemporal budget line. Why was that?
Also, a little note here: For
, we measure his expenditure c1 and c2 on each axis, and then his income
(21 − t 1 , 22 − t 2 ) appears as a fixed point along his budget line. For the government, they are in reverse: We
measure its tax revenue T1 and T2 on each axis, and
then its expenditure (G1 , G2 ) appears as a fixed point
along its budget line. The reason for reversal between
income/revenue and expenditure is that
takes his
income (21 , 22 ) as given and then picks his spending
(c1 , c2 ). On the contrary the government takes their
spending (G1 , G2 ) as given and then picks its revenue
(T1 , T2 ).
9F.13
Ricardian Equivalence Sketch the two budget
constraints on a graph in item 9F.12. Why are they
identical?
What if not?
9F.14
Ricardian Equivalence In item 9F.12, do you expect
to have distinct optimal bundles under two situations?
Explain.
9F.15
Ricardian Equivalence Prove Ricardian equivalence theorem assuming ∆r = 0.
Le maquereau
Capita
Your Paycheck vs GDP per
Canadian GDP per capita was $59,181, coming in as
the 16th in the world in 2017. This is how much we
produced per capita, and according to the incomeexpenditure identity, this is how much we earned as
well. Is this really what’s written on your pay check?
Probably not. Some of you earn more than $59,181
and some, less. Keep in mind that macroeconomics
studies economic aggregates (like GDP) but not individual variables (like your paycheck). Thus, your paycheck does not necessarily match Canadian GDP per
capita. Note also that GDP is the sum of all incomes, which includes not only labor income 4N d (the
paycheck) but also capital income π.
Ricardian Equivalence Repeat item 9F.10 with
B2 = 4 instead.
Rest assured that neither item 9F.10
or item 9F.11 would not actually happen thanks to item 9E.5
on page 24. These two questions should, however, help
you relate the equation to the graph.
See figure 32.
Ricardian Equivalence Assume m = 1. Write
budget constraint under items 9F.5 and 9F.6.
2
T1 (baskets in period 1)
Ricardian Equivalence What does the slope of
the line you found in item 9F.4 represent?
Let’s go
’s budget constraint as a point of referback to the
ence. What does its slope represent there?
Ricardian Equivalence Suppose that in item 9F.2
the government (stupidly) left 4 baskets in period 2 for
the next period (that never comes), i.e., B2 = −4. Overlay this new (stupid) budget constraint in item 9F.4.
See figure 32.
9F.12
20
4
Ricardian Equilibrium Read off from your graph
in item 9F.4 the amount ∆T2 required when you reduce
T1 to 7 [ 1 ] while staying on the budget in item 9F.5.
9F.10
Budget Constraint with B2=4
24
Ricardian Equivalence In item 9F.2, suppose that
currently (T1 , T2 ) = (8, 16). Indicate B1 and (1 + r)B1
on your graph in item 9F.4.
Ricardian Equivalence Put an interpretation on
the x - and 2-intercept in item 9F.4.
Budget Constraint with B2=-4
28
Ricardian Equivalence On a graph with T1 on
the horizontal axis and T2 on the vertical axis, sketch
the set of (T1 , T2 ) that satisfies item 9F.3.
9F.9
9F.11
T2 (baskets in period 2)
Comparative Statics on Tax
9F
9F.16
Ricardian Equivalence Prove Ricardian equivalence theorem.
9F.17
Ricardian Equivalence We assumed that
’s
substitution effect exceeds income effect in magnitude
during the proof of Ricardian equivalence theorem. Does
the theorem not hold if these are in reverse?
PA from the cockpit: Assume that r1 and r p remain constant throughout item 9F.18 to item 9F.23. Also, it is unlikely that I will include
any of item 9F.18 through item 9F.23 in the final, but if you have
extra time to spare, give them a shot nonetheless. These should
help you broaden your understanding of Ricardian equivalence by
way of going through the scenario where it does not apply.
’s
25
18
Chapter 9F
9F.21
Ricardian Equivalence Repeat items 9F.18 to 9F.20
assuming t 1 goes up by ∆t 1 .
9F.22
Ricardian Equivalence Between items 9F.18 to 9F.20
and item 9F.21, which one is more favorable to
and
why?
9F.23
Ricardian Equivalence Redo item 9F.18 assuming, unrealistically,8 that r p < r1 .
9F.24
Ricardian Equivalence Suppose that
ences are represented by
u (c1 , c2 ) = log c1 +
71
Ricardian Equivalence How would c1∗ , c2∗ , s1∗
change in item 9F.19?
4.76
9F.20
Indifference Curves
82
4.7
4.278
2
c1
Figure 33.
9F.27
Ricardian Equivalence How much is the smallest
tax cut in period 1 that would make his credit constraint non-binding in item 9F.25?
Remove the constraint s1 > 0 and find his (usual) optimal consumption
bundle first. Then find out how far you would have to
move his disposable endowment (21 − t 1 , 22 − t 2 ) (by
changing t 1 and t 2 ) to include his optimal bundle as
part of his budget constraint while s1 > 0 is in place.
∆t 1 = −14.
, an IOU Edi-
p
ten on it. For instance, when
issues an IOU, its
buyer, say a bank, will have terms specified. For in, will pay back $100 plus
stance, his IOU might say “I,
5% interest on December 31st of next year”. If you
have an auto loan, mortgage etc., your promissory
note has r written on it.
In contrast, a T-Bill, the government’s IOU (B1 ),
doesn’t say what r applies to it. It simply says, ”I, the
Canadian Government, will pay back $100 on December 31st of next year”. Does this mean B1 has r = 0?
’s
A T-Bill typically has a lower interest rate than
IOU (cf. item 9F.18), but no, the lack of r does not
mean r = 0. A creditor will still earn the interest
from a T-Bill just as he would from
’s IOU. The
thing is, a T-Bill only says the payoff on the maturity
date, but it doesn’t say how much it sells for before
the maturity date. For instance, if a T-Bill that pays
$100 a year from today is traded for $95.24 today,
then the interest rate is r = 100−95.24
≈ 5%. Con95.24
sequently, if the trading price of a T-Bill falls, its
interest rate rises.
and his income is (21 , 22 ) = (28, 52) and (t 1 , t 2 ) =
(18, 0). In addition to his usual intertemporal budget constraint, he also faces credit constraint s1 ≥ 0.
The interest rate is given by r = 1. Write his utility
maximization problem.
You’ve already done this in
item 9B.19 on page 21.
9F.26
Canada v.
Private IOU’s (S1 ) have interest rate r explicitly writ-
1
log c2 ,
2
Ricardian Equivalence Fill in his two constraints in
item 9F.24 in figure 33 and find his optimal bundle.
Ricardian Equivalence Compute and compare
’s
utility levels in items 9F.25 and 9F.26. Explain why the
tax cuts improve his utility level.
Normally they
don’t, due to Ricardian equivalence theorem, which
doesn’t apply to
here. Recall how Cookie and Sensei die as discussed in section 9A.
(part) See the
values on figure 33.
Le maquereau
tion
’s prefer-
9F.25
671
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
Ricardian Equivalence Would Ricardian equivalence theorem apply to item 9F.18? If not, how does his
Recall the case
lifetime income change and why?
discussed in class for the credit constraint that
faces
in reality. Tax cuts work as a substitute for private borrowing for credit-constrained consumers. Tax cuts here
borrows on his own, he
have a similar effect. If
will face r1 , but if his "borrowing" comes in the form
(part) Up by
of tax cuts, what rate will he face?
r1 − r p
∆t 1
baskets (in discounted present value).
1 + rp
72
68
64
60
56
52
48
44
40
36
32
28
24
20
16
12
8
4
0
7
4.2
9F.19
Ricardian Equivalence Suppose that the IOU
that government issues/buys trades at the rate of r1 ,
issues/buys trades at r p (> r1 ).
whereas the one
If t 1 goes down by ∆t 1 , how much should ∆t 2 be
to keep to the government budget constraint? How
have to save in pemany additional baskets does
riod 1 to offset the said ∆t 2 in period 2?
Recall
that if the rates were the same, he should increase his
savings by exactly −∆t 1 in period 1, which pays for
need to save
∆t 2 = −(1 + r)∆t 1 in period 2. Does
that many additional baskets here?
c2
9F.18
8 In reality, IOU’s issued by the government are usually considered to be risk-
free, as opposed to IOU’s issued by you and
s and alike. Consequently, we
usually find ourselves in item 9F.18 than in this one.
26
Chapter 11A
11 Two-Period Model
with Endogenous Output
11A
11A.1
11A.6
Intertemporal Leisure-Consumption Choic Fill in
’s budget constraint in figures 34 and 35 on page 28
Plug the opto confirm your answer to item 11A.5.
timal values you found in item 11A.5 in the variables
other than the pair in question to derive the budget
constraint. For instance, for figure 34(a) replace l2 , C2
with the optimal value and leave l1 , C1 free. Then
sketch the budget constraint. Whatever you do, the line
got to be tangent to the indifference curve at the optimal bundle. If not, then you failed to meet one of the
So
tangency conditions, or the budget constraint.
that you can visualize tangency conditions in this chapter. I can’t draw a four-dimensional object, but I (and
you) can still draw it by reducing the dimension to two,
the familiar11 one. Figures 34 and 35 are the slice of the
four dimensional object cast onto the two-dimensional
coordinates.
11A.7
Intertemporal Leisure-Consumption Choice It
is understandable that r is related to
’s savings dep
cision S1 , and consequently, to his choice of C1 . On
the other hand, r doesn’t seem to be related to his labor supply decision. It does alter his N1s nonetheless.
In class, we went over how increasing r causes income
and substitution effects on l1 (and in turn N1s ). Outline how decreasing r sets off income and substitution
effects on l1 and N1s .
11A.8
Intertemporal Leisure-Consumption Choice If
’s
labor supply function turns out to be downward sloping on the N1s − 41 coordinates, which one of the three
assumptions is different from the one we made in class?
11A.9
Intertemporal Leisure-Consumption Choice When
I went over the third assumption in class, I assumed
that
is a creditor. What if he is a debtor? Do we
still have to assume that the income effect is smaller
than the substitution effect in magnitude to have the
labor supply curve shift to the right with r ? Explain.
Short
No.
Just checking to see if you remember
what happened in chapter 9 when we ran comparative
statics on r .
Utility Maximization Problem
Intertemporal Leisure-Consumption Choice In class,
we calculated the opportunity cost of
(a) C1 in terms of C2
(b) lτ in terms of Cτ (τ = 1, 2)
(c) l1 in terms of l2
as in table 6. Fill the variable in the blank cells and
opportunity cost of
C1
in terms of
C1
C2
l1
1+r [
2 /
NA
1 ]
C2
l1
41 [
l2
1 / hr1 ]
42 [
NA
2 /hr2 ]
NA
41 (1 + r)
[hr2 /hr1 ]
42
l2
NA
Table 6.
give an interpretation of each.9 In particular, explain
why the opportunity cost of l1 in terms of l2 increases
with 41 and r but decreases with 42 .
Just we did in
class, write 2 in terms of x . The coefficient of x is then
the opportunity cost of x measured in terms of 2. This
alone would be
worthy. An interpretation makes
this question
.
11A.2
Intertemporal Leisure-Consumption Choice Supis allotted with h = 30 units of time in pepose that
riod 1 and 2 each. The wage rate is given by 41 = 42 = 4
baskets in each period per unit of time in each period. His non-wage real disposable income is π1 − T1 =
π2 − T2 = 8 baskets in each period. The interest rate is
p
r = 1 [ 2 / 1 ]. Denoting his savings by S1 ,10 write
his budget constraint for
(a) period 1
(b) period 2
(c) the intertemporal one.
11A.3
PA from the cockpit: Items 11A.10 and 11A.11 do not require anything special. Just solve them in the same way you
solved item 11A.3 through item 11A.6 (or any UMP in chapter 11 for that matter), except that I’m leaving 41 variable
in order to derive the labor supply function N1s (41 , 42 , r),
which is a function of 41 . If you fix 41 as in item 11A.3
through item 11A.6, you won’t be able to sketch the labor
supply function in item 11A.12.
Intertemporal Leisure-Consumption Choice In
item 11A.2,
’s preferences are represented by
u (C1 , l1 , C2 , l2 ) ··= log(l1 +2)+log C1 +
1
1
log(l2 +2)+ log C2 .
2
2
Write his utility maximization problem.
11A.4
Intertemporal Leisure-Consumption Choice Derive
’s tangency conditions for item 11A.3 and interpret them.
11A.5
Intertemporal Leisure-Consumption Choice Find
p∗
’s C1∗ , l1∗ , C2∗ , l2∗ and S1 for item 11A.3.
(part)
∗
∗
∗
∗
C1 , l1 , C2 , l2 = (68, 15, 68, 15).
11A.10
Intertemporal Leisure-Consumption Choice Suppose that
has h = 60 in each period to spend on
work and leisure. The real wage in period 2 is 42 = 2
[ 2 / hr2 ]. The non-wage real disposable income is
11 We can actually draw up to three dimensions reasonably well on the twodimensional surface (V1 ® is only available in 2D. At least for now), in which case,
an indifference “curve” becomes an indifference “surface”. I can barely sketch
the indifference curve, much less the surface... My software can though. See
figure 36 on page 29. The flat plane is the budget “plane” rather than “line” and
the curvy one is the indifference “surface” rather than “curve”. This is what I
picture in mind when I solve UMP. Don’t worry. I won’t ask you to sketch this
figure (because I can’t hand draw it either).
9
τ is the number of baskets in period τ.
10 Once again, S p is neither > 0 or < 0. Why is it again? Cf. items 5A.2, 9E.5
2
and 9B.21 on pages 11, 21 and 24.
27
Chapter 11B
128
119
Indifference Curves
791
34
10
17
8
0
0
5
10
15
20
25
11
10
10.5
51
C2
C1
11
1
68
579
10
85
Indifference Curves
10.
102
204
187
170
153
136
119
102
85
68
51
34
17
0
10.5
10
0
30 32
17
34
51
68
85
102
C1
l1
(a) Indifference curves given (l1 , l2 ) = (l1∗ , l2∗ ).
(a) Indifference curves given (C2 , l2 ) = (C2∗ , l2∗ ).
45
Indifference Curves
40
30
68
25
10.5
11
l2
11
51
791
85
5
10.
35
10
102
Indifference Curves
10
128
119
C2
791
20
791
15
34
10
17
8
0
5
10
15
20
25
0
30 32
2.5
5
7.5
10
10 12.5 15 17.5 20 22.5
l1
l2
(b) Indifference curves given (C1 , C2 ) = (C1∗ , C2∗ ).
(b) Indifference curves given (C1 , l1 ) = (C1∗ , l1∗ ).
Figure 35.
Figure 34.
(π1 − T1 , π2 − T2 ) = (60, −120). His preferences are
represented by
u(C1 , l1 , C2 , l2 ) = log C1 + log l1 +
791
5
0
0
10.
5
10
10
1
1
log C2 + log l2 .
2
2
11A.13
Intertemporal Leisure-Consumption Choice Judging by item 11A.12, does
satisfy the first assumption we made in class? Explain. Short
yes.
11A.14
intertemporal Leisure-Consumption Choice In
class we wrote the intertemporal budget constraint in
terms of 1 (this is called discounted present value).
Write the intertemporal budget constraint in terms of
2 (this one doesn’t have a name but I personally call
this inflated future value). Does expressing the budget
constraint in inflated future value change the optimal
bundle? Explain.
Write his utility maximization problem.
11A.11
Intertemporal Leisure-Consumption Choice Find
p
his optimal bundle, N1s , and S1 in item 11A.10.
C1∗ =
A ∗
A
(1
+
r)A
(1
+
r)A
,l =
, C∗ =
, l2∗ =
, where A ··=
3 1
341 2
6
12
20
p∗
60(41 + 1). N1s (41 ) = 40 −
and S1 = 20(41 − 1).
41
11A.12
Intertemporal Leisure-Consumption Choice Sketch
’s labor supply function in item 11A.11 on the N1s −
41 coordinates.
See figure 37 on page 29.
11B
28
Profit Maximization Problem
11B.1
Flow and Stock How can we tell flow and stock
variables apart?
11B.2
Flow and Stock Which ones are flow variables
Chapter 11B
• Property value
• Appreciation
• Rent
• Depreciation
• Mortgage payment
• GDP
• Tax
• Salary
• Wealth
• Capital stock12
• Gross investment
• Net investment
• Currency issued
• Currency in circulation
Figure 36. Indifference curve and budget constraint given l2 = l2∗ . The
tangency condition translates to the indifference curve (now surface)
being just back to back with the budget line (now plane) at the optimal
bundle in this case. Figure 34(a) is this figure sliced along C2 = 68. As
you can see, the cut surface satisfies tangency condition.
11B.3
Two-Period PMP Try item 9E.9 on page 24 if you
This should be clearer by now.
skipped it before.
11B.4
Two-Period PMP Repeat item 11A.14 but for the
firms.
If f (x) takes the highest value at x ∗ , so does
(1 + r) f (x).
11B.5
Law of Motion Was K1 13 a choice variable in
Where does K1 come
section 5? If not, why not?
from, or whose decision is it to set K1 ?
11B.6
Law of Motion Suppose that K1 = 10 and I1 = 2.
Explain why K2 won’t simply be = 12.
11B.7
Law of Motion Can d be > 1? If not, why not?
11B.8
Law of Motion Can d be < 0? If not, why not?
11B.9
Law of Motion Can net investment be equal to gross
investment? If so, under what condition? If not, why
Deep down, this is essentially the same as
not?
item 11B.6.
2
w1
1.5
1
12 Well, obviously...
13 It was simply called K back then because there was only one period and there
was no need to specify the period.
0.5
0
5
10
15
20
25
30
Ns1
Figure 37.
and which ones are stock variables below?
• Account balance
• Deposit and withdrawal
• Debt
• Deficit
• Pumping the gas
• Reading on the fuel gage
• Asset
• Net worth
• Tip
• Tip jar
• Copper production
• Copper deposits
29
Chapter 11B
Le maquereau
oven that he inherited from Cookie Typhon, his late father,
started to wonder how this oven came about.
Write him the law of motion that his predecessor folYou will need K0 somewhere in
lowed to define K1 .
your answer.14
Is GDP Gross?
Ever wondered why we have gross domestic product
rather than just domestic product? Well, there is
nothing disgusting about domestic product. We just
call it so as a reminder that GDP has a secret twin
called net domestic product (NDP). NDP is defined
as GDP minus capital depreciation. We hear a lot
about GDP but barely about NDP. What’s with our
overwhelming preference for GDP over NDP?
Take an airplane (part of I , investment) for example. Unfortunately, your airplane does not last forever. Since you’ll have to replenish your investment
at some point, GDP preempts additional investment
required in the future in the form of capital depreciation d . If you pay for the whole bill upfront, I
(and consequently K ) will go up by the ticket amount
(only) in 2026. However, planes usually last longer
than a year. You’ll keep operating your plane for
years to come. It’s a bit misleading to see K spiking
up only in 2026, when in fact you’ll be still flying the
same airplane, which generates revenue, not only in
2026 but beyond. Instead of dumping all the payment in 2026, we spread the cost over the period
of utilization to keep the one-time payment more
in tune with the multi-years of usage. This practice
is called capital depreciation, which GDP takes into
account but NDP doesn’t. Consequently, a ride on
NDP is bumpier than how our economy drives in effect. A passenger on an NDP ride would probably
barf for too much gyration (now that is gross).
This is why you almost exclusively hear about GDP
but barely about NDP.
If we barely use NDP, why can’t we just call GDP
DP and add ’net’ in front only for (rare) occasions
when we have to refer specifically to NDP? That’s
an approximately 2.2 trillion Canadian dollar question
that I can’t answer. Calling DP GDP is as if calling
a pomegranate a red pomegranate. ’Red’ here is
redundant because I’ve never seen a pomegranate
coming in any different colors.
11B.10
11B.11
11B.12
11B.13
11B.14
11B.15
Dynamic PMP While we are free to select both
N1d and N2d to maximize our profit, we can only do so
with K2 but not with K1 . Why not?
11B.16
Dynamic PMP Suppose that the wage rate is 1
[ 2 /hr2 ] whereas an additional hour of work brings
in 4 more baskets at the ongoing level of N2d . This
∂ π2 (K2 , N2d )
gives the firm the profit margin of
= 4−1
∂ N2d
[ 2 /hr2 ]. While the firm earns the positive profit margin, the ongoing N2d is not a profit maximizer. Why
not?
This is nothing newly introduced in this chapter. I just wanted to check in to see if you remember
how to find the profit maximizer from section F.
11B.17
Dynamic PMP In our model, can
learn? That
is, would his work experience in period 1 make him a
more efficient worker in period 2? If not, what do you
think you would need to model his accumulation of
skill or knowledge over time?
On the contrary, can capital behave like labor? For
instance, can we remove intertemporal dependence by
setting d = 0 for example?
The marginal conditions
for labor are chronologically separable whereas the one
for capital is intertemporal. Understand why by asking
yourself “what if not?”
Law of Motion What is the point of making gross
investment if it takes away the equal amount of output
to make one? Explain the trade off associated with I1
and I2 each.
Two-Period PMP We assumed (and will assume
for the remainder of the semester except in this question) that one basket of output makes one basket of
gross investment in class. What if it takes two baskets
of output to make one basket of gross investment? Update
’s profit in period 1.
Law of Motion We saw that I2 < 0 in class (i.e.,
disinvestment). Can I1 be < 0 as well? If so, how would
you interpret it?
It can, but be sure to understand
what that means.
sets I2 = 0 (the maximum
Law of Motion If
value possible) to maximize his profit, what must have
There are two possible scenarios. They
taken place?
are theoretically possible but practically implausible (and
thus we assume I2 < 0 outside this question). Refer to
the law of motion and examine under what condition
I2 should be zero.
Law of Motion We assume that K1 is exogenous
because it was left by the previous generation from period 0, which is not the part of the periods covered.
While he was baking cookies in period 1, using the
11B.18
Dynamic PMP Derive the marginal conditions for
PMP.
You might want to spell out V N1d , N2d , K2
first before taking the partial derivatives. Use the law
of motion to remove I1 out of V (·).
11B.19
Dynamic PMP Give an economic interpretation
of the marginal condition for K2 in item 11B.18.
11B.20
Dynamic PMP Examine how the profit maximizers
respond to changes in z1 , z2 and K1 using your answers
to item 11B.18.
11B.21
Dynamic PMP What if d = 0? Re-interpret your
answer to item 11B.18.
11B.22
Dynamic PMP What if d = 1? Re-interpret your
answer to item 11B.18.
11B.23
Dynamic PMP What if r = 0? Re-interpret your
answer to item 11B.18.
11B.24
Dynamic PMP Explain why I2 = −(1 − d)K2 .
11B.25
Dynamic PMP I picked K2 as a variable of choice
to maximize the objective function V N1d , N2d , K2 . My
choice was actually arbitrary: Since the law of motion
dictates the relationship between K2 and I1 , I could
14 Strictly speaking, K should be zero just as K should be zero, i.e., Cookie
1
3
Typhon should have disinvested K1 and thrown an epic rave on December 31st to
leave nothing behind when he died. In this class, we assume that he was altruistic
and left some K1 (> 0) for
30
.
Chapter 11C
have as well selected I1 as a choice variable instead of
K2 . If I1 is determined, K2 is uniquely determined as
well, and vice versa. Rewrite your answer to item 11B.18
in terms of I1 rather than of K2 , and give an interpretation of your updated marginal condition.
11B.27
Dynamic PMP Figure 39 plots the capital and investment demand curves in item 11B.27 with varying
values of d (while leaving K1 = 4). Explain how and
1
Dynamic PMP How does I1∗ (or K2∗ if you prefer,
in the spirit of item 11B.25) respond to a change in d ?
Holding everything else constant, an increasing d
needs to be compensated by a high M PK2 (·) (or else,
the marginal revenue from I1 will cut below the interest
payment 1 + r ). To do so, what would you have to do
to I1 and what is the economic meaning of it?
r
11B.26
11B.31
0.9
K2(r; d=1/2)
0.8
K2(r; d=1/4)
0.7
I 1(r; d=1/2)
0.6
I 1(r; d=1/4)
0.5
0.4
Dynamic PMP Suppose production function is
given by
p Ç
z2 F (K2 , N2d ) = z2 K2 N2d .
0.3
0.2
Depreciation rate is d = 21 , TFP is z1 = z2 = 1, and the
∗
firm has inherited either K1 = 2 or 4. N2d = 8 for this
firm. Derive K2 (r) and I1 (r) for each K1 and sketch
For the graph, you can compute the
the functions.
values at r = 0 and .5 and connect them.
(part) See
figure 38.
0.1
0
0
1
r
4
5
6
7
Figure 39.
0.9
K2(r; K1=2)
0.8
K2(r; K1=4)
0.7
I 1(r; K1=2)
0.6
I 1(r; K1=4)
why each K2 (r; d) and I1 (r; d) respond to d the way it
does in figure 39.
Because I ran comparative statics
on K2 and z2 , but not on d in class (not because it is
banned or anything but because if you can run the one
on K2 , then the one on d should be straightforward).
0.5
11B.32
0.4
0.3
Dynamic PMP Labor demand curve is downward sloping because we hire Zedd (with high M PN d (·))
τ
alone when 4τ is high, and hire Zedd and Sensei (with
low M PN d (·)) only when 4τ is low enough. How does
τ
0.2
It should
this compare to investment demand curve?
be analogous to labor demand.
0.1
11B.33
-1
0
1
2
3
4
5
6
7
8
K2 , I 1
Dynamic PMP An increase in z2 implies that an
additional K2 will bring in more output than before.
Put differently, to raise Y2s by the same amount, it only
takes a smaller increase in K2 than before. Why then
does a firm increase I1 in response to a rising z2 ?
11B.30
Dynamic PMP What would happen to your gross
investment demand curve in item 11B.27 if you replace
z1 = 1 with 2, and why? Short
Nothing.
Competitive Equilibrium
11C.1
Competitive Equilibrium Define competitive equilibrium.
11C.2
Y1s Suppose that a firm’s production function in period 1 is given by
Dynamic PMP Explain in words why I1 (r; K1 = 4)
is located to the left of I1 (r; K1 = 2) in item 11B.27.
11B.29
Dynamic PMP Is the economic rate of return
always lower than the accounting rate of return? Why
or why not?
11C
Figure 38. The actual value of I1 doesn’t fall below 0 but I extended
the axis nonetheless to better show you that I1 (r) is a left-shifted version of K2 (r). To actually turn I1 (·) negative in this case, the interest
rate has to be the unlikely rate of whopping 50% or above.
11B.28
3
K2 , I 1
1
0
-2
2
z1 F (K1 , N1d ) = z1
p
K1 +
Ç
N1d
2
,
(10)
with z1 = 1 and K1 = 1. Derive the firm’s labor demand
function N1d (41 ) and fill it in figure 40.
You can pick
41 = 1.5 and 2 to compute N d (41 ), and connect the

‹2
z1
dots.
(part) N1d (41 ) =
K1 .
41 − z1
11C.3
31
Y1s Define output supply curve.
8
Chapter 11C
11C.4
(which one is which?) Derive N1d (41 ; z1 ) for each z1 .
Which one in figure 42(a) corresponds to which case?
Y1s Suppose that labor supply function is given
as in figure 40(a). Trace out the output supply function
You should be able to
in item 11C.2 in figure 40(c).
find the value from (10) but you can use figure 40(b) to
check your answer.
11C.5
Y1s Consider the red point in figure 40(c). Is this
combination part of the equilibria? If not, why not?
11C.6
Y1s Suppose that income effect exceeds substitution
effect in figure 40(a) so that the location of the green
and blue lines are flipped. Trace out the output supply
curve in figure 40(c) in this case.
−1
(part) N1d (41 ; z1 = 2) satisfies 41 = 2(N1d ) 2 , namely,
d
N1 (41 ; z1 = 2) = 44−2
1 .
11C.11
Y1s Green and blue lines in figure 42(a) represents
the labor supply when r = .1 or .2. Which one is which
and why?
11C.12
Y1s Outline the output supply curve in figure 42(c)
for each z1 .
the right.
11C.13
Le maquereau
Are Thunder Bayers
Lazier than Torontonians?
Toronto’s GDP was $181 billion in 2013, making
about 18.6% of Canadian GDP, whereas the population share was around 15.9%.a Thunder Bay’s GDP
was $5.7 billion, .32% of Canadian GDP, whereas the
population share was .31%. In general, per capita GDP
grows with the size of the city. This does not necessarily mean that Thunder Bayers are lazier than
Torontonians. Labor productivity grows with the
size of the city, and accordingly, the nominal wage
also grows with it, the empirical regularity called ’urban wage premium’. There are several reasons behind it (which you’ll learn in Econ 492. See me there
after you finish 302 and 304).
Then why would anyone live in Thunder Bay (or any
small city) on purpose? If a Thunder Bayer moves
to Toronto, she (yes, the same person) can produce and earn more merely because she lives in a big
city. While you do earn urban wage premium in big
cities, they usually come with a steep cost of living
(most notably, rent). In the end, a low cost of living
(among many other location-variant factors such as
congestion) compensates for reduced productivity
in Thunder Bay.
Y1s How about a change in K1 rather than z1 in
item 11C.10? Trace out the output supply curve and
interpret your result.
You can just outline the result but if you want something numerical, leave z1 at 2
and increase K1 from 4 to 16. You’ll get the same result as item 11C.12 (the interpretation will be different
though).
11C.15
Y1s How about a change in z2 instead in item 11C.10?
Does your output supply curve register any difference?
Why or why not?
Recall that labor is chronologically separable.
11C.16
11C.17
11C.18
11C.19
11C.8
Y1s Suppose that G1 went up from 1 to 2 baskets
(other than that, everything is the same as item 11C.2).
Fill in the two output labor supply curves, one each
for G1 ’s in figure 41(c) on page 34.
Locate three
points in figure 41(c) corresponding to the labor market
equilibrium in figure 41(a).
11C.9
11C.10
Y1d Define output demand curve.
Y1d Y1d (r) declines with r . Why?
reasons.
For two
Y1d If the substitution effect is smaller than the in-
come effect, would Y1d (r) be upward-sloping? Explain.
It could but not necessarily. Specify the condition
where it will be upward-sloping.
Y1s
Suppose that K1 or z1 changes in item 11C.2.
How would the output supply in item 11C.4 respond?
Y1s Interpret your result in item 11C.12.
11C.14
a Source: Statistics Canada.
11C.7
If done right, the curve should shift to
Y1d Suppose that optimal consumption and investment are given by
(
C1 (Y1 , r) = 12 Y1 − T1 − r
I1 (r) = 5 − r.
Sketch Y1d (Y1 , r) for a given r on the Y1 − Y1d coordinates.
Y1s Ricardian equivalence implies that the optimal consumption bundle does not change whether the
tax is charged early on or not. Why then does labor
supply change in item 11C.8?
Read the theorem
carefully and you’ll see that this ostensibly reasonable
question in fact does not make much sense.
Y1s We ran a comparative statics on Y1s (·) over G1 in
class. What if z1 changes instead of G1 ? In particular,
assume that the firm’s production function is given by
Ç
z1 F K1 , N1d = z1 K1 N1d ,
11C.20
Y1d Sketch Y1d (Y1 , r) in item 11C.19 on the Y1d − r
coordinates.
11C.21
Y1d In class, we ran comparative statics on Y1d (Y1 , r)
over z2 . How does the output demand curve respond
if we change G1 instead?
You can model after your
class note, or you can plug in two different G1 ’s in
item 11C.19 to compare the resulting output demand
curves.
11C.22
Equilibrium On a graph with output demand and
supply curve today, shade in the following quadrants:
There is
(a) excess supply of goods and excess supply of labor
(b) excess supply of goods and excess demand for
labor
where z1 = 2 or 4. The firm inherited K1 = 4 from
the previous generation. See figure 42(b) on page 35
32
Chapter 11D
(c) excess demand for goods and excess supply of
labor
(d) excess demand for goods and excess demand for
labor.
11C.23
Equilibrium On a graph with output demand
and supply curve today, trace out the following line
segments: There is
(a) excess supply of goods but no excess in the labor
market
(b) excess demand for goods but no excess in the labor market
(c) no excess in the goods market but excess supply
of labor
(d) no excess in the goods market but excess demand
for labor.
11C.24
11D.3
Comparative Statics on G1 The government increased
their spending by 1 basket in item 11D.5. Draw in figure 43 the new Y1d (·) with T1 = 1 (in green) and T1 = 2
(in red) while leaving r = 1. Which one of them does
the economy settle with if r stays at 1?
11D.4
Comparative Statics on G1 In item 11D.3, given the
current r = 1, how much is the output demand?
11D.5
Comparative Statics on G1 What is the ratio between ∆G1 and ∆Y1d in item 11D.3?
11D.6
Comparative Statics on G1 If r = 1 does not constitute an equilibrium in item 11D.4, in which direction
The market tries to
does the goods market move it?
clear the excess demand in this case.
11D.7
Comparative Statics on G1 How does the equilibrium wage 41 respond to the change in r you predicted
in item 11D.6?
11D.8
Comparative Statics on G1 How does the output
supply function respond to the change in the labor
market in item 11D.7?
11D.9
Comparative Statics on G1 Combine items 11D.4
and 11D.8 to gauge the ultimate change in equilibrium
GDP, and examine how the new GDP feeds into the
labor market.
11D.10
Comparative Statics on G1 Is the ratio between ∆G1
and the change in the equilibrium GDP more than or
less than the ratio you found in item 11D.5? Explain.
Explain why ∆G1 crowds out consumption and investment demand.
11D.11
Comparative Statics on G1 Suppose that the government gave a tax cut instead to lift the economy out
of recession in item 11D.3. Would there be any difference in equilibrium GDP?
11D.12
Comparative Statics on z1 Suppose that z1 went up.
How does I1 respond to the change?
11D.13
Comparative Statics on z1 Suppose that ∂ C1 /∂ 41 >
0. How does increased z1 affect the equilibrium GDP,
41 and r ?
Equilibrium On a graph with output demand and
supply curve today, trace out the following point: There
is
(a) no excess in either market.
11C.25
Equilibrium On the last we graph used today, r =
r H (green line) puts the labor market in equilibrium
but the goods market won’t be in equilibrium at this
rate. Explain why.
Anywhere along the output supply curve, the labor market is in equilibrium via the
marginal condition with respect to N1d . But that condition is independent of the marginal condition with
respect to K2 or the tangency condition. Being on the
output supply curve is a necessary condition for an
equilibrium, but that is not sufficient.
11C.26
Equilibrium If your answer to item 11C.15 on page 32
is no, does that mean that a change in z2 has no bearing on equilibrium? Explain.
It does, but through a
different channel than Y1s .
11D
11D.1
11D.2
Comparative Statics
Comparative Statics on G1 I assumed that the
government would finance its spending increase ∆G1
contemporaneously in class, i.e., ∆T1 = ∆G1 . What
if they finance it in the next period? Namely, ∆T1 =
0 and ∆T2 = (1 + r)∆G2 instead. Does the effect on
equilibrium GDP change? Explain.
Comparative Statics on G1 Suppose that
mal consumption is given by
C1 (Y1 , T1 , r) =
’s opti-
1
1
(Y1 − T1 ) − r
2
4
and the gross investment demand is
I1 (r) = 2 −
1
r.
4
Suppose that r = 1 is the current equilibrium rate. Plot
Y1d (Y1 , T1 , r) against Y1 when r = 1, G1 = 1 and T1 = 1.
How much is Y1d in equilibrium?
See figure 43.
33
Chapter 11D
4
4
MPN (K1, N 1)
MPN (K1, N 1)
3.5
1
w 1, MP N (K 1, N 1)
Ns1(r=.1)
3
Ns1(r=.2)
2
1.5
1
1
Ns1(r=.1, G1=1)
3
Ns1(r=.2, G1=1)
2.5
1
2.5
1
w 1, MP N (K 1, N 1)
3.5
Ns1(r=.1, G1=2)
2
1.5
1
0.5
0.5
0
0
0
1
2
3
4
0
5
1
2
(a)
4
5
3
4
5
(a)
10
10
9
9
8
8
7
7
6
6
Ys1
Ys1
3
Ns1, N d1
Ns1, N d1
5
5
4
4
3
3
2
2
1
1
0
0
0
1
2
3
4
5
0
1
2
Nd1
Nd1
(b)
(b)
0.2
0.2
r
0.3
r
0.3
0.1
0.1
0
0
0
1
2
3
4
5
6
7
8
9
10
Ys1
0
1
2
3
4
5
Ys1(r)
(c)
(c)
Figure 40.
Figure 41.
34
6
7
8
9
10
Chapter 11D
MPN (K1, N 1, z 1=?)
1
1
Ns1(r=?)
Ns1(r=?)
1
w 1, MP N (K 1, N 1)
MPN (K1, N 1, z 1=?)
2.8284
2
1.1547
0
0
1
2
3
4
5
Ns1, N d1
(a)
7
Y s1=(one of z1)(K1Nd1 ).5
Y s1=(the other z1)(K1Nd1 ).5
Y d1(Y1, r=1, T1=1, G1=1)
6
Ys1
Yd1(Y1, r)
12
8*sqrt(2)
8
4*sqrt(3)
4
Y d1(Y1, r=1, T1=1, G1=2)
5
Y d1(Y1, r=1, T1=2, G1=2)
4
Y d1=Y1
3
2
1
0
0
1
2
3
4
5
Nd1
0
0
1
2
3
(b)
Figure 43.
0.3
r
0.2
0.1
0
0
4
4*
sq 8
rt(
3)
4
Y1
8* 12
sq
rt(
2)
Ys1(r)
(c)
Figure 42.
35
5
6
7
Chapter 11D
Le maquereau
change and TFP
Diverging Diamond Inter-
As a former Missourian, I’m proud to say that Missouri is the first state in North America to introduce a diverging diamond interchange (DDI). See
figure 44.
DDI can process more traffic (there is no green
left turn arrow) with less traffic accidents. Accordingly, the economy with DDI’s yields more than the
one with alternative crossings, holding everything
else constant. DDI is part of G that the MODOT
purchases from the general contractors, who produce and maintain it according to their production
function Y = zF (K, N d ). Where does this productivity increase fall into? K ? – No. They still use the
same amount of capital (or probably less because it is
usually more space-saving than alternatives). N d ? –
Still no. They still use the same amount of labor (or,
once again, probably less for the same reason). The
only option left is then z , the total factor productivity. The same capital, the same workers but DDI
can squeeze more G out of them.
We tend to focus on the welfare effect of z
through C but TFP improves the utility level through
G just as much. With a DDI, we do not need as much
capital and labor to produce the same G . We can
then spare labor thus saved for production of C or
increase l (or both), through which we can increase
our utility level. The effect would be even more
pronounced if we factor G directly into the utility
function (which we didn’t do for simplicity).
St. Louis has lots to offer (and I left my heart
there) but if you got a chance to visit (and if you
are an urban economics nerd as I am), rent a car
and drive along Dorsett Rd. It might feel a bit
weired because you’ll see your oncoming traffic to
your right while you cross I-270. Don’t freak out.
Everything is clearly marked. Simply follow the signs
and enjoy your ride. Also found along the I-270:
• You might as well travel a bit farther to the
west and continue onto MO 94, an unexpectedly idyllic highway that I think is the most scenic
in Missouri.
• Highway 67, which runs parallel to I-270, is one
of the few highways that drive underneath a
runway. It was here first and then Lambert St.
Louis International expanded on top of it. Do
not poke the tunnel’s ceiling. There’s an active runway with planes taking off and landing
right above your head.
• The I-70, from which the I-270 spins off,
carries the highest exit suffix, the exit 2X
in Kansas City, MO. Exits are numbered by
the mile, and if there is a multiple of them
within the same mileage, an alphabetical suffix
is added for their distinction. In Kansas City,
the exit number 2 is shared by 2A through 2X.
Spaghetti much?
Figure 44. I-270 at Dorsett Rd (Image retrieved from Google Maps).
You are welcome, Missouri, for the unsolicited plug.
Can I get my money now?
This is all me haz. Work hard
and good luck with the final!
Sesame Street ®
36