Econ 100B
Professor: Alonso Villacorta
Problem Set 2
Please upload your answers on Canvas.
Exercise 1. Production function model
Consider an economy “I” with a representative household that consists of 1000 workers and
) = 100). There is a representative firm with a Cobbowns $100 million of capital ("# = 1000, (
Douglas production function that rents capital and hires labor to produce. Assume that the TFP
parameter equals one (A=1), we have * = ( +/- "./- . Markets are competitive.
1. Define an equilibrium in this economy. Follow class notes.
2. Solve for the equilibrium. You should get numbers for (Y,K,L,r,w).
3. Graph the following:
o Graph 1: Plot output per capita (Y-axis) against capital per capita (X-axis). And
show with an “x” the point that characterizes the equilibrium.
§ In this plot output per-capita (y) and capital p.c. (k) are your variables,
while all other are constant and equal to their assumed values.
o Graph 2: Plot wages (Y-axis) against capital per capita (X-axis). And show with an
“x” the point that characterizes the equilibrium.
§ In this plot wages (w) and capital p.c. (k) are your variables.
Edel
Consider another economy “II” with labor equal L=500 and capital equal to K=20. Assume that
this economy also has a TFP parameter that equals one (A=1). First, assume each of these
economies are in autarky, so capital cannot flow across countries.
4. Show in your previous graphs--with a circle--the equilibrium of economy II.
some
steps
as
Assume now that these economies open boundaries. So, capital can freely move across them.
Describe the new equilibrium. What are the new wages and rental rate of capital in each
country? (Assume that there are no transaction costs of capital, so rental rates of capital should
be the same; otherwise there would be an arbitrage opportunity.)
5. In your previous two graphs, show with a square the new equilibrium with open
boundaries.
Exercise 2. The empirical fit of the production model
The table below reports per capita GDP and capital per person in the year 2014 for 10
countries. Your task is to fill in the missing columns of the table.
a) Given the values in column 1 and 2, fill in columns 3 and 4. That is, compute per capita
GDP and capital per person relative to the U.S. values.
1
Define
equilibrium
From
lecture
equilibrium
notes
defined
is
An allocation of
Prices
r w
Such
that
Firms
s
and
K K
LIE
HH
by
LK Y
and price of consumption
markets
clear
100
Met
tooo
clearing conditions
Firm's optimization prob
again
step
zed's
wi
d
1,11 4342 r
tea
1
maximise their profit bychoosing L K
Labor 14 7000 andcapital K 1007
supply their
2
step I
that
we understand
r
K Ld
IiiI
ri
r
off
zit
z jh
k
w
o
Step
Household
hhs
Ls
step
labor andcapital
100
1000
Demand supply
III'd
We have
4
3
Ks L
their entire
supply
I
I
KS
problem
IIÉÉI
r
W
II
ICE
restudy
using eq
w
Solvefor
3
Et
w
s
34
o
310
K
Y
r w
1000 100 110013
g
110094000s
and
e
F
CK
K
5
ooo s
3
1053
3
production
TIL
Function
K's L
Y
KII
y
equilibrium
K's c
K
Y
ill
values
me
k
K'c
and
KIL
EEE
É
When K
E
I
3
4 10.1
0.1
graphy
In port
z
k
W
equilibrium of W and
W
3
Et
W
KIL
36 1
3
k
b
f
K
s
I
k
s
4
Some
steps
as
2
Solve for equilibrium
KF Ki
LEE
É
20
soo
C
FEC
Fbi
Solvefor
using eq
w
3
w
100
s
318
Et
310
economy
K
Es
5
20
11
Equilibrium in this Cese
Allocation
Prices
Li
ta
r Vin W
ki
Ka Ti Ti
Wii
Maximise profit
by choosing
Labor L
Poe
and
i
1000,41 500 and Capital
rich
Kit Kirk
21 IT
1000
Lin L 1 500
K
lil
K
let
1
K
100741
K
K
20
K
100 20
K
120
rn
r
s
ICE
s
II
s
1000
L L
Li Tilt 500
FI 55
K tK2
I 120
KH 12013
K
40
yep
go
80 L
K
1000
O 8 3
Yt
Wages
K
40
80
5
economy
kink
2
rental rent
economy
KY
y'T
I
Eo
40
14
500
k
1
100 Ki
80
É
x
a
6
I
FÉ
gp
Yan
tdm
K
A unknown
C
d
I
colon
Some
populace
y
4
Dg
5
countries
to college
If
K s
Colum 6
can
Support sending
the
ire us
b) In column 5, use the production model (with a capital exponent of 1/3) to compute
predicted per capita GDP for each country relative to the United States, assuming there
are no TFP differences.
c) In column 6, compute the level of TFP for each country that is needed to match up the
model and the data.
d) Why do you think that A is close to one for some countries, but not for others?
I
eking
a
United States
Canada
France
South Korea
Argentina
Kenya
g
Y
In 2014 dollars
(1)
(2)
Capital per Per capita
person
GDP
141,841
51,958
128,667
43,376
162,207
37,360
120,472
34,961
53,821
20,074
4,686
2,971
Relative to the U.S. values (U.S. = 1)
(3)
(4)
(5)
(6)
Capital per Per capita predicted
Implied TFP to
2
0
person
GDP
/
match data 1
1
1
1
1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
e ?
K'Y
E
E
Exercise 3. Human capital
Consider the following variant of the production function that allows for human capital
= 3( +/- (4")./- .
B e * 213
Kus the number
us of workers in the country while 4 represents the
In this equation, " represents
average years of education. Assume that the average years of education of the U.S. is five times
larger than the years of education in Kenya. Using the data from the previous table, calculate
Keen's
again the predicted
Kenvalues of GDP per capita /0 and the TFP needed to fit the data.
Jus
events
Y
United States
Kenya
•
•
Average years
of education
6
1
0.2
Predicted
0
/
1
?
Implied TFP
to match
data
1
?
Is the implied TFP bigger or smaller than in the previous exercise? Why?
What is the wage of U.S. workers relative to the wage in Kenya?
o Hint: To answer this question assume that firms at each country maximize the following
2/3
profits: max:,; Π= = 3= (1/3 (4? ") − B? " − C? (, where ? ∈ {FG, (4HIJ}
I02
Exercise 4. Case Study. Read the following case studies and think about them. (No need to
deliver any answer)
CASE STUDY: LABOR’S SHARE OF OUTPUT ACROSS TIME AND ACROSS
COUNTRIES
We’re going to rely heavily on the Cobb-Douglas equation; in fact, we’re going to treat it as a
basic model of a national economy. If it’s going to be so central, it would be nice to have some
evidence that such a simple equation actually can sum up something as complex as an entire
national economy. So, is there a simple way to check and see if this equation actually makes
some good predictions? Yes, there is. As showed in the lectures, the Cobb-Douglas model,
combined with competitive markets, has a clear prediction about how much of a nation’s income
goes to the workers and how much goes to the firms. It’s surprisingly simple, actually. Recall the
function
* = 3( +/- "./- .
Under competitive markets, Cobb-Douglas makes the following prediction: the exponent on
labor is the fraction of the nation’s income going to workers. That means that in every country in
the world, about two-thirds of the income should go to the workers, and about one-third should
go to owners of capital. In Chapter 2, we showed that in the United States, this share has been
stable for decades. But can this possibly be true around the world?
As the chart below shows, the answer is a rough yes.
Each
dot represents
A Model
of Production
| 29 one country, ranging
from the richest to the poorest. Only in the very poorest countries is there much of a difference
poorest.
Only in predicts.
the very poorest countries is there much of
farm owners stickfrom
to thethe
agreement
they value
two-thirds
our model
a difference from the two-thirds value our model predicts.
1.0
0.9
0.8
0.7
Fraction of GDP
l store? Because they are trapped in a prisconcept many students will have seen in
n introductory political science class, if
over such a topic). Each farm owner hopes
m owners are “honorable” enough to stick
but whether the other farm owners stick to
not, it’s in each farm owner’s self-interest
hers. In competitive markets, firm/farm
g a prisoner’s dilemma against each other.
ll often return to the competitive markets
worth keeping this in mind as we start off.
0.6
0.5
0.4
0.3
0.2
0.1
the farm owners aren’t making any profit?
ey’re not making any profit on their tenth
0.0
0
4,000
8,000
12,000
16,000
20,000
mer is just indifferent between hiring and
Real per capita GDP
rker. But they’re making profit—or more
n on their capital equipment—on each of
Estimates of labor share are derived using an adjustment to
Estimates
of actually
labor shareaccount
are derived
using an adjustment to account for income of self-employed
rkers. How much of
a profit? It’s
for income of self-employed persons and proprietors,
and proprietors,
combined
cross-country
andseries
time-series
data.
The adjustment involves
on this graph. (Justpersons
shift the tangency
line
combined
crosscountry and timedata. The
adjustosses the origin, and
it instantly
involves
assigning
the operating surplus
of private
assigning
thebecomes
operatingment
surplus
of private
unincorporated
enterprises
to labor and capital income
ne.) Now we can see how much (accountunincorporated enterprises to labor and capital income in the
owner makes on each worker at this wage.
same proportions as other portions of GDP.1
mber of workers, the gap between the proIt turns out that the hardest thing to measure when looknd the wage bill line is the profit the farm
ing at these data from different countries is the wages of
e if they hired that many workers.
small-business owners—for the most part individual farmers, people scraping out a bare existence on their own plots
of land. It’s hard to decide how much of a small farmer’s
Y: LABOR’S SHARE OF OUTPUT
income should count as “capital income” and how much as
ME AND ACROSS COUNTRIES
“wage income.” But Gollin sweated the details for years to
in the same proportions as other portions of GDP.
1
It turns out that the hardest thing to measure when looking at these data from different countries
is the wages of small-business owners—for the most part individual farmers, people scraping out
a bare existence on their own plots of land. It’s hard to decide how much of a small farmer’s
income should count as “capital income” and how much as “wage income.” But Gollin sweated
the details for years to create this chart, and in doing so he gave good evidence that for the vast
majority of countries, Cobb-Douglas does a good job predicting how much of GDP gets paid to
workers. Our simple model passes a big test.
This is a surprising result—after all, we often hear in the news about how the power of workers
seems to rise or fall in different countries or in different decades. You might think, for example,
that western Europe, with its strong unions, would have a much higher labor share than the
capitalist-friendly United States. But that isn’t the case; all of the world’s rich countries are right
around the magical two-thirds labor share. Despite these findings, rising wage inequality remains
an important source of increasing income inequality in the United States. The functional income
distribution data does pick up this factor. (For example, see James Galbraith, Created Unequal:
The Crisis in American Pay [New York: Free Press, 1998].)
CASE STUDY: SETTLER MORTALITY AND EXTRACTIVE INSTITUTIONS
In a famous paper, Acemoglu, Johnson, and Robinson tried to find out whether institutions really
do matter. In economics, it’s often hard to separate cause and effect—do countries have good
economies because they have good governments, or is it vice versa? Or does high education
really cause both? Acemoglu, Johnson, and Robinson try to get around these kinds of puzzles by
looking at what happened to countries after 1492, when Europeans started colonizing the rest of
the world.
2
Europeans quickly found that some countries were easier to colonize than others. In some
countries—generally those near the equator—tropical diseases were so deadly that few
Europeans went there. Other places, like North America, Australia, and New Zealand, were
easier for Europeans to settle. Acemoglu, Johnson, and Robinson argue that in places where
colonizers died at high rates, Europeans set up “extractive” government institutions—gold mines
1
Raw data are taken from United Nations (1994). Data on real per capita GDP are taken from the Penn World
Tables, Version 5.6. Douglas Gollin, “Getting Income Shares Right,” Journal of Political Economy 110 (April
2002): 458–74.
2
Daron Acemoglu, Simon Johnson, and James A. Robinson, “The Colonial Origins of Comparative
Development: An Empirical Investigation,” American Economic Review 91, no. 5 (December 2001): 1369–
1401.
and slavery-intensive plantations, for example. These institutions required only a few Europeans
to stick around and endure the deadly environment. In these countries, Europeans generally
didn’t worry about creating incentives for long-term investments in education or about creating
stable property rights. They just needed enough political power to control the mines, plantations,
and other physical sources of wealth—that was all.
By contrast, in places that were less deadly to Europeans, many of them created institutions with
strong property rights, personal freedoms, and mass education. This led, they argue, to centuries
of prosperity for these countries. The combination of disease and power relations that existed
centuries ago appears to have had very real implications for living standards hundreds of years
later.