Macroeconomics: Modern Macroeconomics 1
Katsuya Takii
OSIPP

Katsuya Takii (Institute)

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Introduction

Purpose: The Course is designed to help you understand the basic
concepts and framework of modern macroeconomics. The theories
are supplemented by relevant empirical evidence.
O¢ ce hour: Room 602, 9:50-10:20, 12:15-12:45 on Monday.
Appointment is required for other time.
E-mail Address: takii@osipp.osaka-u.ac.jp

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Introduction

Grading Policy: 42% on assignments and 58% on a …nal exam.
1

2

I will give you 7 assignments. Students must hand them in at thefollowing lecture. If students turn an assignment in by the due date, I
will give them 6 points. If students turn an assignment in late, I will
give them 3 points. If students submit all assignments, you will receive
42 points. Students must write their answers with a pen. I don’tallow the typed answers for this assignment.
The full score of …nal exam is 58 points. I guarantee that 30 points
out of 58 points will come from the assignment. If you hand in all
assignments and you perfectly answer the questions appeared inassignments, you can certainly receive B.

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Introduction

Remarks:
1
2

3

I assume that students have already taken Microeconomics 1.
This course is comparable to the junior or senior undergraduate coursein the economics department.
I will mainly teach this course in Japanese. However, I will not prevent
students from asking questions in English. I can discuss your questions
and comments in Japanese or English at my o¢ ce hour.

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Introduction

Course Outline
1
2
3
4
5
6

The Data of Macroeconomics (2 lectures):
The Framework of Macroeconomics (1 lecture):
Economic Growth and Nation’s income (4 lectures):
Stabilization Policy (6 lectures):
Lucas’s Critique and Micro Foundation (1 lecture): Consumption.
Final Exam (1 lecture).

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What is Macroeconomics?

Macroeconomics is a study to explain the behavior of aggregate datasuch as GDP per capita, in‡ation rate, and unemployment rate.
1

2

3

Observing Statistics, macroeconomists examine the health of oureconomy and make policy suggestions.
For this purpose, we must infer the structure of the economy thatbrings the observable data.
Macroeconomics is the current consensus on the inferences about theeconomic structure.

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The Data of Macroeconomics

Three main statistics
1
2
3

Gross Domestic Product...the measure of richness.
Consumer Price Index...the measure of cost of living
Unemployment Rate..the measure of joblessness

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Gross Domestic Product

De…nition: Gross Domestic Product (GDP) is the gross sum of value
added of each product measured by market prices in a country during
a period.
GDP can be viewed as the total income of the whole economy.
GDP can also be viewed as the total expenditure on the economy’s
outputs of goods and services.
For the economy as a whole, expenditure must equal income. Why?

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Five main features of GDP

The use of market value: GDP evaluates the value of goods andservices by their market value since the prices of goods and services
indicate how much consumers are willing to pay for them. Then
GDP sums up the market value of goods and services in a country.
Example: Suppose that a country produces 5 apples and 10 bananas,
and the price of an apple is 100 yen and the price of a banana is 30
yen. Then
GDP = 100yen

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Five main features of GDP
The value added: GDP is the sum of the value added of eachproduct.
The value added of a …rm equals the value of the …rm’s output minus
the value of intermediate products that the …rm has purchased.
Example: A …rm purchases oranges from a farmer for 60 yen and sells
an orange juice for 100 yen per cup. Then the value added of theorange juice is 40 yen. If a farmer does not buy any intermediate
goods, then the value added of an orange is 60 yen. Therefore thevalue added of the two products equals
40 yen + 60 yen = 100 yen.
If the oranges are imported, GDP is
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Five main features of GDP

GDP vs. GNP:
GDP measures the total income in a country not by residents of thecountry.
Gross National Product (GNP) measures the total income earned byresidents of the country.
The di¤erence is factor payments (wages, pro…ts and rents) fromabroad and factor payments to abroad:
GNP

Katsuya Takii (Institute)= GDP+factor payments from abroad
factor payments to abroad.

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Five main features of GDP

Flow vs. Stock:
1
2

A ‡ow is a quantity measured per unit of time.
A stock is a quantity measured at a given time.

Example:Flow...Annual income, Saving...
Stock....Wealth, Asset

Since GDP measures the total income earned during a period, such asa year, GDP is a ‡ow variable.

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Five main features of GDP

Gross vs. Net:GDP is the gross sum of value added. It does not subtract the
depreciation of capital from the value added- the amount of capital(plants, equipment and residential structures) that wears out over a
period of time.
Net National Product:
NNP = GNP

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Depreciation

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Some details for computing GDP
Used goods: The sale of used goods does not increase the additional
value in a country. Therefore, the sale of used good is not included
in GDP.
Inventories: National Income Accounting system treats inventory as
the sale of goods to themselves during the current period.
Inventories are counted as part of GDP of the period that goods areproduced..
Inventories are not counted as part of GDP of the period that goods
are sold.
It is considered as used goods.

Because of this treatment of inventories, all goods produced arepurchased by somebody. Therefore, total income always equals total
expenditure of a country.

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Some details for computing GDP

Imputations: When some goods are not sold in a market, they do nothave market prices. If GDP includes these goods and services, wemust estimate their value. Such an estimated value is called imputed
value.
1

2

3

4

The Value of Housing: When you rent an apartment, the rent is apart of GDP. When you own a house, you do not pay the rent. GDP
estimates the rent that house owners pay to themselves.
Home Production and Durable Goods: The value of these rental
service and home production is left out of the GDP.
Government Services: The national income accounts estimate thevalue added of government services in the GDP at their cost.
Underground Economy: no imputation is made for the value ofgoods and services sold in the underground economy.

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Comparison across Time Periods
Since GDP is measured by the market prices of goods and services,
GDP increases both when prices increase and when the outputs
increase. In order to exclude the impact of in‡ation, economists
separate real GDP from nominal GDP.
Nominal GDP uses current prices to measure the value of goods andservices.
Real GDP uses a constant set of prices to measure the value of goods
and services. In order to compute real GDP, economists choose thebase year.
Example: Consider a country in which people produce only apples andbananas during 2008 and 2009. Let me choose 2008 is the base year.
RealGDP in2008
RealGDP in2009

Katsuya Takii (Institute)= PA2008= PA2008

QA2008 + PB 2008
QA2009 + PB 2008

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Comparison across Time Periods

GDP de‡ator: The ratio of nominal GDP to real GDP is called GDP
de‡ator:Nominal GDP
GDP De‡ator =Real GDPThe GDP de‡ator captures the movement of the overall price level inthe economy.

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International prices:Di¤erent countries use di¤erent currencies. In
order to compare income across countries, which prices should weuse?
An international price of the goods...The weighted average of the priceof the goods across countries by taking the country’s share of
expenditures as its weight.

Purchasing-Power Parity: Purchasing-Power Parity (PPP) is theratio of nominal GDP to real GDP measured by international prices
PPP =Katsuya Takii (Institute)

Nominal GDP
Real GDP measured by international prices

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International Comparison

GDP per capita vs. . GDP per worker:
GDP per capita=

GDP per worker=

GDP
total populationGDP.
the number of labor force

The production of goods normally made in the factory is mainly donein the household in developing countries. Since GDP cannot measure
the value of home production, GDP per capita may underestimate
well-being of developing countries. Since GDP does not value home
production, it may be reasonable to divide it by labor force.

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Components of Expenditure
GDP or Y , can be divided into consumption of domestic goods andservices, C d , investment in domestic goods and services, I d ,government purchases of domestic goods and services, G d , andexports of domestic goods and services, EX :Y = C d + I d + G d + EX
Consumption, C , investment, I , and government expenditure, G canbe divided into domestic goods or foreign goods:
C = Cd + Cf , I = Id + If , G = Gd + Gf
where superscript f means foreign goods.
GDPY

Katsuya Takii (Institute)= C + I + G + EX

Cf + If + Gf

= C + I + G + EX
= C + I + G + NX

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Consumer Price Index
In order to analyze the changes in the overall cost of living, we need asingle index measuring the overall level of prices.
Consumer Price Index: Economists compute the price of a basketof goods and services purchased by a typical consumer. CPI is theprice of this basket of goods and services relative to the price of thesame basket in some base year.
Example: Consider a country in which typical consumers buy 5 apples
and 10 bananas in a year during 2008 and 2009. Then the basket ofgoods consists of 5 apples and 10 bananas. Let us set 2008 as thebase year. The CPI can be de…ned as follows:
CPI in 2008=1
CPI in2001 =Katsuya Takii (Institute)

5
5PA2009 + 10
PA2008 + 10

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Consumer Price Index
CPI vs. the GDP De‡ator: Both CPI and GDP de‡ator measure
overall price level. But there are three main di¤erences.
1

The GDP de‡ator measures the prices of all goods and services
produced; CPI measures the prices of only the goods and services
bought by consumers.

2

The GDP de‡ator includes only goods produced in a country. It
excludes imported goods. CPI includes imports goods if theconsumers buy such goods.

3

CPI assigns …xed weights to the prices of di¤erent goods, the GDPde‡ator allows the basket of goods to change over time as thecomposition of GDP changes.

Despite these di¤erences, CPI and the GDP de‡ator show a similar
behavior.
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The Unemployment Rate
The unemployment measures the percentages of people who want to work
but who do not have jobs.
Unemployed Workers: People are called unemployed when1
2
3

They do not have a paid job.
They conducted a job seeking activity
If there is a job, they can do it soon (They are available).
Employed Workers: People are called employed if they do have apaid job.
Labor force: The sum of employed workers and unemployed workers
are called labor force.
The unemployment rate:
unemployment rate =Katsuya Takii (Institute)number of unemployed workers
labor force

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The Unemployment Rate

Labor Force Participation rate:labor force participation rate =labor force
adult population

100

where adult population is the number of people 16 years old or more.
If a person is 16 years old or more and he is neither employed not
unemployed, he is not in the labor force.

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Assignment

Students must hand assignment 1 in at the next lecture.

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The Basic Framework of Macroeconomics

This Chapter provides a basic framework of macroeconomics. This is
an application of a general equilibrium analysis in the context ofmacroeconomics.
Using this framework, I will construct the neoclassical growth model
later and ask the following questions.
1
2

Why are some countries rich; others poor?
What is the source of long run growth?

In the chapter 5, I applied this model to the analysis of stabilization
policy.

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The Basic Framework of Macroeconomics

Representative Firm
Produce goods or services using Labor and Capital
Sell goods or services to household

Representative Household
Buy goods or services
Provide …rms with labor and capitalMarkets (Labor Market, Capital Market and Goods Market.)

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Firm

Firms is assumed to maximize its pro…ts given an aggregate
production function:
Π = max fPY
K ,L

Y

WL

RK g

F (K , L)

where P is a price, Y is output, W is a nominal wage rate, L is labor,
R is a nominal rental price and K is capital stock.

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Firm
The property of the aggregate production function: F (K , L) is
constant returns to scale in K and L:
tF (K , L) = F (tK , tL) , for 8t > 0.

(1)

Why should the aggregate production function be constant returns toscale?
CRS means that whatever an individual production function is, if we
use the same production technology twice, the output will be doubled.
This might be a reasonable assumption for the aggregate production
function. For example, assume that an individual plant has a
production function y = φ (l ). Assume that a manager establishes thesame K plants. Then the aggregate output Y is
Y = yK = φ (l ) K
De…ne an aggregate production function production function F suchthat
F (K , L ) φ (l ) K , 8K > 0
where L=lK . Clearly this is constant return to scale in K and L.
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Firm

Constant returns to scale and production possibility setYyF (K , L)
K
,1 L
= F
L
F (k, 1)

where y = YL and k = KL . De…ne ff (k )F (k, 1)ThenyKatsuya Takii (Institute)f (k )

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Firm
Pro…t maximization Problem
Π = max fPY

WL

RK g

= max yL

W
L
P

R
kL P
P

K ,L

k ,L

= max πPL
L

where π = max fy
k

w

rk g

R
where w = W
P and r = P . ThereforeΠ = max πPLL

π = max fykyKatsuya Takii (Institute)

w

rk g

f (k )

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Firm

Assumptions on the aggregate production function
12

f (0) = 0
df (k )
f 0 (k ) = dk > 0.
When the …rm employs more capital per workers, it increases output
per workers.

3f 00 (k ) =d 2 f (k )<0
dk 2This means that the marginal productivity of capital per workers isdiminishing.
4

Inada Conditions: technical conditions.
lim f 0 (k ) = ∞, lim f 0 (k ) = 0.

k !0

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Firm

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Firm

Example: Cobb-Douglas Production Function: f (k ) = k α
f 0 (k ) = αk α 1 > 0 ) α > 0

f 00 (k ) = α (α1) k α 2 < 0 ) α < 1Hence
α 2 (0, 1)

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Firm

Pro…t Maximization with respect to k
π = max fykw

π = max ff (k )krk g , s.t. yw

f (k )

rk g

First Order Conditionsdπ= 0 ) r = f 0 (k ) ) k is determined.
dkπ
π = f (k )Katsuya Takii (Institute)wrk ) π is determined.

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Firm

Optimal Decision
f’(k)

r

k

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Firm

Pro…t maximization with respect to L
Π = max πPL
L

Labor Demand Function
L = 0 if π < 0, w > f (k )rk

L 2 [0, ∞] if π = 0, w = f (k )L = ∞ if π > 0, w < f (k )Katsuya Takii (Institute)

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rkrk

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Firm

Labor Demand Functionw

f(k)-rk

L

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Firm
0 Economic Pro…ts
Π = πPL = 0
When the market is competitive, more entrepreneurs will enter as longas economic pro…ts are positive. Hence, in the long run, economic
pro…t is 0.HenceΠ = 0, L > 0 ) π = 0 ) w = f (k )Y

rk= wL + rK

Accounting Pro…ts: The …rm’s revenue must be divided into wage
payment, capital payment and economic pro…t:
Y = wL + rK + Π
where Π is economic pro…t. But in reality, a …rm’s owner owns
capital also. Hence, we cannot distinguish economic pro…ts from
capital payment. It means
Accounting pro…t = Π + rK
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Firm
Example: Cobb-Douglas Production Function: f (k ) = k α= αk α 1
rk = αk α
rk
rK
rk
α =
=
=
α
k
y
Y
r

= k α rk
= k α αk α
= (1 α ) k α
w
w
wL
=
α =
=
α
k
y
Y

w

1

1=
Katsuya Takii (Institute)rK + wLY

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Household

Representative Households make decisions on
1
2
3

How long they work,
How much they consume today, andWhere to invest.

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Household
How long do they work?: Assume that everybody works one unit oftime no matter what wage rate is. Hence, the supply of labor isequal to total population.
How much do they consume today?
If they do not consume today, they save for future. This is potentiallya di¢ cult question, because the decision depends not only on thecurrent income but on the expected future income. I will leave theanswer to this problem later and at this moment, I take theconsumption per capita, c as given.
Budget Constraint: Nonetheless, chosen consumption has to befeasible. Hence, at least it must satisfy the following budget
constraint.
a+1 + c = (1 + ρ a ) a + w
where c is consumption per capita, ρa is the returns from investment, a
is asset per capita and a+1 is an asset per capita at the next period.
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Household
Where do they invest?:There are several possibilities. They may saveit in a bank. They can invest in …rms’stock. They can alsopurchase investment goods such as house and rent it out.
1

2

So far we assume that a …rm rent capital. So the …rm does not own
capital. Moreover, economic pro…ts of the …rms are 0. Hence, there isno value on a …rm.
It means that household has two choices:1
2

3

to save it in a bank. Then they can earn a safe return, interest rate, ρ.
to purchase investment goods and rent it out. Then it expects to earnr from a unit of investment. In addition, as they are the owner ofcapital, if the capital depreciates, the must bear the cost. Supposethat δ proportion of capital is depreciated. Then real return frompurchasing investment goods is r δ.

Assume that there are lots of investment opportunities so thathousehold hedges idiosyncratic risk. Moreover, we assume that there isno adjustment cost of investment. Then the optimal condition isρa = max fr

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Household

Arbitrage Condition:
1

2

3

If r δ > ρ, everybody buys investment goods. But then, nobody
saves in a bank and the interest rate would become larger.
If r δ < ρ, nobody buys investment goods. But then, everybody
saves in a bank and the interest rate becomes smaller.
In the equilibrium,r δ = ρ = ρaThis is called an arbitrage condition.

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Market Clearing Condition
Labor Market Clearing Condition
L=N
Capital Market Clearing Condition
K = aN
Note that even if banks collect assets from household, the banks must
purchase investment goods. Hence, every assets in the economy isused for investment in capital goods.

Hence capital market clearing condition and labor market clearing
condition implies
K
k=
=a
N
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Equilibrium
Given (c, a), a market equilibrium consists of (y , k, a+1 , ρ, r , w ) which
satis…es1

A Firm’s Pro…t Maximization and the Production Function= f (k )r = f 0 (k )w = f (k ) rky2

A Consumer’s Budget Constraint
a+1 + c = (1 + ρ a ) a + w

3

An Arbitrage Condition
r

4

δ=ρ

Capital and Labor market clearing conditions
k=a
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Goods Market

What happens to a goods market? A goods market is supposed
to equate demand for output and supply of output and determine theRprice of the goods, P. However, note that w = W
P and r = P .
Hence, the wage rate and rental price are not measured by nominal
term, but measured by the unit of output. Even if W , R and P
double, w and r keep the same value. Hence, there is no change in
our economy. In order to make our decisions, we care about therelative prices. We do not need information on the absolute prices.
Hence, the price of a product can be set 1 without a loss ofgenerality. In other words, the market that is supposed to determine
the price is redundant.

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Goods Market
In order to prove the above statement, we derive goods market fromequilibrium conditions. First, we derive a resource constraint.
Second, we derive the ‡ow expression of capital market. Combining
two equations, we derive goods market.
Budget clearing condition implies thata+1 + c=k+1 + c ==yt =(1 + ρa ) a + w(1 + r δ ) k + f (k )f (k ) + (1 δ ) k[ k+1 (1 δ ) k ] + c

rk

De…ne investment, i as
i

k+1

(1

δ) k

Hence,
yt = it + ct ) Yt = It + Ct
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Assignment

Students must hand assignment 2 in at the next lecture.

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Economic Growth and Nations’Income

1

Based on the previous framework, I …rst derive so called, Solow
Model. This model is a useful starting point for the analysis of
economic growth.

2

Solow model predicts that eventually economic growth converges to
0. This is not what we observe in data. In order to match the theory
with data, we introduce an exogenous technology growth andpopulation growth, and compare with the stylized facts about
economic growth observed in many OECD countries.

3

Next, using the extended Solow model, we ask a question: can themodel explain large di¤erences in income across countries?

4

From the above exercises, we recognize the importance of
productivity. Understanding productivity is the issue we discuss later.

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Solow Model
Capital accumulation equation: Assume that xt means x at date t.
Capital accumulation equation is
kt +1 = it + (1δ) kt

= yt ct + (1 δ) kt= f (kt ) ct + (1 δ) kt
Investment makes capital stock bigger, but in order to make large
investment, households cannot enjoy consumption today very much.
This is a basic trade-o¤ in Solow model.
Assumption on ct :ct = (1 s ) yt
Because st = yt

ct is the de…nition of saving,st = syt

The parameter s represents saving rate.
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Solow Model

Substituting the consumption function into the capital accumulation
equation, we havekt +1 = f (kt )= f (kt )(1(1s ) yt + (1δ) kts ) f (kt ) + (1δ) kt

Hence,
kt +1 = sf (kt ) + (1δ) kt .

This is Solow model.

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Solow Model

Solow Model

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Solow Model

De…nition
The steady state is the points at which f(ct , yt , kt )gsatis…es
ct +1 = ct , yt +1 = yt , kt +1 = kt
For any initial capital stock, economy eventually converges to thesteady state. It means
1
2

Economic growth rate eventually converges to 0.
If the steady state is the same across countries, the poor countries
eventually catch up.

A key assumption to derive this result is f 00 (k ) < 0.

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Solow ModelOn the steady state,= sf (k ) + (1sf (k ) = δksk=f (k )δkδ) k

Example: f (k ) = k αKatsuya Takii (Institute)

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The Review of High School Mathematics
Rememberk α+ β = k α k β(k α ) β = k αβ(kl )α = k α l α
1
k 1 =
k
Thereforek α β = k αkklKatsuya Takii (Institute)

α

= kα

β

= kα kβ

1
l

α

1

=

kα
kβ

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Solow Model

Example: f (k ) = k αKatsuya Takii (Institute)

s
δ

=

k
k
=
= (k )1 αf (k )(k ) α

k

=

s
δ

y

= (k ) α =

c

= (1

1
1 α

s
δ

α
1 α

s ) y = (1s)

Macroeconomics: Modern Macroeconomics 1

s
δ

α
1 α

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Solow Model
The impact of s:s’>sKatsuya Takii (Institute)

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Solow Model
The impact of δ:

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Solow Model

An increase in s and a decrease in δ increase k and, therefore, y .
Both changes cannot in‡uence the long run growth rate.

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The Extended Solow Model
Solow model predicts that economic growth rate eventually converges
to 0. This is not what we observe in data.
In order to match the theory with data, we introduce technology
growth and population growth.
Assume thatYt
Tt + 1
Nt + 1= F ( K t , T t Nt )= ( 1 + g ) Tt
= ( 1 + n ) Nt

where Tt and Nt denote technology and population at date t,
respectively. We assume labor augmenting technological progress.
Labor augmenting technological progress means that an increase in thetechnology has the same e¤ect as an increase in population.
We assume the labor augmenting technological progress, because itleads the theory to data.
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The Extended Solow ModelGDP per unit of e¤ective laborYtyet= F ( K t , T t Nt )Kt
= F
, 1 T t Nt
T t Nt= f (ket ) F (ket , 1)

where yet = TYt Nt t and ket = TKt Nt t are GDP per unit of e¤ective laborand capital per unit of e¤ective labor.

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The Extended Solow Model
Capital Accumulation
Kt +1 = It + (1

=
=
=
=

Yt

δ ) KtCt + (1δ ) Kt(1 s ) Yt + (1 δ) Kt
sYt + (1 δ) Kt
syet Tt Nt + (1 δ) ket Tt Nt

Yt

K t + 1 T t + 1 Nt + 1= syet + (1 δ) ket
T t + 1 Nt + 1 T t Nt
ket +1 (1 + g ) (1 + n) = sf (ket ) + (1 δ) ket
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The Extended Solow Model

The Extended Solow Model

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The Extended Solow Model

De…nition
The steady state is the points at which f(ce , ye , ke )gsatis…es
cet +1 = cet , yet +1 = yet , ket +1 = ket
where cet = TCt Nt t , yet = TYt Nt t , and ket = TKt Nt t .
For any initial capital stock, economy eventually converges to thesteady state.
A key assumption to derive this result is also f 00 (k ) < 0.

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The Extended Solow ModelOn the steady state k must satisfyke(1 + g ) (1 + n) ke(1 + g + n + gn) ke

sf (ke ) + (1 δ) ke(1 + g ) (1 + n )= sf (ke ) + (1 δ) ke== sf (ke ) + (1δ) ke

sf (ke ) = (g + n + δ + gn) ke

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The Extended Solow Model

Example f (k e ) = (k e )αs (ke )α = (g + n + δ + gn) kekes
α =g + n + δ + gn
(ke )
s
1 α
=
(ke )
g + n + δ + gn
keKatsuya Takii (Institute)=sg + n + δ + gn

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The Extended Solow Model
The Impact of n :n’>nKatsuya Takii (Institute)

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The Extended Solow Model

An increase in n lowers, ke and therefore, ye .

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The Extended Solow Model
Golden Rule Level of Capital Stocksf (ke ) = (g + n + δ + gn) ke
ce

dce
dke
2
d ce
d (ke )2

= (1 s ) f (ke )= f (ke ) (g + n + δ + gn) ke= f 0 (ke )(g + n + δ + gn) = 0= f 00 (ke ) < 0

ke is the capital stock that maximizes consumption on the steadystate.
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The Extended Solow Model

The Golden Rule Level of Capital Stock

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The Extended Solow Model

Example: f (ke ) = (ke )αα (ke )α 1 = g + n + δ + gn
α(ke )1 α =g + n + δ + gn
ke=

αg + n + δ + gn

Note thatske =g + n + δ + gn

1
1 α

1
1 α

Hence if s = α, the golden rule level of capital stock is attained.

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Kaldor’s Stylized Facts (1963)

Kaldor (1963) pointed out 6 stylized facts of economic growth. These
facts are repeatedly observed by aggregate data of OECD countries.
I would like to examine how the extended Solow growth model
explains these stylized facts.
As the extended Solow model predicts that economy eventuallyconverges to the steady state, we expect that the behavior of real
economy can be approximated by the behavior on the steady state.
Hence, we compare the prediction of the theory on the steady statewith Kaldor’s facts.

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Mathematical Preparation for the Analysis of Growth Rate
De…nitionln k loge k
1 m= 2.71828...It is useful to use e as awhere e = limm !∞ 1 + mbase. The useful properties of e and ln are
de t
dt
If g

d ln k1=dk
k0, ln (1 + g ) g= et ,The review of high school mathematics= ln k + ln l
ln k = α ln k
ln e = 1, ln 1 = 0
ln kl

α

Thereforeln
Katsuya Takii (Institute)k= ln kll1= ln k + ln l 1 = ln k

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Mathematical Preparation for the Analysis of Growth Rate
Lemma
Suppose that the growth rate of x, gx
0, then gx is nearly expressed asfollowsgx
ln xt +1 ln xt

Proof.
Suppose that xt +1 = (1 + gx ) xt ,ln xt +1

Katsuya Takii (Institute)ln xt= ln (1 + gx ) xt ln xt= ln (1 + gx ) + ln xt ln xt= ln (1 + gx )gx if gx0

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Mathematical Preparation for the Analysis of Growth Rate

Remark: If the time interval is not 1 but is almost 0, we can provethat the relationship is exact for any gx :De…ne gx so that xt +∆ = (1 + gx ∆) xt . Then, from the previous
lemma, for any gx there is a small ∆ so that gx ∆ ln xt +∆ ln xt . It
is shown that for any gx ,d ln xt
ln xt +∆ ln xt== gx , where gx
∆
dt
∆ !0
lim

Hence, we treat gx

Katsuya Takii (Institute)ln xt +1

ln xt as if gx = ln xt +1

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dxt
dt

xt
ln xt below.

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Mathematical Preparation for the Analysis of Growth Rate
Lemma
gxy

gx + gy

g

x
y

gx

gx α

αgx

gy

Proof.
gxyln xt +1 yt +1

ln xt yt

= ln xt +1 + ln yt +1 [ln xt + ln yt ]= ln xt +1 ln xt + ln yt +1 ln yt
gx + gy

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Mathematical Preparation for the Analysis of Growth Rate
Proof.
ln xtα+1gx α

ln xtα= α ln xt +1 α ln xt= α [ln xt +1 ln xt ]
αgx
g yx

Katsuya Takii (Institute)= gxy 1
gx + gy 1
gx gy

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Kaldor’s Stylized Facts (1963)

The growth rate of GDP per capita is nearly constant:gy = gye T

gye + gT = gT = g

The growth rate of capital per capita is nearly constant:.
gk = gke T

gke + gT = gT = g

The growth rate of output per worker di¤ers substantially acrosscountries.
Hence in order to …t the theory to data, g must be constant and di¤ers
among countries.

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Kaldor’s Stylized Facts (1963)
The rate of return to capital is nearly constant:rt = f 0 (ke ) = const
The ratio of physical capital to output is nearly constant:
Kt
k T t Ntk= e= e = const
Yt
ye Tt Ntye
The shares of labor and physical capital are nearly constant: Notethat the marginal productivity of capital and labor are
Kt
rt Kt
= rt
= constYt
Yt
Note thatrt Kt + wt Lt
1=
Yt
Hence
wt Lt
rt Kt
=1
= constYt
Yt
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Growth Accounting
Assume thatyet = (ket )α .
Then
kt
Tt

α

yt
Tt

=

yt

= Tt1 α (kt )αHencegyt = gT t1 α ktα(1α) gT t + αgkt

gyt = αgkt + R (t ) , where R (t ) = (1Katsuya Takii (Institute)

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Growth Accounting

Note that α can be estimated as follows:α=r t +1 K t +1
rt K tY t + Y t +1

2

=1

w t +1 N t +1
wt Nt
Yt +
Y t +1

2

So we can estimate
R (t ) = gyt

r t +1 K t +1
rt K tY t + Y t +1

2

gkt = gyt1

w t +1 N t +1
wt Nt
Yt +
Y t +1

2!

gkt

R (t ) is called the Solow residual or the growth rate of the Total
Factor Productivity (TFP). It re‡ects that all sources of growth otherthan the contribution of capital accumulation.

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Growth Accounting

Remarks
1

2
3

The equation for Solow residual can be derived without assumingf (k ) = k α . We don’t need constant returns to scale as well.
This is the simplest case. Usually, we can include other inputs as well.
Growth accounting can be applied for several analysis.
12Young (1994) …nds that the high growth rate of Hong Kong,
Singapore, South Korea, and Taiwan over the past three decades isalmost entirely due to rising investment, increase in labor force
participation, and increase in the level of education, but not to rapid
technological progress.
Productivity slow down puzzle. R (t ) became small after 1970s.

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Solow Model and Income Di¤erences

In order for the Solow model on the steady state to explain the longrun behavior of developed countries, we need to assume that g isconstant and di¤ers across countries.
In this section, we ask a question whether the Solow model can
explain the development facts.
First, I show what the new stylized facts are.
The next, I would like to ask if the Solow model can explain these facts.

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Development Facts

Parente and Prescott (1993) pointed out four main stylized facts.
12
3
4

Income di¤erence across countries is large.
Wealth distribution has shifted up.
Relative Income distribution does not show convergence.
There have been development miracles and disasters.

Durlauf and Quah (1998) also pointed out that1

Relative Income distribution across countries shows two peaks.

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Development Facts

Acemoglu (2009) points out that1

2

The relative ranking of countries has changed little between 1960 and
2000.
The origins of the current cross-country di¤erences in income per capita
occur between the late eighteen century and early twentieth centuries.

Barro and Sala–i-Martin (1995) shows that
1
2

There is no evidence of absolute convergence.
There is evidences of conditional convergence.

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Can the Solow model explain large income di¤erences?

Let me …rst examine whether the Solow model explains the …rst
stylized fact: large income di¤erences.
Let me start with simple exercises.
12

Suppose that T is the same across countries.
Ask whether di¤erences in capital stock per capita alone can explaindi¤erences in income per capita.

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Can the Solow model explain large income di¤erences?
Calibration Exercises 1 (Lucas (1990)):1ye = (ke )α ) ke = (ye ) α
Choose two arbitrary countries, say US and India.
1

yus T1 α
kus T1
kus
yus
=
=
1 )
1
kIndiayindiakIndia T
yindia T1 α

1αus
According to Penn World Table 6.3, yyindia
10 in year 2007 and1
α = 3,kus= 103 = 1000kIndiaIn order to explain large income di¤erences, required di¤erences incapital are too large.
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Can the Solow model explain large income di¤erences?

Why are Required Capital Differences so Large?
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Can the Solow model explain large income di¤erences?
Calibration Exercises 2 (Lucas (1990)):
α 1

MPK = f 0 (ke ) = αkea 1 = α (ye ) α
Choose again US and India
α yus T1
MPKus=
MPKIndia1
α yIndia T

Since yus /yIndia

α 1
α

yus
α 1 =
y
α
india

α 1
α

10 and α = 13 ,
1

1

2

3
3
MPKus
1
1
1
= [10] 3 = [10] 3 = [10] 2 =MPKIndia
100

Attributing di¤erence in output to di¤erence in capital implies a hugevariation in the rate of return on capital.
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Can the Solow model explain large income di¤erences?

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Can the Solow model explain large income di¤erences?

Both examples indicate that
When α = 13 , without productivity di¤erences, the theory cannot
explain large income di¤erences.
If α 1, it may be possible to explain data. This may indicate theexistence of unmeasured capital stock.

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Can the Solow model explain large income di¤erences?
Cross Country Regression (Mankiw, Romer and Weil (1992)): Assume
that every country has the same production function, f (ke ) = keα andthat all countries are on its steady state, then it is shown before thatke=sg + n + δ + gns
g +n+δ

1
1 α

1
1 α

Hence
ye
yt

Katsuya Takii (Institute)= (ke )α )= Ttyt
= (ke )α ) yt = Tt (ke )αTtsg +n+δα1 αMacroeconomics: Modern Macroeconomics 1

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Can the Solow model explain large income di¤erences?
Empirically Testable Equation
ln yt = ln Tt +

α
1

α

α

ln (s )1αln (g + n + δ)

Suppose that t = 0 andln T0 = a + εi
where a is constant and εi is country speci…c shock.
ln y0 = a +α1αln (si )α1αln (ni + g + δ) + εi

Assume that g and δ is constant, 0.05, across countries and s and n
are independent of εi . This assumption implies that there are no
productivity di¤erences other than luck.
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Can the Solow model explain large income di¤erences?

Data: Penn World Table: Non-Oil countries, Non-Oil countries except
for grade D countries and small population countries, OECD
countries.
1

2
3n.. the average growth of working age population over 1960-1985,
where working age is de…ned as 15 to 64.
s...the average share of real investment in real GDP over 1960-1985.
y ...real GDP in 1985 divided by the working age population in that
period.

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Can the Solow model explain large income di¤erences?

The following 4 results are obtained by them.
1

2

3
4

The coe¢ cients on saving and population growth have predicted signs
and 2 of 3 are signi…cant.
The restriction that the absolute values of the coe¢ cients of ln (s ) andln (g + n + δ) are the same cannot be rejected.
High R 2 .
The estimated α is much higher than 1/3.

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Can the Solow model explain large income di¤erences?

Implication from results: Although the Solow model has
qualitatively correct predictions, but quantitatively, α is too small to
explain huge income di¤erences. This indicates the existence of
unmeasured capital.
The problem of the estimation: If productivity di¤erences are notrandom, εi is correlated with s and n. The estimated parameters arebiased upward. It indicates that the true α may be lower than theestimated α.
Conclusion: Including the unmeasured capital as a part of T ,evidence suggests that T must di¤er across countries in order toexplain the large income di¤erences.

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Assignment

Students must hand assignment 3 in at the following lecture.

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Knowledge Accumulation and the Source of the Long run
GrowthAs I have shown, the long run growth rate is determined by g on the
Solow growth model. It is natural to ask what determines g .
This is the main question investigated by endogenous growth models.
There are several di¤erent models: education, learning by doing, R&D
and so on.
But they roughly share the same spirits: the long run productivity
growth is the results of the accumulation of knowledge.
In this section, I review some arguments on the knowledge
accumulation and the long run growth.

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Knowledge Accumulation and the Source of the Long run
GrowthThe Nature of Knowledge:
12

Knowledge is nonrival. Once somebody invents new idea, others canimitate it.
Knowledge is partially excludable. Using legal or possibly informal
mechanisms, we can partially exclude someone to use the knowledge.
1

Ex. Patent and Speci…c Capital

The Implication for Knowledge Accumulation
1

2

Without a reasonable institutional system to prevent others fromcopying a new idea, nobody has an incentive to invent new knowledge.
Knowledge accumulation must have a scale e¤ect.

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Knowledge Accumulation and the Source of the Long run
GrowthScale E¤ect: Many growth models share the following knowledge
accumulation function
Tt +1 = BNtT Tt + Tt
where NtT is the number of workers who work at a knowledge
accumulation sector. This equation implicitly assumes that the larger
the population at the knowledge accumulation sector, the higher the
probability to …nd new invention. Because of externality, pastknowledge Tt , has a positive impact on the creation of new
knowledge.
The above equation implies that

(1 + gT ) Tt = BNtT Tt + Tt ) gT = BNtTThe growth rate of knowledge is proportional to the level of
population. Hence, if a policy increases the number of researchers orengineers, it can increase long run economic growth rate.
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Knowledge Accumulation and the Source of the Long run
GrowthJones Critique (1995): Jones (1995) criticizes that this prediction isagainst evidence in the OECD countries.
World war II, we observe the number of scientists engaged in R&D has
dramatically increased, but the growth rate of TFP is quite stable.
Jones (1995) proposed di¤erent speci…cation:
Tt +1 = BNtT Tt + Tt
βwhere 0 < β < 1. The above model implies(1 + gT ) Tt
gT

= BNtT Tt + Tt
β

=

BNtT
1 β

.

Tt

1 β

The main di¤erence is that we have Tt
makes a big di¤erence. Note thatin the denominator. This

1 β

NtT ") gT t ") Tt +1 ") gT t +1 #Katsuya Takii (Institute)

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Knowledge Accumulation and the Source of the Long run
GrowthJones Critique (1995): We investigate the impacts of Jones
speci…cation on the steady state. On the steady state gT = g .
ggT= g BNtT

1 β
Tt= gB + gN T
0 = gN T (1

gg

g=

nT
1

β

(1 β) gTβ) g

, where nT = gN T .

The new model implies that the growth rate is proportional topopulation growth at the knowledge accumulation sector. A it would
be more di¢ cult for a government to in‡uence the growth rate of
population, the model implies that the long run growth rate is moreor less exogenous.
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Knowledge Accumulation and the Source of the Long run
GrowthJones (2002): Jones (2002) decomposes nT into two parts.
T
Because NtT = ht Nt where ht = NNtt ,nT = gN T = ghN = gh + gN = gh + n
Jones (2002) documented that a rise in educational attainment andresearch intensity can explain 80 % of recent U.S. growth; population
growth explains less than 20 percent. Note that an increase in ht
cannot continue inde…nitely since it is bounded by 1. Hence, hismodel predicts that sooner or later, the world growth rate must
decrease to the level of population growth. Of course, as theknowledge spillover goes beyond a country and many developing
countries will increase the proportion of scientists and engineers infuture, potentially gh can continue to be positive in the middle run.
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Investment Speci…c Technological Change
We still don’t know what the knowledge is. But, there is evidence
that knowledge embodied in investment might be the importantfactor of knowledge accumulation.
The relative price of equipment falls by about 4 % in the U.S..

Apparently, Solow (1957) model cannot explain this evidence, because
the relative price of investment is always equal to 1 by Solow (1957)
model.
Y = C +I
There is another evidence that Solow (1957) model cannot explain.
Productivity growth slows down after 1970s. This evidence is odd
because we observe more new technology after 1970s. It gives aquestion on what Solow residual captures.

Motivated by data, researchers start to pay attention to theinvestment speci…c technological change.
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Investment Speci…c Technological Change
Suppose that T = 1. let us consider the following two sector model.
Consumption Goods Sectormax
π t LctcLt

π ct= max fctkt

wt

rt ktc g , ct = f (ktc )First Order Conditions
rt
wt
wt
rt

Katsuya Takii (Institute)= f 0 (ktc )= f (ktc ) rt ktc= f (ktc ) f 0 (ktc ) ktc

wt
f (ktc ) f 0 (ktc ) ktc=
rt
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Investment Speci…c Technological Change
Investment Goods Sector
max π it Lit
L it

π it

= max pt it
kt

wt

rkti , it = qt f kti

where qt is the investment speci…c productivity.
First Order Conditions

wt
rt

Katsuya Takii (Institute)rt= pt qt f 0 kti

wt= pt qt f kti

rt kti= pt qt f kti
h= pt qt f kti

pt qt f 0 kti kti
i
f 0 kti kti

f kti
f 0 kti kti
wt=
rt
f 0 kti
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Investment Speci…c Technological Change
Equilibriumkt
f (ktc ) f 0 (ktc ) ktc
f 0 (ktc )=f 0 kti kti
f kti
wt
=
rt
f 0 ktiktc = kti

) ktpt

f 0 (kt ) = r = pt qt f 0 (kt ) ) pt qt = 1 ) pt =1
qt

Relative Price and Investment Speci…c Technological Change:
gq = g 1 =p

gpThis means that the declining price of equipment indicates animprovement in the productivity of equipment. Because the pricedrops by 4%, the estimated improvement in the productivity ofequipment is 4%. Hence, there is no slowdown in improvement in q.
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Investment Speci…c Technological Change
There are several issues related to investment speci…c technologicalchange1Creative Destruction:
1
2

3

2

New technology may replace old technology.
If so, workers or …rms that skillfully use old technology may also be
replaced.
This may cause several social problems: resistance, unemployment andso on.

Learning
Because technology is new, there is some leaning period. Therefore,
initially, productivity may slow down.

3

Skill Premium
If new technology replaces unskilled workers and demands skilledworkers, the wage inequality increases.

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Can Human Capital Explain Income Di¤erences?

In order to explain income di¤erences across countries, we need toassume that T di¤ers across countries.
If the knowledge accumulation is the source of increase in T , why thepoor countries do not imitate the knowledge in the developed
countries?
One possible explanation is that the use of knowledge demands
human capital. Let me investigate this possibility.

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Can Human Capital Explain Income Di¤erences?
Hall and Jones (1999): Assume that a country i has the production
function:
Yi = Kiα (Ti Ni )(1 α)where Ti = Ai hi . The variable Ai is the unobserved productivity andhi is the level of human capital. Then
1 =(1 α )

yi

yi

=

Ki
Yi
Ki
Yi

=

Ki
Yi

yi =

Ki
Yi

Hence

Katsuya Takii (Institute)αα

T i NiYi(1 α )( Ti ) ( 1 α )

α
1 α

Ti
α
1 α

Ai hi

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Can Human Capital Explain Income Di¤erences?

yi ...National income and labor force data are taken from Summers
and Heston (1991).
They assume thathi

= exp (0.134= exp (0.134= exp (0.134

E ) , if E

4,4 + 0.101(E 4)) , if 4 < E 8,
4 + 0.068 (E 8)) , if E > 8,

4 + 0.101

where E is average educational attainment. The coe¢ cients, 13.4,
10.1 and 6.8, are taken from previous research. Average educational
attainment is measured in 1985 for the population aged 25 and over,as reported in Barro and Lee (1993).

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Can Human Capital Explain Income Di¤erences?
Capital stock is estimated by the perpetual inventory method:
Kt +1 = It + (1δ ) Ktwhere δ is assumed to be 0.06. We can recursively estimate capitalstock when we know the initial value. Note thatKt + 1

Kt + δKt= It
It
gK + δ =Kt

It
gK + δ
Using this relationship, the initial value at 1960 is estimated by
I1960
K1960 =g +δwhere g is the average geometric growth rate from 1960 to 1970.
The parameter, α, is assumed to be 13 and the variable, Ai , isestimated by the residual.
Kt

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Can Human Capital Explain Income Di¤erences?

Results1

2

3

Output per worker in the …ve richest countries is 31.7 times higher
than output per worker in the …ve lowest countries.
Capital intensity, human capital, productivity in the …ve richest
countries are 1.8, 2.2 and 8.3 times larger than those in the …ve lowest
countries, respectively.
It shows that including human capital does not help much explaining ahuge income di¤erences.

Questions on Hall and Jones (1999):Clearly, school quality, on the job
training, child-rearing, and prenatal care vary across countries. How
can we estimate these e¤ects?

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Can Human Capital Explain Income Di¤erence?
Hendricks (2002): Hendricks use the wage data of immigrants in the
US to estimate relative human capital. Assume that production
function isY
ye = (ke ) )
=
AhN
α

K
AhN

α

) Y = (K )α (Ah)1 α N 1 α

Then
w = MPL = (1α) (K )α (Ah )1 α N

αwhere ke = A usKhusi N us . Let me choose two countries, US and India.
wus
windiahus
hindia=(1 α) (Kus )α (Aus hus )1 α Nusα=(1 α) (Kus )α (Aus hindia )1 α Nusα=

wuswindiaKatsuya Takii (Institute)hushindia(1 α )1
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Can Human Capital Explain Income Di¤erence?
Using this estimate, Hendricks (2002) conduct the same calibration
exercises as Hall and Jones (1999) do.
He concludes that for countries below 40 percent of U.S. output per
worker, less than half of the output gap relative to the United Statesis attributed to human and physical capital.
Remark: Discrimination and a country speci…c skill such as language
strengthen his result because it means that the immigrants may be
more productive in their home countries.

Issue: the selection bias. If the average ability of immigrants arehigher than the average ability of workers in their home countries,
then the wage di¤erence is lower than the average human capitaldi¤erence between the US and their home country.
After considering the selection bias, he concluded that the selection
bias cannot change his result.
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Can Human Capital Explain Income Di¤erence?
Can we dismiss human capital? There are several reasons that humancapital may be still important.
Externality: The exercises assume that there is no externality. But
externality is important, the impact of human capital is larger.
Technology Adoption: Education may be important not only for
production but also for stimulating the adoption of technology. This
point is more rigorously analyzed below.
The In‡uence on Population: Even in the high productive society, if
population grows faster than productivity does, per capita income
cannot increase. As investment in human capital increases the cost ofraising children, it reduces the number of children and reduces
population growth.

Nonetheless, it is less likely that the lacks of physical and humancapital provide a whole story for large income di¤erences. Note thatthe lacks of physical capital and human capital cannot explain
development miracles and disasters. We need to investigate apotential source of A.
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Resource Allocation and Aggregate Productivity
Many macroeconomists currently examine resource allocation as thepotential source of A. If we think that the aggregate production
function must be the result of several micro activities such asn

F (K , AhL) = ∑ Fi (Ki , Ai hi Li )iwhere i implies ith plant, the allocation of resources across plants canin‡uence the aggregate productivity.
Hsieh and Klenow (2009) investigate how much the misallocation of
resources across plants in‡uence the aggregate TFP by using
microdata on manufacturing establishments in China, India and theUnited States. (US: 1977, 1982, 1087, 1992 and 1997, China:
1998-2005, India: 1987-1995.)
Results: When capital and labor are hypothetically reallocated toequate marginal products to the extent observed in the United States,
they calculate manufacturing TFP gains of 30 %~50% in China and40%~60% in India.
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Resource Allocation and Productivity

In reality, misallocation would be everywhere. But, it is di¢ cult for agovernment to dictate better allocation. If it could, many communist
countries could have survived. That is, it is impossible to point out
every ine¢ ciency in a world. For a policy purpose, we need to focus aparticular misallocation that would in‡uence productivity very much.
What kinds of misallocation are important? As we discussed
before, if we believe that the transfer of knowledge is the main sourceof productivity di¤erences, the factors that in‡uence technology
adoption are likely to be important. We can provide two possible
factors which can prevent the reallocation of resources to thetechnology di¤usion.

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Misallocation of Resources and Productivity Di¤erences
Misallocation of talent: There is an indirect way to in‡uence B.
Baumol (1990) investigated historical evidence and provided thefollowing three hypotheses:
12

3

The social system, which determines the relative payo¤s to di¤erent
entrepreneurial activities, changes over time and across regions.
Entrepreneurial behavior changes according to variations in the social
system.
The allocation of entrepreneurship between productive andunproductive activities has a large e¤ect on the innovation oftechnology and dissemination of technological discoveries.
Productive activity...Activity to create new value.
Unproductive activity...Activity to transfer income from somebody toothers.

Murphy, Shleifer and Vishny (1991) provided a formal model thatclari…es Baumol’s hypotheses. It shows that 1) if talented people are
misallocated to a declining industry or the sector that specializing
transferring income, they may reduce the growth rate.
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Misallocation of Resources and Productivity Di¤erences
Resistance to Technology Adoption: Parente and Prescott (1994)
show that the barrier to the adoption of new technology havesubstantial impacts on income di¤erences. Parente and Prescott(1999) argue that monopoly right can be one such barrier. When
government protects a particular company or industry, new comers
cannot enter the market with new technologies. It hampers the
adoption of new technology.
Why do the resistance to technology adoption prevail?
Innovation and technology adoption are accompanied with creative
destruction.
Creative destruction demands the replacement of resources.
In particular, If the incumbent has a large speci…c skill for oldtechnology, the adoption of new technology makes the skill obsolete.

The reallocation of resources brings the con‡icts of interests.
If preventing new entrance is cheaper than adopting new technology,
they may resist new technology.
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Fundamental Determinants of Di¤erences in Economic
Performance

Both arguments suggest that in order for the misallocation to prevail,there must have an institutional barrier to prevent market forces fromreallocating resources.
This point is similar to the arguments by Acemoglu (2009). He said
that investment in physical and human capital, and the reallocation of
resources are proximate causes of income di¤erences. But it does not
answer to the deeper questions? Why do some countries invest morethan others? Why do some countries attain better allocation.
Acemoglu (2009) listed four candidates of fundamental causes. 1)
Luck, 2) Geography, 3) Culture and 4) Institution.

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Fundamental Determinants of Di¤erences in Economic
Performance

Acemoglu, Johnson and Robinson (2001) provide evidence that1

2

3

More than 75 % of the income gap between relatively rich andrelatively poor countries are explained by di¤erences in economic
institution (as proxied by the security of property rights. )Once the e¤ect of institutions is controlled, there appears to be noe¤ect of geographic variables.
Once economic institution is taken into account, cultural variables do
not appear to have a direct e¤ect on economic growth and income percapita.

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Fundamental Determinants of Di¤erences in Economic
Performance
If better institution is important, why do not they choose betterarrangement?
Coase Theorem: If utility is transferable, and there is no liquidity
constraint and no transaction costs, negotiation reaches a Pareto
optimal allocation among parties who join the negotiation.
Example: A company introduces IT system to enhance productivity of
the …rm. It may substitute many clerks, and reduce labor costs. To
do so, managers must …re workers. It is costly for workers. But
manager can negotiate retirement allowance with workers.
1

2

3

4

Because utility is transferable, workers can agree if enough payment iso¤ered.
Because the introduction of IT system is assumed to be productive, a…rm can o¤er enough payment to convince workers.
Because there is no liquidity constraint, even though income is available
in future, the …rm can …nance the payment.
No transaction cost means that both parties cannot tell a lie,negotiation is costless and the contract is perfectly enforced.

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Fundamental Determinants of Di¤erences in Economic
Performance
The previous question can be rewritten as follows: Why cannot Coase
Theorem be applied in a political world?
1

2

3

4

Utility may not be transferable. Maybe, but in this case, enhancing
economic growth is not a Pareto optimal policy. Economists cannot
say much.
There might be a liquidity constraint. Possibly, but there is a
developed international …nancial system. So they can relatively easily
borrow money if the return is huge.
The existence of transaction cost. This is possible. But what kinds oftransaction costs are important?
Coase theorem guarantees Pareto optimal allocation only among
parties who join the negotiation. There might have an external e¤ectof the negotiation.

We discuss the points 3 and 4.
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Fundamental Determinants of Di¤erences in Economic
Performance
The external e¤ect of negotiation: If the results of negotiationin‡uence the third party, it is possible that the parties who can join
the negotiation may steal some income from the third party.
For example, if all policies are determined by the negotiation among
political elite from landlord, they would prevent enhancing
manufacturing sectors, because their peasants or slave may move tomanufacturing sectors. Enhancing manufacturing sectors can be
Pareto improving, but landlords have no incentive to do so because
they can keep their rent at the expense of peasants.

But why does not the third party make an o¤er to negotiate with
them?
There is a cost of organizing an interest group to make a reasonablenegotiation with them. One of di¢ culty in organizing an interest
group is free rider in the group: if some body incurs the cost oforganizing group, others do not need to pay.
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Fundamental Determinants of Di¤erences in Economic
Performance
What kinds of transaction cost are important?
Hold Up Problem: There is no way for politicians to commit theirstatements. Without commitment, Coase theorem does not work.
1

2

For example, if North Korea introduces the property rights and commit
that they maintain investment friendly environment, many …rms may
invest in North Korea and it can potentially grow. However, after
many …rms making investment, they have always an incentive to steal
the return from investment by raising tax or so. Moreover, because
Kim Jong-un and other politicians are making a law, they can easilychange a law. They don’t have any method to commit theirstatements. Because no …rms can trust politicians, they don’t invest.
Similarly, suppose that if some politicians in North Korea can peacefully
replace Kim Jong-un from North Korea, they can potentiallyrestructure the economy. For this purpose, they must commit thatthey provide his families enough income to guarantee their wonderfullife. But, new politicians are in power, they may not keep theirpromise. As new politicians don’t have any commitment device, Kim
Jong-il cannot trust them and they never step down.

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Assignment

Students must hand assignment 4 in at the next lecture.

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Stabilization Policy

Every country experiences boom and recession.
The main question is whether government should or can actively
stabilize aggregate demand for economy.
I would like to investigate how demand stabilization policy a¤ects real
economy.
This was a central issue of policy discussions. A school of economists
advocates that a government should actively stabilize demand for
economy; others disagree with this opinion. I would like to provide
the framework to understand the reasons behind these policy
discussions.

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Stabilization Policy

My lecture in this section is organized as follows.
1

2

3

4

5

6

I introduce money and government to the basic model. For thispurpose, I …rst discuss money market.
Incorporating money and government in the basic model, we discuss
how monetary policy and …scal policy in‡uence GDP in the long run.
I introduce a short run deviation of GDP and the possibility thatdemand stabilization policy has a real impact.
I discuss several explanations that enforce the economy to deviate fromthe long run equilibrium.
I discuss stabilization policy and unemployment. We emphasize theimportance of distinguishing the source of unemployment in the short
run and long run.
Finally, I discuss a di¢ culty of implementing stabilization policy. I
show that how lack of commitment causes undesirable outcome.

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Money Market

What is money? Money is the stock of assets that can be readily
used to make transactions.
Money has three functions
1
2
3

a store of value
a unit of accounta medium of exchange.

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Money Market
Money Supply: Money supply includes both currency in the hands
and deposits at banks that households can use on demand fortransaction:
Ms = C + D(2)
where M s is money supply, C is currency and D is demand deposits
such as ordinary deposits and checking accounts.
Banking System and Money Supply: Banking System in‡uences
the amount of money supply.
Example: Banks are required to keep a proportion of deposit by law inthat depositors can always withdraw money. The deposits that banks
have received but have not lent out are called reserves, denoted by R.
Suppose that you deposit 1000 yen, R = rd 1000 yen where rd is thereserve deposit ratio. Hence, the bank can use (1 rd ) 1000yen to
make loans and (1 rd ) 1000yen goes to public as currency. Sincemoney supply is sum of currency and deposits, money supply is(1 rd ) 1000yen + 1000yen. In this way, the banking systemin‡uences money supply.
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Money Market

Interbank Lending Market: If a bank cannot hold the required
amount of liquidity asset as reserves, it will need to borrow money inthe interbank market to cover the shortfall.
Financial Crisis: During Financial Crisis, there are some di¢ culties
in this interbank system. A bank defaults. Other banks that madeloans to this bank may default too. All banks are worried about therisk of default and the transaction in the interbank lending market
becomes low. It is said that low transaction volume in this market
was a major factor to the …nancial crisis of 2007.

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Money Market
A Model of Money Supply: The total amount of YEN that is
supplied by Bank of Japan is called the monetary base, denoted by B
and the monetary base equals currency plus reserves
B = C +R

(3)

Using equation (2) and equation (3),
Ms
C +D
=
B
C +R
Hence
Ms =

cd + 1B
cd + rd

cd +1
where cd = C /D and rd = R/D. The parameter cd+rd is called
money multiplier, which is a decreasing function of rd.
This equation implies that Bank of Japan can control money supply
by changing monetary base and reserve-deposit ratio.
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Money Market
How does Bank of Japan control B? There are three instruments:
1

2

3

Open-market operations: Bank of Japan can sell or buy government
bonds. When it buys bonds in the market, it pays yen for the bonds.
Therefore, monetary base goes up. On the other hand, when it sells, it
receives yen. Hence monetary base goes down.
Operations at exchange market: Bank of Japan can sell or buy
dollars. When it buys dollars in the market, it pays yen for the dollars.
Therefore, monetary base goes up. On the other hand, when it sells, it
receives yen. Hence monetary base goes down.
Change in the Discount Rate: When commercial banks …nd thatthey do not have enough reserves, they can ask Bank of Japan to
discount their bills. When Bank of Japan reduces the discount rate,commercial banks can easily borrow money from Bank of Japan and it
increases monetary base.

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Money Market

Money Demand: Why do people demand money? There is thebene…ts and costs of holding money.
1

2

The cost of holding money is that you miss interest. Instead ofholding money, you can buy stocks or bonds. If you buy bonds, you
can earn interest. But it you hold money in your pockets, you missthis opportunity. Therefore, the higher the interest rate, the smaller
the demand for money.
Then what is the bene…t of holding money? It makes our transactions
smoother. When you made a deal with your business partners, youmust pay money. Hence we expect that when we have moretransactions, we must demand more money. Since transactions
increase when real GDP is larger, we expect that the demand formoney is an increasing function of real GDP.

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Money Market
The discussions suggest the following money demand function.
Md= L (ρ, Y )P
where M d is the demand for money and P is the aggregate price.
The above equation implies that a decrease in the interest rate andan increase in real GDP raise the demand for money.
For a simple analysis, we assume that the money demand is
proportional to real GDP:
Md
P

= k (ρ) Y , k 0 (ρ) =mdP= k (ρ) y

dk
<0
dρ

d

where md = MN and y = YN , and k (ρ) is called Marshall’s k.
Marshall’s k measures how much money people want to hold forevery income.
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Money Market

Money market ismsmd== k (ρ) yPPs

where ms = MN .

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Government Sector
Government Sector: Assume that government must …nancegovernment expenditure per capita g , and government bond per
capita, bt and transfer payments to households per capita, tr , such as
welfare for the poor and Social Security payments for the elderly, by alump sum tax, with which consumers must pay …xed cost, τ, togovernment.
bt +1 = (1 + ρ) bt + g + tr τ
We assume that government keeps the same amount of debt.
bt +1 = bt .
b = (1 + ρ) b + g + tr
τ

n

τ

τ

tr = ρb + g

where τ n is the net tax.
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Government Sector

Following Ono’s (2006) critique, we assume that θ proportion of gincreases value added. That is, (1 θ ) proportion of g can be seen
as a hidden income transfer, tr h , to somebody:tr h = (1θ) g .

Note that current national account presumes θ = 1.

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Firm
Assumption: Constant returns to scale and the steady state.
Moreover, because we ignore capital accumulation from our following
K = 1 ) k = K = T.
analysis, we simply assume k = TN
N
Note that ignoring capital accumulation simpli…es our explanation, but
makes us impossible to analyze the long run impact of demandstabilization policy through capital accumulation.
Y

F (K , TlN ) = F

1,

TN
l
K

K = F (1, l ) K

De…ne φφ (l )F (1, l )Yφ (l ) K

Then
Moreover,y=Katsuya Takii (Institute)YN

φ (l )K= φ (l ) T
N

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Firm
Pro…t maximization Problem
Π = max fPY
K ,L

WlN

= max Pφ (l ) K
l ,K= max Pφ (l ) K
l ,Kwhere π k

= max π k PKKn= max φ (l )lRwhere w = W
P and r = P .

Katsuya Takii (Institute)

w
l
T

RK g
R
Wl TN
PK
PK
PT K
P
W
R
lPK
PK
PT
P

r

o

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Firm

Assumptions on the aggregate production function
12

φ (0) = 0d φ (l )φ0 (l ) = dl > 0.
When the …rm employs more capital per workers, it increases output
per workers.

3

φ00 (l ) =d 2 φ (l )<0dl 2This means that the marginal productivity of capital per workers isdiminishing.
4

Inada Conditions: technical conditions.
lim φ0 (l ) = ∞, lim φ0 (l ) = 0.

l !0

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Firm

Pro…t maximization with respect to lnπ k = max φ (l )
l

w
l
T

r

First Order Conditionsow= φ0 (l ) ) w = φ0 (l ) T ) l is determined.
T
πk
π k = φ (l )Katsuya Takii (Institute)wlTr = φ (l )φ 0 (l ) lr ) π k is determined.

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Firm

Pro…t maximization with respect to K
Π = max π k PK
K

Capital Demand Function
K
K
K

Katsuya Takii (Institute)= 0 if π k < 0, r > φ (l ) φ0 (l ) l
2 [0, ∞] if π k = 0, r = φ (l ) φ0 (l ) l= ∞ if π k > 0, r < φ (l ) φ0 (l ) lMacroeconomics: Modern Macroeconomics 1

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Firm

0 Economic Pro…tΠ = π k PK = 0
When the market is competitive, more entrepreneurs will enter as longas economic pro…ts are positive. Hence, in the long run, economic
pro…t is 0.HenceΠ = 0, K > 0 ) π k = 0 ) r = φ (l )Katsuya Takii (Institute)

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Household
How long do they work?: Assume that everybody works h (w ) oftime. Hence, the supply of labor is equal to h (w ) N. We assumedh (w )
that h0 ( ) = dw > 0.
How much do they consume today?:
I take the consumption per capita, c as given.
ss= mP . Then
Budget Constraint: Assume that mP+1a+1 + c +msP

= (1 + ρa ) a ++1a+1 + cρams
+ wh (w )P= (1 + ρa ) a + wh (w )= max fr δ, ρg

τ + tr + tr h

τ n + tr hwhere τ is tax rate and tr is income transfer.

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Arbitrage Condition and Market Clearing Condition

Arbitrage Condition implies,rδ = ρ = ρaLabor Market Clearing ConditionlN = h (w ) N ) l = h (w )Capital Market Clearing Condition
kN + bN= aN
b = a kKatsuya Takii (Institute)

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Stabilization Policy in the Long Run
Given (a, c, k (= T ) , M s , g , θ, δ), a market equilibrium withgovernment consists of y , l, a+1 , ρ, r , w , τ n , P, tr h , b which satis…es
A Firm’s Pro…t Maximization and the Production Function determine:y = φ (l ) Twr= φ 0 (l ) T= φ (l ) φ 0 (l ) l

A Consumer’s Budget Constraint
a+1 + c = (1 + ρ) a + wh (w )

τ n + tr h

An Arbitrage Conditionr δ=ρ
Labor market clearing conditions
l = h (w )Capital market clearing conditionsb=a
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Stabilization Policy in the Long Run

Government’s budget constraint determines τ n and tr h :
τn
tr

h

= ρb + g
= (1

θ) g

The money market cleaning condition determines P:
ms= k (ρ) yP

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Stabilization Policy in the Long Run

Supply Side of Equilibrium: Because l = h (w ),= φ (h (w )) T
w = φ0 (h (w )) T
ρ = φ (h (w )) φ0 (h (w )) h (w )yKatsuya Takii (Institute)

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Stabilization Policy in the Long Run
Goods market
a+1 + ck+1 + b + c
k+1 + ck+1 + ck+1 + ck+1 + ck+1 + c(1 + ρ) a + wh (w ) τ n + tr h
(1 + ρ) a + wh (w ) (ρb + g ) + (1 θ ) g(1 + ρ) (a b ) + wh (w ) θg
(1 + r δ) k + wh (w ) θg
φ (l ) φ0 (l ) l + (1 δ) k + φ0 (l ) Tl θgφ (l ) φ0 (l ) l T + (1 δ) k + φ0 (l ) Tl θgφ (l ) T + (1 δ) k θg

=
=
=
=
=
=
=
y

Katsuya Takii (Institute)= c + k+1 ( 1= c + i + θgδ) k + θg

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Stabilization Policy in the Long Run
Supply Side determines (y , w , ρ)

= φ (h (w )) T
w = φ0 (h (w )) T
ρ = φ (h (w )) φ0 (h (w )) h (w )yδDemand Side determines (i, P )ymsP= c + i + θg
= k (ρ) y

Remark: Note that current national income accounting presume thatθ = 1. In facts,
the measured GDP per capita = y + (1Katsuya Takii (Institute)

Macroeconomics: Modern Macroeconomics 1θ) g = c + i + g
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Stabilization Policy in the Long Run

Labor Market Equilibriumw

h l

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Stabilization Policy in the Long Run

Stabilization Policy vs. Supply Side Policy: Both monetary supply,(a change in ms ), and a …scal policy, (a change in g ), cannot
in‡uence GDP per capita, y , the real interest rate, ρ, and the real
wage rate w in the long run.
1

2

Given a labor supply function h (w ), w is determined fromw = φ0 (h (w )) T .
Given w , y = φ (h (w )) T and ρ = φ (h (w )) φ0 (h (w )) h (w )δ

If a government wishes to in‡uence GDP in the long run, the policy
must have an impact on supply side. I will give two examples.

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Stabilization Policy in the Long Run
Suppose that government levies a tax on labor income. In this case,households react to (1 τ l ) w , where τ l is a ‡at labor income tax
rate. Hence, as a reduction in labor income tax increases (1 τ l ) w ,h ((1 τ l ) w ) would be larger.

w

h l

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Stabilization Policy in the Long Run
Suppose that government expenditure increases productivity T .
Because φ0 (h (w )) T implies that the marginal product of labor islarger, …rms have more incentive to employ labor. Hence,y = φ (h (w )) T would be larger.
T’
>T
w

h l

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Stabilization Policy in the Long Run

Fiscal Policy and Crowding Out: How does …scal policy in‡uencean economy in the long run? Consider
y = c + i + θg .
As y is already given supply side, an increase in g must result in
reductions of c + i. It leads to the following theorem.

TheoremA permanent increase in public expenditure (which only a¤ect demandside) will be o¤set by a permanent reduction of private expenditure in thelong run (crowding out).

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Stabilization Policy in the Long Run
Monetary Policy and In‡ation: How does …scal policy in‡uence aneconomy in the long run? Consider
ms= k (ρ) y ,P
Because ρ and y are already given by supply side, equation implies
that an increase in money supply, ms , causes in‡ation (= an increasein the price index, P).
All variables measured in physical units, such as output and relative
prices, are called real variables. Variables expressed in terms ofmoney are called nominal variables. This result implies that money
supply a¤ects nominal variables, but it does not a¤ect real variables.
This is called the neutrality of money.

Theorem
An increase in money supply causes in‡ation, but does not a¤ect realvariables in the long run.
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Stabilization Policy in the Long Run

Aggregate Demand and Aggregate Supply Curve
P

P*

Aggregate Supply Curve

Aggregate Demand Curvey
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Money Market in the Long Run
The quantity theory of money: The quantity theory of money isderived from money market equilibrium.
ms V (ρ) = Py
M s V (ρ) = PY
where V (ρ) = k (1ρ) .
The V (ρ) is called the income velocity of money, which tells us thenumber of times money enters someone’s’income during a given
period of time. It measures the speed of transaction. Note that theincome velocity of money is inverse of Marshall’s k. If people wish to
hold more money given income (= large k), money cannot move
much (= small V (ρ)). If people hold little money in hand, money
can frequently move around.
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Money Market in the Long Run
Note thatgM s VgM s + gV= gPY= gP + gY

Since the supply side determines ρ, ρ is given. If ρ is constant, V isconstant. Hence, gV = 0.
gM s = gP + gY
If an economy is in a steady state, we know thatgY = gyN = gy + gN = gT + gN . we may be able to assume that itis near constant.
Hence, the quantity theory of money suggests that the central bankcan control in‡ation rate gp by changing gM s .
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Money Market in the Long Run
Nominal Interest Rate and Real Interest Rate: Alternativedi¢ culty in controlling in‡ation arises when we realize di¤erence
between nominal interest rate and real interest rate. The real interest
rate is de…ned as nominal interest rate minus the expected in‡ationrate:ρr = ρn gPe(4)where ρr is the real interest rate, ρn is the nominal interest rate andgPe is the expected in‡ation rate.
ρr in‡uences saving decisions and investment decisions.
ρn is considered to be opportunity costs of holding money. Hence,money demand depends on ρn .
Alternative expression:
1 + ρrln (1 + ρr )
ρrKatsuya Takii (Institute)=

1 + ρn1 + gPeln (1 + ρn )
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Money Market in the Long Run
Fisher equation: Rearranging equation
ρn = ρr + gPe
This is called Fisher equation. As Marshall’s k is a decreasing
function of ρn , and therefore the V is an increasing function of ρn .
M s V (ρr + gPe ) = PY , V 0 ( ) > 0.
This equation implies that expected future in‡ation rate can in‡uence
actual in‡ation rate.
gPe ") ρn ") k (ρn ) #) V (ρn ) ") P "Hence, a careless expansionary monetary policy may induce a hyper
in‡ation. So monetary policy must take into account how it
in‡uences people’s expectation. But how?
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Money Market in the Long Run
Note thatln Pt + ln Y
ln Pt

= ln Mt + ln V ρr + gPe ,t= ln Mt + ln V ρr + gPe ,tln Y

Assume that ln V (ρr + gPe ) = γ (ρr + gPe ), and ρr and ln Y areconstant over time. ThengPe ,t(1 + γ) gPe ,tKatsuya Takii (Institute)= E [ln Pt +1 ] ln Pt= E ln Mt +1 + ln V ρr + gPe ,t +1
ln Yreln Mt + ln V ρ + gP ,tln Yre= E ln Mt +1 + γ ρ + gP ,t +1
ln Mt + γ ρr + gPe ,t= E [ln Mt +1 ] ln Mt + γgPe ,t +1
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Money Market in the Long Run

The expected In‡ation rate
gPe ,t =E [ln Mt +1 ]ln Mt + γgPe ,t +1
1+γ

It means that the current expected in‡ation rate depends on the
expectation on the money supply in the next period and the expectedin‡ation rate in the next period. We can also imagine that the
expected in‡ation rate in the next period would depend on the
expectation on the money supply in two period later and so on. It
indicates that in order to control people’s expectation, the commitment
on the future monetary policy is necessary.

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Discussion
Can Bank of Japan really control money supply?: So far we
assume that the Bank of Japan can perfectly control money supply.
Is it true? During Xmas season, we can observe an increase in money
supply and nominal GDP. However, it is di¢ cult to believe that anincrease in money supply during Xmas causes an increase in nominal
GDP. Natural interpretation is opposite: since people transact moreduring Xmas seasons, people demand more money.
Note that there is a relationship between B and M s .
cd + 1
Ms =B
cd + rdwhere cd = C /D and rd = R/D. We so far assume that cd and rd
are constant. But in fact these numbers are endogenous variables.
In Japan, money multiplier has decreased between 1992 and 2002. It
means that although Although Bank of Japan has actively increased
Monetary Base, its impacts on money supply has decline during theperiod.
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Discussion

Interest rate as A Policy Target: People in Bank of Japan typicallybelieve that they rather accurately control a short term nominalinterest rate by changing B. In this case, the quantity theory ofmoney implies alternative e¤ect in the long run:
Note that
M s V (ρn ) = PY , V (ρ) =

1
k (ρ)

If B ") ρn #,B ") ρn #) k (ρn ) ") V (ρn ) #) P #This is di¤erent from

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Discussion

Zero interest rate policy: In fact, in Japan, the nominal interest
rate is set at 0, ρn = 0 between February 1999 and August 2000 and
between March 2001 and July 2006. Since December 16, 2008, theUnited States conducts the same policy. When the nominal interest
rate is 0, the opportunity cost of holding money is 0. There isMs
has increased during the period.
evidence that k (ρn ) = PY

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Discussion

Zero interest rate policy in the long run: Whatever variables that
Bank of Japan can control, zero interest rate policy is likely to inducede‡ation in the long run. Because of Fisher equation,
0 = ρr + gPe
gPe=ρr < 0Key observation is that the real interest rate is not in‡uenced bymonetary policy in the long run. As we discuss later, in the short run,lowering nominal interest rate lowers real interest rate, and, therefore,
encourages investment. But, as real interest rate is constant in thelong run, lowering nominal interest rate simply reduces the expectedin‡ation rate.

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Discussion

Fukuda (2010) claims that the zero interest rate policy couldn’t stopde‡ation during this period. On the other hand, Honda (2011)claims that the zero interest rate policy rather caused the depreciationof the exchange rate and succeeded to stimulate economy. But Saito(2010) claims that it was a just bubble and couldn’t not enhance real
economic growth in the long run.

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Discussion
Friedman Rule: Friedman argues that an optimal nominal interest
rate is 0. In this case, real interest rate is equivalent to de‡ation andsthe money supply must be adjusted to satisfy MP = k (0) Y . Because
the opportunity cost of holding money is 0, as far as the price system
works well, everybody can enjoy the bene…ts of money without cost.
What is wrong with this?
Cost of De‡ation: There are two potential problems on Friedman
rule.
1

2

Friedman rule assumes that everybody accepts a reduction in nominal
wage payment. If not, we should observe more unemployed workers orbankrupt companies.
Because debt is based on nominal value, lower nominal wage meansthat debtors will face a di¢ culty in paying their interest.
In Japan, the largest debtor is government. Hence, government mustsu¤er from the cost of de‡ation, which causes higher tax rate in future.

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Assignment

Students must hand assignment 5 in at the next lecture.

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Adjustment Cost of Investment and Stabilization Policy
So far I assumed that …rms can instantaneously make any investment.
This may be an innocuous assumption in the long run. However, theassumption may be questionable in a short run problem. This section
introduces the adjustment cost of investment.
When there is no adjustment cost of investment, one unit of output istransferred to produce one unit of investment. Hence the value ofcapital is also equal to 1.
When the cost of investment exists, the value of capital deviates fromone. Assume that q is the marginal value of capital and C I f is theadjustment cost of investment. The …rm’s investment problem canbe written as follows.
max qK 0
Iff

K 0 = I + (1I, I = If + C Ifδ) K

where C (0) = C 0 (0) = 0, C 0 I f > 0 and C 00 I f > 0.
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Adjustment Cost of Investment and Stabilization Policy

Adjustment Cost of Investment
C( )Katsuya Takii (Institute)

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Adjustment Cost of Investment and Stabilization Policy
The …rst order condition is
q = C0 If+1

Note that this equation implies that if there is no adjustment cost,the value of capital, q, is equal to 1. The marginal value capital, q,
is called the marginal q.
Because C 0 I f

is large when I f is large,(q1) ") C 0 I f") I f ") I "Hence investment can be written as an increasing function of q
I = ψ (qKatsuya Takii (Institute)1.

1) , ψ 0 ( ) > 0

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Detour to the Q theory of Investment
Q Theory of Investment: With an additional technical conditionon the adjustment cost ( C I f = C I f , K and C is CRSs in I f
and K ), it is known that the marginal q can be estimated by the
Tobin’s Q whereq = Tobin0 s Q
and there is ψ (qMarket Value of a Firm
Replancement Cost of Capital Stock1) such thatI = ψ (q1) = ψ (q1) K .

It means that if the market value of …rm is greater than replacement
cost of capital, then the …rm should invest, if not, it should sell someassets. In this way, Tobin’s Q can be considered as the proxies of theimportance of the expected investment opportunities. This is called
the Q theory of investment.
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Adjustment Cost of Investment and Stabilization Policy
What in‡uences q? Let us maintain the current assumption(C I f does not depend on K ) and discuss what in‡uences the
expected investment opportunities. If you have one unit of capitalstock in your hand, you can sell it and earn qt . If you save qt in yourbank, your account at the next year would be (1 + ρ) qt . On theother hand, if you keep the capital, you expect to receive rental pricerte+1 at date t + 1 and you expect to sell the capital in the next yearby the price qte+1 . Because δ portion of capital is assumed to be
depreciated, you will expect to earn qt +1 (1 δ) by selling thecapital. Hence, an arbitrage condition implies(1 + ρ) qt = rte+1 + qte+1 (1 δ)
rte+1 = E φ (h (wt +1 )) φ0 (h (wt +1 )) h (wt +1 )Hence, the expectation on the future marginal productivity of capitalmust in‡uence the expected investment opportunities.
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Adjustment Cost of Investment and Stabilization Policy

Assume that there is no capital gain and loss for simple analysis andeconomy is stationary:
qt +1 = qt = q, rte+1 = r e = E φ (h (w ))φ0 (h (w )) h (w )Then

(1 + ρ ) q = r eq =rKatsuya Takii (Institute)eδq + qreρ+δ
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Adjustment Cost of Investment and Stabilization Policy
Hence
I =ψ

re
ρ+δ

1 , r e = E φ (h (w ))φ0 (h (w )) h (w )Note that
re
re
re

re
=1)I =0ρ+δreδ > ρ)>1)I >0ρ+δreδ < ρ)<1)I <0ρ+δδ = ρ)

Hence, the long run condition can be seen as the situation that any
adjustment is …nished.
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Adjustment Cost of Investment and Stabilization Policy

De…ne ι ( ) such that
ψ
i=

re
ρ+δ

N

1

ι ( ρ ) , ι (r eδ) = 0, ι0 (ρ) < 0.

Remark: This function ignores that the expected investment
opportunities in‡uence investment decision. Hence, the following
analysis presume that the expected future returns on the investmentr e = E [φ (h (w )) φ0 (h (w )) h (w )], does not in‡uence investment.

Because r e δ 6= ρ in general, the marginal product of capital,φ (h (w )) φ0 (h (w )) h (w ), does not determine the interest rate.

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Adjustment Cost of Investment and Stabilization Policy

Supply Side determines (y , w )= φ (h (w )) T
w = φ0 (h (w )) Ty

Demand Side determines (ρ, P )ymsPKatsuya Takii (Institute)= c + ι (ρ) + θg
= k (ρ) y

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Adjustment Cost of Investment and Stabilization Policy
Supply side conditions still determine GDP and the wage rate.
Hence, …scal policy and monetary policy has no impact on GDP and
the wage rate.
1

2

As an increase in government expenditure cannot change output, it
reduces private expenditure.
Money supply simply raises the price level. Hence, money is stillneutral, too.

The main di¤erence is that …scal policy can change the real interest
rate. Because an increase in government expenditure reduces fund
for the private investment. Hence, the real interest rate increases.
To understand this, we can rewrite

Katsuya Takii (Institute)ι (ρ) = y

c

= y
= y

c

θg
g + (1nτ + ρb + trθ) gh

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Adjustment Cost of Investment and Stabilization Policyg’>gKatsuya Takii (Institute)

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Adjustment Cost of Investment and Stabilization Policy

Since a rise in government expenditure raises the interest rate, it
raises the opportunity cost of money holding. Therefore, it reduces
money demand. It means that money supply becomes larger thanmoney demand. Hence, the price of money, P1 , goes down. In
another word, we have in‡ation.
ρ ") k (ρ) #) k (ρ) y #)Katsuya Takii (Institute)ms#) P "P

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Adjustment Cost of Investment and Stabilization Policy

P

P*

y

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Adjustment Cost of Investment and Stabilization Policy

TheoremA permanent increase in public expenditure (which only a¤ect demandside) will be o¤set by a permanent reduction of private expenditure in thelong run. When the adjustment of investment is slow, it also raises thereal interest rate and causes in‡ation in the long run. An increase inmoney supply causes in‡ation, but does not a¤ect real variables in thelong run.

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Stabilization Policy in the Short Run

I de…ne the short run as the period during which l di¤ers fromequilibrium working hours in the long run equilibrium (h (w )):l 6 = h (w )where w is an equilibrium wage.
We further assume that actually employed hours, e, are an increasing
function of the price level:l = e (P ) , e 0 (P ) > 0
When an increase in demand raises output price, …rms can make more
pro…ts by employing labor.

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Stabilization Policy in the Short Run

Supply Side= φ (e (P )) T
w = φ0 (e (P )) Ty

Demand Sideyms
P

Katsuya Takii (Institute)= c + ι (ρ) + θg
= k (ρ) y

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Stabilization Policy in the Short Run
Aggregate Demand and Aggregate Supply in
the Short Run
P

Aggregate DemandyKatsuya Takii (Institute)

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Stabilization Policy in the Short Run

Suppose P is given. Then the demand side two equations determine
y and ρ.
ISLM

: y = c + ι (ρ) + θg
ms
= k (ρ) y:P

Two equations describe IS curve and LM curve. IS is theabbreviation of Investment and Saving. LM is the abbreviation of
Liquidity and Money.

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Stabilization Policy in the Short Run
Aggregate Demand and Aggregate Supply
when P is given
P

Aggregate DemandyKatsuya Takii (Institute)

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Keynesian Cross
Assume that P and ρ are given. The IS equation determines y .
y = c + i + θg
where i = ι (ρ).
Suppose thathc = c y + ρb + tr hi
τ n , c 0 [ ] 2 (0.1)

where tr h = (1 θ ) g . This assumption implies that consumption
depends only on the current disposable income.
De…ne the planned expenditure, y d as follows.
y d = c [y + ρb + (1θ) g

τ n ] + i + θg

Then IS equation is satis…ed when y = y d .
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Keynesian Cross

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Keynesian Crossg’>gKatsuya Takii (Institute)

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Keynesian Cross
Example,c = c0 + c1

(y + ρb + (1

θ) g

τn )

Then
yd

y

= c0 + c1 (y + ρb + (1 θ ) g τ n ) + i + θg
= c0 + c1 (y + ρb τ ) + i + [c1 (1 θ ) + θ ] g= y d = c0 + c1 (y + ρb τ n ) + i + [c1 (1 θ ) + θ ] g
1=
fc0 + c1 (ρb τ n ) + i + [c1 + (1 c1 ) θ ] g g
1 c1Multiplier E¤ectdyc1=+θ > θ
dg
1 c1
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Keynesian Cross

Note that we have assumed that government keeps the same amountof debt over time, bt +1 = bt , which means τ n = g + ρb. In this case,ydxd= c [y + ρb + (1 θ ) g= c [y θg ] + i + θg
= c [x ] + i(g + ρb )] + i + θg

where x = y θg and x d = y d θg , where x is the disposal income
and x d is the planed private expenditure. It means that goods
market clearing condition is satis…ed when x = x d .

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Keynesian Cross

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Keynesian Cross

Because x = yθg ,y = θg + x

Multiplier E¤ect: (because x is independent of g )
dy
=θ
dg

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IS Curve

Suppose that P is given. Then IS equation determines therelationship between y and ρ.
y = c + ι (ρ) + θg
In order to understand the relationship, note that Keynsian Cross
predicts that an increase in i results in an increase in y .

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IS Curvei’>iKatsuya Takii (Institute)

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IS Curve

Note also thati = ι0 (ρ) < 0, ρ ") i #Henceρ ") i #) y #This relationship can be depicted by IS curve.

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IS Curve

IS CurveyyKatsuya Takii (Institute)

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IS Curvedy
A shift in IS curve. Note that for any given ρ, dg> 0. Hence, anincrease in g shifts the IS curve to the left.

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Money Market
When the price P is given, money market determines interest rate ρ.
LM :

md
ms
=
= k (ρ) yP
P

Money Market in the Short Run

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Money Market
In the short run, an increase in money supply lowers interest rate.

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LM Curve
For example, suppose that k (ρ) = kρ . ThenmsP

= k (ρ) y =

ρ =

ky
ρ

kPy
ms

Hence, there is a negative relationship between ms and ρ.
Note that this relationship is true only if P is given. As we discuss
before, if an increase in money supply results in an increase in P, thenegative relationship may not hold. Moreover, if the expectedin‡ation, gPe , goes up, because ρn = ρr + gPe , the nominal interest
rate may increase.

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LM Curve
When P is given, LM equation also suggest a positive relationshipbetween y and ρ. This relationship is described by LM curve.

LM CurveyKatsuya Takii (Institute)

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LM CurvedρA shift in LM curve: Because dm s < 0, an increase in ms shifts LM
curve down.

yKatsuya Takii (Institute)

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IS-LM analysis
The impacts of a …scal policyg ") y ") md ") ρ ") i #) y # ...

gyKatsuya Takii (Institute)

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IS-LM analysis
The impacts of a monetary policyms ") ρ #) i ") y ") md ") ρ ") i #) y # ...

yKatsuya Takii (Institute)

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IS-LM analysis

Liquidity Trap: When nominal interest rate is near 0, money demandcan be in…nite, k (0) = ∞. In this case, an increase in money supply
cannot reduce nominal interest rate furthermore. Hence, LM curve
becomes ‡at. This situation is called liquidity trap.

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IS-LM analysis

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IS-LM analysis

Liquidity Trap and Fiscal PolicyyyKatsuya Takii (Institute)

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AS-AD analysis

Note that
ρ=

kPy
dρ
)
> 0.
s
m
dP

Hence,
P ") ρ ") i #) y #) md #) ρ #) i ") y " ...
It shows that the overall relationship between P and y is negative.
This relationship is depicted by AD curve.

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AS-AD analysis

P

P’

P
y’ y

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AS-AD analysis

A shift in AD curve. Note that for any given P, IS-LM analysis
suggests thatg ") y ", ms ") y "Hence, increases in government expenditure or/and in money supply
shifts AD curve to the right.

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AS-AD analysis
Theorem
An increase in government expenditure or money supply raises aggregate
demand in general.

PgyKatsuya Takii (Institute)

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AS-AD analysis
Theorem
An increase in government expenditure or money supply causes in‡ation
and raises GDP in the short run.

P

g

y

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The Short Run Aggregate Supply CurveAs I de…ned before, the short run is the period the number of
employed workers di¤ers from its long run equilibrium level. I also
assume that the number of employed workers is increasing in pricelevel in the short run.
1
2

Why does it di¤er from the equilibrium level in the short run?
How can I derive a positive employment function?.

I introduce two prototype models, which illustrate their main idea.

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The Short Run Aggregate Supply Curve
Sticky Nominal Wage Model:
Let me assume that the nominal wage is …xed in the short run:
W = W̄ . There might be several reasons for this assumption. For
example, the bargaining process with labor union may prevent a
reduction of nominal wage. For several contractual reasons, it may
be di¢ cult to change the nominal wage quickly.
Since labor demand curve is a decreasing in the real wage, if thenominal wage is rigid, labor demand must be an increasing in P:
P ") w =W̄#) φ0 (l ) T #) l "P

Note that φ0 (le ) is decreasing in le .
Note that because lowering nominal wage is much more di¢ cult thanincreasing it, …rms have more di¢ culties in adapting to the de‡ation.
This is one of a reason that economists prefer mild in‡ation tode‡ation and no in‡ation.
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The Short Run Aggregate Supply Curve

P’>P

h l

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The Short Run Aggregate Supply Curve
Imperfect Information Model: Another possible reason on an
increasing aggregate supply curve is the imperfection of information.
There are several variants of imperfect information model. I describe
a worker-misperception model to convey its main intuition.
Assume that …rms observe output price P, but workers cannot.
Hence, workers must make their inferences about price. Workers’
expected price is denoted as P e . Then workers respond to PWe .
h

W
Pe

=h

W P
P Pe

=h w

P
Pe

Suppose that the overall price level P goes up. However, workers do
not know a change in aggregate price. Hence, P e stays the same.
Hence, PPe goes up and supply curve shifts to right.
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The Short Run Aggregate Supply Curve

P’>P
w

hl

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From the Short Run to Long Run

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From the Short Run to Long Run

Adjustment under a Worker-Misperception Model
P

w

y

h l

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Discussions

Both models derives the short run aggregate supply curve. However,policy implication may change.
1

2

Sticky nominal wage model brings unemployment; the imperfect
information model does not. When nominal wage is sticky, there areworkers who are willing to work with lower wage. When information isimperfect, all workers and …rms agree on the market price given theirperception.
Policy implication di¤ers. Because there is unemployed workers understicky nominal wage model, an increase in demand can increase GDP by
employing more workers. However, if the imperfect information model
is correct, there are no unemployed workers. If active stabilization
policy itself brings the uncertain movement of price, it may increase
workers’further misperception. It reduces the welfare of an economy.

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Lost Decades

Japanese economy stagnated more than 10 years. It is di¢ cult tobelieve that only sticky price and noisy information can explain such along term stagnation.
There is one possibility that a short run deviation can be longer than
one predicted by the above mechanism: DEFLATIONARY SPIRALS.
Suppose that people start to expect a de‡ation, gPe < 0. Because
investment decisions depend on the real interest rate, IS equation
changes from IS1 to IS2 .
IS1 : y = c + ι (ρn + δ) + θg
IS2 : y = c + ι (ρnKatsuya Takii (Institute)gPe + δ) + θg

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Lost Decades

DEFLATIONARY SPIRALSy’ y

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Lost Decades

Hayashi and Prescott (2002) provide an alternative explanation.
Whatever the reason for the deviation, they believe that 10 years are
too long to justify the deviation from long run equilibrium. They
point out that the growth rate of TFP declines during 90s. That is,they argue that a demand stagnation cannot explain Japanese
problem, but supply side problem can. Following their arguments,
several researchers point out several misallocation of resources inJapanese economy during 90s.

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Lost Decades

P

w

T’>T

y

hl

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Lost Decades
Economists still do not have a consensus on 90s. But, because
Japanese economy is still stagnated after recovery, more economists
feel that it is di¢ cult to deny that Japanese economy has structuralproblems.
Nonetheless, we continuously observe the statement which demands
more expansionary monetary and …sical policy in this 20 years. Thisis understandable because macroeconomics suggests that as far as
there is no in‡ation, expansionary …scal and monetary policies canenhance GDP.
Does this mean that there is no cost of expansionary …scal andmonetary policies when there is no in‡ation?
No. Remember that our current analysis ignore the interaction between
now and future.
1
2
3

An increase in It does not increase Kt +1 .
E [MPKt +1 ] does not in‡uence It .
The future income, Yt +1 does not in‡uence Ct

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Lost Decades
There are several potential mechanisms that an expansionary …scal
and monetary policy can in‡uence the long run growth.
12Expansionary monetary policy lowers real interest rate and stimulate
investment, even if there are no large investment opportunities.
Because MPK declines as more capital is accumulated, the returns
from investment would be lower in the long run. In fact we observe
K and the declining MPK after 1990 (e.g., Fukao
an increase in Y
(2012)). Ando (2002), Hayashi (2006) and Saito (2008) claim that
there are the indications of over-investment.
Similar token, although our current model does not take into accountgovernment investment, if a part of government expenditure is used as
investment and the marginal productivity of government capitaldeclines, expansionary …scal policy reduces the return fromexpansionary …scal policy. This e¤ect is likely to have a similar impact
as a reduction in θ.

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Lost Decades

Because continuous investment without promissing investment
opportunities enforce us to misallocate resources to unproductive use,
it is likely to reduce aggregate productivity. In facts, Nishimura et al.
(2005), Cabarello et al. (2008) and Kwon et all. (2010) found thatthe indication of misallocation.
Note that our long run model suggests that the growth throughinvestment eventually ends and only productivity growth enhances
sustainable growth. Hence, this types of economic growth may not
be desiable in the long run.

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Lost Decades
Moreover, this long run e¤ects can also in‡uence current demand.
Note that an reduction in the expected future productivity reduces
the expected future MPK and Yt +1 . But as Fukao (2012) suggested,
reductions in MPKt +1 and Yt +1 can lower current demand.
1

2

A reduction in E [MPKt +1 ] reduces It . (According to Fukao (2012), a
reduction in TFP and labor are main sources of a reduction in MPK . )A reduction in Yt +1 reduces Ct .

These mechanisms also indicate that we can think about di¤erent
policies that can increase demand.
1

2

If deregulation increases the investment opportunities in future, it
increases E [MPKt +1 ] and It .
If there is a secured pension scheme, household may feel that they canspend more today and increases Ct , ( though the saving rate athousehold sector declines now.).

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Lost Decades

So, even if there is no de‡aton, we cannot say that there is no cost of
expansionary monetary and …sical policies. These considerations
suggest that the expansionary monetary and …sical policy (short run
policies) must be accompanied with the policy that can enhance thefuture productivity (long run policies).
Before a jump into an conclusion, because the existence of
unemployed workers is the main reason for the demand stabilization
policy, we need more careful examination of Japanese labor market.

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Assignment

Students must hand assignment 6 in at the next lecture.

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Stabilization Policy and Unemployment

In order to understand the importance of stabilization policy, we needto understand why unemployment exists.
There are several reasons to believe that unemployed workers exist in
the long run. Economists call the unemployment rate in the long run
the natural rate of unemployment. Unemployment in the short run
can be seen the deviation of unemployment rate from the natural rateof unemployment.
Unfortunately, the natural rate of unemployment may not be
constant. Therefore, it is di¢ cult to distinguish the short run
deviation from the natural rate of unemployment.

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The Natural Rate of Unemployment

Let me provide a model to explain the natural rate of unemployment:
Suppose that s fraction of employed workers are separated every day.
Suppose that m̃ fraction of unemployment workers meets a suitable job
and leave from an unemployment pool.

Assume that m̃ is an increasing function of labor market tightness, Θ:m̃ = mq (Θ) , q 0 (Θ) > 0, Θ =vuwhere m is the productivity of matching, v is the vacancy rate and uis the unemployment rate.

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The Natural Rate of Unemployment
The dynamics of the number of unemployed workers is
mq (Θt ) Ut ,Ut +1 = Ut + sEt

where Ut is the number of unemployed workers at date t and Et is
the number of employed workers at date t.
As unemployed workers and employed workers consists of labor force,
N = Et + U t
where N denotes labor force. Hence
Ut + 1 = Ut + s ( N= Ut + sN

Katsuya Takii (Institute)Ut )mq (Θt ) Ut

(s + mq (Θt )) Ut

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The Natural Rate of Unemployment
Natural Rate of Unemployment: I assume Ut +1 = Ut = U andΘt = Θ in the steady state. ThensN = (s + mq (Θ)) Ut
Hence, the steady state unemployment rate is
U
N

u

=

s
s + mq (Θ)

If matching probability is large, then unemployment rate is small:mq (Θ) ") u #If the separation rate is large, then unemployment rate is large:s ") u "Katsuya Takii (Institute)

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The Natural Rate of Unemployment

Evidence from Japanese labor market suggests that12

mq (Θ) and s has a negative correlation,
while mq (Θ) declined over time, s increased over time.

This evidence suggests that mq (Θ) and s are both in‡uenced by ademand e¤ect and a long run e¤ect.
In fact, an increase in s is coincided with a gradual increase in
non-regular workers during the same period.

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The Natural Rate of Unemployment

What might explain this long run trend?
Adaptation to Changes in Economic Environment: Firms always
face changes in economic environment. The arrival of new technology
may force …rm to …re unskilled workers, …rms may need to relocate theirplants in foreign countries under the pressure of global competition, or
people, especially women, may prefer the ‡exibility more than before.
Whatever the reasons, …rms must adapt to new environment. Hence,
the reallocation of labor is inevitable. They will potentially in‡uenceboth s and mq (Θ). An increase in s, but not mq (Θ) suggests thatJapanese …rms might be struggling to adapt to new environment.

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The Natural Rate of Unemployment

Assuming that m and s are constant, our model predicts therelationship between Θ and u.
u=ss + mq (Θ)

An increasing in labor market tightness reduces the unemployment
rate: Θ ") q (Θ) ") u #.

This relationship brings theoretical predictions on the Beveridge curveon the steady state.

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The Natural Rate of Unemployment
Beveridge curve display the relationship between the vacancy rate andunemployment rate. Because u is a decrease in Θ, Beveridge curve isdownward sloping.

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The Natural Rate of Unemployment
Obtaining data about u, mq (Θ), s and Θ, we can decompose thesources of the unemployment into three factors: m, s and Θ.
For example, compare two years: 1997 and 2007.
1

2

Unemployment rate u, matching probability mq (Θ), and separation
rate s in 2007 are roughly the same as that in 1997.
However, labor market tightness, Θ, in 2007 is much larger than thatin 1997.
Because mq (Θ) is the same but Θ is higher in 2007 than 1997, m
must be lower in 2007 than in 1997.

It would be di¢ cult, if not impossible, to justify a reduction of m inthis period by a change in demand. It is likely that the mismatch has
increased during this period. This means that a frictional
unemployment has increased during this period.
This reasoning is correct only if the data correctly captures Japanese
economy. Abe (2014) questioned this point.
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Frictional Unemployment

Frictional unemployment occurs because …nding a job takes time.
It is not easy to …nd where a job o¤er is, what skill requirement of thejob is and how much he expects to earn. It causes a temporal
unemployment. If only a small number of jobs is open, m would besmaller and frictional unemployment is larger.
Frictional unemployment causes a serious problem when changing
jobs requires changes in skills or locations. This type of
unemployment can be sometimes called structural unemployment.

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Wait Unemployment
Even if m and s are constant, if Θ increases, an unemployment rate
can be reduced. Why do not the vacancies increase in the long run
when there are many unemployed workers?
Sticky real wage can be an alternative reason for unemployment in
the long run. If real wage does not fall down to the equilibrium level,
real wage cannot equate demand to supply. Therefore, we willobserve excess supply of workers. Hence there is some rationing
mechanism there. But what prevents real wage from falling down.
This is called wait unemployment.
Note that this is not sticky nominal wage. Hence, if a real wage islarger than an equilibrium wage, an increase in demand cannot help
reducing unemployment.

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Wait Unemploymentw

hl

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Wait Unemployment

There are several theories to explain why the real wage does not fall
down.
1

2

3

Several protection for workers..For example, if a minimum wage
increases proportional to an increase in price, a real wage may notdecline.
Insider-Outsider Model...Incumbent workers resist their wage fromfalling down.
E¢ ciency wage theory...If high wage induces a high productivity, a …rm
prefers to keep high wage.

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Wait Unemployment
There are several reasons why high wage brings high productivity.
1

2

3

4

5

A high wage can increase workers’food consumption, which makes
workers healthy and therefore productive.
A higher wage can raise workers’incentive to work when managers
cannot monitor it. When wage is higher than an equilibrium wage,
workers in the …rm receive rent. Since workers do not loose this rent,
they will work harder in order to avoid the risk of being …red.
If able person has high reservation wage because he can receive bettero¤er from others, but if a …rm does not know who is able person,o¤ering a high wage raises the average quality of applicants.
A high wage prevents skilled workers from quitting a job. When wage
is higher than an equilibrium wage, skilled workers receives rent.
Therefore, they are less likely to quit the job.
A high wage may induce workers’e¤ort because people are likely totake reciprocal actions.

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Free Entry and Unemployment

Even if incumbents try to keep wage high, new …rms can o¤er lower
wage. Hence, Insider-outsider model and e¢ ciency wage theories
may not explain unemployment if there is a free entry.
Question: if there is a free entry, is it possible to support
unemployment?
Answer: Yes, if …rms must incur sunk costs (= the pre-investment
speci…c to the employed workers) before production. Examples of suchcosts are search costs, training costs or pre-committed rental price.

In other words, I claim that, if there is no minimum wage and entry
barriers, and if there are no sunk costs such as search costs andtraining costs, there is no unemployed workers in the long run and in
the short run.

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Free Entry and Unemployment
In addition, if we cannot write an explicit contract, observed
unemployment is not social desirable because of hold up problems.
Hold Up Problems: When there is the pre-investment, the lack ofcommitment on the returns to investment cause several problems.
When a …rm makes investment speci…c to a particular workers before
production, a …rm must expect to receive enough return from theinvestment. But if we cannot write an explicit contract on theinvestment, after the investment, workers can threaten …rms to leave.
Expecting this possibility, a …rm hesitates to makes enoughinvestment, which brings unemployment.
There are two reasons that we cannot write an explicit contract on
the investment.
1
2

A …rm must invest before meet workers: search costs.
It is impossible or di¢ cult to write a complete contract: training costs.

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Free Entry and Unemployment

A Version of Caballero and Hammour (1996).
A …rm searches and trains a worker by Cf and produces Ah output,
where h is workers’relation speci…c human capital.
The rental cost, training cost and search cost is sunk. But, after
paying this cost, the worker can walk away.

Suppose that if the worker walks away, the worker expects to receive
U, but a …rm cannot receive anything today. Therefore, the surplus
from this match is
S = Ah U

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Free Entry and Unemployment
Assume that the …rm cannot write and commit an wage before
investment and that the wage is determined by the bargaining of two
parties.
On one hand, the worker’s wage w from this bargaining can be
written by
w = βS + U.
If the worker decides to leave, the worker expects to get U. This is
the outside option of workers. The addition to this value, the worker
can receive β part of surplus. We call β the bargaining power ofworkers. Hence, if βS > 0, workers always prefer being employed.
On the other hand, If the worker leaves, a …rm does not receive
anything. Therefore, the …rm expected pro…t from this match wouldbe
J = (1 β) S.
That is, the …rm can receive (1match.
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Free Entry and Unemployment
Before the production, the …rm searches and trains workers. The freeentry condition implies that
Cf = J
Finally, we explain what determines the worker’s outside option, U.
The worker can …nd the similar job with probability e = 1 u where
e is the employment rate and u is unemployment rate. On the other
hand, with probability u, the worker cannot …nd a similar job andreceives unemployment bene…ts or wage from the secondary market.
This reservation value is denoted by z < w .
U = (1u ) w + uz = w

u (w

z)

where is the wage paid by the similar job. It shows that the outside
option of workers is lower than w due to the existence of
unemployment.
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Free Entry and Unemployment
Equilibrium: equilibrium consists of fS, J, U, w , u g that satis…es
De…nition of Surplus

S = AhU

Sharing Rulew
J

= βS + U= (1 β ) S

Free Entry Condition
Cf = J
Worker’s Outside Option
U=w

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u (w

z)

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Free Entry and Unemployment
Surplus: Substituting …rm’s pro…ts into the free entry condition= (1 β ) S
Cf
S =
1 β

Cf

The outside option of workers
S

= Ah

U

U

= Ah

S = AhCf
1βThis means that the worker’s outside option is smaller when thebargaining power of workers is large. If the bargaining power ofworkers is large, …rms cannot expect much pro…ts in the market and…rms are reluctant to enter the market. This must lower the outside
option of workers.
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Free Entry and Unemployment
Wage payment
w

=
=
=
=

βS + UβS + Ah
Ah
Ah

(1
Cf

S
β) S

The equation implies that the wage is independent of the bargaining
power. When the bargaining power is large, because the worker canreceive the large share of surplus, the wage is large. On the other
hand, when the bargaining power of workers is large, the worker’s
outside option is small. In this model, two opposite e¤ects are always
cancelled out and w is independent of β.

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Free Entry and Unemployment
Note that the outside option of workers is
U=w

u (w

z) .

It means that the outside option of workers is small when the wage
payment in the market is small or the number of unemployed workers
is large. Because w is independent of β, but U is increase in β, anincrease in β must increase the number of unemployed workers, u.
Unemployment Ratewu (wz) = Uu (wz) = wu ==Katsuya Takii (Institute)

U
βS

w

(1

z

=

β 1Cf βw z
βCfβ) [Ah Cf

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Free Entry and Unemployment

Unemployment Rate
u=

(1

βCf
β) (Ah Cfz)

Unemployment is positive when a speci…c investment exists, Cf .
Because of this speci…c investment, …rms hesitate to enter the market.
Therefore, job creation is smaller and unemployment is larger.
Unemployment is larger when the bargaining power of worker, β, islarger. If β = 0, no unemployment. Similar to insider and outsidemodel, a strong bargaining power of incumbent workers may increase
unemployment rate.

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Free Entry and Unemployment
Question: If a …rm can optimally choose β, does it choose 0?
Answer: No if the accumulation of relation speci…c investment
requires the e¤ort of workers.

Assume that in order to accumulate h, a worker must pay cost C2w h2 ,
then

max w
h

Cw 2
h
2

= βS + U
S = Ah U

st. w

Hence
max β (AhhKatsuya Takii (Institute)U) + U

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Free Entry and Unemployment

h: First order condition implies
βA = Cw hβAh =Cw
If β = 0, h = 0. In this case, the pro…t of the …rm
J = (1 β) S = (1 β) [Ah U ] = (1 β) U < 0. Therefore,
the …rm must choose β > 0.

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Free Entry and Unemployment
Wage payment: Because the wage payment is expressed as
w = βS + U
and pro…t 0 condition implies
S=

Cf
1

β

> 0.

the positive bargaining power β > 0 implies, the wage payment islarger than outside option, w > U. This is an essence of e¢ ciency
wage theory. Note that Cf > 0 is required for the positive surplus.
That is, the e¢ ciency wage theory also requires a …rm’s speci…cinvestment.

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Unemployment in the Short Run
In the short run the share of actually employed labor may deviate
from the long run equilibrium level. Note that imperfect information
model implies
e=h

W
Pe

=h

W P
P Pe

=h w

P
Pe

Because unemployment rate can be de…ned by u = 1
unemployment rate is an decreasing function of PPte .

e, thetut = uAssume that
u

Pt
Pte

= un

Pt
Pte
α ln

Pt
Pte

where u n is the natural rate of unemployment.
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Unemployment in the Short Run

Then
ut

= un
= un

where gpt = ln Pt

Katsuya Takii (Institute)α [ln Pt
α gpt

ln Pt 1(ln Pteln Pt 1 )]e
gpt

e = ln P e
ln Pt 1 and gpttMacroeconomics: Modern Macroeconomics 1(5)
ln Pt 1 .

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Unemployment in the Short Run
The equation, ut = u n

α gpt

e , describes Phillips curve.
gptPhillips curve shows a trade-o¤ between high in‡ation and highunemployment rate in the short run. When government wishes to killhigh in‡ation, government needs to accept a high unemployment rate.
When government wants to reduce the unemployment rate, they must
accept a high in‡ation rate.
Assuming that Phillips curve is stable in the short run, government
and Central Bank choose the optimal in‡ation rate andunemployment rate.
Unfortunately, Phillips curve cannot be stable in the long run. The
natural rate of unemployment and the expected in‡ation rate may
change in the long run.

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Unemployment in the Short Run

Even if active stabilization policy is desired, economists doubt theability of government to conduct timely stabilization policy.
Politicians have always a pressure from their supporter. Politician
may want to have boom right before their election.
In order to avoid this di¢ culty, many countries give central bank adiscretion of monetary policy. In this way, monetary policy can be
conducted without political intervention. There is a clear negative
relationship between the independence of central banks and in‡ation
rates.

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Unemployment in the Short Run
Even if a central bank has a reasonable discretion.
di¢ cult problem: commitment.

There is anothere , shows that there isNote that Phillips curve, ut = u n α gpt gpt
a trade-o¤ between in‡ation rate and unemployment rate. It meansthat a central bank cannot reduce both the in‡ation rate and theunemployment rate at the same time.

Suppose that Bank of Japan wants to minimize the following loss
function:γ 2.
L (u, π ) = ut + gpt
2
Suppose that Bank of Japan directly controls in‡ation, but not the
unemployment rate. Hence, it chooses gpt in order to minimize theloss function given
ut = u n
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α gpt

e
gpt

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Unemployment in the Short Run
Assume that the central bank announces gpt = 0 and assume thatpublic believes this announcement. Thenut = u nαgpt .

Substituting this equation into the loss function,γ 2
L (u, π ) = u n αgpt + gpt
2
Therefore, the …rst order condition implies
α = γgpt
and the optimal choice isα> 0.
γ
Hence, after announcing gpt = 0, Bank of Japan always have an
incentive to break the promise and to choose a positive in‡ation inorder to reduce unemployment. This is called time inconsistency
problem.
gpt =Katsuya Takii (Institute)

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Unemployment in the Short Run
Once, public knows that Bank of Japan cannot commit theire = α.
announcement and its best strategy is gpt = γα . Hence, gptγ
ut = u n

α gpt

α
,
γ

and
α
γ 2
+ gpt
.
γ
2
In this case the best strategy for Bank of Japan is againαgpt = .
γ
Therefore, this is consistent with Public’s expectation and we cansustain this equilibrium. In this case,L (u, π ) = u n

α gpt

ut = u n .
Hence an active stabilization policy eventually brings a positivein‡ation and the natural rate of unemployment.
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Unemployment in the Short Run

Now consider passive stabilization policy. That is, Bank of Japanannounces the rule of policy and commits the rule. Assume that thiscommitment is possible. In this example, assume that Bank of Japanannounces gpt = 0. Since there is a commitment device, public cane = 0. Then u = u n .
believe that the in‡ation rate is 0: gpttNote that a passive stabilization policy brings the better result for
Bank of Japan. Because of time inconsistency problem, publiccannot fully trust the bank’s announcement unless they can actuallycommit the policy. Without commitment, large bank’s discretionary
power may make a worse situation.

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Lucas’s Critique and Micro Foundation
So far, I assume a consumption function and a money demand
function. Because these functions are the results of individual
behaviors, changes in environment may in‡uence the properties ofthese functions.
Lucas (1976) argues that as people make decisions based on their
expectation on the future economic environment, the expected future
policy change is likely to in‡uence their expectation, and, therefore,
their decisions. It means that the estimated parameters on
consumption functions and money functions are likely to change whena government changes its policy. So we cannot use the estimatedparameters for policy simulations.
Following Lucas’s critique, many macroeconomists pay more attention
to the micro foundation of the consumer’s behavior and …rm’s
behavior, and derives consumption function, investment function andmoney demand function. I would like to review these discussions.
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Lucas’s Critique and Consumption Function
I would like to show the idea of Lucas’s critique by using a simple
consumer’s decision problem. Let me consider the following
consumption function
c = φ0 + φ1 xt + ε
where xt is disposable income.
Once you obtain the parameters φ1 and φ2 , one can conduct a policy
simulation.
Question: how much can we trust these estimated parameters?
Lucas (1976) said that it might be good for a short run prediction.
But it cannot be neither a basis of a policy evaluation nor a long run
prediction.
Let me demonstrate that Lucas’s points here by using a simple
example.
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Lucas’s Critique and Consumption Function

Consider the following household’s decision problem.
max fu (ct ) + βu (ct +1 )g

ct ,ct +1

s.t. at +1 + ct
a + ct + 1
where stn

= ( 1 + ρ ) a + wt
= ( 1 + ρ ) a t + 1 + wt + 1= at + 1 a

Here we assume that household must leave the same amount of asset
a as a bequest for the next generation and household can perfectlypredict future.

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Lucas’s Critique and Consumption Function
Permanent income
a + ct +1 = (1 + ρ) ((1 + ρ) a + wt ct ) + wt +1
wt + 1a + ct + 1= ( 1 + ρ ) a + wt c t +1+ρ
1+ρ
ct + 1awt + 1ct += ρa + wt + a+1+ρ
1+ρ 1+ρρa + wt +1= ρa + wt +
1+ρ
xt +1
= xt +
1+ρ
p= x
where xt = ρa + wt and xtp = xt + x1t++ρ1 . The parameter xtp is calleda permanent income or a life time income.
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Lucas’s Critique and Consumption Function

The budget constraint implies thatct +1 = (1 + ρ) (xtp

ct )Substituting the budget constraints into objective function, theoriginal problem is rewritten as
max fu (ct ) + βu [(1 + ρ) (xtpctKatsuya Takii (Institute)

Macroeconomics: Modern Macroeconomics 1ct )]g .

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Lucas’s Critique and Consumption Function

Lemma
Suppose that y = g (x ) and z = f (y ). Then
dz
dz dy
=
= f 0 (y ) g 0 (x )
dx
dy dx

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Lucas’s Critique and Consumption Function
First Order Condition: Applying the lemma to u (ct +1 ) andct +1 = (1 + ρ) (xtp ct )0 = u 0 ( ct )β ( 1 + ρ ) u 0 [ ct + 1 ]For my simple analysis, assume that u (c ) = γc
γ

ηct

ηct
i
2
η β ( 1 + ρ ) + 1 ct
h

γ

ct

Katsuya Takii (Institute)η 22c= β (1 + ρ ) [ γ= β (1 + ρ ) [ γηct +1 ]= γβ (1 + ρ)

β (1 + ρ ) [ γ= (1=η (1 + ρ) (xtpct )]
η (1 + ρ) xtp ]β (1 + ρ)) γ + βη (1 + ρ)2 xtp

β (1 + ρ )2
(1 β (1 + ρ)) γ
h
i+h
i xtp
2
2
η β (1 + ρ ) + 1
β (1 + ρ ) + 1

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Lucas’s Critique and Consumption Function

Consumption Function:
ct = A + Bxtp
where
A =

B

Katsuya Takii (Institute)=γ [1 β (1 + ρ)]hiη 1 + β (1 + ρ )2β (1 + ρ )2

1 + β (1 + ρ )2

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The Alternative Derivation of Consumption Function withHigh School Math
For my simple analysis, assume that u (c ) = γcβu [(1 + ρ) (xtp==
where x

ct )]

βη
2
[(1 + ρ) (xtp ct )]2
βη (1 + ρ)2 p2pβγ (1 + ρ) (xtct )(xt ct )2iη (1 + ρ )2 h p 2
βγ (1 + ρ) (xtp ct )2xtp ct + ct2
(xt )
2
h
i
βη (1 + ρ)2 2
ct
x + β η (1 + ρ)2 xtp γ (1 + ρ) ct
2
h
i
ρ )2
p 2
= β γ (1 + ρ) xtp η (1 +
x
.
(
)
t
2

= βγ ((1 + ρ) (xtp
=

η 2
2c .

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The Alternative Derivation of Consumption Function withHigh School Math

The maximization Problem
max fu (ct ) + βu [(1 + ρ) (xtp ct )]g .
ct(ηγct 2 ct2 + x i+
h
= max
ct
β η (1 + ρ)2 xtp γ (1 + ρ) ct

The …rst order condition ish
0 = γ ηct + β η (1 + ρ)2 xtp

Katsuya Takii (Institute)γ (1 + ρ )Macroeconomics: Modern Macroeconomics 1

iβη (1 +ρ)2 2
ct
2

)

.

βη (1 + ρ)2 ct

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The Alternative Derivation of Consumption Function withHigh School Math
Consumption Function: From the …rst order condition,hiη 1 + β ( 1 + ρ ) 2 ct

β (1 + ρ)] + βη (1 + ρ)2 xtp .

= γ [1Hence
ct = A + Bxtp
where
A =

B
Katsuya Takii (Institute)=γ [1 β (1 + ρ)]hiη 1 + β (1 + ρ )2β (1 + ρ )2

1 + β (1 + ρ )2

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Lucas’s Critique and Consumption Function

Let me assume that ρa + wt +1 = (1xtp

Katsuya Takii (Institute)τ x ) (ρa + wt ).

ρa + wt +1
1+ρ
(1 τ x ) (ρa + wt )= ρa + wt +
1+ρ
1 τx
= 1+
[ρa + wt ]1+ρ2 + ρ τx=xt
1+ρ

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Lucas’s Critique and Consumption Function
The empirically testable equation isct= A + Bxtp = A + B

2 + ρ τxxt ,1+ρ= φ0 + φ1 xt + εtih
2 +ρ τ x.
where φ0 + εt = A and φ1 = B
1 +ρNote that τ x can change φ1 . Based on the estimated φ0 and φ1 , a
policy maker can simulate the impact of tax policy on aggregate
consumption. But in fact, the future tax change a¤ects φ1 itself.
Hence we cannot trust any estimations φ1 for a basis of a policy
evaluation. If we consider a general equilibrium e¤ect, the results are
more devastated. Since most likely policy changes will a¤ect ρ, itchanges the parameters φ0 and φ1 . In other word, the parametersthat are estimated from the reduced form estimation are not robust.

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Lucas’s Critique and Consumption Function

After Lucas’s critique, macroeconomists agree that it is important toexamine the micro foundation of macroeconomics.
Macro economists assume that changes in policy cannot in‡uence
preference and technology such as γ, β and η. Hence, once weidentify γ, β and η, we may be able to conduct a policy simulation
based on these parameters. For this purpose, macroeconomic model
must be based on a reasonable micro foundation.
Currently, most macro economists use the dynamic stochastic
general equilibrium model as a foundation of macroeconomics.

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Permanent Income Hypothesis
The previous model derives current consumption as a function of apermanent income.
ct = A + Bxtp
The permanent income hypothesis has a strong implication: thetemporal movement of income has little impact on consumption.
Since the marginal bene…t from consumption is decreasing, consumers
prefer stable consumption to unstable consumption. Therefore, if it
is possible, consumers have always incentive to smooth theirconsumption. When consumption depends on the permanent
income, it is possible to smooth consumption by exchanging
consumption today and tomorrow.
Note that this implies that the even if a tax cut increases disposal
income, it may fail to increase private consumption, and therefore,
aggregate demand.
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Permanent Income Hypothesis

To see this implication formally, note the …rst order condition of theprevious problem implies
β (1 + ρ ) [ γηct +1 ] = γ

ηctAssume that ρt +1 = ρ is constant, which is satis…ed by steady state.
Then the equation implies that current consumption is the best
predictor of the future consumption. Once we observe current
consumption, we do not need any other information to predict ct .

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Permanent Income Hypothesis

In particular, if β = 1 +1 ρ , thenct + 1 = ct .
In this extreme case, the expected consumption is the same over time.
That is, the predicted value of the future consumption is just current
consumption, and a change in consumption is unpredictable.
∆ct = 0where ∆ct = ct +1

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Permanent Income Hypothesis
Evidence: Hall (1978) tested this observation. He cannot reject the
hypothesis that lagged values of either income and consumption
cannot predict a change in consumption. This result supports theimplication of the permanent income hypothesis. After Hall (1978),much empirical research was conducted for this issue. For example,
did the following regression: Campbell and Mankiw (1989)∆ct
where vt

= λ∆xt + vt ,= (1 λ ) εt ,where εt is the change in consumers’prediction of their permanent
income. Hence, if the permanent income hypothesis is right, λ isclose to 0, and if the traditional theory is correct, λ is close to 1.
They found λ = 0.42~0.52. This result indicates that consumption
partially responds to disposable income.
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Permanent Income Hypothesis
If a change in current income has an impact on a change in
consumption, what explains this behavior.
1

Serially Correlated Income: When today’s income is highly correlatedwith the future income, consumers can easily predict the steam of the
future income based on the current income. Remember that assuming
ρa + wt +1 = (1 τ x ) (ρa + wt ), I show that
xtp =

2

2 + ρ τxxt .
1+ρIt shows that a rise in current income increases the permanent income,and therefore consumption.
Liquidity Constraint: If some consumers cannot borrow enough money,
they cannot buy consumption goods more than their income.
Therefore, current disposable income limits consumption:
ct

xt .

If the optimal consumptions of many household are constrained bycurrent income, and if current income increases, then obviously,
consumers will increase their consumption.
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Ricardian Equivalence and Government Debt

One of the interesting application of the permanent income
hypothesis is government debt. When government …nances its
expenditure by taxes, we don’t see any multiplier e¤ect even in the
short run. What if government issues its bond. In this way, it does
not increase tax burden today.
However, if consumers care about their permanent income, they are
worried not only about today’s income, but tomorrow’s income.
Since today’s debt can be seen as the future tax burden, it does notchange their permanent income. Therefore, consumers do not change
their consumption decision. This is called Ricardian Equivalence.

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Ricardian Equivalence and Government Debt
Let me formally analyze Ricardian Equivalence. Assume that
population is 1 for a simple analysis. In order to …nance government
expenditure, gt , government either imposes a lump sum tax: τ t or
issues a bond, bt . Assume that when government issues a bond atdate t, it must pay back to consumers at date t + 1. , thegovernment’s budget constraint is
bt +1 = gt
gt +1 = τ t +1

τt ,(1 + ρ) bt +1 .

We can derive a government’s intertemporal budget constraint.
gt +1 = τ t +1 (1 + ρ) (gtτ t +1gt +1
τt += gt +
.
1+ρ
1+ρ

τt )

This equation shows that the present value of tax revenue must be
equal to the present value of government expenditure.
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Ricardian Equivalence and Government Debt
Let wt = wt
ct +

τ t . Following the previous argument, it is shown thatct + 11+ρ

ρa + wt +1
1+ρ
ρa + wt τ t +1
= ρa + wt τ t +1+ρ
ρa + wt
τ t +1= ρ t a + wt +
τt +
1+ρ
1+ρ
ρa + wt
gt +1= ρ t a + wt +
gt +
1+ρ
1+ρ

= ρat a + wt +Note that the budget constraint does not depend on neither tax nor
bond. That is, issuing bond does not change consumers’permanent
income, therefore it does not change consumption.
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Ricardian Equivalence and Government Debt
Deviation from Ricardian Equivalence: In reality, Ricardianequivalence does not perfectly hold. There are several reasons that
Ricardian Equivalence may not hold.
12

3

Liquidity Constraint: If there is liquidity constraints, the reduction oftax can increase disposable income and raises consumption. More
importantly, if a government faces liquidity constraint because of thefear of default, by de…nition, government cannot issue a bond to…nance expenditure.
Transfer across generations: If parents do not care about their children,
parents can enjoy low tax today and enforce their children to pay forthem. It means that if they care their children like themselves,
Ricardian equivalence can still hold. However, if they do not care theirchildren, an increase in government debt might induce their demand.
Distortional tax: If a change in a tax in‡uences the marginal bene…t or
cost of consumption, it a¤ects consumer’s decision and consumption.
In particular, if government raises capital income tax, consumers are
discouraged to save and increases consumption. In this case, obviously
tax schedule matters.

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Assignment

Students must hand assignment 7 in at the next lecture.

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Final Words

Congratulation. This the end of this lecture. You are now standing
in front of the door of a graduate level macroeconomics. It uses the
dynamic stochastic general equilibrium model as a basic tool. If you…nd it interesting, see you again somewhere.

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