Macroeconomics:A

Growth Theory Perspective

Mark Huggett

August 14,2013

Contents

1 Introduction 3

2 Measurement of Output and Prices 5

2.1 Output and Income Accounting...................5

2.2 Simple Example Economies.....................9

2.3 Comparing GDP Across Countries.................11

2.4 Price Indices.............................12

2.5 Cost-of-Living Index.........................14

2.6 Deﬁnitions...............................17

3 Growth Theory 18

3.1 Production Function.........................18

3.2 Solow Growth Model.........................23

3.2.1 The Basic Solow Model...................23

3.2.2 The Full Solow Model....................28

3.3 Explaining Kaldor’s Growth Facts.................30

3.4 Evaluating Solow Growth Theory..................33

3.5 Golden Rule..............................35

3.5.1 Bad Allocations........................36

3.5.2 Observable Implications of Bad Allocations........38

3.6 Growth Accounting..........................41

3.6.1 Growth Accounting:Theory................41

3.6.2 US Economy:1909-49....................43

3.6.3 The Asian Growth “Miracle”................47

3.7 Deﬁnitions...............................49

4 Dynamic Consumer Theory 50

4.1 Static Consumer Theory:Two Good Case.............50

4.2 Dynamic Consumer Theory:Two Periods.............52

4.3 Dynamic Consumer Theory:Many Periods............55

4.4 Some Uses Of The Model......................57

4.4.1 Consumption Patterns Over The Life Cycle........58

4.4.2 Temporary vs Permanent Shocks..............60

4.4.3 Savings Rate Observations.................61

1

4.5 Overview...............................63

4.6 Deﬁnitions...............................63

5 Life-Cycle Model 64

5.1 Benchmark Model..........................65

5.2 How the Benchmark Model Works.................66

5.3 Analyzing a One-Time Shock....................69

5.4 Can Model Allocations Be Improved?...............70

5.5 Review of Marginal Conditions...................73

5.6 Deﬁnitions...............................75

6 Business-Cycle Fluctuations 76

6.1 Business-Cycle Facts.........................76

6.2 Outlines of an Unsuccessful Theory.................81

6.3 Technology Shocks and Business Cycles..............84

6.3.1 A Model with Technology Shocks..............85

6.4 The Keynesian View.........................87

6.4.1 A Simple Keynesian Model.................89

6.5 Smoothing Out the Business Cycle.................91

6.5.1 Expected Utility Theory...................91

6.5.2 Gain to Eliminating Business Cycles............93

6.6 Deﬁnitions...............................96

7 Fiscal Policy 97

7.1 Accounting Framework........................98

7.2 Present-Value Constraint......................99

7.3 Fiscal Policy in the Life-Cycle Model................102

7.4 Three Ways to Finance a War....................104

7.5 Multipliers..............................105

7.6 Social Security Systems.......................107

7.6.1 Social Security:Theory...................108

7.6.2 Social Security:Some US Facts...............111

7.7 Overview...............................112

7.8 Deﬁnitions...............................113

2

Chapter 1

Introduction

To get an idea of what macroeconomics is about,it is useful to list some ques-

tions that macroeconomists try to answer:

1.What explains growth in gross domestic product (GDP) per capita?

2.What explains business-cycle ﬂuctuations?

3.If the government provides a temporary tax cut,then what will happen

to GDP?

4.Should the government try to smooth out business-cycle ﬂuctuations?

A ﬁrst take on these questions is that some are positive and some are norma-

tive.Speciﬁcally,questions 1-3 are positive in that they ask one to explain what

is or what will be.While answering these questions will require some theoreti-

cal framework,they do not necessarily require a system of values.In contrast,

question 4 is normative.An answer will rely at least implicitly upon a system

of values.To see this,suppose that two economists agree on how a particular

proposal to smooth out aggregate ﬂuctuations will impact the economy.Even if

they agree on this as a matter of positive economics they could still disagree on

the issue of staying with the status quo or adopting the policy proposal.The

reason is simply that they could still value the consequences quite diﬀerently.

Asecond take on these questions is that they require a theoretical framework

that is capable of addressing how the economy as a whole works.This is because

the questions do not deal only with a small part of the economy.For this reason,

partial equilibrium methods which are common in microeconomics will be of

doubtful relevance.

1

Instead,general equilibrium methods will be key.Such

1

One might think that partial equilibriummethods could be useful for understanding how

a freeze in Florida will impact the price of navel oranges in the supermarket.An argument

could be based on the idea that the quantity demanded depends mainly on price,preferences,

national income and the prices of other goods.Near-term supply depends mainly on price,

weather and the stock of mature orange trees.If developments in the navel orange market

have a negligible impact on demand side variables such as preferences,national income and

3

methods determine how all relevant variables (i.e.prices and quantities of all

goods and services) are simultaneously determined.This is one reason why

macroeconomics is diﬃcult.

The approach to answering these questions in this book will be almost en-

tirely based on microeconomic principles.A key assumption employed widely in

microeconomics is that individual consumers and ﬁrms are maximizers.Thus,

in microeconomics consumers pick the best choices that are within their budget

sets and ﬁrms choose inputs to maximize proﬁt.An implication of the assump-

tion that consumers are making best choices is that persuasive arguments for

how government action can produce welfare gains will need to be more subtle

than a layman may at ﬁrst appreciate.

In making best choices,we will also assume that consumers and ﬁrms are

forward looking.Consumers are forward looking in that they have a reasoned

view about the possible shocks that may impact the economy and on how shocks

impact variables (e.g.wages and interest rates) that shape budget sets.This

assumption is called rational expectations in the macroeconomic literature.This

assumption is not the same as saying that these agents are clairvoyant.As we

will see later on,the response of such forward looking consumers to shocks

anticipated to be temporary can diﬀer greatly from the response to shocks of

the same magnitude that are anticipated to be permanent.The assumption

of forward looking agents will be especially important in contemplating the

eﬀects of alternative government policies.The analysis will be based on the

assumption that consumers understand the eﬀects of new policies and make

reasoned decisions based on how the world works under the new policies.This

use of the forward looking hypothesis in this context seems much stronger than

in others as past observations may be of limited relevance for forecasting the

future.

A long-standing view in economics is that if one had a tolerably-good theory

of long-run growth then such a framework would be a useful starting point for

addressing a wide range of macroeconomic questions.The structure of this book

reﬂects this view.Thus,the book starts out by developing a theory of long-run

growth.The models used in this book are variants of the neoclassical growth

model.Most undergraduate-level,macroeconomic textbooks do not integrate

standard growth theory and traditional macroeconomic issues to the degree to

which it is carried out here.

the prices of other goods,then the change in the price of navel oranges might usefully be

analyzed holding all items other than weather ﬁxed.This is a verbal description of the basis

for ﬁxing a demand curve while shifting the supply curve.This approach is unlikely to be

useful for addressing questions about the aggregate economy exactly because the other things

equal assumption is bound to be a poor assumption.

4

Chapter 2

Measurement of Output

and Prices

2.1 Output and Income Accounting

The National Income and Product Accounts (NIPA) are a conceptual frame-

work for organizing data on the production of goods and services and data on

the incomes received by factors of production.We give a brief sketch of this

accounting framework.

We start out with the concept of nominal Gross Domestic Product (GDP).

Nominal GDP can be deﬁned as the total value of all ﬁnal goods and services

produced domestically at current-year prices.We will often be more interested

in the concept of real GDP.We deﬁne real GDP as the total value of all ﬁnal

goods and services produced domestically at base-year prices.

We will now act as humble GDP accountants and develop three accounting

approaches to compute nominal GDP,which is denoted Y below.Thus,we

need to develop some ways of adding observable stuﬀ up so that the sum equals

GDP.In what follows

i

denotes the operation of summation over all the items

indexed by the symbol i.

1

1.Final Sales (Expenditure) Method

Y =

i

p

i

y

i

p

i

and y

i

are the price and quantity produced of ﬁnal good i.

This method involves ﬁrst getting a list of all the ﬁnal goods.Imagine that

these are numbered so that i indicates the number of the good.The method

1

For example,if i indexes four items i = 1,2,3,4 then the symbol

i

x

i

is a short-hand

notation for the sum x

1

+x

2

+x

3

+x

4

.In short,

i

x

i

= x

1

+x

2

+x

3

+x

4

.

5

states that we ﬁgure out the total expenditure p

i

y

i

on ﬁnal good i and then

add up these expenditures across all the goods on this comprehensive list.For

this method to work we do not need to observe how much is the ﬁnal goods

output y

i

and the price p

i

of good i but just the product of these which is the

expenditure.In some parts of the economy prices and physical quantities are

easily determined.For example,for oil producers output can be measured in

units of barrels of oil of a given quality grade and prices are stated in dollars

per barrel.However,in the legal sector one can observe the total expenditure

on legal services but the units in which the output of these legal services can be

measured is unclear.

2.Value Added Method

Y =

i

V A

i

V A

i

is the value added of ﬁrm i

V A

i

= Sales

i

−Intermediate Goods Purchased

i

The Value Added method involves creating a list of all the ﬁrms in the

economy.We can index a ﬁrm by its number i on this list.The method then

computes the value added of each ﬁrm and adds the value added of all the

diﬀerent ﬁrms together.In the simple case in which a ﬁrm produces just a

single good,then the dollar value of the sales of the ﬁrm could be thought of as

equaling p

i

y

i

- the product of the price of the good and the quantity produced.

The term Intermediate Goods Purchased

i

in the formula above stands for the

value of all the intermediate goods that are purchased by ﬁrm i.

An intermediate good is a good produced by a ﬁrm that is sold to another

ﬁrm to be used in production.Thus,an intermediate good does not leave the

ﬁrm sector.A ﬁnal good is a good produced by a ﬁrm and sold directly to a

household.An example of an intermediate good is the corn which is produced

by a farmer and sold to Kelloggs to be converted into Kelloggs Corn Flakes.The

Corn Flakes produced and sold to households counts as a ﬁnal good.Any corn

which is produced by a farmer and sold directly to households is considered to be

a ﬁnal good.Thus,some part of the total production of corn is an intermediate

good and some part is a ﬁnal good.

3.Factor Income Method

Y = 1 +2 +3 = NationalIncome +2 +3 = NetDomesticProduct +3

1 = Wages +Proprietor

sIncome +Corp.Profit +Interest +Rent

2 = IndirectTaxes −NetForeignFactorIncome

3 = Depreciation

The Factor Income method is not as easily stated or as easily explained as the

other methods.However,the basic idea behind this method is not complicated.

6

The basic idea is that all the value of the ﬁnal production of the ﬁrm sector of

the economy has to be paid out to the owners of the capital and labor (factors

of production) that produce this output.Thus,instead of keeping track of the

value of ﬁnal goods produced we could just count up all the incomes (factor

payments) paid to factors of production.

Figure 2.1:Plumbing Diagram

This simple idea is often illustrated using the Plumbing Diagram in Figure

2.1.The intuition is that the payments from households to ﬁrms for ﬁnal goods

7

can be viewed as one ﬂow of water through a pipe and,in the absense of any

leakages,this amount of water must also ﬂow from ﬁrms to households in the

form of payments to owners of the labor L and capital K used in production.

The next step is to try to make this idea work in practice.This is where

things get messy.First,make a list of distinct types of payments to factors of

production.This would include (i) wages paid to employees,(ii) all the income

paid to sole Proprietors - those who are self employed,(iii) all the corporate

proﬁt paid out by ﬁrms organized as corporations,(iv) the net interest paid by

ﬁrms to lenders and (v) the rent paid by ﬁrms or individuals for using capital

(e.g.using a building or using machinery for a period of time).This does not

sound so diﬃcult,but some tricky issues arise in practice (e.g.do homeown-

ers eﬀectively pay rent to themselves?YES!).This thinking accounts for the

items listed under item 1 above,which is labeled National Income in the NIPA

accounts.You may think that income accountants might be done at this stage

and this would be true if there were no leakages or other complications to deal

with.

One important leakage is due to government.Speciﬁcally,governments

sometimes take away some part of a ﬁrm’s income before the ﬁrm has had

a chance to pay out this income to labor and capital.Sales taxes in the United

States are an example of this.So income accountants need to add this leakage

back in (this accounts for the term labeled Indirect Taxes in the Income ap-

proach) so that both the income and expenditure approach will count the same

thing.Thus,it is apparent that in practice Figure 2.1 needs to be modiﬁed to

capture this important leakage.

Depreciation is an important complication that must be addressed.This is

at ﬁrst mysterious.Here the issue is not whether or not physical capital wears

out over time or by use.Instead,the Depreciation term is added back in in the

GDP formula for the factor income method simply because income accountants

end up using data on corporate proﬁts produced by following corporate tax

laws.In a stylized calculation of corporate proﬁts,a corporation starts with

total revenue and then subtracts wages and depreciation to get to corporate

proﬁts.The upshot is that the sum of wages and corporate proﬁt does not

equal the value of such a ﬁrm’s sales of ﬁnal goods- its revenue.One would be

missing the ”depreciation” subtracted.Thus,the depreciation calculated by the

corporate accountants needs to be tacked back on so that all revenue of the ﬁrm

is accounted for in payments to owners of the factor inputs used by the ﬁrm.

This accounts for why a Depreciation term is added in term 3 of the formula

for the Income method.I told you that this would be messy.

Further Comments:

1.Equivalence of the Final Sales and the Value Added Method

In essence,the value added approach must produce the same number as the

ﬁnal sales approach because the value of intermediate goods production enters

positively in a calculation of one ﬁrm’s value added but enters negatively,and

of equal value,in the calculation of value added of other ﬁrms.Thus,the value

8

added approach amounts to a tricky way of counting only the value of the ﬁnal

goods and services produced.Of course,this is exactly what needs to be true if

both approaches are equivalent.

2.Diﬀerence between GDP and GNP

GNP (Gross National Product) used to be (in the 1980’s or so) the stan-

dard measure of income highlighted in newspapers and in government policy

discussions in the United States.Nominal GNP is deﬁned as the total value of

income paid to all nationally owned factors of production in a period of time at

current prices.Unlike GDP,GNP is not a geographic concept.It adds together

income paid to nationally owned factors of production regardless of where in

the world they are located.GDP is based on geographic borders of a country

as it focuses on the income paid to domestically located factors of production

or,alternatively,the value of all ﬁnal goods produced domestically.Given the

deﬁnition above,it is clear that GNP equals GDP plus net foreign factor in-

come.The term net foreign factor income adds in all the wage payments and

capital payments to labor and capital owned by U.S.nationals located abroad

but subtracts the wage and capital paymenbts to foreign labor or foreign owned

capital located within the geographic conﬁnes of the United States.For the US

economy,GDP and GNP do not diﬀer dramatically.

2.2 Simple Example Economies

We examine how to do GDP accounting in simple example economies.The

example economies are highly stylized so as to highlight as clearly as possi-

ble how GDP accounting handles three important issues:intermediate goods,

depreciation and the distinction between GDP and GNP.

Example 1:Highlight Intermediate Goods

A Farmer produces y

w

=10 wheat w/labor

A Miller produces y

f

=10 ﬂour w/labor and 10 wheat

A Baker produces y

b

= 10 bread w/labor and 10 ﬂour

Price Data:p

w

= 1,p

f

= 2,p

b

= 4 per unit of each good

Three methods are employed to compute GDP:

Y =

i

p

i

y

i

= p

w

y

w

+p

f

y

f

+p

b

y

b

= 1 ×0 +2 ×0 +4 ×10 = 40

Y =

i

V A

i

= V A

1

+V A

2

+V A

3

= 10 +(20 −10) +(40 −20) =40

Y = Wages +Profit +Prop.Income = 0 +0 +40 = 40

To apply the expenditure approach it is critical to keep in mind the distinc-

tion between ﬁnal goods production,intermediate goods production and the

total production of a good.In this example,the total production of wheat is

10 units but all of this is sold to the Miller and converted into ﬂour.Thus,

of the 10 units of total production of wheat 10 counts as intermediate goods

9

production and 0 counts as ﬁnal goods production.Recall that the production

of some amount of a good is considered to be ﬁnal goods production if it is sold

directly to households,the production of some amount of a good which is sold

to another goods producer is considered to be intermediate goods production.

It is useful to look again at the Plumbing Diagram in Figure 2.1 to see this

distinction.

To apply the value added approach,we simply calculate the value added for

each ﬁrm.In this example,the Farmer,Miller and Baker are considered to be

diﬀerent ﬁrms.The value added of each ﬁrm is measured in a common unit of

account which is taken to be dollars in this example.One could use a diﬀerent

but common unit of account,such as bread,if one wanted to do so.There is

nothing wrong with doing GDP accounting in a diﬀerent unit of account.

To apply the income approach,we need to make some assumption on how

the Farmer,Miller and Baker are organized.They could be organized as sole

proprietors or as ﬁrms which pay out wages and proﬁts.The text of this example

did not provide this information.If we assume that each is a sole proprietor

then the income of each is simply value added in which case the Farmer’s and

the Miller’s income are both 10,whereas the Baker’s income is 20.This is the

assumption used above.

Example 2:Highlight the Treatment of Depreciation

Firm 1 produces 20 dollars of a consumption good w/labor and capital.

Profit

1

= output

1

−wages

1

−depreciation

1

= 20 −10 −5 =5

Firm 2 produces 10 dollars of a capital good w/labor

Profit

2

= output

2

−wages

2

−depreciation

2

= 10 −10 −0 =0

Three methods are employed to compute GDP:

Y = C +I +G = 20 +10 +0 = 30

Y =

i

V A

i

= V A

1

+V A

2

= (20 −0) +(10 −0) = 30

Y = Wages +Corp Profit +Depreciation =20 +5 +5 = 20

The accounting equation Y = C +I +G,commonly taught in introductory

courses,is simply a version of the expenditure approach.When we use this equa-

tion (rather than Y =

i

p

i

y

i

) what we are doing is grouping the expenditure

on ﬁnal goods and services by category - C for consumption I for investment and

G for government spending - and then adding across these categories.Of course,

one needs to be careful to realize that government purchases of goods and ser-

vices that enter into the G term above are not to be confused with government

transfer payments (e.g.government social security checks).The government

purchases G that enter the NIPA accounts in the equation Y = C +I +G are

actual expenditures on goods and services such the expenditure on elementary

school education by local governments.

10

If one reﬂects on example 2 it is clear that the calculation of GDP is not

sensitive to how ”depreciation” is calculated.The basic idea of the income

approach is that as long as the value of the output of all ﬁrms is paid to factors

of production (i.e.owners of capital and labor),then the sumof factor payments

must equal the value of this ﬁnal output.To clarify this point,suppose that

the corporate accountants (or the corporate tax laws) change what counts as

”depreciation”.Suppose to be concrete that depreciation for ﬁrm 1 in example

2 is now 10 rather than 5.GDP computed using the income method will still

be 20.The reason is that corporate proﬁts shrink by 5 but depreciation grows

by 5.GDP,as measured by the income approach,is unchanged.

Example 3:Highlight GDP vs GNP

A small country produces $10 worth of vacation services with labor and

capital.Labor receives 5 and capital receives 5.The countries’ nationals do not

own any capital.Under these assumptions,what is GDP and GNP?

GDP = wages +profit =5 +5 = 10

GNP = GDP +net FOREIGN factor income = 10 −5 = 5

2.3 Comparing GDP Across Countries

For less developed countries it is typically the case that GDP per capita ex-

pressed as a ratio to GDP per capita in the US is much smaller when one con-

verts a countries GDP into US dollars using exchange rates than when making

these comparisons by purchasing power parity (PPP) type methods.I presented

a Table in class showing that India’s GDP per capita relative to US GDP per

capita increases by a factor of 5 when one uses PPP methods as opposed to

exchange rate based methods.Thus,the economic and geopolitical signiﬁcance

of the developing world becomes vastly more important when GDP statistics

are calculated and viewed in this way.

Below we highlight the exchange rate comparison (method 1) versus two

PPP methods.The ﬁrst PPP method (method 2) works in two steps.First,

one computes GDP in each country in each countries currency.Then one deﬁnes

a common basket of goods for which it is possible to ﬁnd the cost of purchasing

the basket in each country.Denote the basket by quantities (x

1

,x

2

,...,x

n

) of

the n distinct goods in the basket.Clearly,one has a diﬀerent PPP exchange

rate for each basket.The Economist magazine regularly provides its Big Mac

index as an example of how this method works.This index is simple as the

basket consists of one widely available and uniform good world wide:the Big

Mac.Given a choice of a common basket,the PPP exchange rate is just the US

cost of the basket as a ratio to the Indian cost of the basket.

Method 3 is a very diﬀerent comparison that is based on calculating GDP

across countries using a common set of “world relative prices”.There is a

literature on how to weight country speciﬁc relative prices to get the world

11

relative prices for pairs of goods,but we will not get into the merits of diﬀerent

schemes to assign these weights.The comparisons based on method 3 are the

basis for a large empirical literature in economics that is based on a longstanding

research project to improve international comparisons.

Method 1:Exchange Rates e

Compare GDP

US

to eGDP

India

e- exchange rate in units of Dollars per Indian Currency

Method 2:PPP Exchange Rate e

∗

Compare GDP

US

to e

∗

GDP

India

PPP exchange rate e

∗

=

i

p

US

i

x

i

/

i

p

India

i

x

i

Method 3:PPP (Penn World Tables) Comparisons

Compare GDP

US

=

i

p

i

y

US

i

to GDP

India

=

i

p

i

y

India

i

p

i

- “world relative price” of good i

Comment:You can download data from the Penn World Tables by going

to the following web site (http://pwt.econ.upenn.edu/).This is the standard

data source that economists use to make cross-country GDP comparisons.

2.4 Price Indices

Two standard price indices are presented below.The ﬁrst is a “ﬁxed basket”

index.The standard example of this type of index is the Consumer Price Index

(CPI),which is widely reported on in the press.In this type of price index

one constructs a weighted average of the price of goods over time where the

“weights” one uses are the ﬁxed quantities x

i

of the diﬀerent goods i in the

basket.We normalize the CPI by dividing by

i

p

∗

i

x

i

,which is the cost of the

basket in the base year.

CPI

t

=

i

p

it

x

i

i

p

∗

i

x

i

• x

i

is the quantity of good i in the basket of goods.

• p

it

is the price of good i at time t.

• p

∗

i

is the price of good i in the base year.

• The numerator of the CPI is the cost of the basket in year t,whereas the

denominator is the cost of the same basket in the base year.Note that as

deﬁned above the CPI is equal to 1.0 in the base year.In some textbooks,

the CPI as deﬁned above is multiplied by 100 so that the index equals 100

in the base year rather than 1.

12

The second price index is a “time-varying basket” index.The standard

example of this type of index is the GDP Deﬂator.This type of price index is

also a weighted average of prices.In this case the weights y

it

are the quantities

of the diﬀerent ﬁnal goods produced in year t.From a mathematical point of

view,the key diﬀerence between the two indices is that in one the weights do

not change but in the other the weights change over time.

DEFLATOR

t

=

i

p

it

y

it

i

p

∗

i

y

it

• y

it

is the quantity of ﬁnal good i in the basket in year t.

• p

it

is the price of good i at time t.

• p

∗

i

is the price of good i in the base year.

• The numerator of the GDP DEFLATOR is nominal GDP in year t.This

follows if the weights y

it

are the quantities of ﬁnal good i produced in

year t.The denominator is real GDP in year t.The index thus tells one

the cost of buying up current year ﬁnal output in current years prices

compared to buying it up in base year prices.

Why is it important to measure price indices such as the CPI accurately?

Here are some standard answers:

1.The CPI is used to index nominal social security retirement payments.

Thus,any systematic bias will be compounded year after year.If the CPI

is biased upwards as a measure of the “cost of living” in the sense that

the CPI tends to grow faster than a true cost of living index,then social

security payments will eﬀectively bear interest.This can turn out to be

in aggregate ﬁnancial terms a very big deal.

2.The federal income tax code in the United States links tax brackets to

the CPI.Thus,absent a change in legislation,if the inﬂation rate is 10

percent then the level of income at which a given tax rate applies is also

raised by 10 percent.Thus,any systematic bias in measured inﬂation can

increase or decrease real tax revenue as inﬂation occurs.

3.The price data collected by the Bureau of Labor Statistics in the U.S.is

used to compute GDP.Speciﬁcally,one can ﬁgure out the quantity of the

diﬀerent ﬁnal goods that are produced each year by dividing expenditures

on these goods by prices.So if the prices are too high,then the quantities

produced,which one infers from price and expenditure data,are too low.

This can be important in computing GDP growth rates.Suppose it is the

case that year by year the calculated inﬂation rate in a speciﬁc good is

too high in the sense of higher than true.Then the growth rate of real

GDP (using base year prices) will be too low.

13

2.5 Cost-of-Living Index

One of the important uses of actual price indices is as an empirical measure of

the “cost of living”.To an economist this term means something quite diﬀerent

fromwhat a layman might think that it means.To an economist,a cost-of-living

index is a theoretical concept.Acost-of-living index measures the minimumcost

of achieving a ﬁxed level of utility over time as prices change.Thus,the “cost of

living” is well deﬁned within consumer theory.Within the theory,(i) consumers

have preferences over diﬀerent bundles of goods,(ii) preference rankings among

bundles can be represented by a utility function,(iii) the utility function does

not change over time even though goods prices and consumer incomes may be

changing and (iv) the consumer picks the best consumption bundle which is in

the budget constraint,deﬁned by prices and consumer income.This is standard

vanilla consumer theory from introductory-level economics.

Figure 2.2 illustrates this idea in the case where there are only two goods:

burgers and beer.Here the consumer chooses a point (x

∗

1

,x

∗

2

) which is a best

choice,given prices and income.Note that the consumer’s indiﬀerence curve

passing through (x

∗

1

,x

∗

2

) is tangent to the budget line which also passes through

(x

∗

1

,x

∗

2

).Tangency reﬂects the idea that any bundle giving higher utility (located

northeast of (x

∗

1

,x

∗

2

)) is not in the budget set - not aﬀordable given prices and

consumer income.

Now suppose that the people in charge of computing the consumer price

index (CPI) go out and observe the actual year 1 consumption choices (x

∗

1

,x

∗

2

)

and corresponding (base year) prices (p

∗

1

,p

∗

2

).These consumption choices are

now assumed to be the basis for the ﬁxed weights,discussed in the last section,

for calculating the CPI.Furthermore,now suppose that in year 2 the CPI folk

observe new prices (p

12

,p

22

) that diﬀer from base year prices.They could then

compute the CPI

2

for year 2 as follows:

CPI

2

=

p

12

x

∗

1

+p

22

x

∗

2

p

∗

1

x

∗

1

+p

∗

2

x

∗

2

Lastly,let us suppose that the consumers in this world are simply collecting

social security retirement beneﬁt checks,issued by the U.S.government,as their

sole source of income.Social security in this world ties the value of these checks

to the CPI.Thus,if the CPI goes up,then the amount of dollars on the check

goes up proportionally to the CPI measure of the increase in the “cost of living”.

The interesting question is then to ask whether using the CPI in this way

over compensates,under compensates or correctly compensates these retirees

for changes in the cost of living.Thus,does the CPI serve as a cost-of-living

index as we deﬁned it above?The answer is NO.Speciﬁcally,using the CPI

in this way will in theory over compensate in the sense that it gives too much

money to the retirees so that the retirees will be able to get strictly more utility

in year 2 than in year 1!

This over compensation always occurs when two conditions hold.The ﬁrst

condition is that between year 1 and year 2 there is a change in relative prices

so that

p

12

p

22

=

p

∗

1

p

∗

2

.The second condition is that the indiﬀerence curve through

14

Figure 2.2:

the original point (x

∗

1

,x

∗

2

) is “smooth” in that there is no kink so that the

indiﬀerence curve has a unique tangent line at this point.As long as both these

hold,then the new budget line in year 2 will still run through the point (x

∗

1

,x

∗

2

)

but will cut through the old indiﬀerence curve.The upshot is that because the

new budget line cuts through the old indiﬀerence curve there will be a better

consumption choice in year 2 than the choice (x

∗

1

,x

∗

2

) that was optimal in year

15

1 at year 1 prices and incomes.

2

Figure 2.3:

Figure 2.3 illustrates this idea.Figure 2.3 is consistent with a rise in the beer

price but no change in the burger price.This implies a lower relative price for

2

It is fairly clear that there is nothing special about the choice of illustratingthis theoretical

point in the case of exactly two consumption goods.If there are three goods,then the idea

is that there is always over compensation when the plane describing the “budget line” cuts

through the indiﬀerence surface.With more than three goods the same ideas apply but

visualization is diﬃcult.

16

burgers.Thus,the new budget line is ﬂatter and consuming a bit more burgers

and a bit less beer will be a way to increase utility.This is illustrated by the

movement from point A to point B in Figure 2.3.

The economics of this over compensation in using a ﬁxed weight price index

such as the CPI as a cost-of-living index has been understood at a theoretical

level for well over half a century.There have been several literature surveys in

recent years which have discussed the likely empirical magnitude of the annual

over compensation due to the “substitution eﬀect” highlighted in this section.

3

2.6 Deﬁnitions

An intermediate good is a good which is produced but sold to another

producer and embodied in some other (intermediate or ﬁnal) good.

A ﬁnal good is a good which is produced and then sold to the household

sector.

Real GDP is the value of all ﬁnal goods and services produced domes-

tically over a period of time,where the value is measured using base-year

prices.

Nominal GDP is the value of all ﬁnal goods and services produced

domestically over a period of time,where the value is measured using

current-year prices.

The value added for a ﬁrm equals the sales of the ﬁrm less the cost of

the intermediate goods purchased by that ﬁrm.

A cost-of-living index measures the minimum cost of achieving a ﬁxed

level of utility over time as prices change.

3

See Moulton (1996),Journal of Economic Perspectives,vol.10,159- 77 for a discussion

of (1) details of how the CPI is computed in the U.S.,(2) plausible magnitudes of any bias

and (3) what statistical agencies were doing to allow the CPI to more accurately mimic a

cost-of-living index.

17

Chapter 3

Growth Theory

This chapter lays out the basic elements of a theory of economic growth that es-

sentially all professional economists have learned.To a large degree,the growth

theory that economists use to this day are extensions of this theory.Much

of this theory is based on the work of Robert Solow.Robert Solow received

the Nobel prize in economics in 1987 for two important papers on economic

growth.

1

Solow’s ﬁrst contribution was to provide a simple theoretical model

of economic growth that could confront some stylized facts of growth.Solow’s

second contribution was to provide a method of accounting for the sources of

growth in aggregate output and then to apply it to U.S.data.The sections that

follow will present each of these contributions as well as the closely related work

on the Golden rule.Edmond Phelps was an important contributor to work on

the Golden Rule.He received the Nobel Prize in 2006 partly for his work on

the Golden Rule.

2

3.1 Production Function

The growth theory to be presented is based on elementary properties of pro-

duction functions.This section reviews these properties.A typical abstraction

used in growth theory is that output Y

t

in time period t is produced using pre-

cisely two factor inputs:capital K

t

and labor L

t

.With this abstraction all

capital is the same or homogeneous as is all labor.The production function

Y

t

=A

t

F(K

t

,L

t

) describes the output level Y

t

which is technologically feasible

from given quantities of capital K

t

and labor L

t

.The variable A

t

describes the

level of the technology at time t.Higher values of A

t

describe better technologies

in the sense that more output can be produced with the same inputs.Within

1

See Solow (1956),A Contribution to the Theory of Economic Growth,Quarterly Journal

of Economics,70,65-94 and Solow (1957),Technical Change and the Aggregate Production

Function,Review of Economics and Statistics,39,312-20.

2

See Phelps (1961),The Golden Rule of Accumulation:A Fable for Growthmen,American

Economic Review,51,638- 43.

18

this framework output can change over time because of changes in factor inputs

(K

t

,L

t

) or because of a change in the technology A

t

.

Y

t

= A

t

F(K

t

,L

t

)

Y

t

- output at time t

K

t

- capital at time t

L

t

- labor at time t

A

t

- technology at time t

Standard properties of production functions are now discussed.

Constant Returns to Scale

A production function Y = AF(K,L) is constant returns to scale provided that

when all factor inputs are scaled up or down by a common factor,then output is

also scaled by the same factor.For example,when all factor inputs are doubled

it must be true that output is also doubled if the production function is constant

returns to scale.Similarly,when all factor inputs are halved it must be true

that output is also halved if the production function is constant returns to scale.

This deﬁnition can be described in mathematical terminology below,where λ

is the scaling factor and the symbol ∀ means “for all”.Note that by setting

the factor λ = 2 the expression below implies that output doubles when inputs

double.

λY =AF(λK,λL),∀λ >0

An important implication of constant returns to scale is that the ratio of

output to labor is determined,technology held ﬁxed,solely by the ratio of

capital to labor.This result can be expressed more boldly as follows.Imagine

that there are two countries with the same constant returns to scale technology.

Then the size of the two countries is irrelevant for determining which country

is richer in the sense of a larger GDP per unit of labor input.The only thing

that is relevant is which country has the larger capital-labor ratio K/L.Thus,

with the same constant returns technology the size of a country,in terms of

its labor force,is irrelevant for determining which country is richer.This logic

is expressed below in mathematical terms.It follows from the deﬁnition of

constant returns to scale upon setting λ = 1/L.

Y/L = AF(K/L,L/L) = AF(K/L,1)

19

Diminishing Marginal Products

A production function Y = F(K,L) has a diminishing marginal product of

capital provided that the marginal product of capital falls or decreases as the

quantity of capital is increased,holding other inputs constant.

3

Similarly,a

production function Y = F(K,L) has a diminishing marginal product of labor

provided that the marginal product of labor falls as the quantity of labor is

increased,holding other inputs constant.In what follows it will often be helpful

to have some notation to denote the marginal products of capital and labor.

The notation adopted here is to use a subscript K or L to denote that one is

talking about the marginal product of capital or labor.

4

F

K

(K,L) - marginal product of capital

F

L

(K,L) - marginal product of labor

Implications of Proﬁt Maximization

Elementary microeconomic theory implies that when a ﬁrm takes output and

input prices as given then proﬁt maximization implies that factors of production

are paid their marginal products.The logic behind this claimis presented below.

Proﬁt equals total revenue less total cost or F(K,L) −WL−RK.Speciﬁcally,

revenue equals F(K,L) when the price of output is normalized to equal 1 and

cost equals WL+RK,which is composed of labor costs (L units of labor times

the wage W) plus capital costs (K units of capital times the rental rate of capital

R).Here it is assumed that since the ﬁrm does not own its capital it must rent

each period all capital (e.g.buildings and machines).

Proﬁt Maximization Problem:

maxF(K,L) −WL −RK

implication:(1) F

L

(K,L) = W

implication:(2) F

K

(K,L) = R

The theory implies that if the ﬁrm is maximizing proﬁt then implications

(1)-(2) above hold.The reason is that if either of these conditions did not hold

then the ﬁrm could increase proﬁt.For example,suppose that F

L

(K,L) > W

so that the marginal product of labor is above the wage.Then the ﬁrm could

not be maximizing proﬁt.The reason is that the ﬁrm could hire one more unit

of labor and produce extra revenue F

L

(K,L) which exceeds the extra cost of

3

The marginal product of a factor input is the extra output produced by an extra unit of

the factor input,holding other inputs constant.Mathematically,the marginal product of an

input is the relevant slope of a production function at a point.In mathematics,this ”slope”

is simply a partial derivative as discussed in any elementary calculus course.

4

Since the marginal products will also depend on the quantities of capital and labor em-

ployed,the notation indicates that the relevant marginal products are functions of these factor

inputs.

20

the labor W.This would earn the ﬁrm more proﬁt,since F

L

(K,L) −W > 0,in

contradiction to the claim that the ﬁrm was maximizing proﬁt.Similar argu-

ments can be made when F

L

(K,L) < W.Thus,proﬁt can only be maximized

when F

L

(K,L) = W.The same reasoning establishes that F

K

(K,L) = R must

also hold if the ﬁrm maximizes proﬁt.

Cobb-Douglas Production Function

The Cobb-Douglas production function is very useful.It is constant returns

to scale and has diminishing marginal products.Furthermore,it implies that

capital’s share of output is always equal to a constant β independent of the

quantities of capital and labor employed.

Let’s develop some of the story behind this production function.Douglas

was interested in ﬁguring out the class of production functions that have two

properties.The ﬁrst property is constant returns to scale.The second property

is that the fraction of output paid to capital and labor,respectively,are constant.

This second property is motivated by looking at U.S.data on factor shares.

Douglas is well known for documenting that factor shares in U.S.data have no

strong trend movements,although they ﬂuctuate at business cycle frequencies.

Thus,economists view constant factor shares as a useful approximation.

The result is the Cobb-Douglas production function below.

5

Y = F(K,L) = AK

β

L

1−β

,where 0 < β < 1 and A > 0

F

K

(K,L) = βAK

β−1

L

1−β

=βA(L/K)

1−β

F

L

(K,L) = (1 −β)AK

β

L

−β

= (1 −β)A(K/L)

β

We note that for this production function the following two properties hold:

Y = F(K,L) =F

K

(K,L)K +F

L

(K,L)L

1 =

F

K

(K,L)K

Y

+

F

L

(K,L)L

Y

= β +(1 −β)

One interpretation of the ﬁrst property is that all of output is paid to factors

and thus economic proﬁt is zero!This is an interesting point that is traditionally

discussed in a principles of microeconomics course.It turns out that this is true

of constant returns to scale production functions - not just Cobb-Douglas.This

ﬁrst property was given a formal proof long ago by Leonhard Euler - a very

famous Swiss mathematician from the 1700’s.The second property is that the

fraction of output paid to capital is always β,regardless of the values of factor

inputs.

5

See Cobb,C.W.;Douglas,P.H.(1928).”A Theory of Production”.American Economic

Review 18 (Supplement):139 165.

21

Types of Technological Change

The literature on technological change distinguishes between embodied and dis-

embodied technological change.A disembodied technological improvement oc-

curs when existing capital and labor beneﬁt from the new technology.In con-

trast,an embodied technological improvement occurs when only new capital

beneﬁts from the new technology.An example of an embodied technological

improvement is the invention of a faster computer chip.Without the purchase

of the new chip it is not possible to take advantage of the new technology.This

type of technological change is quite common as new technologies are often

embodied in new buildings,new cars and new machines.

Examples of disembodied technological improvements are less common.An

example is the invention of new ways of organizing production that do not

require new investment.Adam Smith’s famous description of the “pin” factory

is such an example.

6

Smith observed that in the making of pins the separate

jobs (straightening the wire,cutting the wire,sharpening the point,attaching

the head and putting the pins into a box) are done by separate people.In this

way a group of people could make many more pins in a given amount of time

than if every person made one pin at a time.This is related to the change

in organization well known in the automobile industry as the creation of the

assembly line.The “technological” improvement required,at least in the case

of the pin factory,little new investment to implement.

In growth theory it is usually the case that the focus is on disembodied

technological change purely for simplicity.The reason is that one need only

keep track of the total stock of capital rather than the quantities of all the dis-

tinct types or vintages of capital.When dealing with disembodied technological

change the literature distinguishes three cases:neutral,labor augmenting and

capital augmenting.These cases are distinguished by whether technological

improvements (increases in A) act to increase both eﬀective capital (KA) and

labor (LA),which is the neutral case,or act to increase solely eﬀective labor

(the labor augmenting case) or solely capital (the capital augmenting case).For

technical reasons,which we will not get into here,Solow growth theory requires

labor augmenting technological change so that the model produces a constant

growth rate in steady state.However,in any of these three cases a technologi-

cal improvement occurs when at least as much or strictly more output can be

produced with the same factor inputs.

Y = AF(K,L) or Y = F(KA,LA) - Neutral

Y = F(K,LA) - Labor Augmenting

Y = F(KA,L) - Capital Augmenting

6

See Adam Smith’s (1776),The Wealth of Nations.

22

3.2 Solow Growth Model

It is helpful to have some facts in mind when one theorizes.The following

six facts are helpful for thinking about growth and are attributed to Nicholas

Kaldor.

Kaldor’s Growth Facts:

1.Output per capita grows over time.

2.Capital per capita grows over time.

3.The capital-output ratio is approximately constant over time.

4.Capital and labor’s share of output is approximately constant over time.

5.The return to capital does not have a strong trend.

6.Levels of output per capita vary widely across countries.

A key issue is the degree to which these “facts” describe the behavior of

particular countries over particular time periods.We will not address this issue,

but we will take the ﬁrst ﬁve of these “facts” to be descriptive of the experience

of the richest countries for the last two to three hundred years.This is the

time period that is often refered to as the period of modern economic growth.

Before this time period,average growth rates over long time periods for the

most advanced economies are believed to be approximately zero.During the

Industrial Revolution,Angus Maddison calculates that the average growth rate

of GDP per man hour in the UK from 1700-1780 was 0.3 percent!

7

Thus,even

during the Industrial Revolution the economic growth rate was quite small by

modern standards.

3.2.1 The Basic Solow Model

We will build up to the most general formulation of Solow’s growth model in two

steps.The model introduced in the ﬁrst step will be called the “basic” Solow

growth model.The model created in the second step,the full Solow model,

will be able to explain Kaldor’s growth facts 1-5.A satisfactory explanation

of Kaldor’s sixth fact is still an open problem in growth theory.If you have a

convincing explanation for the magnitude of the observed diﬀerences in output

per capita across countries,then you will receive the Nobel prize!

The basic Solow growth model is described in the four equations below.

The ﬁrst equation says that output Y

t

can be divided into consumption C

t

and

investment I

t

and that output Y

t

is a function of the factor inputs.In particular,

the equation Y = F(K,L) describes the technological possibilities for producing

7

See Table 2.2 in Angus Maddison’s (1991) work “Dynamic Forces in Capitalist Develop-

ment”,Oxford University Press.

23

output Y using inputs of capital K and labor L.The second equation describes

how the aggregate capital stock evolves over time.A subscript indicates the

model period in which the variable is being measured.Thus,the second equation

indicates that the capital stock available for use in period t +1 is determined

by the sum of the investment I

t

in capital goods in period t and the capital

remaining from period t,K

t

(1 −δ).Here it is assumed that capital depreciates

at a constant rate δ.The most natural interpretation of depreciation is that

capital goods such as buildings and machines deteriorate over time.

C

t

+I

t

=Y

t

=F(K

t

,L

t

)

K

t+1

=I

t

+(1 −δ)K

t

I

t

=sF(K

t

,L

t

)

L

t

=L

The third equation is the assumption that consumers save a constant frac-

tion s of output each period.This assumption is behavioral,whereas the ﬁrst

two assumptions were technological.Normally in economics,consumption and

saving decisions are assumed to be chosen optimally by households and ﬁrms.

Solow’s model is quite simple exactly because of this simplifying assumption.

The fourth equation is the simplifying assumption that there is no growth in

population.We will relax this assumption soon enough.

In the analysis of the Solow growth model the function F(K,L) is assumed

to display several properties you have seen in your ﬁrst courses in economics.In

particular,we assume that (i) the production function displays constant returns

to scale,(ii) there is diminishing marginal products of capital and (iii) the

marginal product of capital becomes arbitrarily small when capital is increased

suﬃciently,holding labor constant.

The dynamics of all variables in the simplest version of the Solow model is

completely determined by the pattern in the accumulation of capital K or the

capital-labor ratio k.Using the second and third equations above and simplify-

ing we get the equation below which describes how capital changes over time.

This equation can also be written as the second line below simply by dividing

the ﬁrst line by L

t

.I adopt the notational convention that small lettered vari-

ables denote ratios to labor input (i.e.k

t

≡ K

t

/L

t

,y

t

≡ Y

t

/L

t

,i

t

≡ I

t

/L

t

,c

t

≡

C

t

/L

t

).Either of these equations can be used to determine the behavior of ag-

gregate capital or the capital-labor ratio over time.Once these key variables are

determined then all other variables (e.g.output,consumption and investment)

are easily determined.

K

t+1

= sF(K

t

,L

t

) +(1 −δ)K

t

k

t+1

= sF(k

t

,1) +(1 −δ)k

t

24

Steady States

It is useful to think of the notion of a steady state in the Solow model.In the

simplest model a steady state can be thought of as a situation in which none

of the variables in the Solow model change over time.Later on we will use a

diﬀerent and more general notion of steady state in which all variables grow

at constant rates.Using this ﬁrst notion,we can rearrange the previous two

equations as indicated below.Note then that the economy is in steady state

precisely when investment (sF(K

t

,L

t

)) exactly equals the amount of capital

that depreciates (δK

t

).

K

t+1

−K

t

=sF(K

t

,L

t

) −δK

t

k

t+1

−k

t

=sF(k

t

,1) −δk

t

Figure 3.1 is very useful for summarizing the behavior of the variables in the

Solow model.Figure 3.1 graphs output,investment and depreciation per unit

of labor input all as functions of the capital-labor ratio k.Steady state occur

precisely where the investment and depreciation graphs cross.There are two

such points in Figure 1 but we will focus on the positive steady state capital-

labor ratio denoted as k

∗

and corresponding to it is a steady state output y

∗

,

investment i

∗

and consumption c

∗

all stated as a fraction of labor input.

In Figure 3.1 there is a maximum feasible steady state capital-labor ratio

k

∗∗

.This is a steady state if the fraction of output saved is one hundred percent

(i.e.s = 1).At this steady state all resources are used to replace depreciated

capital.Consumption is exactly zero.It is intuitively clear that the people

living in this model are not very happy in this steady state despite having a

high output level.

The next thing to note about Figure 3.1 is that the steady state k

∗

is a point

of attraction.More precisely,given any initial k > 0 the economy will converge

over time to k

∗

.To see this note that if k < k

∗

then over time the capital-

labor ratio will grow since investment is greater than depreciation.Further,

if k > k

∗

then the capital-labor ratio will shrink back to k

∗

as depreciation

exceeds investment.It should also be clear that all other variables such as

output,investment and consumption converge to steady state values over time.

All these properties of the Solow model (unique positive capital steady state

for a given savings rate,maximumsteady state and convergence to steady state)

are quite general.They rely on the depreciation rate δ being positive and on the

marginal product of capital tending to zero as capital is increased.Economists

typically regard positive depreciation rates and diminishing marginal products

as quite realistic assumptions.

A Numerical Example

So far the analysis of the basic Solowmodel has been at an abstract level.To see

how the model works at a mechanical level it is useful to consider a numerical

25

Figure 3.1:Basic Solow Model

Solow Model—Steady State

0

k

dk

y =F(k,1)

i=sF(k,1)

k *

i*

y *

26

example.Speciﬁcally,we will describe a particular production function and

particular values of all parameters (e.g.the depreciation rate,the savings rate

and all parameters describing the production function).Once this is done,we

will use the basic equation of the Solow model to compute how values of the

capital-labor ratio and other variables change over time.All calculations can

be done with a standard spread sheet,with any programming language or with

some pain by hand calculator.

EXAMPLE:

Y = F(K,L) = AK

β

L

1−β

,where A = 1.0,β = 0.3

δ =.04 - depreciation rate

s = 0.1 - savings rate

The key equation of the Solow model which describes dynamics is given in

the ﬁrst equation below.This equation is written in terms of the capital-labor

ratio.To use this equation we have to express the production function in terms

of ratios to the labor input.This can be done for the Cobb-Douglas function

Y = F(K,L) = AK

β

L

1−β

simply by dividing both sides by labor L to get that

y = Ak

β

.The second equation below then follows fromthe ﬁrst by substituting

y = F(k

t

,1) = Ak

β

into the ﬁrst equation.

k

t+1

= sF(k

t

,1) +(1 −δ)k

t

k

t+1

= sAk

β

t

+(1 −δ)k

t

Table 3.1 uses the above equation to calculate time proﬁles for a number

of variables in the Solow model.Table 1 is based on the assumption that the

initial capital-labor ratio equals 1 (i.e.k

0

= 1.0).Proﬁles are calculated for a

number of periods and the ﬁnal steady state quantities are also indicated at the

bottom of the table.

Table 3.1:Time Paths in the Basic Solow model

Capital Output Investment Consumption

y = Ak

β

i = sy c = (1 −s)y

k

0

= 1.0 y

0

= 1.0 i

0

=.10 c

0

=.90

k

1

= 1.060 y

1

= 1.017 i

1

=.101 c

1

=.915

k

2

= 1.119 y

2

= 1.034 i

2

=.103 c

2

=.930

k

∞

= 3.69 y

∞

=1.47 i

∞

=.147 c

∞

= 1.323

[NOTE:s = 0.1,δ = 0.04,A= 1.0,β = 0.3]

27

3.2.2 The Full Solow Model

The most general version of the Solowmodel is that in which both the population

and the technology are allowed to grow at constant rates over time.This full

Solow model covers all others (i.e.those with no technology growth or with no

population growth) as special cases.Thus,once you see how this model works

it is best to think in terms of this model rather than the basic model that we

started out discussing.

The key modiﬁcation of the previous model is the addition of technological

change and population growth.The variable A

t

can be interpreted as the level

of technology available in period t.There are two things to be noted about

the way Solow introduced technological change.First,technological change

is called labor augmenting as improvements in the technology act to increase

“eﬀective” labor.The quantity L

t

A

t

will be called eﬀective labor.Second,when

there is a technological improvement (i.e A

t

increases) ALL capital- old or new-

equally beneﬁts.This comes from the assumption that technological change is

disembodied as discussed earlier in chapter 2.The growth rate of technology

and population are denoted with the symbols g and n,respectively.

C

t

+I

t

=Y

t

=F(K

t

,L

t

A

t

)

K

t+1

=I

t

+(1 −δ)K

t

I

t

=sF(K

t

,L

t

A

t

)

L

t+1

=L

t

(1 +n)

A

t+1

= A

t

(1 +g)

We can now proceed as before to examine the dynamics of the Solow model.

The key equations are listed below.The ﬁrst equation accounts for the ac-

cumulation of physical capital K

t

.To get to the second equation,we di-

vide both sides of the ﬁrst equation by eﬀective labor L

t

A

t

.We let all lower

case variables denote ratios of the upper case variable to eﬀective labor so

k

t

≡ K

t

/L

t

A

t

,y

t

≡ Y

t

/L

t

A

t

,c

t

≡ C

t

/K

t

/L

t

A

t

,i

t

≡ I

t

/L

t

A

t

.There are two

little tricks that should be mentioned in deriving the second equation.First,

K

t+1

/L

t

A

t

= K

t+1

(1+n)(1+g)/L

t+1

A

t+1

= k

t+1

(1+n)(1+g).This accounts

for the left-hand-side of the second equation.Second,sF(K

t

,L

t

A

t

)/L

t

A

t

=

sF(K

t

/L

t

A

t

,1) = F(k

t

,1) using the fact that the aggregate production func-

tion F() is by assumption constant returns to scale.

K

t+1

= sF(K

t

,L

t

A

t

) +K

t

(1 −δ)

k

t+1

(1 +n)(1 +g) = sF(k

t

,1) +k

t

(1 −δ)

A steady state in the full Solow model is a situation in which all ratios are

constant.Expressed with the last equation,the requirement is that k

t+1

−k

t

=

sF(k

t

,1)−k

t

(δ +n+g +ng) = 0.Thus,in a steady state (k,y,c,i) do not grow

28

Figure 1: Solow Model—Steady State

0

k

k(d+n+g+ng)

y=F(k,1)

i=sF(k,1)

k*

i*

y*

Figure 3.2:Full Solow Model

but capital K,output Y,consumption C and investment I all grow.Speciﬁcally,

since k = K/LA then it is clear that in steady state K must grow at the same

rate as eﬀective labor LA.Now,since eﬀective labor grows at rate g +n +ng,

or approximately rate n +g,K also grows at his rate.

The second equation above is the key equation for the full Solow model.We

can use it mechanically to ﬁgure out the implied behavior of the model over

time.One simply plugs in a starting value (say k

1

= 1.0) for the capital per

eﬀective labor ratio into the right-hand side of the equation and ﬁnd out what

is the value next period of k.Repeating this process,one ﬁnds out how the

system behaves.We will now see that there is an easier way to ﬁgure out the

qualitative behavior of this model.

We can analyze the Solow model with a graph that is essentially identical

to that used before.The idea is to rearrange the last equation above to read

that in steady state 0 = k

t+1

−k

t

= sF(k

t

,1) −k

t

(δ +n +g +ng).Thus,in

a steady state investment must be suﬃcient keep the capital per eﬀective labor

ratio constant and thus to oﬀset the eﬀects of physical depreciation,population

growth and technological change.

Figure 3.2 is the key graph for the Solow model.Here as before k

∗

is a steady

state since this is where the investment graph cross the “depreciation” graph.

8

8

This graph can also be thought of as describing the amount of investment needed for a

given k to maintain the capital-labor ratio constant.

29

You will see that the addition of labor augmenting technological change has

not changed most of the key features of the Solow model without technological

change.In particular,there is still a unique steady state that is attractive in

the sense that the capital per eﬀective labor ratio converges to this steady state

over time.The next section examines how this simple model handles Kaldor’s

growth facts.

3.3 Explaining Kaldor’s Growth Facts

The best way to review the merits of our theory of growth is to see whether the

model is capable of producing Kaldor’s facts.Three predictions of the Solow

model for growth are listed below.These three predictions come from Figure

3.2.First,observe in Figure 3.2 the steady state capital per eﬀective labor

ratio k

∗

is constant.This implies that total capital K

t

grows at the growth

rate of eﬀective labor input which is g + n + ng.This follows,as discussed

in the last subsection,since in steady state k = K/LA is constant and,thus,

aggregate capital must grow at the same rate as the denominator.By the same

reasoning total output,consumption and investment also grow at this rate in

steady state.This is summarized in point 1 below.Second,the model predicts

that in steady state all these aggregate variables as a ratio to labor grows at

rate g.This is point 2 below.This in essence follows from point 1.Third,the

capital-output ratio K/Y is constant in steady state.This is point 3 below.

Note that this follows from point 1 since output and capital grow at the same

rate.Summarizing,the model is able to produce Kaldor’s ﬁrst three facts if and

only if there is positive technological progress (i.e g > 0).

1.(Y

t

,K

t

,I

t

,C

t

) all grow at rate n +g +ng in steady state

2.(Y

t

/L

t

,K

t

/L

t

,I

t

/L

t

,C

t

/L

t

) all grow at rate g in steady state

3.In steady state K

t

/Y

t

is constant.

The next three of Kaldor’s facts are listed below:

4.Capital and labor’s share of output is approximately constant over time.

5.The return to capital does not have a strong trend.

6.Levels of output per capita vary widely across countries.

To see how the Solow model can explain at least some of these will require

some careful thinking.Let’s consider fact 4.According to the basic principles

of GDP accounting,output is equal to the sum of all payments to factor inputs.

In the context of the Solow model there are two factors of production:labor

and capital.Thus,output must equal these payments.Further we could assume

in the model that factors are paid at competitive rental prices,where W is the

rental price of labor (the wage) and R is the rental price of capital.

30

Y = labor income +capital income = WL+RK

1 = (labor income)/Y +(capital income)/Y = WL/Y +RK/Y

To prove that capital’s share of output is constant in steady state of the Solow

model amounts to proving that capital’s share of income RK/Y is constant.It is

clear that capital’s share will be constant if the rental price R is constant since

we already know from prediction 3 that the ratio K/Y is constant in steady

state.Now if we assume that factor rental prices are competitively determined

then we know frombasic microeconomics that the rental price R must equal the

marginal product of capital F

K

(K,LA).To ﬁnish oﬀ the argument simply note

that in Figure 3.2 the marginal product of capital is the slope of the production

function and that the slope does not change over time in a steady state.

9

This last argument also proves that the Solow model predicts that the return

to capital is constant in steady state as we have just argued that R is constant

in steady state and the net return to a unit of capital in this model is just the

rental price R less depreciation δ.This was point 5 above.

The only remaining fact that the Solow model has yet to explain is Kaldor’s

sixth fact.Here the Solow model can explain diﬀerences in steady state output

per capita across countries.These diﬀerences are due to diﬀerences in saving

rates s and population growth rates n across countries.This assumes that the

technology is that same across countries but that the steady states are diﬀerent

only because s and n diﬀer.

The situation in which two countries diﬀer in saving rates is illustrated in

Figure 3.3.The theory predicts that two countries that are alike in all respects

but the savings rate will have diﬀerent steady state output levels at any point

in time.Speciﬁcally,the country with the larger savings rate will have a higher

level of Y/L at any point in time compared to the country with the lower savings

rate.The theory also predicts that all countries have the same growth rate of

Y/L in steady state despite having diﬀerent savings rates,given the maintained

assumption that technology is common across countries.This was initially a

surprising ﬁnding for the pioneers of growth theory.

We ﬁnish this section by discussing what,according to Solow growth theory,

are the consequences of increasing the saving rate.Figure 3.3 is useful.It

shows that if a country is initially in steady state with savings rate s

1

and then

increases the savings rate permanently to s

2

,then the country will converge

over time to a higher steady state.In transition to this new steady state Y/L

must grow faster than before.However,a key point is that this higher growth

is temporary.As the country gets closer to the new steady state growth slows

down and converges to exactly the old steady-state growth rate!In summary,

9

To see that the rental price R really is constant we can use calculus.By deﬁnition,

the marginal product of capital equals MP

K

= dF(K,LA)/dK = dLAF(K/LA,1)/dK =

F

1

(K/LA,1) = F

1

(k,1),which clearly is constant in steady state by looking at Figure 3.2.

31

Figure 3.3:Solow Model With Two Diﬀerent Savings rates

32

within the Solow model a change in the savings rate leads to a level eﬀect on

output per unit of labor input but not a change in the steady state growth rate.

3.4 Evaluating Solow Growth Theory

Up to this point,we have focused mostly on developing the logic of how the

Solow model works.However,the interesting issue is whether or not broadly

the model seems to make sense of data.The previous section argued that the

model in steady state can produce Kaldor’s facts 1-5.What is not clear is the

extent to which the model addresses Kaldor’s sixth fact:Y/L diﬀers widely

across countries at a point in time.The US has a level of GDP per capita that

is approximately 30 times that of some very poor countries.

10

While a careful quantitative analysis of the degree to which the Solow model

is in agreement with such facts is beyond the scope of this book,it may be

useful to lay out some facts and some opinions about the state of the literature.

First,cross-country data does ﬁnd that countries with high measured Y/L also

typically have a high measured K/L.This is good news for a theory that requires

that Y/L =F(K/L,A) and that maintains as a provisional assumption that Ais

common across countries.Second,countries with high measured Y/L typically

have a high measured investment rates I/Y over long time periods.This also

seems to be good news as within the Solow model a high investment rate (i.e.a

high s) in steady state is the means of attaining high K/L and high Y/L,given

the assumption that technology is common across countries.

Now one can approach this second fact froma diﬀerent angle,to see if this is

really good news for the Solow model.One could ask ﬁrst what are the savings

or investment rates at the high and low end of the distribution.One can ﬁnd

very low investment countries averaging s

L

=.05 and very high investment

countries averaging s

H

=.30 for a few decades.One could then ask whether

such diﬀerences lead to big steady state diﬀerences in Y/L or Y/LA,holding

technology A,depreciation δ,population growth n and technological growth g

constant across countries.

We now carry out this analysis using a Cobb-Douglas production function

y = F(k,1) = Ak

β

.The ﬁrst equation below is the steady state condition

that requires that the investment curve in the Solow model crosses the straight

line describing possible steady-state k values.The second equation ﬁgures out

steady-state capital k,given the use of the Cobb-Douglas production function.

The third equation ﬁgures out steady state y.The fourth ﬁgures out the ratio

of steady-state incomes corresponding to the highest savings rate s

H

and the

lowest s

L

:

10

Facts relating to diﬀerences in GDP per capita across countries at a point in time and

across time periods are presented in Parente and Prescott’s paper entitled “Changes in the

Wealth of Nations”,Federal Reserve Bank of Minneapolis Quarterly Review,1994.In that

work diﬀerences in GDP per capita across countries at a point in time are measured using a

common set of world prices.

33

sF(k,1) =k(δ +n +g +ng)

sAk

β

= k(δ +n +g +ng) ⇒k =[

sA

(δ +n +g +ng)

]

1

1−β

y =Ak

β

⇒y = A[

sA

(δ +n +g +ng)

]

β

1−β

y

H

y

L

=

A[

s

H

A

(δ+n+g+ng)

]

β

1−β

A[

s

L

A

(δ+n+g+ng)

]

β

1−β

= [

s

H

s

L

]

β

1−β

The ratio of steady state y = Y/L in high saving rate to low saving rate

countries is simply

y

H

y

L

= [

s

H

s

L

]

β

1−β

= [

.3

.05

]

β

1−β

.This uses the posited diﬀerences

in savings-investment rates.We will have our answer if we take a stand on the

parameter β.If β =.30 - a ball-park number for capital’s share in the US -

then the ratio is 6

.3

.7

≈ 2.15!This is a tiny ratio compared to the factor of 30

diﬀerences observed in cross-country data.Even if β =.5,then the ratio is

6.The upshot is that steady-state diﬀerences in output implied by measured

diﬀerences in saving rates alone are quite small compared to output diﬀerences

measured in data.Thus,we conclude that something other than savings through

physical capital must be very important.

There is an additional problem with this type of mechanism for producing

diﬀerent output-labor ratios across countries based on diﬀerent capital-labor

ratios.Assuming that technology is common across countries,poor countries

(those with low capital-labor ratios) should have high marginal products of

capital.If so,there should be strong incentives to locate more physical capital in

such countries.Although there are some notable exceptions,it is clear that there

are not dramatic capital ﬂows to all of the poorest countries.This suggests that

other assumptions,such as the technology held equal assumption,are strongly

violated.

One implicit assumption in applying the model to interpret cross-country

data is that workers are the same quality across countries.This assumption

seems likely to be strongly violated.We now indicate how in principle one

might go about trying to account for real GDP per worker diﬀerences across

countries in a framework that allows for quality diﬀerences in workers across

countries.The framework described below allows country i’s GDP per worker

denoted Y

i

to be determined by the technology A

i

and the per worker input

of capital K

i

and the per worker quality adjusted labor L

i

in country i via

an aggregate production function as highlighted below.It is typical in this

literature to use a Cobb-Douglas production function and an empirical estimate

of capital’s share β.

Y

i

= F(K

i

,L

i

,A

i

) = K

β

i

(L

i

A

i

)

1−β

34

A

i

= [Y

i

/K

β

i

L

1−β

i

]

1/(1−β)

The basic idea is then to measure (Y

i

,K

i

,L

i

) in a cross section of countries

and then to back out technology A

i

.The literature which does this is surveyed

by Caselli (2005).

11

A key issue is then to have a measure of worker quality.In

practice economists use data on the distribution of the workforce by experience

(years worked) and by years of schooling.The idea is that in cross-section data

earnings increase with both experience and schooling and thus workers with

high experience and schooling are more productive and,hence,are of higher

quality.To the degree that rich countries have a distribution of workers with

higher experience and higher schooling than poor countries,then these are the

proximate reasons providing empirical support for rich countries having larger

quality adjusted labor input L

i

per worker and,thus,higher output per worker.

A typical ﬁnding from this literature (see Caselli (2005)) is that rich coun-

tries (i.e.countries with high Y ) have relatively high technology A,capital per

worker K and labor quality L.Thus,variation in measured factor inputs (K,L)

accounts for some of the output diﬀereces across countries but do not by them-

selves explain all of the output per worker Y variation across countries.Rich

countries are infered to have higher technology than poor countries and this is

a quantitatively important source of GDP diﬀerences.

Some recent work by Lagakos,Moll,Porzio and Qian (2012) argues that

better measurement of labor quality diﬀerences across countries substantially

reduces the importance of technology diﬀerences.

12

They argue that diﬀerences

in capital and labor quality explains approximately two-thirds of the measured

ratio of GDP per capita of the country at the 90th percentile of the distribution

compared to GDP per capita of the country at the 10th percentile.If this result

proves to be widely supported in the data,then the key question in the literature

is what accounts for such measured diﬀerences in labor quality across countries.

Of course,the Solow growth model is silent on the sources of these diﬀerences

as it is not a theory of worker quality diﬀerences.The dominant body of work

on such quality diﬀerences is the literature on human capital accumulation.

13

3.5 Golden Rule

Within the context of the technology for production used in the Solow growth

model it is natural to try to address normative questions.Recall that normative

questions deal with what should be or what ought to be according to some set

of values.Thus,a set of values allows one to describe allocations which are

“good” versus those that are “bad” in some theoretical world.This section

11

See Francesco Caselli (2005),Accounting for Cross-Country Income Diﬀerences,Hand-

book of Economic Growth,Chapter 9.

12

Lagakos,Moll,Porzio and Qian (2012),Experience Matters:Human Capital and Devel-

opment Accounting.

13

Gary Becker received the Nobel Prize in 1992 in part for his work on human capital.

35

seeks to answer the two questions below.Early theoretical work on these issues

was done by Edmund Phelps.

14

Question 1:In the context of growth theory,which allocations are clearly

bad allocations?

Question 2:What are the observable implications of these bad allocations?

3.5.1 Bad Allocations

To answer the ﬁrst question,let us ﬁrst ask the question of which steady state

of the Solow model is the best steady state to live in.To answer this question,I

will put forward the assumption that people living in this world care only about

the path of consumption over time.This is where we use a ”set of values”.In

particular,I will assume that consumption paths that have higher consumption

at each date are prefered to those with lower consumption at each date.With

this assumption,the best steady state is then the steady state k that gives

maximum consumption.Economists call this steady state the Golden Rule

steady state.

The Golden Rule steady state is easy to describe both with a graph and with

simple mathematics.First,consider the mathematics.The problemof choosing

a steady state k to maximize consumption is written in the ﬁrst line below.

The ﬁrst term in the maximization problem is output and the second term is

steady state investment.Thus,the diﬀerence is consumption.The solution to

this problem is written in the second line below.The second line notes that the

maximum should have the property that there is no gain (in consumption) to

having a little more or a little less capital.Thus,the derivative or slope of the

ﬁrst line should be precisely zero at the Golden rule capital-labor ratio.

Max F(k,1) −k[(1 +g)(1 +n) −(1 −δ)]

⇒F

k

(k,1) −[(1 +g)(1 +n) −(1 −δ)] =0

This situation is graphed in Figure 3.4.The Golden Rule steady state k

GR

occurs at the capital level k where the distance between the production function

and the steady state investment line is greatest.Geometrically,this can be de-

termined by shifting the steady state investment line up vertically until the line

is just tangent to the production function.Figure 3.4 highlights this geometric

description of the Golden Rule steady state.Note that the geometry amounts

to the claim that the slope of the production function equals the slope of the

steady state investment line.

We are now ready to answer Question 1.The answer is that any allocation

where the sequence of capital stock always remains strictly above the Golden

14

See Phelps (1961),The Golden Rule of Accumulation:A Fable for Growthmen,American

Economic Review,51,638- 43.Edmund Phelps received the Nobel Prize in 2006 partly for

his work on the Golden Rule.

36

Rule steady state capital stock is a bad allocation.The reason why such an

allocation is bad is that one can come up with a feasible alternative allocation

that allows for comparatively more aggregate consumption in all periods.

To be concrete,assume that the economy is at a steady state above the level

k

GR

.Then there is a “free lunch” that can be had simply by decreasing the

capital stock to the Golden Rule level and maintaining it there forever.Clearly,

this is possible since consumption at the Golden Rule is larger than at any

capital level above the Golden Rule.In summary,any steady state above the

Golden rule steady state is bad since,paradoxically,the economy suﬀers from

having too much investment.

37

Figure 3.4:Golden Rule Steady State

3.5.2 Observable Implications of Bad Allocations

Now that we have a theory describing which allocations are “bad” it is natural

to ask what are the observable implications of these bad allocations.This might

allow us to say whether or not actual economies suﬀer from being “above the

Golden Rule”.To do this,consider the four equations below.Each of these

is a simple rewriting of the ﬁrst equation below which says that the capital-

38

labor ratio k is above the Golden Rule level.The ﬁrst equation follows from the

equation deﬁning the Golden Rule capital stock or,alternatively,from Figure

3.4.This equation is based on the geometry in Figure 3.4 in that the slope of

the production function is smaller than the slope of the straight line deﬁning

steady-state investment.

F

k

(k,1) < [(1 +g)(1 +n) −(1 −δ)]

1 +F

k

(k,1) −δ < (1 +g)(1 +n)

kF

k

(k,1) < k[(1 +g)(1 +n) −(1 −δ)]

k(F

k

(k,1) −δ) < k[(1 +g)(1 +n) −1]

These equations are useful as they have simple interpretations in terms of

observables.The second equation can be interpreted as stating that the gross

interest rate (i.e 1 +r ≡ 1 +F

k

(k,1) −δ) is less than the steady state growth

rate of aggregate output (i.e (1 +g)(1 +n)).

15

Both of these quantities can be

measured.The third equation says that aggregate payment to capital kF

k

(k,1)

is less than aggregate investment k[(1 +g)(1 +n) −(1 −δ)].Once again,each

of these quantities can be measured.The fourth equation says that aggregate

net payment to capital is less than aggregate net investment.

These interpretations were related to data in a well-known paper by Abel,

Mankiw,Summers and Zeckhauser (1989).

16

They ﬁrst note that relating the

gross interest rate to the gross growth rate of output is problematic.The reason

that this is problematic is that there are many interest rates and returns that

can be calculated fromdata in actual economies.For example,one could choose

the average real interest rate on US Treasury Bills or,alternatively,the average

real return on the US stock market.The average real return on Treasury Bills

and Treasury bonds are about 1 and 2 percent,respectively,and the average

real return on the US stock market is about 6 percent over long time periods.

17

One of these returns is larger than the 3 percent average growth rate of real

output in the US over long time periods and the other two are smaller.Thus,

using average returns one could conclude either that the US economy is well

above the Golden rule or well below,depending on which asset one chooses to

look at!

The problem with the second equation is evidently that the model is too

simple.Treasury bills and stock diﬀer enormously in risk characteristics and,

15

Recall that in a steady state of the Solow model output grows at a gross rate which is

approximately equal to the population growth rate plus the growth rate of the technology.

16

Abel et.al.(1989),Dynamic Eﬃciency:Theory and Evidence,Review of Economic

Studies,Volume 56,1-20.

17

See Jeremy Siegel (2002,Table 1.1 and 1.2) ”Stocks for the Long Run” Third Edition,

McGraw Hill.

39

as a result,have diﬀerent average returns.The theory abstracts from risk,has

a single real interest rate and therefore provides no help in deciding which asset

return to use and how to use it.To respond to this issue one needs a theory

that incorporates risk.While this type of analysis is done in the literature it is

too advanced for a useful discussion at the level of this book.

Abel et.al.(1989) argue that the third and fourth equation above can

be related to data in a manner which does not lead to ambiguity.Following

the discussion above,they compute the gross payment to capital and the gross

investment in the US as a ratio to GNP.These are empirical proxies for the un-

derlying theoretical concepts in the third equation above.Some of the empirical

results of their paper for the payment to capital and investment as a ratio to

GNP are contained in Figure 3.5.

Figure 3.5:Investment and Payment to Capital in the US

0.2

0.3

0.4

t

ionofGNP

USData1929 85

0

0.1

1920 1930 1940 1950 1960 1970 1980 1990

Frac

t

Year

Investment/Y

PaymenttoCapital/Y

They ﬁnd that the gross payment to capital is always well above gross in-

vestment in the US.Their measure of gross payment to capital varies from a

40

low of about 23 percent in 1945 to a high of 32 percent in 1929.By comparison,

gross investment varies from a low of 1.9 percent in the Great Depression to

a high of 19 percent in 1950.Thus,investment is always below the payment

to capital.This pattern also holds for a number of European countries plus

Japan.Based on this evidence,Abel et.al.(1989) conclude that the advanced

economies appear to all be below the Golden Rule.Thus,there appears to be

no free lunch to be had from growth theory.Stated diﬀerently,the advanced

economies of the world may have many problems but one problem that they do

not suﬀer from is having accumulated too much physical capital.

3.6 Growth Accounting

Growth accounting is a tool for dividing up output growth into distinct sources.

This tool can be used to answer two types of questions.The ﬁrst type asks what

portion of observed output growth in a country (or even a ﬁrm) over some period

of time can be accounted for by changes in technology versus the portion that

can be accounted for by changes in factor inputs.The second type of question

asks what would be the eﬀect on output growth of a change in the technology

or a change in some speciﬁc factor input,other things equal.

In questions of the ﬁrst type,growth accounting tells one where growth

comes from.However,it does not tell one why the economy functions in this

way.Here,the analogy with ﬁnancial accounting is apt.An accountant may

be able to tell you where the income of a ﬁrm or government comes from but

at the same time an accountant may not have any theory explaining why it is

the case that income comes from these distinct sources.To answer the latter

question one needs a theory and not merely an accounting framework.

3.6.1 Growth Accounting:Theory

We will now lay out the theory behind growth accounting.Solow assumed that

there is an aggregate production function Y

t

= A

t

F(K

t

,L

t

).Thus,aggregate

output Y

t

is produced when the technology level equals A

t

and the factor inputs

of capital and labor are K

t

and L

t

,respectively.

Solow next took the time derivative of this production function.The re-

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