Growth facts. Solow-Swan modelApplied Macroeconomics: Lecture 6

Marcin BieleckiSpring 2018

University of Warsaw

1

In case you need a review of basic concepts

https://www.mruniversity.com/courses/principles-economics-macroeconomics

Chapters 1-3

2

GDP per capita and welfare: consumption

500 1000 2000 5000 10000 20000 50000 100000

GDP per capita (2011 USD)

500

1000

2000

5000

10000

20000

50000

100000

Con

sum

pti

onp

erca

pit

a(2

011

US

D)

GDP vs consumption per capita in 2014

3

GDP per capita and welfare: life duration

500 1000 2000 5000 10000 20000 50000 100000

GDP per capita (2011 USD)

45

50

55

60

65

70

75

80

85

90

Lif

eex

pec

tan

cyat

bir

thin

2014

(yrs

)GDP per capita vs life expectancy in 2014

4

GDP per capita and welfare: life satisfaction

Stevenson and Wolfers (2013)Economic Growth and Subjective Well-Being: Reassessing the Easterlin Paradox

5

Growth in total income vs income of bottom 40%

Dollar, Kleineberg and Kraay (2016)Growth still is good for the poor

6

Growth fact 1

There is enormous variation in GDP per capita across economies.

0 20 40 60 80 100

GDP per capita relative to USA

0

20

40

60

80

100

Cu

mu

lati

vew

orld

pop

ula

tion

(%)

World population by GDP per capita in 2014

7

Growth fact 1

There is enormous variation in GDP per capita across economies.

500 1000 2000 5000 10000 20000 50000 100000

GDP per capita (2011 USD)

0.00

0.01

0.02

0.03

0.04

Pop

ula

tion

den

sity

GDP per capita population-weighed density

1960

1985

2014

8

Growth fact 1

There is enormous variation in GDP per capita across economies.The poorest countries have per capita incomes that are less than5 percent of per capita incomes in the richest countries.

• Income per capita (or GDP per capita) is not the sole measureof well-being, but it’s a useful summary statistic.

• Income per capita ignores distribution of incomewithin a country.

• Comparing income per capita across countries is not trivial:• You have to convert between currencies.• Countries have different relative prices for goods.• Large uncertainty in comparing income across countriesand over time: Johnson et al. (2013) Is newer better?Penn World Table Revisions and their impact on growth estimates.

9

Growth fact 2

Rates of economic growth vary substantially across countries.

−4 −2 0 2 4 6 8

Annual GDP per capita growth rate, 1970-2014 (%)

0

5

10

15

20

25

30

Nu

mb

erof

cou

ntr

ies

Histogram of GDP per capita growth rates

Developed countries

Lagging behind

Catching up

10

Growth fact 2

Rates of economic growth vary substantially across countries.

• We will try to distinguish whether these are long-termdifferences or just transitional differences.

• If they are long-term, then eventually some countrieswill be infinitely rich compared to others.

• We think most differences are transitional.

Small differences in rates of growthtranslate to big differences in incomes over time:

Rate Initial Income after ... yearsof growth income 25 50 70 1001.0% 100 128 164 201 2701.5% 100 145 211 284 4432.0% 100 164 269 400 7242.5% 100 185 344 563 11813.0% 100 209 438 792 1922 11

Rule of 70

A way to estimate the number of years it takesfor a certain variable to double.

Find T for which xT = 2x0, assuming constant (annual) growth rate g

xT = x0 · (1+ g)T

2x0 = x0 · (1+ g)T

2 = (1+ g)T | lnln 2 = T · ln (1+ g)0.7 ≈ T · g

T ≈ 0.7g =

70100 · g

If g is expressed in percent, the number of years for a variableto double is approximately equal to 70 divided by g(expressed in percentage points).

12

There are both “growth miracles” and “growth disasters”GDP per capita GDP per worker Emp. rate GDP per capita Avg. growth Years to

Country 2014 2014 2014 1970 1970-2014 double

United States 52 292 112 517 0.46 23 608 1.8 38

United Kingdom 40 242 83 612 0.48 15 176 2.2 31

France 39 374 95 498 0.41 16 436 2.0 35

Japan 35 358 68 989 0.51 12 956 2.3 30

Singapore 72 583 117 472 0.62 5 814 5.9 12

Hong Kong 51 808 100 467 0.52 7 613 4.5 16

Taiwan 44 328 92 979 0.48 4 738 5.2 13

South Korea 35 104 67 247 0.52 2 100 6.6 10

Botswana 16 175 37 637 0.43 798 7.1 10

China 12 473 21 394 0.58 1 285 5.3 13

Indonesia 9 707 21 853 0.44 995 5.3 13

India 5 224 13 261 0.39 1 282 3.2 21

Zimbabwe 1 869 4 384 0.43 2 429 -0.6 -117

Madagascar 1 237 2 833 0.44 1 479 -0.4 -171

Dem. Rep. of Congo 1 217 3 757 0.32 2 536 -1.7 -42

Niger 852 2 397 0.36 1 395 -1.1 -6213

Growth fact 3

World growth rates have increased sharply in the twentieth century.

1500 1600 1700 1800 1900 2000

Year

500

1000

2000

5000

10000

GD

Pp

erca

pit

a(1

990

US

D)

0.04

0.20

1.11

2.24

Evolution of world GDP per capita

14

Growth fact 3

For individual countries, growth rates also change over time.

1880 1900 1920 1940 1960 1980 2000

Year

1000

2000

5000

10000

20000

50000

GD

Pp

erca

pit

a(2

011

US

D)

GDP per capita in Japan

15

Growth fact 3

For individual countries, growth rates also change over time.

1950 1960 1970 1980 1990 2000 2010

Year

500

1000

2000

5000

10000

GD

Pp

erca

pit

a(2

011

US

D)

GDP per capita in India

16

Growth fact 3

Growth rates are not generally constant over time.For the world as a whole, growth rates were close to zero over mostof history but have increased sharply in the twentieth century.For individual countries, growth rates also change over time.

• The big changes in growth rates over history are frompre-Industrial Revolution (close to 0% growth) to modern times(roughly 1.85% growth per year for developed countries).

• The big changes in growth rates within countries tend to be asthey transition from poor to rich (e.g. Japan), after which growthslows down.

17

Growth fact 4

Countries can go from being “poor” to being “rich”.

1950 1960 1970 1980 1990 2000 2010

Year

500

1000

2000

5000

10000

20000

50000

100000

GD

Pp

erca

pit

a(2

011

US

D)

Evolution of GDP per capita: “growth miracles”

USA

UK

Japan

South Korea

Botswana

China

India

Indonesia

10:90 percentile range

18

Growth fact 4

Countries can go from being “rich” to being “poor”.

1950 1960 1970 1980 1990 2000 2010

Year

200

500

1000

2000

5000

10000

20000

50000

100000

GD

Pp

erca

pit

a(2

011

US

D)

Evolution of GDP per capita: “growth disasters”

USA

UK

Niger

Nigeria

Djibouti

Zimbabwe

Madagascar

Dem. Rep. of Congo

10:90 percentile range

19

Growth fact 4

A country’s relative position in the world distribution of per capitaincomes is not immutable. Countries can go from being “poor”to being “rich”, and vice versa.

• The “growth miracles” in 1960 were very poor.Now they are catching up to the rich countries.

• The “growth disasters” were in 1960 richer than East Asia.Now they are well behind.

20

Kaldor's stylized facts

In the USA (and other developed countries):

1. Per capita output grows over time,and its growth rate does not tend to diminish.

2. Physical capital per worker grows over time.3. The rate of return to capital is not trending.4. The ratio of physical capital to output is nearly constant.5. The shares of labor and physical capital in national incomeare nearly constant.

6. Real wage grows over time.

21

Kaldor's stylized fact 1

Per capita output grows over time,and its growth rate does not tend to diminish.

1880 1900 1920 1940 1960 1980 2000

Year

2000

5000

10000

20000

50000

100000

GD

Pp

erca

pit

a(2

011

US

D)

GDP per capita in USA

22

Kaldor's stylized fact 2

Physical capital per worker grows over time.

1950 1960 1970 1980 1990 2000 2010

Year

105

2× 105

3× 105

4× 105

Cap

ital

per

wor

ker

(201

1U

SD

)

Capital per worker in USA

23

Kaldor's stylized fact 3

The rate of return to capital is not trending.

1880 1900 1920 1940 1960 1980 2000

Year

−20

−10

0

10

20

Exp

-pos

tre

alin

tere

stra

te(%

)

Real interest rate in USA and UK

United States

United Kingdom

24

Kaldor's stylized fact 3

The rate of return to capital is not trending.

DeLong (2015)blogpost

25

Kaldor's stylized fact 3

The rate of return to capital is not trending.

Gomme, Ravikumar and Rupert (2015)Secular Stagnation and Returns on Capital

26

Kaldor's stylized fact 4

The ratio of physical capital to output is nearly constant.

1950 1960 1970 1980 1990 2000 2010

Year

2.0

2.5

3.0

3.5

4.0

4.5

Rat

io

Capital to GDP ratio in USA

27

Kaldor's stylized fact 5

The shares of labor and physical capital in national incomeare nearly constant.

1950 1960 1970 1980 1990 2000 2010

Year

50

55

60

65

70

75

Sh

are

inin

com

e(%

)

Labor share in income in USA

28

Kaldor's stylized fact 6

Real wage grows over time.

1950 1960 1970 1980 1990 2000 2010

Year

101

2× 101

3× 101

4× 101

Mea

nh

ourl

yco

mp

ensa

tion

(200

9U

SD

) Real mean hourly compensation in USA

29

Explaining growth

We want to explain:

• Why some countries are poor and other rich?• Why some countries that were previously poor become rich?• Why not all poor countries catch up to rich countries?• Why do rich countries still grow?

30

Solow-Swan model

Developed by Robert Solow (1956) and Trevor Swan (1956).

Growth in income per capita comes from two sources:

• Capital accumulation (endogenous).• Improvements in technology (exogenous).

But capital accumulation alone cannot sustain growthin the absence of technology improvements.

Does not explain “deep” sources of economic growth.

Departure point for growth theory.

31

Simplifications and assumptions

• Closed economy.• No government.• Single, homogenous final good with its price normalized to 1in each period (all variables are expressed in real terms).

• Two types of representative agents:• Firms.• Households.

32

Production

Real GDP is produced according to a neoclassical prod. function:

Yt = F (Kt,AtLt)

where Y is real GDP, F is a neoclassical prod. function, K is capitalstock, A is the technology level and L is the number of workers.

Technology grows at a rate g > 0 and increases productivityof labor (otherwise Kaldor's stylized facts would be violated):

At+1 = (1+ g)AtVery often we use a Cobb-Douglas production function:

Yt = Kαt (AtLt)

1−α

Like other neoclassical prod. functions, it exhibits constant returnsto scale – doubling inputs K and L doubles the amount produced:

(λKt)α(At · λLt)1−α

= λαλ1−αKαt (AtLt)

1−α= λYt

33

Firms

Perfectly competitive firms maximize their profit:

maxKt,Lt

Kαt (AtLt)

1−α − rkt Kt − wtLt

where rk denotes the rental rate on capital.

FOCs:

Kt : αKα−1t (AtLt)1−α − rkt = 0 −→ rkt = α

YtKt

Lt : (1− α) Kαt A1−α

t L−αt − wt = 0 −→ wt = (1− α)

YtLt

Total factor payments are equal to GDP:

rkt Kt + wtLt = αYtKtKt + (1− α)

YtNtLt = αYt + (1− α) Yt = Yt

34

Factor shares

Calculate the fraction of GDP that is paid to each factor:

wtLtYt

=(1− α)

YtLt

· Lt

Yt= (1− α) and rkt Kt

Yt=

αYtKt

· Kt

Yt= α

Cobb-Douglas function implies constant sharesof labor and physical capital in income.

Confronting with the US data, we can obtain α ≈ 13 and (1− α) ≈ 2

3 .

35

Households

Own factors of production (capital and labor)and earn income from renting them to firms.

Each households supplies one unit of labor: Lt = Ntand population grows at a rate n:

Nt+1 = (1+ n)Nt

Capital accumulates from investment It and depreciates at rate δ > 0:

Kt+1 = It + (1− δ) Kt

Income of households is consumed or saved (invested):

Yt = wtLt + rkt Kt = Ct + St = Ct + It

Households don't optimize, save a constant fraction s of income:

It = sYt and Ct = (1− s) Yt

36

GDP per worker

Usually we are most interested in GDP per worker (or per capita), y:

yt ≡ YtLt

=Kαt (AtLt)

1−α

Lt= At

(KtAtLt

)α

≡ Atkαt

where k is capital K divided per effective unit of labor (AL).

Clearly, GDP per worker increases due to improvements intechnology and due to capital accumulation.

The production function exhibits diminishing marginal returns tocapital. GDP per worker increases with k, but the size of the increasefalls with k.

It is also useful to define output per effective unit of labor y:

yt =ytAt

= kαt

37

Capital accumulation

Capital accumulates according to:

Kt+1 = sYt + (1− δ) Kt

And capital per effective labor according to:

Kt+1 = sYt + (1− δ) Kt | : AtLtLt+1Lt

At+1At

Kt+1At+1Lt+1

= s YtAtLt+ (1− δ)

KtAtLt

(1+ n) (1+ g) kt+1 = syt − δkt

kt+1 =skα

t + (1− δ) kt1+ n+ g+ ng ≈ skα

t + (1− δ) kt1+ n+ g

The growth rate of capital per effective labor equals:

kt+1 − kt =skα

t − (δ + n+ g) kt1+ n+ g

gk ≡ ∆kt+1kt

=kt+1 − kt

kt=skα−1

t − (δ + n+ g)1+ n+ g 38

Balanced growth path (steady state)

Variables per effective labor converge to their steady state values.If kt+1 = kt = k∗ (or, equivalently, gk = 0) then:

s(k∗)α−1

= δ + n+ g

k∗ =

(s

δ + n+ g

) 11−α

y∗ =

(s

δ + n+ g

) α1−α

Along the balanced growth path (BGP) variables per workergrow together with increases in technology:

y∗t = Aty∗ −→

∆y∗t+1y∗t

=∆At+1At

= g

And aggregate variables like aggregate capital and GDP growat the sum of rates of increase in population and technology.

39

Comparative statics

Solow-Swan model predicts that the BGP level of GDP per worker:

y∗t = At

(s

δ + n+ g

) α1−α

is higher in countries with higher investment share of GDP sand higher technology level A,and lower in countries with higher population growth rate n.

40

Investment share of GDP s vs GDP per worker y

0 10 20 30 40 50

Average investment share of GDP, 1950-2014 (%)

1000

2000

5000

10000

20000

50000

100000

200000

GD

Pp

erw

orke

rin

2014

(2011

US

D)

Investment rate vs GDP per worker

41

Population growth rate n vs GDP per worker y

−2 −1 0 1 2 3 4 5

Average population growth rate, 1950-2014 (%)

1000

2000

5000

10000

20000

50000

100000

200000

GD

Pp

erw

orke

rin

2014

(2011

US

D)

Population growth rate vs GDP per worker

42

Transitional dynamics

We are also interested in the determinants of growth ratesin GDP per worker outside of the BGP.

Start with growth rates of GDP per effective labor:

∆yt+1yt

≈ log(yt+1yt

)= log

(kαt+1

kαt

)= α log

(kt+1kt

)≈ α

∆kt+1kt

∆yt+1yt

≈ α[skα−1

t − (δ + n+ g)]

To obtain the growth rate of GDP per worker,add the growth rate of technology g:∆ytyt

= α[skα−1

t − (δ + n+ g)]+ g = α

[skα−1

t − (δ + n)]+ (1− α)g

An increase in s or a decrease in n temporarily increases the growthrate of GDP per worker. Note that even if higher g decreases k∗,it increases the rate of growth of GDP per worker. 43

Investment share of GDP s in “growth miracle” countries

1950 1960 1970 1980 1990 2000 2010

Year

0

10

20

30

40

50

60

70

Inve

stm

ent

shar

eof

GD

P(%

)Investment rates in “growth miracle” countries

Taiwan

South Korea

Singapore

Hong Kong

44

Factor payments once again

Using k∗ as capital per effective labor along the BGP,let us revisit factor prices:

rk∗t = αKα−1

t (AtLt)1−α= α

(k∗)α−1

w∗t = (1− α) Kα

t A1−αt L−α

t = (1− α)At(k∗)α

The model predicts that along the BGP the interest rates are constantwhile hourly wages increase at the same rate as GDP per hour:

1950 1960 1970 1980 1990 2000 2010

102

6× 101

2× 102

3× 102

4× 102

Ind

ex19

60=

100

Real compensation vs GDP per hour in USA

Real mean hourly compensation

Real GDP per hour

45

Convergence

Solow-Swan model predicts that if countries have access to thesame technology and share the same steady state, then ones thatare poorer should grow faster:

0

γk

k∗ k

46

Convergence: USA

We can observe convergence across individual states in USA:

Barro and Sala-i-Martin (1991)Convergence across States and Regions

47

Convergence: “West”

We can observe convergence across “Western” countries (+ Japan):

1000 2000 5000 10000

GDP per capita in 1870 (2011 USD)

1.25

1.50

1.75

2.00

2.25

2.50

An

nu

algr

owth

rate

,18

70-2

013

(%)

AUSBEL

CAN

CHE

DEUDNK

ESP

FIN

FRA

GBR

ITA

JPN

NLD

NOR

PRT

SWE

USA

Convergence across “Western” countries

48

Convergence: OECD

We can observe convergence across initial OECD members:

104 2× 104 3× 104 4× 104

GDP per worker in 1960 (2011 USD)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

An

nu

algr

owth

rate

,19

60-2

014

(%)

AUT

BEL

CAN

DNK

FRA

DEU

GRC

ISL

IRL

ITA

LUX

NLD

NORPRT

ESP

SWE

CHE

TUR

GBR

USAAUS

FIN

JPN

NZL

Convergence across initial OECD members

49

Convergence: conditional/club, but not absolute

In general it is not true that poorer countries grow faster:

1000 2000 5000 10000 20000 50000

GDP per worker in 1960 (2011 USD)

−2

0

2

4

6

8

An

nu

algr

owth

rate

,19

60-2

014

(%)

ARG

AUS

AUTBEL

BFA

BGD

BOL

BRA

BRB

CAN

CHECHL

CHN

CIVCMR

COD

COLCRI

CYPDEU

DNK

DOM

ECU

EGY

ESP

ETH

FINFRA

GBR

GHA

GRC

GTM

HKGIDN

IND

IRL

IRNISL

ISR

ITA

JAM

JOR

JPN

KEN

KOR

LKA

LUX

MAR

MDG

MEX

MLI

MLT

MOZ

MWI

MYS

NER

NGA

NLD

NOR

NZL

PAK

PERPHL

PRT

ROU

SEN

SGP

SWE

SYR

THA

TTO

TUN TUR

TWN

TZAUGA URY

USA

VEN

ZAFZMB

No convergence across all countries

50

Conditional convergence

But countries grow faster the further away they are from their ownsteady state:

DZA

ARG

AUS

AUT

BGD

BRB

BEL

BENBOL

BWA

BRA

BDICMR

CAN

CAF

CHL

CHN

COLCOG

CRICIV

CYP

DNKDOM

ECU

EGY

SLV FJI

FINFRA

GAB

GMBGHA

GRC

GTM

HTI

HND

HKG

ISL

INDIDN

IRN

IRL

ISR

ITA

JAM

JPN

JORKEN

KOR

LSOLUX

MWI

MYS

MLI

MRTMUS

MEX

MAR

MOZ

NAMNPL

NLD

NZL

NIC

NER

NORPAKPANPNG

PRY PER

PHL

PRT

ROM

RWASEN

SGP

ZAF

ESP

LKA

SWE

CHESYR

TWN

TZA

THA

TGO

TTO

TUR

UGA

GBRUSA URY

VENZMB

ZWE

−0.02

0.00

0.02

0.04

0.06G

row

th r

ate

of G

DP

per

wor

ker,

196

0−−

2008

0.25 0.50 0.75 1.00 1.25 1.50Deviation from steady state in 1960, log scale

Jones and Vollrath (2013)Introduction to Economic Growth

51

Speed of convergence

The model implies a relationship between the distance from steadystate and the current rate of growth:

∆yt+1yt

≈ (1− α) (δ + n+ g)︸ ︷︷ ︸β

(log y∗t − log yt)

Econometric studies both on individual countries and states withinUSA find that β ≈ 0.02, meaning that it takes about 35 years to closehalf of the gap between the current income level and its steady state.

Given sensible parameter values: α = 0.33, δ = 0.05, n = 0.01,g = 0.02, the model generates β = 0.053, implying that it would takeabout 13 years to close half of the gap, a very unrealistic number.

Adding human capital allows the model to assign lower weightto raw labor and be consistent with slow convergence.

52

Human capital augmented Solow model

The production function that accounts for human capital:

Yt = Kαt (AtHt)

1−α

Ht = h (ut) Lt

where u are average years of schooling. Benchmark empiricalestimates on returns to schooling are expressed via the h function:

lnh (u) =

0.134 · u if u ≤ 40.134 · 4+ 0.101 · (u− 4) if 4 < u ≤ 80.134 · 4+ 0.101 · 4+ 0.068 · (u− 8) if u > 8

The estimates capture the empirical regularity that schooling boostsindividuals' wages. Wages contain not only rewards to raw labor, butalso to human capital. Empirical estimates of the income share of``broad capital'' are consistent with convergence factor β ≈ 0.02.

53

Human capital augmented Solow model

The production function that accounts for human capital:

Yt = Kαt (AtHt)

1−α

Ht = h (ut) Lt

Human capital generates level effects for GDP per workeralong the BGP:

yt = Ath (ut)(

sδ + n+ g

) α1−α

54

Human capital per capita h vs real GDP per worker y

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Index of human capital per worker in 2014

1000

2000

5000

10000

20000

50000

100000

200000

GD

Pp

erw

orke

rin

2014

(2011

US

D)

Human capital level vs GDP per worker

55

Fit of human capital-augmented Solow model

Suggests that poor countries ``should'' be richer:

DZA ARG

AUS

AUT

BGD

BRBBEL

BEN

BOL

BWA

BRA

CMR

CAN

CHLCHN

COL

COG

CRI

CIV

CYPDNK

DOM

ECU

EGYSLV

FJIFINFRA

GAB

GMB

GHA

GRC

GTMHTI

HND

HKGISL

IND

IDNIRN

IRLISRITAJAM

JPN

JOR

KEN

KOR

LSO

MYS

MLI

MRT

MUSMEX

MAR NAM

NPL

NLDNZL

NIC

NER

PAK

PAN

PNG

PRYPER

PHLPRT

ROM

RWA

SEN

SGP

ZAF

ESPLKA SWECHE

SYR

TWN

TZA

THA TTO

TUR

UGA

GBR

USA

URY

VENZMB

0.02

0.05

0.10

0.20

0.30

0.50

0.75

1.00

Pre

dict

ed s

tead

y st

ate

valu

e of

rel

ativ

e Y

/L

0.02 0.05 0.10 0.20 0.30 0.50 0.75 1.00Relative Y/L

Jones and Vollrath (2013)Introduction to Economic Growth

56

Solow residual: accounting for technology differences

There are also significant differences in technology across countries:

DZA

ARG

AUS

AUT

BGD

BRB

BEL

BEN

BOL

BWABRA

CMR

CAN

CHL

CHN

COL

COG

CRI

CIV

CYP

DNK

DOM

ECU

EGYSLV

FJI

FINFRA

GAB

GMB

GHA

GRC

GTM

HTI

HND

HKGISL

IND

IDN

IRN

IRL

ISR

ITA

JAM

JPN

JOR

KEN

KOR

LSO

MYS

MLI

MRT

MUS

MEX

MAR

NAM

NPL

NLD

NZL

NIC

NER

PAK

PAN

PNG

PRY

PER

PHL

PRT

ROM

RWA SEN

SGP

ZAF

ESP

LKA

SWECHE

SYR

TWN

TZA

THA

TTO

TUR

UGA

GBR

USA

URYVEN

ZMB

0.05

0.10

0.20

0.30

0.50

0.75

1.001.25

1.75

Rel

ativ

e A

0.02 0.05 0.10 0.20 0.30 0.50 0.75 1.00Relative Y/L

Jones and Vollrath (2013)Introduction to Economic Growth

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Takeaway

• Long run growth stems from improvements in technology.• Countries can achieve higher balanced growth paths if theyaccumulate more physical and human capital.

• Just as important as accumulation is technology adoption.• Did not touch on “deep” causes of growth – we treated manychoice variables as exogenous parameters:

• Countries with low s may not protect private ownership properlyor have underdeveloped financial system.

• Countries with high n may have high mortality rates incentivizingfamilies to have many children in hopes that at least some surviveinto adulthood to be able to support their (then old) parents.

• Countries may have high u because they are already richand children do not have to work there.

• Groups of interest within a country may obstruct technologyadoption if they have a monopoly over the old technology.

58