the facts of growth – the long run

Economics 302 Growth 1

The Facts of Growth – The Long Run



Annual Growth Rate Real Output per CapitaOutput per Capita (%) (1992 dollars)

Ratio of Real OuputPer Capita

1950-1973 1973-1998 1950 1998 1998/1950

France 4.2 1.6 5,150 19,158 3.7

Germany 4.9 1.8 4,356 20,059 4.6

Japan 8.1 2.5 1,820 19,907 10.9

United Kingdom 2.5 1.9 6,870 19,005 2.8

United States 2.2 1.5 11,170 25,890 2.3

Average 4.4 1.9 5,872 20,804 3.5



Constructing Output NumbersConstructing Output Numbers

Population

Output• Output Per Capita =

Purchasing Power Parity: Adjusts for differences in exchange rates and prices



ObservationsObservations

• Strong growth 1950-2001

• Growth rates have decreased since the mid 1970s1950-1978 4.4% (GDP/capita doubles every 16 years)

1973-2001 1.9% (GDP/capita doubles every 37 years)

• Convergence in output/capita across countries



Convergence in Output/CapitaConvergence in Output/Capita



What do you think… What do you think…

Could the finding of convergence beinfluenced by the way the countries areselected?



A Broader Look Across Time and SpaceA Broader Look Across Time and Space

Looking across two millennia

• From the end of the Roman Empire to 1500, no output per capita growth in Europe

• 1500-1700 -- Small growth in output per capita (0.1%/year and 0.2%/year 1700 to 1820)

• 1820-1950 -- Modest growth (U.S. = 1.5%)

• The high-growth of the 1950s and 1960s is unusual



A Broader Look Across Time and SpaceA Broader Look Across Time and Space

Looking across two millennia (Continued)

• 1st Millennium to the 15th century, China had the highest output/capita

• Leaders in output/capita change frequently:ItalyNetherlandsU.K.



The Reality of Growth: A Workingman’s Budget in 1851The Reality of Growth: A Workingman’s Budget in 1851Amount Percent of

Item of Expenditure (Dollars) Total

Butcher’s meat (2 lb a day) 72.80 13.5Flour (6 1/2 lb a year) 32.50 6.0Butter (2 lb a week) 32.50 6.0Potatoes (2 pk a week) 26.00 4.8Sugar (4 lb a week) 16.64 3.0Coffee and tea 13.00 2.4Milk 7.28 1.4Salt, pepper, vinegar, starch, soap, yeast, cheese, eggs 20.80 3.9 Total expenditure for food 221.52 41.0Rent 156.00 29.0Coal (3 tons a year) 15.00 2.8Charcoal, chips, matches 5.00 0.9Candles and oil 7.28 1.4Household articles (wear, tear, and breakage) 13.00 2.4Bedclothes and bedding 10.40 1.9Wearing apparel 104.00 19.3Newspapers 6.24 1.2 Total expenditures other than food 316.92 58.9



Looking Across Countries – Convergence Not the RuleLooking Across Countries – Convergence Not the Rule



Looking Across Countries – A Closer LookLooking Across Countries – A Closer Look




• OECD countries are converging• Asian countries are converging• African countries are not converging

Three Conclusions:




• Miracles: Japan + Asian Tigers (South Korea, Taiwan, Singapore, Hong Kong)(average annual growth of over 5% 1960-1990)• Disasters: Argentina (average income in 1900Similar to those of world leaders; now middle of World income distribution)• Disaster: Sub-saharan Africa (Chad, Ghana,Mozambique) extremely poor throughout history

Growth Miracles vs Growth Disasters



A SummaryA Summary

1. Growth is not a historical necessity

2. Convergence of OECD countries to the U.S. may be the prelude to leapfrogging

3. The rapid post WWII growth was atypical



Thinking About Growth: A Primer (The Solow Model)Thinking About Growth: A Primer (The Solow Model)

The Aggregate Production Function

Y = F (K, N)Y = Aggregate OutputK = CapitalN = Labor



The Aggregate Production Function

Y = F (K, N)

F: Depends on technologyF: Depends on technology



Returns to Scale and Returns to FactorsReturns to Scale and Returns to Factors

Constant returns to scale: 2Y = F(2K,2N)xY = F(xK,xN)

Decreasing returns to factors (capital & labor):

•Increases in K and N lead to smaller andsmaller increases in output



Output and Capital per workerOutput and Capital per worker

)1,(

1

),(

n

KF

n

YN

x

xNxKfxY



Output and Capital per workerOutput and Capital per worker

n

KY)F(K,

N

Yn

Kn

Y

ondepends :1

rkercapital/wo

keroutput/wor



Output and Capital per workerOutput and Capital per workerO

utp

ut

pe

r w

ork

er,

Y/N

Capital per worker, K/N

Y/N = (K/N, 1)

A

A´

B´

B

C´

C

D´

D


F(K/N, 1)

F(K/N, 1)


The Sources of GrowthThe Sources of Growth

• An improvement in technology shifts the production function up

Ou

tpu

t p

er

wo

rke

r, Y

/N


A

A´

B´




• Increases in occur when technology

shifts the production functionsn

Y




• Capital Accumulation•Cannot sustain growth because of

diminishing returns to capital•Capital accumulation requires savings,

therefore, what is the appropriate savingsrate?




• Technological Progress•Required for sustained growth•What determines the rate of technological

progress?


Saving, Capital Accumulation, and OutputThe Long Run

Observation:Observation:

)GDP

Savings(The savings rate since 1950

U.S. 18.6%Germany 24.6%Japan 33.7%

What do you think…

Would increasing the U.S. savings rate lead to sustained higher U.S. growth in the future?Would increasing the U.S. savings rate lead to sustained higher U.S. growth in the future?


Interactions between Output and Capital

Two long-run relations between output and capitalTwo long-run relations between output and capital

The amount of capital determines the amount of output being produced

The amount of output determines the amount of savings and investment and, thus, the

amount of capital



Capital, Output, and Saving/InvestmentCapital, Output, and Saving/Investment



The Effects of Capital on OutputThe Effects of Capital on Output

NK

fNY

NY

output per worker

NK capital per worker



The Effects of Capital on OutputThe Effects of Capital on Output

NK

fNY tt

Therefore:

Two assumptions:

1. Employment (N) is constant

2. There is no technological progress (f is constant)



The Effects of Output on Capital AccumulationThe Effects of Output on Capital Accumulation

Three assumptions:

1. Closed economy: I = S + (G-T)

2. I = S if G = T = 0, therefore (G-T)=0

3. S = sY: Private saving is proportional to

incomes = Savings rate (between 0 &

1)

Output & Investment




Observations

1. The savings rate does not appear to systematically increase or decrease as Y increases

2. Richer countries do not appear to have systematically higher savings rates than poorer ones

3. Investment is proportional to output

Output & Investment It = sYt




The evolution of the capital stock:

Investment and Capital Accumulation

Kt+1 = (1- ) Kt + It

= the depreciation rate




The relation between output and capital accumulation


Kt+1 = (1- ) Kt + It and It = sYtKt+1 = (1- ) Kt + sYt and ÷ N

NY

sNK

N

K ttt )1(1






N

Ys

N

K

N

K ttt )1(1

Capital/worker in t+1 = Capital/Worker in t, adjusted for depreciation and investment

Investment/worker = Savings rate x Output/worker in t

Capital/worker in t+1 = Capital/Worker in t, adjusted for depreciation and investment

Investment/worker = Savings rate x Output/worker in t






N

Ys

N

K

N

K ttt )1(1

NK

NY

sNK

N

K tttt 1Reorganizing:

Savings/worker - depreciationChange inN

K



A SummaryA Summary

The Production Side

Capital-Output Relation:

N

Kf

N

Y tt

N

K

N

Ys

N

K

N

K tttt 1

Output-Capital Relation:


Implications of Alternative Saving Rates

Dynamics of Capital and OutputDynamics of Capital and Output

Given:

NK

NY

sNK

N

K tttt 1

N

Kf

N

Y ttAnd:

N

K

N

Ksf

N

K

N

K tttt

1




Given:

N

K

N

Ksf

N

K

N

K ttt

1

-Change in capitalfrom year t to year t+1

= Invest-ment during year t

depreciationduring year t




N

Kt

N

Ksf

N

K

N

K tt

1

The change in capital/worker from t to t+1 depends on the difference between:

Investment/Worker & Depreciation/WorkerInvestment/Worker & Depreciation/Worker




Capital/Worker increases:

Investment/Worker > Depreciation/Worker

Capital/Worker decreases:

Investment/Worker < Depreciation/Worker


Output per worker f(Kt/N)

Depreciation per worker Kt/Nδ

Investment per worker sf(Kt/N)


Dynamics of Capital and Output GraphicallyDynamics of Capital and Output GraphicallyO

utp

ut

pe

r w

ork

er,

Y/N


A

B

Y*/N

C

K*/N

D

(Ko/N)

AB = Output/worker

AC = Investment/worker

AD = Depreciation

AC > AD




The evolution of & over timeN

K

N

Y

Assume:

N

Kis low, therefore I > depreciation ( )

N

KIf I > , increases until I =

When I = : & remain constant @N

K

N

Y

N

Y

Long-run equilibrium



Steady-State Capital and OutputSteady-State Capital and Output

Steady-State Value of Capital/Worker

N

K

N

Ksf

N

K **:

*

Steady-State Value of Output/Work

N

Kf

N

Y **



The Saving Rate and OutputThe Saving Rate and Output

What are the effects of the saving rate on the rate of output per worker?

1. The saving rate has no effect on the long run growth rate of output/worker, this is equal to zero.

2. The saving rate determines the level of output/worker in the long run.

3. An increase in the saving rate will lead to a higher growth of output/worker for some time, but not forever.


Investment s0f(Kt/N)

(K0/N)


The Effects of Different Saving RateThe Effects of Different Saving RateDepreciation per worker Kt/Nδ

Ou

tpu

t p

er

wo

rke

r, Y

/N



K1/N




Ou

tpu

t p

er

wo

rke

r, Y

/N


Depreciation per worker Kt/Nδ


The Effects of Different Saving RateThe Effects of Different Saving Rate

Output per worker f(Kt/N)Y1/N

B

A

K1/N(K0/N)

Y0/N

D

C

I > δ


Ou

tpu

t p

er

wo

rke

r, Y

/N

Time



Y1/N

Y0/N

t

Associated with saving rate s0

Associated with saving rate s1 > s0

(No technological progress)


Ou

tpu

t p

er

wo

rke

r, Y

/N(l

og

sc

ale

)

Time



t

Associated with saving rate s0

Associated with saving rate s1 > s0

(Technological progress)



The Savings Rate and the Golden RuleThe Savings Rate and the Golden Rule

Does an increase in saving lead to an increase inconsumption in the long run?

Two Scenarios:Two Scenarios:

•Saving Rate = 0•Capital = 0•Output = 0•Consumption = 0

•Saving Rate = 1•Consumption = 0

•Output replaces depreciation




The Golden-Rule Level of Capital:

The value of saving that yields the highestlevel of consumption in steady state.The value of saving that yields the highestlevel of consumption in steady state.



Co

ns

um

pti

on

pe

r w

ork

er,

C/N

Saving rate, s

sG

0 1

Maximum steady stateConsumption per worker


Question for DiscussionQuestion for Discussion

Can a low saving/investment rate explain why the U.S. growth rate has been so low since 1950?

Can a low saving/investment rate explain why the U.S. growth rate has been so low since 1950?

Getting a Sense of Magnitudes




Are countries likely to have too muchcapital?

The value of saving that yields the highestlevel of consumption in steady state.The value of saving that yields the highestlevel of consumption in steady state.

Consider:

What do you think…What do you think…



Some Questions:Some Questions:

1. How large is the effect of a change in the saving rate on output in the long run?

2. For how long and by how much does an increase in the saving rate affect growth?

3. How far is the U.S. from the Golden Rule of Level of Capital?


Calculating the AnswersCalculating the Answers

Assume: NKY (Constant return to scale and decreasing returns to either capital or labor)

N

K

N

K

N

NK

N

Y



N

K

N

Y

Replace withN

K

N

Kf t

Recall:

N

K

N

Ksf

N

K

N

K tttt

1

N

K

N

Ks

N

K

N

K tttt 1



N

K

N

Ks

N

K

N

K tttt 1

The Effects of the Saving Rate on Steady-State OutputThe Effects of the Saving Rate on Steady-State Output

In steady-state is constant and the left side= 0 and:

NK

NK

NK

s t





N

K

N

Ks tt : square both sides

222

N

K

N

Ks tt : Divide by and reorganize

N

K

2

s

N

K



Steady-State Output/Worker:

ss

N

K

N

Y

2

ObservationObservation

Higher saving rate and lower depreciation both

lead to higher and in the long run.n

Y

n

K




4toincreases%;20&%10N

Ks

Assume: 1statesteady%;10&%10 s

2toincreasesN

Y



The Dynamic Effects of an Increase in the Saving RateThe Dynamic Effects of an Increase in the Saving Rate

Assume:


•The saving rate has always been equal to 0.1

•Then the saving rate increases to 0.2 forever

Then:

11)1.0/1.0( 220 N

K

N

K

N

Ks

N

K

N

K 0001



1.11 N

K

]1)1.0[()]1)(2.0[(1/1 NK

Continuing for each year yields…




The U.S. Saving Rate and the Golden RuleThe U.S. Saving Rate and the Golden Rule

What saving rate that would maximize steady-state consumption?

In Steady-State:

N

K

N

Y

N

C

2)( ssN

C

)1( ss



The Saving Rate and the Steady-State Levels of Capital, Output, and Consumption per WorkerThe Saving Rate and the Steady-State Levels of Capital, Output, and Consumption per Worker

Capital Output ConsumptionSaving Rate, per Worker, per Worker, per Worker,

s K/N Y/N C/N

0.0 0.0 0.0 0.00.1 1.0 1.0 0.90.2 4.0 2.0 1.60.3 9.0 3.0 2.10.4 16.0 4.0 2.10.5 25.0 5.0 2.50.6 36.0 6.0 2.4.. .. .. ..

1.0 100.0 10.0 0.0



The U.S. Saving Rate and the Golden RuleThe U.S. Saving Rate and the Golden Rule

Observation

If s < .50: increasing s will increase long-runconsumption

In the U.S., s < 20%


Physical Versus Human Capital

Human Capital:Human Capital:

The set of skills of the workers in theeconomy.



ObservationsObservations

OECD Countries:

•100% of children get a primary education

•90% of children get a secondary education

•38% of children get a higher education

•Literacy rate above 95%



Extending the Production FunctionExtending the Production Function

Measuring the Impact of Human Capital:

),(

,

NH

NK

fNY

rkerCapital/Wo HumanN

H



Human Capital, Physical Capital, and OutputHuman Capital, Physical Capital, and Output

How does including human capital impact our analysis?

Investment now includes physical and humancapital

In the U.S.:

Education spending = 6.5% of GDPInvestment = 16.0% of GDP



Human Capital, Physical Capital, and OutputHuman Capital, Physical Capital, and Output

Some Complications:

• Education is partly consumption• Education cost should include the opportunity

cost• On-the-job training is not included• Should compare investment rates net of

depreciation



Endogenous GrowthEndogenous Growth

What do you think…

To what extent does growth depend on the savings rate, education, and technological change?

the facts of growth – the long run

Documents