The Facts of Growth – The Long Run
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Economics 302 Growth 1
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
Economics 302 Growth 1
The Facts of Growth – The Long Run
Constructing Output NumbersConstructing Output Numbers
Population
Output• Output Per Capita =
Purchasing Power Parity: Adjusts for differences in exchange rates and prices
Economics 302 Growth 1
The Facts of Growth – The Long Run
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
Economics 302 Growth 1
The Facts of Growth – The Long Run
Convergence in Output/CapitaConvergence in Output/Capita
Economics 302 Growth 1
The Facts of Growth – The Long Run
What do you think… What do you think…
Could the finding of convergence beinfluenced by the way the countries areselected?
Economics 302 Growth 1
The Facts of Growth – The Long Run
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
Economics 302 Growth 1
The Facts of Growth – The Long Run
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.
Economics 302 Growth 1
The Facts of Growth – The Long Run
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
Economics 302 Growth 1
The Facts of Growth – The Long Run
Looking Across Countries – Convergence Not the RuleLooking Across Countries – Convergence Not the Rule
Economics 302 Growth 1
The Facts of Growth – The Long Run
Looking Across Countries – A Closer LookLooking Across Countries – A Closer Look
Economics 302 Growth 1
The Facts of Growth – The Long Run
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:
Economics 302 Growth 1
The Facts of Growth – The Long Run
Looking Across Countries – A Closer LookLooking Across Countries – A Closer Look
• 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
Economics 302 Growth 1
The Facts of Growth – The Long Run
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
Economics 302 Growth 1
The Facts of Growth – The Long Run
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
Economics 302 Growth 1
The Facts of Growth – The Long Run
The Aggregate Production Function
Y = F (K, N)
F: Depends on technologyF: Depends on technology
Economics 302 Growth 1
The Facts of Growth – The Long Run
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
Economics 302 Growth 1
The Facts of Growth – The Long Run
Output and Capital per workerOutput and Capital per worker
)1,(
1
),(
n
KF
n
YN
x
xNxKfxY
Economics 302 Growth 1
The Facts of Growth – The Long Run
Output and Capital per workerOutput and Capital per worker
n
KY)F(K,
N
Yn
Kn
Y
ondepends :1
rkercapital/wo
keroutput/wor
Economics 302 Growth 1
The Facts of Growth – The Long Run
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
Economics 302 Growth 1
F(K/N, 1)
F(K/N, 1)
The Facts of Growth – The Long Run
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
Capital per worker, K/N
A
A´
B´
Economics 302 Growth 1
The Facts of Growth – The Long Run
The Sources of GrowthThe Sources of Growth
• Increases in occur when technology
shifts the production functionsn
Y
Economics 302 Growth 1
The Facts of Growth – The Long Run
The Sources of GrowthThe Sources of Growth
• Capital Accumulation•Cannot sustain growth because of
diminishing returns to capital•Capital accumulation requires savings,
therefore, what is the appropriate savingsrate?
Economics 302 Growth 1
The Facts of Growth – The Long Run
The Sources of GrowthThe Sources of Growth
• Technological Progress•Required for sustained growth•What determines the rate of technological
progress?
Economics 302 Growth 1
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?
Economics 302 Growth 1
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
Economics 302 Growth 1
Interactions between Output and Capital
Capital, Output, and Saving/InvestmentCapital, Output, and Saving/Investment
Economics 302 Growth 1
Interactions between Output and Capital
The Effects of Capital on OutputThe Effects of Capital on Output
NK
fNY
NY
output per worker
NK capital per worker
Economics 302 Growth 1
Interactions between Output and Capital
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)
Economics 302 Growth 1
Interactions between Output and Capital
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
Economics 302 Growth 1
Interactions between Output and Capital
The Effects of Output on Capital AccumulationThe Effects of Output on Capital Accumulation
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
Economics 302 Growth 1
Interactions between Output and Capital
The Effects of Output on Capital AccumulationThe Effects of Output on Capital Accumulation
The evolution of the capital stock:
Investment and Capital Accumulation
Kt+1 = (1- ) Kt + It
= the depreciation rate
Economics 302 Growth 1
Interactions between Output and Capital
The Effects of Output on Capital AccumulationThe Effects of Output on Capital Accumulation
The relation between output and capital accumulation
Investment and Capital Accumulation
Kt+1 = (1- ) Kt + It and It = sYtKt+1 = (1- ) Kt + sYt and ÷ N
NY
sNK
N
K ttt )1(1
Economics 302 Growth 1
Interactions between Output and Capital
The Effects of Output on Capital AccumulationThe Effects of Output on Capital Accumulation
The relation between output and capital accumulation
Investment and Capital Accumulation
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
Economics 302 Growth 1
Interactions between Output and Capital
The Effects of Output on Capital AccumulationThe Effects of Output on Capital Accumulation
The relation between output and capital accumulation
Investment and Capital Accumulation
N
Ys
N
K
N
K ttt )1(1
NK
NY
sNK
N
K tttt 1Reorganizing:
Savings/worker - depreciationChange inN
K
Economics 302 Growth 1
Interactions between Output and Capital
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:
Economics 302 Growth 1
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
Economics 302 Growth 1
Implications of Alternative Saving Rates
Dynamics of Capital and OutputDynamics of Capital and Output
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
Economics 302 Growth 1
Implications of Alternative Saving Rates
Dynamics of Capital and OutputDynamics of Capital and Output
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
Economics 302 Growth 1
Implications of Alternative Saving Rates
Dynamics of Capital and OutputDynamics of Capital and Output
Capital/Worker increases:
Investment/Worker > Depreciation/Worker
Capital/Worker decreases:
Investment/Worker < Depreciation/Worker
Economics 302 Growth 1
Output per worker f(Kt/N)
Depreciation per worker Kt/Nδ
Investment per worker sf(Kt/N)
Implications of Alternative Saving Rates
Dynamics of Capital and Output GraphicallyDynamics of Capital and Output GraphicallyO
utp
ut
pe
r w
ork
er,
Y/N
Capital per worker, K/N
A
B
Y*/N
C
K*/N
D
(Ko/N)
AB = Output/worker
AC = Investment/worker
AD = Depreciation
AC > AD
Economics 302 Growth 1
Implications of Alternative Saving Rates
Dynamics of Capital and OutputDynamics of Capital and Output
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
Economics 302 Growth 1
Implications of Alternative Saving Rates
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 **
Economics 302 Growth 1
Implications of Alternative Saving Rates
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.
Economics 302 Growth 1
Investment s0f(Kt/N)
(K0/N)
Implications of Alternative Saving Rates
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
Capital per worker, K/N
Investment s1f(Kt/N)
K1/N
Economics 302 Growth 1
Investment s0f(Kt/N)
Investment s1f(Kt/N)
Ou
tpu
t p
er
wo
rke
r, Y
/N
Capital per worker, K/N
Depreciation per worker Kt/Nδ
Implications of Alternative Saving Rates
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
Economics 302 Growth 1
Ou
tpu
t p
er
wo
rke
r, Y
/N
Time
Implications of Alternative Saving Rates
The Effects of Different Saving RateThe Effects of Different Saving Rate
Y1/N
Y0/N
t
Associated with saving rate s0
Associated with saving rate s1 > s0
(No technological progress)
Economics 302 Growth 1
Ou
tpu
t p
er
wo
rke
r, Y
/N(l
og
sc
ale
)
Time
Implications of Alternative Saving Rates
The Effects of Different Saving RateThe Effects of Different Saving Rate
t
Associated with saving rate s0
Associated with saving rate s1 > s0
(Technological progress)
Economics 302 Growth 1
Implications of Alternative Saving Rates
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
Economics 302 Growth 1
Implications of Alternative Saving Rates
The Savings Rate and the Golden RuleThe Savings Rate and the Golden Rule
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.
Economics 302 Growth 1
Implications of Alternative Saving Rates
Co
ns
um
pti
on
pe
r w
ork
er,
C/N
Saving rate, s
sG
0 1
Maximum steady stateConsumption per worker
Economics 302 Growth 1
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
Economics 302 Growth 1
Implications of Alternative Saving Rates
The Savings Rate and the Golden RuleThe Savings Rate and the Golden Rule
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…
Economics 302 Growth 1
Getting a Sense of Magnitudes
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?
Economics 302 Growth 1
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
Getting a Sense of Magnitudes
Economics 302 Growth 1
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
Getting a Sense of Magnitudes
Economics 302 Growth 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
Getting a Sense of Magnitudes
Economics 302 Growth 1
Implications of Alternative Saving Rates
The Effects of the Saving Rate on Steady-State OutputThe Effects of the Saving Rate on Steady-State Output
N
K
N
Ks tt : square both sides
222
N
K
N
Ks tt : Divide by and reorganize
N
K
2
s
N
K
Economics 302 Growth 1
The Effects of the Saving Rate on Steady-State OutputThe Effects of the Saving Rate on Steady-State Output
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
Getting a Sense of Magnitudes
Economics 302 Growth 1
The Effects of the Saving Rate on Steady-State OutputThe Effects of the Saving Rate on Steady-State Output
4toincreases%;20&%10N
Ks
Assume: 1statesteady%;10&%10 s
2toincreasesN
Y
Getting a Sense of Magnitudes
Economics 302 Growth 1
The Dynamic Effects of an Increase in the Saving RateThe Dynamic Effects of an Increase in the Saving Rate
Assume:
Getting a Sense of Magnitudes
•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
Economics 302 Growth 1
The Dynamic Effects of an Increase in the Saving RateThe Dynamic Effects of an Increase in the Saving Rate
1.11 N
K
]1)1.0[()]1)(2.0[(1/1 NK
Continuing for each year yields…
Getting a Sense of Magnitudes
Economics 302 Growth 1
The Dynamic Effects of an Increase in the Saving RateThe Dynamic Effects of an Increase in the Saving Rate
Getting a Sense of Magnitudes
Economics 302 Growth 1
The Dynamic Effects of an Increase in the Saving RateThe Dynamic Effects of an Increase in the Saving Rate
Getting a Sense of Magnitudes
Economics 302 Growth 1
Getting a Sense of Magnitudes
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
Economics 302 Growth 1
Getting a Sense of Magnitudes
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
Economics 302 Growth 1
Getting a Sense of Magnitudes
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%
Economics 302 Growth 1
Physical Versus Human Capital
Human Capital:Human Capital:
The set of skills of the workers in theeconomy.
Economics 302 Growth 1
Physical Versus Human Capital
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%
Economics 302 Growth 1
Physical Versus Human Capital
Extending the Production FunctionExtending the Production Function
Measuring the Impact of Human Capital:
),(
,
NH
NK
fNY
rkerCapital/Wo HumanN
H
Economics 302 Growth 1
Physical Versus Human Capital
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
Economics 302 Growth 1
Physical Versus Human Capital
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
Economics 302 Growth 1
Physical Versus Human Capital
Endogenous GrowthEndogenous Growth
What do you think…
To what extent does growth depend on the savings rate, education, and technological change?