Growth - ECON 30020: Intermediate Macroeconomicsesims1/growth_slides_spring2018.pdfI Via assumptions...
Transcript of Growth - ECON 30020: Intermediate Macroeconomicsesims1/growth_slides_spring2018.pdfI Via assumptions...
GrowthECON 30020: Intermediate Macroeconomics
Prof. Eric Sims
University of Notre Dame
Spring 2018
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Readings
I GLS Ch. 4 (facts)
I GLS Ch. 5-6 (Solow Growth Model)
I GLS Ch. 7 (cross-country income differences)
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Economic Growth
I When economists say โgrowth,โ typically mean average rateof growth in real GDP per capita over long horizons
I Long run: frequencies of time measured in decades
I Not period-to-period fluctuations in the growth rate
I โOnce one begins to think about growth, it is difficult to thinkabout anything elseโ โ Robert Lucas, 1995 Nobel Prize winner
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US Real GDP per capita
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
50 55 60 65 70 75 80 85 90 95 00 05 10 15
Real GDP Per Capita Linear Trend
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Summary Stats
I Average (annualized) growth rate of per capita real GDP:1.8%
I Implies that the level of GDP doubles roughly once every 40years
I Growing just 0.2 percentage points faster (2% growth rate):level doubles every 35 years
I Rule of 70: number of years it takes a variable to double isapproximately 70 divided by the growth rate
I Consider two countries that start with same GDP, but countryA grows 2% per year and country B grows 1% per year. After100 years, A will be 165% richer!
I Small differences in growth rates really matter over longhorizons
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Key Question
I What accounts for this growth?I In a mechanical sense, can only be two things:
I Growth in productivity: we produce more output given thesame inputs
I Factor accumulation: more factors of production help usproduce more stuff
I Two key factors of production on which we focus are capitaland labor
I Labor input per capita is roughly trendless โ empirically not asource of growth in per capita output
I The key factor of production over the long run is capital:physical stuff that must itself be produced that in turn helpsus produce more stuff (e.g. machines).
I Which is it? Productivity or capital accumulation? What arepolicy implications?
I What accounts for differences in standards of living acrosscountries? Productivity or factor accumulation?
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Stylized Facts: Time Series
1. Output per worker grows at an approximately constant rateover long periods of time picture
2. Capital per worker grows at an approximately constant rateover long periods of time picture
3. The capital to output ratio is roughly constant over longperiods of time picture
4. Laborโs share of income is roughly constant over long periodsof time picture
5. The return to capital is roughly constant over long periods oftime picture
6. The real wage grows at approximately the same rate as outputper worker over long periods of time picture
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Stylized Facts: Cross-Section
1. There are large differences in income per capita acrosscountries table
2. There are some examples where poor countries catch up(growth miracles), otherwise where they do not (growthdisasters) table
3. Human capital (e.g. education) strongly correlated withincome per capita table
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Solow Model
I Model we use to study long run growth and cross-countryincome differences is the Solow model, after Solow (1956)
I Model capable of capturing stylized factsI Main implication of model: productivity is key
I Productivity key to sustained growth (not factor accumulation)I Productivity key to understanding cross-country income
differences (not level of capital)
I Important implications for policy
I Downside of model: takes productivity to be exogenous.What is it? How to increase it?
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Model Basics
I Time runs from t (the present) onwards into infinite future
I Representative household and representative firm
I Everything real, one kind of good
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Production Function
I Production function:
Yt = AtF (Kt ,Nt)
I Kt : capital. Must be itself produced, used to produce otherstuff, does not get completely used up in production process
I Nt : labor
I Yt : output
I At : productivity (exogenous)
I Think about output as units of fruit. Capital is stock of fruittrees. Labor is time spent picking from the trees
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Properties of Production Function
I Both inputs necessary: F (0,Nt) = F (Kt , 0) = 0
I Increasing in both inputs: FK (Kt ,Nt) > 0 andFN(Kt ,Nt) > 0
I Concave in both inputs: FKK (Kt ,Nt) < 0 andFNN(Kt ,Nt) < 0
I Constant returns to scale: F (qKt , qNt) = qF (Kt ,Nt)
I Capital and labor are paid marginal products:
wt = AtFN(Kt ,Nt)
Rt = AtFK (Kt ,Nt)
I Example production function: Cobb-Douglas:
F (Kt ,Nt) = K ฮฑt N
1โฮฑt , 0 < ฮฑ < 1
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Consumption, Investment, Labor Supply
I Fruit can either be eaten (consumption) or re-planted in theground (investment), the latter of which yields another tree(capital) with a one period delay
I Assume that a constant fraction of output is invested,0 โค s โค 1. โSaving rateโ or โinvestment rateโ
I Means 1โ s of output is consumed
I Resource constraint:
Yt = Ct + It
I Abstract from endogenous labor supply โ labor suppliedinelastically and constant
I Current capital stock is exogenous โ depends on past decisions
I Capital accumulation, 0 < ฮด < 1 depreciation rate:
Kt+1 = It + (1โ ฮด)Kt
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Equations of Model
Yt = AtF (Kt ,Nt)
Yt = Ct + It
Kt+1 = It + (1โ ฮด)Kt
Ct = (1โ s)Yt
It = sYt
wt = AtFN(Kt ,Nt)
Rt = AtFK (Kt ,Nt)
I Six endogenous variables (Yt ,Ct ,Kt+1, It ,wt ,Rt) and threeexogenous variables (At , Kt , Nt)
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Central Equation
I First four equations can be combined into one:
Kt+1 = sAtF (Kt ,Nt) + (1โ ฮด)Kt
I Define lowercase variables as โper worker.โ kt =KtNt
. Inper-worker terms:
kt+1 = sAt f (kt) + (1โ ฮด)kt
I One equation describing dynamics of kt . Once you knowdynamic path of capital, you can recover everything else
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Plot of the Central Equation of the Solow Model
๐๐ก+1
๐๐ก
๐๐ก+1 = ๐๐ก
๐๐ก+1 = ๐ ๐ด๐ก๐(๐๐ก) + (1 โ ๐ฟ)๐๐ก
๐โ
๐โ
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The Steady State
I The steady state capital stock is the value of capital at whichkt+1 = kt
I We call this kโ
I Graphically, this is where the curve (the plot of kt+1 againstkt) crosses the 45 degree line (a plot of kt+1 = kt)
I Via assumptions of the production function along withauxiliary assumptions (the Inada conditions), there exists onenon-zero steady state capital stock
I The steady state is โstableโ in the sense that for any initialkt 6= 0, the capital stock will converge to this point
I โOnce you get there, you sit thereโ
I Since capital governs everything else, all other variables go toa steady state determined by kโ
I Work through dynamics
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Algebraic Example
I Suppose f (kt) = kฮฑt . Suppose At is constant at Aโ. Then:
kโ =
(sAโ
ฮด
) 11โฮฑ
y โ = Aโkโฮฑ
cโ = (1โ s)Aโkโฮฑ
iโ = sAโkโฮฑ
Rโ = ฮฑAโkโฮฑโ1
w โ = (1โ ฮฑ)Aโkโฮฑ
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Dynamic Effects of Changes in Exogenous Variables
I Want to consider the following exercises:I What happens to endogenous variables in a dynamic sense
after a permanent change in Aโ (the constant value of At)?I What happens to endogenous variables in a dynamic sense
after a permanent change in s (the saving rate)?
I For these exercises:
1. Assume we start in a steady state2. Graphically see how the steady state changes after the change
in productivity or the saving rate3. Current capital stock cannot change (it is
predetermined/exogenous). But kt 6= kโ. Use dynamicanalysis of the graph to figure out how kt reacts dynamically
4. Once you have that, you can figure out what everything else isdoing
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Permanent Increase in Aโ
๐๐๐ก๐ก+1
๐๐๐ก๐ก
๐๐๐ก๐ก+1 = ๐๐๐ก๐ก
๐๐๐ก๐ก+1 = ๐ ๐ ๐ด๐ด0,๐ก๐ก๐๐(๐๐๐ก๐ก) + (1 โ ๐ฟ๐ฟ)๐๐๐ก๐ก
๐๐๐ก๐ก = ๐๐0โ
๐๐๐ก๐ก+1
๐๐๐ก๐ก+1 = ๐ ๐ ๐ด๐ด1,๐ก๐ก๐๐(๐๐๐ก๐ก) + (1 โ ๐ฟ๐ฟ)๐๐๐ก๐ก
๐๐๐ก๐ก+2
๐๐๐ก๐ก+1 ๐๐1โ
๐๐1โ
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Impulse Response Functions: Permanent Increase in Aโ
๐๐ก ๐ฆ๐ก
๐๐ก ๐๐ก
๐ค๐ก ๐ ๐ก
๐ก๐๐๐ ๐ก๐๐๐
๐ก๐๐๐ ๐ก๐๐๐
๐ก๐๐๐ ๐ก๐๐๐
๐0โ ๐ฆ0
โ
๐0โ
๐0โ
๐ค0โ
๐ 0โ
๐1โ
๐ฆ1โ
๐1โ ๐1
โ
๐ค1โ
๐ก ๐ก
๐ก ๐ก
๐ก ๐ก
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Permanent Increase in s
๐๐๐ก๐ก+1
๐๐๐ก๐ก
๐๐๐ก๐ก+1 = ๐๐๐ก๐ก
๐๐๐ก๐ก+1 = ๐ ๐ 0๐ด๐ด๐ก๐ก๐๐(๐๐๐ก๐ก) + (1 โ ๐ฟ๐ฟ)๐๐๐ก๐ก
๐๐๐ก๐ก = ๐๐0โ
๐๐๐ก๐ก+1
๐๐๐ก๐ก+1 = ๐ ๐ 1๐ด๐ด๐ก๐ก๐๐(๐๐๐ก๐ก) + (1 โ ๐ฟ๐ฟ)๐๐๐ก๐ก
๐๐๐ก๐ก+2
๐๐๐ก๐ก+1 ๐๐1โ
๐๐1โ
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Impulse Response Functions: Permanent Increase in s
๐๐ก ๐ฆ๐ก
๐๐ก ๐๐ก
๐ค๐ก ๐ ๐ก
๐ก๐๐๐ ๐ก๐๐๐
๐ก๐๐๐ ๐ก๐๐๐
๐ก๐๐๐ ๐ก๐๐๐
๐ก ๐ก
๐ก ๐ก
๐ก ๐ก
๐0โ ๐ฆ0
โ
๐0โ ๐0
โ
๐ค0โ ๐ 0
โ
๐1โ ๐ฆ1
โ
๐1โ
๐1โ
๐ค1โ
๐ 1โ
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Discussion
I Neither changes in Aโ nor s trigger sustained increases ingrowth
I Each triggers faster growth for a while while the economyaccumulates more capital and transitions to a new steady state
I Arenโt we supposed to be studying growth? In the long run,there is no growth in this model โ it goes to a steady state!
I Weโll fix that. You can kind of see, however, that sustainedgrowth must come from increases in productivity. Why?
I No limit on how high A can get โ it can just keep increasing.Upper bound on s
I Repeated increases in s would trigger continual decline in Rt ,inconsistent with stylized facts
I Bottom line: sustained growth must be due to productivitygrowth, not factor accumulation. You canโt save your way tomore growth
I Key assumption: diminishing returns to capital
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Golden Rule
I What is the โoptimalโ saving rate, s?
I Utility from consumption, not outputI Higher s has two effects โ the โsize of the pieโ and the
โfraction of the pieโ:I More capital โ more output โ more consumption (bigger
size of the pie)I Consume a smaller fraction of output โ less consumption (eat
a smaller fraction of the pie)
I Golden Rule: value of s which maximizes steady stateconsumption, cโ
I s = 0: cโ = 0I s = 1: cโ = 0
I Implicity characterized by Aโf โฒ(kโ) = ฮด. Graphical intuition.
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Growth
I Wrote down a model to study growth
I But model converges to a steady state with no growth
I Isnโt that a silly model?
I It turns out, no
I Can modify it
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Augmented Solow ModelI Production function is:
Yt = AtF (Kt ,ZtNt)
I Zt : labor-augmenting productivityI ZtNt : efficiency units of laborI Assume Zt and Nt both grow over time (initial values in
period 0 normalized to 1):
Zt = (1 + z)t
Nt = (1 + n)t
I z = n = 0: case we just didI Zt not fundamentally different from At . Mathematically
convenient to use Zt to control growth while At controls levelof productivity
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Per Efficiency Unit Variables
I Define kฬt =Kt
ZtNtand similarly for other variables. Lower case
variables: per-capita. Lower case variables with โhatsโ: perefficiency unit variables
I Can show that modified central equation of model is:
kฬt+1 =1
(1 + z)(1 + n)
[sAt f (kฬt) + (1โ ฮด)kฬt
].
I Practically the same as before
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Plot of Modified Central Equation
๐๐๏ฟฝ๐ก๐ก+1
๐๐๏ฟฝ๐ก๐ก
๐๐๏ฟฝ๐ก๐ก+1 = ๐๐๏ฟฝ๐ก๐ก
๐๐๏ฟฝ๐ก๐ก+1 =1
(1 + ๐ง๐ง)(1 + ๐๐)๏ฟฝ๐ ๐ ๐ด๐ด๐ก๐ก๐๐๏ฟฝ๐๐๏ฟฝ๐ก๐ก๏ฟฝ + (1 โ ๐ฟ๐ฟ)๐๐๏ฟฝ๐ก๐ก๏ฟฝ
๐๐๏ฟฝโ
๐๐๏ฟฝโ
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Steady State GrowthI Via similar arguments to earlier, there exists a steady state kฬโ
at which kฬt+1 = kฬt . Economy converges to this point fromany non-zero initial value of kฬt
I Economy converges to a steady state in which per efficiencyunit variables do not grow. What about actual and per capitavariables? If kฬt+1 = kฬt , then:
Kt+1
Zt+1Nt+1=
Kt
ZtNt
Kt+1
Kt=
Zt+1Nt+1
ZtNt= (1 + z)(1 + n)
kt+1
kt=
Zt+1
Zt= 1 + z
I Level of capital stock grows at approximately sum of growthrates of Zt and Nt . Per capita capital stock grows at rate ofgrowth in Zt
I This growth is manifested in output and the real wage, butnot the return on capital
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Steady State Growth and Stylized FactsI Once in steady state, we have:
yt+1
yt= 1 + z
kt+1
kt= 1 + z
Kt+1
Yt+1=
Kt
Yt
wt+1Nt+1
Yt+1=
wtNt
Yt
Rt+1 = Rt
wt+1
wt= 1 + z
I These are the six time series stylized facts!31 / 52
Understanding Cross-Country Income Differences
I Solow model can reproduce time series stylized facts if it isassumed that productivity grows over time
I Letโs now use the model to think about cross-country incomedifferences
I What explains these differences? Three hypotheses for whycross-country income differences exist:
1. Countries initially endowed with different levels of capital2. Countries have different saving rates3. Countries have different productivity levels
I Like sustained growth, most plausible explanation forcross-country income differences is productivity
I Consider standard Solow model and two countries, 1 and 2.Suppose that 2 is poor relative to 1
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ConvergenceI Suppose two countries are otherwise identical, and hence have
the same steady state.I But suppose that country 2 is initially endowed with less
capital โ k2,t < k1,t = kโ
๐๐๐๐,๐ก๐ก+1
๐๐๐๐,๐ก๐ก
๐๐๐๐,๐ก๐ก+1 = ๐๐๐๐,๐ก๐ก
๐๐๐๐,๐ก๐ก+1 = ๐ ๐ ๐ด๐ด๐๐๏ฟฝ๐๐๐๐,๐ก๐ก๏ฟฝ + (1 โ ๐ฟ๐ฟ)๐๐๐๐,๐ก๐ก
๐๐1,๐ก๐ก = ๐๐โ
๐๐โ
๐๐2,๐ก๐ก
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Catch Up
I If country 2 is initially endowed with less capital, it shouldgrow faster than country 1, eventually catching up withcountry 1
๐ ๐ ๐ ๐ 0 0
๐๐๐๐,๐ก๐ก+๐ ๐
๐๐1,๐ก๐ก = ๐๐โ
๐๐2,๐ก๐ก ๐๐1,๐ก๐ก๐ฆ๐ฆ = 0
๐๐๐๐,๐ก๐ก+๐ ๐ ๐ฆ๐ฆ
๐๐2,๐ก๐ก๐ฆ๐ฆ
Country 1 Country 2
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Is There Convergence in the Data?
0
2
4
6
8
10
12
14
16
0 2,000 6,000 10,000 14,000
Y1950
Y201
0/Y1
950
Correlation between cumulative growthand initial GDP = -0.18
I Correlation between growth and initial GDP is weakly negativewhen focusing on all countries
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Focusing on a More Select Group of Countries
2
4
6
8
10
12
14
2,000 4,000 6,000 8,000 10,000 14,000
Y1950
Y201
0/Y19
50
Correlation between cumulative growthand initial GDP = -0.71
I Focusing only on OECD countries (more similar) story looksmore promising for convergence
I Still, catch up seems too slow for initial low levels of capital tobe the main story
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Pseudo Natural Experiment: WWII
0
0.2
0.4
0.6
0.8
1
1.2
1950 1960 1970 1980 1990 2000 2010
US
Germany
UK
Japan
I WWII losers (Germany and Japan) grew faster for 20-30 yearsthan the winners (US and UK)
I But donโt seem to be catching up all the way to the US:conditional convergence. Countries have different steadystates
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Differences in s and Aโ
I Most countries seem to have different steady states
I For simple model with Cobb-Douglas production function,relative outputs:
y โ1y โ2
=
(A1
A2
) 11โฮฑ(s1s2
) ฮฑ1โฮฑ
.
I Question: can differences in s plausibly account for largeincome differences?
I Answer: no
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Differences in s
I Suppose Aโ the same in both countries. Suppose country 1 is
US, and country 2 is Mexico:y โ1y โ2
= 4. We have:
s2 = 4ฮฑโ1
ฮฑ s1
I Based on data, a plausible value of ฮฑ = 1/3. Meansฮฑโ1
ฮฑ = โ2
I Mexican saving rate would have to be 0.0625 times US savingrate
I This would be something like a saving rate of one percent (orless)
I Not plausible
I Becomes more plausible if ฮฑ is much bigger
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What Could It Be?
I If countries have different steady states and differences in scannot plausibly account for this, must be differences inproductivity
I Seems to be backed up in data: rich countries are highlyproductive
0
20,000
40,000
60,000
80,000
100,000
120,000
140,000
160,000
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
TFP (relative to US = 1)
GDP
per
wor
ker (
2011
US
Dolla
rs)
Correlation between TFP and GDP = 0.82
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Productivity is King
I Productivity is what drives everything in the Solow model
I Sustained growth must come from productivity
I Large income differences must come from productivity
I But what is productivity? Solow model doesnโt say
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Factors Influencing Productivity
I Including but not limited to:
1. Knowledge and education2. Climate3. Geography4. Institutions5. Finance6. Degree of openness7. Infrastructure
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Policy Implications
I If a country wants to become richer, need to focus on policieswhich promote productivity
I Example: would giving computers (capital) to people insub-Saharan Africa help them get rich? Not without theinfrastructure to connect to the internet, the knowledge ofhow to use the computer, and the institutions to protectproperty rights
I Also has implications when thinking about poverty within acountry
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Output Per Worker over Time
1950 1960 1970 1980 1990 2000 2010 2020
Year
10.2
10.4
10.6
10.8
11
11.2
11.4
11.6
log(
RG
DP
per
Wor
ker)
Actual SeriesTrend Series
go back
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Capital Per Worker over Time
1950 1960 1970 1980 1990 2000 2010 2020
Year
11.4
11.6
11.8
12
12.2
12.4
12.6
log(
Cap
ital p
er W
orke
r)Actual SeriesTrend Series
go back
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Capital to Output Ratio over Time
1950 1960 1970 1980 1990 2000 2010 2020
Year
2.9
3
3.1
3.2
3.3
3.4
3.5
K/Y
go back
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Labor Share over Time
1950 1960 1970 1980 1990 2000 2010 2020
Year
0.62
0.63
0.64
0.65
0.66
0.67
0.68
Labo
rs S
hare
go back
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Return on Capital over Time
1950 1960 1970 1980 1990 2000 2010 2020
Year
0.095
0.1
0.105
0.11
0.115
0.12
0.125R
etur
n on
Cap
ital
go back
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Real Wage over Time
1950 1960 1970 1980 1990 2000 2010 2020
Year
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
log(
Wag
es)
Actual SeriesTrend Series
go back
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Income Differences
GDP per Person
High income countriesCanada $35,180Germany $34,383Japan $30,232
Singapore $59,149United Kingdom $32,116United States $42,426
Middle income countriesChina $8,640
Dominican Republic $8,694Mexico $12,648
South Africa $10,831Thailand $9,567Uruguay $13,388
Low income countriesCambodia $2,607
Chad $2,350India $3,719Kenya $1636Mali $1,157Nepal $1,281
go back
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Growth Miracles and Disasters
Growth Miracles1970 Income 2011 Income % change
South Korea $1918 $27,870 1353Taiwan $4,484 $33,187 640China $1,107 $8,851 700
Botswana $721 $14,787 1951Growth Disasters
Madagascar $1,321 $937 -29Niger $1,304 $651 -50
Burundi $712 $612 -14Central African Republic $1,148 $762 -34
go back
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