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)

2 / 52

Economic Growth

I When economists say “growth,” 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 “Once one begins to think about growth, it is difficult to thinkabout anything else” – 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 – 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. Labor’s 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

8 / 52

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 αt N

1−αt , 0 < α < 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 ≤ s ≤ 1. “Saving rate” or “investment rate”

I Means 1− s of output is consumed

I Resource constraint:

Yt = Ct + It

I Abstract from endogenous labor supply – labor suppliedinelastically and constant

I Current capital stock is exogenous – depends on past decisions

I Capital accumulation, 0 < δ < 1 depreciation rate:

Kt+1 = It + (1− δ)Kt

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Equations of Model

Yt = AtF (Kt ,Nt)

Yt = Ct + It

Kt+1 = It + (1− δ)Kt

Ct = (1− 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) + (1− δ)Kt

I Define lowercase variables as “per worker.” kt =KtNt

. Inper-worker terms:

kt+1 = sAt f (kt) + (1− δ)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

𝑘𝑡+1

𝑘𝑡

𝑘𝑡+1 = 𝑘𝑡

𝑘𝑡+1 = 𝑠𝐴𝑡𝑓(𝑘𝑡) + (1 − 𝛿)𝑘𝑡

𝑘∗

𝑘∗

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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 k∗

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 “stable” in the sense that for any initialkt 6= 0, the capital stock will converge to this point

I “Once you get there, you sit there”

I Since capital governs everything else, all other variables go toa steady state determined by k∗

I Work through dynamics

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Algebraic Example

I Suppose f (kt) = kαt . Suppose At is constant at A∗. Then:

k∗ =

(sA∗

δ

) 11−α

y ∗ = A∗k∗α

c∗ = (1− s)A∗k∗α

i∗ = sA∗k∗α

R∗ = αA∗k∗α−1

w ∗ = (1− α)A∗k∗α

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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 A∗ (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= k∗. 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 A∗

𝑘𝑘𝑡𝑡+1

𝑘𝑘𝑡𝑡

𝑘𝑘𝑡𝑡+1 = 𝑘𝑘𝑡𝑡

𝑘𝑘𝑡𝑡+1 = 𝑠𝑠𝐴𝐴0,𝑡𝑡𝑓𝑓(𝑘𝑘𝑡𝑡) + (1 − 𝛿𝛿)𝑘𝑘𝑡𝑡

𝑘𝑘𝑡𝑡 = 𝑘𝑘0∗

𝑘𝑘𝑡𝑡+1

𝑘𝑘𝑡𝑡+1 = 𝑠𝑠𝐴𝐴1,𝑡𝑡𝑓𝑓(𝑘𝑘𝑡𝑡) + (1 − 𝛿𝛿)𝑘𝑘𝑡𝑡

𝑘𝑘𝑡𝑡+2

𝑘𝑘𝑡𝑡+1 𝑘𝑘1∗

𝑘𝑘1∗

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Impulse Response Functions: Permanent Increase in A∗

𝑘𝑡 𝑦𝑡

𝑐𝑡 𝑖𝑡

𝑤𝑡 𝑅𝑡

𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒

𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒

𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒

𝑘0∗ 𝑦0

∗

𝑐0∗

𝑖0∗

𝑤0∗

𝑅0∗

𝑘1∗

𝑦1∗

𝑐1∗ 𝑖1

∗

𝑤1∗

𝑡 𝑡

𝑡 𝑡

𝑡 𝑡

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Permanent Increase in s

𝑘𝑘𝑡𝑡+1

𝑘𝑘𝑡𝑡

𝑘𝑘𝑡𝑡+1 = 𝑘𝑘𝑡𝑡

𝑘𝑘𝑡𝑡+1 = 𝑠𝑠0𝐴𝐴𝑡𝑡𝑓𝑓(𝑘𝑘𝑡𝑡) + (1 − 𝛿𝛿)𝑘𝑘𝑡𝑡

𝑘𝑘𝑡𝑡 = 𝑘𝑘0∗

𝑘𝑘𝑡𝑡+1

𝑘𝑘𝑡𝑡+1 = 𝑠𝑠1𝐴𝐴𝑡𝑡𝑓𝑓(𝑘𝑘𝑡𝑡) + (1 − 𝛿𝛿)𝑘𝑘𝑡𝑡

𝑘𝑘𝑡𝑡+2

𝑘𝑘𝑡𝑡+1 𝑘𝑘1∗

𝑘𝑘1∗

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Impulse Response Functions: Permanent Increase in s

𝑘𝑡 𝑦𝑡

𝑐𝑡 𝑖𝑡

𝑤𝑡 𝑅𝑡

𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒

𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒

𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒

𝑡 𝑡

𝑡 𝑡

𝑡 𝑡

𝑘0∗ 𝑦0

∗

𝑐0∗ 𝑖0

∗

𝑤0∗ 𝑅0

∗

𝑘1∗ 𝑦1

∗

𝑐1∗

𝑖1∗

𝑤1∗

𝑅1∗

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Discussion

I Neither changes in A∗ 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 Aren’t we supposed to be studying growth? In the long run,there is no growth in this model – it goes to a steady state!

I We’ll 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 – 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 can’t save your way tomore growth

I Key assumption: diminishing returns to capital

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Golden Rule

I What is the “optimal” saving rate, s?

I Utility from consumption, not outputI Higher s has two effects – the “size of the pie” and the

“fraction of the pie”:I More capital → more output → more consumption (bigger

size of the pie)I Consume a smaller fraction of output → less consumption (eat

a smaller fraction of the pie)

I Golden Rule: value of s which maximizes steady stateconsumption, c∗

I s = 0: c∗ = 0I s = 1: c∗ = 0

I Implicity characterized by A∗f ′(k∗) = δ. 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 Isn’t that a silly model?

I It turns out, no

I Can modify it

26 / 52

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

27 / 52

Per Efficiency Unit Variables

I Define k̂t =Kt

ZtNtand similarly for other variables. Lower case

variables: per-capita. Lower case variables with “hats”: perefficiency unit variables

I Can show that modified central equation of model is:

k̂t+1 =1

(1 + z)(1 + n)

[sAt f (k̂t) + (1− δ)k̂t

].

I Practically the same as before

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Plot of Modified Central Equation

𝑘𝑘�𝑡𝑡+1

𝑘𝑘�𝑡𝑡

𝑘𝑘�𝑡𝑡+1 = 𝑘𝑘�𝑡𝑡

𝑘𝑘�𝑡𝑡+1 =1

(1 + 𝑧𝑧)(1 + 𝑛𝑛)�𝑠𝑠𝐴𝐴𝑡𝑡𝑓𝑓�𝑘𝑘�𝑡𝑡� + (1 − 𝛿𝛿)𝑘𝑘�𝑡𝑡�

𝑘𝑘�∗

𝑘𝑘�∗

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Steady State GrowthI Via similar arguments to earlier, there exists a steady state k̂∗

at which k̂t+1 = k̂t . Economy converges to this point fromany non-zero initial value of k̂t

I Economy converges to a steady state in which per efficiencyunit variables do not grow. What about actual and per capitavariables? If k̂t+1 = k̂t , 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

30 / 52

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 Let’s 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 – k2,t < k1,t = k∗

𝑘𝑘𝑗𝑗,𝑡𝑡+1

𝑘𝑘𝑗𝑗,𝑡𝑡

𝑘𝑘𝑗𝑗,𝑡𝑡+1 = 𝑘𝑘𝑗𝑗,𝑡𝑡

𝑘𝑘𝑗𝑗,𝑡𝑡+1 = 𝑠𝑠𝐴𝐴𝑓𝑓�𝑘𝑘𝑗𝑗,𝑡𝑡� + (1 − 𝛿𝛿)𝑘𝑘𝑗𝑗,𝑡𝑡

𝑘𝑘1,𝑡𝑡 = 𝑘𝑘∗

𝑘𝑘∗

𝑘𝑘2,𝑡𝑡

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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

𝑠𝑠 𝑠𝑠 0 0

𝑘𝑘𝑗𝑗,𝑡𝑡+𝑠𝑠

𝑘𝑘1,𝑡𝑡 = 𝑘𝑘∗

𝑘𝑘2,𝑡𝑡 𝑔𝑔1,𝑡𝑡𝑦𝑦 = 0

𝑔𝑔𝑗𝑗,𝑡𝑡+𝑠𝑠𝑦𝑦

𝑔𝑔2,𝑡𝑡𝑦𝑦

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 don’t 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 A∗

I Most countries seem to have different steady states

I For simple model with Cobb-Douglas production function,relative outputs:

y ∗1y ∗2

=

(A1

A2

) 11−α(s1s2

) α1−α

.

I Question: can differences in s plausibly account for largeincome differences?

I Answer: no

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Differences in s

I Suppose A∗ the same in both countries. Suppose country 1 is

US, and country 2 is Mexico:y ∗1y ∗2

= 4. We have:

s2 = 4α−1

α s1

I Based on data, a plausible value of α = 1/3. Meansα−1

α = −2

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 α 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 doesn’t 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

44 / 52

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

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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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Education and Income Per Capita

67

89

1011

log(

RG

DP

per P

erso

n)

1 1.5 2 2.5 3 3.5Index of Human Capital

go back

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