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Economics 122

Macroeconomics in the long run:Economic growth

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Agenda

• Introductory background • Essential aspects of economic growth • Aggregate production functions • Neoclassical growth model• Simulation of increased saving experiment • Debt, deficits, and growth

Great divide of macroeconomics

Aggregate demandand business cycles

Aggregate supplyand “economic growth”

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Review of aggregate production functionYt = At F(Kt, Lt)

Kt = capital services (like rentals as apartment-years)

Lt = labor services (hours worked)

At = level of technology

gx = growth rate of x = (1/xt) dxt/dt = Δ xt/xt-1 = d[ln(xt))]/dt

gA = growth of technology = rate of technological change = Δ At/At-1

Constant returns to scale: λYt = At F(λKt, λLt), or all inputs increased by λ means output increased by λ

Perfect competition in factor and product markets (for p = 1):MPK = ∂Y/∂K = R = rental price of capital; ∂Y/∂L = w = wage rate

Exhaustion of product with CRTS:MPK x K + MPL x L = RK + wL = Y

Alternative measures of productivity:Labor productivity = Yt/Lt

Total factor productivity (TFP) = At = Yt /F(Kt, Lt)

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Review: Cobb-Douglas aggregate production function

Remember Cobb-Douglas production function:

Yt = At Kt α Lt

1-α

or ln(Yt)= ln(At) + α ln(Kt) +(1-α) ln(Lt)

Here α = ∂ln(Yt)/∂ln(Kt) = elasticity of output w.r.t. capital; (1-α ) = elasticity of output w.r.t. labor

MPK = Rt = α At Kt α-1 Lt

1-α = α Yt/Kt

Share of capital in national income = Rt Kt /Yt = α = constant. Ditto for share of labor.

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Note 1. Cobb-Douglas p. f. implies that factor shares in national income are constant.

Note 2. Cobb-Douglas p. f. implies that growth in wages = growth in labor productivity (gw = gY/L)

The MIT School of Economics

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Paul Samuelson

Robert Solow

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Basic neoclassical growth modelMajor assumptions:1. Basic setup:

- full employment- flexible wages and prices- perfect competition- closed economy

2. Capital accumulation: ΔK/K = sY – δK; s = investment rate = constant3. Labor supply: Δ L/L = n = exogenous 4. Production function

- constant returns to scale- two factors (K, L)- single output used for both C and I: Y = C + I- no technological change to begin with- in next model, labor-augmenting technological change

5. Change of variable to transform to one-equation model: k = K/L = capital-labor ratio

Y = F(K, L) = LF(K/L,1) y = Y/L = F(K/L,1) = f(k), where f(k) is per capita production fn.

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Major variables:Y = output (GDP)L = labor inputsK = capital stock or servicesI = gross investmentw = real wage rater = real rate of return on capital (rate of profit)E = efficiency units = level of labor-augmenting technology (growth of E is

technological change = ΔE/E) L~= efficiency labor inputs = EL = similarly for other variables with

“~”notation)

Further notational conventionsΔ x = dx/dtgx = growth rate of x = (1/x) dx/dt = Δxt/xt-1=dln(xt)/dt s = I/Y = savings and investment ratek = capital-labor ratio = K/Lc = consumption per capita = C/Li = investment per worker = I/Lδ = depreciation rate on capitaly = output per worker = Y/Ln = rate of growth of population (or labor force)

= gL = Δ L/Lv = capital-output ratio = K/Yh = rate of labor-augmenting technological change

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We want to derive “laws of motion” of the economy. To do this, start with:

5. Δ k/k = Δ K/K - Δ L/LWith some algebra, this becomes:5’. Δ k/k = Δ K/K - n Y Δ k = sf(k) - (n + δ) k which in steady state is:6. sf(k*) = (n + δ) k* In steady state, y, k, w, and r are constant. No growth in real

wages, real incomes, per capita output, etc.

ΔK/K = (sY – δK)/K = s(Y/L)(L/K) – δΔk/k = ΔK/K – n = s(Y/L)(L/K) – δ – nΔk = k [ s(Y/L)(L/K) – δ – n] = sy – (δ + n)k = sf(k) – (δ + n)k

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k

y = Y/L

y = f(k)

(n+δ)k

y*

i* = (I/Y)*

k*

i = sf(k)

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Predictions of basic model:– “Steady state”– constant y, w, k, and r

– gY = n

Uniqueness and stability of equilibrium. – Equilibrium is unique– Equilibrium is stable

(meaning k → k* as t → ∞ for all initial

k0).

Results of neoclassical model without TC

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k

y = Y/ L

y = f(k)

(n+δ)k

y*

i* = (I/ Y)*

k*

i = sf(k)

Now let’s step back a moment to consider the meaning of economic growth in the broader context.

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1. Growth involves potential output- Potential GDP = output if unemployment rate at

NAIRU- Distinguish from business cycles, which is utilization

of potential- Implicit assumption about business cycle: policy

keeps economy near full employment

2. Most growth theories deal with dynamic version of full-employment, “classical”-type economy- This is closest thing to “consensus macroeconomics”

General concepts

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Actual and potential output

14,000

13,000

12,000

11,000

10,000

9,000

8,000

7,00090 92 94 96 98 00 02 04 06 08 10

Actual GDPPotential GDP (CBO)

Bill

ion

s of

20

05 $

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3. “Productivity isn't everything, but in the long run it is almost everything.”

- As we will see, in the long run, real wages and per capita income grow (almost) proportionally with labor productivity (= GDP per hour worked).

General concepts

Growth in per capita income (average, percent per year)

0.0

0.5

1.0

1.5

Gro

wth

in p

er c

apita

inco

me

up to 1700

1700-1830

1830 to present

Data for Western Europe from Angus Maddison

U.S. Productivity Growth in the 20th Century

0

0.5

1

1.5

2

2.5

3

3.5

1899-2005 1899-19481948-19731973-19951995-2008

Gro

wth

in

ou

tpu

t p

er

hou

r (%

per

year)

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4. Economic growth involves:- increase in quantity (bushels of wheat)- improved quality of goods and services- new goods and services

General concepts

Mining in rich and poor countries

Canada

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D.R. Congo

Farming the rocks, Morocco, 2001

Medicine in rich and poor countries

Scan for lung cancer

22African medicine man

Economic growth and improved

health status:

Eradication of polio

Health: Disappearance of polio:A benefit of growth that is not captured

in the GDP statistics!

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Historical Trends in Economic Growth in the US since 1800

1. Strong growth in Y

2. Strong growth in productivity (both Y/L and TFP)

3. Steady “capital deepening” (increase in K/L)

4. Strong growth in real wages since early 19th C; g(w/p) ~

g(X/L)

5. Real interest rate and profit rate basically trendless

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Share of compensation in national income

Labor share of national income in US

- Slow increase over most of century- Tiny decline in recent years as profits rose- Big rise in fringe benefit share (and decline in wage share)

.45

.50

.55

.60

.65

.70

1930 1940 1950 1960 1970 1980 1990 2000

Compensation/ National incomeWages & salaries/ National income

Fringe benefits: (health, retirement, social insurance)

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Growth trend, US, 1948-2008

0.0

0.4

0.8

1.2

1.6

2.0

50 55 60 65 70 75 80 85 90 95 00 05

ln(K)ln(Y)ln(hours)

gX = xxx/60;

So:

gY = 3.3%

gK = 2.9%

gH=1.5%

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Predictions of basic model without TC:– “Steady state” property misses major trends of growth

in y, w, and k.– Missing element is technological change

Results of neoclassical model without TC

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Introducing technological change

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Introducing technological change

First model omits technological change (TC). Let’s see if we can fix up the problems by introducing TC.What is TC?

- New processes that increase TFP (assembly line, fiber optics)

- Improvements in quality of goods (plasma TV)- New goods and services (automobile, telephone,

iPod)Analytically, TC is

- Shift in production function.

k

y = Y/ L

new f(k)

k*

old f(k)

The greatest technological change in history

Thomas arithmometer, 1870 IBM/LANL Roadrunner

(10-16 petaflops) (1.10 petaflops)

[petaflop = 1015 floating point operations per second]

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Introducing technological change

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Introducing technological change

We take specific form which is “labor-augmenting technological change” at rate h.

For this, we need new variable called “efficiency labor units” denoted as E

where E = efficiency units of labor and ~ indicates efficiency units.

New production function is then

Note: Redefining labor units in efficiency terms is a specific way of representing TC that makes everything work out easily. Other forms will give slightly different results.

k

y = Y/ L

new f(k)

k*

old f(k)

, ( , )

  , / ,1

( ),   /

/ ( )

Y F K EL F K L

F K L L F K L

L f k wherek K L

y Y L f k

L = E L

T.C. for the Cobb-Douglas

In C-D case, labor-augmenting TC is very simple:

So for C-D, labor augmenting is “output-augmenting” with a scalar adjustment of the labor elasticity.

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10

0

10

(1 ) 10 0

A

A ( )

A

t t t

htt t t t

t t t t

h tt t t

Y K L

e

Y K

Y E e K

L = E L = E L

E L

L

3333

i*=I*/Y*

Impact with labor-augmenting TCy Y/L

y* y ( )f k

( )n k

i ( )sf k

k* k=K/ L

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For C-D case,

Unique and stable equilibrium under standard assumptions:Predictions of basic model:

– Steady state: constant – Here output per capita, capital per capita, and wage rate

grow at h.– Labor’s share of output is constant.– Hence, capture the basic trends!

Results of neoclassical model with labor-augmenting TC

*0 0

0 0

*

1/ (1 )*

/ (1 )

( *) ( ) *

( *) A

Set A 1 for simplicity,

( ) *

/ ( )

* ( *) / ( )

sf k n h k

f k E k

E

sk n h k

k s n h

y f k s n h

y, w, k, and r

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ln (Y), etc.

time

ln (L)

Time profiles of major variables with TC

ln (K)

ln (Y)

ln( )L

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Sources of TC

Technological change is in some deep sense “endogenous.”But very difficult to modelSubject of “new growth theory”:

– explains TC as return to investment in research and human capital.

– Major difference from conventional investment is “public goods” nature of knowledge

– I.e., social return to research >> private return because of spillovers (externalities)

Major policy questions:- research and development policy- intellectual property rights (such as patents for drugs)- big $ and big economic stakes involved

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20,000

2,000

200

20

275 00 25 50 75 00 25 50 75 00 25 50

Now let’s move on to applications of economic growth theory

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Several “comparative dynamics” experiments

• Change growth in labor force (immigration or retirement policy)

• Change in rate of TC

• Change in national savings and investment rate (tax changes, savings changes, demographic changes)

Here we will investigate only a change in the national savings rate.

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Two faces of saving

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Government debt and deficits and the economy:

What is the effect of deficit reduction on the economy?

1. A. In short run, with constant real interest rate: • contractionary (straight Keynesian effect in IS-LM

analysis)B. In short run, with full monetary response: (neoclassical synthesis, Samuelson-Tobin policy)• first have fiscal tightening• then have monetary response with output/inflation

targets that offsets fiscal contraction• no impact on U or short-run Q• have higher public and national savings rate

3. In long-run, neoclassical growth model- higher s rate leads to a higher trajectory for K, Y,

w, etc.- numerical example with Cobb-Douglas

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k

y = f(k)

(n+δ)k

y*

(I/Y)*

k*

Impact of Higher National Saving

k**

y**

i = s2f(k)

i = s1f(k)

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Numerical Example of 1993 Budget Act in Neoclassical Growth Model

Assumptions:1. Production is by Cobb-Douglas with CRTS2. Labor plus labor-augmenting TC:

1. n = 1.5 % p.a.; h = 1.5 % p.a.

3. Full employment; constant labor force participation rate.

4. Savings assumption:a. Private savings rate = 18 % of GDPb. Initial govt. savings rate = minus 2 % of GDPc. In 1992, govt. changes fiscal policy and runs a

surplus of 2 % of GDPd. All of higher govt. S goes into national S (i.e.,

constant private savings rate)

5. “Calibrate” to U.S. economy

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Impact of Increased Government Saving on Major Variables

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

35%

1990 1995 2000 2005 2010

Pe

rce

nt

ch

an

ge

fro

m b

as

eli

ne

Consumption per capitaGDP per capitaCapital per capitaNNP per capita

- Note that takes 10 years to increase C-Political implications- Must C increase?- No if k>kgoldenrule

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Results on Growth Rates:

- Modest impact on growth in short run- Consumption down then up- No impact on growth in long run- GDP v NNP (remember that GDP excludes earnings on

foreign assets)

Growth rates of Potential

NNP per GDP per ConsumptionNNP capita capita per capita

1982-1992 3.02% 1.50% 1.50% 1.50%

1992-2002 3.28% 1.75% 1.97% 1.47%

2002-2012 3.11% 1.59% 1.69% 1.69%

2012-2052 3.03% 1.51% 1.53% 1.53%

2052-2092 3.02% 1.50% 1.50% 1.50%

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What if savings in an open economy?

• For small open economy– What happens if savings rate increases?– In this case the marginal investment is abroad!– Therefore, same result, but impact is upon net

foreign assets, investment earnings, and not on domestic capital stock and domestic income.

– No diminishing returns to investment (fixed r)– Will show up in NNP not in GDP! (Most macro models get this wrong.)

• Large open economy like US:– Somewhere in between small open and closed.– I.e., some increase in domestic I and some in

increase net foreign assets

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Open economy with mobile financial capital ( r = world r = rw)NX = S - I

I(r)

r = rw

S r = realinterest rate

I, S, NX

NX deficit

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With higher saving in small open economy

I(r)

r = rw

S1

r = realinterest rate

I, S, NX0

Final NX surplus

Original NX deficit

S0

Higher saving:1. No change I2. No change GDP3. Higher foreign saving4. Increase GNP, NNP

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Conclusions on Fiscal Policy and Economic Growth

• Fiscal policy affects economic growth through impact of government surplus through national savings rate

• Increases potential output through:– higher capital stock for domestic investment– higher income on foreign assets for foreign

investment

• Consumption decreases at first then catches up after a decade or so

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Growth Accounting

Growth accounting is a widely used technique used to separate out the sources of growth in a country relies on the neoclassical growth model

DerivationStart with production function and competitive assumptions. For simplicity, assume a Cobb-Douglas production function with labor-augmenting technological change:

(1) Yt = At Kt α Lt

1-α

Take logarithms:(2) ln(Yt) = ln(At )+ α ln(Kt) + (1 - α) ln(Lt )

Now take the time derivative. Note that ∂ln(x)/∂x=1/x and use chain rule:(3) ∂ln(Yt)/∂t= g[Yt] = g[At] )+ α g[Kt] + (1 - α) g[Lt ]

In the C-D production function (see above), ) α is the competitive share of K sh(K); and (1 - α) the competitive share of labor sh(L).

(4) g[Yt] = g[At] + sh(K) g[Kt] + (1 – sh(L)) g[Lt ]

From this, we estimate the rate of TC as:(5) g[At] = g[Yt] –{sh(K) g[Kt] + (1 – sh(L)) g[Lt ]}

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(1) Yt = At Kt α Lt

1-α

Take logarithms:(2) ln(Yt) = ln(At )+ α ln(Kt) + (1 - α) ln(Lt )

Now take the time derivative. Note that ∂ln(x)/∂x=1/x and use chain rule:

(3) ∂ln(Yt)/∂t= g[Yt] = g[At] )+ α g[Kt] + (1 - α) g[Lt ]

In the C-D production function (see above), ) α is the competitive share of K sh(K); and (1 - α) the competitive share of labor sh(L).

(4) g[Yt] = g[At] + sh(K) g[Kt] + (1 – sh(L)) g[Lt ]

while growth in per capita output is:(5) g[Yt/Lt] = g[At] + sh(K) (g[Kt] - g[Lt ])

From this, we estimate the rate of TC as:(5) g[At] = g[Yt] –{sh(K) g[Kt] + (1 – sh(L)) g[Lt ]}

Note that this is a very practical formula. All terms except h are

observable. Can be used to understand the sources of growth in different times and places.

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Some applications

1. Clinton’s growth policy (see above)2. U.S. growth since 19483. China in central planning and reform period4. Soviet Union growth, 1929 - 1965

The very rapid (measured) growth in the Soviet economy came primarily from growth in inputs, not from TFP growth.

5. Japanese growth, 1950-75 Japan had very large TFP growth after WWII. Wide variety of sources, including adoption of foreign

6. Supply-side economics (Reagan 1981-89; Bush II 2001-2009)

- To follow

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Business sector of US Period

Growth in: 1948-73 1973-95 1995-2002

Output 4.01 3.08 3.74

Output per hour 3.30 1.50 2.96

Total factor productivity 2.10 0.55 1.21

Source: BLS,

http:/ / www.bls.gov/ news.release/ prod3.t01.htm

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Source: Source: G. Chow, Accounting for Growth in Taiwan and Mainland China. Assumes Cobb-Douglas aggregate production function with elasticity of K = 0.4.

GDP: China Period:

Growth in: 1952-78 1978-98

Output 5.82 9.27

Capital 7.13 9.02

Labor 2.54 2.78

Combined inputs 4.38 5.27

Total factor productivity 1.44 3.99

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TFP, Soviet Union

Source: William Easterly and Stanley Fischer,”The Soviet economic decline : historical and republican data,” World Bank Research Working Paper no. 1284, 1994.

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Promoting Technological Change

Much more difficult conceptually and for policy:- TC depends upon invention and innovation- Market failure: big gap between social MP and private

MP of inventive activity- No formula for discovery analogous to increased saving

Major instruments:- Intellectual property rights (create monopoly to reduce

MP gap): patents, copyrights- Government subsidy of research (direct to Yale; indirect

through R&D tax credit)- Rivalry but not perfect competition in markets (between

Windows and Farmer Jones)- For open economy, openness to foreign technologies

and management