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Actual and potential output
8,000
9,000
10,000
11,000
12,000
13,000
14,000
15,000
1996 1998 2000 2002 2004 2006 2008 2010
Actual GDPPotential GDP
Rea
l GD
P (bi
llion
s of
200
5 $)
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Agenda• Introductory background • Essential aspects of economic growth • Aggregate production functions • Neoclassical growth model• Simulation of increased saving experiment • Then in the last week: deficits, debt, and
economic growth
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Great divide of macroeconomics
Aggregate demandand business cycles
Aggregate supplyand “economic growth”
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How do they fit together?
Examples of why Keynesian Classical
In long run, prices and wages are flexible.In long run, expectations are accurate.In long run, entry and exit make economy more
competitive.
All these make long-run look more classical than short run.
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π
Y
AD
Yt* = potential output
AS short run
AS-AD in short, medium, long run
πt
AS medium run
AS long run
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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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The MIT School of Economics
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Paul Samuelson (1915-2009)
Robert Solow (1924 - )
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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 = constant
3. 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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Mathematical noteWe will use the following math fact:
Define z = y/xThen(1) (Growth rate of z) = (growth rate of y) – (growth rate of x)Or gz = gy - gx
Proof:Using logs:ln(zt) = ln(yt – ln(xt)
Taking time derivative:[dzt/dt]/zt = [dyt/dt]/yt - [dxt/dt]/xt
which is the desired result.
Note that we sometimes use the discrete version of (1), as we did in the last slide. This has a small error that is in the order of the size of the time step or the growth rates. For example, if gy = 5 % and gx = 3 %, then by the formula gz = 2 %, while the exact calculation is that gz = 1.9417 %. This is close enough for most purposes. 12
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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)
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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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Simplest model misses major trends of growth in y, w, and k.
Missing element is technological change
Problem with neoclassical model without TC
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The greatest technological change in history
Thomas arithmometer, 1870 China’s Tianhe-1A
(10-16 petaflops) (2.5 petaflops)
[petaflop = 1015 floating point operations per second]
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1. Growth involves potential output (not business cycles)
2. Growth theories deal with dynamic version of full-employment, “classical”-type economy
3. Economic growth involves:- increase in quantity (bushels of wheat)- improved quality (safer cars)- new goods and services (computer replaces typewriter)
Further thoughts on growth
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Mining in rich and poor countries
Canada
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D.R. Congo
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Farming the rocks, Morocco, 2001
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Medicine in rich and poor countries
Scan for lung cancer
23African medicine man
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Economic growth and improved
health status:
Eradication of polio
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Disappearance of polio:A benefit of growth that is not captured
in the GDP statistics!
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Introducing technological change
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)
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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
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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
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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: • Higher savings is contractionary • Mechanism: lower S, lower AD, lower Y, inflation
(straight Keynesian effect in dynamic AS-AD analysis that we did last week)
2. In long-run, neoclassical growth model• Higher savings leads to higher potential output • Mechanism: higher I, K, Y, w, etc. (through
neoclassical growth model)
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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