Transcript Review of aggregate production function

Economics 122
Macroeconomics in the long run:
Economic growth
1
Agenda
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Introductory background
Essential aspects of economic growth
Aggregate production functions
Neoclassical growth model
Simulation of increased saving experiment
Debt, deficits, and growth
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Great divide of macroeconomics
Aggregate demand
and business cycles
Aggregate supply
and “economic growth”
Review of aggregate production function
Yt = 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:
or
Yt = At Kt α Lt 1-α
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)
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The MIT School of Economics
Robert Solow
Paul Samuelson
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Basic neoclassical growth model
Major 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 inputs
K = capital stock or services
I = gross investment
w = real wage rate
r = 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 conventions
Δ x = dx/dt
gx = growth rate of x = (1/x) dx/dt = Δxt/xt-1=dln(xt)/dt
s = I/Y = savings and investment rate
k = capital-labor ratio = K/L
c = consumption per capita = C/L
i = investment per worker = I/L
δ = depreciation rate on capital
y = output per worker = Y/L
n = rate of growth of population (or labor force)
= gL = Δ L/L
v = capital-output ratio = K/Y
h = 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/L
With 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) – δ
Δk/k = ΔK/K – n = s(Y/L)(L/K) – δ – n
Δk = k [ s(Y/L)(L/K) – δ – n]
= sy – (δ + n)k = sf(k) – (δ + n)k
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y = Y/L
y*
y = f(k)
(n+δ)k
i = sf(k)
i* =
(I/Y)*
k*
k
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Results of neoclassical model without TC
y = Y/L
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).
y*
y = f(k)
(n+δ)k
i = sf(k)
i* =
(I/Y)*
k*
k
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Now let’s step back a moment to consider the meaning of
economic growth in the broader context.
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General concepts
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”
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Actual and potential output
14,000
13,000
Billions of 2005 $
12,000
11,000
10,000
9,000
8,000
7,000
Actual GDP
Potential GDP (CBO)
90
92
94
96
98
00
02
04
06
08
10
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General concepts
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).
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Growth in per capita income
Growth in per capita income
(average, percent per year)
1.5
1.0
up to 1700
1700-1830
0.5
0.0
Data for Western Europe from Angus Maddison
1830 to present
U.S. Productivity Growth in the 20th Century
Growth in output per hour (% per year)
3.5
3
2.5
2
1.5
1
0.5
0
1899-2005
1899-1948 1948-1973 1973-1995 1995-2008
General concepts
4. Economic growth involves:
- increase in quantity (bushels of wheat)
- improved quality of goods and services
- new goods and services
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Mining in rich and poor countries
D.R. Congo
Canada
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Farming the rocks, Morocco, 2001
Medicine in rich and poor countries
Scan for lung
cancer
African medicine man
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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!
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
.70
Fringe benefits:
(health, retirement,
social insurance)
Labor share of
national income in US
.65
- 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)
.60
.55
.50
Compensation/National income
Wages & salaries/National income
.45
1930
1940
1950
1960
1970
1980
1990
2000
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Growth trend, US, 1948-2008
2.0
gX = xxx/60;
ln(K)
ln(Y)
ln(hours)
1.6
So:
gY = 3.3%
1.2
gK = 2.9%
gH=1.5%
0.8
0.4
0.0
50
55
60
65
70
75
80
85
90
95
00
05
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Results of neoclassical model without TC
Predictions of basic model without TC:
– “Steady state” property misses major trends of growth in y, w,
and k.
– Missing element is 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.
y = Y/L
new f(k)
old f(k)
k*
k
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The greatest technological change in history
Thomas arithmometer, 1870
(10-16 petaflops)
IBM/LANL Roadrunner
(1.10 petaflops)
[petaflop = 1015 floating point operations per second]
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.
L=EL
New production function is then
y = Y/L
new f(k)
Y  F  K , EL   F(K , L )
old f(k)
 F  K , L   L F  K/L , 1 
 L f ( k ), where k  K / L
y
k*
k
 Y / L  f (k)
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.
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T.C. for the Cobb-Douglas
In C-D case, labor-augmenting TC is very simple:
Y t  A 0 Kt L t1 
L t = E t L t= E 0 e ht L t
Y t  A 0 Kt (E t Lt )1 
Y t  A 0 E 0 e h(1  )t K t Lt 1 
So for C-D, labor augmenting is “output-augmenting” with
a scalar adjustment of the labor elasticity.
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y  Y/L
Impact with labor-augmenting TC
y  f (k)
y*
(n   )k
i*=I*/Y*
i  sf ( k )
k*
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k=K/L
Results of neoclassical model with labor-augmenting TC
For C-D case,
sf ( k *)  (n  h   ) k *
f ( k *)  A 0 E 0 k * 
Set A 0 E0  1 for simplicity,
sk *  (n  h   )k *
k *   s / (n  h   ) 1/(1  )
y *  f (k *)   s / (n  h   )  /(1  )
Unique and stable equilibrium under standard assumptions:
Predictions of basic model:
– Steady state: constant y, w, k, and r
– 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!
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ln (Y), etc.
Time profiles of major variables with TC
ln (Y)
ln (K)
ln(L)
ln (L)
time
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Sources of TC
Technological change is in some deep sense “endogenous.”
But very difficult to model
Subject 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
Now let’s move on to applications of
economic growth theory
200
20
2
75
00
25
50
75
00
25
50
75
00
25
50
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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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y**
Impact of Higher National Saving
y = f(k)
y*
i = s2f(k)
i = s1f(k)
(I/Y)*
(n+δ)k
k
k*
k**
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Numerical Example of 1993 Budget Act in
Neoclassical Growth Model
Assumptions:
1. Production is by Cobb-Douglas with CRTS
2. 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 GDP
b. Initial govt. savings rate = minus 2 % of GDP
c. In 1992, govt. changes fiscal policy and runs a surplus of 2 %
of GDP
d. 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
1990
1995
2000
2005
2010
35%
Percent change from baseline
30%
25%
20%
15%
Consumption per capita
GDP per capita
Capital per capita
NNP per capita
- Note that takes 10
years to increase C
-Political
implications
- Must C increase?
- No if k>kgoldenrule
10%
5%
0%
-5%
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-10%
Results on Growth Rates:
Growth rates of Potential
1982-1992
1992-2002
2002-2012
2012-2052
2052-2092
-
NNP
NNP per
capita
GDP per
capita
Consumption
per capita
3.02%
3.28%
3.11%
3.03%
3.02%
1.50%
1.75%
1.59%
1.51%
1.50%
1.50%
1.97%
1.69%
1.53%
1.50%
1.50%
1.47%
1.69%
1.53%
1.50%
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)
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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
S
r = real
interest rate
r = rw
NX deficit
I(r)
I, S, NX
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With higher saving in small open economy
r = real
interest rate
S1
S0
Higher saving:
1. No change I
2. No change GDP
3. Higher foreign saving
4. Increase GNP, NNP
Final NX
surplus
r = rw
Original
NX
deficit
I(r)
0
I, S, NX
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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
Derivation
Start with production function and competitive assumptions. For
simplicity, assume a Cobb-Douglas production function with laboraugmenting 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 1948
3. China in central planning and reform period
4. 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
Growth in:
Period
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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GDP: China
Growth in:
Output
Period:
1952-78
1978-98
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
Source: Source: G. Chow, Accounting for Growth in Taiwan and Mainland
China. Assumes Cobb-Douglas aggregate production function with elasticity
of K = 0.4.
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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
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