Download Review of aggregate production function

Actual and potential output
Real GDP (billions of 2005 $)
15,000
14,000
13,000
12,000
Actual GDP
Potential GDP
11,000
10,000
9,000
8,000
1996
1998
2000
2002
2004
2006
2008
2010
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Agenda
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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 supply
and “economic growth”
Aggregate demand
and business cycles
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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AS-AD in short, medium, long run
AS long run
π
AS medium run
AS short run
πt
AD
Yt* = potential output
Y
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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The MIT School of Economics
Robert Solow (1924 - )
Paul Samuelson (1915-2009)
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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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Mathematical note
We will use the following math fact:
Define z = y/x
Then
(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.
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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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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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Problem with neoclassical model without TC
Simplest model misses major trends of growth in
y, w, and k.
Missing element is technological change
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Further thoughts on growth
1. Growth involves potential output (not business cycles)
2. Growth theories deal with dynamic version of fullemployment, “classical”-type economy
3. Economic growth involves:
- increase in quantity (tons of coal)
- improved quality (safer cars)
- new goods and services (computer replaces typewriter)
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The greatest technological change in history
Thomas arithmometer, 1870
(10-16 petaflops)
China’s Tianhe-1A
(2.5 petaflops)
[petaflop = 1015 floating point operations per second]
Mining in rich and poor countries
D.R. Congo
Canada
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Farming the rocks, Morocco, 2001
Capital deepening in agriculture
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Technological change in medicine
Scan for lung
cancer
African medicine man
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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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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:
Set A 0  E 0  1 for simplicity.
Y t  K t ( e ht Lt )1 
y t  Y t / L t  K t ( e ht Lt )1  / L t  Kt L t1  / L t
y t  Kt L t = k t
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Results of neoclassical model with labor-augmenting TC
For C-D case,
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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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
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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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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The major non-cycle economic issue of
today: The federal budget deficit and debt
Important documents:
National Academy Sciences, Our Fiscal Choices
National Commission on Fiscal Responsibility and Reform,
Co-Chairs’ Proposal
Both will be studied in coming weeks.
Issue for us: Since reduced deficit is lower government
savings, what is the effect on output:
- short run
- long run
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http://www.fiscalcommission.gov/sites/fiscalcommission.gov/files/
documents/CoChair_Draft.pdf
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Why you can’t get anywhere without Econ 122:
An example from the Co-chairs’ proposal
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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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Basics of the deficit and growth
Assume a closed economy
I = T– G + [Y – T – C(Y-T, r)]
= Budget surplus + Sp
At full employment and assuming that private C does not
respond significantly to r, for deficits that reduce G:
ΔI = ΔS = Δ (Budget surplus)
So this is the motivation of deficit reduction for long-run growth.
Question: what is the effect?
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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 Deficit Reduction
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 = 22% of GDP
b. Initial govt. savings rate = minus 6 % of GDP
c. In 2012, govt. changes fiscal policy to a deficit of minus 2 %
of GDP.
d. All of higher govt. S goes into national S (i.e., constant
private savings rate) and closed economy
5.
“Calibrate” to U.S. economy
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Impact of Lower Govt Deficit on Major Variables
2010
2015
2020
2025
2030
35%
Consumption per capita
Percent change from baseline
30%
GDP per capita
Capital per capita
25%
20%
NNP per capita
- Note that takes 10
years to increase C
-Political
implications
- Must C increase?
- No if k>kgoldenrule
15%
10%
5%
0%
-5%
-10%
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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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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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