Download Review of aggregate production function

The first known anthro-capital
Olduvai stone chopping tool, 1.8 – 2 million years BP
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Report from the Fed
• Hello to Econ 122 from Vice-chair Yellen
• Report from former Yale TA William Nelson to the
Chairs of the Regional Federal Reserves
… on forward guidance
• Valediction for Chairman Bernanke
“character, integrity, competence”
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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
Actual and potential output
Economic growth:
Studies growth of
potential output
Congressional Budget Office, March 2013, cbo.gov.
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Potential and actual real output in the long run
20,000
16,000
14,000
12,000
Potential output
Actual output
10,000
8,000
6,000
4,000
2,000
1950
1960
1970
1980
1990
2000
2010
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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 (rational?).
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
Growth trend, US, 1948-2008 (pre-recession)
2.0
ln(K)
ln(Y)
ln(hours)
1.6
1.2
0.8
0.4
0.0
50
55
60
65
70
75
80
85
90
95
00
05
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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, total factor
productivity)
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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Capital deepening in agriculture
Food for Commons
Morocco, 2002
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Mining in rich and poor countries
D.R. Congo
Canada
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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 = 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 (math on next slide):
5. Δ k/k = Δ K/K - Δ L/L
With some algebra,* this becomes:
5’. Δ k/k = Δ K/K - n Y, or
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.
*The algebra of the derivation:
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 expository 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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But does it match the growth trends from the historical
record?
No.
What is missing from the NCM up to now?
Technological change!
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Economic growth (II): Technological change
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World fastest supercomputer, 2012 (Tianhe-2 in NUDT, Guangzhou, China ) at 33.8
petaflops (34x1015 floating point operations per second , up from 16.2 last year)
Economic Growth (II): Technological Change
Yale
Abacus master
(.03 flops)
IBM 1620, circa 1960
(104 flops)
High-end PC, 2013
(1011 flops)
Tianhe-2, top supercomputer, 2013
(34x1015 flops)
[flop = floating point operations per second, e.g., 1011011011001001+0010110100010101]
Review from last time
Neoclassical growth model is classical model + time
Major dynamic equation:
Δ kt = sf(kt) - (n + δ) kt
Without technological change (TC):
- Long-run growth is gY = gK = n; gw = gk = gy = 0
- Change in n, s affect level, not growth in the long run
NCM without TC cannot explain basic trends.
So now introduce TC.
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Introducing technological change
First model omits technological change (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, Twitter)
Analytically, TC is
- Shift in production function.
y = Y/L
new f(k)
old f(k)
k*
k
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The Wealth of Nations
Why do countries, regions, people have such divergent
incomes?
• People in US: top 1% to bottom 1 %: 717,000/$1000?
• US states: top county to bottom county: $132,700/$5213
(Wyoming/South Dakota)
• Top to bottom country: $88,900/$375 (Qatar/Congo)
Most of the differences among regions are technological, so
need to understand the economics of technological change.
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Technological change in medicine
Scan for lung
cancer
African medicine man
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Disappearance of polio:
A benefit of growth that is not captured
in the GDP statistics!
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 of TC will give slightly different results.
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The math with technological change is this:
y
 Y / L  f (k )
k = K /L
 k = s f (k )  (n  h   ) k
Test the long-run equilibrium of  k  0 :
s f ( k ) = (n  h   ) k
In Cobb-Douglas case: Set A0  E 0  1 for simplicity
without loss of generality.
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 *   s /( n  h   )  /(1 )
Unique and stable equilibrium under standard assumptions:
Predictions of basic model:
– Steady state: constant y, w, k, and r ; these all grow at h.
– Labor’s share of output is constant.
– Hence, captures 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); gY = n+h
ln (K); gK = n+h
ln( L); gL  n  h
ln (L) ); gL = n
time
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What are the contributors to growth? Growth Accounting
Growth accounting is a widely used technique used to separate
out the sources of growth in a country; it relies on the
neoclassical growth model
Assume Cobb-Douglas, and calculate growth rates:
Take logarithms and time derivative:
g(Yt) = g(At )+ α g(Kt) + (1 - α) g(Lt )
Or if we assume competitive α = sh(K) and (1- α) = sh(L)
g(Y) = g(A) + sh(K) g(K) + sh(L) g(L )
Subtract growth of labor to get per capita growth:
g(Y)- g(L ) = g(y ) = g(A) + sh(K) g(k)
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Per capita output
g(y) = g(A) + sh(K) (g(k))
g(A) = g(y) - sh(K) (g(k))
Historical data:
g(y) = 1.7% p.y.; sh(K) = ¼; g(k) = 2.4% p.y.
g(A) = 1.7 % – 0.25 2.4% = 1.1%
So contribution to p.c. y growth is
Of A:
1.1%, or 65% of total
Of k:
¼ x 2.4, or 35% of total
This is the very surprising results that technology contributes most of the
rise in per capita output over the period. Similar in other
countries/periods.
Source: BLS, multifactor productivity page.
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Understanding Technological Change
TC is conceptually much more difficult than investment:
- 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)
- Openness to foreign technologies and management practices
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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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Jobs today … or …
consumption tomorrow?
Two faces of saving
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National savings and fiscal policy
The major long-run policy to affect national savings is fiscal
policy. Remember:
S = Sp + Sg = Sp + T – G
We show the analysis today, and return to the
controversies of fiscal policy at the end of the course.
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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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Summary
• Neoclassical model is workhorse of long-run
growth theory.
• NCM + TC does good job of explaining major
trends.
• Make sure you understand the impacts of
changing n, s, and h.
• Changes in fiscal policy affect mainly national
savings.
• Innovation policy is a critical and poorly
understood aspect of economic policy.
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