Transcript Economic growth

Economic growth
Olduvai stone chopping tool, 1.8 – 2 million years BP
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Actual and potential output
Billions of 2005$
16,000
14,000
12,000
Actual GDP
Economic growth:
10,000
Studies growth of
potential output
Potential GDP
8,000
6,000
1980
1985
1990
1995
2000
2005
2010
2015
2020
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Congressional Budget Office, Sept 2012, cbo.gov.
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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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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Notes on the readings
Chapters 4 and 5 are core and consensus economics
Chapter 6 is controversial and one point of view.
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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)
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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 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 not have growth predictions of historical record, so
need further work (next lecture).
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Economic growth (II)
World fastest supercomputer, 2012 (IBM Sequoia at 16.2 petaflops)
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Agenda
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Continue analysis of NGM
Discuss problem sets and ground rules
Savings experiment
On Monday, do economics of technological change and
real business cycle model
- On Wednesday, pset 1 due at the beginning of class.
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Review from last time
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Neoclassical growth model: classical + dynamics
Growth involves potential output (not business cycles)
Key assumptions: fixed s, labor growth at n
Law of motion of economy: Δ k/k = sf(k) - (n + δ) k
Unique and stable equilibrium: sf(k*) = (n + δ) k*
Alas, does not capture major trends.
Concluded need to include technological change (TC).
• Economic growth involves:
- increase in quantity (number of cars)
- improved quality (safer cars)
- new goods and services (computer replaces typewriter)
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Growth trend, US, 1948-2008
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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Near-constancy of labor’s share of national income
80%
Labor share of
national income in US
70%
60%
- 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)
50%
40%
30%
Share of compensation
Share of wages
20%
10%
0%
1929
1939
1949
1959
1969
1979
1989
1999
2009
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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
Derivation
Start with production function and competitive assumptions. For
simplicity, assume a Cobb-Douglas production function with 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, 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 T.C. as:
(5) g[At] = g[Yt] –sh(K) g[Kt] - sh(L) g[Lt ]}
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Apply to U.S. private non-farm business, 1987-2011
g[Yt] = g[At] + sh(K) g[Kt] + sh(L) g[Lt ]
We have data on everything but g[At].
According to U.S. Bureau of Labor Statistics:
g[Yt] = 2.8 % p.y.; sh(K) = ¼; g[Kt] =3.6%; g[Lt ] = 1.1%
So contribution to growth is
K:
0.9% per year
L:
0.8 % per year
g[At]: 1.1 % per year
So technology contributes 55% and capital 45% to growth in output per
hour.
Source: BLS, multifactor productivity page.
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Added after class (correct the mistake)
What are the contributions to per capita output growth?
g[Y/L] = g(y) = g[A] + sh(K) g[K] + sh(L) g[L ] –g[L]
= g[A] + sh(K) g[K] - sh(K) g[L ]
= g[A] + sh(K) (g[K] - g[L ])
= g[A] + sh(K) (g[k])
Since g[y] = 1.7 % p.y.; sh(K) = ¼; g[k] =2.5%
So contribution to growth is
Of A:
1.1%, or 65% of total
Of k:
¼ x 2.5, 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.
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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, iPod)
Analytically, TC is
- Shift in production function.
y = Y/L
n ew f(k )
o ld f(k )
k
k*
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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!
The greatest technological change in history
Abacus master, 1945
(.03 flops)
IBM 1620, circa 1960
(104 flops)
Sony Laptop, 2010
(1010 flops)
IBM Sequoia, top supercomputer, 2012
(16x 1015 flops), 2x 2011 tops
[flop = floating point operations per second, e.g., 1011011011001001/0010110100010101]
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
n ew f(k )
Y  F  K , EL   F (K , L )
 F  K,L
 L f ( k ),
y

o ld f(k )
L F  K /L , 1 
w here k  K / L
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
T est th e lo n g -ru n eq u ilib riu m o f  k  0 :
s f (k ) = (n  h   ) k
The equilibrium is unique and globally stable. It has exactly the
same properties as earlier one, except:
• Note that the growth term includes h (rate of tech. change).
• All natural variables are growing at h: wage rates, per capita
output, capital-labor ratio etc. are growing at h rather than 0.
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T.C. for the Cobb-Douglas
In C-D case, labor-augmenting TC is very simple:
S et A 0  E 0  1 fo r sim p licity .
Y t K

t
( e ht L t ) 1  
yt  Y t/ Lt  K
yt

Kt L

t

t

( e ht L t ) 1   / L t  K t L
1 
t
/L
t

= kt
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Results of neoclassical model with labor-augmenting TC
For C-D case,
s k *  ( n  h   ) k *
k *   s / (n  h   ) 
1 /( 1   )
 /( 1   )
y *  f ( k *)   s / ( n  h   ) 
Unique and stable equilibrium under standard assumptions:
Predictions of basic model:
–
–
–
–
Steady state: constant y , w , k , a n d r
Here output per capita, capital per capita, and wage rate 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*
k = K33/ L
ln (Y), etc.
Time profiles of major variables with TC
ln (Y); gY = n+h
ln (K); gK = n+h
ln ( L ); g L  n  h
ln (L) ); gL = n
time
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Sources of TC
Technological change is in some deep sense “endogenous.”
The underlying theory will be discussed on Monday.
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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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