Download Introducing technological change

The first known anthro-capital
Olduvai stone chopping tool, 1.8 – 2 million years BP
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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 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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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)
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 greatest technological change in history
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]
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
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:
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, captures the basic trends!
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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 )
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 )
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Per capita output
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.
Source: BLS, multifactor productivity page.
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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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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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