Download Review of aggregate production function

Economics 122
Macroeconomics in the long run:
Economic growth
1
Agenda
•
•
•
•
•
•
Introductory background
Essential aspects of economic growth
Aggregate production functions
Neoclassical growth model
Simulation of increased saving experiment
Debt, deficits, and growth
2
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)
4
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.
5
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)
6
The MIT School of Economics
Robert Solow
Paul Samuelson
7
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.
8
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
9
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
10
y = Y/L
y*
y = f(k)
(n+δ)k
i = sf(k)
i* =
(I/Y)*
k*
k
11
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
26
12
Now let’s step back a moment to consider the meaning of
economic growth in the broader context.
13
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”
14
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
15
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).
16
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
19
Mining in rich and poor countries
D.R. Congo
Canada
20
Farming the rocks, Morocco, 2001
Medicine in rich and poor countries
Scan for lung
cancer
African medicine man
22
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
25
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
26
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
27
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
28
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
29
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.
31
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.
32
y  Y/L
Impact with labor-augmenting TC
y  f ( k)
y*
(n   )k
i*=I*/Y*
i  sf ( k )
k*
33
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  )
 /(1  )
y *  f (k *)   s / (n  h   ) 
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!
34
ln (Y), etc.
Time profiles of major variables with TC
ln (Y)
ln (K)
ln(L)
ln (L)
time
35
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
36
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
37
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.
38
Two faces of saving
39
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
40
y**
Impact of Higher National Saving
y = f(k)
y*
i = s2f(k)
i = s1f(k)
(I/Y)*
(n+δ)k
k
k*
k**
41
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
42
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%
43
-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)
44
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
45
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
46
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
47
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
48
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 ]}
50
(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.
51
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
52
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
53
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.
54
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.
55
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
56