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
1
Agenda
•
•
•
•
•
•
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
2
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.
4
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)
6
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.
7
The MIT School of Economics
Robert Solow (1924 - )
Paul Samuelson (1915-2009)
8
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.
9
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
10
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
11
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.
12
y = Y/L
y*
y = f(k)
(n+δ)k
i = sf(k)
i* =
(I/Y)*
k*
k
13
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
14
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
15
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
16
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
17
Problem with neoclassical model without TC
Simplest model misses major trends of growth in
y, w, and k.
Missing element is technological change
18
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)
19
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
21
Farming the rocks, Morocco, 2001
Capital deepening in agriculture
23
Technological change in medicine
Scan for lung
cancer
African medicine man
24
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
25
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.
26
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
27
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!
28
y  Y/L
Impact with labor-augmenting TC
y  f (k)
y*
(n   )k
i*=I*/Y*
i  sf ( k )
k*
29
k=K/L
ln (Y), etc.
Time profiles of major variables with TC
ln (Y)
ln (K)
ln(L)
ln (L)
time
30
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
31
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.
32
Two faces of saving
33
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
34
http://www.fiscalcommission.gov/sites/fiscalcommission.gov/files/
documents/CoChair_Draft.pdf
35
Why you can’t get anywhere without Econ 122:
An example from the Co-chairs’ proposal
36
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)
37
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?
38
y**
Impact of Higher National Saving
y = f(k)
y*
i = s2f(k)
i = s1f(k)
(I/Y)*
(n+δ)k
k
k*
k**
39
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
40
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%
41
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
42
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
43