Transcript Chapter 4

Trade Growth and
Inequality
Professor Christopher Bliss
Hilary Term 2004
Fridays 10-11 a.m.
Ch. 4 Convergence in
Practice and Theory
• Cross-section growth empirics starts with
Baumol (1986)
• He looks at β-convergence
• β-convergence v. σ-convergence Friedman (1992)
• De Long (1988) – sampling bias
Barro and Sala-i-Martin
• World-wide comparative growth
• “Near complete” coverage (Summers-Heston
•
•
•
data) minimizes sampling bias
Straight test of β-convergence
Dependent variable is growth of per-capita
income 1960-85
Correlation coefficient between growth and
lnPCI60 for 117 countries is .227
Table 4.1 Simple regression
result N=117 F=6.245
Variable
Coefficient
t-value
Constant
-.0135
-.998
LnPCI60
.0046
2.50
Correlation and Causation
• Correlation is no proof of causation
• BUT
• Absence of correlation is no proof of the
absence of causation
• Looking inside growth regressions
perfectly illustrates this last point
The spurious correlation
• A spurious correlation arises purely by
chance
• Assemble 1000 “crazy” ordered data sets
• That gives nearly half a million pairs of
such variables
• Between one such pair there is bound to
be a correlation that by itself will seem to
be of overwhelming statistical significance
Most correlations encountered in
practice are not “spurious”
• But they may well not be due to a simple
causal connection
• The variables are each correlated causally
with another “missing” variable
• As when the variables are non-stationary
and the missing variable is time
Two examples of correlating nonstationary variables
• The beginning econometrics student’s
consumption function
ct = α + βyt + εt
• But surely consumption is causally
connected to income
• ADt = α + βTSt + εt
where TS = teachers’ salaries
AD = arrests for drunkeness
Regression analysis and missing
variables
• A missing variable plays a part in the DGP
and is correlated with included variables
• This is never a problem with Classical
Regression Analysis
• Barro would say that the simple regression
of LnPCI60 on per capita growth is biassed
by the exclusion of extra “conditioning”
variables
Table 4,2 Growth and extra
variables
Sources * Barro and Sala-i-Martin (1985)
* Barro-Lee data set
Variable
Definition
Mean
Standard deviation
Growth*
Growth rate per
capita income
1960-85
.0226
.0161
LnPCI60*
Log of PCI 1960
7.5201
,8930
bmp**
Forex black market
premium
.1188
,1675
govsh4**
Gov. con. / GDP
.1571
.0656
geerec**
Public exp. On
Edu./GDP
.0245
,0103
I/Y*
Invest./
GDP ratio
.0968
.1893
pinstab**
Political instability
.1916
,0859
Table 4.3 Regression result
N = 73 F = 8.326 R2 = .4308
Variable
Coefficient
t-value
Partial Rsquared
Constant
.0698
3.83
.1821
LnPCI60
-.01133
-3.89
.1863
Bmp
.0035
.345
.0018
Govsh4
-.0419
-1.66
.0400
Geerec
.4922
2.71
.0999
Pinstab
.0003
.029
.000
I/Y
.1673
6,02
.3545
Table 4.4 Regression with One
Conditioning Variable
Variable
Coefficient
t-value
Partial R2
Constant
.0281
2.17
.0403
LnPCI60
-.0048
-2.33
.0463
I/Y
.1502
7.08
.3092
Looking Inside Growth Regressions
I
g is economic growth
ly is log initial per capita income
z is another variable of interest, such as I/Y,
which is itself positively correlated with
growth.
All these variables are measured from their
means.
Inside growth regressions II
We are interested in a case in which the
regression coefficient of g on ly is near
zero or positive. So we have:
v{gly}≥0
where v is the summed products of g and ly
Inside Growth regressions III
Thus v{gly} is N times the co-variance
between g and ly.
Now consider the multiple regression:
g=βly+γz+ε
(3)
Inside Growth Regressions IV
Inside Growth Regressions V
So that:
vglY=βvgg + γvgz
(5)
Then if vglY ≥ 0 and vg > 0, (5) requires that either β or
γ, but not both, be negative. If vglY > 0 then β and γ may
both be positive, but they cannot both be negative.
One way of explaining that conclusion is to say that a
finding of β-convergence with an augmented regression,
despite growth and log initial income not being
negatively correlated, can happen because the additional
variable (or variables on balance) are positively
correlated with initial income.
A Growth Regression with one
additional variable
g
ly
I/Y
g
.00034
.00384
.00921
ly
I/Y
.82325
.05216
.00780
Growth Regression with I/Y
Variable
Coefficient
t-value
Partial R2
Constant
LnPCI60
.0281
2.17
.0403
-.0048
-2.33
.0463
I/Y
.1502
7.08
.3092
N=117
R2=.346
F=29.57
One additional variable
regression
From (5) and the
variance/covariance matrix above:
.00384 = .82325β + .05216γ
Now if γ is positive, β must be negative
This has happened because the added
Adding the Mystery Ingredient
L
g=βly+γL+ε
The correlation matrix is:
g
ly
L
g
1
.17480
.32184
(7)
ly
L
1
.73373
1
Growth Regression with L
Variable
Coefficient
.016141
Constant
LnPCI60 -.00083
L
.000435
t-value
1.03
Partial
R2
.0092
-.348
.0011
3.24
.0893
N=117 R2=.346 F=29.57
Correlation and Cause
1. The Barro equation is founded in a
causal theory of growth
2. The equation with L cannot have a
causal basis
3. What is causality anyway?
4. Granger-Sims causality tests. Need time
series data. Shocks to causal variables
come first in time
Causality and Temporal Ordering
1. An alarm clock set to ring just before
sunrise does not cause the sun to rise.
2. If it can be shown that random shocks to
my alarm setting are significantly
correlated with the time of sunrise, the
that is an impressive puzzle
3. Cause is a (an optional) theory notion
Convergence Theory
The Solow-Swan Model
Solow-Swan Model II
The model gives convergence in two important
cases:
1. Several isolated economies each with the same
saving share. Only the level of per capita
capital distinguishes economies
2. There is one integrated capital markets
economy and numerous agents with the same
saving rate. Only the level of per capita capital
attained distinguishes one agnet from another.
Solow-Swan Model III
If convergence is conditional on various
additional variables, how precisely do
these variables make their effects felt?
For country I at time t income is:
AiF[Ki(t),Li(t)]
A measures total factor productivity, so will
be called TFP
Determinants of the Growth
Rate
The growth rate is larger:
• The larger is capital’s share
• The larger is the saving share
• The larger is the TFP coefficient
• The smaller is capital per head
• The smaller is the rate of population
growth
Mankiw, Romer and Weil (1992)
• 80% of cross section differences in growth
rates can be accounted for via effects 2
and 5 by themselves
• The chief problem for growth empirics is
to disentangle effects 3 and 4
Convergence: The Ramsey
Model
Ramsey (1928) considered a many-agent
version of his model (a MARM)
He looked at steady states and noted the
paradoxical feature that if agents discount
utility at different rates, then all capital will
be owned by agents with the lowest
discount rate
Two different cases
Just as with the Solow-Swan model the cases are:
• Isolated economies each one a version of the
same Ramsey model, with the same utility
discount rate. The level of capital attained at a
particular time distinguishes one economy from
another
• One economy with a single unified capital
market, and each agent has the same utility
function. The level of capital attained at a
particular time distinguishes one agent from
another
Isolated Economies
Chapter 3 has already made clear that there
is no general connection between the level
of k and (1/c)(dc/dt).
The necessary condition for optimal growth
is:
{[-c(du/dc)]/u}{(1/c)(dc/dt)}=F1[k(t),1]-r
(20)
Where u is U1[c(t)]
Determinants of the Growth of
Consumption
The necessary condition for optimal growth is:
{[-c(du/dc)]/u}{(1/c)(dc/dt)}=F1[k(t),1]-r When
k(t) takes a low value the right-hand side of (20)
is relatively large. If the growth rate of
consumption is not large, the elasticity of
marginal utility
[-c(du/dc)]/u
Must be large.
The idea that β-convergence follows from optimal
growth theory is suspect.
Growth in the MARM
• With many agents the optimal growth
condition (20) becomes:
[-d(du/dc)/dt]/u]=F1[Σkii(t)),1]-r (23)
In steady state (23) becomes:
F1[Σkii(t)),1]=r
Note the effect of perturbing one agent’s
capital holding
A non-convergence result
In the MARM:
1. Non-converging steady states are
possible
2. Strict asymptotic convergence can never
occur
3. Partial convergence (or divergence) clubs
are possible depending on the third
derivative of the utility function
What does a MARM maximize?
Any MARM equlibrium is the solution to a problem of the
form:
Max ΣN1∫0∞U[ci(t)]dt
Non-convergence is hsown despite the assumptions that:
• All agents have the same tastes and the same utility discount
rate
• All supply the same quantity of labour and earn the same wage
• All have access to the same capital market where they earn the
same rate of return
• All have perfect foresight and there are no stochastic effects to
interfere with convergence
Asymptotic and β-convergence
• For isolated Ramsey economies we have seen
that we need not have β-convergence, but we
must have asymptotic convergence
• On the other hand we may have β-convergence
without asymptotic convergence
lnyI = aI - b/t+2 lnyII = aII - b/t+1
aI< aII
Country I has the lower income and is always
growing faster
Strange Accumulation Paths can be
Optimal
In the Mathematical Appendix it is shown
that:
Given a standard production function and a
monotonic time path k(t) such that k goes
to k*, the Ramsey steady state value, and
the implied c is monotonic, there exists a
“well-behaved” utility function such that
this path is Ramsey optimal
Optimal Growth with Random
Shocks
Bliss (2003) discusses the probability density of income
levels when Ramsey-style accumulation is shocked each
period with shocks large on absolute value
Two intuitive cases illustrate the type of result available:
• Low income countries grow slowly, middle income
countries rapidly and rich countries slowly. If shocks are
large poverty and high income form basins of attraction
in which many countries will be found. Compare Quah
(1997)
• If shocks are highly asymmetric this will affect the
probability distribution of income levels, even if the
differential equation for income is linear. Earthquake
shocks.
The BMS Model
Barro, Mankiw and Sala-i-Martin (1995)
Human capital added which cannot be used as
collateral
One small country converges on a large world in
steady state (existence is by exhibition).
A more general case is where many small
countries have significant weight. Then if they
differ some may leave the constrained state
before others and poor countries may not be
asymptotically identical
Concluding Remarks
1. There is no simple statistical association between initial
2.
3.
4.
income and subsequent growth, hence no support for
β-convergence from a basic two-variable analysis
With multivariate analysis the hypothesis of a causal
connection between initial income and subsequent
growth on an other things equal basis is not rejected
Theoretical models with common technology often
confirm the β-convergence hypothesis
Surprisingly the literature neglects “catching-up”