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### Trade Growth and Inequality

### One additional variable regression

### Convergence Theory

Professor Christopher BlissHilary Term 2004Fridays 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

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 non-stationary variables

- The beginning econometrics student’s consumption functionct = α + βyt + εt
- But surely consumption is causally connected to income
- ADt = α + βTSt + εtwhere TS = teachers’ salariesAD = 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 variablesSources * Barro and Sala-i-Martin (1985) * Barro-Lee data set

LookingInside 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)

InsideGrowth 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.

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 variable is positively correlated with g

Correlation and Cause

- The Barro equation is founded in a causal theory of growth
- The equation with L cannot have a causal basis
- What is causality anyway?
- Granger-Sims causality tests. Need time series data. Shocks to causal variables come first in time

Causality and Temporal Ordering

- An alarm clock set to ring just before sunrise does not cause the sun to rise.
- 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
- Cause is a (an optional) theory notion

The Solow-Swan Model

Solow-Swan Model II

The model gives convergence in two important cases:

- Several isolated economies each with the same saving share. Only the level of per capita capital distinguishes economies
- 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:

- Non-converging steady states are possible
- Strict asymptotic convergence can never occur
- 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

- There is no simple statistical association between initial 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”

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