6. Towards a non-aggregative growth theory

6.1. Anillustration

The presumption of neo-classical growth theory was that being a citizen of a poor coun­try gives one access to many exciting investment opportunities, which eventually lead on to convergence.

What can we say about the long-run evolution of an economy where there are re­warding opportunities that are not necessarily exploited? In this section we will explore this question under the assumption that the only source of inefficiency in this econ­omy comes from limited access to credit. The goal is to illustrate what non-aggregative growth theory might look like, rather than to suggest an alternative canonical model.

The model we have in mind is as follows: There are individual production functions associated with every participant in this economy that are assumed to be identical and a function of capital alone (F(K)) but otherwise quite general. In particular, we do assume that they are concave. Individuals maximize an intertemporal utility function of the form:

People are forward-looking and at each point of time they choose consumption and savings to maximize lifetime utility. However, the maximum amount they can borrow is linear and increasing in their wealth and decreasing in the current interest rate: An individual with wealth w can borrow up to λ(r_t)w. Credit comes from other members of the same economy and the interest rate clears the credit market. We do not assume that everyone starts with the same wealth, but rather that at each point of time there is a distribution of wealth that evolves over time.

This model is a straightforward generalization of the standard growth model. What it tells us about the evolution of the income distribution and efficiency depends, not surprisingly, on the shape of the production function.

The simplest case is that of constant returns in production. In this case, inequality remains unchanged over time, and production and investment is always efficient.

With diminishing returns, greater inequality can lead to less investment and less growth, because the production function is concave. However, inequality falls over time and in the long run no one is credit constrained, although we do not necessarily get full wealth convergence. The long run interest rate converges to its first best level, and hence investment is efficient. To see why this must be the case, note first that because of diminishing returns the poor always have more to gain from borrowing and investing than the rich. In other words, the rich must be lending to the poor. As long as the poor are credit constrained, they will earn higher returns on the marginal dollar than their lenders, i.e., the rich (that is what it means to be credit constrained). As a result, they will accumulate wealth faster than the rich and we will see convergence. This process will only stop when the poor are no longer credit constrained, i.e., they are rich enough to be able to invest as much as they want.

With increasing returns, inequality increases over time; we converge to a Gini coeffi­cient of 1. Wealth becomes more and more concentrated with only the richest borrowing and investing. Because there are increasing returns, this is also the first best outcome. The logic of this result is very similar to the previous one: Now it is the rich who will be borrowing and the poor who will be lending, with the implication that the rich are the ones who are credit constrained and the ones earning high marginal returns. Therefore, they will accumulate wealth faster and wealth becomes increasingly concentrated.

Finally we consider the case of “S-shaped” production functions, which are produc­tion functions that are initially convex and then concave. The Cobb-Douglas with an initial set-up cost discussed at length in Section 5.2 is a special case of this kind of technology.

What happens in the long run in this model depends on the initial distribution of in­come. When the distribution is such that most people in the economy can afford to invest in the concave part of the production function, the economy converges to a situation that is isomorphic to the diminishing returns case, with the entire population “escaping” the convex region of the production function.

The more unusual case is the one where some people start too poor to invest in the concave region of the production function. The poorer among such people will earn very low returns if they were to invest and therefore will prefer to be lenders. Now, as long as the interest rate on savings is less than 1 ∕δ, they will decumulate capital (since the interest is less than the discount factor) and eventually their wealth will go to zero. On the other hand, anyone in this economy who started rich enough to want to borrow will stay rich, even though they are also dissaving, in part because at the same time they benefit from the low interest rates. The economy will converge to a steady state where the interest rate is 1∕δ, those who started rich continue to be rich and those who started poor remain poor (in fact have zero wealth).

This is classic poverty trap: Moreover, since no one escapes from poverty, nor falls into it, there is a continuum of such poverty traps in this model. This kind of multiplicity is, however, fragile with respect to the introduction of random shocks that allow some of the poor to escape poverty and impoverish some of the rich.

Even in a world with such shocks there can be more than one steady state: The reason is that the presence of lots of poor people drives down interest rates, and low interest rates make it harder for the poor to save up to escape poverty even with the help of a positive shock.

The key to this multiplicity is the endogeneity of the interest rate. It is the pecuniary externality that the poor inflict on other poor people that sustains it. This is why such poverty traps are sometimes called collective poverty traps, in contrast to the individual poverty traps described above.

The investigation of the evolution of income distribution in models with credit constraints and endogenous interest rates goes back to Aghion and Bolton (1997). Matsuyama (2000, 2003) and Piketty (1997) emphasize the potential for collective poverty traps in a variant of this model, without the forward-looking savings deci­sions.

This class of models is a part of a broader group of models which study the simulta­neous evolution of the occupational structure, factor prices and the wealth distribution in a model with credit constraints. Loury (1981) studied this class of modelsand showed that in the long run the neo-classical predictions tend to hold as long as the production function is concave. Dasgupta and Ray (1986) and Galor and Zeira (1993) provide ex­amples of individual poverty traps in the presence of credit constraints and S-shaped production functions. Banerjee and Newman (1993) show the possibility of a collective poverty trap in a model with a S-shaped production function which is driven by the en­dogeneity of the wage - essentially high wages allow workers to become entrepreneurs easily, which keeps the demand for labor, and hence wages, high. Recent work by Buera (2003) shows that the multiplicity results in Banerjee and Newman survive in an envi­ronment where savings is based on expectations of future returns.^[303] Ghatak, Morelli and Sjostrom (2001, 2002) and Mookherjee and Ray (2002, 2003) explore related but slightly different sources of individual and collective poverty traps.

5.3. Can we take this model to the data?

Models like the one we just developed (as well as political economy models that we do not discuss here^[304]) have been invoked as motivation for a large empirical literature on the relationship between inequality and growth in cross-country data.

In this section, we evaluate whether, given these concerns, estimating the relationship between inequality and growth in a cross-country data set remains useful. Having con­cluded that it has, at best, very limited use, we discuss an alternative approach based on calibrating non-aggregative models using micro data.

6.2.1. What are the empirical implications of the above model?

Functional form issues With constant returns to scale, distribution is irrelevant for growth. With diminishing returns, an exogenous mean-preserving spread in the wealth distribution in this economy will reduce future wealth and, by implication, the growth rate. However, the impact depends on the level of wealth in the economy: Once the economy is rich enough that everyone can afford the optimal level of investment, in­equality should not matter. The estimated relationship between inequality and growth should therefore allow for an interaction term between inequality and mean income. Moreover, an economy closer to the steady state has both lower inequality and lower growth. This has two implications for the estimation of the inequality growth relation­ship.

Controlling properly for the effect of mean wealth (or mean income), is therefore vital for getting meaningful results. The usual procedure is to control linearly (as in most other growth regressions) for the mean income level at the beginning of the period. It is, however, not clear that there is any good reason why the true effect should be linear. Moreover, it seems plausible that different economies will typically have different λs, and therefore will converge at different rates.

The model also tells us that while initial distribution matters for the growth rate, it only matters in the short run. Over a long enough period, two economies starting at the same mean wealth level will exhibit the same average growth rate. In other words, the length of the time period over which growth is measured will affect the strength of the relationship between inequality and growth.

The preceding discussion assumed that the interest rates converged. As we noted, that does not need to be the case. If we do not assume it, variants of the simple concave economy may no longer converge, even in the weaker sense of the long-run mean wealth being independent of the initial distribution of wealth. Intuitively, poor economies will tend to have high interest rates, and this in turn will make capital accumulation difficult (note that λ' < 0) and tend to keep the economy poor.^[305] This effect reinforces the claim made above that inequality matters most in the poorest economies.^[306] This economy can have a number of distinct steady states that are each locally isolated. This means that small changes in inequality can cause the economy to move towards a different and further away steady state, making it more likely that the relationship will be non-linear.

With increasing returns, growth rates increase with a mean preserving spread in income. As the economy grows, it also becomes more unequal. Interpreting the relation­ship between inequality and growth is difficult even after controlling for convergence.

In the S-shaped returns case, the relationship between inequality and growth can be negative or positive depending on the initial distribution, and the size of the increase. For example, if everybody is very poor (on the left of the convex zone), a small increase in inequality will reduce growth, but increasing inequality enough may push more peo­ple to the point where they are able to take advantage of the more efficient technology, and increases in inequality will increase growth. The relation between inequality and growth delivered by this model is clearly non-monotonic. Moreover, the strong conver­gence property does not hold in general. In other words, the growth rate of wealth may jump up once the economy is rich enough, with the obvious implication that economies with higher mean wealth will not necessarily grow more slowly. In other words, the effect of mean wealth, that is the so-called convergence effect, may not be monotonic in this economy. Linearly controlling for mean wealth therefore does not guarantee that we will get the correct estimate of the effect of inequality. It is worth noting that this econ­omy will have a connected continuum of steady states. This means that after a shock the economy will not typically return to the same steady state. However, since it does converge to a nearby steady state, this is not an additional source of non-linearity.

Identification issues Even if we could agree on a specification that is worth estimating, it is not clear how we can use cross-country data to estimate it. Countries, like individ­uals, are different from each other. Even in a world of perfect capital markets, countries can have very different distributions of wealth because, for example, they have different distributions of ability. There is no causal effect of inequality on growth in this case, but they could be correlated for other reasons. For example, cultural structures (such as a caste system) may restrict occupational choices and therefore may not allow individu­als to make proper use of their talents, causing both higher inequality and lower growth. Conversely, if countries use technologies that are differently intensive in skilled labor, those countries using the more skill intensive technology can have both more inequality and faster growth.

As we discussed in detail above, countries have different kinds of financial institu­tions, implying differences in the λ's in our model. Ourbasic model would predict that the country with the better capital markets is likely both to be more equal and to grow faster (at least once we control for the mean level of income). The correlation between inequality and growth will therefore be a downwards-biased estimate of the causal pa­rameter, if the quality of financial institutions differs across countries.^[307]

If these country specific effects were additive, one could control for them by includ­ing a country fixed-effect in the estimated relationship (or by estimating the model in first difference). This strategy will be valid only under the assumption that changes in inequality are unrelated to unobservable country characteristics that are correlated with changes in the growth rate. While this is a convenient assumption, it has no reason to hold in general. For example, skill-biased technological progress will lead both to a change in inequality and a change in growth rates, causing a spurious positive corre­lation between the two. To make matters worse, we have to recognize the fact that λ itself (and therefore the effect of inequality on growth at a given point in time) may be varying over time as a result of monetary policies or financial development, and may itself be endogenous to the growth process.^[308]

The more general point that comes out of the discussion above is that unless we as­sume capital markets are extremely efficient (which, in any case, removes one of the important sources of the effect of inequality), changes in inequality will be partly en­dogenous and related to country characteristics which are themselves related to changes in the growth rate. Identifying the effect of inequality by including a country fixed-effect would not necessarily solve all the endogeneity problems. Moreover, as we discussed above, the theory suggests that the specification should allow for non-linear functional forms, and interaction effects, which will be difficult to accommodate with a fixed effect specification.

5.3.2. Empirical evidence

The preceding discussion suggests that empirical exercises using aggregate, cross­country data to estimate the impact of inequality and growth will be extremely difficult to interpret. The results are also likely to be sensitive to the choice of specification. This may explain the variety of results present in the literature. A long literature [see Benabou (1996) for a survey] estimated a long run equation, with growth between 1990 and 1960 (say) regressed on income in 1960, a set of control variables, and inequality in 1960. Estimating these equations tended to generate negative coefficients for inequality. As the discussion in the previous subsection suggests, there are many reasons to think that this relationship may be biased upward or downwards. To address this problem, Li and Zou (1998) and Forbes (2000) used the Deininger and Squire data set to focus on the impact of inequality on short run (5 years) growth, and introduced a linear fixed effect.^[309] The results change rather dramatically: The coefficient of inequality in this specification is positive, and significant. Finally, Barro (2000) used the same short fre­quency data (he is focusing on ten-year intervals), but does not introduce a fixed effect. He finds that inequality is negatively associated with growth in the poorer countries, and positively in rich countries.

Banerjee and Duflo (2003) investigate whether there is any reason to worry about the non-linearities that the theory suggests should be present. They find that when growth (or changes in growth) is regressed non-parametrically on changes in inequality, the relationship is an inverted U-shape. There is also a non-linear relationship between past inequality and the magnitudes of changes in inequality. Finally, there seems to be a negative relationship between growth rates and inequality lagged one period. These facts taken together, and in particular the non-linearities in these relationships (rather than the variation in samples or control variables), account for the different results obtained by different authors using different specifications.

Townsend and Ueda (2003) illustrate very clearly that this diversity of results is likely to come from the functional form and identification problems we just discussed. They simulate the 30 year evolution for 1,000 economies based on a model similar to the ones we describe in this section, with non-linear individual production function and credit constraints. The economies start in 1976, with a distribution of wealth calibrated to match the Thai economy in the same year. They then introduce aggregate and indi­vidual level shocks, and run regressions similar to the regressions run in the literature. Using the 1985 year as the “base year”, they replicate the findings of the long run re­gressions. Using 1980 as the base year, they do not replicate those results. A regression similar to that of Forbes (2000) finds either a positive or negative relationship, depend­ing on sampling decisions. This exercise clearly shows that aggregate cross-country regressions are the wrong tool to evaluate the pertinence of this class of models.

6.3. Where do we go from here?

The discussion on functional form and identification, coupled with the empirical ev­idence of non-linearities even in very simple exercises, suggests that cross-country regressions are unlikely to be able to shed any meaningful light on the empirical rel­evance of models that integrate credit constraints and other imperfections of the credit markets. This is made worse by the poor quality of the aggregate data, despite the con­siderable efforts to produce consistent and reliable data sets. This contrasts with the increased availability of large, good quality, micro-economic data sets, which allow for testing specific hypotheses and derive credible identifying restrictions from theory and exogenous sources of variation. Throughout this chapter, we quoted many studies using micro-economic data which tested the micro-foundations for the models we discussed in this section.

Even a series of convincing micro-empirical studies will not be enough to give us an overall sense of how, together, they generate aggregate growth, the dynamics of income distribution, and the complex relationships between the two. The lessons of develop­ment economics will be lost to growth if they are not brought together in an aggregate context. In other words, it is not enough to use them to loosely motivate cross-sectional growth regression exercises - the discussion in this section is but an example of the misleading conclusions to which this can lead.

An alternative that seems likely to be much more fruitful is to try to build macroeco­nomic models that incorporate the features we discussed, and to use the results from the microeconomic studies as parameters in calibration exercises. The exercise we per­formed in Section 5 of this chapter is an illustration of the kind of work that we can hope to do. There are a number of recent papers that in some ways go further in this direction than we have gone. In particular, Quadrini (1999) and Cagetti and De Nardi (2003), for the U.S., and Paulson and Townsend (2004), for Thailand, try to calibrate a model with credit constraints to understand the correlation between wealth and the probability ofbecoming an entrepreneur. The paper by Buera (2003), mentioned above, emphasizes the fact that the long run correlation between wealth and entrepreneurship is weaker than the short run correlation, because, as noted by Skiba (1978), Deaton (1992), Aiyagari (1994) and Carroll (1997), those who are credit constrained now but want to invest in the future have a very strong incentive to save. This, Buera points out, reduces the ultimate efficiency cost of imperfect credit markets, though in spite of this, the person with the median ability level and the median starting wealth loses about 18% of lifetime welfare because of the credit constraints. Caselli and Gennaioli (2002) offer a slightly different calibration: Like Buera, they are worried about the fact that with credit constraints the biggest firms may not be run by the best entrepreneurs. This can be a source of very large productivity losses in the short run. However, since the best entrepreneurs will make the most money, in the long run their firms would necessarily become the largest, unless they died young. They show that even with this limiting fac­tor, reasonable death rates would imply a 20% loss of productivity when we compare an economy without credit constraints with one that has them.

The calibrations so far have not attempted to see if the path of wealth distribution that results from calibrating this type of model matches the data. Our exercise above, for example, tries to match the distribution of firm sizes at a point of time, but says nothing about the path, while Buera does not try to match the data. The one exception is the papers by Robert Townsend and his collaborators based on Thai data [Jeong and Townsend (2003), Townsend and Ueda (2003)].

These papers, as well as those mentioned in the previous paragraphs, start from the assumption that every firm has a single, usually strictly concave, production technol­ogy. The only fixed cost comes from the fact that the firm needs an entrepreneur. As we saw above, this model does not do very well in terms of explaining the cross-sectional variation in the firm sector or the overall productivity gap, as compared to a model with a small number of alternative technologies and varying fixed costs. More generally, we need both a better empirical understanding of where the most important sources of inef­ficiency lie and better integration of this understanding when we assess the predictions of growth theory.

And perhaps above all, we need better growth theory: Our exercise at the beginning of this section was intended to advertise the possibility of a growth theory that does not assume aggregation. While we attempted to link the results to some relatively general properties of the production function, our analysis relies heavily on the fact that the in­efficiency we assumed was in the credit market and that this took the form of a credit limit that was linear in wealth. One can easily imagine other ways for the credit market to be imperfect and other results from such models. Moreover, while the class of pro­duction technologies covered by our model was broader than usual, it does not include the (multiple-fixed-cost) technology that the previous section advocates.

There are, of course, other types of non-aggregative models: There are some exam­ples of non-aggregative growth models that build on the inefficiency that comes from poorly functioning insurance markets.^[310] There are also interesting attempts to build growth models that emphasize the fact that some people are favored by the govern­ment while others are not, and especially the fact that this changes over time in some predictable way (see Roland Benabou’s contribution to this volume). Some interesting recent work has been done on the dynamic interplay between growth and political insti­tutions (see the chapter by Acemoglu, Johnson and Robinson in this volume) as well as between growth and social institutions [see Oded Galor’s contribution to this volume, as well as Cole, Mailath and Postlewaite (1992, 1998, 2001)]. However, even more than in the case of the literature on credit markets and growth, it is not clear how much the insights from these models rely on specific details of how the environment or the im­perfection was modeled and to what extent they can be seen as robust properties of this entire class of models.

There are also areas where growth theory has not really reached: We have no models that, for example, incorporate reputation-building or learning into growth theory. The same can be said about the entire class of behavioral models of underinvestment.

Finally, there is the open question of whether we gain anything by building grand models that incorporate all these different reasons for inefficiency in a single model. To answer this we would need to assess whether the fact that different forms of inefficiency interact with each other has empirically important consequences.

This is an exciting time to think about growth. We are beginning to see the contours of a new vision, both more rooted in evidence and more ambitious in its theorizing.

Acknowledgements

The authors are grateful to Pranab Bardhan, Michael Kremer, Rohini Pande, Chris Udry and Ivan Werning for helpful conversations and Philippe Aghion and Seema Jay- achandran for detailed comments. A part of this material was presented as the Kuznets Memorial Lecture, 2004, at Yale University. We are grateful for the many comments that we received from the audience.

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