Inequality and Unsustainable Growth: Two Sides of the Same Coin?

Abstract

The relationship between income inequality and economic growth is complex and the evidence mixed. This paper focuses on the connections between income inequality and the fragility of economic growth. We find that longer growth spells—periods of strong, healthy, per capita growth—are robustly associated with more equality in the income distribution, even when controlling for a range of other standard determinants. A key implication is that it would be a gamble to think that distribution will take care of itself provided policy makers steadfastly pursue growth. Over longer horizons, avoiding excessive inequality and sustaining economic growth may be two sides of the same coin.

Appendix: Calculation of Breaks and Spells

We apply a variant of a procedure proposed by Bai and Perron (1998, 2003) to test for multiple structural breaks in time series when both the total number and the location of breaks are unknown.¹¹ Our approach differs from the Bai–Perron approach in that it uses sample-specific critical values that take into account heteroskedasticity and sample size as opposed to asymptotic critical values; and in that it extends Bai–Perron’s algorithm for sequential testing of structural breaks, as described below.

We set the minimum “interstitiary period:” the minimum number of years, h, between breaks to 8.¹² We next employ an algorithm that sequentially tests for the presence of up to m breaks in the GDP growth series. The first step is to test for the null hypothesis of zero structural breaks against the alternative of one or more structural breaks (up to the preset maximum m). The location of potential breaks is decided by minimizing the sum of squared residuals between the actual data and the average growth rate before and after the break. Critical values are generated through Monte Carlo simulations using bootstrapped residuals that take into account the properties of the actual time series (i.e., sample size and variance). We choose these critical values so as to reject true nulls at the 10 percent level.

We define growth spells as periods of time

• beginning with a statistical upbreak followed by a period of at least g percent average growth; and

• ending either with a statistical downbreak followed by a period of less than g percent average growth (“complete” growth spells) or with the end of the sample (“incomplete” growth spells).

Since growth in our definition means per capita income growth, growth of as low as 2 percent might be considered a reasonable threshold; we thus focus on the g = 2 case.¹³

1.1 Specification of the Hazard Model

The analysis of growth spells and their determinants consists of estimating proportional hazard models to relate the probability that a growth spell will end to a set of time-varying covariates. Let $T$ denote survival time (duration), a random variable with a cumulative distribution function $F(t) = \Pr (T \le t)$. The survival function $S(t)$ is the complement of the distribution function $S(t) = \Pr (T > t) = 1 - F(t)$. Methods for analyzing survival data often focus on modeling the hazard rate which assesses the instantaneous risk of demise at time t, conditional on survival up to that time

$$ h(t) = \mathop {\lim }\limits_{\Delta t \to 0} \frac{\Pr [(t \le T \le t + \Delta t)|T \ge t]}{\Delta t} = \frac{f(t)}{S(t)}. $$

(1)

Duration is modeled by parametrizing the hazard rate and estimating the relevant parameters. Assuming a proportional hazards model and that the relationship between the hazard and covariate vector $X_{i}$ is log linear, the “benchmark hazard” takes a particular functional form for subject $i$ with covariate vector as

$$ \lambda (t|X_{i} ) = \lambda_{0} (t)\exp (X_{i} \beta ) $$

(2)

where $\lambda_{0} (t)$ is the benchmark hazard at time $t$ and $\beta$ is a vector of unknown parameters.

One potential problem in estimating (2) arises from the feedback of spell duration to the covariates. For example, a covariate may depend on whether or not a spell has ended or is still ongoing. It is possible to estimate (2) consistently if we assume that the hazard at time t conditional on the covariates at time t depends only on the lagged realizations of those covariates (Wooldridge, Woolridge 2002). Thus, we define the hazard at time t (now in discrete time) as the probability that the spell will end during period t + 1 conditional on time-varying covariates up to and including time t, including some variables measured at the beginning of the spell. This does not rule out all sources of endogeneity (e.g., through expectations that the end of a spell is imminent), but it should prevent bias through standard feedback from the end of a spell to potential determinants.

We assume that $\lambda_{0} (t)$ follows a Weibull distribution, i.e., $\lambda_{0} (t) = pt^{p - 1}$. The parameter p, which is estimated, determines whether duration dependence is positive (p > 1) or negative.¹⁴