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Why progressive redistribution can hurt the poor
Journal of Public Economics 92 (2008) 738 – 747 www.elsevier.com/locate/econbase
Why progressive redistribution can hurt the poor ☆ Reto Foellmi, Manuel Oechslin ⁎ University of Zurich, Institute for Empirical Research in Economics, Bluemlisalpstrasse 10, CH-8006 Zürich, Switzerland Received 31 October 2006; received in revised form 1 August 2007; accepted 26 August 2007 Available online 31 August 2007
Abstract Recent macroeconomic research discusses credit market imperfections as a key channel through which inequality retards growth: With convex technologies, progressive transfers increase aggregate output because marginal returns become more equalized across investment opportunities. We argue that this reasoning may not hold in general equilibrium. Since the investment functions are concave in wealth, reducing inequality increases capital demand and the interest rate. Hence, through the impact on capital costs, shifting wealth from the rich to the middle class depletes the poorest investors' access to credit. But because the poor face the highest marginal returns, the net effect on output may be negative. We find, however, that redistributing towards the bottom-end of the distribution has a clear positive impact. Finally, we discuss the implications of our theoretical findings for future empirical research. © 2007 Elsevier B.V. All rights reserved. JEL classification: O11; D31; H20 Keywords: Capital market imperfections; Inequality; Growth; Efficiency
1. Introduction Recent macroeconomic research has brought up credit market imperfections as a key channel through which inequality affects aggregate output and growth. With an imperfect credit market, access to credit depends on individual wealth so that marginal returns are not necessarily equalized across investment opportunities. Based on this reasoning it has been prominently argued in the literature that – if technologies are convex – lower inequality should be associated with better economic performance because investment returns are less heterogeneous.1 This paper, however, shows that the above intuition does in general not hold true once the credit market is not completely “turned off” (as in, e.g., Bénabou, 1996) but only imperfect. Specifically, we demonstrate that even with a globally convex technology and limited borrowing a reduction in inequality may decrease aggregate output. At the heart of this result is the interest rate's endogenous response to more equality. To see this, consider the example of a progressive transfer from the top end of the distribution towards the middle class. Since the investors' borrowing ☆
This paper is previously circulated under the title “Equity and Efficiency under Imperfect Credit Markets”. ⁎ Corresponding author. Tel.: +41 44 634 36 09; fax: +41 44 634 49 07. E-mail addresses: [email protected] (R. Foellmi), [email protected] (M. Oechslin). 1 Empirical papers on the inequality-growth nexus explicitly highlighting this channel include, among others, Perotti (1996), Barro (2000), and Banerjee and Duflo (2003). 0047-2727/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jpubeco.2007.08.006
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capacity is an increasing but concave function of wealth, such redistribution leads to an outward-shift of the capital demand schedule and hence drives up the interest rate. Holding endowments constant, a higher interest rate tightens the borrowing-constraints of all investors but particularly those of the poorest agents. As a result, the latter have to downsize their investments substantially. But since the poorest investors face the highest marginal returns, this indirect negative effect may dominate the direct positive effect of the progressive transfer.2 Thus, our result comes from the distinction between partial and general equilibrium: The initially cited intuition may be misleading because it ignores a general equilibrium effect that is absent in models with a fixed interest rate or a completely closed credit market. We are, of course, not the only to point out that more inequality in the presence of credit market imperfections may have a positive effect on output. If the production function is not globally concave, the relationship between inequality and output or economic development is ambiguous as well (see, e.g., Galor and Zeira, 1993; Banerjee and Newman, 1993). However, we emphasize and illustrate by means of an example that the link may be non-monotonic even if nonconvexities are entirely absent. Moreover, the example shows that a positive association is not only a local phenomenon but may extend over a wide inequality-range. Hence, also with decreasing returns and significant borrowing-constraints, the theory offers no clear prediction whether a summary statistic for overall inequality (such as the Gini coefficient) should enter growth regression with a positive or negative sign.3 In this light, it is perhaps less of a surprise that the empirical literature in this field is anything but conclusive. While in the present model the relationship between inequality and output is ambiguous in general, we find, however, that a specific type of inequality is clearly bad for efficiency. Reducing inequality by redistributing towards the poorest part of the population unambiguously increases aggregate output. In this case, the poorest investors may clearly scale up their investments, and the negative interest rate effect falls on richer agents with lower marginal products. So what the decreasing-returns framework really points to is a negative link between bottom-end inequality and the GDP.4 Our findings have implications for future empirical research. We suggest that it is more promising to rely on quantile shares rather than on an overall inequality statistic to evaluate the heterogeneous-returns argument. In accordance with the model, such an approach allows inequality coming from different parts in the distribution to have a different impact on economic performance. The remainder of this paper is organized as follows. Section 2 presents the model. The set-up follows Bénabou (1996) with the exception of assuming an imperfect rather than a completely closed credit market. In Section 3, we derive our main results and illustrate that the relationship between inequality and output may be non-monotonic even in a simple example. The implications for future empirical research are discussed in Section 4. Section 5 concludes. 2. The basic model 2.1. Assumptions 2.1.1. Preferences, endowments, and technology We consider a closed and static economy that is populated by a continuum of individuals of measure 1. The individuals derive utility from consumption of a single output good; marginal utility is strictly positive. The agents are heterogeneous with respect to their initial capital endowment (“wealth”), ωi, i ∈ [0, 1]. Initial capital is distributed according to the distribution function G(ω). Each individual runs a single firm which uses capital to produce the homogeneous output good. The amount of capital invested by agent i is denoted by ki. The technology, which is identical across firms, is given by y = f (k), where f (·) is increasing, strictly concave, and f (0) = 0. The price of the output good is normalized to unity. 2.1.2. The credit market Individuals may borrow and lend capital in a competitive but imperfect credit market. The credit market is competitive in the sense that the individuals take the equilibrium interest rate, ρ, as given. It is imperfect, however, 2 Note that the seminal theoretical contributions by, e.g., Aghion and Bolton (1997) and Piketty (1997) rely also on an endogenous borrowing rate. However, these papers do not explicitly address the impact of changes in inequality on contemporaneous output. 3 An additional complication is that a higher Gini coefficient does not necessarily mean higher inequality in the Lorenz sense. 4 Legros and Newman (1996) make a related point. In their model of endogenous firm formation, it is poverty (rather than inequality) that gives rise to inefficient firm organization.
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since the agents face borrowing-constraints due to the possibility of default at low cost. Following Matsuyama (2000), we assume that agent i loses only a fraction λ ∈ (0, 1] of the firm revenue, f (ki), in the event of default on the repayment obligation which is given by the amount of credit times ρ.5 The parameter λ can be interpreted as the degree of legal protection of creditors. A low λ means poor creditor protection since a borrower may default on the loan without incurring a substantial cost, and vice versa. We further assume that each borrower i defaults whenever it is in his interest do so. Taking these incentives into account, a lender will give credit only up to λf (ki) / ρ so that the borrower just pays back; awarding a larger amount of credit would induce the latter to break the contract and hence leave the lender without any income out of this credit relationship. Note that default does not occur in equilibrium. Borrowing is limited because it is possible to default. 2.2. Investment decision Agent i chooses ki to maximize income, f (ki) − ρ (ki − ωi). Thereby, he is limited by the borrowing-constraint qðki xi Þ V kf ðki Þ;
ð1Þ
stating that the repayment obligation cannot exceed the cost of default. Note that for each endowment level ωi there exists a unique level of maximal investment k¯(ωi), implicitly defined by Eq. (1) when holding with equality. Since initial capital is the only source of heterogeneity across individuals, we will drop the index for individuals in what follows. Lemma 1. Let ρ / λ N f ′(∞) ≡ limx→∞ f ′(x). Then, the maximum firm size k¯(ω) is strictly increasing and strictly concave in the capital endowment, ω. Proof. Equation ρ(k − ω) = λf (k) defines a unique k¯(ω) with ρ N λf ′(k¯(ω). Implicit differentiation gives dk¯ / dω = ρ/ ( ρ − λf ′(k¯) N 0, dk¯ / dω is decreasing since f ′(k¯) is falling. □ The maximum firm size rises in ω for two reasons. To see this, we write the derivative of k¯ with respect to initial capital as d¯ k kf Vð¯ kÞ ¼1þ : dx q kf Vð¯k Þ
ð2Þ
The first term on the right-hand side captures simply that – for a given amount of credit – the feasible level of investment increases one-to-one in the entrepreneur's capital endowment. The second term mirrors the higher borrowing capacity of richer investors. Intuitively, since punishment is a fraction of total output (which is produced from borrowed funds and own capital), richer individuals can offer more “collateral.” But since the technology exhibits decreasing returns, the positive impact of an additional endowment unit on the entrepreneur's borrowing capacity falls in ω. Consider now the individuals' decision problem. We refer to e k as the investment that equates the marginal product of capital and the interest rate: e ¼ q: f VðkÞ
ð3Þ
Obviously, an agent with endowment ω ≥ e k invests e k capital units in his own firm and lends the rest, ω − e k , on the e credit market. Otherwise, if ω b k , the agent borrows as much as he can in order to close the gap between e k and ω. Denote by ω e the level of ω allowing to invest exactly e k capital units and thus separating credit-constrained entrepreneurs from entrepreneurs being able to implement the unconstrained optimum.6 Inserting Eq. (3) into the borrowing-constraint (1) and rearranging terms yield ek kf ðk eÞ=f Vðk eÞ if k b aðk eÞ ð4Þ x e¼ eÞ ; 0 if k z aðk
5 In a dynamic setting, as suggested by Genicot and Ray (2006), one could think of this cost as the loss associated with the exclusion from the credit market in the future. 6 Here, an entrepreneur is said to be credit-constrained if and only if the amount he would optimally like to raise exceeds his credit limit.
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where α(ke) ≡ e k f ′(ke) / f (ke) b 1. Eq. (4) states that, given ρ and G(ω), the mass of credit-rationed agents decreases in creditor protection, λ, and goes to zero as λ approaches α(ke), the output elasticity with respect to capital. The discussion so far suggests that, for a given interest rate, the optimal incentive-compatible firm size, k (ω), is given by ¯ k ðxÞ if x bx e kðxÞ ¼ : ð5Þ ke if x zx e According to Lemma 1, k(ω) increases in ω if ω b ω e and stays constant thereafter. 2.3. Aggregate equilibrium In equilibrium, the interest rate has to equate aggregate (gross) capital supply, KS, and aggregate Rl R l (gross) capital S demand. K is exogenously given and equals Ku 0 xdGðxÞ. Aggregate capital demand, K D u 0 kðxÞdGðxÞ, is obtained by adding up the individual investments, k(ω). Using Eq. (5), KD reads Zl Z xe Z xe ek dGðxÞ ¼ ¯ ¯k ðxÞdGðxÞ þ ð1 GðxÞÞ k ðxÞdGðxÞ þ ð6Þ K D ðqÞ ¼ e ek : 0 0 x e We now establish that the equilibrium borrowing rate is uniquely determined. Proposition 1. There is a unique credit market equilibrium with ρ⁎ N λf ′ (∞). Proof. Note first that limρ →f ′(∞) KD = ∞ because limρ →f ′(∞) k¯(ω) = limρ →f ′ (∞) e k = ∞. Note further that both ωe and e k D go to 0 as ρ approaches f ′(0). Hence, limρ Yf ′(0) K = 0. To determine the slope of the KD -schedule we have to calculate dk¯(ω) / dρ. Implicit differentiation of ρ(k − ω) = λf (k) gives dk¯(ω) / dρ = −(k¯ − ω) / ( ρ − λf ′ (k¯ )) b 0. Combining this with dke / dρ b 0 (from Eq. (3)) we have dKD / dρ b 0. Thus, since KS is perfectly inelastic, there must be a unique equilibrium interest rate ρ⁎ N f ′(∞) N λf ′(∞). □ In view of Eq. (6), it is obvious that the equilibrium interest rate depends on the distribution of initial capital endowments, G(ω). 3. Redistribution and efficiency This section analyzes the impact of inequality on aggregate output. We start by asking under what condition the firstbest outcome is reached even with limited borrowing and an uneven distribution. 3.1. The first-best output Suppose for the moment that there is no heterogeneity in initial endowments, i.e., suppose that ωi = K. Then, we must have ki = K, and the equilibrium interest rate adjusts to f′(K) so that it is indeed optimal to run firms of size K. Rl Finally, aggregate output, Y u 0 f ðkðxÞÞdGðxÞ, is given by Y ¼ f ðKÞ: Note that Y takes its first-best value because no agent is credit-constrained in equilibrium and the marginal productivity of capital is equalized across firms. Perfect equality, however, is not a necessary condition for aggregate output to be at its maximum. Y may achieve the first-best level even with inequality and imperfect enforcement of credit contracts if either the degree of creditor protection is not “too low” or the distribution of initial endowments is not “too unequal.” To see this, assume that there are no credit-rationed individuals so that ki = e k = K and hence ρ = f ′(K). This situation will be the equilibrium allocation
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if the poorest agent's endowment, denote it by ω e . Formally, from Eq. (4), we have Y = f (K ) if _ ≥ 0, is not lower than ω the condition k 1 x KVP aðKÞ
ð7Þ
holds. Condition (7) is satisfied independently of the level of ω _ if λ ≥ α(K), i.e., in case of relatively strong creditor protection; otherwise, if λ b α(K), ω must be at least as high as ω e = (1 − λ / α(K)) K b K. Put differently, the distribution _ becomes relevant only if the credit market imperfection is sufficiently strong. 3.2. Redistribution and aggregate output Suppose now that condition (7) is violated. Specifically, assume that a positive mass of individuals own less than (1 − λ / α(K)) K capital units. Then, there must be credit-constrained individuals in equilibrium and, consequently, aggregate output is lower than its first-best level. How does aggregate output react to more inequality in such a situation? Consider a redistributive program “taxing” a positive mass of poorer individuals and distributing the proceeds among a set of richer individuals. Assume further that the poorer (i.e., “taxed”) individuals are credit-constrained while the richer (i.e., “subsidized”) individuals may or may not be. Proposition 2. Redistribution from the poor to the rich as defined above decreases the equilibrium interest rate, ρ. Proof. From Lemma 1 and Eq. (5) we know that k(ω) is strictly concave for ω b ω e. Hence, for a given ρ, taxing creditconstrained agents and redistributing the proceeds towards richer entrepreneurs decreases capital demand, and the claim immediately follows. □ The intuition behind Proposition 2 can be seen by looking at Eqs. (2) and (5). An additional unit of own capital increases a beneficiary's maximum amount of investment only to a low extent while a poor individual's maximum investment decreases strongly (d2k¯ / dω2 b 0). Moreover, given the interest rate, rich agents already investing e k units do not invest more in response to an increase in own resources and a higher borrowing limit. As a result, the KD-schedule shifts to the left and the borrowing rate has to fall in order to restore the equality of capital demand and capital supply. The fact that the interest rate falls in response to regressive redistribution is the reason why an unambiguous prediction with respect to output is in general not possible. The only exception is when the poorest individuals are affected. To see this, consider a regressive redistributive program involving a positive measure of the poorest agents in the economy. Specifically, assume that these agents are equally endowed with capital and that they are all taxed by the same amount. Then, according to Proposition 2, the interest rate must fall, and since dk¯ (ω) / dρ b 0 and d k˜ / dρ b 0, the individuals belonging to the remaining part of the population (i.e., the subsidized agents and those not directly affected) increase their amount of capital invested; because aggregate gross capital supply has not risen, the taxed individuals must invest less in the new equilibrium. Finally, since each of the downsized firms had (and has) a higher marginal productivity of capital than each of the other firms, aggregate output falls. For all other types of regressive redistributive programs, however, we may not reach such a clear-cut prediction. If we redistribute away from a set of credit-constrained agents not belonging to the poorest part of the population, aggregate output may well increase. Due to the lower interest rate, the poorest agents (who are not involved into the transfer by assumption) have better access to credit and will run larger firms. Put differently, redistribution from individuals with higher marginal returns to individuals with lower marginal returns does not necessarily reduce output because the lower interest rate softens the borrowing-constraint for other high-return firms.7 To summarize, Proposition 3. Let a positive measure of individuals be endowed with ω _ N 0. Taxing each of these poorest agents by an equal amount and distributing the proceeds to richer agents unambiguously reduces aggregate output, Y. Other types of regressive redistributive programs may increase Y. 7 A related argument can be found in Banerjee and Newman (1998). In their dual-economy model, the borrowers in the traditional sector – facing comparatively low returns but only weak incentives to default – induces the lenders to charge high interest rates — which tightens the borrowingconstraints in the high-return (i.e., modern) sector.
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Proof. The first part of the proposition has been proven in the text. We will prove the second part by using an example (see Subsection 3.3). □ It is worth emphasizing that Proposition 3 states a sufficient but not necessary condition for a negative relationship between relative poverty and output. The relationship remains negative in case we redistribute away from the set of the, say, γ poorest – but not necessarily equally poor – agents if the differences in endowments within this set are sufficiently small. Finally note that for λ = 0, i.e., when the credit market is completely closed, the second part of Proposition 3 no longer holds. In this limiting case, each entrepreneur simply invests his initial capital endowment, ω, R land the sizedistribution of firms coincides with the wealth distribution. Aggregate output is then given by Y ¼ 0 f ðxÞdGðxÞ which is, due to the strict concavity of f (·) and Jensen's inequality, smaller than f (K), and it is clear that each regressive transfer unambiguously lowers aggregate output further. Intuitively, the relationship between inequality and output is monotonic because the interest rate effect is absent when the credit market is inexistent. 3.3. A simple example We now demonstrate by means of an example that the relationship between inequality (in the Lorenz sense) and output may be non-monotonic even if borrowing is limited and the technology is convex. Let the production function be of the Cobb–Douglas type: f (k) = kα, with 0 b α b 1. Assume further that there are three types of individuals. A measure βP of the population is “poor”, βM individuals belong to the “middle class”, and the remaining 1 − βP − βM agents are “rich”. The individuals' endowments are given by θiK, i ∈ {P, M, R}, θP b θM b θR, P and θR = (1 − βPθP − βMθM) / (1 − βP − βM) because 3i¼1 bi hi ¼ 1. Finally, we choose θP and θM sufficiently low so that the poor and the members of the middle class are credit-constrained in equilibrium. According to Proposition 3, a reduction of θP diminishes aggregate output since this redistributive program involves the set of the poorest individuals. By contrast, decreasing θM and redistributing towards the rich does not take anything away from the poor. Hence, the impact of such a regressive redistributive program on output is a priori unclear. Fig. 1 illustrates these predictions (with βP = 0.65, βM = 0.3, θP = 0.2, and θM = 0.9 as default values). Panel a shows aggregate output, Y, as a function of θP (with θM constant). In line with Proposition 3, making the poorest even poorer clearly reduces Y. Panel b shows the impact of a change in θM (with θP constant) on aggregate output and on the Gini coefficient. Again in line with Proposition 3, we see that the equality–output relationship is no longer monotonic but hump-shaped. More equality is good at lower levels of θM but the impact becomes negative as θM increases. Eventually, if θM N 1.47, aggregate output is independent of θM because the members of the middle class are no longer creditconstrained.8 This inverted-U pattern is not an artefact of the present example but remains qualitatively unchanged with more than three groups. For instance, in a four-group economy (including a “lower” and an “upper” middle class with relative endowments θPM and θRM), there is a hump-shaped link between output and both θPM and θRM. The decreasing part, however, is more pronounced in case of θRM — which supports the basic notion that inequality coming from higher parts of the distribution is less harmful than inequality at lower parts (simulations available upon request). Fig. 1 conveys yet another interesting message. In the present example, it is even the case that output is higher when the middle-class individuals invest the same amount as the poor do (i.e., with θM = θP = 0.2) as compared to the case where they invest the same as the rich (i.e., with θM ≥ 1.47). This observation can be stated in a slightly different way. In a two-group economy (with only poor and rich agents) output would improve as the fraction of the poor population increases from 0.65 to 0.95.9 To get an intuition for this result, let us consider the simpler case of a two-group economy in some more detail. Denote the size of the single poor group by βP. Again, an increase in βP does not affect the capital endowment of a poor agent. A higher βP means that some of the rich agents get even richer and that some of them lose and end up at a 8 It is important to note that dY / dθM must turn negative as θM approaches 1.47 where θMK = ω e. In general, regressive redistribution “taxing” agents with an endowment of exactly ω e (and hence k(ω e ) = ke ) necessarily increases Y. By lowering ρ, such redistribution exerts a positive first-order effect on output since all poorer agents (with a “high” marginal product) increase their investments. The investments by the richer agents – which are also given by ke – rise as well but since the richer agents face the same marginal return as the “taxed” individuals, this reallocation effect is only of second order. 9 With θM N 1.47, our three-group economy generates the same output as a two-group economy with βP (= 0.65) poor individuals and 1 − βP rich individuals since the middle class agents invest the same as the rich. Similarly, with θM = 0.2, output in the three-group economy is at the same level as in a two-group economy with βP + βM (= 0.95) poor agents.
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Fig. 1. A three-group society.
level similar to that of a poor agent. Fig. 2 plots output against βP. As discussed above, output rises as βP increases from 0.65 to 0.95. So why does efficiency improve as βP rises and the society becomes very unequal? Notice that in order to satisfy the equilibrium condition on the credit market, bP ¯k ðhP K; qÞ þ ð1 bP Þða=qÞ1=ð1aÞ ¼ K;
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Fig. 2. A two-group society.
the interest rate must remain strictly positive. In particular, ρ cannot be lower than λK α − 1 / (1 − θP) since k¯(θPK; λK α − 1 / (1 − θP)) = K.10 But this implies that the firm size of the rich, e k = (α/ρ)1/(1−α), is finite. Consequently, the firm size of the poor, k¯(θPK; ρ), must approach K as βP tends towards unity. Put differently, the interest rate has to fall to a level that allows the poor to invest exactly the first-best amount. Hence, the social optimum is reached in the limit. 3.4. Discussion To judge the generality of the present analysis, we briefly discuss the two key conditions for our results to hold. The first important feature of the model is the shape of the investment function, k¯(ω; ρ). The investment function must be globally concave in the initial endowment (and strictly concave at lower levels), and the size of the investment must be decreasing in the interest rate. Here, these properties are implied by fairly general assumptions. Specifically, the function's attributes are due to the combination of a strictly convex technology and limited sanctions in case of default: With decreasing returns and a positive interest rate, the optimal investment size is bounded so that the investment function must eventually become concave in wealth. The strict concavity at lower levels follows from our assumptions on sanctions in case of default. Since punishment is a fraction of output, wealthier individuals can offer more “collateral” and therefore borrow more. Yet, with decreasing returns, the impact of additional wealth on the size of the “collateral” falls as the wealth level rises. Note, however, that a wider range of micro-foundations of limited borrowing leads to such properties. Consider, for instance, the case of non-enforceable effort supply. Also here a lower interest rate or a higher endowment allow for larger investments because the incentives to supply effort are stronger. But the impact of additional wealth is decreasing because the cost of effort is convex or marginal utility from consumption is falling.11 In such a framework, progressive transfers to the middle class would also increase the interest rate and hence give the poor weaker incentives to supply effort. Again, this interest rate effect may dominate the positive direct effect of lower inequality in the middle. The model's second important feature is the shape of the capital supply schedule, KS. For the interest rate to rise in response to higher capital demand, the KS-curve must have a positive slope. Essentially, this means that the borrowing rate must not be fixed by world market conditions. 4. Implications for empirical research Our model highlights that even with convex technologies and limited borrowing the relationship between inequality and the GDP is complex. Higher inequality at the expense of the poorest agents clearly reduces aggregate output while 10
For a more general production function this result holds when the marginal product of capital is sufficiently decreasing as k goes to infinity. In k f ðKÞ particular, Inada conditions are sufficient but not necessary. To see this, note that for the poor to invest K, the interest rate must equal q ¼ 1h K P which is strictly positive. 11 Such a continuous-effort model is presented in Piketty (1992), for instance.
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regressive transfers from the middle to the top of the distribution can have a positive impact. An immediate corollary is that it may be misleading to evaluate the heterogeneous-returns argument by relying on a single summary statistic for overall inequality such as the Gini index. The present analysis does, however, provide some guidance on how future empirical work could proceed to uncover the impact of inequality when borrowing is limited. In particular, it underlines that future work should account separately for inequality arising from different parts in the distribution: Holding the wealth shares of the middle quantiles constant, the model suggests a monotonically positive link between output and the share of the lowest quantile while, holding the bottom-end constant, it predicts a hump-shaped relationship between output and each of the quantile shares in the middle. Thus, the general suggestion is to control for each of these quantile shares separately and to allow for non-monotonic links between the quantile shares and economic performance in growth regressions. 12 To consider a concrete example, suppose that the wealth distributions are approximated by four quartile shares. Then, the suggestion is to choose an expression like h(θQ1, θQ2, θQ3) to control for the distribution, where θQ1, θQ2 and θQ3 denote, respectively, the wealth shares of the first, the second, and the third quartile. The fourth quartile share is implicitly determined by the first three so that a rise in θQj mirrors a progressive transfer from the top end to quartile j. The function h(·) should be parameterized to allow for a non-monotonic association in the second and the third quartile share so that, in accordance with our simulations, there may be a hump-shaped relationship between economic performance and both θQ2 and θQ3.13 It may further be interesting to allow the impact of θQj to vary with the degree of integration into the world credit market, I. With a stronger integration (i.e., with a larger I), the rise in capital costs as a response to more equality and hence higher demand should be lower. One way to capture this is to use a modified h-function, hb(θQ1, θQ2, θQ3,I), so that ∂2hb / (∂θQj ∂I) N 0, j = 1, 2, 3. 5. Summary and conclusions Many empirical studies on the determinants of growth refer to credit market imperfections as a key channel through which inequality retards economic performance: With limited access to credit and convex technologies, redistributing from rich investors to poorer agents improves economic performance because the latter face higher marginal returns. The present paper, however, argues that this reasoning misses an important point. In particular, it ignores that – except for a completely closed credit market or a perfectly elastic credit supply – progressive transfers drive up the interest rate. As a result, progressive redistribution towards the middle may have a negative overall effect: Through the impact on capital costs, such redistribution reduces access to credit of the poorest investors who face the highest marginal returns. It is only when the transfers go to the least affluent individuals that the model points to a clear positive association between a reduction in inequality and economic performance. Thus, even with limited borrowing and convex technologies, we should not expect an unambiguous association between inequality in the Lorenz sense and economic performance but only one between inequality coming from the bottom-end of the distribution and aggregate output. Our analysis has two main implications. First, concerning future research on the inequality-growth nexus, the model suggests to use quantile shares to control for the distribution. This approach accounts for the fact that higher inequality at the expense of the poorest fraction of the population may be more harmful than inequality coming from an uneven distribution between the (upper) middle class and the rich. Recently, the literature has taken some steps towards this direction. Voitchovsky (2005) finds that growth is negatively related to measures of relative poverty and positively to measures for top-end inequality.14 Although Voitchovsky does not exactly use the empirical measures proposed here, her results are consistent with the implications of our model. The second insight concerns the design of redistributive programs with the objective to dampen the adverse effects of a malfunctioning financial system. The model highlights the importance of not only including the middle class but also the least affluent individuals — in particular if the local credit market is not well integrated into the world market. 12
Note that the model does not support a monotonic relationship between economic performance and quantile share ratios (such as the third quintile to the first) which are occasionally used to capture bottom-end inequality (see, e.g., Voitchovsky, 2005). 13 Relying on the Gini coefficient (which is given by 3/4 − (3/8)θQ1 − (2/8)θQ2 − (1/8)θQ3) rather than on h(·) would clearly be too restrictive since the Gini index imposes a monotonic relationship between output and each of the first three quartile shares. 14 In addition, using a cross-section of U.S. Metropolitan Statistical Areas, Bhatta (2001) finds that the fraction of the population below the poverty line is negatively associated to future growth.
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Acknowledgments We thank Thomas Piketty, the editor, and two anonymous for valuable suggestions. We are grateful to Josef Falkinger, Rafael Lalive, Adriano Rampini, Josef Zweimüller and seminar participants at MIT, the DEGIT conference, and the EEA Meeting in Vienna for helpful comments. Reto Foellmi acknowledges support from the SNF Research Grant No. PA001-105068. References Aghion, Philippe, Bolton, Patrick, 1997. A theory of trickle-down growth and development. Review of Economic Studies 64 (2), 151–172. Banerjee, Abhijit, Duflo, Esther, 2003. Inequality and growth: what can the data say? Journal of Economic Growth 8 (3), 267–299. Banerjee, Abhijit, Newman, Andrew, 1993. Occupational choice and the process of development. Journal of Political Economy 101 (2), 274–298. Banerjee, Abhijit, Newman, Andrew, 1998. Information, the dual economy, and development. Review of Economic Studies 65 (4), 631–653. Barro, Robert, 2000. Inequality and growth in a panel of countries. Journal of Economic Growth 5 (1), 5–32. Bhatta, Saurav Dev, 2001. Are inequality and poverty harmful for economic growth: evidence from metropolitan areas in the United States. Journal of Urban Affairs 23 (3–4), 335–359. Bénabou, Roland, 1996. Inequality and growth. In: Bernanke, B., Rotemberg, J. (Eds.), NBER Macroeconomics Annual 1996. MIT Press, Cambridge, MA. Galor, Oded, Zeira, Joseph, 1993. Income distribution and macroeconomics. Review of Economic Studies 60 (1), 35–52. Genicot, Garance, Ray, Debraj, 2006. Bargaining power and enforcement in credit markets. Journal of Development Economics 79 (2), 398–412. Legros, Patrick, Newman, Andrew, 1996. Wealth effects, distribution, and the theory of organization. Journal of Economic Theory 70 (2), 312–341. Matsuyama, Kiminori, 2000. Endogenous inequality. Review of Economic Studies 67 (4), 743–759. Perotti, Roberto, 1996. Growth, income distribution, and democracy: what the data say. Journal of Economic Growth 1 (2), 149–187. Piketty, Thomas, 1992. Imperfect capital markets and persistence of initial wealth inequalities. Discussion Paper, vol. TE/92/255. The Suntory Centre, LSE. Piketty, Thomas, 1997. The dynamics of the wealth distribution and the interest rate with credit rationing. Review of Economic Studies 64 (2), 173–189. Voitchovsky, Sarah, 2005. Does the profile of income inequality matter for economic growth. Journal of Economic Growth 10 (3), 273–296.
Why progressive redistribution can hurt the poor ☆ Reto Foellmi, Manuel Oechslin ⁎ University of Zurich, Institute for Empirical Research in Economics, Bluemlisalpstrasse 10, CH-8006 Zürich, Switzerland Received 31 October 2006; received in revised form 1 August 2007; accepted 26 August 2007 Available online 31 August 2007
Abstract Recent macroeconomic research discusses credit market imperfections as a key channel through which inequality retards growth: With convex technologies, progressive transfers increase aggregate output because marginal returns become more equalized across investment opportunities. We argue that this reasoning may not hold in general equilibrium. Since the investment functions are concave in wealth, reducing inequality increases capital demand and the interest rate. Hence, through the impact on capital costs, shifting wealth from the rich to the middle class depletes the poorest investors' access to credit. But because the poor face the highest marginal returns, the net effect on output may be negative. We find, however, that redistributing towards the bottom-end of the distribution has a clear positive impact. Finally, we discuss the implications of our theoretical findings for future empirical research. © 2007 Elsevier B.V. All rights reserved. JEL classification: O11; D31; H20 Keywords: Capital market imperfections; Inequality; Growth; Efficiency
1. Introduction Recent macroeconomic research has brought up credit market imperfections as a key channel through which inequality affects aggregate output and growth. With an imperfect credit market, access to credit depends on individual wealth so that marginal returns are not necessarily equalized across investment opportunities. Based on this reasoning it has been prominently argued in the literature that – if technologies are convex – lower inequality should be associated with better economic performance because investment returns are less heterogeneous.1 This paper, however, shows that the above intuition does in general not hold true once the credit market is not completely “turned off” (as in, e.g., Bénabou, 1996) but only imperfect. Specifically, we demonstrate that even with a globally convex technology and limited borrowing a reduction in inequality may decrease aggregate output. At the heart of this result is the interest rate's endogenous response to more equality. To see this, consider the example of a progressive transfer from the top end of the distribution towards the middle class. Since the investors' borrowing ☆
This paper is previously circulated under the title “Equity and Efficiency under Imperfect Credit Markets”. ⁎ Corresponding author. Tel.: +41 44 634 36 09; fax: +41 44 634 49 07. E-mail addresses: [email protected] (R. Foellmi), [email protected] (M. Oechslin). 1 Empirical papers on the inequality-growth nexus explicitly highlighting this channel include, among others, Perotti (1996), Barro (2000), and Banerjee and Duflo (2003). 0047-2727/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jpubeco.2007.08.006
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capacity is an increasing but concave function of wealth, such redistribution leads to an outward-shift of the capital demand schedule and hence drives up the interest rate. Holding endowments constant, a higher interest rate tightens the borrowing-constraints of all investors but particularly those of the poorest agents. As a result, the latter have to downsize their investments substantially. But since the poorest investors face the highest marginal returns, this indirect negative effect may dominate the direct positive effect of the progressive transfer.2 Thus, our result comes from the distinction between partial and general equilibrium: The initially cited intuition may be misleading because it ignores a general equilibrium effect that is absent in models with a fixed interest rate or a completely closed credit market. We are, of course, not the only to point out that more inequality in the presence of credit market imperfections may have a positive effect on output. If the production function is not globally concave, the relationship between inequality and output or economic development is ambiguous as well (see, e.g., Galor and Zeira, 1993; Banerjee and Newman, 1993). However, we emphasize and illustrate by means of an example that the link may be non-monotonic even if nonconvexities are entirely absent. Moreover, the example shows that a positive association is not only a local phenomenon but may extend over a wide inequality-range. Hence, also with decreasing returns and significant borrowing-constraints, the theory offers no clear prediction whether a summary statistic for overall inequality (such as the Gini coefficient) should enter growth regression with a positive or negative sign.3 In this light, it is perhaps less of a surprise that the empirical literature in this field is anything but conclusive. While in the present model the relationship between inequality and output is ambiguous in general, we find, however, that a specific type of inequality is clearly bad for efficiency. Reducing inequality by redistributing towards the poorest part of the population unambiguously increases aggregate output. In this case, the poorest investors may clearly scale up their investments, and the negative interest rate effect falls on richer agents with lower marginal products. So what the decreasing-returns framework really points to is a negative link between bottom-end inequality and the GDP.4 Our findings have implications for future empirical research. We suggest that it is more promising to rely on quantile shares rather than on an overall inequality statistic to evaluate the heterogeneous-returns argument. In accordance with the model, such an approach allows inequality coming from different parts in the distribution to have a different impact on economic performance. The remainder of this paper is organized as follows. Section 2 presents the model. The set-up follows Bénabou (1996) with the exception of assuming an imperfect rather than a completely closed credit market. In Section 3, we derive our main results and illustrate that the relationship between inequality and output may be non-monotonic even in a simple example. The implications for future empirical research are discussed in Section 4. Section 5 concludes. 2. The basic model 2.1. Assumptions 2.1.1. Preferences, endowments, and technology We consider a closed and static economy that is populated by a continuum of individuals of measure 1. The individuals derive utility from consumption of a single output good; marginal utility is strictly positive. The agents are heterogeneous with respect to their initial capital endowment (“wealth”), ωi, i ∈ [0, 1]. Initial capital is distributed according to the distribution function G(ω). Each individual runs a single firm which uses capital to produce the homogeneous output good. The amount of capital invested by agent i is denoted by ki. The technology, which is identical across firms, is given by y = f (k), where f (·) is increasing, strictly concave, and f (0) = 0. The price of the output good is normalized to unity. 2.1.2. The credit market Individuals may borrow and lend capital in a competitive but imperfect credit market. The credit market is competitive in the sense that the individuals take the equilibrium interest rate, ρ, as given. It is imperfect, however, 2 Note that the seminal theoretical contributions by, e.g., Aghion and Bolton (1997) and Piketty (1997) rely also on an endogenous borrowing rate. However, these papers do not explicitly address the impact of changes in inequality on contemporaneous output. 3 An additional complication is that a higher Gini coefficient does not necessarily mean higher inequality in the Lorenz sense. 4 Legros and Newman (1996) make a related point. In their model of endogenous firm formation, it is poverty (rather than inequality) that gives rise to inefficient firm organization.
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since the agents face borrowing-constraints due to the possibility of default at low cost. Following Matsuyama (2000), we assume that agent i loses only a fraction λ ∈ (0, 1] of the firm revenue, f (ki), in the event of default on the repayment obligation which is given by the amount of credit times ρ.5 The parameter λ can be interpreted as the degree of legal protection of creditors. A low λ means poor creditor protection since a borrower may default on the loan without incurring a substantial cost, and vice versa. We further assume that each borrower i defaults whenever it is in his interest do so. Taking these incentives into account, a lender will give credit only up to λf (ki) / ρ so that the borrower just pays back; awarding a larger amount of credit would induce the latter to break the contract and hence leave the lender without any income out of this credit relationship. Note that default does not occur in equilibrium. Borrowing is limited because it is possible to default. 2.2. Investment decision Agent i chooses ki to maximize income, f (ki) − ρ (ki − ωi). Thereby, he is limited by the borrowing-constraint qðki xi Þ V kf ðki Þ;
ð1Þ
stating that the repayment obligation cannot exceed the cost of default. Note that for each endowment level ωi there exists a unique level of maximal investment k¯(ωi), implicitly defined by Eq. (1) when holding with equality. Since initial capital is the only source of heterogeneity across individuals, we will drop the index for individuals in what follows. Lemma 1. Let ρ / λ N f ′(∞) ≡ limx→∞ f ′(x). Then, the maximum firm size k¯(ω) is strictly increasing and strictly concave in the capital endowment, ω. Proof. Equation ρ(k − ω) = λf (k) defines a unique k¯(ω) with ρ N λf ′(k¯(ω). Implicit differentiation gives dk¯ / dω = ρ/ ( ρ − λf ′(k¯) N 0, dk¯ / dω is decreasing since f ′(k¯) is falling. □ The maximum firm size rises in ω for two reasons. To see this, we write the derivative of k¯ with respect to initial capital as d¯ k kf Vð¯ kÞ ¼1þ : dx q kf Vð¯k Þ
ð2Þ
The first term on the right-hand side captures simply that – for a given amount of credit – the feasible level of investment increases one-to-one in the entrepreneur's capital endowment. The second term mirrors the higher borrowing capacity of richer investors. Intuitively, since punishment is a fraction of total output (which is produced from borrowed funds and own capital), richer individuals can offer more “collateral.” But since the technology exhibits decreasing returns, the positive impact of an additional endowment unit on the entrepreneur's borrowing capacity falls in ω. Consider now the individuals' decision problem. We refer to e k as the investment that equates the marginal product of capital and the interest rate: e ¼ q: f VðkÞ
ð3Þ
Obviously, an agent with endowment ω ≥ e k invests e k capital units in his own firm and lends the rest, ω − e k , on the e credit market. Otherwise, if ω b k , the agent borrows as much as he can in order to close the gap between e k and ω. Denote by ω e the level of ω allowing to invest exactly e k capital units and thus separating credit-constrained entrepreneurs from entrepreneurs being able to implement the unconstrained optimum.6 Inserting Eq. (3) into the borrowing-constraint (1) and rearranging terms yield ek kf ðk eÞ=f Vðk eÞ if k b aðk eÞ ð4Þ x e¼ eÞ ; 0 if k z aðk
5 In a dynamic setting, as suggested by Genicot and Ray (2006), one could think of this cost as the loss associated with the exclusion from the credit market in the future. 6 Here, an entrepreneur is said to be credit-constrained if and only if the amount he would optimally like to raise exceeds his credit limit.
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where α(ke) ≡ e k f ′(ke) / f (ke) b 1. Eq. (4) states that, given ρ and G(ω), the mass of credit-rationed agents decreases in creditor protection, λ, and goes to zero as λ approaches α(ke), the output elasticity with respect to capital. The discussion so far suggests that, for a given interest rate, the optimal incentive-compatible firm size, k (ω), is given by ¯ k ðxÞ if x bx e kðxÞ ¼ : ð5Þ ke if x zx e According to Lemma 1, k(ω) increases in ω if ω b ω e and stays constant thereafter. 2.3. Aggregate equilibrium In equilibrium, the interest rate has to equate aggregate (gross) capital supply, KS, and aggregate Rl R l (gross) capital S demand. K is exogenously given and equals Ku 0 xdGðxÞ. Aggregate capital demand, K D u 0 kðxÞdGðxÞ, is obtained by adding up the individual investments, k(ω). Using Eq. (5), KD reads Zl Z xe Z xe ek dGðxÞ ¼ ¯ ¯k ðxÞdGðxÞ þ ð1 GðxÞÞ k ðxÞdGðxÞ þ ð6Þ K D ðqÞ ¼ e ek : 0 0 x e We now establish that the equilibrium borrowing rate is uniquely determined. Proposition 1. There is a unique credit market equilibrium with ρ⁎ N λf ′ (∞). Proof. Note first that limρ →f ′(∞) KD = ∞ because limρ →f ′(∞) k¯(ω) = limρ →f ′ (∞) e k = ∞. Note further that both ωe and e k D go to 0 as ρ approaches f ′(0). Hence, limρ Yf ′(0) K = 0. To determine the slope of the KD -schedule we have to calculate dk¯(ω) / dρ. Implicit differentiation of ρ(k − ω) = λf (k) gives dk¯(ω) / dρ = −(k¯ − ω) / ( ρ − λf ′ (k¯ )) b 0. Combining this with dke / dρ b 0 (from Eq. (3)) we have dKD / dρ b 0. Thus, since KS is perfectly inelastic, there must be a unique equilibrium interest rate ρ⁎ N f ′(∞) N λf ′(∞). □ In view of Eq. (6), it is obvious that the equilibrium interest rate depends on the distribution of initial capital endowments, G(ω). 3. Redistribution and efficiency This section analyzes the impact of inequality on aggregate output. We start by asking under what condition the firstbest outcome is reached even with limited borrowing and an uneven distribution. 3.1. The first-best output Suppose for the moment that there is no heterogeneity in initial endowments, i.e., suppose that ωi = K. Then, we must have ki = K, and the equilibrium interest rate adjusts to f′(K) so that it is indeed optimal to run firms of size K. Rl Finally, aggregate output, Y u 0 f ðkðxÞÞdGðxÞ, is given by Y ¼ f ðKÞ: Note that Y takes its first-best value because no agent is credit-constrained in equilibrium and the marginal productivity of capital is equalized across firms. Perfect equality, however, is not a necessary condition for aggregate output to be at its maximum. Y may achieve the first-best level even with inequality and imperfect enforcement of credit contracts if either the degree of creditor protection is not “too low” or the distribution of initial endowments is not “too unequal.” To see this, assume that there are no credit-rationed individuals so that ki = e k = K and hence ρ = f ′(K). This situation will be the equilibrium allocation
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if the poorest agent's endowment, denote it by ω e . Formally, from Eq. (4), we have Y = f (K ) if _ ≥ 0, is not lower than ω the condition k 1 x KVP aðKÞ
ð7Þ
holds. Condition (7) is satisfied independently of the level of ω _ if λ ≥ α(K), i.e., in case of relatively strong creditor protection; otherwise, if λ b α(K), ω must be at least as high as ω e = (1 − λ / α(K)) K b K. Put differently, the distribution _ becomes relevant only if the credit market imperfection is sufficiently strong. 3.2. Redistribution and aggregate output Suppose now that condition (7) is violated. Specifically, assume that a positive mass of individuals own less than (1 − λ / α(K)) K capital units. Then, there must be credit-constrained individuals in equilibrium and, consequently, aggregate output is lower than its first-best level. How does aggregate output react to more inequality in such a situation? Consider a redistributive program “taxing” a positive mass of poorer individuals and distributing the proceeds among a set of richer individuals. Assume further that the poorer (i.e., “taxed”) individuals are credit-constrained while the richer (i.e., “subsidized”) individuals may or may not be. Proposition 2. Redistribution from the poor to the rich as defined above decreases the equilibrium interest rate, ρ. Proof. From Lemma 1 and Eq. (5) we know that k(ω) is strictly concave for ω b ω e. Hence, for a given ρ, taxing creditconstrained agents and redistributing the proceeds towards richer entrepreneurs decreases capital demand, and the claim immediately follows. □ The intuition behind Proposition 2 can be seen by looking at Eqs. (2) and (5). An additional unit of own capital increases a beneficiary's maximum amount of investment only to a low extent while a poor individual's maximum investment decreases strongly (d2k¯ / dω2 b 0). Moreover, given the interest rate, rich agents already investing e k units do not invest more in response to an increase in own resources and a higher borrowing limit. As a result, the KD-schedule shifts to the left and the borrowing rate has to fall in order to restore the equality of capital demand and capital supply. The fact that the interest rate falls in response to regressive redistribution is the reason why an unambiguous prediction with respect to output is in general not possible. The only exception is when the poorest individuals are affected. To see this, consider a regressive redistributive program involving a positive measure of the poorest agents in the economy. Specifically, assume that these agents are equally endowed with capital and that they are all taxed by the same amount. Then, according to Proposition 2, the interest rate must fall, and since dk¯ (ω) / dρ b 0 and d k˜ / dρ b 0, the individuals belonging to the remaining part of the population (i.e., the subsidized agents and those not directly affected) increase their amount of capital invested; because aggregate gross capital supply has not risen, the taxed individuals must invest less in the new equilibrium. Finally, since each of the downsized firms had (and has) a higher marginal productivity of capital than each of the other firms, aggregate output falls. For all other types of regressive redistributive programs, however, we may not reach such a clear-cut prediction. If we redistribute away from a set of credit-constrained agents not belonging to the poorest part of the population, aggregate output may well increase. Due to the lower interest rate, the poorest agents (who are not involved into the transfer by assumption) have better access to credit and will run larger firms. Put differently, redistribution from individuals with higher marginal returns to individuals with lower marginal returns does not necessarily reduce output because the lower interest rate softens the borrowing-constraint for other high-return firms.7 To summarize, Proposition 3. Let a positive measure of individuals be endowed with ω _ N 0. Taxing each of these poorest agents by an equal amount and distributing the proceeds to richer agents unambiguously reduces aggregate output, Y. Other types of regressive redistributive programs may increase Y. 7 A related argument can be found in Banerjee and Newman (1998). In their dual-economy model, the borrowers in the traditional sector – facing comparatively low returns but only weak incentives to default – induces the lenders to charge high interest rates — which tightens the borrowingconstraints in the high-return (i.e., modern) sector.
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Proof. The first part of the proposition has been proven in the text. We will prove the second part by using an example (see Subsection 3.3). □ It is worth emphasizing that Proposition 3 states a sufficient but not necessary condition for a negative relationship between relative poverty and output. The relationship remains negative in case we redistribute away from the set of the, say, γ poorest – but not necessarily equally poor – agents if the differences in endowments within this set are sufficiently small. Finally note that for λ = 0, i.e., when the credit market is completely closed, the second part of Proposition 3 no longer holds. In this limiting case, each entrepreneur simply invests his initial capital endowment, ω, R land the sizedistribution of firms coincides with the wealth distribution. Aggregate output is then given by Y ¼ 0 f ðxÞdGðxÞ which is, due to the strict concavity of f (·) and Jensen's inequality, smaller than f (K), and it is clear that each regressive transfer unambiguously lowers aggregate output further. Intuitively, the relationship between inequality and output is monotonic because the interest rate effect is absent when the credit market is inexistent. 3.3. A simple example We now demonstrate by means of an example that the relationship between inequality (in the Lorenz sense) and output may be non-monotonic even if borrowing is limited and the technology is convex. Let the production function be of the Cobb–Douglas type: f (k) = kα, with 0 b α b 1. Assume further that there are three types of individuals. A measure βP of the population is “poor”, βM individuals belong to the “middle class”, and the remaining 1 − βP − βM agents are “rich”. The individuals' endowments are given by θiK, i ∈ {P, M, R}, θP b θM b θR, P and θR = (1 − βPθP − βMθM) / (1 − βP − βM) because 3i¼1 bi hi ¼ 1. Finally, we choose θP and θM sufficiently low so that the poor and the members of the middle class are credit-constrained in equilibrium. According to Proposition 3, a reduction of θP diminishes aggregate output since this redistributive program involves the set of the poorest individuals. By contrast, decreasing θM and redistributing towards the rich does not take anything away from the poor. Hence, the impact of such a regressive redistributive program on output is a priori unclear. Fig. 1 illustrates these predictions (with βP = 0.65, βM = 0.3, θP = 0.2, and θM = 0.9 as default values). Panel a shows aggregate output, Y, as a function of θP (with θM constant). In line with Proposition 3, making the poorest even poorer clearly reduces Y. Panel b shows the impact of a change in θM (with θP constant) on aggregate output and on the Gini coefficient. Again in line with Proposition 3, we see that the equality–output relationship is no longer monotonic but hump-shaped. More equality is good at lower levels of θM but the impact becomes negative as θM increases. Eventually, if θM N 1.47, aggregate output is independent of θM because the members of the middle class are no longer creditconstrained.8 This inverted-U pattern is not an artefact of the present example but remains qualitatively unchanged with more than three groups. For instance, in a four-group economy (including a “lower” and an “upper” middle class with relative endowments θPM and θRM), there is a hump-shaped link between output and both θPM and θRM. The decreasing part, however, is more pronounced in case of θRM — which supports the basic notion that inequality coming from higher parts of the distribution is less harmful than inequality at lower parts (simulations available upon request). Fig. 1 conveys yet another interesting message. In the present example, it is even the case that output is higher when the middle-class individuals invest the same amount as the poor do (i.e., with θM = θP = 0.2) as compared to the case where they invest the same as the rich (i.e., with θM ≥ 1.47). This observation can be stated in a slightly different way. In a two-group economy (with only poor and rich agents) output would improve as the fraction of the poor population increases from 0.65 to 0.95.9 To get an intuition for this result, let us consider the simpler case of a two-group economy in some more detail. Denote the size of the single poor group by βP. Again, an increase in βP does not affect the capital endowment of a poor agent. A higher βP means that some of the rich agents get even richer and that some of them lose and end up at a 8 It is important to note that dY / dθM must turn negative as θM approaches 1.47 where θMK = ω e. In general, regressive redistribution “taxing” agents with an endowment of exactly ω e (and hence k(ω e ) = ke ) necessarily increases Y. By lowering ρ, such redistribution exerts a positive first-order effect on output since all poorer agents (with a “high” marginal product) increase their investments. The investments by the richer agents – which are also given by ke – rise as well but since the richer agents face the same marginal return as the “taxed” individuals, this reallocation effect is only of second order. 9 With θM N 1.47, our three-group economy generates the same output as a two-group economy with βP (= 0.65) poor individuals and 1 − βP rich individuals since the middle class agents invest the same as the rich. Similarly, with θM = 0.2, output in the three-group economy is at the same level as in a two-group economy with βP + βM (= 0.95) poor agents.
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Fig. 1. A three-group society.
level similar to that of a poor agent. Fig. 2 plots output against βP. As discussed above, output rises as βP increases from 0.65 to 0.95. So why does efficiency improve as βP rises and the society becomes very unequal? Notice that in order to satisfy the equilibrium condition on the credit market, bP ¯k ðhP K; qÞ þ ð1 bP Þða=qÞ1=ð1aÞ ¼ K;
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Fig. 2. A two-group society.
the interest rate must remain strictly positive. In particular, ρ cannot be lower than λK α − 1 / (1 − θP) since k¯(θPK; λK α − 1 / (1 − θP)) = K.10 But this implies that the firm size of the rich, e k = (α/ρ)1/(1−α), is finite. Consequently, the firm size of the poor, k¯(θPK; ρ), must approach K as βP tends towards unity. Put differently, the interest rate has to fall to a level that allows the poor to invest exactly the first-best amount. Hence, the social optimum is reached in the limit. 3.4. Discussion To judge the generality of the present analysis, we briefly discuss the two key conditions for our results to hold. The first important feature of the model is the shape of the investment function, k¯(ω; ρ). The investment function must be globally concave in the initial endowment (and strictly concave at lower levels), and the size of the investment must be decreasing in the interest rate. Here, these properties are implied by fairly general assumptions. Specifically, the function's attributes are due to the combination of a strictly convex technology and limited sanctions in case of default: With decreasing returns and a positive interest rate, the optimal investment size is bounded so that the investment function must eventually become concave in wealth. The strict concavity at lower levels follows from our assumptions on sanctions in case of default. Since punishment is a fraction of output, wealthier individuals can offer more “collateral” and therefore borrow more. Yet, with decreasing returns, the impact of additional wealth on the size of the “collateral” falls as the wealth level rises. Note, however, that a wider range of micro-foundations of limited borrowing leads to such properties. Consider, for instance, the case of non-enforceable effort supply. Also here a lower interest rate or a higher endowment allow for larger investments because the incentives to supply effort are stronger. But the impact of additional wealth is decreasing because the cost of effort is convex or marginal utility from consumption is falling.11 In such a framework, progressive transfers to the middle class would also increase the interest rate and hence give the poor weaker incentives to supply effort. Again, this interest rate effect may dominate the positive direct effect of lower inequality in the middle. The model's second important feature is the shape of the capital supply schedule, KS. For the interest rate to rise in response to higher capital demand, the KS-curve must have a positive slope. Essentially, this means that the borrowing rate must not be fixed by world market conditions. 4. Implications for empirical research Our model highlights that even with convex technologies and limited borrowing the relationship between inequality and the GDP is complex. Higher inequality at the expense of the poorest agents clearly reduces aggregate output while 10
For a more general production function this result holds when the marginal product of capital is sufficiently decreasing as k goes to infinity. In k f ðKÞ particular, Inada conditions are sufficient but not necessary. To see this, note that for the poor to invest K, the interest rate must equal q ¼ 1h K P which is strictly positive. 11 Such a continuous-effort model is presented in Piketty (1992), for instance.
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regressive transfers from the middle to the top of the distribution can have a positive impact. An immediate corollary is that it may be misleading to evaluate the heterogeneous-returns argument by relying on a single summary statistic for overall inequality such as the Gini index. The present analysis does, however, provide some guidance on how future empirical work could proceed to uncover the impact of inequality when borrowing is limited. In particular, it underlines that future work should account separately for inequality arising from different parts in the distribution: Holding the wealth shares of the middle quantiles constant, the model suggests a monotonically positive link between output and the share of the lowest quantile while, holding the bottom-end constant, it predicts a hump-shaped relationship between output and each of the quantile shares in the middle. Thus, the general suggestion is to control for each of these quantile shares separately and to allow for non-monotonic links between the quantile shares and economic performance in growth regressions. 12 To consider a concrete example, suppose that the wealth distributions are approximated by four quartile shares. Then, the suggestion is to choose an expression like h(θQ1, θQ2, θQ3) to control for the distribution, where θQ1, θQ2 and θQ3 denote, respectively, the wealth shares of the first, the second, and the third quartile. The fourth quartile share is implicitly determined by the first three so that a rise in θQj mirrors a progressive transfer from the top end to quartile j. The function h(·) should be parameterized to allow for a non-monotonic association in the second and the third quartile share so that, in accordance with our simulations, there may be a hump-shaped relationship between economic performance and both θQ2 and θQ3.13 It may further be interesting to allow the impact of θQj to vary with the degree of integration into the world credit market, I. With a stronger integration (i.e., with a larger I), the rise in capital costs as a response to more equality and hence higher demand should be lower. One way to capture this is to use a modified h-function, hb(θQ1, θQ2, θQ3,I), so that ∂2hb / (∂θQj ∂I) N 0, j = 1, 2, 3. 5. Summary and conclusions Many empirical studies on the determinants of growth refer to credit market imperfections as a key channel through which inequality retards economic performance: With limited access to credit and convex technologies, redistributing from rich investors to poorer agents improves economic performance because the latter face higher marginal returns. The present paper, however, argues that this reasoning misses an important point. In particular, it ignores that – except for a completely closed credit market or a perfectly elastic credit supply – progressive transfers drive up the interest rate. As a result, progressive redistribution towards the middle may have a negative overall effect: Through the impact on capital costs, such redistribution reduces access to credit of the poorest investors who face the highest marginal returns. It is only when the transfers go to the least affluent individuals that the model points to a clear positive association between a reduction in inequality and economic performance. Thus, even with limited borrowing and convex technologies, we should not expect an unambiguous association between inequality in the Lorenz sense and economic performance but only one between inequality coming from the bottom-end of the distribution and aggregate output. Our analysis has two main implications. First, concerning future research on the inequality-growth nexus, the model suggests to use quantile shares to control for the distribution. This approach accounts for the fact that higher inequality at the expense of the poorest fraction of the population may be more harmful than inequality coming from an uneven distribution between the (upper) middle class and the rich. Recently, the literature has taken some steps towards this direction. Voitchovsky (2005) finds that growth is negatively related to measures of relative poverty and positively to measures for top-end inequality.14 Although Voitchovsky does not exactly use the empirical measures proposed here, her results are consistent with the implications of our model. The second insight concerns the design of redistributive programs with the objective to dampen the adverse effects of a malfunctioning financial system. The model highlights the importance of not only including the middle class but also the least affluent individuals — in particular if the local credit market is not well integrated into the world market. 12
Note that the model does not support a monotonic relationship between economic performance and quantile share ratios (such as the third quintile to the first) which are occasionally used to capture bottom-end inequality (see, e.g., Voitchovsky, 2005). 13 Relying on the Gini coefficient (which is given by 3/4 − (3/8)θQ1 − (2/8)θQ2 − (1/8)θQ3) rather than on h(·) would clearly be too restrictive since the Gini index imposes a monotonic relationship between output and each of the first three quartile shares. 14 In addition, using a cross-section of U.S. Metropolitan Statistical Areas, Bhatta (2001) finds that the fraction of the population below the poverty line is negatively associated to future growth.
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Acknowledgments We thank Thomas Piketty, the editor, and two anonymous for valuable suggestions. We are grateful to Josef Falkinger, Rafael Lalive, Adriano Rampini, Josef Zweimüller and seminar participants at MIT, the DEGIT conference, and the EEA Meeting in Vienna for helpful comments. Reto Foellmi acknowledges support from the SNF Research Grant No. PA001-105068. References Aghion, Philippe, Bolton, Patrick, 1997. A theory of trickle-down growth and development. Review of Economic Studies 64 (2), 151–172. Banerjee, Abhijit, Duflo, Esther, 2003. Inequality and growth: what can the data say? Journal of Economic Growth 8 (3), 267–299. Banerjee, Abhijit, Newman, Andrew, 1993. Occupational choice and the process of development. Journal of Political Economy 101 (2), 274–298. Banerjee, Abhijit, Newman, Andrew, 1998. Information, the dual economy, and development. Review of Economic Studies 65 (4), 631–653. Barro, Robert, 2000. Inequality and growth in a panel of countries. Journal of Economic Growth 5 (1), 5–32. Bhatta, Saurav Dev, 2001. Are inequality and poverty harmful for economic growth: evidence from metropolitan areas in the United States. Journal of Urban Affairs 23 (3–4), 335–359. Bénabou, Roland, 1996. Inequality and growth. In: Bernanke, B., Rotemberg, J. (Eds.), NBER Macroeconomics Annual 1996. MIT Press, Cambridge, MA. Galor, Oded, Zeira, Joseph, 1993. Income distribution and macroeconomics. Review of Economic Studies 60 (1), 35–52. Genicot, Garance, Ray, Debraj, 2006. Bargaining power and enforcement in credit markets. Journal of Development Economics 79 (2), 398–412. Legros, Patrick, Newman, Andrew, 1996. Wealth effects, distribution, and the theory of organization. Journal of Economic Theory 70 (2), 312–341. Matsuyama, Kiminori, 2000. Endogenous inequality. Review of Economic Studies 67 (4), 743–759. Perotti, Roberto, 1996. Growth, income distribution, and democracy: what the data say. Journal of Economic Growth 1 (2), 149–187. Piketty, Thomas, 1992. Imperfect capital markets and persistence of initial wealth inequalities. Discussion Paper, vol. TE/92/255. The Suntory Centre, LSE. Piketty, Thomas, 1997. The dynamics of the wealth distribution and the interest rate with credit rationing. Review of Economic Studies 64 (2), 173–189. Voitchovsky, Sarah, 2005. Does the profile of income inequality matter for economic growth. Journal of Economic Growth 10 (3), 273–296.