WPS8005 Policy Research Working Paper 8005 Trade Policy and Redistribution When Preferences Are Non-Homothetic Quy-Toan Do Andrei A. Levchenko Development Research Group Poverty and Inequality Team March 2017 Policy Research Working Paper 8005 Abstract This paper compares redistribution through trade restric- wisdom, domestic lump-sum transfers are not necessarily tions versus domestic lump-sum transfers. When preferences superior to distortionary trade policy. The paper devel- are non-homothetic, even domestic lump-sum transfers ops this argument in the context of the food export bans affect relative prices. Thus, contrary to the conventional imposed by many developing countries in the late 2000s. This paper is a product of the Poverty and Inequality Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at abonfield@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Trade Policy and Redistribution When Preferences Are Non-Homothetic Quy-Toan Do∗ Andrei A. Levchenko† Keywords: trade restrictions, redistribution, non-homothetic preferences. JEL Codes: F13, Q17, O24 ∗ World Bank (qdo@worldbank.org). † University of Michigan, NBER, and CEPR (alev@umich.edu). 1 Introduction Should restrictive trade policy be used as a means of redistribution within a country? Conven- tional wisdom in trade policy analysis offers an unequivocal “no” answer. The reason is that trade policy distorts prices and thus generates misallocation of resources. According to this line of reasoning purely domestic redistribution policies, ideally in the form of lump-sum transfers, are superior to trade restrictions as they generate fewer or no distortions, either domestically or internationally. This paper shows that the presence of non-homothetic preferences importantly qualifies or even erases the superiority of domestic redistribution over trade restrictions. This is because under non-homothetic preferences, even lump-sum transfers between individuals/groups with different income elasticities of consumption have an impact on prices, making them more similar to the distortionary trade restrictions. We develop this argument in the context of the food export bans that were imposed by a num- ber of developing countries during the last commodity price boom. Sharma (2011) estimates that 33 countries imposed some forms of export restrictions on grains and other food commodities between 2007 and 2011. These policies met strong resistance from the international community. Then World Bank president Robert Zoellick urged countries “to remove export bans and restric- tions. These controls encourage hoarding, drive up prices, and hurt the poorest people around the world who are struggling to feed themselves.”1 A few years later, then U.S. Secretary of State Hillary Clinton issued a similar call: “[s]ome policies that countries enacted with the hope of mit- igating the crisis, such as export bans on rice, only made matters worse. (...) And that sounder approach includes (...) abstaining from export bans no matter how attractive they may appear to be, using export quotas and taxes sparingly if at all (...).”2 At first glance this setting should be a textbook application of the conventional wisdom. The trade policy implemented by these countries was an export quota, widely considered the least efficient type of trade restriction. While the redistributionary objective of these export bans was widely recognized in the global policy making circles, these countries were advised to achieve that objective by means of domestic policies. Our analysis will consider the best kind of domestic policy, namely a lump-sum transfer. In our model, there are N countries, two commodities, Food and Garments, and two types of 1 2008 High-Level Conference on World Food Security: http://go.worldbank.org/BUEP7C3NC0 2 http://www.state.gov/secretary/rm/2011/05/162795.htm 2 individuals, rich and poor. Preferences are Stone-Geary in Food and Garments, with a minimum Food consumption requirement. We analyze the case in which the poor consume only Food, and compare the effects of two policies in an individual country. The first is a binding Food export quota, corresponding to the Food export bans implemented in many developing countries. The second is a lump-sum redistribution policy from the rich to the poor. When preferences are ho- mothetic, it is immediate that a lump-sum transfer achieves the government’s redistributionary objective without distorting the world market for Food. Thus, the first-best outcome obtains. However, under non-homothetic preferences, even lump-sum redistribution affects prices. Be- cause the poor have a higher propensity to consume Food than the rich – indeed, in the case we analyze all of their extra income goes to Food consumption – a transfer of one dollar of income from the rich to the poor increases demand for Food, and therefore the Food price. Thus this purely domestic lump-sum transfer has, at least qualitatively, a similar effect on the price of Food as the export quota. In the limit, as non-homotheticity becomes extreme – that is, the income elasticity of Food consumption of the rich goes to zero while the poor continue consuming only Food – the domestic lump-sum redistribution policy converges to the export ban in its effect on the global Food price. Our paper makes contact with two literatures. The first is the older debate on how trade policy compares to other policy instruments (Haberler 1950, Hagen 1958, Bhagwati and Ramaswami 1963, Corden 1974). The main thrust of this literature is that trade restrictions are not normally the most efficient way to achieve a given policy objective. We show that this conclusion must be qualified if preferences are non-homothetic. Epifani and Gancia (2009) show that countries have an incentive to increase the size of government as doing so improves the country’s terms of trade. Because government expenditure is mainly on non-tradeables, increasing it pulls labor out of the tradeable sector, reducing export supply and thus raising the prices of the country’s exports. Ours is a qualitatively different mechanism, as it does not rely on non-tradeability of government- provided goods, or indeed on the existence of the government expenditure on goods or services. The policy we consider is a pure lump-sum transfer. The second is the line of research on trade policy in the presence of preference non-homotheticity. While a number of recent influential papers model non-homothetic preferences and the impact of trade opening on agents at the different points on the income distribution (Fajgelbaum, Grossman and Helpman 2011, Fajgelbaum and Khandelwal 2016), there is comparatively less theoretical or empirical work on trade policy in particular. Porto (2006) and and Faber (2014) analyze the dis- 3 tributional impact of individual trade agreements. Gresser (2002) documents that in the United States, tariffs are strongly anti-poor. None of these papers compare tariff reductions to alternative policies to benefit the poor. Most closely related to ours is the paper by Glazer and Ranjan (2007), who develop a theoretical framework in which an import tariff lowers the poor’s marginal utility of income and makes redistribution to them less efficient, a different interaction between trade policy and redistribution than the one we consider here. The rest of the paper is organized as follows. Section 2 presents the model and derives the main results. Section 3 concludes. 2 Baseline model 2.1 Preliminaries Consider an endowment economy with two goods – Food and Garments – and N equal-sized countries that trade with one another. Country c has an endowment profile (Φc , Γc ) of Food and Garments, respectively. Each country is populated with two types of agents: the rich and the poor, who are assumed to be in equal number. The poor have endowment (λc Φ, λc Γ) while the rich own the remaining ((1 − λc ) Φ, (1 − λc ) Γ). Agents have non-homothetic preferences and we assume they are of the Stone-Geary form: uc (f, g ) = (f − φc )αc g 1−αc , where f and g are respectively the amounts of Food and Garments consumed by an individual in country c and φc is the minimum level of Food consumption to be met before households diversify and consume other goods.3 Stone-Geary preferences imply that agents with income below a given threshold will spend it all on Food, while agents with income above will also consume Garments. Trading of Garments and Food takes place on a spot market. Denote by pc and qc the prices of Food and Garments in country c. Trade is free of physical impediments but may be restricted by policy. Without loss of generality, we henceforth consider policies in country c = 1, and restrict the analysis to the case in which only this country implements a policy. The two available policies are a lump-sum redistribution scheme and an export quota. Denote by τ1 a lump-sum transfer from the rich to the poor in country c = 1. Without loss of generality, 3 From an empirical perspective, the higher propensity of the poor to consume food is referred to as Engel’s law and has ample empirical support; see e.g. Houthakker (1957). 4 assume that transfers are made in units of Food. The stated objective of trade insulation practices is to protect domestic net Food consumers from high international Food prices. The second policy ˙ 1 in country 1. we thus consider is a Food export quota X i={rich,poor} An equilibrium of the world exchange economy is a set of consumption allocations Fci , Gi c c=1,...N and relative prices {pc , qc }c=1,...N such that (i) agents maximize their utility, and (ii) markets clear. 2.2 Laissez-faire equilibrium As there are no trade costs, the laissez-faire equilibrium prices for Food and Garments are the same for every country and denoted by (¯ ¯). The poor dedicate their entire endowment (λc Φc , λc Γc ) to p, q Food consumption, while the rich spend a fraction αc of their adjusted income on Food. Denoting µc = λc + αc (1 − λc ), we can write aggregate Food consumption in country c as ¯ ¯c = µc Φc + (1 − αc )φc + q F µc Γc . ¯ p Global market clearing conditions pin down equilibrium prices. Without policy interventions, the equilibrium price is: 1 ¯ q N c [(1 − µc ) Φc − (1 − αc )φc ] = 1 . (1) ¯ p N c µc Γc Prices are the usual preference-weighted ratio of aggregate endowments adjusted for the Stone- Geary parameters. Since we restrict the analysis to policies aimed at protecting the poor (and resulting in an ¯ q increase in Food consumption by the poor), p ¯ will be the upper-bound for international prices, ¯min q while one lower-bound, which we denote ¯min p , corresponds to the case where country c = 1 consumes food only (α1 = 1), i.e. ¯min q 0+ c>1 [(1 − µc ) Φc − (1 − αc )φc ] min = . (2) ¯ p Γ1 + c µc Γc To ensure that throughout the analysis, the property that the poor in every country only con- sume Food, while the rich diversify their consumption, it is sufficient to ensure that for every c, q ¯ ¯min q λc Φc + Γc ≤ φc and (1 − λc ) Φc + Γc ≥ φc . p ¯ ¯min p To that end, we make the following assumption, which we will henceforth refer to as the Stone- 5 Geary conditions: A1: Stone-Geary conditions. For every country c, φc < Φc (3)    φc φc  λc < min ¯ q ;1 − ¯min q . (4)  Φc + p¯ Γc Φc + p Γ  ¯min c Note that prices are functions of λc , so we need to verify that the set of parameters λc that satisfy (4) is not empty. To see this, note that the first argument in the bracket is positive, and (3) implies that the second also is for every value of λc . Finally, the level of Food exports by country c under the laissez-faire equilibrium is given by ¯ ¯ c = [(1 − µc )Φc − (1 − αc )φc ] − q X µc Γc . (5) ¯ p 2.3 Trade insulation ¯ 1 , where X ˙1 < X Now consider a binding export quota X ¯ 1 is the laissez-faire level of exports from country 1. In the trade insulation equilibrium, prices in country 1 will now differ from prices in p ˙ 1) p ˙ 1 (X ˙ 1) ¨(X every other country in the world. The equilibrium thus features a set of two prices q ˙1 (X˙ 1) , q ˙ 1) ¨(X , p ˙ 1) ˙ 1 (X ¨ p where ˙1 (X q ˙ 1) is the domestic relative price of Food in country c = 1 and ¨ q is the international ˙ 1 applies uniformly, country c = 1 derives relative price of Food. Assuming that export quota X p ˙ 1) ¨(X ˙ 1 units of Food at international price its income from selling X ˙ 1 at and the remainder, Φ1 − X q ˙ 1) ¨(X p ˙ 1) ˙ 1 (X domestic price q ˙1 (X˙ 1) . Aggregate consumption of Food in country c = 1 is thus ˙ 1) − p ˙ 1) ˙ 1) F ¨(X ˙ 1 ) = µ1 Φ 1 + p ˙ 1 (X ˙1 (X ˙1 (X ˙ 1 + (1 − α1 )φ1 + q X µ Γ , p ˙ 1) ˙ 1 (X p ˙ 1) 1 1 ˙ 1 (X q ˙ 1) ¨(X Other countries face international price p ˙ 1) ¨(X and consume ˙ 1) F ¨(X ˙ 1 ) = [µc Φc + (1 − αc )φc ] + q ˙ c (X µΓ. p ˙ 1) c c ¨(X 6 International prices International prices clear the international market for Food, i.e. 1 q ˙ 1) ¨(X 1 1 1 [(1 − µc )Φc − (1 − αc )φc ] = µc Γc − µ1 Γ1 + ˙1 , (1 − µ1 )Φ1 − (1 − α1 )φ1 − X N p ˙ 1) ¨(X N N N c c which can be solved to derive the expression for the relative price: q ˙ 1) ¨(X q ¯ 1 1 ¯1 − X X ˙1 = 1− 1 , (6) p ˙ 1) ¨(X ¯ p N 1 − γ1 c [(1 − µc )Φc − (1 − αc )φc ] N where γ1 measures the relative size of country 1 with respect to the rest of the world, i.e. 1 µ1 Γ1 γ1 = 1 . (7) N N c µc Γ c Similarly, we define ¯1 − X ¨(X ) = 1 θ 1 X (8) N 1 − γ1 1 N c [(1 − µc )Φc − (1 − αc )φc ] in order to express (6) as q ˙ 1) ¨(X ¯ q = ¨(X 1−θ ˙ 1) . (9) p ˙ ¨(X1 ) ¯ p ¨(.) is decreasing in X : as export restrictions are lifted, the upward pressure on Food The function θ ˙1 = X prices is released. Naturally, when X ¨(X ¯ 1 , the export restriction no longer binds and θ ¯ 1 ) = 0. Domestic prices The domestic market clearing condition on the other hand is given by q ˙ 1) ˙1 (X p ˙ 1) − p ¨(X ˙ 1) ˙ 1 (X (1 − µ1 )Φ1 − (1 − α1 )φ1 = µ1 Γ1 + 1 + µ1 ˙ 1, X (10) ˙1 (X p ˙ 1) p ˙ 1) ˙1 (X which after rearranging becomes q ˙ 1) ˙1 (X ¯ q = ˙1 (X 1+θ ˙) (11) p ˙ ˙1 (X1 ) ¯ p where ¨(X ) θ ¯1 − X ) − (X 1−θ¨(X ) µ1 X ˙1 (X ) = θ . (12) ¯1 + (1 − µ1 )Φ1 − (1 − α1 )φ1 − X 1 ¨(X ) µ1 X 1−θ ˙1 (.) is decreasing: the domestic price of Food increases as more Food Conversely, the function θ is being exported. We summarize these results in Proposition 1 below; the proof is in the appendix. 7 Proposition 1: Trade insulation equilibrium Suppose that assumption A1 holds. There exists ¯ 1 , such that for any quota X ˙m < X an export quota X ˙ m , the equilibrium prices of the ˙1 > X 1 1 economy are characterized by (9) and (11). The implication of Proposition 1 is that by imposing an export quota on Food, country 1 im- proves its terms of trade, negatively affecting its average trading partner. By lowering the domes- ˙ 1 benefits the poor in country c = 1. However, it is accom- tic price of Food, an export quota X panied by increased international prices as reflected in (9). This effect is behind the significant amount of opposition to export bans outside of the countries imposing them. We next demon- strate that in the presence of non-homothetic demand, even purely domestic lump-sum transfers have a similar effect. 2.4 Redistribution Since domestic lump-sum transfers do not impede international exchange, in the equilibrium with only domestic redistribution the prices are equalized in all countries. We denote the equilibrium ˆ(τ1 ) p world price ratio by ˆ(τ1 ) , q indexing it explicitly by the amount of redistribution. Equilibrium consumption of Food in country c = 1 is given by ˆ(τ1 ) ˆ1 (τ1 ) = µ1 Φ1 + (1 − α1 )φ1 + q F µ1 Γ1 + (1 − α1 ) τ1 . ˆ(τ1 ) p Redistributing resources from the rich to the poor induces income to be transferred from agents with a low propensity to consume Food towards agents with a high propensity to do so. Aggregate Food consumption in country c = 1 hence increases. In other countries, consumption remains equal to ˆ(τ1 ) ˆn (τ1 ) = µn Φn + (1 − αn )φn + q F µn Γ n . ˆ(τ1 ) p Equalizing aggregate consumption and aggregate endowment yields q ˆ(τ ) q ¯ 1 (1 − α1 )τ1 = 1− 1 . (13) ˆ(τ ) p ¯ p N N c [(1 − µc )Φc − (1 − αc )φc ] As above, we define ˆ(τ ) = 1 (1 − α1 )τ θ 1 , (14) N N c [(1 − µc )Φc − (1 − αc )φc ] 8 which is increasing with τ ; we can rewrite (13) as ˆ(τ1 ) q ¯ q ˆ(τ1 ) . = 1−θ (15) ˆ(τ1 ) p ¯ p We summarize the results in the Proposition below, the proof is in the appendix: Proposition 2: Domestic redistribution equilibrium When the Stone-Geary conditions hold, m , such that any redistribution τ ∈ (0, τ m ) from the rich to the poor in there exists a positive τ1 1 1 country c = 1 leads to an equilibrium relative price characterized by (15). A redistributive domestic policy has international price implications when preferences are non-homothetic. In equilibrium domestic redistribution τ1 leads to higher Food prices as the size of the transfer from the rich to the poor increases. Thus, this policy also improves country 1’s terms of trade as long as it is a net exporter of Food. 2.5 Trade insulation as redistribution To compare the effect of lump-sum transfers and export quotas on the world relative price of Food, ˙ 1 that keeps the poor in let’s consider a redistribution policy τ1 and choose a level of export quota X ˙ 1 will henceforth be said to be the pro-poor- country c at an identical welfare level. Export quota X ˙ 1 and τ1 are characterized by equal welfare equivalent of the social protection policy τ1 : the values X levels of the poor. Thus, choosing τ1 low enough so that the characterization of the equilibrium in ˙ 1 is also sufficiently large Proposition 2 holds implies that its pro-poor-equivalent export quota X for the equilibrium to be properly characterized by Proposition 1. Thus, F ˙ 1) = F ˙1 (X ˆ1 (τ1 ) implies τ1 λ1 ¨(X θ ˙1 (X ˙ 1) + θ ˙ 1) ¯˙ ˙ q 1 − γ1 (1 − α1 ) = ˙1 X + θ 1 (X1 )Γ1 , (16) λ1 µ1 1−θ ¨(X˙ 1) ¯ p ¨(.) are defined by (12) and (8), respectively. Both left-hand and right-hand sides ˙1 (.) and θ where θ of equation (16) are strictly increasing in their respective arguments, which implies a one-to-one ˙. correspondence between τ and X Next we want to compare the price implications of using export quotas in lieu of redistribution in order to assess the inefficiencies of export restrictions. Let’s consider τ a redistribution parame- ter in country c = 1 and X its pro-poor-equivalent export quota. The world Food price difference 9 between the redistribution and the export quota equilibria is: 1 ¯ − X ) − (1 − α1 )τ q ˆ(τ ) q¨(X ) q ¯ ¨ ˆ(τ ) = 1 1−γ1 (X1 − = θ(X ) − θ 1 . (17) ˆ(τ ) p p ¨(X ) ¯ p N N c µc Γ c We can thus linearize (16) and establish the following proposition: Proposition 3: Distortions from trade insulation m For large N and for any redistribution τ1 < τ1 ˙ 1 will generate increased adopted in country c = 1, its pro-poor-equivalent export restriction X Food prices such that q ˆ(τ1 ) q ˙ 1) ¨(X 1 α1 τ1 1 − ≈ 1 ≡ ∆(τ1 ). (18) p ˆ(τ1 ) p ˙ ¨(X1 ) N λ1 N c µc Γc N Export quotas distort prices, which induces the rich in country c = 1 to over-consume Food. Export quotas could thus be viewed as a poorly targeted social transfer program that leads to negative terms-of-trade effects for the rest of the world. To further inform the comparison between lump-sum redistribution and export quotas, we next show that export-quota-related distortions vanish as preference for Food among the rich goes to zero. Proposition 4: Trade insulation as redistribution Consider a sequence of economies indexed n } characterizes consumer preferences in country c = 1. Denote by {∆n (τ )} by n ≥ 1, where {α1 the associated price distortions induced by an export quota pro-poor equivalent to redistribution n = 0, then for every τ , lim τ . If limn→∞ α1 n n→∞ ∆ (τ ) = 0. As n increases, consumption patterns in country c = 1 become fully polarized: the poor con- sume Food only and the rich consume Garments only (except for the minimum required φc ), so that an export quota becomes akin to a lump-sum transfer from the rich to the poor. The substi- tution effect no longer operates: when α goes to zero, domestic demand for Food becomes price inelastic. There is no longer any distortion because the rich do not increase their consumption of Food when domestic prices drop. Put another way, the distortion created by an export quota is lower when the commodity being targeted for export quotas is an inferior good, the demand for which has higher income elasticity. For the rest of the world, we have just shown that redistribution is always preferred to the ex- port ban by country 1, as it worsens its terms of trade by less. For country 1 which of these policies is preferred is ambiguous, as it trades off the greater terms of trade improvement under the export 10 ban against the deadweight loss of a distorting policy. We can show that a small country (N → ∞) will always prefer redistribution, as the terms-of-trade effect is zero in that case. But when the country is not small, the comparison between an export quota and redistribution is ambiguous. A fuller evaluation of this tradeoff should also acknowledge that both policies likely involve sub- stantial additional frictions, such as rent-seeking to capture export quota rents (Krueger 1974), as well as inefficiencies and leakages that plague domestic redistribution schemes (see e.g. Murgai, Ravallion and van de Walle 2013). 3 Conclusion When preferences are non-homothetic, even lump-sum redistribution will affect equilibrium prices. We explore the trade consequences of this phenomenon, in the context of food export bans intro- duced by developing countries during the last commodity price boom. Export bans indeed raise the world price of food and improve the export-banning countries’ terms of trade. What has been underappreciated is that in this context, even purely domestic redistribution policies might have a qualitatively similar effect. The terms-of-trade improvement under domestic lump-sum redis- tribution is in the limit as large as under export quotas. 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Porto, Guido G, “Using survey data to assess the distributional effects of trade policy,” Journal of International Economics, 2006, 70 (1), 140–160. Sharma, Ramesh, “Food Export Restrictions: Review of the 2007-2011 Experience and Consider- ations for Disciplining Restrictive Measures,” FAO Commodity and Trade Policy Research Working Paper 32, Food and Agriculture Organization May 2011. A Appendix: Proofs of Propositions A.1 Proof of Proposition 1 We have left to find the conditions such that in every country, the poor do not consume any Gar- ments, while the rich consume both. First, note that aggregate income in country c = 1 is given by p ˙ 1) + p ˙ 1 )(Φ1 − X ˙ (X ˙1 + q ˙ 1 )X ¨(X ˙ 1 )Γ1 . ˙(X 12 p ¨(X˙ 1) Since p ˙ 1) > p ¨(X ˙ 1 ), Φ1 + ˙ (X ˙ 1 > Φ1 so that, by virtue of Stone-Geary condition (3), −1 X p ˙ 1) ˙ (X ˙ 1 . They thus consume both the income of the rich in country 1 exceeds φ1 , for any export quota X goods. For the poor in country c = 1, consumption of Food only would imply ˙ 1) ˙ 1) ¨(X ˙ 1) + p λ1 (Φ1 − X ˙(X ˙1 + q X Γ < φ1 , p ˙ 1) ˙ (X p ˙ 1) 1 ˙(X which can be rewritten as q ¯ ¨(X p ˙ 1) q ˙ 1) q ˙(X ¯ λ1 Φ1 + Γ1 + ˙1 + −1 X − Γ1 < φ1 . (19) p ¯ p ˙ 1) ˙ (X ˙ ˙(X1 ) p p ¯ ¯ ¯ 1 , the left-hand side of equation (19) converges to λ1 Φ1 + q ˙ 1 goes to X As X ¯ Γ1 < φ1 by virtue of p ¯ 1 such that the poor consume Food ˙m X only under trade insulation regime X 1 A.2 Proof of Proposition 2 As for the case of Proposition 1, we have left to show that in country c = 1, the poor only consume Food, while consumption of both Food and Garments is taking place for the rich. The Stone-Geary conditions imply that this is indeed the case in countries c > 1. For consumers in country c = 1, the conditions for the poor to consume Food only and the rich to consume both are ˆ(τ1 ) q ˆ(τ1 ) q λ1 [Φ1 + Γ1 ] + τ1 ≤ φ1 and (1 − λ1 )[Φ1 + Γ1 ] − τ1 ≥ φ1 , ˆ(τ1 ) p ˆ(τ1 ) p which is equivalent to ˆ(τ1 ) q ˆ(τ1 ) q τ1 < min φ1 − λ1 Φ1 + Γ1 ; (1 − λ1 ) Φ1 + Γ1 − φ1 . (20) ˆ(τ1 ) p ˆ(τ1 ) p Note that Stone-Geary condition (4) implies ˆ(τ1 ) q ¯ q φ1 − λ 1 Φ1 + Γ1 > φ1 − λ1 Φ1 + Γ1 >0 ˆ(τ1 ) p ¯ p 13 and q ˆ(τ1 ) ¯min q (1 − λ1 ) Φ1 + Γ1 − φ1 > (1 − λ1 ) Φ1 + Γ1 − φ1 > 0. p ˆ(τ1 ) ¯min p m > 0 such that (20) holds for every τ < τ m . Thus, there exists τ1 1 1 A.3 Proof of Proposition 3 Considering equation (17), the first-order term in 1/N can be obtained by taking the limit: ˙ 1) 1 ¯ ˙ 1 ) − (1 − α1 )τ1 −X q ˆ(τ1 ) q¨(X 1−γ1 (X1 ∆(τ1 ) = lim N · − = lim 1 (21) N →∞ ˆ(τ1 ) p p ˙ 1) ¨(X N →∞ c µc Γc N First, note that we have: • from equation (7): limN →∞ γ1 = 0 ¨(X • from expression (8): limN →∞ θ ˙ 1) = 0 • from expression (12): X¯1 − X˙1 1 X ˙1 ¯1 − X ˙1 (X lim θ ˙ 1) = = ¯ 1 + µ1 X ˙1 ¯ q µ1 p ˙ N →∞ (1 − µ1 )Φ1 − (1 − α1 )φ1 − X ¯ Γ1 + X1 which allows us to substitute using (16) to determine the limit: ˙ 1 − (1 − α1 )τ1 = ¯1 − X µ1 α1 lim X − (1 − α1 ) τ1 = τ1 . (22) N →∞ λ1 λ1 Substituting in equation (21) leads to expression (17). A.4 Proof of Proposition 4 For every n, τ1 n ∆n (τ ) = α1 1 n (23) λ1 N µ1 Γ1 + c>1 µc Γc n = 0, we have by continuity and limn→∞ α1 lim ∆n (τ ) = 0 (24) n→∞ for every τ . 14