Economic Theory (2008) 34: 359–382 DOI 10.1007/s00199-006-0171-x

R E S E A R C H A RT I C L E

Thomas Gall

Lotteries, inequality, and market imperfection: Galor and Zeira go gambling

Received: 7 August 2006 / Revised: 4 October 2006 / Published online: 15 November 2006 © Springer-Verlag 2006

Abstract This paper analyzes a simple extension to the work of Galor and Zeira (Rev Econ Stud 60:35–52, 1993). Allowing for endowment lotteries alters the dynamics of the model fundamentally: the poverty trap found in the original work vanishes for a wide class of parameters. Moreover, it turns out that in the presence of lotteries the relationship between the severity of credit market imperfections and long run aggregate income may be non-monotonic. We identify cases such that reducing the scope for moral hazard on the capital market decreases aggregate utility and may create a poverty trap and persistent income inequality in the economy. Keywords Poverty trap · Credit market imperfections · Investment indivisibility · Lotteries JEL Classification Numbers O16 · D51 · D31 1 Introduction Rigidities in credit markets are widely held to be obstacles for sustainable growth. This is a feature common to most of the recent literature inquiring into cause and effect of wealth and income inequality (this is the case in Aghion and Bolton 1997; Banerjee and Newman 1993; Galor and Zeira 1993; Piketty 1997; Matsuyama I am grateful for many helpful comments to Hans-Peter Grüner and an anonymous referee and for valuable discussion to Stefan Behringer and Petr Zemcík, ˇ and to seminar participants at Mannheim University and ENTER Jamboree 2003, Tilburg. Financial support from DFG is gratefully acknowledged. T. Gall Economic Theory II, University of Bonn, Lennestr. 37, 53113 Bonn, Germany E-mail: [email protected]

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2000 among others). These models build mainly on two ingredients to analyze the effects of income distributions: imperfect capital markets and indivisible investments with locally increasing returns to scale. It follows from the first ingredient as an immediate corollary that reducing market imperfections improves efficiency of the economic allocation. In this paper, however, we show that this need not be the case and an improvement of credit market conditions may have a non-monotonic effect on aggregate income. It is well-known that investment indivisibilities may induce even risk averse agents to demand fair lotteries, an idea that was put forward as early as in Friedman (1953). The indivisibility gives rise to a jump in lifetime income with respect to initial wealth which in turn leads to a non-concavity in lifetime utility. It is possible to render the lifetime utility function concave, however, by introducing a suitable lottery, as was pointed out by Ng (1965) in the context of indivisible consumption choice. As a consequence, agents in the neighborhood of the non-concavity, although being averse to risk, prefer participating in the lottery to their certain income. Indeed, substantial amounts of income are invested in gambling activities in countries all over the world (see Garrett 2001). Moreover, it appears that lottery consumption is prevalent especially among the poor (see for instance the studies on state lotteries byWorthington 2001; Scott and Garen 1993). Besley et al. (1993) find evidence for so called random rotating saving and credit associations (random ROSCAs).1 Lotteries may also be interpreted as a subtle shifting of individual behavior towards riskier activities, the lack or incompleteness of available insurance for agents operating in informal markets, or the presence of rent-seeking contests. This paper introduces endowment lotteries into the framework of Galor and Zeira (1993). Credit market imperfections are captured by the spread between debt and deposit interest rates reflecting the efficiency of a lender’s monitoring. This may be interpreted as the degree of transparency of the market for loans, the scope for moral hazard in the economy or the severity of markets imperfections. As the scope for moral hazard on the borrowers’ side rises, so does the spread between the lending and borrowing interest rate increasing the opportunity cost of capital. Hence more agents prefer gambling to borrowing in spite of the risk. As market conditions deteriorate, lotteries crowd out loans and the credit market eventually breaks down. This may lead to a different dynamic behavior of the model economy and in particular to a different set of stable steady states. The introduction of lotteries always leads to an instantaneous Pareto improvement and we provide a condition to ensure that it is sustainable. Analyzing the steady states we find that the poverty trap emerging in the original work vanishes if the wage differential between skilled and unskilled work is sufficiently small or the rate of return of human capital acquisition sufficiently high, which depends on sufficient scope for moral hazard in the economy. In contrast to the original results, in these cases improving credit market conditions may reduce aggregate welfare and generate the poverty trap. This is because lotteries constitute an alternative means of capital allocation that may be dynamically more efficient than the credit market. Only minor improvements of 1 The random ROSCAs considered by Besley et al. (1993) are lotteries over the timing of investment rather than over endowments, since every participant gets paid out eventually. In this paper agents have an incentive to participate in random ROSCAs even when commitment is not feasible and winners are free to walk away from the scheme.

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the credit market may then reduce the attractiveness of lotteries and promote the use of the less efficient credit market. That is, such policy may indeed put in danger long term growth prospects. In an extension we note that opportunities for randomization over different income levels may consist in predatory or criminal activities, or rent seeking contests. For instance, agents may attempt to expropriate other agents, and the outcome of such a conflict has a random component, such that the probability of success depends positively on an agent’s endowments devoted to conflict. This opportunity to engage in expropriation is effectively interpretable as a lottery. Yet conflict typically wastes resources. Extending the model to allow for costly lotteries yields similar conclusions provided lotteries are not too wasteful. This paper is related to the work of Garratt and Marshall (1994) who examine a static setting and find that the optimal social contract to provide education has the form of a lottery, similar to the one emerging in this paper. However, they assume the absence of a market for credits to finance human capital acquisition. Our results show that in the Galor and Zeira (1993) framework there exist cases where allowing people to gamble leads to a complete breakdown of the credit market and that this may actually be desirable. To some extent this provides a motivation for their presumption. The development economics literature provides a similar finding in the contribution of Ghatak et al. (2001). They look at a Banerjee and Newman (1993) style model where young agents may exert effort to overcome borrowing constraints in order to earn entrepreneurial rents when old. Both the labor market determining effort choice and the capital market are imperfect in the sense that there is moral hazard and transaction cost. Increasing transaction costs may then lead to higher aggregate income in the long run. Intuitively, greater income inequality due to more severe market imperfections increases incentives for young agents to choose high effort. As there is under-investment in effort due to the labor market imperfections, greater scope for moral hazard in the credit market may partially offset the underinvestment problem and improve welfare. That is, their result follows from the theorem of the second best whereas this paper does not have multiple imperfect markets. This paper proceeds by presenting the model framework and some preliminary results in Sects. 2 and 3. Section 4 analyzes the dynamic implications of lotteries and Sect. 5 conducts a welfare analysis. An application of the model is considered in Sect. 6, while Sect. 7 concludes. 2 The model We begin by briefly restating the underlying model of Galor and Zeira (1993). Agents live for two periods in overlapping generations. When young, an individual may acquire human capital at a cost h or work unskilled for a low wage wn . When old, an agent either works skilled earning a high wage ws > wn if he invested in human capital, or he works unskilled. Agents have ‘warm-glow’ preferences that give rise to the utility function u = α ln c + (1 − α) ln b, where c denotes second period consumption and b the amount bequeathed to their offspring. Let x denote the initial endowment of an agent at the beginning of his life.

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In the beginning of the economy these endowments are continuously distributed on [0, L] with L > h. Agents can deposit or borrow in order to finance human capital acquisition on a credit market. The deposit world market interest rate 1 + r is determined exogenously as agents live in a small open economy.2 There exists moral hazard on the side of the borrowers, namely the possibility to default on purpose. Lenders may monitor borrowers at cost z, but borrowers may choose to default deliberately even if monitored—although facing a cost βz, β > 1. This can be interpreted as a lack of contract enforceability due to the institutional, in particular legal, framework. There exists an incentive compatible debt contract making borrowers indifferent between defaulting and paying back the debt. It requires the borrowing interest rate to exceed the deposit interest rate, that is 1 + i > 1 + r . In the capital market β equilibrium (1 + i) = (1 + r ) β−1 . This creates a rigidity in the credit market. We define the interest rate spread as γ ≡

β . β −1

It is natural to interpret γ as the scope for moral hazard in the economy. As β approaches 1 and the cost of purposeful defaulting decreases, γ increases and as γ approaches 1 the moral hazard problem disappears. Utility maximization of agents shows that individuals separate into three categories. The first are agents who prefer to work unskilled both periods thereby earning (1 + r )x + (2 + r )wn . Agents are born with x and earn wn in the first period of their life and lend on the capital market. In the second period they earn wn , consume and bequeath. There exists an endowment f such that lifetime income from unskilled working coincides with the lifetime income from borrowing in the first period and working skilled in the second. Equating lifetime incomes yields f =

1 [(2 + r )wn + (1 + i)h − ws ] . i −r

Thus all individuals with x < f prefer unskilled working. The second category of individuals consists of agents with f ≤ x < h who borrow and work skilled earning (1 + i)(x − h) + ws . Finally, all agents with endowments x ≥ h need not borrow to acquire human capital, work skilled, and have lifetime income (1 +r )(x − h) + ws . This fully characterizes agents’ equilibrium behavior: in any given period t every old individual bequeathes a fixed share (1−α) of his lifetime income to his offspring because of ‘warm-glow’ preferences. These bequests become the next generation’s endowments xt+1 . Then there may arise an interesting case permitting Fig. 1. An individual’s bequest function in period t depending on this period’s endowments xt has two stable steady states. One at x n and another one at x s .3 In the former, people work unskilled and choose not acquire education because of the credit market rigidity. In the latter agents work skilled, as they receive sufficient 2 This exogeneity is not crucial to our findings. If decreasing demand for loans induces a decline in the lending interest rate and there exists a costless storage technology, this would render the results of this model even more pronounced. 3 We will refer to the first one using the term unskilled worker equilibrium or low income steady state, and to the second by skilled worker equilibrium or high income steady state.

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Fig. 1 The bequest function without lotteries

bequests from their parents and do not need to borrow. A third, unstable, steady state g separates the two growth paths and is given by g=

(1 − α)[(1 + i)h − ws ] . (1 + i)(1 − α) − 1

(1)

It now appears appropriate to state the assumptions that are needed in the original work to achieve this dynamic behavior of the economy. Assumption 1 In the original work it is assumed that the following holds: (i) (2 + r )wn + (1 + r )h ≤ ws < (1 + r )γ h, (ii) (1 − α)(1 + r ) < 1 < (1 − α)(1 + i), and (iii) x s > g > x n > 0. The first part is needed to ensure that agents separate into three groups. The remaining two permit the dynamics in Fig. 1 and ensure that the lifetime income curve intersects the 45◦ line three times thus yielding three steady states x n , g, and x s . Note that (iii) implies g > f > x n and (1 − α)ws > h. Note that the case Galor and Zeira analyze is characterized by a fairly high scope for moral hazard γ and a sufficiently high bequest ratio (1 − α). Below we will be interested in the relation between individuals’ well being and the scope for moral hazard. To provide a benchmark for the case without lotteries, it suffices to examine endowment points f and g. An agent endowed with f is indifferent between borrowing and investing in human capital and unskilled working. f increases in the scope for moral hazard, that is the measure of people earning an unskilled worker’s wage increases as f shifts to the right. Also the unstable steady state g, which separates the two growth paths, increases in γ . That is, the measure of individuals choosing the growth path towards the unskilled worker equilibrium increases in the scope for moral hazard.4 4

Both statements can be verified by inspection of the first derivatives.

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Fig. 2 An interior solution tangential lottery

Therefore the fraction of agents who find it profitable to acquire human capital and work skilled and those who choose the growth path to the high income steady state decreases. The former implies that the aggregate income curve shifts downwards in a period of an increase of the degree of moral hazard. The latter implies that aggregate income from steady state endowments decreases as probability mass is shifted to the low income steady state. Thus, without lotteries the presence of moral hazard is unambiguously socially wasteful. 3 Preliminary results Let us now introduce endowment lotteries in this setting. That is, we are looking for agents’ preferred lotteries in the set of actuarially fair lotteries over initial endowments. Intuitively, an adequate lottery, i.e. a convex combination of different utility levels, should render the lifetime utility function concave as shown in Fig. 2. Geometrically, we are interested in the convex hull of the lifetime utility function and particularly in the linear part of the convex hull. This is the optimal actuarially fair lottery for all agents.5 Between the points of tangency, say between endowments x ∗ and x ∗+y ∗ , the convex combination of the respective utility levels is higher than utility from certain income. Hence, individuals with endowments between the points of tangency prefer to have a certain income of x ∗ and to invest their remaining wealth in the lottery. Individuals outside the interval [x ∗ , x ∗ + y ∗ ] prefer not to gamble. The Pareto optimal lottery is determined by the tangent to both Un (x) = ln((2 + r )wn + (1 + r )x) on 0 < x ≤ f and Us (x) = ln(ws + (1 + i)(x − h)) on f < x < h. For x ≥ h the lifetime utility function changes into ln((1 + r )(x − h) + ws ) which is flatter than Us and Un . That means the second point of tangency cannot correspond to endowments greater than h, i.e. x + y ≤ h. The second property of the tangent is that it connects the points Un (x ∗ ) and Us (x ∗ + y ∗ ). That is, it must hold that: 5

A thorough analysis of the general underlying problem including a result on Pareto optimality of the tangential lottery can be found in Marshall (1984) or in Garratt and Marshall (1994).

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Un (x ∗ ) +

∂Un (x ∗ ) ∂x ∂Un (x ∗ ) ∂x

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y ∗ = Us (x ∗ + y ∗ ) =

∂Us

(x ∗ +y ∗ ) ∂x

and

.

(2)

These are two equations in two unknowns having a unique solution. Uniqueness follows from strict concavity of the utility function and existence additionally from inequality of the income slopes (1 + r ) < (1 + i). The first case, which we will call the interior solution, is that the unique solution to (2), (x ∗ , y ∗ ), lies in the feasible set defined by:
 2+r ∗ ∗ (x , y ) ∈ (x, y) : − (3) wn < x ≤ f ; 0 ≤ x + y ≤ h . 1+r Solving the system yields the expressions for x ∗ and y ∗ , and x ∗ + y ∗ ≤ h a condition for the optimal lottery to be interior. Lemma 1 The optimal lottery is an interior solution if and only if (1 + r )h + (2 + r )wn <

1 + ln(γ ) ws . γ

It is given by 2+r ws wn − and 1+r 1+i (1 + i)h − ws + (1 − ln(γ ))γ (2 + r )wn x∗ = (1 + i) ln(γ ) y∗ = h +

with the property f < x ∗ + y ∗ ≤ h. This fully characterizes the optimal lottery and the set of agents who want to participate. It is a raffle paying out a fixed prize y ∗ and participants buy their desired winning probability. Individuals invest an amount x of their initial endowments in lottery tickets and receive in return the probability x/y of winning the prize y.6 The raffle is preferred by all agents with endowments x ∈ (x ∗ , x ∗+y ∗ ), see Fig. 2. These individuals wish to invest x − x ∗ of their endowments in lottery tickets. Since (1 +i)h > ws by Assumption 1(i), it follows that y ∗ > 2+r 1+r wn > 0. This implies that there always exists a variety of endowment levels such that agents are better off participating in the lottery, namely all x ∈ (max{x ∗ ; 0}; x ∗ + y ∗ ). Note that x ∗ may be negative implying that all the poor prefer the lottery. Additionally, it holds that x ∗ < f and x ∗ + y ∗ > f , so all lottery winners invest in human capital and become skilled workers and all losers become unskilled workers. If the unique solution of the equation system is not contained in the feasible set (3), the point of tangency to Us (x) is to the right of endowment h. In this case, which we will call the corner solution,7 it must hold that 6 This lottery is analogous to the one described in Friedman (1953), footnote 13. Note that this lottery is equivalent to one paying out x ∗ to the losers and x ∗ + y ∗ to the winners and minimum investment of x ∗ . Then all individuals with x ∈ (x ∗ , y ∗+x ∗ ) invest all their wealth in the lottery. 7 There is another corner solution at x ∗ = 0. Since this case does not alter our results at all, for notational convenience we will limit our attention without loss of generality to cases where x ∗ ≥ 0.

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x ∗ + y∗ = h and (1 + r )y ∗ + ln((1 + r )x ∗ + (2 + r )wn ) = ln(ws ) (1 + r )x ∗ + (2 + r )wn which can be rearranged to yield Lemma 2 The corner solution optimal lottery is given by   ws (1+r )x ∗ + (2+r )wn ∗ with x ∗ + y ∗ = h. y = ln (1+r )x ∗ + (2+r )wn 1+r That is, in case of a corner solution winners’ endowments after the lottery are equal to h and the market for educational loans collapses as all agents in need of capital choose to gamble. Thus lotteries may crowd out loans up to a complete breakdown of the credit market. Clearly, in this case the interval (x ∗ , x ∗ + y ∗ ) is non-empty, too. This means that there are always endowment levels for which it is profitable to participate in the lottery. Then some agents are better off gambling while the other agents behave as in the original work, so we can state the following preliminary. Lemma 3 In the setting of Galor and Zeira there always exists a fair lottery with a prize y ∗ such that a non-empty set of agents characterized by their initial wealth x ∈ (x ∗ ; x ∗ + y ∗ ) participates. This leads to a Pareto improvement from an ex ante point of view. Hence, although lotteries are available the poor still tend to prefer the certain income from unskilled working and the rich to pay the acquisition cost of human capital still and earn the skilled worker’s income with certainty. Yet some individuals with intermediate endowments prefer to gamble. Their endowments are almost or just sufficient to find investing in human capital profitable without lotteries but they face a high cost of capital due to the moral hazard. Note that this behavior appears to be consistent with the data collected by Garrett (2001). He finds in a cross section analysis that gambling expenditure as a percentage of GDP is highest for intermediate income countries. Elasticity of demand for gambling is positive and appears to increase for low and intermediate countries whereas it is negative for high income countries. Finally, the solution to (2) has a property that will be very useful in the further course of the paper: the corner solution is the limit of the interior solution as the degree of moral hazard γ increases. Lemma 4 An interior solution (x ∗ , y ∗ ) of the system of Eq. (1) converges to the corner solution as the scope for moral hazard increases. In the case of an interior solution x ∗ and y ∗ increase with moral hazard. Proof In Appendix. That is, introducing lotteries generates an alternative means of capital allocation. Human capital acquisition may now be financed by buying a lottery ticket which pays partly for the education for the winners. As noted by Garratt and Marshall (1994) this bears certain resemblance to a private and a public schooling sector. Publicly funded schools are financed by parental income or bequest tax giving each contributor a probabilistic right of admission with the success probability increasing in parental income.

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4 Lotteries and intergenerational dynamics Having shown that lotteries enable an instantaneous Pareto improvement it is of interest whether also the long run dynamics of the economy is affected. Two concepts will be useful in order to analyze this. Sustainability is used to indicate that the lottery winners’ increase in utility carries over to the next generation putting the entire dynasty on the growth path to the skilled worker equilibrium. Winning the lottery at the beginning of a period increases the winners’ endowments from xt = x to xt = x ∗+y ∗ . All agents with endowments x > g are situated on a growth path converging to x s . Hence, sustainability is defined as follows. Definition 1 Sustainability is said to hold if x ∗ + y ∗ > g. Obviously, this implies also that whenever sustainability holds, the unstable steady state g ceases to exist as people in g prefer gambling. We will be able to provide a condition to ensure that lottery winners’ bequests are sufficiently high to put their offspring on the desirable growth path. The second concept relevant for long run dynamics is the elimination of the poverty trap, that is whether x n is a steady state. To ensure that it is not, individuals with endowment x n must prefer the optimal lottery to certain income from unskilled working. This is the case if and only if x n > x ∗ motivating the following definition. Definition 2 Elimination of the low income steady state is said to hold if x ∗ < x n . Potential benefits from introducing endowment lotteries are greatest when both sustainability and elimination of the low income steady state hold. Then the presence of lotteries not only yields a Pareto improvement but also erases the long run poverty trap at endowments x n (see Fig. 1). Key to the long run dynamics is the individual bequest function. In the presence of lotteries, any agent with initial wealth x ∈ [x ∗ ; x ∗ + y ∗ ] prefers to participate in the lottery, as stated in Lemma 3. The remaining agents’ behavior is unaffected. This allows us to state the individual bequest function as follows. ⎧ ((2+r )wn + (1+r )x) for 0 ≤ x < x ∗ ⎪
⎪ ∗ ⎪ ⎨ ((2+r )wn + (1+r )x ) or for x ∗ ≤ x < x ∗ + y ∗ (ws + (x ∗ + y ∗ − h)(1+i)) xt+1 = (1 − α) ⎪ ⎪ for x ∗ + y ∗ ≤ x < h ⎪ ⎩ (ws + (x − h)(1 + i)) (ws + (x − h)(1 + r )) for h ≤ x· 4.1 Sustainability Analyzing the individuals’ bequest behavior in the presence of lotteries in the Galor and Zeira setting we are able to state our first result about the intergenerational dynamics. Proposition 1 For any set of parameters there exists some γˆ , such that for all γ ≥ γˆ sustainability holds. If γ < γˆ , (i) there exists an interval (h 1 , h 2 ] such that for the cost of acquiring human capital h ∈ (h 1 , h 2 ] sustainability holds, and (ii) there exists an interval [w1 , w2 ) such that for skilled workers’ wage ws ∈ [w1 , w2 ) sustainability holds.

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Proof Assume parameters h, ws , wn , r , α and γ such that Assumption 1 holds. Using Lemma 1 and the definition of the point g from (1), the condition x ∗ + y ∗ − g > 0 can be restated for interior solutions8 as: 

 (1−α)(1 + i) ws 1 2+r ln γ + 1 − h− + wn > 0 ln γ (1−α)(1 + i) − 1 1+i ln γ 1+r

(1−α)(1 + i) − (ln γ + 1) ws 1 2+r ⇔ h− + wn > 0 ln γ [(1−α)(1 + i) − 1] 1+i ln γ 1+r  

ln γ ws 2+r ⇔ 1− (4) h− + wn > 0. (1−α)(1 + i) − 1 1+i 1+r For x ∗ + y ∗ > g to hold it suffices that the first term in the last expression is positive. This is the case if: ln γ + 1 < (1 − α)(1 + r )· γ This shows, as the LHS decreases with γ , that for any set of parameters there exists always some γˆ sufficiently large such that the above condition holds for all ws γ ≥ γˆ .9 The constraints on γ stated in Assumption 1, in particular (1+r )h < γ 1 and (1−α)(1+r ) < γ , are obviously consistent with the sufficient condition and the implication. This in turn implies that if there is sufficient scope for moral hazard, inequality (4) holds and lottery winners’ endowments x ∗ + y ∗ > g after the lottery took place. Thus, winners are set on the growth path to x s . As stated in Lemma 4, an increase in γ drives the interior solution towards the corner solution where sustainability holds trivially. It is thus always the case that there exists some γ sufficiently great such that lottery winnings are sustainable. To prove the second part of the proposition we proceed with the analysis of 1 inequality (4). Let ln1γ < (1−α)(1+i)−1 and the sufficient condition does not hold. Then x ∗ + y ∗ > g is equivalent to:   ln γ (2 + r )γ wn > (5) − 1 ((1 + i)h − ws ) (1 − α)(1 + i) − 1 where both sides are positive. It cannot be concluded whether this condition generally holds from the Assumption 1 alone. However, it can be shown by manipulating ws ws (5) that (i) given all other parameters there exists hˆ > 1+i such that for 1+i < ˆ consistent with Assumption 1, condition (5) holds, and that (ii) given all h ≤ h, other parameters there exists 0 < wˆs < (1 + i)h such that for all (1 + i)h > ws ≥ wˆs , consistent with Assumption 1, condition (5) holds. On the other hand, if γ happens to be very close to its lower bound, then the inequality need not hold.10 That means there are parameter values such that a lottery winner’s income is not sustainable, but there exists always some γ sufficiently high such that inequality (4) holds.  8 Note that this holds by definition if there is a corner solution to the maximization problem, i.e. x ∗ + y ∗ = h. 9 Note also that then the LHS of inequality (4) increases with γ . 10 An example where γ is consistent with Assumption 1 but sufficiently small not to induce sustainability can be found in the Appendix.

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That is, in case of an interior solution the long run effect on lottery winners’ income paths is ambiguous. If the scope for moral hazard γ is sufficiently high, however, sustainability always holds, since sustainability trivially holds in case of a corner solution. In this case winners’ endowments after the lottery h are sufficient for human capital acquisition.

4.2 The low income steady state The next issue is the robustness of the unskilled worker steady state x n . Lotteries might change the behavior of the agents situated in that steady state, namely inducing them to gamble for a better life. Formal analysis shows that this indeed may be the case and the low income steady state may disappear. To facilitate the exposition denote the rate of return of human capital acquisition by R = ws / h and the wage differential between skills by W = ws /wn . Note that 1 + r < R < (1 + r )γ by Assumption 1. Proposition 2 Allowing for lotteries in the setting of Galor and Zeira the dynamics of the model has the following properties: (i) In case of an interior solution, x n ceases to be a steady state if and only if   (1 − k)W (1 + r )γ −1 < γ (ln γ − (1−k)) , R 2+r where k = (1 − α)(1 + r ). This condition tightens in γ for interior solutions. (ii ) In case of a corner solution, x n ceases to be a steady state if and only if   (1 − k)W 1+r (1 − k)W < ln + k, R 2+r 2+r where k = (1 − α)(1 + r ). Condition (ii) implies condition (i). (iii) If condition (ii) holds, then there always exists a γ sufficiently high, such that lotteries both achieve sustainability and eliminate the low income steady state. Proof Writing down Definition 2 as x n − x ∗ > 0 leaves us with:   1 ws 2+r (1 − α)(2 + r )wn − h− + (1 − ln γ ) wn > 0 1 − (1 − α)(1 + r ) ln γ 1+i 1+r   1 − ln γ (1 − α) ln γ ws − h + (2 + r )wn − > 0. ⇔ 1+i 1 − (1 − α)(1 + r ) 1+r This can be simplified and rearranged to yield the condition in number (i):   2+r ws ln γ − [1 − (1 − α)(1 + r )] −h+ wn >0 (1 + r )γ 1+r 1 − (1 − α)(1 + r )   ln γ (1 + r )γ h − ws <γ −1 . ⇔ (2 + r )wn 1 − (1 − α)(1 + r )

(6)

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By closer inspection we find that the RHS of this inequality is increasing in γ for sufficiently small γ .11 Note that for sufficiently high γ the corner solution prevails. For interior solutions, however, by Lemma 4 x ∗ strictly increases in γ . This implies that inequality (6) unambiguously tightens as γ increases for interior x ∗ . For corner solutions x ∗ and x ∗ remain constant. Turning to the case of a corner solution note that x ∗ < x n does not automatically hold. It is, however, equivalent to the following inequality ∂Un (x n ) (h − x n ) + Un (x n ) < Us (h) ∂x   (1+r )(h − x n ) ws ⇔ − ln < 0. (1+r )x n + (2+r )wn (1 + r )x n + (2 + r )wn

(7)

Note that ln(·) must be strictly positive by Assumption 1. Applying the definition (1−α)(2+r ) of x n = 1−(1−α)(1+r ) wn to (7) and rearranging we obtain   [1 − (1−α)(1+r )]ws [1 − (1−α)(1+r )](1+r )h − (1−α)(1+r ) < ln . (8) (2+r )wn (2+r )wn Let k = (1−α)(1+r ), then (8) becomes

  (1 − k)W 1+r (1 − k)W − k < ln . R 2+r 2+r

This allows us to state condition (ii) in the proposition. If this inequality holds the poverty trap vanishes in case of a corner solution. By Lemma 4 we know that x ∗ strictly increases in γ for interior solutions. This means that x ∗ is maximal in case of the corner solution. Moreover, x n does not depend on γ which implies that if x ∗ ≤ x n for the corner solution which is equivalent to the last inequality, this must hold for interior solutions as well. Hence inequality (8) implies inequality (6). To show the last part of Proposition 2, note first that the corner solution condition (ii) implies the interior one (i) and, by Lemma 4, the interior solution converges to the corner solution as γ increases. Together this means (ii) implies that x n ceases to be a steady state for all levels of moral hazard. Now it is possible to apply Proposition 1.  In essence, Proposition 2 states that the low income steady state vanishes if the rate of return of human capital acquisition is sufficiently high or the wage differential sufficiently low. Note that condition (i) becomes less likely to hold as γ increases. This means there now appears a trade-off between sustainability and the elimination of x n as a steady state. The critical condition is number (ii) stating a condition such that the corner solution lottery induces agents endowed with x n to gamble. The following corollary provides sufficient conditions. Corollary 1 Sufficient conditions for the elimination of the low income steady state are (i) W is sufficiently close to 1, or (ii) both R and γ are sufficiently great. 11 In fact, one can easily show by differentiating the RHS of (6) that it has a maximum for ‘small’ γ which may or may not be smaller than the LHS of (6). As γ goes out of bounds the RHS approaches minus infinity.

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The first part is implied by the properties of the ln function and k < 1 and R > 1 + r . The second part follows from the fact that ln(·) > 0 and R < (1 + r )γ . The corollary states that a corner solution lottery eliminates the low income steady state if the wage differential between skilled and unskilled workers is sufficiently small or the rate of return on human capital acquisition is sufficiently great. Since the rate of return of human capital investment is bounded by the market interest rate for loans this is a condition on the severity of market imperfections as well. Intuitively, this requires the cost of human capital acquisition to be small compared to wages so that x n is close to the kink at f .12 Second, a high rate of return implies a large increase in income for lottery winners, thus providing strong incentives to gamble. This means condition (ii) in Proposition 2 can be expected to hold in economies where rents obtainable by wealthy agents are large and credit market imperfections are correspondingly severe.

5 Welfare analysis Given the results in the last section the implications of the scope for moral hazard for long run dynamics and aggregate income appear ambiguous. This section provides a sufficient condition for aggregate utility to increase in finite time when market imperfections deteriorate discretely. It suffices to focus on individuals who choose to participate in the lottery, since utility of those who do not is unaffected. Call individuals who prefer to participate in the lottery given some scope for moral hazard γ gamblers. To make the analysis meaningful, we assume that these individuals have positive measure at the beginning of our analysis. At the beginning of each period the gamblers invest all their wealth surpassing x ∗ in lottery tickets, so ∗ ∗ ∗ that the individual winning probability is given by x−x y ∗ . Winners receive x + y , ∗ whereas losers keep x . Losers prefer to work unskilled, whereas winners invest in human capital and work skilled by Lemma 1. This means the expected lifetime income I G (x) of a gambler with endowment x is E[I G (x)] =

x − x∗ ((1 + i)(x ∗ + y ∗ − h) + ws ) y∗   x − x∗ + 1− ((1+r )x ∗ + (2+r )wn ). y∗

Applying Jensen’s inequality a useful implication follows immediately. Lemma 5 The individual expected income allowing for lotteries is strictly greater than the certain income without lotteries for almost all endowment levels among gamblers. Proof In Appendix. Lemma 5 holds for any distribution of endowments and thus for any period t. This implies that if lotteries are allowed for in any given period, within that period probability mass is shifted to the right on the endowment line. If sustainability For instance, using (1 − α)ws > h and an adequate first order Taylor approximation one can show that any 1 < W ≤ 9 implies condition (ii) independently of all other parameters. 12

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Fig. 3 Increasing the scope for moral hazard

holds, this carries over to the following periods. As individual expected lifetime income is greater with lotteries, so is aggregate lifetime income because the lotteries are actuarially fair and there is no aggregate uncertainty. By Proposition 1 sustainability can always be obtained by a sufficiently large increase in the scope for moral hazard. How is the individuals’ well-being be affected by this? Note first that all individuals with endowments x ∈ (x ∗ , h) obviously face a utility decrease as γ increases. By the increase, some agents are made sufficiently worse off in certain income, so that they prefer a convex combination of utilities, which they did not before the increase. Let there be an exogenous deterioration of market conditions in the economy in some period t, that can be captured by a change in the scope for moral hazard γ to γˆ > γ . As shown in Fig. 3, lotteries crowd out borrowing as the scope for moral hazard rises. An increase of γ to γˆ shifts f to fˆ,13 so that the optimal lottery before the increase, given by the flatter dotted line, translates into the steeper dotted line, characterized by xˆ ∗ and yˆ ∗ . The interval of endowments such that gambling is preferred extends and the interval of endowments such that borrowing is preferred shrinks. Because of ‘warm glow’-preferences an increase in utility of the following generations does not enter the present generation’s utility function. Hence, we have to focus on future period aggregate utility. Suppose the increase in the scope for moral hazard is such that at γˆ sustainability just holds, i.e. xˆ ∗ + yˆ ∗ = gˆ + ε, ε > 0. Denote the gamblers’ initial conditional distribution of initial endowments in period t by Ft (x) := Ft (xt | xt∗ ≤ xt ≤ xˆt∗ + yˆt∗ ). 13

All endogenous variables are denoted by hats for the high moral hazard case.

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Let Ft (x) be integrable. Aggregate utility in any period t + k is given by x s Ut+k =

EU (I (x))dFt+k (x) x∗

where EU (I (x)) denotes the expected utility of an agent depending on endowments and Ft+k (x) the period-t + k-distribution of endowments of period-t gamblers. Although an increase of γ strictly lowers utility in the same period and may lower aggregate income, in the long run, however, in particular in the steady states, aggregate utility may be higher if there is more scope of moral hazard. We are able to state the following. Proposition 3 Assume given γ there is an interior solution lottery, such that sustainability does not hold, i.e. x ∗ + y ∗ < f . Then increasing γ to γˆ , with γˆ sufficiently great such that sustainability holds, raises both aggregate utility and income in the economy after finitely many periods if (i) condition (ii) in Proposition 2 holds, or (ii) condition (i) in Proposition 2 does not hold, but it holds that ∗ + yˆ ∗ xˆ

(Ft (g) − Ft (x))dx > 0· xˆ ∗

Proof In Appendix. Proposition 3 states in its first part that aggregate utility increases after finitely many periods for all periods thereafter in the scope for moral hazard when a corner solution lottery induces agents with endowment x n to gamble (this case is shown in Fig. 3). Note that this does not depend on the endowment distribution in the initial period. The second part of the proposition provides another sufficient condition namely that there is sufficient probability mass to the left of the point g. That is, there have to be many agents preferring to gamble with endowments less than g who benefit from sustainability in the long run. Loosely speaking, this happens whenever the poor outnumber the middle income class individuals. Note that whenever elimination of the low income steady holds but sustainability does not (see the Appendix for a discussion of this case) there emerges a lottery poverty trap. This lottery poverty trap consists of agents with endowments in [x ∗ , x ∗ + y ∗ ] who gamble but do not win enough to be put on the growth path towards x s . Reversing Proposition 3 yields an interesting conclusion. Given an economy with a corner solution lottery that satisfies condition (ii) in Proposition 2, an improvement in credit market conditions, that is a decrease in γ , leads to the emergence of a lottery poverty trap if the new scope for moral hazard fails sustainability. If the improvement of market conditions is sufficiently large, for instance resulting in perfect credit markets, x s remains the only stable steady state. That is, long run economic growth may — at least locally — depend positively on the scope for moral hazard. Reducing market imperfections by too little may place the economy on a growth path towards less aggregate utility, or even

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create the poverty trap found by Galor and Zeira (1993). In case of a corner solution, when market imperfections are sufficiently severe to cause a credit market breakdown, a neat connection to the work of Garratt and Marshall (1994) appears who analyze a case where the credit market to finance human capital acquisition is assumed not to exist. In an otherwise similar setting they find that an optimal social contract involves a fixed prize lottery. This paper might offer a theoretical motivation for their assumption, explaining why an economy might experience a market breakdown for educational loans once lotteries are taken into account.

6 Application A natural opportunity to gamble that does not rely on third party enforcement is conflict. Typically the outcome of conflict between two parties does reflect the relative strength of each party, yet the outcome is subject to a random component. The choice of weapons affects success probabilities but victory is not certain. Incorporating the possibility of costly conflict in the framework, the model predicts a certain segregation within the economy. While at least some of the poorer agents find it desirable to engage in conflict, the rich do not. Complementarily, agents choosing to fight do not target too wealthy agents. This draws a picture of socioeconomic segregation within an economy where poverty and criminal activity are highly correlated. Some evidence for this is provided, for instance, by Cullen and Levitt (1999). They find that although migration into urban areas does not appear to be affected by changes in crime rates, richer people tend to be very responsive to higher crime rates and to leave areas where crime rates have increased. One could argue that socioeconomic segregation is largely due to social interaction, e.g. via negative role models, and that this in turn determines criminal activity. The study by Ludwig et al. (2001) casts some doubt on this argument finding that in a randomized housing experiment property crime rates after relocation are positively affected by being treated. The main departure of an economy with conflict from our model lies in the waste of resources due to fighting. This section aims to provide an illustration of how conflict may serve as a means to provide agents with an opportunity to convexify their consumption sets. Suppose two agents fight and the winner has the opportunity to expropriate the loser. Suppose further the result is affected by the agents’ individual strength represented by the amount of endowments they carry with them. However, the outcome of conflict is not deterministic. Being stronger than one’s opponent does not ensure victory, it increases the chances of winning. Therefore, we use a stochastic contest technology, the ratio form contest success function (see Hirshleifer 1989; Skaperdas 1996; Neary 1997).14 14 Under the ratio form contest success function a party that does not dedicate any endowments to conflict loses with certainty. Hirshleifer (2000) argues that the difference form appears to be more consistent with fighting as a non-resisting losing party need not lose everything due to real-world frictions. In our case we assume that conflicting parties are able to hide away part of their initial endowment they do not want to risk in a fight before conflict thus incorporating the desired property.

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To be precise, an agent i uses part of his initial endowment xi − x in the conflict. Let y denote the sum of all endowments used by parties in the conflict. Agent i’s probability of winning the conflict is given by π(x, y) =

xi − x y

(9)

The winner’s prize in the conflict is determined by the sum of endowments used in the conflict y. However, during the conflict part of the endowments is wasted such that the winner conflict obtains the prize δy where δ < 1 is a cost parameter. This means that for an agent i a conflict is equivalent to a costly ex-ante lottery (x, y) over endowment levels x + δy with probability π(x, y) and x with probability 1 − π(x, y). In this setting we are able to state the following proposition. Proposition 4 Agents with initial endowments xi in the neighborhood of f prefer a costly lottery to their certain endowment if (i) the scope for moral hazard is sufficiently high, or if (ii) the cost of lotteries is sufficiently low. Agents with endowments xi < x ∗ or xi > x ∗ + y ∗ prefer their certain endowment to any costly lottery. For the prize x + δy of a costly lottery that is the optimal choice for some agent it holds that x + δy ∈ ( f, x ∗ + y ∗ ]. Proof In Appendix. The first statement of the proposition is quite intuitive. Agents in the neighborhood of f choose to gamble if lotteries are not too wasteful and capital markets sufficiently inefficient. Agents willing to engage in conflict are a subset of the gamblers of the sections above. Moreover, there is an upper bound on a winner’s endowment depending on the cost of conflict. Combined with the fact that winners will become skilled workers, this implies that for sufficiently low cost of conflict the analysis of the previous sections goes through as well. On the other hand, sufficiently rich agents are not threatened by conflict as fighting occurs mainly among the poor. This appears to be consistent with the empirical evidence. 7 Conclusion This paper finds that allowing for lotteries in the setting of Galor and Zeira (1993) leads unambiguously to an instantaneous Pareto improvement. Consistent with existing literature it is found that mainly agents with intermediate incomes want to gamble whereas rich individuals never do. For wide classes of parameters the dynamics of the model economy changes fundamentally, erasing the possibility of a poverty trap emerging in the original work. Intuitively, agents previously caught in the poverty trap now become gamblers and are thus able to escape towards the high income steady state with positive probability independently of the initial distribution of endowments. Hence, the finding of persistent inequality due to the presence of capital market frictions may be reversed once the possibility of gambling is taken into account. Moreover, the effect of reducing credit market imperfections may be non-monotonic. Starting out from an economy where market imperfections are severe and the rent to the wealthy high, too small a reduction in the scope for moral hazard and in the interest rate spread may decrease aggregate welfare by generating a

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poverty trap. The underlying mechanism of the model, namely the crowding out of inefficient means to accumulate sufficient capital, seems to apply to far more general settings of dynamic inefficiency. Introducing a less inefficient accumulation instrument to circumvent barriers to extra rents, in particular when these rents are high and market imperfections severe, may more generally lead to similar findings. Finally, we considered an application where lotteries are given by the possibility to engage in conflict or rent seeking contests. If conflict wastes sufficiently few resources, agents of low and intermediate wealth engage in contests and the previous results carry over. This may be interpreted e.g. as property crime, predicting that higher population shares of poor agents induce higher crime rates, seemingly consistent with empirical observations. Appendix Proof of Lemma 4 Taking the derivative of y ∗ + x ∗ as defined in Lemma 1 with respect to γ yields ∂(x ∗ + y ∗ ) = ∂γ

1+ln γ +(ln γ )2 ws γ

− (1 + r )h − (2 + r )wn

(1 + r )γ (ln γ )2

.

Inspection of the derivative shows that (x ∗+y ∗ ) has a global maximum in γ , strictly increases before attaining it, and strictly decreases thereafter. But it is also the case that lim (x ∗ + y ∗ ) = h +

γ →∞

2+r wn > h. 1+r

This implies that the derivative with respect to γ must be strictly positive whenever the function value (x ∗ + y ∗ ) is smaller than its limit. Hence, given an interior solution with (x ∗ + y ∗ ) < h, increasing γ also increases (x ∗ + y ∗ ) until reaching the boundary h. Concerning the second part of the lemma, y ∗ is obviously increasing in γ . To show that x ∗ also is, take any γ1 , γ2 with γ1 > γ2 within the definition range. Then it can easily be checked that ∂Us (x) ∂Us (x) > ∀ x· ∂ x γ 1 ∂ x γ 2 On the other hand it holds that Us (h) = ln(ws ) ∀ γ · This implies by monotonicity that Us (x)|γ1 < Us (x)|γ2

∀ x < h.

(10)

This in turn implies that any tangent to Us (x)|γ2 lies strictly above the area under Us (x)|γ1 for all x < h.

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Now take some point z > Us (x)|γ2 , x < h. Let T2 (x) be any tangent to Us (x)|γ2 with the property that it contains the point z. Denote by T1 (x) a tangent to Us (x)|γ1 with the property that it also contains z. Let the respective points of tangency ti be implicitly defined by Ti (ti ) = Us (ti )|γi , i = 1, 2. Then it must hold that T2 (t1 ) > T1 (t1 ) as long as t1 , t2 lie to the left of h implying that that the slope of T2 is greater than the slope of T1 . This must be true because T2 (x) > Us (x)|γ2 ∀ x due to concavity of Us and because of inequality (10). This implies that the tangent to both Un (x) at x = x1 and Us (x)|γ2 must be steeper than the tangent to both Un (x) at x = x2 and Us (x)|γ1 for all tangency points to the left of h. This follows from monotonicity of Un (x). This and strict concavity then also imply that x1 > x2 . This means x ∗ increases in γ as long as an interior solution is the case.  Sustainability Sustainability holds if h is sufficiently close to wn Using part (i) of Assumption 1, inequality (5) is implied by   ln γ − 1 ((i − r )h − (2 + r )wn ) (2 + r )γ wn > (1 − α)(1 + i) − 1    ln γ (1 + r )(γ − 1) h −1 . ⇔γ > −1 (1 − α)(1 + i) − 1 (2 + r ) wn Taking some γ > 1 with 1 < small compared to wn .

ln γ (1−α)(1+i)−1 ,

this condition holds for h sufficiently

A case where sustainability fails Looking at a rewritten inequality (5) we find    ln γ ws −1 (1 + r )h − . (2 + r )wn > (1 − α)(1 + r )γ − 1 γ It is now possible to find wn sufficiently small, such that this inequality does not hold. Let W = ws /wn , R = ws / h and k = (1−α)(1+r ). In general, sustainability fails if kγ − 1 W 1+r W < − . ln γ − (kγ − 1) 2+r R (2 + r )γ Now turn to elimination of the low income steady state. Using condition (ii) of Proposition 2 and rearranging yields   kγ − 1 (1 − k)W k 1 W − < ln − . ln γ − (kγ − 1) 1 − k 1−k 2+r (2 + r )γ The left hand side is negative if k < 1/(γ − ln γ ) which is compatible with the assumption 1/γ < k < (1 + ln γ )/γ . Hence, a tuple (k, γ ) satisfying 1/γ < k < 1/(γ − ln γ ) and 1/γ < ln[(1 − k)W/(2 + r )]/[(1 − k)W/(2 + r )] satisfies condition (ii) of Proposition 2 and implies failure of sustainability. It can be checked algebraically that there exist a positive measure of such tuples in R2 .

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Proof of Lemma 5 First note that every gambler weakly prefers the lottery because the convex combination of the corresponding utility levels lies above the certain utility level for gamblers. Expected utility from gambling is given by E[U (I G (x))] =

x −x ∗ ln((1+i)(x ∗ + y ∗ − h) + ws ) y∗   x −x ∗ + 1− ∗ ln((1+r )x ∗ + (2+r )wn ). y

That means that for all x ∈ (x ∗ , x ∗ + y ∗ ) it must hold that ∀x ∈ (x ∗, f ) : E(U G (x)) > ln((1+r )x + (2+r )wn ) > ln((1+i)(x −h) +ws ) ∀x ∈ [ f, x ∗+y ∗ ) : E(U G (x)) > ln((1+i)(x −h) +ws ) > ln((1+r )x + (2+r )wn ). Yet Jensen’s inequality implies E(U G (I G (x))) ≤ ln(E[I G (x)]). So that ∀x ∈ (x ∗ , f ) : E[I G (x)] > (1 + r )x + (2 + r )wn and ∀x ∈ [ f, x ∗ + y ∗ ) : E[I G (x)] > (1 + i)(x − h) + ws . This establishes the lemma. 

Proof of Proposition 3 Step 1: Note that aggregate utility levels Ut+k and Uˆ t+l , k, l ∈ N are converging sequences in R. To see this, write down e.g. Ut+k as a function of endowments x s Ut+k =

EU (I (x))dFt+k (x). 0

So the dynamic behavior of Ut+k is entirely governed by the distribution of initial endowments among period-t-gamblers in period t + k. But we know that Ft+k (x) converges. Hence, so does Ut+k . An analogous argument applies to Uˆ t+k . Step 2: We show that if Uˆ ≡ liml→∞ Uˆ t+l > U ≡ limk→∞ Ut+k , then there exists an N < ∞ such that Uˆ t+N > U N +l . Because of convergence, the sequences {Uˆ t+k , k ∈ R} and {Ut+k , k ∈ R} have the properties ∀ ε > 0 ∃ K : |Ut+k − U | < ε ∀ k ≥ K ∀ ε > 0 ∃ L : |Uˆ t+l − Uˆ | < ε ∀ l ≥ L·

and

Let 2ε = |U − Uˆ |, then it must hold for all n ≥ N ≡ max{K , L} that Uˆ t+n > Ut+n . Step 3: If condition (ii) in Proposition 2 holds, then Uˆ > U trivially since sustainability holds, thus establishing the first part of the proposition.

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Step 4: Assume condition (i) in Proposition 2 does not hold. Then only in the first period agents are willing to gamble both under γ and γˆ . Steady state income under γ is given by: g ISS (x) =

dFt (x)[x n (1 + r ) + wn (2 + r )] 0 ∗ + yˆ ∗ xˆ

dFt (x)[(x s − h)(1 + r ) + ws ],

+ g

and steady state income under γˆ by: ⎡ ∗ ⎤ ∗ + yˆ ∗ xˆ xˆ ⎢ ⎥ dFt (x) − Pth ⎦ [x n (1 + r ) + wn (2 + r )] IˆSS (x) = ⎣ dFt (x) + xˆ ∗

0

+Pth [(x s − h)(1 + r ) + ws ], where Pth is the fraction of agents among gamblers who escape the poverty trap under γˆ defined by ∗ + yˆ ∗ xˆ

Pth = xˆ ∗

x − xˆ ∗ dFt (x). yˆ ∗

Integration by parts yields 1 Pth = Ft (xˆ ∗ + yˆ ∗ ) − ∗ yˆ

∗ + yˆ ∗ xˆ

Ft (x)dx. xˆ ∗

The sign of the change in steady state aggregate income after an increase of γ is given by the sign of the net fraction of agents that is placed on the high income growth path, so that IˆSS (x) − ISS (x) can be written as: Pth − Ft (xˆ ∗ + yˆ ∗ ) + Ft (g) > 0 ⇔ yˆ ∗ Ft (g) >

∗ + yˆ ∗ xˆ

Ft (x)dx. xˆ ∗

Using the fact that Ft (g) is a constant, the LHS can be rearranged: yˆ ∗ Ft (g) = (xˆ ∗ + yˆ ∗ )Ft (g) − yˆ ∗ Ft (g) ∗ + yˆ ∗ xˆ

Ft (g)dx.

= xˆ ∗

(11)

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Inserting this into (11) yields: ∗ + yˆ ∗ xˆ

[Ft (g) − Ft (x)]dx > 0. xˆ ∗

By definition the last inequality means that if Ft (x) second order stochastically dominates any lottery [(xˆ ∗ − c), (xˆ ∗ + yˆ ∗ + c), Ft (g), (1 − Ft (g))] with c > 0 on the interval [xˆ ∗ , xˆ ∗ + yˆ ∗ ], then aggregate utility from the steady state endowments is higher under γˆ .  Proof of Proposition 4 Consider an agent with endowment 0 < xi < h. Suppose the agent chooses a symmetric lottery (x, y) such that x +1 = xi and δy = 1 +2 and x < f < x +δy. This lottery is preferred to the certain endowment xi , if xi − x x + y − xi Us (x + δy) + Un (x) ≥ U (xi ). y y That is δ1 [Us (xi + 2 ) − Un (xi − 1 )] ≥ U (xi ) − Un (xi − 1 ). 1 + 2 Let xi = f − ε, ε > 0 sufficiently small, such that Us ( f − ε + 2 ) − Un ( f − ε) ≥ Us ( f − ε + 2 )2 . Using the fact that U (.) is piecewise concave, a sufficient condition can then be obtained by an adequate first order approximation:   1 γ 2 1 + 2 δ + . (12) ≥ K1 K 2 + (1 + i)2 K 1 − (1 + r )1 with K 1 = (2 + r )wn + (1 + r )( f − ε) and K 2 = ws + (1 + i)( f − ε − h). Closer inspection shows that for ratios 2 1−δ > , 1 γδ − 1 γ δ > 1, there exist 1 , 2 > 0 sufficiently small such that (12) holds. By piecewise concavity and continuity of U (.), for any 2 > 0 there exists 0 < ε < 2 such that Us ( f − ε + 2 ) − Un ( f − ε) ≥ Us ( f − ε + 2 )2 and the approximation is valid. Hence, for any δ > γ1 there exists a neighborhood to the left of f preferring some lottery to a certain endowment xi . Let now xi = f + ε, 1 > ε > 0 sufficiently small such that Us ( f + ε) − Un ( f + ε − 1 ) ≤ Un ( f − 1 )1 . Such an ε exists due to continuity of U (.). Using the fact that U (.) is piecewise concave, an adequate first order approximation to obtain a sufficient condition is now   1 γ 2 γ (1 + 2 ) δ + . (13) ≥ K1 K 2 + (1 + i)2 K 1 − (1 + r )1

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with K 1 = (2 + r )wn + (1 + r ) f and K 2 = ws + (1 + i)( f + ε − h). Note that condition (13) coincides with condition (12) so that the first part of the proposition is verified. To prove the second part, recall the definitions of x ∗ and y ∗ from Lemmata 1 and 2. Denote the convex hull of U (.) by U C (.). By definition all points U (x) with x ≤ x ∗ or x ∗ + y ∗ ≤ x lie on the convex hull of U (.). For these x it must hold  that U C (x) ≤ 0 by piecewise concavity of U (.). This means that for these x there exists no convex combination λU C (x + 2 ) + (1 − λ)U C (x − 1 ) > U (x) with 1 1 λ = 1+ . For λ = δ , δ < 1, the strict inequality becomes weak. That is, 2 1 +2 for endowment levels xi < x ∗ and xi > x ∗ + y ∗ the certain income from xi is preferred to any costly endowment lottery. Conversely, for every costly lottery that is utility-maximizing for some agent it must hold that x ≥ x ∗ and x + δy ≤ y ∗ . For any optimal costly lottery it must hold for y that Us (x + δy) − Un (x) ∂Us (x + δy) = . ∂ x + δy δy

(14)

That is, given x, δy has to be chosen such as to maximize the slope of the lottery. Us (.) decreases in its argument, so that an x that maximizes the corresponding δy is associated to the flattest lottery given (14). That is the lottery that is tangent to Un (.). This lottery is given by x ∗ and x ∗ + y ∗ . Hence, x + δy ≤ x ∗ + y ∗ . To conclude the proof, suppose that x + δy ≤ f . Then due to concavity of Un (x) for x ∈ [0, f ] and Jensen’s inequality there exists no xi ∈ [x, x + δy] such that a costly lottery with δy > 0 is preferred to Un (xi ).
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Neary, H.M.: Equilibrium structure in an economic model of conflict. Econ Inq 35(3), 480–494 1997 (1997) Ng, Y.K.: Why do people buy lottery tickets? Choices involving risk and the indivisibility of Expenditure. J Polit Econ 73(5), 530–535 (1965) Piketty, T.: The dynamics of the wealth distribution and the interest rate with credit rationing. Rev Econ Stud 64, 173–189 (1997) Scott, F., Garen, J.: Probability of purchase, amount of purchase, and the demographic incidence of the lottery tax. J Public Econ 54, 121–143 (1993) Skaperdas, S.: Contest success functions. Econ Theory 7, 283–290 (1996) Worthington, A.C.: Implicit finance in gambling expenditures: Australian evidence on socioeconomic and demographic tax incidence. Public Finance Rev 29(4), 326–342 (2001)