A Theory of Progressive Lending Ch 1

A Theory of Progressive Lending Abstract We characterize Pareto efficient long term ‘relational’ lending contracts with one-sided lender commitment, to finance specific investments and smooth intertemporal consumption of a borrower who cannot commit to repaying loans. The borrower can save and accumulate assets, and has strictly concave preferences over consumption. For borrowers with initial wealth above a threshold, the first-best is sustainable with a stationary c ontract. For poorer agents, there is perpetual but shrinking underinvestment which disappears in the long run. Borrowing, investment and wealth grow and converge to the first-best t hreshold. Optimal allocations can be implemented by backloaded ‘progressive’ lending: a sequence of one period loans of growing size. Increased borrower bargaining power raises short run consumption and investment, with no long term effects. We discuss extensions to incorporate random productivity shocks. 1 Introduction This paper studies Pareto efficient lending contracts between a lender that can commit, and a borrower (lacking collateralizable wealth) who cannot commit to a long term relationship. The borrower prefers to smooth consumption intertemporally, and maximize present value of her lifetime utility by accumulating assets via: (a) saving at a constant interest rate equal to her intertemporal discount rate, and/or (b) investing in a strictly concave production technology. Loans help finance investments and smooth consumption. Following any loan default, the borrower’s autarkic outside option is increasing in wealth accumulated so far. Productive investments are at least partly ‘specific’ in the sense that the investment technology in autarky generates a lower marginal return, owing to withdrawal of access (following any default) to technology or marketing assistance provided by the lender.1 In the absence of such specificity, it is well known (e.g., Bulow and Rogoff (1989), Rosenthal (1991)) that no lending contract exists that would enable the lender to break even and the borrower to be better off compared with autarky. The model is well-suited to contexts where financial institutions (banks or microfinance institutions (MFIs)) lend to poor borrowers that lack collaterizable wealth, by building long-term lending relationships. The assumption of lender commitment seems reasonable for financial institutions with established reputations. The model is also applicable to long-term employment relationships between established firms and workers, where the firm finances training of workers in specific skills and the relationship has to be structured to provide incentives to workers to not leave the firm after receiving training. In both contexts, the agent is poor, seeks to smooth consumption and finance investment to accumulate assets, and cannot commit to defaulting on loans (or quitting). The partial specificity of investments to the relationship with the lender helps sustain some lending. To overcome the ex post moral hazard problem, lenders need to limit lending initially and ‘backload’ borrower incentives by offering larger loans in the future conditional on repayment of current loans. The extent of such backloading is limited by the borrower’s preference for consumption smoothing. Hence consumption smoothing preferences of the agent reduce the scope of loans to generate higher asset growth and welfare. The aim of this paper is to study the resulting short run and long run implications of these distortions. In particular we address questions such as the possibility of emergence of debt traps in 1 Similar results would obtain in a setting where specificity arises instead from the ability of the lender to expropriate a fraction of the agent’s output following a default. But we do not pursue that extension here, as it would require a slightly more complicated model. which poor borrowers may be stuck in perpetuity. Previous analyses of such contexts (Thomas and Worrall (1994), Albuquerque and Hopenhayn (2004)) have confined attention to the case where the borrower has linear utility and subject to limited liability. They show that optimal lending strategies enable poor borrowers to gradually accumulate capital and eventually attain first best investments. In the absence of preferences for intertemporal consumption smoothing, optimal contracts involve maximal ‘backloading’ of incentives: the agent is not permitted to consume until investment distortions disappear. Such extreme backloading is no longer possible when the borrower has strictly concave utility, as optimal contracts need to trade off consumption smoothing and backloading. Ray (2002) considers a model of self-enforcing contracts with one-sided commitment and non-transferable utility, in a repeated game context where the agent cannot save or invest. Hence consequences for investment and growth are not explored in his paper. The recent work of Thomas and Worrall (2017) studies a context with two agents with concave utility who contribute effort to a common joint output. Their setting differs from ours by incorporating lack of commitment of both parties, and not allowing either agent to save or invest. We show that the lender can without loss restrict attention to a sequence of one-period loans offered conditional on repayment of past loans and achievement of stipulated investment targets. Moreover, loan contracts are conditioned on a single state variable, a measure of net wealth of the borrower equal to value of current output, less debt repayments due. We obtain a recursive representation of Pareto efficient contracts maximizing welfare of an agent with arbitrary initial net wealth, subject to a minimum profit target for the lender, and no-default incentive constraints. This representation simplifies the analysis by enabling us to represent the incentive constraints in a tractable manner.2 We use this recursive representation to characterize the dynamics of the agent’s investment, wealth and consumption. Our main results are the following. Agents with initial wealth above some threshold can achieve first-best welfare, in a steady state where investment is first-best and consumption perfectly smoothed. Hence we focus thereafter on poor agents, who start with a wealth below this threshold. We show that the optimal strategy can be implemented by progressive lending, where loan amounts are strictly increasing in net wealth. Investment and consumption grow over time, though allocations are distorted at every date. Our main result is that these distortions eventually vanish: the allocation converges 2 Specifically, the value of continuing the relationship the next period onwards should not fall below the outside option corresponding to the Ramsey autarkic value starting with an endowment equal to the current output, but with the inferior autarkic technology. to the steady state corresponding to the first-best wealth threshold, irrespective of initial conditions or consumption smoothing preferences. Hence a debt trap never arises, irrespective of the extent of the principal’s bargaining power, concavity of the borrower’s payoff or the rate of time discount. The dynamic is qualitatively similar to that in a Ramsey model of autarkic self-financing, except that the steady state involves higher welfare, and convergence is faster (i.e., consumption, productivity and wealth are higher at every date). Here is a sketch of the underlying argument. (i) The recursive representation of the contracting problem implies optimal investment is a function of the agent’s current net wealth. Owing to concavity of utility in current consumption, wealthier agents face a lower marginal cost of investing. Hence investment is non-decreasing in wealth. This implies that the sequence of wealths is monotonically increasing or decreasing over time, and thus must converge. (ii) Since net wealth converges, so must consumption — for large enough t, the agent’s consumption must be smoothed nearly perfectly. This implies the investment distortion must also vanish, since the agent can always self-finance some extra investment. A first-best allocation must therefore be attained in the limit. And the agent’s limiting wealth must be at least w∗ , the first-best threshold. (iii) However a first-best allocation cannot be achieved in finite time, because this would result in a consumption distortion without a co-existing investment distortion.3 Hence the agent’s wealth must be strictly less than w∗ at all dates. This is only possible if wealth is rising and converging to w∗ . (iv) The argument for optimality of progressive lending (i.e., loan sizes are increasing in net wealth) is somewhat more involved, and is based on showing that consumption grows faster on the equilibrium path than in the counter-factual event of default. Intuitively, this is because the technology available to the agent on the equilibrium path has a higher rate of return than the autarkic technology.4 3 Specifically, if the first-best could be attained at some finite date T , consumption would be smooth after T , and strictly higher than at T − 1. Moreover output at T would be first-best, which requires first-best investment at T − 1. In other words, the distribution of consumption between T and T − 1 would be distorted, while investment is not. This cannot be optimal: the agent could reduce investment slightly at T − 1 so as to reduce the consumption distortion, while the reduction in output at T would have a zero first order welfare effect. 4 Since the incentive constraint binds, the present value of consumption is the same on and off the equilibrium path. Comparative dynamics with respect to increased ‘aid’ granted by a benevolent not-for-profit lending institution (or more generally, a decrease in the lender’s profit target or bargaining power) yields short-run increases in investment and consumption, leaving the long-run allocation unaffected. Hence the benefits of aid are entirely front-loaded, with no long-term consequences. In the context of intrafirm skill training, the model predicts lifetime productivity age-profiles shift upward and become flatter when workers’ bargaining power improves (e.g., as a result of unions or increased competition among firms for workers).5 The last section of the paper extends the model to incorporate productivity shocks. The recursive representation continues to apply, and investments increase in net wealth. Moreover, conditional on stationary (or nondecreasing) productivity shocks, the agent’s net wealth, investment and output increase over time, with underinvestment disappearing once net wealth crosses some threshold. Of course in this setting wealth could also contract, owing to the possibility of declining productivity shocks. If wealth falls from one period to the next owing to an adverse shock, it continues to fall thereafter if the shock persists or becomes worse over time. Hence the wealth dynamic continues to be qualitatively similar to the neoclassical growth model.6 Our model therefore provides a rationale for progressive lending both in a positive and normative sense. Loan sizes increase, conditional on past repayment, and loans are repaid on the equilibrium path, broadly consistent with observed practice of MFIs.7 These strategies provide repayment incentives efficiently, ensure that the lender’s profit targets are met, and enable poor borrowers to escape poverty and accumulate wealth (conditional on absence of adverse shocks).8 Section 2 discusses relation to existing literature in more detail. Sections 3 and 4 provide analyses Therefore current consumption must be lower on the equilibrium path. This implies a higher marginal welfare impact of increasing wealth on the equilibrium path, i.e. wealthier borrowers can credibly commit to repaying larger loans. 5 The empirical evidence on this appears to be somewhat mixed, as discussed in Acemoglu and Pischke (1998). 6 We conjecture that results concerning ergodicity of wealth dynamics also continue to hold, but do not pursue this further. 7 Table 1 shows that MFIs experience remarkably low default rates, while Table 2 provides evidence of use of progressive lending strategies by many large MFIs in Bangladesh, India and Vietnam. 8 This is in an ‘ideal’ setting with rational agents with dynamically consistent preferences, convex technologies, and lender commitment to exclusive long-term credit contracts.Exclusivity clauses prevent borrowers from switching to a new lender after defaulting. These can arise owing to lender’s coordinating on strategies, or borrowers being institutionally prevented from borrowing simultaneously from multiple lenders. See Pauly (1974), Arnott and Stiglitz (1986), Bizer and DeMarzo (1992) or Kahn and Mookherjee (1998). To the extent that the observed anti-poverty impact of microcredit so far have been less successful than the model predicts (e.g., see Banerjee et al. (2015)), our results suggest that the problem does not stem from any inherent defect in progressive lending policies per se. Instead they must lie in violation of one or more of the other ideal assumptions. for the deterministic and uncertainty contexts respectively. Proofs are collected in the Appendix. 2 Related Literature As mentioned in the Introduction, Thomas and Worrall (1994) and Albuquerque and Hopenhayn (2004) study the case where the agent has no preferences for consumption smoothing, and the extent of backloading is limited by a limited liability constraint. In an extension of their benchmark model, Thomas and Worrall (1994) show their main results extend when the utility function is ‘not too’ concave. Our paper reinforces and extends this result to any concave utility function for the borrower, upon applying a different approach to the optimal contracting problem. With concave utility, the analysis is considerably more complicated owing to the need to characterize consumption dynamics via an Euler equation respecting incentive constraints. Owing to the maximal backloading with linear utility, the first best wealth threshold is strictly lower than for any concave utility. The effects of increased aid are different: the agent would continue to consume nothing initially, and the benefits would accrue later. With concave utility, the optimal contract needs to strike a balance between front-loading and back-loading, so the benefits of greater aid appear also at the beginning. In both cases, however, the agent would converge ‘faster’ to the first best threshold. Ray (2002) considers a general model of constrained Pareto efficient self-enforcing contracts in the context of a repeated game between a principal and agent with limited transferability of utility, where the agent cannot save nor commit. He shows that all such contracts back-load in the sense the allocation of surplus tends progressively to the agent’s favor at later dates, converging to the one that maximizes the agent’s continuation payoff, which may or may not involve a distortion. Our model imposes more structure on preferences and technology, but incorporates investment and wealth accumulation, and shows that distortions disappear eventually. We also obtain a backloading result analogous to his: the optimal allocation involves an initial adjustment of the agent’s initial wealth with a lump-sum transfer to the principal, followed by a contract that maximizes the agent’s continuation utility subject to a breakeven constraint for the principal. Thomas and Worrall (2017) study optimal relational contracts between two agents neither of whom can commit, both contribute effort to a common joint output, and cannot save or invest. They consider both the case where the agents have strictly concave utility, and where they have linear utility and subject to limited liability. In the former case, their results turn out similar to ours: over-investment never occurs; and convergence to the first-best is monotone if the first-best is sustainable. Dasgupta and Roy Chowdhury (2018) provide an alternative explanation of progressive lending in a framework with a nonconvexity, where the borrower (with linear utility) has an opportunity to graduate to a higher occupation or productive activity which requires a minimum investment. The lender provides a savings as well as borrowing opportunity which enables the borrower to accumulate wealth and graduate at an endogenous finite date. Over time, savings banked with the lender increase, which the borrower forfeits in case of default. The endogenous growth of collateral permits the lender to extend larger loans which are repaid. In a context with a similar nonconvexity, Liu and Roth (2020) present a model of a debt poverty trap that arises when the lender is profit-maximizing and cannot commit to long term contracts. Mookherjee and Ray (2002) present a model where poverty traps can arise without any technological nonconvexity, with a profit-maximizing lender with limited commitment, and a borrower with concave utility subject to ex ante moral hazard. Comparing our results with these papers highlights the role of lender commitment in preventing debt traps. A number of papers on dynamic lending in microfinance focus on unobserved borrower heterogeneity, and the possible role of progressive lending in screening borrowers. In all of these models, defaults necessarily occur on the equilibrium path. This limits their relevance to the microfinance setting where repayment rates are near 100%. For instance, Ghosh and Ray (2016) show how progressive lending can help screen out bad borrowers who always default from good borrowers, by providing small initial loans followed by larger ones after the bad borrowers have been eliminated. Egli (2004) shows that progressive lending may fail to identify a “bad” type, since a bad borrower may camouflage herself as a “good” borrower (who always repays) in order to get a higher amount of loan later on which she defaults with certainty. Shapiro (2015) examines a framework with uncertainty over borrowers’ discount rates. He shows that even in the efficient equilibrium almost all the borrowers default. An earlier paper with the same feature is Aghion and Morduch (2000), where in a two period context the borrower repays the first period loan to get a higher amount of loan in the second period on which she subsequently defaults. In contrast to these papers, we provide a theory of progressive lending without any equilibrium default and without any unobserved heterogeneity. 3 Model with Deterministic Technology Consider an agent with endowment e, in an infinite horizon discrete time framework. Her current payoff is given by u(c) where c denotes consumption. u(· ) is time-stationary, twice differentiable, strictly increasing, strictly concave and satisfying Inada conditions. The agent’s objective is to maximize the P t present discounted value of her lifetime utility: ∞ t=0 δ u(ct ); where δ ∈ (0, 1) is the discount factor. 3.1 Autarky The agent always has access to (i) a neoclassical production technology g(· ) which is strictly increasing, strictly concave and satisfies the Inada conditions: g 0 (0) = ∞ and g 0 (∞) = 0 and (ii) a linear savings opportunity at a constant rate of return r = 1 δ − 1. Together, these imply that the agent has access to a transformation possibility of resources from any date t to t + 1 at a rate of return bounded below by r = 1δ − 1. This possibility is represented by the function φ(k) which provides total resources available at the next date if the agent invests a total of k, shown in Figure 1. The budget constraint at t is then ct + kt+1 ≤ φ(kt ), where ct , kt denote capital stock at t. Return φ(k) (1 + r)k g(k)  φ(k) = kδA Investment (k) g(k) g(kδA ) + (1 + r)(k − kδA ) i f k ≤ kδA otherwise. Figure 1: Autarky Technology This formulation assumes that capital depreciates entirely after one period. As is well known, it is easy to extend the model to incorporate less than 100% depreciation, by redefining the production function. Suppose kt+1 = (1−d)kt +it , where d is the proportion of capital stock that depreciates in one period, it denotes investment, kt denotes capital stock, and output yt = y(kt ) at date t, where y(.) is a strictly increasing, concave function satisfying y(0) = 0 and y 0 (0) = ∞.The agent’s budget constraint at t is then ct + it ≤ y((1 − d)kt−1 + it−1 ), which can be rewritten as ct + kt+1 ≤ φ(kt ) ≡ y(kt ) + (1 − d)kt , with φ(0) = 0, φ 0 (0) = ∞. In the remainder of the paper, we will therefore simplify notation by assuming full depreciation. Letting kδA denote the solution to δg 0 (k) = 1, in autarky the problem of the agent with endowment e > 0 is to   Maximize u e − k1 + ∞ {kτ+1 }τ=0 ∞ X   δτ−t u φ(kτ ) − kτ+1 τ=1 Subject to: k1 ≤ e and kτ+1 ≤ φ(kτ ), ∀τ ≥ 1. We denote the solution to this standard Ramsey optimal growth problem by V A (e). It is well known 0 0 that V A is differentiable, with V A (e) = u (e − kA (e)), where kA (e) is the optimal investment rule which is nondecreasing in e. 3.2 The Lender The lender provides the agent with access to a more productive technology z(k), as well as to loans. The cost of capital of the lender is r, equal to the agent’s rate of return on savings. The production function z(.) dominates g(.) in terms of both absolute and marginal returns to investment: z(k) > g(k) and z 0 (k) > g 0 (k) for all k. This can represent a higher TFP, or a higher price at which the output can be sold. Alternatively, it could be a different production opportunity subject to decreasing returns, which enables the agent to raise returns by allocating investment between the autarkic technology and the new one.9 Let kδ denote the first-best investment, which solves δy 0 (k) = 1. And let y(k) denote the return on investment k which is optimally allocated between production and savings, so that  z(k) i f k ≤ kδ y(k) = z(kδ ) + (1 + r)(k − kδ ) otherwise. 3.2.1 Credit Contracts The agent is subject to ex post moral hazard and cannot commit to repaying loans. The lender on the other hand can commit to a long-term contract providing access to the improved technology, stipu9 If the production function in the new opportunity is represented by a strictly concave twice-differentiable function f (k) satisfying Inada conditions, the agent will optimally allocate a total investment of k into x(k) in the new technology and 0 0 k − x(k) in the old, such that f (x(k)) = g (k − x(k)). This results in total production z(k) = f (x(k)) + g(k − x(k)) which dominates the old one in the sense described. lated investments and financial transfers to the agent, as a function of past history which includes past investments, loans and repayments (all of which are verifiable). Standard arguments imply that attention can be restricted without loss of generality to contracts where all loans are repaid on the equilibrium path, and any default is followed by suspension of technology and loan access at all future dates. Hence we can focus attention on incentive compatible contracts. A contract is a sequence p ≡ {p0 , p1 , ...} and k ≡ {k0 , k1 , ...} of stipulated net transfers and investments at each date t = 0, 1, .., conditional on absence of any past defaults. When pt < 0, the borrower effectively obtains a loan, while pt > 0 denotes payments made by the borrower. Let e denote the initial endowment of the borrower, and π an arbitrary profit target for the lender. A Pareto optimal contract then solves the following problem. D max ∞ {pt }∞ t=0 ,{kt }t=0 E V0 ≡ [u(e + p0 − k0 ) + ∞ X δt u(y(kt−1 ) + pt − kt )] (1) t=1 subject to: Lender’s Profit Constraint ∞ X δt pt ≤ −π LPC : p0 + t=1 Incentive Compatibility Constraints ∞ X δτ−t u(y(kτ−1 ) + pτ − kτ ) ≥ V A (y(kt−1 )), ICt : Vt ≡ ∀t ≥ 1. τ=t Borrower Participation Constraint BPC : V0 ≥ V A (e). We do not include non-negativity constraints on the agent’s consumption owing to Inada conditions. The borrower has the option to default at any date t ≥ 1; hence the incentive compatibility constraint requires that continuation payoff does not fall below the autarkic payoff corresponding to an endowment equal to the current output y(kt−1 ). The participation constraint requires an analogous condition at t = 0, where the borrower’s outside option corresponds to the autarkic payoff starting with an endowment e. We start with a preliminary observation which enables the problem to be simplified. Lemma 1. (a) There exists an upper bound π̄(e) ∈ (0, ∞) to the profit that can be earned by the lender in any feasible contract. (b) Given any feasible profit target π ≤ π̄(e), the optimal contract solves the following competitive equilibrium (CE) problem with initial endowment w ≡ e − π ≥ −π̄(0): V(w) ≡ D max ∞ {pt }∞ t=0 ,{kt }t=0 E[u(w + p0 − k0 ) + ∞ X δt u(y(kt−1 ) + pt − kt )] (2) t=1 subject to: Lender’s Breakeven Constraint ∞ X LBC : p0 + δt pt ≤ 0 t=1 Incentive Compatibility Constraints ∞ X ICt : Vt ≡ δτ−t u(y(kτ−1 ) + pτ − kτ ) ≥ V A (y(kt−1 )), ∀t ≥ 1. τ=t (3) (a) states there is a positive upper bound π̄(e) to the profit that the lender can earn by contracting with a borrower with initial endowment e > 0: any higher profit would violate the borrower’s participation constraint. Any lower profit than π̄(e) allows the existence of a feasible contract. The upper bound is positive since there is always a feasible contract if π = 0: access to the more profitable technology allows the borrower to attain a higher payoff compared to autarky in the absence of any transfers (i.e., if pt = 0 for all t). Hence the borrower would be willing to pay a positive fee for such access, even if it is not bundled with any loans. Bundling with loans will further increase the scope for earning profits. Part (a) therefore implies we can focus attention on the case where π ≤ π̄(e). The borrower participation constraint can then be dropped. Part (b) says moreover that the optimal contracting problem can be simplified as follows. First modify the borrowers initial endowment from e to w = e − π, i.e., via a lump-sum transfer of π from the borrower to lender.10 Then the lender provides transfers pt at successive dates to the borrower, subject to a break-even constraint, besides the repayment incentive constraints. It therefore suffices to focus on the problem in (b), which corresponds to the CE contract maximizing payoff of a borrower with initial endowment w = e − π. In the remainder of the paper, we therefore study the CE problem (b). We shall refer to w = e − π as the borrower’s initial wealth and treat this as a parameter. Observe that π̄(e) = π̄(0) + e, so there is a feasible contract and V(w) is defined for all w ≥ −π̄(0), i.e., including some negative values of borrower net wealth. 10 It is evident that the borrower’s CE payoff V(w) is strictly increasing in her initial wealth, since this permits the borrower to consume more at t = 0 without disturbing any of the incentive constraints (which pertain to t ≥ 1). Moreover, this payoff must strictly exceed her outside option V A (w) corresponding to an initial endowment of w, because it is always feasible for a lender to break even by providing the borrower with access to the technology y(.) unaccompanied by any financial transfers. Moreover, the optimal contract can be implemented by a sequence of one period loans at the competitive interest rate r, which satisfy a ‘no Ponzi’ scheme condition: Lemma 2. (a) V(.) is strictly increasing, and V(w) > V A (w) for all w. (b) The optimal contract can be implemented by a sequence of one period loan contracts lt , t = 1, 2, .. charging the interest rate r, which are always repaid on the equilibrium path, where l0 = p0 , lt = pt + (1 + r)lt−1 , ∀t (4) lim δT lT ≤ 0. (5) and T →∞ (b) follows from observing that the payment sequence pt is equivalently represented by the oneperiod loan sequence lt defined by (4), so pt = lt − lt−1 . δ We can interpret the transfer pt at any date as the composition of a fresh loan lt that is offered, after the agent has repaid the previous loan. Observe P that Tt=0 δt pt = δT lT . Hence the break-even constraint (LBC) reduces to (5). 3.3 First-best Contracts Consider the optimal contract when all the incentive constraints are dropped. It involves full consumption smoothing (via choice of transfers pt ) and efficient investment kt = kδ . The constant consumption c∗ (w) is obtained by the requirement that the resulting present value of consumption δ present value of endowment/output minus investment: w − kδ + [y(kδ ) − kδ ], so 1−δ c∗ (w) = (1 − δ)w + δy(kδ ) − kδ . The borrower then attains welfare V ∗ (w) = u(c∗ (w)) . 1−δ c∗ (w) 1−δ equals the (6) As the first best contract is stationary, all the incentive constraints collapse to a single constraint c∗ (w) ≥ (1−δ)y(kδ )+δg(kδA )−kδA .11 The first-best is incentive compatible if and only if the borrower’s wealth exceeds the following threshold: w∗ ≡ y(kδ ) − (δy(kδ ) − kδ ) − (δg(kδA ) − kδA ) 1−δ (7) Since the y(.) technology dominates g(.), it follows that the threshold w∗ is smaller than the firstbest output y(kδ ). An agent with wealth in the interval [w∗ , y(kδ )) obtains a loan at the first date of δ[y(kδ ) − w] and then repays (1 − δ)[y(kδ ) − w] at every subsequent date. These repayments are motivated by the lender threatening a defaulter with loss of access to the more productive technology, which allows the borrower to earn additional surplus (δy(kδ ) − kδ ) − (δg(kδA ) − kδA ) at every date in the future. Observe that δT lT = δT +2 [y(kδ ) − w] which converges to 0 as T → ∞, so the ‘no Ponzi scheme’ condition (5) is satisfied. This is sustained by a stationary sequence of one-period loans of size δ[y(kδ ) − w] at every date. If w falls below w∗ the loan is too large, resulting in a repayment obligation that would motivate the borrower to default. For agents starting with wealth above y(kδ ), there is no need to borrow to achieve the first-best allocation: they invest kδ in production, and supplement this by saving δ[w − y(kδ )] at every date. Note that the incentive problem arises only for intermediate ranges of the discount factor. If δ approaches 1, the first best can be sustained for any initial w, as the threshold w∗ goes to minus infinity. While if δ approaches zero, the threshold approaches zero (as in this case, the efficient investment approaches zero), and the demand for loans vanishes. It follows the first-best contract is incentive compatible if and only if the borrower is wealthy enough to start with: w ≥ w∗ . 3.4 Second-best Contracts for Poor Borrowers Now we focus on poor borrowers with w < w∗ and characterize the features of the optimal contract. Let ct = y(kt ) + pt − kt denote the agent’s consumption at date t ≥ 0. Lemma 3. ct ≥ ct−1 for all t. 11 If the borrower were to default, she would enter autarky with an initial wealth of y(kδ ). Since this exceeds g(kδA ) the autarkic steady state output, the agent would smooth consumption perfectly, and attain a per period consumption of cA = (1 − δ)y(kδ ) + δg(kδA ) − kδA . Proof. Suppose otherwise, and ct < ct−1 for some t. Lower pt−1 slightly, and raise pt correspondingly to keep pt−1 + δpt unchanged. This smooths consumption, raising Vl for every l ≤ t, while leaving it unchanged for every l > t. Hence LBC and all incentive constraints are preserved, while raising borrower welfare. To make further progress we use Lemma 1 to obtain a recursive formulation of the problem in terms of one-period loans. We study the ‘relaxed’ problem where the asymptotic breakeven constraint (5) is ignored. This relaxed problem can be stated as ∞  X   lt−1 t Maximize u(w + l − k ) + δ u y(k ) − + l − k 0 0 t−1 t t ∞ h{lt }∞ δ t=0 ,{kt+1 }t=0 t=1 subject to ∞ X     lτ−1 δτ−t u y(kτ−1 ) − IC: Vt ≡ + lτ − kτ ≥ V A y(kt ) δ τ=t ∀t ≥ 1. (8) We will show later that the solution to this relaxed problem will end up automatically satisfying the breakeven constraint (5). Hence the optimal contract can be characterized by the solution to the relaxed problem. What makes this problem tractable is that the relaxed problem admits the following convenient recursive representation. Observe that starting from any date t, the effect of past history is summarized in the single state lt−1 variable wt ≡ y(kt−1 )− , the borrower’s net wealth which is the value of current output less inherited δ debt. So the contracting problem can be restated as follows. Lemma 4. The maximum attainable welfare V(w) for a borrower with initial wealth w must satisfy   l l V(w) = max[u(w + l − k) + δV(y(k) − )] subject to: V(y(k) − ) ≥ V A y(k) (9) l,k δ δ Denote target wealth by Ω(l, k) ≡ y(k) − δl . Then (9) can be restated as: V(w) = max[u(w + l − k) + δV(Ω(l, k)] l,k subject to: V(Ω(l, k)) ≥ V A (y(k)) (10) This problem can be broken into two stages. At the first stage, given any ‘target’ wealth Ω for the next date, select (l, k) to minimize the net investment cost, i.e, the sacrifice of current consumption k − l, subject to the incentive constraint V(Ω) ≥ V A (y(k)). Let the resulting minimized cost be denoted by C(Ω). Formally, C(Ω) = min(k − l) l,k subject to: y(k) − l =Ω δ and V(Ω) ≥ V A (y(k)). (11) Then at the second stage, select the optimal target wealth Ω(w) for the next date, given current wealth w. We summarize this as follows. Lemma 5. The maximum attainable welfare V(w) for a borrower with initial wealth w must satisfy V(w) = max[u(w − C(Ω)) + δV(Ω)] Ω (12) Let us start with the first stage cost minimization problem. Given target wealth Ω and capital choice k, the associated current loan must be l(Ω, k) = δy(k) − δΩ. Hence we can simplify (11) and reduce it to choice of investment alone as follows: C(Ω) = δΩ + min(k − δy(k)) k subject to: V(Ω) ≥ V A (y(k)) (13) So when Ω ≥ w∗ , the agent invests kδ from the very first period and hence C(Ω) in that case becomes δΩ − [δy(kδ ) − kδ ]. While if Ω < w∗ , the incentive constraint binds and in particular   V(Ω) = V A y(k) (14)  −1    so the resulting investment size is k(Ω) = y−1 V A (V(Ω)) . Since V(Ω) < V(w∗ ) = V A y(kδ ) , and V A , V and y are increasing, hence when Ω < w∗ there will be underinvestment: k(Ω) < kδ . We summarize this in the following lemma. Lemma 6. For target wealths Ω smaller than w∗ , investment k(Ω) is smaller than the efficient level kδ , and equal to the efficient level otherwise. So given target wealth Ω, optimal investment is uniquely determined:  kδ if Ω ≥ w∗  −1  k(Ω) ≡ y−1 V A (V(Ω)) otherwise and the cost function is C(Ω) = δΩ + k(Ω) − δy(k(Ω)) (15) which is continuous and strictly increasing. Clearly the marginal cost of target wealth is δ for wealthy borrowers (w > w∗ ) and larger than δ for poor borrowers. The second-stage problem involves choosing the target wealth Ω. Observe first that the set of attainable target wealths is bounded above by wδ , since the marginal cost of target wealth is bounded below by δ, and current consumption must be non-negative owing to the Inada conditions. It is also bounded below, e.g., by the incentive constraint which requires V(Ω) ≥ V A (y(0)). Hence there always exists an optimal target wealth.12 The optimal target wealth however may be non-unique. In what 12 This relies on the upper semi-continuity of the value function, which follows from a standard recursive argument. follows we consider any policy function which is a measurable selection from the optimal target wealth correspondence. Since V is strictly increasing, a higher target wealth is always valuable. The borrower must trade off a higher target wealth against the current cost. The concavity of u implies wealthier borrowers incur a lower marginal cost (in terms of sacrifice of utility from current consumption) of achieving higher future wealth. Hence those currently wealthier will remain wealthier in future. Lemma 7. Ω(w) is nondecreasing in w. One useful consequence of this result is the following. Lemma 8. The value function V(w) is differentiable almost everywhere, with derivative equal to u 0 (c(w)), where c(w) ≡ w − k(w) + l(w) denotes the agent’s consumption policy. More generally, at every w, the right hand derivative of V is bounded below by u 0 (c(w)). We continue to focus on the case where initial wealth is below the threshold w∗ . We are now ready to present our first main result, showing the absence of any poverty t rap: the wealth of every poor agent will converge to the first-best threshold w∗ . Proposition 1. If w < w∗ , the sequence of net wealths wt is strictly increasing, strictly smaller than w∗ at every t, and converges to the first best threshold w ∗ as T → ∞ . The corresponding investment sequence kt is nondecreasing and converges to kδ , and consumption ct is nondecreasing and converging to c∗ (w∗ ). The underlying intuition is the following. The sequence of net wealths is monotone, hence must converge. This implies that near the limit, consumption is almost perfectly stationary, i.e., the consumption distortion vanishes asymptotically. Then there cannot be any underinvestment in the limit. Otherwise it is feasible for the agent to increase welfare by raising investment slightly and finance it with a combination of altered borrowing and self-financing which preserves the incentive constraint. A similar logic (but reversed) ensures that convergence cannot be achieved in finite t ime (because that would imply absence of a production distortion and presence of a consumption distortion at the previous date). Observe also that the no Ponzi scheme condition (5) holds for the optimal contract for any poor agent starting below w∗, because the wealth and behavior of such agents eventually are arbitrarily close to those of someone who starts with w∗ . Hence as T → ∞, δT lT converges to l(w∗ )limT →∞ δT = 0, and LBC is automatically satisfied at the solution to the relaxed problem in which it was dropped. Our next main result is that the optimal strategy involves progressive lending: when w < w∗ optimum loan size increases over time. Proposition 2. Starting with any w < w∗ , the borrower obtains a loan l(w) which is strictly positive and strictly increasing in w. The reasoning is based on observing that the optimal loan size is characterized by the binding l(w) ) = VA (y(k(w)) for all w < w∗ . The loan size l(w) is rising in w δ exceeds VA0 (y(k(w)) = u 0 (cA (y(k(w)))), where cA (w 0 ) denotes the optimal incentive constraint: V(y(k(w)) − if V 0 (Ω(w)) = u 0 (c(Ω(w)) consumption of an agent in autarky when starting with wealth w 0 . In other words, consumption on the equilibrium path lies below consumption on the corresponding autarkic outside option which generates equal welfare. This in turn holds because the borrower has access to a more productive technology on the equilibrium path, implying a faster rate of consumption growth. Since the present value of consumption is the same on and off the equilbrium path, current consumption on the equilibrium path must be lower. We conclude this section by considering the effect of lowering π, the profit target of the lender. This corresponds either to a higher bargaining power of the borrower, or an increase in ‘aid’ disbursed by a non-profit lender. The latter interpretation corresponds to an external aid donor providing aid to facilitate lending by a non-profit MFI that seeks to design contracts to maximize the present value utility of the representative borrower, subject to a break-even constraint. If the external aid donor provides aid a > 0 per borrower, the MFI’s profit target is lowered from 0 to −a. Proposition 3. Suppose the borrower has initial endowment e < w∗ + π, and the profit target of the lender is lowered from π to π − a with a > 0. This raises (weakly) the net wealth, borrowing and investment at every date, while long run wealth, borrowing, investment and consumption are unaffected. Hence the effects of aid are entirely ‘front-loaded’, just as in a neoclassical growth model. Proposition 3 follows straightforwardly from our preceding results. Given the initial endowment e of the borrower, this raises her initial net wealth from e − π to e − π + a. Owing to the monotonicity of the Ω(w), k(w), l(w) functions, this results in a (weak) increase in the net wealth, borrowing and investment of the borrower at the next date, and thereafter at every subsequent date. However in the long run the borrower’s wealth must converge to the same limit w∗ , and so must all the other outcomes.