A Theory of Progressive Lending Ch 2

4 Extension to Uncertain Productivity Shocks In autarky, the output of the agent is now g(k; s), where s is an i.i.d. shock with a CDF J(.) over a finite support [s, s̄], and g is twice differentiable, satisfying gk > 0 > gkk , g s > 0, gks ≥ 0 besides Inada conditions. This includes both the case of additive (g(k; s) = g̃(k) + s) and multiplicative shocks (g(k; s) = sg̃(k)). In the presence of the lender, the agent has a production function y(k; s) with higher output and marginal product of capital than the autarkic technology at any (k; s) and satisfying all other analogous properties of g() mentioned above. We assume that at any date t, the shock is observed before investment decisions are made. This is analogous to the model of Albuquerque and Hopenhayn (2004)). If the shock is observed after investments are made, it can be verified that all the results continue to apply if the shocks are additive.13 4.1 Autarky The autarkic value function is for a state with current wealth e which is the value of output realized from investment at the previous date, and shock s applying to production at the current date: VA (e, s) = max [u(e − k) + δE s 0 {VA (g(k; s), s 0 )}] 0≤k≤w (16) Standard arguments imply that optimal investment kA (e, s) is non-decreasing in both arguments. This implies that wealth follows a monotone Markov process (Hopenhayn and Prescott (1992)): the distribution of wealth at the next date conditional on current wealth w first order stochastically dominates that conditional on a lower current wealth w 0 < w. Note that unlike Hopenhayn and Prescott (1992) we have not imposed any exogenous upper bound on capital stocks, so the state space is not compact and their results concerning invariant or limiting distributions do not necessarily apply. 13 The main complication arises from the need to keep track of a single dimensional measure of wealth the value of which can be targeted while making current investment decisions. If the current shock is known, the borrower and lender can predict the former’s output that will result next period. If the shock is unknown, the borrower’s output cannot be predicted with certainty. However, in this case if the shock is additive, the mean output can be predicted which serves to pin down the distribution of output as well as wealth next period by their corresponding certainty equivalents (CE). That is, with production function ỹ(k) + s, the wealth next period from choice of (k, l) today will be w̃ + s where w̃ = ỹ(k) − δl . Hence future CE wealth (w̃ + Es) is determined by current decisions, and a higher CE wealth results in a wealth distribution that is ‘larger’ in the sense of first order stochastic dominance. Hence we can extend the theory to this case where we replace target wealths by target CE wealths. 4.2 Access to Lender A lender provides access to the superior technology y(k; s), and a lending contract featuring a sequence of a state-contingent investments kt (ht ) and transfers pt (ht ) where ht ≡ (s0 , . . . , st ), provided the borrower has not defaulted at any previous date. As in the deterministic case, the analysis of Pareto efficient contracts with a minimum profit constraint π for the lender, reduces to the analysis of CE contracts with a lender breakeven constraint and a borrower initial wealth of w = e − π. Such a contract generates welfare for an agent starting with wealth w and initial shock s0 : u(w − k0 + p0 ) + E s1 ,s2 ,.. ∞ hX δt u(y(kt−1 , st−1 ) − kt + pt ) i (17) t=1 The lender’s breakeven constraint is E[ ∞ X δt pt ] ≤ 0 (18) t=0 while the incentive constraint requires at any date t and following any history ht : E ∞ hX i δ j u(y(kt−1+ j ), st+ j ) − kt+ j + pt+ j )|ht ≥ E s 0 [VA (y(kt−1 ), st ), s 0 )] (19) j=0 A contract is feasible if it satisfies (18) and (19). Let welfare V(w, s0) denote the maximum value of (17) subject to the two feasibility constraints. Analogous to the deterministic case, we obtain: Lemma 9. Any feasible contract can be implemented by a sequence of one period state-contingent loans satisfying l0 = p0 , lt (ht ) = pt (ht ) − lt−1 (ht−1 ) , δ which (i) are always repaid on the equilibrium path, and (ii) satisfy the break-even condition: E[ lim δT lT ] ≤ 0 T →∞ (20) In other words, the lender provides a fresh one period loan at each date-history pair, which provides the required net transfer after allowing for repayment of the previous loan. The break-even condition (20) reduces to a ‘no-Ponzi’ scheme requirement which holds in expectation. To ensure this, we shall assume there is a finite lower bound w on the borrower’s net wealth imposed by law or MFI policy. This amounts to a limit on loan size that depends on anticipated current output: lt ≤ δ[y(kt, st) − w] (21) As in the deterministic case, we shall ignore constraint (20) and then show that the solution to the ‘relaxed’ problem (which incorporates constraint (21) instead) satisfies it automatically We then obtain a recursive representation of the optimal contracting problem: l V(w, s) = max[u(w − k + l) + δE s 0 [V(y(k, s) − , s 0 )] k,l δ (22) subject to the incentive constraint l V(y(k, s) − , s 0 ) ≥ VA (y(k, s), s 0 ), ∀s 0 ∈ [s, s̄] δ (23) l ≤ δ[y(k, s) − w] (24) and borrowing constraint This is similar to the case of certainty, except that there is a separate incentive constraint for each possible realization of the shock at the next date, while the objective function involves only the corresponding expected value of the continuation utility. The measure of net wealth is now w ≡ y(k, s) − δl , which is bounded below by w and unbounded above. And we can continue to break down the recursive contracting problem into two steps. First, given current shock s and a target net wealth Ω ≥ w in the next period, minimize the cost in terms of foregone current consumption: C(Ω, s) ≡ min(k − l) subject to: (23) and y(k, s) − l =Ω δ (25) Then at the second step, select the target wealth Ω(w, s) for next period: V(w, s) = max[u(w − C(Ω, s)) + δE s 0 V(Ω, s 0 )] Ω≥w (26) As in the deterministic case, optimal target wealths are well-defined, and we consider any policy function which is a measurable selection from the optimal policy correspondence. Let an optimal investment and financing policy be denoted by k(w, s), l(w, s) respectively. Our main result for the case of uncertainty is the following. Proposition 4. (a) Wealth next period w 0 = Ω(w; s) is non-decreasing in current wealth w and productivity shock s. (b) k(w, s) ≤ kδ (s) for all w, where δy 0 (kδ (s); s) = 1. (c) Along any history, the no-Ponzi scheme condition limT →∞ δT lT ≤ 0 holds, and the lender breaks even. (d) k(w; s) is non-decreasing in w. (e) Conditional on wt ≥ wt−1 and a sequence of nondecreasing productivity shocks: st+ j ≥ st+ j−1 , ∀ j = 0, 1, 2.. (27) net wealth wt+ j and investment kt+ j are nondecreasing in j. (f) For any s, there exists wealth threshold w∗ (s) such that k(w, s) = kδ (s) and l(w, s) ≤ 0 for all w ≥ w∗ (s). Part (a) implies the evolution of wealth follows a monotone Markov process. (b) ensures there is never any under-investment, while (c) implies the break-even condition for the lender along every history. Part (d) says that investment is non-decreasing in wealth. These results imply part (e): conditional on a nondecreasing sequence of productivity shocks (combined with wealth rising initially), the wealth and investment of the borrower rises monotonically over time. This shows that the main result of Albuquerque and Hopenhayn (2004)) regarding the effect of rising ‘age’ of the relationship (conditional on productivity shock) continues to hold in this setting. Once wealth rises sufficiently, (f) states that investment levels become first-best. As wealth rises further, the agent becomes a lender rather than borrower and attains first-best investment. Of course, welfare is not first-best, owing to lack of insurance and resulting consumption distortions. Wealthy agents may suffer a string of negative productivity shocks and subsequent declines in wealth and forced to borrow again. We have not provided any results concerning invariant or limiting wealth distributions. Such results could be obtained upon imposing exogenous upper bounds on lending and capital investment, as in (Hopenhayn and Prescott (1992)), which ensure a compact state space. Such bounds would be arbitrary and ad hoc, so we avoid imposing them. Whether such results can be obtained despite the absence of such bounds, remains an interesting open question. Table 1: MFI Portfolio at Risk Average, By Region Regions14 Percentage of Total Borrowers Africa East Asia and the Pacific (EAP) Eastern Europe and Central Asia (ECA) Latin America and the Carribean (LAC) Middle East and North Africa (MENA) South Asia 5% 1% 3% 19% 2% 57% Percentage of Gross Loan Portfolio (GLP) 9% 16% 11% 42% 1% 20% Portfolio at Risk> 30 Days (PAR) 10.60% 3.40% 10.00% 5.40% 3.60% 2.60% “Portfolio at Risk (PAR)”: is one of the indicators of repayment rate. Portfolio at Risk [xx] days is defined as the value of all loans outstanding that have one or more installments of principal past due more than [xx] days. This includes the entire unpaid principal balance, including both the past due and future installments, but not accrued interest. It also includes loans that have been restructured or rescheduled. “Gross Loan Portfolio (GLP)”: All outstanding principals due for all outstanding client loans. This includes current, delinquent, and renegotiated loans, but not loans that have been written off. Source MIX (2017): Global Outreach and Financial Performance Benchmark Report 2015. Table 2: MFI Clienteles, Loan Portfolio and Progressive Lending Practices in Three Countries Country Number of Active Borrowers 0 000 Gross Loan Portfolio (GLP)(USD)m India 38,097.6 11,640.8 Bangladesh Vietnam 23,977.7 7,533.9 5,753.7 7,351.9 Financial Service Provider (FSP) Bandhan Janalakshmi Bharat Financial (SKS) Share SKDRDP Grameen Bank ASA BRAC BURO Bangladesh TMSS VBSP CEP Coopbank TYM MOM Top three countries by active borrowers and top five MFIs from each of them. Source MIX (2017): Global Outreach and Financial Performance Benchmark Report 2015 and respective websites. Number of Active Borrowers 0 000 5,888.75 5,323.06 3,740.00 3,612.43 7,180.00 6,207.69 5,356.52 917.46 736.98 6,863.04 288.49 111.93 97.42 40.17 Gross Loan Portfolio (GLP)(USD)m 2,352.66 1,973.48 1,413.30 251.68 754.60 1,294.65 1,533.97 1,768.61 311.61 188.95 6,434.69 108.28 726.19 45.45 7.26 Progressive Lending? Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Appendix: Proofs Proof of Lemma 1: Denote the original problem (1) by P(e, π) and note that it can be rewritten as follows with a change of variable from transfers pt to consumption ct ≡ y(kt−1 ) + pt − kt , t ≥ 1 and c0 ≡ e + p0 − k0 : D max ∞ {ct }∞ t=0 ,{kt }t=0 E V0 ≡ ∞ X δt u(ct ) (28) t=0 subject to: ∞ ∞ ∞ X X X LPC : δt ct ≤ e − π − δt kt + δt y(kt−1 ) t=0 ICt : t=0 ∞ X Vt ≡ t=1 δτ−t u(cτ ) ≥ V A (y(kt−1 )), ∀t ≥ 1. τ=t BPC : V0 ≥ V A (e). Similarly refer to the CE problem (2) by PC (w), and using the same change of variables this can be written as: V(w) ≡ D ∞ X max ∞ {ct }∞ t=0 ,{kt }t=0 E δt u(ct ) (29) t=0 subject to: ∞ ∞ ∞ X X X t t LBC : δ ct ≤ w − δ kt + δt y(kt−1 ) t=0 ICt : Vt ≡ t=0 ∞ X t=1 δτ−t u(cτ ) ≥ V A (y(kt−1 )), ∀t ≥ 1. τ=t (30) We claim that the feasible set in P(e, π) is non-empty if and only if the feasible set in PC (e − π) is non-empty and V(e − π) ≥ V A (e). To show the ‘if’ part, observe that the solution to PC (e − π) is feasible in P(e, π). For the converse, take any feasible contract {^ct , ^kt } in P(e, π). This is feasible in P t PC (e − π). Hence the solution to the latter problem yields a payoff V(e − π) of at least ∞ t=0 δ u(ct ), which in turn is at least V A (e) owing to (BPC). Next observe that V(.) is strictly increasing, since any increase in w can be accompanied by an equivalent increase in c0 without violating either LBC or any ICt . Define π̄(e) ≡ sup{π|V(e − π) ≥ V A (e)}. It is evident that π̄(e) < ∞, since V A (e) > 0 for any e > 0. Also π̄(e) > 0, since V(e) > V A (e), as the access to superior technology combined with financial autarky ensures a higher payoff to the borrower. Since V(.) is strictly increasing, it follows that a feasible contract in P(e, π) exists if and only if π ≤ π̄(e). Finally, observe that given any π ≤ π̄(e), BPC is redundant in P(e, π), and the two problems P(e, π), PC (e − π) are the same as they involve the same objective function and constraint set. Proof of Lemma 2: Already provided in the text. Proof of Lemma 7: If this is false, there exist w1 < w2 with Ω1 ≡ Ω(w1 ) > Ω(w2 ) ≡ Ω2 . Then V(Ω1 ) > V(Ω2 ) and u(w2 − C(Ω2 )) − u(w2 − C(Ω1 )) ≥ δ[V(Ω1 ) − V(Ω2 )] > 0 (31) which implies C(Ω1 ) > C(Ω2 ). On the other hand, δ[V(Ω1 ) − V(Ω2 )] ≥ u(w1 − C(Ω2 )) − u(w1 − C(Ω1 )) (32) u(w2 − C(Ω2 )) − u(w2 − C(Ω1 )) ≥ u(w1 − C(Ω2 )) − u(w1 − C(Ω1 )) (33) which implies This contradicts the concavity of u. Proof of Lemma 8: Consider any w and a slightly higher wealth w +  > w. Since the incentive constraint in (9) does not depend on w, the policies (k(w), l(w)) and (k(w + ), l(w + )) are feasible for both agents with starting wealth w and w + . Therefore: V(w + ) ≡ u(w +  − C(Ω(w + ))) + δV(Ω(w + )) ≥ u(w +  − C(Ω(w))) + δV(Ω(w)) V(w) ≡ u(w − C(Ω(w))) + δV(Ω(w)) ≥ u(w − C(Ω(w + ))) + δV(Ω(w + )) which implies u(w +  − C(Ω(w + ))) − u(w − C(Ω(w + ))) V(w + ) − V(w) ≥   u(w +  − C(Ω(w))) − u(w − C(Ω(w))) ≥  Take limits as  → 0+. Since Lemma 7 implies C(Ω(w)) is nondecreasing in w, it is continuous almost everywhere. At any continuity point of C(Ω(w), it follows that the right-hand derivative of V exists and equals u 0 (w − C(Ω(w))). A parallel argument for the case of  < 0 with direction of inequalities reversed in (34) holds, implying the left-hand derivative of V also exists and equals u 0 (w − C(Ω(w))). Finally observe that for any  > 0, it is always feasible to let the agent consume the incremental wealth immediately, so right hand derivative of V is everywhere bounded below by u 0(c(w)). Proof of Proposition 1: Consider any w < w∗. If Ω(w) ≤ w, Lemma 7 implies that starting from w the sequence of net wealth is monotonically nonincreasing. Conversely, if Ω(w) > w, the sequence is monotonically nondecreasing. Hence either way, the sequence of net wealths must converge. This implies that the sequence of consumption and investments must also converge. Next we show that the limiting wealth w∞ cannot be smaller than w∗ . Suppose otherwise. Then we claim there is a variation on the contract which is feasible and raises the borrowers welfare. Since w∞ < w∗ , for all large t we have wt < w∗ , and there is underinvestment in the limit (kδ > k∞ ). So [δy 0 (kt ) − 1] is positive and bounded away from zero for all large t. For all large t, the incentive constraint binds, hence V(y(kt ) − lδt ) = V A (y(kt )). Consider an increase in kt by  > 0, and let lt change by ∆t  where 0 ∆t = δy 0 (kt )[1 − V A (y(kt )) ] u 0 (ct+1 ) (34) For  sufficiently small, the IC is preserved because: ∂V(y(kt + ) − ∂ lt δ ≥ u 0 (ct+1 )[y 0 (kt ) − − ∆t ) δ |=0+ ∆t ] δ 0 = V A (y(kt ))y 0 (kt ) (35) where the last equality follows from construction of ∆t (equation (34)), and the preceding inequality follows from Lemma 8. The resulting borrower’s welfare at t is Vt () ≡ u(wt + lt + ∆t  − kt − ) + δV(y(kt + ) − lt + ∆t  ) δ (36) implying that at  = 0, the rate of rise of Vt is at least: −u 0 (ct )[1 − ∆t ] + u 0 (ct+1 )[δy 0 (kt ) − ∆t ] = u 0 (ct )[δy 0 (kt ) − 1] + [u 0 (ct+1 ) − u 0 (ct )][δy 0 (kt ) − ∆t ] (37) For t sufficiently large (37) is positive, because [δy 0(kt) − 1] is positive and bounded away from zero, 0 [u 0(ct+1) − u 0(ct)] converges to zero, and [δy 0(kt) − ∆t)] = δy 0(kt) V A (y(kt )) 0(u c ) is t+1 positive and converges to a finite number as t → ∞. Hence w∞ ≥ w∗. Since Ω(w∗) = w∗, Lemma 7 implies that Ω(w) ≤ w∗ for any w < w∗. Hence w∞ ≤ w∗, and it follows that w∞ = w∗. Next, we show that first-best w ealth w ∗ c annot b e a chieved a t a ny fi nite da te. Ot herwise, there exists some date t with wt < w∗ and Ω(wt ) = wt+1 = w∗ . From Lemma 6 it follows that k( wt ) = kδ , and hence δy 0 (kt ) = 1, while ct+1 = c(wt+1 ) = c∗ (w∗ ). And ct must be strictly lower than c∗ (w∗ ), otherwise the agent achieves welfare at least V(w∗ ) at a wealth wt < w∗ . So there is a consumption distortion resulting in u 0 (ct+1 ) − u 0 (ct ) < 0. Now we can consider a sequence of reverse perturbations analogous to that constructed above, with  < 0 and converging to 0 from below. From (37) it is evident that this will raise welfare for  close enough to zero. Finally, the sequence of wealths must be strictly increasing at every date (otherwise wt+1 = Ω(wt ) = wt , and wealth will remain at wt < w∗ for ever). Proof of Proposition 2: Whenever w < w∗ the IC binds, hence V(Ω(w)) ≡ V(y(w) − l(w) ) = V A (y(w)) δ implies Ω(w) < y(w) since V(w ) > V (w ) for all w . Hence l(w) ≡ δ[y(w) − Ω(w)] > 0. 0 A 0 0 Next, we show that l(w) must be strictly increasing for all w < w∗ . Claim 1: To establish this, it suffices to show that c(Ω(w)) < cA (y(w)) for all w < w∗ , i.e., optimal consumption on the equilibrium path is smaller than the optimal consumption off the equilibrium path in the first period of any d eviation. This is because the right hand derivative of V at Ω(w) is bounded below by u 0 (c(Ω(w))), while the slope of V A (y(w)) equals u 0 (cA (y)) (where c(y), cA (y) respectively denote the optimal consumptions at the first date along the equilibrium path and in autarky respectively, starting with wealth y). Hence c(Ω(w)) < cA (y(w)) implies the slope of V at Ω(w) is larger than V A (y(w)). So Ω(w) must rise moreAslowly in w than y(w). Next, observe that if k(w) ≥ kδ , off-equilibrium-path consumption is stationary. Then Claim 1 0 follows from the IC: V(Ω(w)) = V A(y(w)), since consumption is growing on the equilibrium path (an argument similar to that used in Proposition 1 rules out the possibility that consumption is stationary on the equilibrium path). 0 0 Let ctA,t 0 denote optimal consumption at date t in autarky, resulting from a deviation at t < t . A,t+1 Claim 2: If at any date t: ctA,t ≤ ct , then the same must be true at t + 1: ct+1 ≤ ct+1 . A,t+1 To establish Claim 2, suppose otherwise that at some date t: ctA,t ≤ ct and ct+1 > ct+1 . Then ∆t ≡ δy 0 (kt )[1 − A,t+1 u 0 (ct+1 ) ]>0 0 u (ct+1 ) and so u 0 (ct )[1 − ∆t ] < u 0 (ct ) ≤ u 0 (ctA,t ) = δ ∂ A [V (g(ktA,t ))] ∂k (38) Since the IC at t binds, we have lt u(ct ) + δV(yt+1 − ) = u(ctA,t ) + δV A (g(ktA,t )) δ so ctA,t ≤ ct implies lt V A (g(ktAA,t )) ≥ V(yt+1 − ) ≥ V A (yt+1 ) δ where the last inequality again uses the IC at t. Hence g(ktA,t ) ≥ yt+1 . Since V A is concave, this implies ∂ A δ ∂k∂ [V A (g(ktA,t ))] ≤ δ [V ∂k (yt+1)]. Hence (38) implies u 0 (ct )[1 − ∆t ] < δ ∂ A [V (yt+1 )] ∂k Using an argument similar to that used in Proposition 1 , it is feasible to increase investment slightly on the equilibrium path and raise the borrower’s welfare, contradicting optimality of the original contract. Hence Claim 2 holds. 0 Finally, Claim 2 implies by induction that ctA,t ≤ ct 0 for all t 0 > t. Since kt converges to the first-best 0 capital stock kδ which strictly exceeds kδA , there exists some date t 0 > t when kt 0 = k(wt 0 ) > kδA and we would obtain a contradiction to the argument in the previous paragraph. This completes the proof. Proof of Proposition 4: (a) The same argument as in the deterministic case ensures Ω(w; s) is non-decreasing in w. To show that it is non-decreasing in s, it suffices to show that C and marginal cost of wealth Cw (Ω, s) are both non-increasing in s. Observe that C(Ω, s) = δΩ − δZ(Ω, s) (39) where k(y, s) ] subject to: δ and k(Y, s) denotes the solution for k in y(k, s) = Y. Z(Ω, s) ≡ max[y − y≥0 VA (y, s 0 ) ≤ V(Ω, s 0 )∀s 0 (40) Since (using standard arguments) VA(y, s 0) is strictly increasing and continuous in y, we can define Y(Ω) as the largest (or supremum) y satisfying the incentive constraint (IC) in (40). The IC can then be replaced by y ≤ Y(Ω). Standard arguments also imply V(w, s) is increasing in w, hence Y(Ω) is increasing. Therefore Z(Ω, s) ≡ max[y − y≥0 k(y, s) ] δ subject to: y ≤ Y(Ω) (41) As k(y, s) is non-increasing in s, it follows that Z is non-decreasing in s, and C is non-increasing in s. The unconstrained solution to (41) involves setting y(Ω, s) = yδ (s) ≡ y(kδ (s), s)), and the IC binds iff yδ (s) > Y(Ω). Hence y(Ω, s) = Y(Ω) if yδ (s) > Y(Ω) and yδ (s) otherwise. It follows that C(Ω, s) = δΩ − y(kδ (s), s)) + 1δ y(kδ (s), s)) in the former case, and δΩ − Y(Ω, s) + 1δ k(Y(Ω, s), s) otherwise. Hence the marginal cost of a higher wealth target Cw (Ω, s) equals δ if Ω is large enough that IC does not bind, so does not depend on s. When the IC does bind, the marginal cost is defined a.e. (whenever Y 0 (Ω) exists), in which case it equals δ − Y 0 (Ω)[1 − 1δ yk (k(Y(Ω), s), s)]. This is non-increasing in s because yk (k, s) is decreasing in k and increasing in s, k(Y, s) is decreasing in s, and Y 0 (Ω) > 0. (b) Suppose δy 0 (k(w, s)) < 1. Consider a small reduction in both capital and borrowing: k = k(w, s) −  and l = l(w, s) − , which then results in an increase in y(k, s) − δl . The IC constraint (23) and borrowing constraint (24) continue to hold, continuation utility E s 0 [V(y(k, s) − δl ), s 0 ] rises and current consumption is unchanged. So the original contract could not have been optimal. (c) From (b) and (24) we have lt (w, s) ≤ δ[yδ (s) − w] ≤ δ max s [yδ (s) − w], so the size of loans is uniformly bounded above, implying limT →∞ δT lT ≤ 0 along any history. (d) If this is false, there exists s and wealths w > w 0 with k(w, s) < k(w 0 , s) ≤ kδ (s). From (a), y(k(w, s), s) − l(w,s) δ ≥ y(k(w 0 , s), s) − l(w 0 ,s) , δ which implies l(w 0 , s) − l(w, s) ≥ δ[y(k(w 0 , s), s) − y(k(w, s), s)] (42) Define l 00 ≡ [k(w 0 , s) − k(w, s)] + l(w, s). Since k(w, s) < k(w 0 , s) ≤ kδ (s), the Intermediate Value Theorem implies δ[y(k(w 0, s), s) − y(k(w, s), s)] > k(w 0, s) − k(w, s) = l 00 − l(w, s). (43) (42) then implies that l(w 0 , s) > l 00 . Therefore: y(k(w 0 , s), s) − l(w 0 , s) l 00 > y(k(w 0 , s), s) − δ δ (44) implying that the agent with wealth w could feasibly borrow l 00 and invest k(w 0 , s) while preserving the incentive and borrowing constraints. This would entail the same current consumption: k(w, s) − l(w, s) = k(w 0, s) − l 00, and generate higher continuation utility because of (43), so we obtain a contradiction. (e) Note to start with that wt+1 = Ω(wt, st) ≥ wt = Ω(wt−1, st−1) upon using (a), combined with wt ≥ wt−1 and st ≥ st−1 . Now use the same argument inductively at all subsequent dates. (f) If this is false, there is some s for which k(w, s) < kδ (s) for all w. Observe first that this implies l(w, s) ≥ 0 for all w, since otherwise l(w, s) < 0 for some w and the IC (23) does not bind for an agent with wealth w. This agent can increase k and l by the same small amount and raise continuation utility strictly, while keeping current consumption unchanged. Since y 0(k(w, s), s) > 1 > 1, the borrowing δ constraint is also preserved. So the contract could not have been optimal. Since l(w, s) ≥ 0 for all w, (b) implies that net wealth next period Ω(w, s) ≡ y(k(w, s), s)) − l(w,s) δ ≤ yδ (s) for all w, i.e., is bounded above. Consumption in the next period is therefore bounded above because investment is nonnegative and borrowing is bounded above by max s 0 yδ (s 0 ) − w. On the other hand, current consumption for the agent is bounded below by w−kδ (s) since borrowing is non-negative for all w. Hence as w → ∞, current consumption goes to ∞, while consumption in the next period is bounded above. Since the utility function satisfies Inada conditions, the marginal utility of current consumption must be lower than discounted marginal utility of consumption in the following period, for sufficiently large w. Such an agent can reduce l slightly below l(w, s) which preserves borrowing and incentive constraints, and raises welfare. So the contract cannot be optimal for large enough w.