Information and Enforcement in Informal Credit Markets Appendix

21 9. Appendix Proof of Proposition 1: For the purpose of this proof as well as later ones, it will be convenient to represent the equilibrium as the fixed point of a certain single valued function. For this purpose, we define a mapping φ(x) as follows. Arbitrarily assign a value x to VN , the N -Phase borrower’s lifetime utility. Now, taking this value as given, solve the O-Phase problem. Notice that the sign of the loan is determined exclusively by the effective discount rate, i.e, δ(1 − θ), and the cost of funds r. It is given by
the condition δ(1 − θ)F (LO ) = 1 + r. Denote this value by L∗O and note that it is independent of x. The repayment amount, RO is, however, dependent on x, and can be expressed as follows (from the fact that the incentive constraint is binding), using (9). RO (x) = δ(1 − θ)[F (L∗O ) − x] (35) Next, we express the value of VO in terms of x, utilizing the solution to the O-Phase problem listed above. Using (4), we have VO (x) = (36) 1−δ δθ [F (L∗O − RO (x)] + x 1 − δ(1 − θ) 1 − δ(1 − θ) On substituting the expression for RO (x) obtained above and simplifying, we have VO (x) = (1 − δ)F (L∗O ) + δx (37) It is also useful to write down the difference in expected lifetime utilities between the old and new phases, i.e, the expression for VO − VN , as a function of x. This is as follows: VO (x) − x = (1 − δ)[F (L∗O ) − x] (38) We now turn to the N -Phase problem, taking, once again, VN = x and VO = VO (x). The maximization problem can be written as: (39) max F (LN ) − RN LN ,RN subject to RN ≤ δ(1 − θ)[F (L∗O ) − x] (40) and 1+r β(1 − θ) LN − [δ(1 − θ)F (L∗O ) − (1 + r)L∗O − δ(1 − θ)x] π 1 − β(1 − θ) Closely following our previous notation, but introducing influential variables more explicitly, let L̂N (x, π), R̂N (x, π) denote the coordinates of the point where both constraints bind, and let LN (π), RN (π; x) be defined as before (i.e, the latter denotes the solution to the maximization problem if the incentive constraint is ignored). The full solution is then given by (41) RN ≥ 22 (42) LN (x) = min{LN (π), L̂N (x)} (43) RN (x) = min{RN (π; x), R̂N (x)} The function φ(x) is now defined as follows. It is the value of VN , computed from (5), after substituting the solutions to the N and O-Phase problems calculated above (as functions of x). In other words, we calculated lender’s choices of loan size and repayment amounts, taking an arbitrary value x as a defaulting borrower’s continuation payoff. This, in turn, gives rise to a new continuation payoff, which we denote by φ(x). Using (5), we can explicitly write down the expression for φ as follows: (44) φ(x) = δ(1 − θ)[{1 − δ(1 − θ)}F (L∗O ) + δ(1 − θ)x] + [1 − δ(1 − θ)][F (LN (x)) − RN (x)] Rational expectations requires that φ(x) = x in equilibrium. Thus, the fixed point of φ(.) pins down the equilibrium. The proof now proceeds by establishing certain properties of the φ function. Possible non-existence, then, arises from a discontinuity in φ(x), whereas uniqueness arises from the fact that the slope of the φ function is everywhere less than 1.






Lemma 1. There exist x and x (x < x ) such that for x ≤ x , LN (x) = LN (π) (i.e, the


N -Phase incentive constraint does not bind), for x < x ≤ x , LN (x) = L̂N (x) (the constraint

binds), and for x > x , the N -Phase feasible set is empty, so that lenders cease to provide loans and φ(x) = 0. Proof: Consider any value of x for which LN (x) = LN (π). Consider a lower value of x. Notice that both the constraints (40) and (41) are relaxed, hence the feasible set is strictly larger for this value. In particular, L̂N (x) is higher as a result. Recalling the definition of LN (x), it is
clear that LN (x) = LN (π) for this lower value of x. The existence of the switch point x is thus proved. Next, observe that the right hand side of the incentive constraint in (41) (the upper bound on RN ) decreases monotonically and unboundedly as x grows large, while the RHS of the break even constraint in (41) (the lower bound on RN ) increases monotonically and unboundedly as

x increases. This proves the existence of the second threshold x beyond which the feasible set

is empty. The value of x can be explicitly calculated by letting LN = 0 and setting the two bounds on RN mentioned above, equal.




Lemma 2. φ(x) is continuous, except at x = x , differentiable except at x and x , and has a slope less than one wherever defined. Proof: It is easy to see (since objective functions and continuous in x). In differentiability is similar.

that the loan and repayment choices are continuous for x < x are continuous, and the feasible sets for each problem are convex addition, VN is a continuous function of these variables. Proving

The discontinuity at x arises because, as no loans are provided in 23 the N -Phase, borrower’s type is not revealed, and hence, the loan size in the O-Phase, too, abruptly falls to zero.13
For the result on the slope, first take the case: x ≤ x . In this case, we have φ(x) = δ(1 − θ)[{1 − δ(1 − θ)}F (L∗O ) + δ(1 − θ)x] + [1 − δ(1 − θ)][F (LN (π)) − RN (x; π)] Hence, φ (x) = δ 2 (1 − θ)2 − δ(1 − θ).∂RN /∂x. It is easy to see that the last derivative is
positive, so that φ (x) < δ 2 (1 − θ)2 < 1.
Now turn to the case: x > x , where the N -Phase IC is binding, so that LN (x) = L̂N (x) and RN (x) = R̂N (x). Observe, by comparing (10) and (15), that the incentive constraints in the N and the O-Phase are essentially the same, so that R̂N (x) = RO (x). Using this and the expression for RO (x) in the expression for φ(x), we obtain
φ(x) = δ(1 − θ)x + [1 − δ(1 − θ)]F (L̂N (x)) As argued in the proof of the previous lemma, the feasible set shrinks as x increases, so that L̂N (x) is decreasing in x. Hence, in this case, φ(x) < δ(1 − θ) < 1. That establishes the result. For the next result on equilibrium existence, as well as the later comparative static result on the effect of changes in the composition of the borrower pool, we need to know how the function φ behaves with respect to changes in the parameter π. For this purpose, introduce π explicitly in the function: φ = φ(x; π)..

Lemma 3. φ(x; π) is increasing in π, and x is independent of π. Proof: Observe that, for a fixed x, the feasible set of the N -Phase problem strictly expands with an increase in π. This is because the break even constraint (examine (41)) is now more easily satisfied, due to the decrease in lending risk. Given that the objective function is strictly monotone in both LN and RN , it follows that the maximized value F (LN (x))−RN (x) is strictly increasing when π increases. On the other hand, the first term in the expression for φ(x) is independent of π (see (44)), since the O-Phase choices, made under full information about borrower’s type, do not directly depend on π. It follows that φ(x, π) is an increasing function of π; with a parametric increase in the latter, the φ(x) schedule jumps up.

For the last part of the claim, observe how x is defined. It is obtained by equating the right hand sides of (40) and (41), putting LN = 0. Clearly, π plays no role in the determination of this value. The proof now follows easily from the preceeding lemmas. Given some π, it is not hard to see



that a necessary and sufficient condition for the existence of a fixed point is that φ(x ) < x , i.e, at the point of discontinuity, the function lie below the 45 degree line. This follows from the intermediate value theorem, using the fact that φ(0) > 0, and φ is a continuous function in the

range [0, x ]. Now, suppose a fixed point exists for π = π1 . Consider some π2 < π1 . By Lemma



3, φ(x , π2 ) < φ(x , π1 ). Since an equilibrium exists for π1 , by the preceeding argument, it 13This is not made explicit in the mathematical formulation of the problem, but is clearly the sensible depiction. 24





follows that φ(x , π1 ) < x . Combining the last two inequalities, we have φ(x , π2 ) < π2 , which implies existence for π = π2 . Uniqueness follows from the fact that the slope of φ(x) is everywhere less than one. Proof of Proposition 2: It can be easily checked that, in equilibrium, the incentive constraints of the two optimization problems define identical bounds on the repayment amounts, RO and RN . This is because, on default, the borrower loses the same payoff from termination– (VO − VN ), appropriately discounted, since on default in any phase, the borrower returns to the beginning of a new relationship. Since the incentive constraint in the O-Phase is always ∗ ≤ R∗ . Now, if possible, let L∗ ≥ L∗ . Then, from (6), V ∗ − V ∗ = binding, it follows that RN O N O O N ∗ ∗ ∗ − R∗ }] ≤ 0, which is impossible, since the incentive constraint (1 − δ)[{F (LO − F (LN )} + {RN O cannot be satisfied. ∗ = R∗ , implying R∗ /L∗ > R∗ /L∗ . That If the N -Phase incentive constraint is binding, RN O N N O N proves the last part of the result. Proof of Proposition 3: The proof for part (i) follows straight from Lemma 3. Since the function φ(x) undergoes an upward shift as π increases, it follows that the value of the fixed point x∗ goes up. However, x∗ is nothing but the equilibrium value of VN . For part (ii), utilize the fact that the incentive constraint in the O-Phase problem is binding. Thus, utilizing (37) at the equilibrium fixed point, we have (45) VO (x∗ ) = (1 − δ)F (L∗O ) + δx∗ Since L∗O is independent of π (as argued previously), and x∗ increases as π increases, the equilibrium value of VO given by VO (x∗ ) also increases as a result. For part (iii), suffice it to notice from (18) that L∗O is determined entirely by the “effective” discount rate, δ(1 − θ). Further, since the incentive constraint (9) binds in equilibrium, and ∗ as shown above, the equilibrium value of VN increases as π goes up, we conclude that RO decreases with an increase in π. Proof of Proposition 4: We start with the following lemma: Lemma 4. Construct the φ(x) function exactly as before, with the cash constraint imposed as an additional requirement. Then, there exists x̄ such that for x ≤ x̄, the new function is identical to the old one, whereas for x > x̄, φ(x) = 0. Proof: Examine the N -Phase problem from the previous section, i.e, without the cash constraint. Notice that F (LN (x)) − RN (x) is decreasing in x (since the feasible set shrinks as x increases; recall the proof of Lemma 1). Choose x̄ such that F (LN (x̄)) = RN (x̄). The proof is now obvious. Redefine the function φ(x) by introducing the rationing parameter q as follows. For any x and q, redefine the N and O-Phase problems exactly as before, with the term qx replacing x in the incentive constraints. For ease of understanding, introduce q explicitly in all functional forms; thus LN = LN (x, q), RN = RN (x, q) RO = RO (x, q) and in particular, defineΦ(x, q) as follows (46) Φ(x, q) = 1−δ [{1−δ(1−θ)}{F (LN (x, q)−RN (x, q)}+δ(1−θ){F (L∗O )−RO (x, q)}] 1 − δ + (1 − q)δθ 25 The above expression is merely the value of VN , calculated by assuming that on termination of a contract, a borrower faces a lifetime utility of x, if rehired immediately, the probability of finding a new credit source in any date being captured in the parameter q. Now, for any given x, define q(x) as follows. q(x) = 1 (47) if F (LN (x, 1)) ≥ RN (x, 1) = q̃ ∈ (0, 1) if F (LN (x, q̃)) = RN (x, q̃) This formulation incorporates the consistency requirement imposed above. The nature of q(x) is summarised in the following lemma. Lemma 5. For every x, there exists an unique q(x), which can be characterized as follows. For x ≤ x̄, q(x) = 1. For x > x̄, q(x) = x̄/x. Proof: Fix x. Notice that as q is reduced, both the incentive and the break even constraints in the N -Phase problem are relaxed. Hence, the maximized value of the objective function F (LN (x, q)) − RN (x, q) is nondecreasing as q increases. This guarantees uniqueness. Next, assume q(x).x = x̄. It has to checked that the consistency requirement is satisfied. Notice that since all constraints in the two maximization problems depend only on the term qx, the solutions to these problems would be unchanged if q(x) were replaced by 1, and x by x̄. Now, by definition of x̄, F (LN (x̄)) = RN (x̄). Hence, F (LN (x, q(x))) = RN (x, q(x)), implying that the consistency requirement is satisfied for the above formulation of the function q(x). That completes the proof. Now, redefine the function φ(x) of the previous section as follows: φ(x) = Φ(x, q(x)). The following lemma is crucial in guaranteeing existence. Lemma 6. The function φ(x), as defined above, is continuous at all points, and almost everywhere differentiable. For x ≤ x̄, the function takes the form exactly as defined in the previous section. For x > x̄, φ(x) has a negative slope. Proof: For x ≤ x̄, q = 1 satisfies the consistency requirement (see Lemma 4). From the uniqueness of q(x) (Lemma 5), it follows that in this range, φ(x) is identical to that in the previous section. Next, turn to the case where x > x̄. Notice that all the constraints in the N and the OPhase problems depend only on the factor qx. Since xq(x) = x̄ from Lemma 5, it follows that LN (x, q(x)) = LN (x̄, 1), RN (x, q(x)) = RN (x̄, 1) and RO (x, q(x)) = RO (x̄, 1). Using these observations, and the fact that q(x) = x̄/x by Lemma 5, we have (48) Φ(x, q(x)) = (1 − δ)x .Φ(x̄, 1) (1 − δ)x + δθ(x − x̄) Recalling the new definition of φ(x), we have (49) φ(x) = (1 − δ)x .φ(x̄) (1 − δ)x + δθ(x − x̄) 26 On examination of the above equation, it becomes clear that φ(x) is continuous and negatively sloped in the interior of this range. Further, observe from above that lim φ(x) = φ(x̄) (50) x→x̄− which proves continuity throughout. The existence result follows trivially from the previous two lemmas. Proof of Proposition 5: As before, let RO (x), L∗O denote the solution to the O-phase problem when the defaulting borrower’s continuation payoff is x. Also, let ΠO (x) denote the resultant lifetime expected profit of the lender. Using (13) and (35), we have: (51) ΠO (x) = β [δ(1 − θ)F (L∗O ) − (1 + r)L∗O − δ(1 − θ)x] 1 − β(1 − θ) It is easy to see that the function is continuous and differentiable in x. On taking derivatives: (52)
ΠO (x) = −δ(1 − θ). 1−β 1 − β(1 − θ) We now turn our attention to a different maximization problem. Consider the problem of devising an N -phase contract (L̃N (x), R̃N (x)) which maximizes the lender’s lifetime expected profit, subject to the borrower’s usual incentive constraint (assuming the continuation payoff to be x in case of termination), and a participation constraint which requires the N -borrower’s lifetime utility to be at least as great as that of an O-borrower. Notice that this problem is slightly different from the dual of the problem described in (39) through (41). The participation constraint of that dual problem requires an individual lender’s contract to provide at least as much expected payoff to the N -borrower as that provided by any other lender in the N -phase. The reason for this difference will be clear in a moment. The problem described above can be written as follows: (53) max πRN − (1 + r)LN RN ,LN subject to (54) (1 − δ)[F (LN ) − RN ] + δ(1 − θ)VO (x) + δθx ≥ (1 − δ)F (LN ) + x and (55) [F (LN ) − RN ] ≥ F (L∗O ) − RO (x) The incentive constraint, equation (54) above, can be rewritten, using the value of VO (x) from (37), to obtain (56) RN ≤ δ(1 − θ)[F (L∗O ) − x] 27 Denote by (L̃N (x), R̃N (x)) the optimal solution to this problem, and let Π̃N (x) denote the corresponding expected payoff of the lender (using the value of ΠO (x) derived above in (51)). Thus, we have (57) Π̃N (x) = (1 − β)[π R̃N (x) − L̃N (x)] + πβ(1 − θ)ΠO (x) Let ψ(x) = π R̃N (x) − L̃N (x). Then
(58)

Π̃N (x) = (1 − β)ψ (x) + πβ(1 − θ)ΠO (x) Subtracting (58) from π times (52), we have
(59)

πΠO (x) − Π̃N (x) = −(1 − β)[ψ (x) + πδ(1 − θ)]

Lemma 7. πΠO (x) = Π̃N (x) for all x. Proof: Note that the participation constraint in the N -phase problem above always binds. If not, the value of the objective function can be increased by reducing LN , keeping RN fixed (the incentive constraint continues to be satisfied). The incentive constraint may or may not bind at the optimum. Thus, two possible cases arise. We deal with each of them separately. Case 1: (The incentive constraint is slack). In this case, the first order conditions require that the lender’s isoprofit curve be tangent to the participation constraint at the optimum point.
This implies L̃N (x) = LN (π) (where πF (LN (π)) = 1 + r) and is independent of x. From the binding participation constraint, we have: (60) R̃N (x) = F (LN (π)) − [1 − δ(1 − θ)]F (L∗O ) − δ(1 − θ)x Using this in the definition of ψ(x) and on differentiating, we obtain: (61)
ψ (x) = −πδ(1 − θ) Using this in (58), we have (62)

πΠO (x) − Π̃N (x) = 0 which establishes the result for this case. Case 2: (Both constraints bind). Treating (54) and (55) as equalities, and on eliminating RN , we have F (LN ) = F (L∗O ) − RO (x) + δ(1 − θ)[F (L∗O ) − x] = F (L∗O ], implying L̃N (x) = L∗O . Then, from (55), we have R̃N (x) = RO (x) = δ(1 − θ)[F (L∗O ) − x]. Using these values in the definition of ψ(x), we have (63)
ψ (x) = −δ(1 − θ) as in the previous case. Therefore the lemma holds in this case too. Denote, as before by x∗ , the fixed point of the equilibrium correspondence, i.e, φ(x∗ ) = x∗ . 28 Lemma 8. (i) Π∗O = ΠO (x∗ ) (ii) Π∗N = 0 > Π̃N (x∗ ) (iii) π Π̃O − Π̃N = π Π̃O (0) − Π̃N (0) Proof: Part (i) follows from definition. For (ii), observe that at the fixed point, VO∗ > VN∗ , ∗ > F (L∗ ) − R∗ , so that at x = x∗ , the participation constraint which implies F (L∗O ) − RN N N in the variant of the N -phase problem described in (53) through (55) is strictly tighter than that in the dual of the true N -phase problem outlined in (39) through (41). Consequently, the value of the objective function is strictly lower, i.e, Π̃N (x∗ ) < Π∗N . Coming to part (iii), we show that π Π̃O > πΠO (0) and Π̃N > Π̃N , but by the same margin.14 Turning to the O-phase first, utilizing the fact that L∗O is the optimal loan size in either problem, and the fact that the respective incentive constraints are binding, we obtain using (32), R̃O = δF (L∗O ), and from (56), R̃O (0) = δ(1−θ)F (L∗O ). Using these values in the definition of ΠO from (13), we obtain (64) Π̃O − ΠO (0) = δθ(1 − β) · F (L∗O ) 1 − β(1 − θ) Next, observe by comparing (32) with (54), and by noting the difference in value between R̃O and RO (0), that for any given value of LN , both the incentive and the participations constraints of the true N -phase problem (as defined in (31) through (33)) are higher than the corresponding ones defined in (54) and (55), by an identical magnitude given by δθF (L∗O ). It is easy to see from the diagrammatic depiction of the N -phase problem, that L̃N = L̃N (0), while R̃N − R̃N (0) = R̃O − R̃O (0) = δθF (L∗O ). Using this in the definition of ΠN from (14), we obtain (65) Π̃N − Π̃N (0) = πδθ(1 − β) · F (L∗O ) 1 − β(1 − θ) Comparison of (64) and (65) yields part (iii) of the Lemma. The proof of Proposition 5 is now straightforward. From Lemma 7, πΠO (0) − Π̃N (0) = πΠO (x∗ ) − Π̃N (x∗ ). Using Lemma 8, it follows that π Π̃O − Π̃N > πΠ∗O − Π∗N , which completes the proof. 14The difference arises because while the true incentive constraints ((30) and (32)) assume that a non- defaulting borrower gets an identical contract elsewhere on exogenous termination, the incentive constraints in (35) and (54) (with x = 0) assume that the borrower receives zero subsequent payoff on exogenous termination, even if he has no history of default.