Each entrepreneur has the opportunity to invest a fixed amount (normalized at unity) in one project at most. Letebe the entrepreneur’s initial wealth. For simplicity, it is assumed that initial wealth cannot be used to invest in the project, and yields zero return: after one period, initial wealth is stille. Therefore, to make the investment, the entrepreneur has to borrow from a lender. The entrepreneur can choose between two possible techniques – good and bad – to undertake the investment. If the project is successful, it yields a return ofIj, where the subscriptj =g or
brepresents the good or bad technique, respectively, andIg < Ib. If the project is unsuccessful,
it has a return of zero. The probability of success of the project ispj. The project with the good
technique is more likely to succeed – i.e,pg > pb. Furthermore, it is assumed that the expected
return from the project with the good technique is higher than that from the project with the bad technique – i.e.,pgIg > pbIb. This assumption implies that it is not efficient to finance a project
with the bad technique.
A loan contractEconsists of two elements: the repayment requirementR(R= 1 +r, where
r is the interest rate on the loan) and the amount of collateral C. The probability of obtaining a loan is given byγ (0≤ γ ≤ 1), whereγ indicates the fraction of the entrepreneurs who can obtain the loans at the specified contract.
At the end of time 1, given technique j, the entrepreneur’s wealth is Y1
j = e +Ij −R
if the project is successful, or Yj0 = e−C = Y0 if it is unsuccessful. It is assumed that all entrepreneurs have the same utility functionU (U increases in wealth at a decreasing rate – that is U0(Y) > 0andU00(Y) < 0). It is assumed there is decreasing absolute risk aversion – that isdA/dY <0, whereAis the absolute risk aversion (A=−U00(Y)/U0(Y)).2 Thus, as will be shown later, the entrepreneur with more initial wealth is less risk-averse and tends to invest with
2This assumption has been made in Stiglitz and Weiss [1992] and Coco [1999], and has been proved to be critical
the bad technique, compared with the one with less initial wealth. At the end of period 1, the entrepreneur’s expected utility from making a loan application under contactEis
EUj(E, γ) =
γ[p
jU(e+Ij −R) + (1−pj)U(e−C)] + (1−γ)U(e) if C < R,
γ[pjU(e+Ij −R) + (1−pj)U(e−R)] + (1−γ)U(e) if C ≥R.
If the loan is fully secured (C ≥ R), the entrepreneur always pays off the loan. In this situation, the entrepreneur’s utility decreases withR and is independent of C. As a result, the indifference curve is vertical inR−C space.
However, a full collateral requirement (C ≥R) results in limited credit provision, especially to low wealth entrepreneurs. As a result, the focus is put on the situation where only partial collateral is required (C < R). When the loan is partially secured (C < R), the indifference curve inR−Cspace is downward sloping,
−dC
dR|U =
pjU0(e+Ij −R)
(1−pj)U0(e−C)
>0, (4.1)
since the entrepreneur requires a reduction in the collateral requirementCto compensate for an increase in the repayment requirementR. The indifference curve is also concave inRandC as shown in panelaof Figure 4.1.3Notice thatγdoes not affect the shape of the indifference curve. In panel a of Figure 4.1, curvesU¯g andU¯b are the indifference curves associated with the
good technique and the bad technique, respectively, givenγ. The two curves are drawn on the assumption that, with the given technique, the entrepreneur is indifferent between the contracts on curveU¯gandU¯b. CurveU¯gis steeper than curveU¯bat any point(R, C). This is because, with
the good technique, the entrepreneur is more likely to succeed and repay the loan, and thus is less likely to lose the collateral. Therefore, when he/she invests with the good technique, the
3The slope of the indifference curve decreases inR, since
d2C dR2|U¯ =− pj 1−pj U0(e+Ij−R) U0(e−C) −U 00(e+I j−R) U0(e+I j−R) +U 00(e−C) U0(e−C) dC dR <0.
a. With no choice of techniques. b. With choice of techniques R Ub Ug SL R C R=C C R=C ¯ U good technique is used bad technique is used 1 O O
Figure 4.1: Entrepreneur’s indifference curves and switch line.
entrepreneur is willing to accept a larger increase in the collateral requirement C in exchange for a given reduction in the repayment requirementRthan when he/she uses the bad technique:
−dC dR| g U = pgU0(e+Ig −R) (1−pg)U0(e−C) > pbU 0(e+I b−R) (1−pb)U0(e−C) =−dC dR| b U. (4.2)
It is assumed that the entrepreneur is rational, and that he/she chooses the contract and the technique that gives him/her the highest level of expected utility. The locus of the contracts at which the entrepreneur is indifferent between the two techniques is called the switch line and is graphed as an upward sloped curveSLinR−Cspace in panelbof Figure 4.1. This switch line is upward sloping, since
dC
dR|Ug=Ub =
pbU0(e+Ib−R)−pgU0(e+Ig−R)
(pb−pg)U0(e−C)
>0. (4.3)
Assumption 4.1. When full collateral is required, i.e., R = C, the entrepreneur prefers the good technique; when zero collateral is required, i.e.,C = 0, the entrepreneur prefers the bad technique since defaulting is costless.
This assumption guarantees that, for the same γ, the entrepreneur’s indifference curves associated with different techniques (curvesU¯g and U¯b) cross and cross only once in the area
whereR > C (see panel a in Figure 4.1). The switch lineSLis thus an upward sloping curve in this area (see panel b in Figure 4.1), and it is truncated by lineR= 1sinceR≥1.
ForR andC combinations that lie above the switch lineSL, the entrepreneur chooses the good technique. Thus, a relatively low repayment requirementRand a relatively high collateral requirement C will give rise to the choice of the good technique. Below SL, R is high and
C is low, and thus the bad technique is chosen. Therefore, by specifyingR and C, the lender can indirectly control the entrepreneur’s choice of techniques. Notice thatγdoes not affect the switch line.
The indifference curves are not concave if the entrepreneur can choose the technique. As shown in panelbin Figure 4.1, the indifference curve is depicted by curveU – the upper envelop of the indifference curves associated with the good technique and the bad technique, respectively.
The following additional assumptions about the entrepreneur’s behaviour are made:
• If the entrepreneur is indifferent between the two techniques, he/she chooses the good technique; and
• If the entrepreneur is indifferent between two contracts, he/she chooses the one that is designed for his/her type.
Assume that there are two types of entrepreneurs – wealthy entrepreneurs (r) (hereafter called rich entrepreneurs) and low wealth entrepreneurs (p) (hereafter called poor entrepreneurs). To simplify the illustration, the rich entrepreneur hereafter is referred to as a male and the poor entrepreneur is referred to as a female. The rich entrepreneur has more initial wealth than the poor entrepreneur – i.e., er > ep. Let βi denote the proportion of typei entrepreneurs in the
market, wherei=r, pandP
iβi = 1.
The assumption of decreasing absolute risk aversion implies that a difference in initial wealth results in a difference in the indifference curves: the poor entrepreneur’s indifference curve at any point inR−Cspace is flatter than that of the rich entrepreneur, given that the same technique is employed. This result occurs because the rich entrepreneur is less risk-averse. He thus requires a smaller reduction in the repayment requirementRto compensate for a certain increase in the collateral requirementCthan does a poor entrepreneur:
∂dCdR ∂e =− pj 1−pj U0(e+Ij−R) U0(e−C) U00(e+Ij −R) U0(e+I j−R) − U 00(e−C) U0(e−C) <0. (4.4)
The assumption of decreasing absolute risk aversion also implies that initial wealth has an impact on the entrepreneur’s choice of techniques. The bad technique is more attractive to a rich entrepreneur than it is to a poor individual given the same contract. This can be seen from the switch lines: the rich entrepreneur’s switch line lies above the poor entrepreneur’s switch line. This is because when the poor entrepreneur is indifferent between the two techniques, the rich entrepreneur must prefer the bad technique, and thus an increase in the collateral requirement is necessary to make him indifferent between the two techniques. A switch from the good to the bad technique can be considered as “a mean utility preserving change” and it would cause “a reduction in one’s expected utility for a more risk averse individual” (Stiglitz and Weiss [1992], p. 170). The switch linesSLrandSLpare depicted in Figure 4.2. Because the rich entrepreneur’s
switch line (curveSLr) is steeper than the poor entrepreneur’s (SLp), the two curves never cross.
R SLr SLp O C R=C I II III 1
Figure 4.2: Entrepreneurs’ switch linesSLrandSLp.
In Figure 4.2, three areas can be identified. Any contract that falls in area I results in all entrepreneurs investing with the good technique. In area II, the rich entrepreneur invests with the bad technique and the poor entrepreneur invests with the good technique. In area III, all entrepreneurs invest with the bad technique. Moreover, at any point in areaI andIII, the indifference curve of the rich entrepreneur is steeper than that of the poor entrepreneur. However, at any point in areaII, the indifference curve of the rich entrepreneur may be either steeper or flatter than that of the poor entrepreneur.
Assumption 4.2. At any contractE(R, C) in area II, ∆EUpg < ∆EUrb, where ∆EUpg =
pgU(ep +Ig −R) + (1 −pg)U(ep −C)−U(ep) and ∆EUrb = pbU(er +Ib −R) + (1 −
pb)U(er−C)−U(er).
According to Assumption 4.2, at any contract in area II, the expected utility gain of the rich entrepreneur is greater than that of the poor entrepreneur. This assumption means that if a poor entrepreneur finds a contract between the switch lineSLrandSLpdesirable, so does a rich
entrepreneur.