A property of the equilibrium is that investors’ beliefs about λG are cor- rect. Note also that the probability that the entrepreneur invests in her productivity, i.e., λG, depends on the security, which in turn, depends on λG through investors’ beliefs. Hence, the optimal security and the optimal value ofλG, are determined jointly in equilibrium.
8Note the entrepreneur offers the securityafterinvesting in her productivity. However,
as long ascis not verifiable, the equilibrium security is the same independently of whether the entrepreneur offers the security before or after investing in her productivity.
We solve the game backwards. We start from the contracting stage, by taking investors’ beliefs aboutλG as given. Then, we proceed with the investment-in-productivity stage, by taking the security gas given.
3.5.1.1 Contracting stage
Recall that the entrepreneur’s decision to invest in her productivity is un- observable. Thus, this stage is identical to the contracting stage at Section 3.4, apart from a critical modification: probability λG is now replaced by the beliefs of investors about the probability that the entrepreneur has in- vested in her productivity, denoted by ˜λG. Hence, by Proposition 2, the security which maximizes the entrepreneur’s expected utility, subject to the constraint thatλG =λG˜ , is captured in Lemma 4.
Lemma 4 g∗(F) = F , ¯g∗ = I−pB(S−g∗(S, ˜λG)) g∗(S, ˜λG) = ˜ λG(I−F(1−pG)) +λB˜ pBS ˜ λGpG+λB˜ pB 3.5.1.2 Investment-in-productivity stage
When the entrepreneur considers investing in her productivity, she antici- pates that the payments in the financing stage will be given by Lemma 4. Thus the entrepreneur’s expected utility in each case is:
EU(invest) = pG(S−g∗(S, ˜λG)) + (1−pG)(F−g∗(F))−c
Sinceg∗(F) = F, the entrepreneur invest in her productivity as long as: c |{z} cost ≤(pG−pB)(S−g∗(S, ˜λG)) | {z } benefit ≡cˆ (3.11)
where ˆc can be interpreted as the investment threshold. Figure 3.1 repre- sents the relationship between ˜λG (on the horizontal axis) and the benefit of investing in productivity (dashed curve), for four different distributions ofc. As expected, there is a negative relation between the probability ˜λG, and the payment in case of success that investors are willing to accept to finance an entrepreneur. In addition, the higher the ˜λG, the stronger the entrepreneur’s incentive to invest in productivity. This is captured in equa- tion (3.11).
3.5.1.3 Equilibrium existence and uniqueness of interior equilibrium
We start the analysis by focusing on equilibria for which ˜λ∗G ∈ (0, 1), there- after, “interior equilibria”. We analyze the equilibrium conditions when the project implementation is at the entrepreneur’s discretion- similar in- tuition applies when the entrepreneur is obliged to implement the project. We considermonotone or threshold equilibria in which the investment strat- egy is monotonic in c. In an interior threshold equilibrium, the following conditions need to hold:
ˆ
c∗ = (pG−pB)(S−g∗(S, ˜λ∗G)) (3.12)
0.2 0.4 0.6 0.8 1 0.5 1 1.5 E A (pG−pB)(S−g∗(S, ˜λG)) Φ−1(˜ λG) Panel A 0.2 0.4 0.6 0.8 1 0.5 1 1.5 E (pG−pB)(S−g∗(S, ˜λG)) Φ−1(˜ λG) Panel B 0.2 0.4 0.6 0.8 1 0.5 1 1.5 E (pG−pB)(S−g∗(S, ˜λG)) Φ−1(˜ λG) Panel C 0.2 0.4 0.6 0.8 1 0.5 1 1.5 E (pG−pB)(S−g∗(S, ˜λG)) Φ−1(λ˜G) Panel D
Figure 3.1. Cost and benefit of investing in productivity.
Condition (3.12) provides the critical value ˆc for which the entrepreneur is indifferent between investing and not investing. Condition (3.13) relates to the fact that investor’s beliefs should be correct. In particular, the probabil- ity that investors attribute to the entrepreneur being of good type coincides with the probability that the entrepreneur’s cost is below ˆc. Besides, for in- vestors’ beliefs to be correct, the entrepreneur must prefer investing when
c ≤ cˆ, and not investing otherwise. An implication of a threshold equilib- rium is that ˆc depends on g∗(S, ˜λG), which in turn, depends on ˆc. Thus, the optimal values ˆc∗ and g∗(S, ˜λ∗G), are determined jointly in equilibrium. Recall that Pr(c ≤ cˆ∗) ≡Φ(cˆ∗). In order to illustrate graphically the
equilibrium existence, it is more informative to use the inverse ofΦ(cˆ):
Φ−1(˜
λG) = cˆ
where Φ(cˆ) is invertible as a continuous and strictly increasing function. Hence, every interior equilibrium in the investment-in-productivity stage satisfies the following condition:
Φ−1(˜
λG) = (pG−pB)(S−g∗(S, ˜λG)) (3.14)
where Φ−1(λG˜ ) is depicted by the solid curve in Figure 3.1. Note that
Φ−1(˜
λG) is strictly increasing in ˜λG, as a consequence of the fact that Φ(cˆ) is strictly increasing in ˆc. Thus, relation (3.14) characterizes value of ˜λG where the dashed and the solid curve intersect. Relation (3.14), however, is not a sufficient condition for an interior equilibrium to exist. For ex- ample, point A in panel A of Figure 3.1 satisfies (3.14) but it can not be an equilibrium; for cost ˆcA+e, with e > 0, and as long as the beliefs are consistent, the expected benefit exceeds the cost. Hence, for cost ˆcA+e the entrepreneur has incentive to invest in productivity, which contradicts the definition of a threshold equilibrium: in a threshold equilibrium, the en- trepreneur invests in her productivity only ifc ≤ cˆA. Thus, for an interior equilibrium to exist, the solid curve should cross the dashed curve from below. Following that, the unique interior equilibrium is given by pointE
in panel A of Figure 3.1.
Note that there could be more than one combinations of ˜λG and
from below. If this the case, the unique interior equilibrium is the one which corresponds to the maximum ˜λG. This is because the expected util- ity of the entrepreneur is increasing in ˜λ∗G, due to the impact of the latter on g∗(S, ˜λ∗G).
We conclude this subsection with the case where there is no interior equilibrium. If it is very costly to invest in productivity, the only equilib- rium in the investment stage is for ˜λG =0, for which the market collapses (Panel C). In contrast, if the cost of investing is very low, the unique equilib- rium is for ˜λG =1 (Panel D). Lemma 5 presents the sufficient conditions for interior equilibrium to exist, where the payment g∗(S, ˜λG) is determined in Lemma 4.
Lemma 5: Interior and corner equilibrium - sufficient conditions
• If there is no c ∈ [c, ¯c] which satisfies:
c ≤(pG−pB)(S−g∗(S, ˜λG =Φ(c))) (3.15)
then, the unique equilibrium is forλG˜ =0, for which the market collapses.
• If c satisfies:¯
¯
c >(pG−pB)(S−g∗(S, ˜λG =Φ(c¯)) (3.16)
then, the unique equilibrium is forλ˜G =1.
• If (3.16)is violated, and there exists c∈ [c, ¯c]which satisfies(3.15), then the unique interior equilibrium is characterized by the maximum λG˜ ∈ (0, 1)
which solves:
Φ−1(˜
λG) = (pG−pB)(S−g∗(S, ˜λG)) (3.17)
Proof. See Appendix C.