Firms use external funds to acquire inputs for their technologies. Let R denote the gross interest rate for obtaining one unit of external funds. One unit of input costs one unit the output. The firm’s optimization problem can be written as
max
S (1 − AS/e)[S − R] (2)
The first order necessary condition is(−A/e) · S +(1 −AS/e) = (−A/e) · R, which can be rewritten as
H(S, e) ≡ S +1− AS/e
(−A/e) = R, (3)
R
H(S, e)
S∗(R, e) R′
S∗(R′, e) S Figure 1: Loan Rates and the Choice of S
H(S, e) is the risk(Ps)-adjusted marginal contribution of an additional unit of out-put to expected outout-puts, and the right hand side of (3) is the risk-adjusted marginal cost of increasing outputs. One can easily show that the left hand side of (3) is increasing in S, (i.e., HS(S, e) > 0 for e ∈ [e, ¯e]), while the right hand side is con-stant over S. When the risk-adjusted marginal cost (R) increases, the firm should respond by increasing the scale of outputs (S). Let S∗(R, e) be the solution. When P(S, e) = 1 − AS/e, S∗ = (e + AR)/(2A). Figure 1 describes how a change in R affect S∗. As S increases, the probability of success decreases while the expected outputs increases due to the output increase. Thus the optimal response to an in-crease in R is to choose a greater S. A higher costs of funds (higher R) results in the choice of more risky project (a higher S and a lower probability of success).
The Ps-adjusted marginal contribution of additional output H(S, e) is
decreas-R
H(S, e)
S∗(R, e)
H(S, e′)
S∗(R, e′) e′> e
S
Figure 2: The Choice of S and Heterogeneity of Firms
ing in e, He(S, e) < 0. Given the cost of funding, R, firms with different e choose different levels of production. A firm with a higher e indicates that his technology more likely succeed and leads him to choose a higher S for a higher return. In Figure 2a greater e (e′) corresponds to a lower curve of H and results in the opti-mal choice of a greater S. This shows that how heterogeneity of firms affects the choice of risky technologies. In sum, S∗(R, e) has properties: S∗R> 0 and S∗e > 0.
Next we define the value function of the firm’s optimization (2), denoted by V(R, e), as
V(R, e) = P(S∗(R, e), e)[S∗(R, e) − R] =(e − AR)2 4Ae ;
Note that R is affordable only if S∗(R, e) = (e + AR)/(2A) ≥ R, or e > AR. The value function V(R, e) is increasing in e:
Ve= e2− A2R2 4Ae2 > 0.
and is decreasing in R:
VR(R, e) = −e− AR
2e < 0. (4)
Moreover VRe = −2 eAR2 < 0. Some important result about the choice of projects and the probability of success are summarized in the following proposition:
Proposition 1 Facing the same borrowing rate R,
[1] Firms with greater skill levels of management chooses more productive projects:
S∗e(R, e) > 0.
[2] the default risk decreases as the skill level of management increases:
dP(S∗(R, e))
de = AR
8 e2 > 0
Firms with higher e choose technologies of higher productivity than firms with lower e; however, their failure probabilities are lower. The greater productiv-ity (S) indicates a higher failure probabilproductiv-ity (Ps(S, e) < 0); however a higher skill level of management indicates a higher success probability (P(S, e1) > P(S, e2)) for all S. Two impacts work in the opposite directions. However, the direct effect dominates the indirect effect.
Two forms of external funds
Firms have two alternatives to raise their funds, borrowing loans from banks or issuing securities to the market. Let RL denote the (gross) loan rate and RU the rate in the direct lending market. In addition to RU, a firm has to pay an amount
φ of upfront underwriting fees for issuing securities. This upfront cost can be thought of as underwriting expenses.2 The firm’s optimization problems of ob-taining funds in these two alternatives are
maxS P(S, e) · [S − RL], and
maxS P(S, e) · [S − RU] −φ,
respectively.3 The decision of funds-raising is transformed into the following problem:
max {V (RL, e),V (RU, e) −φ}.
Recall that the value function V(R, e) has properties:
Ve(R, e) > 0, VR(R, e) < 0, and VRe(R, e) < 0. (5)
Consistent with empirical findings, we will only discuss the equilibrium out-come in which the direct lending is less costly than loan-finance (not considering upfront costs); i.e., RU < RL. (Later on we will show that, in our parametric
2For simplicity, we assume that a firm has some resources to pay upfront costs, but they do not use them to finance their inputs. The results of our analysis is insensitive to this innocuous assumption. This upfront cost setup follows from Kanatas and Qi (1998, 2003). Puri (1999) has a similar setup.
3Here we assume that firms are price-takers in the borrowing markets. In a seller’s market firms do not have any influence over the price he pays. Later on when we analyze the underwriting activities, we will relax this assumption. When a firm chooses the type of project, it affects what he pays through the riskiness he chooses.
e1
Figure 3: Underwriting Fee and Fund-Raising
example, when the screening cost of loan business (c) is large enough, the equi-librium in our model economy does have this property.) Then the the property of VR(R, e) < 0 implies that V (RU, e) > V (RF, e) for all e. We will use the properties of (5) to show that as long as the underwriting fee φ is large enough and RU and RL falls apart enough distances, high e firms will issue securities to raise their funds, while low e firms will raise their funds through loans.
We use Figure 3 to illustrate this result: when φ is large enough and e1 is greater than e2 by a significant gap, it can be the case in which V(RU, e2) −φ >
V(RL, e2), while V (RU, e1) −φ < V (RL, e1). This can happen because the third property of (5). As a result, type e1 firms choose market finance, while type e2 firms choose loan finance. Moreover, from Proposition 1, one can easily verify
that projects financed by markets have greater probabilities of success than those financed by loans. We summarize these results in the following proposition.
Proposition 2 When RL> RU andφ is large enough,
[1] a very high e firm chooses market finance and a very low e firm chooses loan finance;
[2] markets-financed projects succeed with a greater probability than banks-financed projects: