Adverse selection and financing of innovation: is there a need for R&D subsidies?

Abstract

We study the interaction of private and public funding of innovative projects in the presence of adverse-selection based financing constraints. Government programs allocating direct subsidies are based on ex ante screening of the subsidy applications. This selection scheme may yield valuable information to market-based financiers. We find that under certain conditions, public R&D subsidies can reduce the financing constraints of technology-based entrepreneurial firms. First, the subsidy itself reduces the capital costs related to the innovation projects by reducing the amount of market-based capital required. Second, the observation that an entrepreneur has received a subsidy for an innovation project provides an informative signal to the market-based financiers. We also find that public screening works more efficiently if it is accompanied with subsidy allocation.

Appendices

Appendix: Proofs of the results

Proof

(Proposition 1) Since a low-type entrepreneur never wants to reveal her type, both types offer the same repayment to the financier when (6) holds, and so either all entrepre-neurs get funding or no-one gets when $A \, < \, \hat{A}.$ As (5) gives the threshold value of initial capital needed to get financing, when the financier anticipates all the entrepreneurs to seek financing, the entrepreneurs with $A \,< \, {\min}\{ \hat{A},\bar{A}\}$ do not receive funding. From (6) and (5) we observe that $\hat{A}$ is independent of p whereas $\bar{A}$ is decreasing in p. Solving $\hat{A} \, < \, \bar{A}$ for p gives $p<{\frac{I-\lambda_{L}R_{L}} {\lambda_{H}R_{H}-\lambda_{L}R_{L}}},$ i.e., for $p\geq{\frac{I-\lambda_{L}R_{L}} {\lambda_{H}R_{H}-\lambda_{L}R_{L}}}$ (5) is the binding constraint. From (5) we obtain that $\bar{A}\geq0$ when $p\leq{\frac{I-\lambda_{L}R_{H}} {\left(\lambda_{H}-\lambda_{L}\right)R_{H}}}.$ □

Proof

(Lemma 1) High-type entrepreneurs always apply if $E(\Uppi_{AP}^{E,H}) > 0$ holds in equilibrium. Substituting (15) and (16) (with i = H) for (17) shows that $E(\Uppi_{AP}^{E,H})> 0$ if $\alpha_{SC}+\alpha_{NSC,S}> {\frac{c} {\lambda_{H}(R_{H}-F^{S})-A+c}}.$ Similarly, low-type entrepreneurs always apply (μ = 1) if $E(\Uppi_{AP}^{E,L})> 0,$ which can be rewritten after substituting (15) and (16) (with i = L) for (18) as $\alpha_{NSC,S}> {\frac{c} {\lambda_{L}(R_{L}-F^{S})-A+c}}.$ Correspondingly, low types never apply (μ = 0) if $E(\Uppi_{AP}^{E,L})<0,$ i.e., if $\alpha_{NSC,S}< {\frac{c}{\lambda_{L}(R_{L}-F^{S})-A+c}}.$ It immediately follows that if $\alpha_{NSC,S}={\frac{c} {\lambda_{L}(R_{L}-F^{S})-A+c}},$ low-types are indifferent between applying and not $(E(\Uppi_{AP}^{E,L})=0)$ and use a mixed strategy (μ, 1 − μ). □

Proof

(Lemma 2) If $E(\Uppi_{SC}^{G})> \max\{ E(\Uppi_{NSC,S}^{G}),-c\},$ it is optimal for Government to choose pure strategy SC (α _SC  = 1). If $E(\Uppi_{NSC,S}^{G})> \max\{ E(\Uppi_{SC}^{G}),-c\},$ then s ^G = (NSC, S) (α_NSC,S = 1) is optimal for Government and if both $E(\Uppi_{SC}^{G})$ and $E(\Uppi_{NSC,S}^{G})$ are smaller than −c, pure strategy $s^{G}=(NSC,NS) \left(\alpha_{SC}=\alpha_{NSC,S}=0\right)$ is optimal. Inserting (19) into (20) and (21) shows that $E(\Uppi_{SC}^{G}) > E(\Uppi_{NSC,S}^{G})$ if $\mu>\underline{\mu}\equiv\left({\frac{p} {1-p}}\right)\left({\frac{\sigma} {I+gS-\lambda_{L}R_{L}-\sigma}}\right).$ Similarly, by using (19) and (20) it turns out that $E(\Uppi_{SC}^{G})>-c$ if $\mu \,< \,\overline{\mu}\equiv\left({\frac{p} {1-p}}\right)\left({\frac{\lambda_{H}\left(R_{H}+W\right)-I-gS-\sigma} {\sigma}}\right).$ As a result, Government sets α _SC  = 1 if $\mu\in\left(\underline{\mu},\overline{\mu}\right).$ It immediately follows that if $\mu \, < \, \underline{\mu} \,< \, \overline{\mu}, E(\Uppi_{NSC,S}^{G}) > E(\Uppi_{SC}^{G})>-c,$ meaning that Government chooses α_NSC,S = 1, and that if $\mu>\overline{\mu}>\underline{\mu}, E(\Uppi_{NSC,S}^{G}) \,< \, E(\Uppi_{SC}^{G})< -c,$ implying α _SC  = α_NSC,S = 0.

Government is indifferent between pure strategy SC and pure strategy (NSC, NS) and hence uses a mixed strategy α _SC and 1 − α _SC when $E(\Uppi_{SC}^{G})=-c,$ i.e., when $\mu=\overline{\mu}.$ In turn, Government randomizes between SC and (NSC, S) with probabilities α _SC and α_NSC,S = 1−α _SC when $E(\Uppi_{SC}^{G})=E(\Uppi_{NSC,S}^{G}),$ i.e., when $\mu=\underline{\mu}.$ □

Proof

(Proposition 2) Lemma 2 implies that screening is a plausible strategy if $\mu\in\left[\underline{\mu},\overline{\mu}\right].$ This is a non-empty set if $\sigma\leq{\frac{(I+gS-\lambda_{L}R_{L})(\lambda_{H}\left(R_{H}+W\right)-I-gS)} {\lambda_{H}\left(R_{H}+W\right)-\lambda_{L}R_{L}}}.$ We also need to make sure that $\underline{\mu}\leq1,$ i.e., that σ ≤ (1 − p)(I + gS − λ _L R _L ). □

Proof

(Proposition 3) Let’s first prove that there is no pure strategy equilibrium in this game. Note from (15) that $\Uppi_{S}^{E,i}>0$ implies that ${\frac{c} {\lambda_{i}(R_{i}-F^{S})-A+c}}\in(0,1).$ If a low-type entrepreneur always applied (μ = 1), Lemma 1 shows that $\alpha_{NSC,S}>{\frac{c}{\lambda_{L}(R_{L}-F^{S})-A+c}}$ should hold. However, Lemma 2 suggests that if μ = 1, it would be optimal for Government either to choose (NSC, NS) or (SC) implying that α_NSC,S = 0. If a low-type entrepreneur never applied (μ = 0), Lemma 1 implies $\alpha_{NSC,S}< {\frac{c} {\lambda_{L}(R_{L}-F^{S})-A+c}}$ should hold. But if μ = 0, it would be optimal for Government to set α_NSC,S = 1 (Lemma 2), which is larger than ${\frac{c}{\lambda_{L}(R_{L}-F^{S})-A+c}}.$ A similar argument shows that an equilibrium where Government randomizes between (NSC, NS) and (SC), implying that low types will not apply, does not exists.

Lemma 1 shows that for a low-type to be willing to use a mixed strategy 0 < μ < 1, α_NSC,S must be equal to ${\frac{c}{\lambda_{L}(R_{L}-F^{S})-A+c}}.$ Given that α_NSC,S > 0, Lemma 2 shows that the only possible mixed strategy for Government is to randomize between SC and (NSC, S) with probabilities $\alpha_{NSC,S}={\frac{c} {\lambda_{L}(R_{L}-F^{S})-A+c}}$ and $\alpha_{SC}=1-\alpha_{NSC,S}={\frac{\lambda_{L}(R_{L}-F^{S})-A} {\lambda_{L}(R_{L}-F^{S})-A+c}}.$ This Government strategy satisfies $\alpha_{SC}+\alpha_{NSC,S}>{\frac{c} {\lambda_{H}(R_{H}-F^{S})-A+c}},$ which prompts high-type entrepreneurs always apply by Lemma 1. When Government randomizes between SC and (NSC, S), Lemma 2 dictates that a low-type entrepreneur applies with probability $\mu=\underline{\mu}=\left({\frac{p} {1-p}}\right)\left({\frac{\sigma} {I+gS-\lambda_{L}R_{L}-\sigma}}\right).$ Inserting this into (19) shows that Government’s belief is given by $\theta={\frac{I+gS-\lambda_{L}R_{L}-\sigma} {I+gS-\lambda_{L}R_{L}}}.$ □

Proof

(Proposition 4)

• (i) $\bar{A}> \bar{A}^{S}\Leftrightarrow\left({\frac{\lambda_{H}} {\lambda_{H}-\bar{\lambda}}}\right)(I-\bar{\lambda}R_{H})> \left({\frac{\lambda_{H}} {\lambda_{H}-\hat{\lambda}}}\right)(I-S+c-\hat{\lambda}R_{H}) \Leftrightarrow(\hat{\lambda}-\bar{\lambda})(\lambda_{H}R_{H}-I) +(\lambda_{H}+\bar{\lambda})(S-c)> 0.$ From the last inequality, we can see that it holds if $\hat{\lambda}\geq\bar{\lambda}.$ High-type projects are economically viable, therefore λ _H R _H  − I > 0. Since we are analyzing entrepreneurs that have been granted an R&D subsidy, $(\lambda_{H}+\bar{\lambda})(S-c)> 0,$ if S > c and $\bar{A}> \bar{A}^{S}$ even if $\hat{\lambda}=\bar{\lambda}.$

• (ii) $\hat{\lambda}=\hat{p}\lambda_{H}+(1-\hat{p})\lambda_{L}> \bar{\lambda},$ if $\hat{p}> p.$ Knowing that $\hat{p}=\alpha_{SC}+(1-\alpha_{SC})\theta$ gives us that for $\hat{p}> p, \alpha_{SC}$ must satisfy $\alpha_{SC}>{\frac{p-\theta}{1-\theta}}.$ This is true since $p< \theta={\frac{p}{p+\mu(1-p)}}< 1\; \left(0 < p < 1\, \hbox{and}\, 0 < \mu < 1\right).$ □

Proof

(Proposition 5) $\bar{A}> \bar{A}^{S}$ must hold for a specific value of $\hat{p},$ if the subsidy program reduces financial constraints. It can be shown that $\bar{A}> \bar{A}^{S}\Leftrightarrow\hat{p}\geq{\frac{I-S+c-\lambda_{L}R_{L}} {\lambda_{H}R_{H}-\lambda_{L}R_{L}}}=\underline{\hat{p}}.$ Proposition 1 gives that in the funding gap region $p< {\frac{I-\lambda_{L}R_{H}} {\left(\lambda_{H}-\lambda_{L}\right)R_{H}}}=\bar{p}.$ It can be shown that $\bar{p}> \underline{\hat{p}}.$ In addition, we know from Proposition 4 that for a given $p, \hat{p}> p,$ so the lower bound of p is smaller than $\underline{\hat{p}}.$ Substituting ${\frac{p}{p+\mu(1-p)}}$ for θ in $\hat{p}=\alpha_{SC}+(1-\alpha_{SC})\theta$ gives the implicit form for p as a function of $\hat{p,\alpha_{SC}}$ and μ that is $p={\frac{(\hat{p}-\alpha_{SC})\mu} {(1-\hat{p})+(\hat{p}-\alpha_{SC})\mu}}.$ Substituting $\underline{\hat{p}}$ for $\hat{p}$ gives the lower bound of p in the implicit form and the interval in Proposition 5.□

Comparative statics of government screening

We briefly sketch the comparative statics of Government screening probability α _SC in the full game where the entrepreneur’s repayment obligation F ^S is endogenous. After tedious algebra, it turns out that the partial derivatives of α _SC with respect to σ, c, A, and S are given by

$$ {\frac{\partial\alpha_{SC}} {\partial\sigma}}={\frac{\lambda_{L}\left({\frac{(\lambda_{H}-\lambda_{L})\alpha_{SC}} {I^{S}-\lambda_{L}R_{L}}}\right)(I-A-S+c)c} {[\hat{\lambda}(\lambda_{L}(R_{L}-F^{S})-A+c)]^{2}-(\lambda_{H} -\lambda_{L})(1-\theta)(I-A-S+c)c}}, $$

$$ {\frac{\partial\alpha_{SC}}{\partial c}}=-{\frac{\hat{\lambda}^{2}\lambda_{L}\left(\hat{\lambda}(R_{L}-F^{S})+c\right)} {[\hat{\lambda}(\lambda_{L}(R_{L}-F^{S})-A+c)]^{2}-(\lambda_{H}- \lambda_{L})(1-\theta)(I-A-S+c)c}}, $$

$$ {\frac{\partial\alpha_{SC}}{\partial A}}=-{\frac{\hat{\lambda}\left(\lambda_{L}+\hat{\lambda}\right)c} {[\hat{\lambda}(\lambda_{L}(R_{L}-F^{S})-A+c)]^{2}-(\lambda_{H} -\lambda_{L})(1-\theta)(I-A-S+c)c}}, $$

and

$$ {\frac{\partial\alpha_{SC}}{\partial S}}=-{\frac{\lambda_{L}\left[\hat{\lambda}+(\lambda_{H}- \lambda_{L})(1-\alpha_{SC})\left({\frac{g\sigma} {(I_{S}-\lambda_{L}R_{L})^{2}}}\right)(I-A-S-c)\right]c} {[\hat{\lambda}(\lambda_{L}(R_{L}-F^{S})-A+c)]^{2}-(\lambda_{H} -\lambda_{L})(1-\theta)(I-A-S+c)c}}. $$

If the denominator is positive then ${\frac{\partial\alpha_{SC}} {\partial\sigma}}< 0, {\frac{\partial\alpha_{SC}}{\partial c}}< 0, {\frac{\partial\alpha_{SC}}{A}}< 0$ and ${\frac{\partial\alpha_{SC}}{S}}> 0.$ Note first that in equilibrium θ is given by the exogenous parameters as $\theta=1-{\frac{\sigma}{I+gS-\lambda_{L}R_{L}}}.$ It can then be shown that when θ = 1 the denominator is positive. Moreover, it can be shown that the denominator reaches it’s minimum, which is negative, at a negative value of θ. As a function of θ, the denominator is an upward opening parabola, so by continuity there must be an interval of $\theta\in[\hat{\theta},1],$ where the denominator is positive. The restrictions imposed on σ and p imply that in the funding gap region $\theta\in\left[{\frac{I-gS-\lambda_{L}R_{L}} {\lambda_{H}(R_{H}+W)-\lambda_{L}R_{L}}},1\right].$ Consequently, if θ is sufficiently close to unity, there are financially constrained high-type entrepreneurs and the denominator of the partial derivatives is positive.