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.

1 Introduction

Previous research suggests asymmetric information about the quality of an innovation project between an entrepreneur and a financier leads to a higher cost of external than internal capital, creating a funding gap. This funding gap may prevent especially small and new technology-based entrepreneurial firms from undertaking economically viable innovation projects. This observation has provided grounds for government intervention aimed at reducing the financing constraints of technology-based start-ups. One widely used policy tool is direct subsidies to corporate R&D. However, the theoretical literature linking financing constraints and R&D subsidies is scant. ¹ We develop a theoretical model to analyze how a governmental R&D subsidy program works in the presence of financial constraints created by asymmetric information.

Since Akerlof (1970), huge literature singles out adverse selection stemming out from informational asymmetries between entrepreneurs and financiers as a major source of financing constraints. These informational problems are acknowledged to be particularly severe in the financing of R&D projects (Alam and Walton 1995; Hubbard 1998). R&D activities typically involve soft information that is hard to verify. Hence, if adverse selection related financing constraints exist, they should be especially relevant to science and technology-based start-ups whose main assets are founders’ human capital and intellectual property. Moreover, the standard solutions provided to the adverse-selection problem—signaling, reputation and financial intermediation—are more likely to fail in the case of science and technology-based start-ups. ² Such firms cannot have acquired reputation nor assets that can be offered as collateral and credit worthiness of these firm is difficult to assess. Even venture capital and related organizations that are often regarded as the solution to innovation financing may fail to provide an adequate solution to the financing of science and technology-based start-ups (see, e.g., Hall 2002; Lerner 1998, 2002). ³ There is indeed abundant empirical evidence that R&D investment are sensitive to cash flow, at least in the case of newly established, small, technology-based firms (e.g., Hall 1992; Hao and Jaffe 1993; Himmelberg and Petersen 1994; Bond et al. 2003; Bougheas et al. 2001). ⁴

A major objective of governmental R&D subsidy programs is to reduce these constraints on the financing of innovation. In contrast to some other innovation policy tools such as R&D tax credits, government programs allocating direct subsidies are based on ex ante screening of the applications. Despite the wide use of R&D subsidy programs the functioning of these programs and the related screening process has been little explored in the theoretical literature.

We model a situation in which capital constrained entrepreneurs can try to tap a public agency for funding in addition to private funding sources and analyze whether R&D subsidy policies can reduce adverse-selection based financing constraints. Our aim is thus to provide a positive analysis of the application and allocation of R&D subsidies rather than normative welfare analysis of R&D subsidies. More specifically, we address two questions: (i) When is it worthwhile for the government to invest in ex ante screening of subsidy applications? (ii) Could a subsidy provided by a public agency act as a certification for an unknown entrepreneur and ease her possibilities to secure funding from market-based financiers? While the idea of certification by a trusted financial intermediary is pervasive in the corporate finance literature, to the best of our knowledge, it has not been previously applied to the public funding of corporate R&D (but see Lerner 2002 for an informal discussion). Certification hypothesis emphasizes the role of government screening activities that are inherent in R&D subsidy programs but overlooked by the literature.

The main results are summarized as follows. First, government screening activities are more valuable if the proportion of entrepreneurs with economically non-viable projects is non-negligible and the screening costs are low enough. The former condition is needed to guarantee the existence of adverse-selection based financing constraints, while the latter prevents unsustainable screening activities. Second, the government’s incentives to screen are increasing in the subsidy amount. As entrepreneurs anticipate that screening increases with the subsidy amount, larger subsidies can deter the entrepreneurs with low quality projects from applying. In other words, government project screening is more credible if it is accompanied with subsidy allocation. Third, it turns out that the provision of R&D subsidies and the related screening may help financially constrained entrepreneurs to finance their projects. The effect comes through two channels. First, the subsidy itself reduces the capital costs related to the innovation projects by reducing the amount of market-based funding needed. Second, the observation that an entrepreneur has received a subsidy for an innovation project provides an informative signal to the market-based financier. Finally, the findings suggest that under certain conditions R&D subsidy policy may be welfare improving.

Our modeling framework builds on Holmström and Tirole (1997), which has subsequently been used to study entrepreneurial finance, e.g., by Repullo and Suarez (2000) and Da Rin et al. (2005). These papers highlight the role of interim monitoring by informed financiers (banks or venture capital organizations) in mitigating moral hazard problem and in bringing along less well-informed investors. Instead of moral hazard, we focus on adverse selection created by ex ante informational asymmetries, and the role of screening and signaling by a public funding agency in reducing financing constraints. Our starting point is that banks are not informed enough and venture capital markets do not function well enough to eliminate the financing constraints of small, innovative firms. We analyze under which circumstances R&D subsidies allocated by a public agency could improve the situation.

While the theoretical literature linking R&D subsidy programs and financial constraints is limited, much more work has been done on the need to subsidize entrepreneurs or their finance in the presence of asymmetric information arising from the influential contributions by Stiglitz and Weiss (1981) and De Meza and Webb (1987). As summarized by Boadway and Keen (2005), the results depend on what are assumed about the project return distributions. In particular, adverse selection may generate too much lending to entrepreneurs rather than financing constraints. In our model, too, the beneficial effects of subsidies are more limited if the problem caused by adverse selection is overinvestment rather than financing constraints. This literature, however, abstracts from the signaling role of subsidies as well as from the social benefits of R&D.

The design and the institutional setting of the R&D subsidy program modeled in this paper are inspired by the Finnish institutional environment, but the situation we describe is common in many countries where public R&D subsidy programs are in place and the markets for private start-up finance are imperfect. ⁵

Section 2 describes the model. Section 3 identifies the funding gap by analyzing entrepreneurs’ possibilities to fund their innovation projects in the absence of subsidies. Section 4 presents a dynamic game of incomplete information describing the subsidy application and allocation process. The section concludes with the equilibrium strategies of both the public agency and the entrepreneurs. Section 5 links public- and market-based financiers to analyze the effects of subsidies on the funding gap. Section 6 concludes the paper.

2 The model

The model has three types of risk-neutral agents: (potential) entrepreneurs, market-based financiers, and a public financier. As will be specified below, entrepreneurs have some initial wealth but are nonetheless capital constrained and need to seek funding from external financiers to be able to launch their projects. The entrepreneurs are heterogeneous in terms of their type (“talent”), which determines the productivity of their projects. Following the convention in the literature (see, e.g., De Meza and Webb 1987; Boadway and Keen 2005), we assume that the entrepreneur’s type is her private information but the level of her initial wealth is common knowledge (or at least verifiable). We proceed as if entrepreneurs first tried to seek public funding before turning to private sources but we could equally well assume that entrepreneurs first contacted market-based financiers who would make their funding decisions contingent on the public funding decision. We will look for perfect Bayesian equilibria (PBE), which require that at each stage of the game, the agents’ strategies are optimal given their beliefs, and the beliefs are obtained from equilibrium strategies and observed actions by using Bayes’ rule.

2.1 Entrepreneurs

There is a continuum of entrepreneurs who have access to an innovation project requiring an investment of size I. The projects have a two-point return distribution: A fraction p of the entrepreneurs are high (H) types having access to a positive net-present value (NPV) project, the rest (1 − p) are low (L) types with a negative NPV project. Let λ _i and R _i denote the project success probability and the project return conditional on the success of an entrepreneur of type $i,i\in\{H,L\}.$ A failed project yields zero irrespective of the entrepreneur’s type. Following Holmström and Tirole (1997), we assume that $\lambda_{H}> \lambda_{L}$, $R_{L}> R_{H}$, $\lambda_{H}R_{H}> I>\lambda_{L}R_{L}$.⁶

Entrepreneurs differ in the amount of their initial capital (cash) A, which is distributed across entrepreneurs according to a cumulative distribution function G(A), and it is independent of the entrepreneur’s type. No entrepreneur has more than I of initial wealth, so G(A) is defined on interval [0, I]. A project is initiated only when an entrepreneur invests all her initial capital in her own project and manages to raise the rest of the required funds I − A from other sources.⁷

2.2 Public financier

One source of external finance is a public funding agency which is called Government in the following. The public funding is a pure subsidy that needs not to be paid back but it needs to be applied for. To apply for the public funding, an entrepreneur needs to incur a fixed cost of c. In practice, an application process involves both monetary and non-monetary costs, such as the costs of filling and filing the application form and providing necessary supplementary data, the opportunity costs of time and effort that the application process consumes. Since allowing for both monetary and non-monetary costs would unnecessarily complicate the analysis, we assume that c is a monetary cost.⁸ This means that if the entrepreneur applies for a subsidy, the total size of the project will be I + c instead of I.

For simplicity, we assume that Government can give a fixed subsidy (S) to any project to which public funding is applied for. Government’s budget constraint does not bind, but the use of public funds involves an opportunity cost of 1 + g (0 < g < 1).⁹ A successful project may generate social benefit to Government beyond the private return R _i . Such social benefit covers the externalities generated by the project including, e.g., spillovers and consumer surplus. More specifically, we assume that private and social benefits are positively correlated: a successful project of a high-type entrepreneur generates a social benefit W to Government whereas a low-type entrepreneur’s project generates no social benefit irrespective of its success.

As will be clear later, assuming that only successful high-type projects generate social benefits is not crucial for any of the main qualitative results of the paper. For example, by letting W = 0 our results immediately generalize to the usual case analyzed in the literature of entrepreneurial finance where no project yields social benefits beyond private returns. We could also equally well assume that a low-type entrepreneur’s project generates social benefits in so far the net welfare of the low-type’s project remains negative. Similarly, we could assume that failed projects generate social benefits in so far such benefits are small enough. While we think that positive correlation between private and social returns is both realistic and theoretically sound, this assumption could also be relaxed. Such a change or assuming a positive net welfare of the low-type entrepreneur’s project would modify the welfare implications of the model but not its basic structure.

Government does not observe the types of entrepreneurs but has an access to a screening technology. If Government receives an application for a subsidy from an entrepreneur, Government can learn the type of the entrepreneur by screening the application. For simplicity, we assume that screening is costly but perfect: by incurring a screening cost σ, Government can verify the entrepreneur’s true type. A major task of the personnel in the public funding agencies is to evaluate project proposals and they are classified in many dimensions. Such screening is obviously costly. While the cost of screening per application is fixed in our model, in equilibrium Government will screen an application with some positive probability and this probability measures the intensity of screening.¹⁰

2.3 Market-based financier

Entrepreneurs can also try to tap private sources for funding. Private funding involves no application costs but entrepreneurs need to pay the market rate for such funding. Private sector financiers have access to unlimited supply of financial capital. They are competitive and the required expected rate of return on investor capital is exogenous and normalized to unity.

The market-based financiers posses no screening technology and only know the share of high-type entrepreneurs in the population. When contemplating whether to extend funding to an entrepreneur or not, market-based financiers observe whether the entrepreneur has received a subsidy from Government or not, and they know Government’s objective function. If the entrepreneur applied for the subsidy, the market-based financiers do not observe whether Government screened the entrepreneur or not. Nor do they observe whether an entrepreneur without a subsidy actually applied for the subsidy but in equilibrium this is immaterial.

Our assumption that project screening is optimal for Government but not for external financiers is of course strong, but it is not essential for our results. We only need to assume that Government’s subsidy decisions are not completely random so that the subsidy decision contains some valuable information to the market.¹¹ If one wants to take the assumption literally, it could be motivated by a benevolent public financier’s interest in the aggregate welfare generated by the project. That is, the public financier’s objective function should also include the externalities generated by the project besides the financial return. In contrast, market-based financiers only care about the financial return. This is what we assume: a successful high-type project generates a social benefit W to Government beyond the private return R _i . Therefore, the assumption that the public financier is better motivated to screen projects than private financiers is in line with our model.

Moreover, there are several factors that may dilute the incentives of market-based financiers to engage in screening activities, especially in the case of a small country like Finland. First, the public financier is often granting project specific funding, whereas private financiers, especially those using debt finance, typically operate at the firm level. Second, since screening is a public good, private financiers can suffer from a free-riding problem. A public screening agency can offer a solution to the free-riding problem, but at the same time reduce the incentives of the private sector to engage in screening. Third, a subsidy in itself should also reduce the stake the private financiers need to take in the project, which further dilutes their incentives to screen. So if a public screening and funding agency exists for some reason, it may worsen the free-riding problem by giving an additional incentive for the private financiers to economize on screening investments.¹² Fourth, in the Finnish case at least, the public financier constitutes a centralized screening device that has massive resources to screening. It receives a large amount of applications that it can compare against each other. As a result, the public financier could be expected to have quite a good overview about the state of the art in each relevant field. Fifth, according to the so called competition-stability tradeoff, competition in banking sector can reduce banks’ information surplus and thereby their incentives to gather information (e.g., Keeley 1990).¹³ Information reusability can also be hampered by intertemporal volatility of borrower credit risks (Chan et al. 1986). Moreover, financial innovation has enabled the intermediaries to transfer credit risk off their balance sheets, which may have undermined their incentives to screen new borrowers.

Following Myers and Majluf (1984) pecking-order hypothesis, we assume that to the extent an entrepreneur’s initial wealth and her public funding is insufficient, the entrepreneur issues debt to market-based financiers. We consider risky debt contracts that give a financier fixed payment in the case of success and zero in the case of failure. In principle, this does not require all entrepreneurs to have the same repayment obligation. Since the market-based financiers are uninformative, our focus on debt financing is not entirely implausible. Moreover, such risky debt contracts are optimal when project success is verifiable but returns are not,¹⁴ and we restrict our attention to a “realistic” subspace of contracts where (i) parties are protected by limited liability; (ii) markets must clear¹⁵; and (iii) the financial contract cannot specify a positive reward for an entrepreneur to refrain from investing.¹⁶

3 Innovation finance without public funding

The case without public support for innovation reduces to a fairly standard model of entrepreneurial finance under incomplete information. In our setup, the entrepreneurs differ in the amount of initial capital they posses, and our focus is to determine how the composition of entrepreneurs receiving market-based financing depends on the amount of their initial capital.

In the absence of public funding, there are three periods beyond the initial determination of types.

• 0. Nature draws a type $i \in\{L,H\}$ for an entrepreneur. Probabilities of a high type and a low type are p and 1 − p, 0 < p < 1.

• 1. The entrepreneur observes her type and decides whether to seek external funding.

• 2. Financiers decide whether to give funding under the terms proposed by the entrepreneur, and the funded projects are executed.

• 3. Project returns are realized, successful entrepreneurs compensate their financiers according to the contract terms.

In the last stage of the game, an entrepreneur and her financier(s) split the return from a successful project so that

$$ R_{i}=R_{i}^{E}+R_{i}^{F} $$

where $R_{i}^{E}$ is the share received by an entrepreneur of type i and $R_{i}^{F}$ is her financier’s share.

Since the rate of return on capital is assumed to be equal to one, the entrepreneur’s participation constraint reads as

$$ \lambda_{i}R_{i}^{E}\geq A,$$

(1)

which simply means that an entrepreneur is willing to launch the project if her expected profit from the project (in the left-hand side of the equation) is at least as much as the entrepreneur would get from investing the initial capital into alternative sources. When (1) binds,

$$ R_{i}^{F\max}=R_{i}-R_{i}^{E}=R_{i}-\frac{A} {\lambda_{i}} $$

(2)

captures the i-type entrepreneur’s pledgeable income, that is, the maximum amount an entrepreneur of type i could credibly promise to pay back to a financier in the case of success. Note that even if $R_{L}^{F\max}\geq R_{H}^{F\max}$ holds and a low-type entrepreneur could offer the financier a larger return, she will not do so as it is not in her interest to reveal her type.

Financiers, who are assumed to be competitive and break even, are willing to invest in a project if the expected return from investing equals the market value of funds supplied, I − A. They do not observe the type of the entrepreneur they are facing, but know the proportions of high and low types (p and 1 − p, respectively) in the population and, consequently, put prior probability $\bar{\lambda}=p\lambda_{H}+(1-p)\lambda_{L}$ on the success of a project by an average entrepreneur in the population. The minimum repayment F that a financier requires to invest in a project of an average quality is then given by $\bar{\lambda}F=I-A$ or, equivalently, by

$$ F={\frac{I-A}{\bar{\lambda}}}. $$

(3)

As a result, projects can get market-based funding as long as $F\leq R_{H}^{F\max}$ where $R_{H}^{F\max}$ is the maximum repayment that a high-type entrepreneur, and by implication, a low-type entrepreneur, are willing to offer to the financier. Using (2) and (3), we can write the inequality as

$$ {\frac{I-A}{\bar{\lambda}}}\leq R_{H}-{\frac{A}{\lambda_{H}}}. $$

(4)

Equation 4 is the financier’s participation constraint when all entrepreneurs apply for funding. Solving (4) for A gives

$$ A\geq\bar{A}\equiv{\frac{\lambda_{H}\left(I-\bar{\lambda}R_{H}\right)} {\lambda_{H}-\bar{\lambda}}} $$

(5)

In (5), $\bar{A}$ gives the threshold value of initial capital needed to get financing, when the financier anticipates all the entrepreneurs to seek financing.

Another important threshold value of initial capital comes from the condition $R_{L}^{F\max}\geq R_{H}^{F\max}.$ Using (2) to solve this inequality for A gives

$$ A\leq\hat{A}\equiv{\frac{\lambda_{L}\lambda_{H}\left(R_{L}-R_{H}\right)} {\lambda_{H}-\lambda_{L}}}. $$

(6)

When (6) holds, the maximum repayment a high-type entrepreneur is willing to promise to the financier if the project succeeds is never higher than what a low-type entrepreneur could promise. This means that when (6) holds, a high-type entrepreneur has no means to truthfully signal her quality even if she had an incentive to do so. We are ready to state our first result (all the proofs of the results of this paper are relegated to the Appendix).

Proposition 1

High-type entrepreneurs with$A < {\min}\{\hat{A},\bar{A}\},$where$\bar{A}\geq0$when$p\leq {\frac{I-\lambda_{L}R_{H}} {(\lambda_{H}-\lambda_{L})R_{H}}},$suffer from the funding gap that prevents them from undertaking economically viable innovation projects.

Proposition 1 shows that no entrepreneur with $A < {\min}\{ \hat{A},\bar{A}\}$ receives funding. In other words, there is a funding gap. As the low-type entrepreneurs’ projects have a negative NPV, we are only interested in the financing difficulties encountered by high types. Quite naturally, the funding gap emerges when entrepreneurs have relatively little initial wealth and an average project has a negative NPV. When (6) holds, high-type entrepreneurs cannot separate themselves from low-type entrepreneurs and (5) then determines the level of initial capital which is required to get funding when low types are pooling with high types. The latter condition is irrelevant when the share of high types in the population is high enough $(p>{\frac{I-\lambda_{L}R_{H}} {(\lambda_{H}-\lambda_{L})R_{H}}})$ so that an average project in the population has a positive NPV.

We next describe what will happen when entrepreneurs are wealthier, i.e., when $A\geq{\min}\{\hat{A},\bar{A}\}$. From the above analysis, we know that all entrepreneurs receive funding when $A\in[\bar{A},\hat{A}].$ However, for some parameter values all entrepreneurs will continue to be funded even when $A>\hat{A}.$ To see this note that when $A>\hat{A}, R_{H}^{F\max}> R_{L}^{F\max},$ a high-type entrepreneur could truthfully signal her quality, but it is not necessarily in her interest to do so. Given the assumption of competitive financial markets, a financier continues to require F as given by (3) to invest in a project of unknown entrepreneurial quality, as long as also low-type entrepreneurs can afford offering that amount to the financier. This happens when $R_{L}^{F\max}\geq F$ or, equivalently, when

$$ {\frac{(I-A)}{\bar{\lambda}}}\leq R_{L}-{\frac{A}{\lambda_{L}}}. $$

(7)

The left-hand side of (7) is the minimum repayment the financier requires to invest in a project in a pooling equilibrium and the right-hand side is the maximum repayment a low-type entrepreneur is willing to promise to the financier. Solving (7) for A gives us

$$ A\leq\dot{A}\equiv{\frac{\lambda_{L}(\bar{\lambda}R_{L}-I)} {p\left(\lambda_{H}-\lambda_{L}\right)}}. $$

(8)

A high-type entrepreneur has no incentive to separate herself from a low-type: she should offer at least $R_{L}^{F\max}$ to a financier to credibly signal her type, but only F is needed to ensure funding.

When A > $\dot{A}$, a low-type entrepreneur can no longer offer F to a financier and will drop out with this interest rate. However, if the financier knew that the entrepreneur seeking funding is of a high-type, ${\frac{I-A}{\lambda_{H}}}$ would be a large enough repayment for the financier to be willing to invest in her project. But because $\lambda_{H}>\bar{\lambda},$ a low-type entrepreneur can offer the financier ${\frac{I-A}{\lambda_{H}}}$ for some values of A greater than $\dot{A}.$ Solving the inequality ${\frac{I-A} {\lambda_{H}}}\leq R_{L}-{\frac{A}{\lambda_{L}}}$ for A gives

$$ A\leq{\frac{\lambda_{L}(\lambda_{H}R_{L}-I)} {\lambda_{H}-\lambda_{L}}}\equiv\ddot{A}. $$

(9)

Only when $A>\ddot{A},$ the financier knows that only high-type entrepreneurs remain in the pool of loan applicants and are willing to accept ${\frac{I-A}{\lambda_{H}}}.$ If $\dot{A}< A\leq\ddot{A,}$ a low-type entrepreneur can pretend to be of high-type by offering ${\frac{I-A}{\lambda_{H}}}$ to the financier. Therefore, when $\dot{A} < A \leq \ddot{A},$ there is a semi-separating equilibrium in which all the high-type entrepreneurs and a share of low-type entrepreneurs are funded.

Figure 1 summarizes different funding regions. Given that $\bar{A}$ and $\dot{A}$ depend on the share of high-type entrepreneurs in the population (p), the different regions are presented with coordinates $(p,A), p\in[0,1], A\in[0,I].$ Note from (6), (5) and (8) that $\hat{A}$ is independent of p whereas $\bar{A}$ is decreasing and $\dot{A}$ is increasing in p, and that the three lines cross each others when $p={\frac{I-\lambda_{L}R_{L}} {\lambda_{H}R_{H}-\lambda_{L}R_{L}}}.$ When $A < \min\{ \hat{A},\bar{A}\},$ market-based financiers are willing to fund no projects, as shown in Proposition 1. When $A\in[\bar{A},\dot{A}]$ all entrepreneurs are funded. When $A\in]\max\{ \hat{A},\dot{A}\} ,\ddot{A}],$ all the high-type entrepreneurs and a share of low-type entrepreneurs are funded. When $A > \ddot{A},$ only high-type entrepreneurs are funded.

Fig. 1

Market-based financing with different values of initial capital

Let us compare the outcome in each region of Fig. 1 to the outcome under complete information. With complete information, a high-type entrepreneur will receive funding by offering ${\frac{I-A} {\lambda_{H}}}$ to the financier, since the rate of return required by the financier is normalized to unity and the NPV of the project is positive. In contrast, the low-type entrepreneurs’ projects have a negative NPV, which raises the cost of external funding for low types so high that no low type is willing to launch her project. Because all projects by high-type entrepreneurs but no projects by low types will be executed, the market for entrepreneurial finance is efficient, and there is no need for Government intervention.

The region 4 in Fig. 1 where $A > \ddot{A}$ corresponds to the complete information outcome. Only high-type entrepreneurs are financed, and the financier gets ${\frac{I-A}{\lambda_{H}}},$ if the project succeeds.

In region 2 where $A\in[\bar{A},\dot{A}]$ and in region 3 where $A\in]\max\{\hat{A},\dot{A}\} ,\ddot{A}],$ all the high-type entrepreneurs are financed so there is no social inefficiency related to the financing of high-type entrepreneurs. But also at least some low-type entrepreneurs are financed, which creates a social loss compared with the complete information case where no low types are financed. In other words, under incomplete information there is excessive financing in regions 2 and 3, as in De Meza and Webb (1987). In equilibrium, high-type entrepreneurs also cross-subsidize low types and receive lower share from a successful project than what they would get under complete information.

In region 1 $(A < \min\{\hat{A},\bar{A}\})$, no entrepreneur is financed as shown by Proposition 1. From the social point of view, it is efficient that low-type entrepreneurs do not get financing. High types should, however, obtain funding as in the complete information case. Financial constraints that prevent high-type entrepreneurs with $A < \min\{\hat{A},\bar{A}\}$ from undertaking economically viable innovation projects create a social loss. Since this paper is about financing constraints, we in what follows focus on region 1 where the funding gap exists.

4 R&D subsidy application and allocation

In this section, we solve the subgame where entrepreneurs contemplate applying for subsidies and Government decides on screening and awarding a subsidy, abstracting from the funding decisions of the market-based financiers. We will proceed under the assumption that receiving a subsidy is both a necessary and sufficient condition to secure the additional external funding from private sources.¹⁷ In other words, we assume that with the subsidy an entrepreneur can launch an innovation project that she could not undertake otherwise. In the next section, we verify the parameter values when this constitutes an equilibrium of the full game where the funding decisions of the market-based financiers are explicitly taken into account.

Because the subgame considered in this section is more complicated than the standard adverse selection model outlined in the previous section, it is useful to specify the timing of actions and agents’ strategies.

• 0. Nature draws a type $i\in\{H,L\}.$ Probabilities of a high type and a low type are p and 1 − p, 0 < p < 1.

• 1. The entrepreneur observes her type and then chooses an action $a^{E}\in A^{E}$  = {apply for a subsidy (AP), do not apply (NAP)} where A ^E is the action space of the entrepreneur.

• 2. Government receives the application, but does not observe the type of the entrepreneur. Government chooses an action $a_{1}^{G}\in A_{1}^{G}$ = {screen the application (SC), do not screen (NSC)} where $A_{1}^{G}$ is the Government’s action space at this stage.

• 3. Government chooses an action $a_{2}^{G}\in A_{2}^{G}$ = {give a subsidy (S), do not give (NS)}, where $A_{2}^{G}$ is the Government’s action space at this stage.

• 4. The entrepreneurs with the subsidy execute their projects, and payoffs are realized as shown below.

Since the entrepreneur’s action in the last stage of the game is straightforward, the entrepreneur’s only strategic decision is to whether to apply for a subsidy or not in stage 1. Hence, we can write that the entrepreneur chooses a pure strategy s ^E from her pure-strategy space Σ^E = A ^E = {AP, NAP}. If Government screens and finds out the true type of the entrepreneur in stage 2, it gives a subsidy to a high-type entrepreneur but not to a low-type entrepreneur in stage 3. Government’s pure-strategy space is hence

$$ \Upsigma^G = \{SC, \;S \quad \hbox{if}\, i=H, \;NS \quad \hbox{if}\, i=L),(NSC, S),(NSC, NS)\}. $$

In the following, we refer to the first strategy as SC so

$$\Upsigma^{G}=\{SC,(NSC, S),(NSC, NS)\}. $$

As we focus on PBE, Government’s updated belief θ about the entrepreneur’s type in the non-singleton information sets is determined by Bayes’ rule using the prior probabilities and the equilibrium strategies. Figure 2 shows the extensive-form representation of the subgame.

Fig. 2

Extensive-form representation of the application process with perfect screening

Let $\Uppi_{s^{G}}^{G,i}$ refer to Government’s payoff from choosing a pure-strategy $s^{G}\in\Upsigma^{G}$ when the entrepreneur applies for a subsidy and the type of the entrepreneur is $i\in\{ H,L\}.$ When Government decides to screen (s ^G = SC) and the entrepreneur is of a high-type (i = H), Government’s payoff is given by

$$ \Uppi_{SC}^{G,H}=\lambda_{H}(R_{H}+W)-I-gS-c-\sigma. $$

(10)

Upon finding out that the entrepreneur is of a high-type, Government grants a subsidy S to the entrepreneur who can then secure the rest of the required funds, I + c − A − S, from the private sector financiers and is able to launch her project. Recall that the total size of the project is I + c after the monetary cost of applying for the subsidy (c) is taken into account. The entrepreneur’s and her private financiers’ joint expected payoff is then λ _H R _H  − I − c + S. Since Government’s objective function includes the private sector agents’ payoffs as arguments, the net cost of the subsidy to Government consists of the shadow cost of public funds gS. In (10), σ is the screening cost and W is the social externality generated by a successful project of a high-type entrepreneur.

Similarly, if Government decides to economize on screening costs, but nonetheless grants a subsidy and the applicant is of a high-type, Government’s payoff is

$$ \Uppi_{NSC,S}^{G,H}=\lambda_{H}(R_{H}+W)-I-gS-c, $$

(11)

which is identical to (10) save the cost of screening σ. Government’s payoff from the same strategy s ^G = (NSC, S) when the applicant is of a low type is given by

$$ \Uppi_{NSC,S}^{G,L}=\lambda_{L}R_{L}-I-gS-c. $$

(12)

In this case, there are no societal benefits associated to the low-type entrepreneur’s project even if it succeeds.

When the applicant is of a low type, Government’s payoff to screening is

$$\Uppi_{SC}^{G,L}=-c-\sigma. $$

(13)

After screening and realizing that the entrepreneur is of a low type, Government does not give a subsidy. Hence, under our assumptions, the entrepreneur cannot execute her project. For the same reason, Government’s payoff in case Government does not screen and does not give a subsidy is simply

$$ \Uppi_{NSC,NS}^{G,i}=-c $$

(14)

irrespective of the entrepreneur’s type.

Let us next consider the entrepreneurs’ payoffs from applying for a subsidy to any given Government’s pure strategy $s^{G}\in\Upsigma^{G}.$ A high-type entrepreneur gets a subsidy if Government follows the strategy s ^G = SC or s ^G = (NSC, S), but does not get a subsidy if Government follows the strategy s ^G = (NSC, NS). Similarly, if Government follows the strategy s ^G = SC or s ^G = (NSC, NS), a low-type entrepreneur does not get a subsidy, but if s ^G = (NSC, S), she gets a subsidy. Since from the entrepreneur’s point of view the only payoff-relevant decision of Government is whether it gives a subsidy or not, we will use $\Uppi_{a_{2}^{G}}^{E,i}$ to denote the payoff of an entrepreneur of type $i\in\{ H,L\}$ to Government’s second action $a_{2}^{G}\in A_{2}^{G}.$ As a result, the entrepreneur’s payoff to an accepted subsidy application $(a_{2}^{G}=S)$ is given by

$$ \Uppi_{S}^{E,i}=\lambda_{i}\left(R_{i}-F^{S}\right)-A, $$

(15)

and to a rejected application $\left(a_{2}^{G}=NS\right)$ by

$$ \Uppi_{NS}^{E,i}=-c. $$

(16)

In (15), F ^S is the entrepreneur’s repayment obligation to the market-based financier if the entrepreneur has received a subsidy and her project succeeds. For the moment, we take it as given and assume that it is small enough to render $\Uppi_{S}^{E,i}> 0$ and the problem interesting. F ^S will be determined as part of equilibrium in Sect. 5.

4.1 Equilibria

Since a pure-strategy equilibrium is an equilibrium in degenerate mixed strategies, we focus on mixed strategies. As we are interested in Government’s screening activities, we seek PBE where screening is a viable strategy choice for Government, a high-type entrepreneur always applies, and a low-type entrepreneur chooses a mixed strategy $\mu_{s^{^{E}}}\in\Updelta\Upsigma^{E}$ where ΔΣ^E denotes the set of probability distributions over pure strategies, and $\mu_{s^{E}}$ is the probability assigned to a pure strategy $s^{E}\in\Upsigma^{E}=\{AP,NAP\}.$ ¹⁸ Similarly, Government chooses a mixed strategy $\alpha_{s^{G}}\in\Updelta\Upsigma^{G}$ over pure strategies $s^{G}\in\Upsigma^{G}=\{SC,(NSC,S),(NSC,NS)\}.$ As $\mu_{s^{^{E}}}$ and $\alpha_{s^{G}}$ are probability distributions we will write that μ _AP  = μ, μ _NAP  = 1 − μ, and $\alpha_{NSC,NS}=1-\alpha_{SC}-\alpha_{NSC,S}\; (\mu,\alpha_{SC},\alpha_{NSC,S}\geq0),$ meaning that a low-type entrepreneur applies with probability μ and Government randomizes between strategies SC, (NSC, S) and (NSC, NS) with probabilities α _SC , α_NSC,S and 1 − α _SC  − α_NSC,S.

We first consider the entrepreneurs’ optimal strategies. Given Government’s mixed strategy $\alpha_{s^{G}},$ the expected payoff of a high-type entrepreneur from applying is

$$ E(\Uppi_{AP}^{E,H})=(\alpha_{SC}+\alpha_{NSC,S})\Uppi_{S}^{E,H}+ (1-\alpha_{SC}-\alpha_{NSC,S})\Uppi_{NS}^{E,H}.$$

(17)

and the low-type entrepreneur’s expected payoff from applying is

$$ E(\Uppi_{AP}^{E,L,})=(1-\alpha_{NSC,S})\Uppi_{NS}^{E,L} +\alpha_{NSC,S}\Uppi_{S}^{E,L}. $$

(18)

In (17) and (18), $\Uppi_{S}^{E,i}$ and $\Uppi_{NS}^{E,i}, i\in\{ H,L\},$ are specified by (15) and (16). The entrepreneurs who do not apply for a subsidy obtain zero payoff.

Lemma 1

(Optimal strategies of entrepreneurs)

• If $\alpha_{SC}+\alpha_{NSC,S}> {\frac{c}{\lambda_{H}(R_{H}-F^{S})-A+c}},$ the best strategy for a high-type entrepreneur is to apply.

• If$\alpha_{NSC,S}> {\frac{c}{\lambda_{L}(R_{L}-F^{S})-A+c}},$the best strategy for a low-type entrepreneur is to apply (μ = 1).

• If$\alpha_{NSC,S}\,< \,{\frac{c}{\lambda_{L}(R_{L}-F^{S})-A+c}},$the best strategy for a low-type entrepreneur is not to apply (μ = 0).

• If$\alpha_{NSC,S}={\frac{c}{\lambda_{L}(R_{L}-F^{S})-A+c}},$a low-type entrepreneur randomizes between applying and not with probabilities μ and (1 − μ).

Lemma 1 describes the entrepreneurs’ optimal application strategies as function of Government’s screening and subsidy allocation strategies. High types get a subsidy unless Government picks strategy (NSC, NS) according to which it does not screen nor give subsidies. Low types care in turn only whether Government is using strategy (NSC, S) as this is their only chance to obtain subsidies, given our assumption of perfect screening.

Let us next turn to Government’s strategy choices. Since a low-type entrepreneur is using a mixed strategy (μ, 1 − μ), Government’s belief θ that an applicant is of a high type is given by Bayes’ rule as

$$ \theta={\frac{p}{p+\mu(1-p)}}. $$

(19)

Government’s expected payoff from choosing pure strategy screening (α _SC  = 1) is $E(\Uppi_{SC}^{G})=\theta\Uppi_{SC}^{G,H}+(1-\theta)\Uppi_{SC}^{G,L}$ which, by using (10) and (13), can be written as

$$ E(\Uppi_{SC}^{G})=\theta\left[\lambda_{H}(R_{H}+W) -I-gS\right]-c-\sigma. $$

(20)

Similarly, using (11) and (12) we see that Government’s expected payoff from α_NSC,S = 1 is

$$ E(\Uppi_{NSC,S}^{G})=\theta\left[\lambda_{H}(R_{H} +W)\right]+(1-\theta)\lambda_{L}R_{L}-I-gS-c. $$

(21)

Finally, from choosing pure-strategy s ^G = (NSC, NS) (α _SC  = α_NSC,S = 0), Government’s payoff is simply given by (14) as $E(\Uppi_{NSC,NS}^{G})= -c.$

As (20) and (21) indicate, Government’s strategy choice depends on the low-type entrepreneurs’ application strategies (value of μ, incorporated in θ). Let us define $\underline {\mu}\equiv\left({\frac{p}{1-p}}\right) \left({\frac{\sigma}{I+gS-\lambda_{L}R_{L}-\sigma}}\right)$ and $\bar{\mu}\equiv\left({\frac{p} {1-p}}\right)\left({\frac{\lambda_{H}\left(R_{H}+W\right)-I-gS-\sigma} {\sigma}}\right).$ We proceed under the assumption that $\underline {\mu}\leq\bar{\mu}$ and verify later that it holds in the equilibria we are focusing on.

Lemma 2

(Optimal strategies of Government)

• If${\mu}< \underline{\mu},$the best strategy for Government is (NSC, S) (α_NSC,S = 1).

• If$\mu > \bar{\mu},$the best strategy for Government is (NSC, NS) (1 − α _SC  − α_NSC,S = 1).

• If$\underline {\mu}< \mu < \bar{\mu,}$the best strategy for Government is SC (α _SC  = 1).

• If$\mu=\bar{\mu},$Government randomizes between SC and (NSC, NS) with probabilities α _SC and 1 − α _SC .

• If$\mu=\underline{\mu},$Government randomizes between SC and (NSC, S) with probabilities α _SC and α_NSC,S = 1 − α _SC .

Lemma 2 suggests that Government would screen applicants only with intermediate values of μ. If low types were not likely to apply (μ is small), Government would give subsidies to all applicants without screening. In contrast, if low types would apply with a high probability (μ is high), Government would not give subsidies to any one. This conclusion is, however, based on the assumption that $\bar{\mu}$ is smaller than one. If $\bar{\mu}> 1,$ the strategy (NSC, NS) is not a plausible option for Government and Government would always screen if $\mu > \underline{\mu}.$ Figure 3 summarizes the optimal strategies for Government as a function μ. The figure is drawn under the assumption that $\bar{\mu}< 1.$

Fig. 3

Optimal strategies for Government with different values of μ

As we aim at analyzing Government’s screening activities, we focus on the parameter range where screening remains a viable option for Government.

Proposition 2

Screening is a plausible strategy for Government only if $\sigma\leq \min\left\{{\frac{(I+gS-\lambda_{L}R_{L})(\lambda_{H} \left(R_{H}+W\right)-I-gS)}{\lambda_{H}\left(R_{H}+W\right) -\lambda_{L}R_{L}}},(1-p)(I+gS-\lambda_{L}R_{L})\right\} .$

Proposition 2 suggests that screening is viable strategy choice for Government only if screening costs and the proportion of high-type entrepreneurs in the population are low enough. Obviously, if screening is costly enough or its benefits—the probability of finding a low type project—are low enough, screening is no longer profitable.

Generally, the set of sensible Government strategies depends on the values of σ and p, as shown by Fig. 4.¹⁹ The figure displays four different regions. In regions 1 and 2, screening is a plausible strategy, whereas in regions 3 and 4 the combinations of p and σ are such that screening is never optimal. In region 3, it is always optimal for Government to grant a subsidy without screening. In other words, the screening costs are so high compared to the relatively high share of high-type entrepreneurs in the population that it is optimal for Government just to grant a subsidy to every applicant. In region 4, Government chooses between strategies (NSC, S) and (NSC, NS). Note that the figure is restricted to the parameter space where financing constraints exist, $p\leq{\frac{I-\lambda_{L}R_{H}} {\left(\lambda_{H}-\lambda_{L}\right)R_{H}}}$ (see Proposition 1).

Fig. 4

Plausible Government strategies with different values of screening costs (σ) and different share of high-type entrepreneurs in the economy (p)

Propositions 1 and 2 identify the parameter region which is the focus of this paper.²⁰ Consequently, we consider regions 1 and 2 from Fig. 4. Note that if $\bar{\mu}> 1,$ the strategy (NSC, NS) is not a plausible option for Government and we are in region 2 of Fig. 4. Since $\bar{\mu}\equiv\left({\frac{p} {1-p}}\right)\left({\frac{\lambda_{H}\left(R_{H}+W\right)-I-gS-\sigma} {\sigma}}\right), \bar{\mu}< 1$ if σ > p(λ _H (R _H  + W) − I − gS). The intuition is that (NSC, NS) is a plausible strategy for Government only if screening costs are high relative to the share of high-type entrepreneurs in the population (region 1 of Fig. 4). For p low enough, it is possible that $\bar{\mu} \, < \,1$ and Proposition 2 hold simultaneously.

We can now state the main result of this section.

Proposition 3

In a PBE of the game

• A high-type entrepreneur always applies.

• A low-type entrepreneur applies with probability $\mu=\underline {\mu}\equiv\left({\frac{p}{1-p}}\right)\left({\frac{\sigma} {I+gS-\lambda_{L}R_{L}-\sigma}}\right).$

• Government randomizes between (NSC, S) and SC with probabilities $\alpha_{NSC,S}={\frac{c} {\lambda_{L}(R_{L}-F^{S})-A+c}}$ and $\alpha_{SC}=1-\alpha_{NSC,S}={\frac{\lambda_{L}(R_{L}-F^{S})-A} {\lambda_{L}(R_{L}-F^{S})-A+c}}$ and Government’s belief that the applicant is of a high type is determined by $\theta=1-{\frac{\sigma}{I+gS-\lambda_{L}R_{L}}}.$

In this PBE low types have an incentive to apply because Government does not screen all applications but just gives a subsidy without screening with some positive probability. In this way Government can save some screening costs that offsets the welfare loss caused by subsidies directed to low type projects.

Government’s mixed strategy can be interpreted as Government deciding on the intensity of screening. The higher is the probability of screening versus automatically granting a subsidy, the higher is the screening intensity and the higher is the probability of finding out the true type of the project. Only if the probability of screening is equal to one, screening is truly perfect and Government finds out the true type of the project for sure.²¹

The PBE identified by Proposition 3 is based on the assumption that the subsidy program is in place and Government chooses whether to screen or not, i.e, the possibility to just close the program is not taken into account. If Government chooses to close the whole program, the payoff is zero to both entrepreneur and Government (ignoring the costs related to the closing of the program). If the strategy profile characterized by Proposition 3 generates a strictly positive payoff to Government then it is an equilibrium, even taking into account the possibility of closing the subsidy program. It can be shown that the above strategy profile remains an equilibrium with minor modifications to the restriction imposed on σ in Proposition 2.²²

Comparative statistics of Government screening probability would be straightforward if we took the entrepreneur’s repayment obligation, F ^S, as fixed. However, in an equilibrium of the full game, determined in the next section, the parameters of F ^S will include S, c, α _SC and θ. As a result, in an equilibrium of the full game, the formula for α _SC given in Proposition 3 is in an implicit form. In the end of the Appendix, we derive the partial derivatives of the screening probability with respect to σ, c, A and S when F ^S is endogenous. If the parameters are such that Government is relatively confident that an application comes from a high-type entrepreneur (the equilibrium value of θ is sufficiently high, e.g., because σ is very low), the results are intuitive: the screening probability is decreasing in the screening cost, in the application cost, and in the initial wealth, and increasing in the level of the subsidy.

Fortunately, comparative statics of the low-type’s optimal strategy are easier. An increase in the screening cost (σ), in the share of high types (p) in the population, or in the NPV of the low-type entrepreneurs’ projects (λ _L R _L  − I) increases low-type’s application probability as could be expected, but an increase in the subsidy decreases the application probability. The latter outcome may seem counterintuitive, but it is explained by Government’s screening incentives, which are increasing with S. If S increases, low-type entrepreneurs anticipate tighter screening and are less likely to apply. Hence, public screening works more efficiently in discouraging low-type entrepreneurship if it is accompanied with subsidy allocation.

5 Public and private funding of innovations

We now analyze the full model where the entrepreneurs can first apply for an R&D subsidy from Government, and then seek market-based financing from other sources. For brevity, we assume that in the funding gap region that we are focusing on, entrepreneurs need external market-based financing besides the subsidy to be able to undertake the innovation project (A − c + S < I) and have enough initial wealth to apply for a subsidy (A > c).

As mentioned, we assume that the private financier observes whether the entrepreneur has received an R&D subsidy or not, and it knows how Government funding policy works. The subsidy observation provides additional information to the market-based financier about the type of the project. Then, if the entrepreneur has been granted a subsidy, market-based financiers’ participation constraint reads as

$$ I-A-S+c\leq\hat{\lambda}F^{S}, $$

(22)

where $\hat{\lambda}$ is the updated success probability when the entrepreneur has received an R&D subsidy, and it is determined by Bayes’ rule as

$$ \hat{\lambda}=P(H|S)\lambda_{H}+[1-P(H|S)]\lambda_{L}. $$

(23)

In (23), $P(H\vert S)$ is the conditional probability that the entrepreneur is of a high-type, given that she has received an R&D subsidy from Government. In equilibrium, Government randomizes between SC and (NSC, S) with probabilities α _SC and 1 − α _SC . This means that $P(H|S)=\hat{p}=\alpha_{SC}+(1-\alpha_{SC})\theta$ where θ and α _SC are given by Proposition 3. Since in this equilibrium high-type entrepreneurs always apply, the financier knows for sure that an entrepreneur without a subsidy is a low type. Given that financiers must break-even, Eq. 22 holds with equality and the share of a successful project given to a financier is

$$ F^{S}={\frac{I-A-S+c}{\hat{\lambda}}}. $$

(24)

The entrepreneur’s participation constraint remains $\lambda_{i}R_{i}^{E}\geq A,$ since to receive an R&D subsidy the entrepreneur has to invest her initial wealth in the project (where now the application and investment constitute the project). The pledgeable income that can be offered to the financier is $R_{i}^{F\max}=R_{i}-R_{i}^{E}=R_{i}-{\frac{A}{\lambda_{i}}}$ as before. As a result, an entrepreneur with a subsidy can get market-based financing if

$$ {\frac{I-A-S+c}{\hat{\lambda}}}\leq R_{H}-{\frac{A}{\lambda_{H}}}. $$

(25)

The right-hand side of Eq. 25 is the pledgeable income that a high-type entrepreneur is willing to offer to the financier, and it is the same as without a subsidy program. Solving Eq. 25 for A shows that if the entrepreneur has been granted an R&D subsidy, the private financiers grant funding if

$$ A\geq\bar{A}^{S}\equiv{\frac{\lambda_{H}} {\lambda_{H}-\hat{\lambda}}}[I-S+c-\hat{\lambda}R_{H}]. $$

Proposition 4

(i) Entrepreneurs with an R&D subsidy can get market-based financing with less initial capital, i.e., $\bar{A}> \bar{A}^{S},$ if $\hat{\lambda}\geq\bar{\lambda}.$ (ii) Due to Government screening, the fact that an entrepreneur has received an R&D subsidy provides an informative signal to the financier, i.e., $\hat{\lambda}> \bar{\lambda}.$

Figure 5 shows how the funding gap region presented in Fig. 1 changes as a result of the introduction of a subsidy program. From Eq. 6, we know that $\hat{A}\equiv{\frac{\lambda_{L}\lambda_{H}\left(R_{L}-R_{H}\right)}{\lambda_{H}-\lambda_{L}}}$ and it does not change when a subsidy program is introduced, since the participation constraint of an entrepreneur remains the same. What happens is that the $\bar{A}$-curve shifts downward. Whether the shift reduces financial constraints depends on the value of $\hat{p}.$

Fig. 5

Change in Region 1, when a subsidy program is introduced

Proposition 5

R&D subsidy program reduces financial constraints, when$p\in\left[{\frac{(\hat{p}-\alpha_{SC})\mu}{(1-\underline {\hat{p}})+(\underline {\hat{p}}-\alpha_{SC})\mu}},{\frac{I-\lambda_{L}R_{H}} {\left(\lambda_{H}-\lambda_{L}\right)R_{H}}}\right]$where α _SC and μ are the equilibrium strategies and$\underline {\hat{p}}={\frac{I-S+c-\lambda_{L}R_{L}} {\lambda_{H}R_{H}-\lambda_{L}R_{L}}}.$

Propositions 4 and 5 summarize the main result. R&D subsidies and the related screening process can help financially constrained entrepreneurs to get external financing for their innovation projects, if the share of high-type entrepreneurs in the population is sufficiently high. Two different channels generate this effect. Part (i) of Proposition 4 presents a trivial one: a subsidy reduces the amount of external capital needed, thus reducing capital costs. The more interesting channel is depicted in the second part of Proposition 4: subsidy observation provides additional information to market-based financiers about the quality of the project. With this additional information, market-based financiers are willing to fund entrepreneurs with a subsidy with a lower rate of return and this reduces the funding gap.

The expected total welfare effect of R&D subsidies to a society, with p belonging to the interval stated in Proposition 5, depends on the distribution of initial wealth. The initial wealth required to get financing from private sources becomes smaller, i.e., $\bar{A}$ is transformed to $\bar{A}^{S}.$ Figure 6 presents the pledgeable incomes of a low and high type entrepreneurs, $R_{L}^{F\max}$ and $R_{H}^{F\max},$ and the share of a successful project that a financier requires to invest in the project with and without a subsidy, F ^S and F. When a subsidy program is introduced the repayment required by a financier declines from F to F ^S and, as a result, the funding gap region reduces from $[0,\bar{A}]$ to $[0,\bar{A}^{S}].$

Fig. 6

Change in the funding gap region as a subsidy program is introduced

The expected net benefit to society from one project that has received a subsidy is

$$ E(\Uppi^{G})=\alpha_{SC}E(\Uppi_{SC}^{G})+(1-\alpha_{SC})E(\Uppi_{NSC,S}^{G}). $$

In equilibrium, Government is indifferent between the strategies SC and NSC, S, which implies that the expected payoffs from these two strategies are equal. This gives

$$ \begin{aligned} E(\Uppi^{G})&=E(\Uppi_{SC}^{G})=E(\Uppi_{NSC,S}^{G})=\\ &{\frac{(I+gS-\lambda_{L}R_{L})\left[\lambda_{H}\left(R_{H}+W\right) -I-gS-c)\right]-\sigma\left[\lambda_{H}\left(R_{H}+W\right) -\lambda_{L}R_{L}\right]}{I+gS-\lambda_{L}R_{L}}}. \end{aligned} $$

Depending on the value of σ this can be either positive or negative. If

$$ \sigma< {\frac{(I+gS-\lambda_{L}R_{L})\left[\lambda_{H}\left(R_{H}+W\right) -I-gS-c)\right]}{\lambda_{H}\left(R_{H}+W\right)-\lambda_{L}R_{L})}} $$

then E(Π^G) is positive.²³

The expected total net benefit to society depends on the share of entrepreneurs whose initial wealth is in the interval $[\bar{A}^{S},\bar{A}].$ Clearly the outcome is not the first-best: also some low-type entrepreneurs are financed. However, if the total net benefit to society is positive, the subsidy program improves the market outcome under asymmetric information.

6 Conclusions

Financial constraints are one of the rationales used to justify government intervention in the form of R&D subsidies. This study provides insights into the questions of whether and how R&D subsidies could be expected to alleviate financial constraints. The following conclusions can be drawn:

• Asymmetric information about the quality of R&D projects creates financing constraints for collateral-poor firms, if there is non-negligible share of non-viable projects within the economy.

• R&D subsidy policies that involve screening of projects are sustainable, if the screening costs are low enough.

• The higher the expected loss generated by low-quality projects and the lower the share of high-quality projects in the economy, the higher the screening costs can be without rendering screening activities unsustainable.

• R&D subsidies can reduce financing constraints for two different reasons: (1) The subsidy in itself reduces the cost of external capital because the need for market-based financing diminishes. (2) If market-based financiers can observe that a project has received a subsidy from the public agency, the subsidy provides an informative signal about the quality of the R&D project. Such a subsidy-observation increases the success probability of the project anticipated by the market-based financier. This reduces the cost of external capital for subsidized projects.

These findings highlight that the screening activities typically embedded into R&D subsidy policies can have a role of their own in reducing financial constraints. Instead of merely allocating subsidies, the public agency could have a certification role and yet reduce the financing constraints. This raises the question of whether, in terms of financial constraints, it would suffice to reduce the asymmetry of information merely through screening. We find, however, that granting funding besides screening not only strengthens the leverage effect, but also makes screening more efficient in discouraging low-quality entrepreneurship.

While this paper is more a positive analysis of application and allocation of R&D subsidies rather than normative welfare analysis of R&D subsidies, the findings suggest that under certain conditions R&D subsidy policies may be welfare improving. However, we focus on the range of parameter values where all entrepreneurs suffer from financing constraints. If the high-type entrepreneurs not suffering from financing constraints get subsidies, this limits the welfare-improving prospects of subsidy policies. Nonetheless, the screening activities of the public financier may prevent some low-type entrepreneurs from getting such market-based financing they would obtain in the absence of public funding. But even in the funding gap region the outcome is not fully efficient—also some low-quality projects are funded, and future work should consider optimal policy.

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.