Skip to main content

Investor protection and optimal contracts under risk aversion and costly state verification

Abstract

We present a model of firm finance that encompasses imperfect investor protection, risk aversion and costly state verification. We characterize optimal contracts and study the conditions under which standard debt is optimal. Under suitable assumptions about the structure of the problem, standard debt contracts (SDCs) are optimal if and only if investor protection is sufficiently low. On the other hand, low investor protection results in higher funding costs and bankruptcy probabilities. In our setting, this implies that when SDCs are optimal, lowering investor protection reduces the entrepreneur’s welfare. Numerical examples show that moderate changes in investor protection can have large effects on the terms of the contract and on the entrepreneur’s welfare. Finally, we study the role of leverage and consider the welfare consequences of suboptimally implementing standard debt contracts.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. A recent survey of the literature on the finance-growth links is in Fernandez and Tamayo (2014).

  2. Townsend’s model has been extended in several dimensions and is now the workhorse of dynamic macroeconomic models with financial frictions. A partial list of extensions is as follows: economies with production (Gale and Hellwig 1985), heterogeneous borrowers (Williamson 1987), multiple investors (Winton 1995), limited commitment (Krasa and Villamil 2000), non-contractible verification (Hvide and Leite 2010), and imperfect monitoring (Greenwood et al. 2010). The use of CSV in macroeconomics was launched by Bernanke and Gertler (1989).

  3. In related but independent work, Galindo and Micco (2005) also model investor protection in this fashion. Our model contrasts with Sevcik (2012) where protection is captured by monitoring costs only. See also Castro et al. (2004).

  4. As pointed by Galindo and Micco (2004), the problem of limited creditor protection is widespread in developing countries and particularly acute in Latin America.

  5. As in Krasa et al. (2008), \(b\) captures the fraction of inputs (i.e., working capital) that are financed by the investor. For simplicity, our analysis abstracts from any setup investment or fixed costs, which can be interpreted in two ways: either setup investments/costs are zero or they are fully recovered under any contract. When nonzero setup costs are brought into the analysis (and subject to the legal constraints introduced below), our results will hold as long as imperfect investor protection impairs the distribution of cash flows more than the distributions of the initial investment [see Gennaioli and Rossi (2010)].

  6. We assume full commitment since we want to study in isolation the effects of exogenous imperfect investor protection (e.g., estate exemptions in Chapter 7). For a CSV model in which stochastic verification is optimal see Hvide and Leite (2010).

  7. The assumption that \(\underline{s}>0\) is also in Townsend (1979) and allows us to better study reward functions that satisfy Inada conditions.

  8. For an appropriately chosen weight vector, the solution to the program in (\(\mathcal {FB}.2\))–(\(\mathcal {FB}.3\)) is also a solution to a problem that maximizes the weighted average of the payoffs of the match subject to individual rationality constraints (IRC) for both. For a setup that maximizes the investor’s payoff subject to the entrepreneur’s IRC see Krasa et al. (2008).

  9. Throughout the paper, inequalities and equations involving random variables are assumed to hold almost everywhere (a.e.). We also assume that it is legitimate to differentiate under the integral sign.

  10. Krasa et al. (2008) consider a model that imposes sequential rationality in the players’ strategies (the equilibrium contract must be a PBE), so the CSV is a special case of their model.

  11. In Krasa et al. (2008), risk neutrality (of the borrower) implies that a SDC is the only optimal contract, so their comparative statics results naturally apply to SDCs only.

  12. This condition is satisfied, for example, by any monotonically increasing transformation of the normal distribution and by most relevant instances of the log-normal distribution.

  13. Among the first to consider seriously, the possibility that strong investor protection may discourage debt issuance are Rajan and Zingales (1995) although their analysis was far from conclusive. More recent studies provide sharper inference on the negative relationship between strong creditor rights and firm debt [e.g., Acharya et al. (2011), Ghoul et al. (2012)]. This is in contrast with the supply-side hypothesis of credit provision bourne out of the strong association observed in the data between investor protection and measures such as credit-to-GDP [see, e.g., Porta and Lopez-de (1998) and Djankov et al. (2007)].

  14. The effect of increasing \(\gamma \) on the borrower’s welfare is always negative and is independent of optimality considerations, so we concentrate on \(\eta \).

  15. Separability of preferences over consumption and effort/cost is a common assumption in the asymmetric information literature.

  16. Heuristically, these conditions are that the entrepreneur’s continuation payoffs induced by a contract after each history are common knowledge and that a boundedness condition on the continuation payoffs is satisfied. Green (1987) and Spear and Srivastava (1987) were the first to show that under these conditions, there exists a recursive formulation for the maximization problem faced by the principal in models of repeated moral hazard.

  17. Given the heterogeneity of bankruptcy cases, we would very much like to work with the median rather than the average. However, averages allow us to recover some figures from the original data and ensure internal consistency which the median does not. For instance, in Table III of Bris et al. (2006) the average optimistic recovery rate before expenses is 80 % of total assets. This is consistent with the average recovery rate of 51 % after expenses reported on their table XIII and with the average costs being 38 and 8 % of bankruptcy assets and total assets, respectively. On the other hand, their reported median recovery rate before expenses is 38 %, and after expenses is 70 %, something altogether problematic for our purposes.

  18. Interestingly, the parameter values for the benchmark specification in Krasa et al. (2008) lie outside these ranges as they use \(\eta =0.1\) and \(\gamma =0.1\), implying bankruptcy costs of 25 % of distributed assets.

  19. An alternative source of estimates for \(\eta \) is Blazy et al. (2010) who repor recovery rates \((\eta )\) between 76 % (senior creditors) and 10 % (junior creditors) in Germany and between 31 and 6 % in the UK.

  20. For instance, \(u\left( c\right) =\ln c\) requires \(\eta \ge 0.3064\), resulting in a very tight constraint, with \(\lambda ^{*}\ge 18.3\).

  21. In the current environment, the revelation principle still holds, and therefore, we can reduce the message space to the set \(\tilde{\varSigma }\) without loss of generality. A formal proof be found in Monnet and Quintin (2002).

  22. This is actually a sufficient condition. In fact, the necessary condition is \(\lim _{t\rightarrow \infty }\beta ^{t}\mathbb {E}_{0}w_{t}\left( h^{t}\right) =0\), which is a condition equivalent to (7) in Theorem 9.2 of Stokey et al. (1989) (pp. 246).

References

  • Acharya, V.V., Amihud, Y., Litov, L.: Creditor rights and corporate risk-taking. J. Financ. Econ. 102(1), 150–166 (2011)

    Article  Google Scholar 

  • Bernanke, B., Gertler, M.: Agency costs, net worth, and business fluctuations. Am. Econ. Rev. 79(1), 14–31 (1989)

    Google Scholar 

  • Blazy, R., Petey, J., Weil, L.: Can bankruptcy codes create value? Evidence from creditors recoveries in France, Germany, and the UK. Working Paper, Laboratoire de Recherche en Gestion et Economie (LaRGE), Universit de Strasbourg (France) (2010)

  • Bris, A., Welch, I., Zhu, N.: The costs of bankruptcy: chapter 7 liquidation versus chapter 11 reorganization. J. Finance 61(3), 1253–1303 (2006)

    Article  Google Scholar 

  • Caputo, M.: Foundations of Dynamic Economic Analysis: Optimal Control Theory and Applications. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  • Castro, R., Clementi, G., MacDonald, G.: Investor protection, optimal incentives, and economic growth. Q. J. Econ. 119(3), 1131–1175 (2004)

    Article  Google Scholar 

  • Cheng, S., Shiu, C.: Investor protection and capital structure: international evidence. J Multinatl Financ Manag 17, 30–44 (2007)

    Article  Google Scholar 

  • Clementi, G., Hopenhayn, H.: A theory of financing constraint and firm dynamics. Q. J. Econ. 121, 229–265 (2006)

    Google Scholar 

  • Cressy, R.: Credit rationing or entrepreneurial risk aversion? An alternative explanation for the evans and jovanovic finding. Econ. Lett. 66, 235–240 (2000)

    Article  Google Scholar 

  • Djankov, S., McLiesh, C., Shleifer, A.: Private credit in 129 countries. J. Financ. Econ. 84, 299–329 (2007)

    Article  Google Scholar 

  • Fang, H., Nofsinger, J.: Risk aversion, entrepreneurial risk, and portfolio selection. J. Entrep. Financ. 13, 25–55 (2009)

    Google Scholar 

  • Fernandez, A., Tamayo, C.: From institutions to financial development and growth: what are the links? Research Department Working Paper, Inter-American Development Bank (2014)

  • Gale, D., Hellwig, M.: Incentive compatible debt contracts: the one period problem. Rev. Econ. Stud. 52(4), 647–663 (1985)

    Article  Google Scholar 

  • Galindo, A., Micco, A.: Creditor protection and financial markets: empirical evidence from Latin America. Econ. Rev. FRB of Atlanta. 2nd Quarter, 29–37 (2004)

  • Galindo, A., Micco, A.: Creditor protection and credit volatility. Research Department Working Paper No. 528, Inter-American Development Bank (2005)

  • Gennaioli, N., Rossi, S.: Judicial discretion in corporate bankruptcy. Rev. Financ. Stud. 23, 4078–4114 (2010)

    Article  Google Scholar 

  • Ghoul, S.E., Guedhami O., Cho, S., Suh, J.: Creditor rights and capital structure: evidence from international data. In: Preceedings of the Annual Meetings, European Financial Management Association (2012)

  • Green, E.: Lending and the smoothing of uninsurable income. In: Prescott, E., Wallace, N. (eds.) Contractual Arrangements for International Trade. University of Minnesota Press, Minneapolis (1987)

    Google Scholar 

  • Greenwood, J., Sanchez, J.M., Wang, C.: Financing development: the role of information costs. Am. Econ. Rev. 100, 1875–1891 (2010)

    Article  Google Scholar 

  • Hvide, H.K., Leite, T.E.: Optimal debt contracts under costly enforcement. Econ. Theor. 44(1), 149–165 (2010)

    Article  Google Scholar 

  • Kalemli-Ozcan, S., Sorensen, B., Yesiltas, S.: Leverage across firms, banks and countries. NBER Working Paper No. 17354 (2011)

  • Krasa, S., Villamil, A.P.: Optimal contracts when enforcement is a decision variable. Econometrica 68(1), 119–134 (2000)

    Article  Google Scholar 

  • Krasa, S., Sharma, T., Villamil, A.P.: Bankruptcy and firm finance. Econ. Theor. 36, 239–266 (2008)

    Article  Google Scholar 

  • La Porta, R., Lopez-de-Silanes, F., Shleifer, A., Vishny, R.W.: Legal determinants of external finance. J. Finance 52(30), 1131–1150 (1997)

    Article  Google Scholar 

  • La Porta, R., Lopez-de-Silanes, F., Shleifer, A., Vishny, R.W.: Law and finance. J. Polit. Econ. 106(6), 1113–1155 (1998)

    Article  Google Scholar 

  • Lewellen, K.: Financing decisions when managers are risk averse. J. Financ. Econ. 82, 551–589 (2006)

    Article  Google Scholar 

  • Monnet, C., Quintin, E.: Optimal contracts in a dynamic costly state verification model. ECB Working Paper No. 126 (2002)

  • Monnet, C., Quintin, E.: Optimal contracts in a dynamic costly state verification model. Econ. Theor. 26, 867–885 (2005)

    Article  Google Scholar 

  • Panousi, V., Papanikolaou, D.: Investment, idiosyncratic risk, and ownership. J. Finance 67, 1113–1148 (2012)

    Article  Google Scholar 

  • Pereira, P.F., Ferreira, M.A.: Capital structure and law around the world. J. Multinatl. Financ. Manag. 21, 119–150 (2011)

    Article  Google Scholar 

  • Rajan, R., Zingales, L.: What do we know about capital structure? Some evidence from international data. J. Finance 50, 1421–1460 (1995)

    Article  Google Scholar 

  • Schmid, T., Ampenberger, M., Kaserer, C., Achleitner, A.-K.: Family firms, agency costs and risk aversion: empirical evidence from diversification and hedging decisions. CEFS Working Paper No. 2008-13, (2008)

  • Sevcik, P.: Financial contracts and the political economy of investor protection. Am. Econ. J. Macroecon. 4, 163–197 (2012)

    Article  Google Scholar 

  • Smith, A., Wang, C.: Dynamic credit relationships in general equilibrium. J. Monet. Econ. 53, 847–877 (2006)

    Article  Google Scholar 

  • Spear, S., Srivastava, S.: On repeated moral hazard with discounting. Rev. Econ. Stud. 54, 599–617 (1987)

    Article  Google Scholar 

  • Stokey, N., Lucas, R.E., Prescot, E.: Recursive methods in economic dynamics. Harvard University Press, Cambridge (1989)

    Google Scholar 

  • Townsend, R.: Optimal contracts and competitive markets with costly state verification. J. Econ. Theory 21(2), 265–293 (1979)

    Article  Google Scholar 

  • Wang, C.: Dynamic costly state verification. Econ. Theor. 25, 887–916 (2005)

    Article  Google Scholar 

  • Williamson, O.: Corporate finance and corporate governance. J. Finance 43, 567–591 (1988)

    Article  Google Scholar 

  • Williamson, S.: Costly monitoring, loan contracts, and equilibrium credit rationing. Q. J. Econ. 102, 135–146 (1987)

    Article  Google Scholar 

  • Winton, A.: Costly state verification and multiple investors: the role of seniority. Rev. Financ. Stud. 8(1), 91–123 (1995)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Cesar E. Tamayo.

Additional information

My special thanks go to Roberto Chang, Todd Keister and Colin Campbell for their guidance and helpful comments. I would also like to thank Cristina Fuentes-Albero, Arturo Galindo, Richard P. McLean, Tomas Sjöström, Oscar Valencia, Anne Villamil and participants in the 2013 Midwest Macro Meetings at the University of Illinois, Urbana-Champaign for useful discussions. The paper was greatly improved by suggestions from two anonymous referees. The information and opinions presented are entirely those of the author, and no endorsement by the Inter-American Development Bank, its Board of Executive Directors or the countries they represent is expressed or implied.

Appendix

Appendix

Proof of Theorem 1

Proof

(Lemma 1) The UPC implies that unverified payments must only depend on the message, that is, \(R\left( \hat{s},s\right) =R\left( \hat{s}\right) \) for any \(\hat{s}\notin B\). Therefore, the entrepreneur will choose \(\tilde{s}=\arg \min _{\hat{s}\notin B}R\left( \hat{s}\right) \), so the contract may as well set \(R\left( \tilde{s}\right) =\bar{R}\). \(\square \)

Proof

(Lemma 2) For any \(\hat{s},s\in B\), the assumption \(u^{\prime }>0\) implies that a necessary and sufficient condition for (\(\mathcal {P}.3\)) to be satisfied is that \(R\left( \hat{s},s\right) =R\left( s,s\right) \). Hence, in the verification region, \(R\left( \hat{s},s\right) =\hat{R}\left( s\right) \) for some \(\hat{R}\left( \cdot \right) \). Now, for \(s\in B\), \(\hat{R}\left( s\right) >\bar{R}\) can never be optimal since in this case the entrepreneur will prefer to misreport and pay \(\bar{R}\) (the ICC is not satisfied). Next, if \(\hat{R}\left( s\right) =\bar{R}\) on a set of positive measure, then the investor will inefficiently pay verification costs when she does not need to so the contract cannot be optimal. This implies that \(\hat{R}\left( s\right) =\bar{R}\) can hold only for a zero-measure event (i.e., a single point). Therefore, \(\hat{R}\left( s\right) <\bar{R}\) a.e. and \(\hat{R}\left( s\right) \le \bar{R}\) everywhere. \(\square \)

Proof

(Lemma 3) We first show that \(B\) is a connected set. This part of the proof is constructive and is a special case of item (iii) of Proposition 1 in Winton (1995). Without loss of generality, suppose that the contract has as verification set a disjoint interval \(B=\left[ \underline{s},x\right] \cup \left[ s_{1},s_{2}\right] \) for some \(\bar{s}>s_{2}>s_{1}>x> \underline{s}\), and repayment \(\hat{R}\left( s\right) \) for \(\hat{s}\in B\) and \(R\left( \hat{s},s\right) =\bar{R}\) for \(\hat{s}\notin B\). The investor’s payoff from this contract is:

$$\begin{aligned} V=\int \limits _{\underline{s}}^{x}\hat{R}\left( s\right) \hbox {d}H\left( s\right) +\int \limits _{s_{1}}^{s_{2}}\hat{R}\left( s\right) \hbox {d}H\left( s\right) +\bar{R}-(\bar{R}-\gamma )\left[ H\left( s_{2}\right) -H\left( s_{1}\right) +H\left( x\right) \right] \end{aligned}$$

and the entrepreneur’s payoff from the contract is given by:

$$\begin{aligned} U&= \int \limits _{\underline{s}}^{x}u[s-\hat{R}\left( s\right) ]\hbox {d}H\left( s\right) +\int \limits _{x}^{s_{1}}[us-\bar{R}]\hbox {d}H\left( s\right) +\int \limits _{s_{1}}^{s_{2}}u[s-\hat{R}\left( s\right) ]\hbox {d}H\left( s\right) \\&\quad +\int \limits _{s_{2}}^{\bar{s}}u[s-\bar{R}]\hbox {d}H\left( s\right) \end{aligned}$$

Now, incentive compatibility then requires that for \(s\in \left[ s_{1},s_{2}\right] \), \(\bar{R}\ge \hat{R}\left( s\right) \). If \(\bar{R}=\hat{R}\left( s\right) \), there is nothing to prove so suppose that \(\bar{R}>\hat{R}\left( s\right) \). Now construct a new contract \(\left( \vartriangle \right) \). To do so, notice that \(\bar{R}>\hat{R}\left( s\right) \) implies \(\int _{\underline{s}}^{x}\hat{R}\left( s\right) \hbox {d}H\left( s\right) +\bar{R}\left[ 1-H\left( x\right) \right] -\gamma H\left( x\right) >V\). Hence, there exists a contract with \(B^{\vartriangle }=\left[ \underline{s},x\right] ,R^{\vartriangle }\left( s,s\right) =\hat{R}\left( s\right) \) for \(\hat{s}\in B^{\vartriangle }\) and \(R^{\vartriangle }\in [\hat{R}\left( s\right) ,\bar{R})\) satisfying:

$$\begin{aligned} V^{\vartriangle }=\int _{\underline{s}}^{x}\hat{R}\left( s\right) \hbox {d}H\left( s\right) +R^{\vartriangle }\left[ 1-H\left( x\right) \right] -\gamma H\left( x\right) =V \end{aligned}$$

Such a contract is feasible since in the initial contract, for \(s\in \left[ s_{1},s_{2} \right] \), \(\left( 1-\eta \right) s\ge \hat{R}\left( s\right) > \bar{R}>R^{\vartriangle }\). It is also incentive compatible since the repayment function is constant \(\forall \, \hat{s}\notin B\) and satisfies \(R^{\vartriangle }\ge \hat{R}\left( s\right) \). Under such a contract, the concavity of \(u\) guarantees that:

$$\begin{aligned} U^{\vartriangle }=\int _{\underline{s}}^{x}u\left[ s-\hat{R}\left( s\right) \right] \hbox {d}H\left( s\right) +\int _{x}^{\bar{s}}u\left[ s-R^{\vartriangle }\right] \hbox {d}H\left( s\right) \ge U \end{aligned}$$

Thus, we have found a contract that is feasible, incentive compatible and that weakly improves the entrepreneur’s welfare, while leaving the investor as well off. Summarizing, when the contract specifies \(B\) as a disjoint interval, the contract fails to be optimal. We now show that \(B\) is in fact a lower interval. It suffices to show that \(B\ne \varnothing \Rightarrow \underline{s}\in B\) and we proceed by contradiction. Suppose that \(B\ne \varnothing \) but \(\underline{s}\notin B\). Since \(\underline{s}\notin B\), we have \(R\left( \underline{s},s\right) =\bar{R}\), while incentive compatibility requires \(\bar{R}\ge \hat{R}\left( s\right) \). On the other hand, limited liability requires \(\left( 1-\eta \right) \underline{s}\ge \bar{R}\). Since \(\left( 1-\eta \right) \underline{s}\le \left( 1-\eta \right) s\ \forall \ s\in \varSigma \), it follows that \(\bar{R}=\hat{R}\left( s\right) =\left( 1-\eta \right) \underline{s}\) which in turn implies that \(B=\varnothing \), a contradiction. \(\square \)

Proof

(Theorem 1) For the reformulated problem (\(\mathcal {PP}.1\))–(\(\mathcal {PP}.3\)), Lemma 5 below shows that the constraint qualification holds. Thus, the problem is equivalent to problem (43) on page 102 of Caputo (2005) with no differential constraints. The Maximum Principle (e.g., Theorem 4.4 in of Caputo (2005)) then implies that there exist constants \(\lambda >0,\ \phi \ge 0\) and nonnegative, continuous functions \(\mu _{1}\left( \cdot \right) ,\ \mu _{2}\left( \cdot \right) \) such that the following conditions hold:

$$\begin{aligned} -\mu _{1}\left( s\right)&=\left\{ u^{\prime }\left[ s-\hat{R}\left( s\right) \right] -\lambda \right\} h\left( s\right) -\mu _{2}\left( s\right) \quad \forall \ s\le x \end{aligned}$$
(3)
$$\begin{aligned} -\phi&=\int _{x}^{\bar{s}}u^{\prime }\left[ s-\bar{R}\right] \hbox {d}H\left( s\right) -\lambda \left[ 1-H\left( x\right) \right] \end{aligned}$$
(4)
$$\begin{aligned} \frac{-\phi \left( 1-\eta \right) }{h\left( x\right) }&=u\left[ x-\hat{R}\left( x\right) \right] -u\left[ x-\bar{R}\right] +\lambda \left[ \hat{R}\left( x\right) -\bar{R}-\gamma \right] \end{aligned}$$
(5)

along with complementary slackness conditions:

$$\begin{aligned} 0&=\lambda \left\{ \int _{\underline{s}}^{x}\hat{R}\left( s\right) \hbox {d}H\left( s\right) +\bar{R}\left[ 1-H\left( x\right) \right] -\gamma H\left( x\right) -b\right\} \end{aligned}$$
(6)
$$\begin{aligned} 0&=\phi \left[ \left( 1-\eta \right) x-\bar{R}\right] \end{aligned}$$
(7)
$$\begin{aligned} 0&=\mu _{1}\left( s\right) \left[ \left( 1-\eta \right) s-\hat{R}\left( s\right) \right] \end{aligned}$$
(8)
$$\begin{aligned} 0&=\mu _{2}\left( s\right) \hat{R}\left( s\right) \end{aligned}$$
(9)

where \(\lambda ,\phi ,\mu _{1}\left( \cdot \right) ,\mu _{2}\left( \cdot \right) \) are, respectively, the multipliers on the IRC and LLCs. Now suppose that the seven-tuple \(\{\bar{R}^{*},\hat{R}^{*}\left( \cdot \right) ,x^{*},\lambda ^{*},\phi ^{*},\mu _{1}^{*}\left( \cdot \right) ,\mu _{2}^{*}\left( \cdot \right) \}\) is a solution to the system comprising (3)–(9). Then, the triplet \(\{\bar{R}^{*},\hat{R}^{*}\left( \cdot \right) ,x^{*}\}\) achieves the unique maximum of (\(\mathcal {PP}.1\)). To see this, notice that the constraint set is convex and the “maximized Hamiltonian” of the problem above \(H(s,\hat{R}\left( s\right) ,\bar{R}^{*})\) is strictly concave in \(\hat{R}\) for every \(s\in \varSigma \). Thus, Arrow’s Sufficiency Theorem (see, e.g., Theorem 6.4 in Caputo (2005)) immediately applies. We now classify optimal contracts into families:

  1. (i)

    First, the optimal contract is standard debt if and only if \(\mu _{1}^{*}\left( s\right) >0\, \forall \ s\le x^{*}\). In turn, \(\mu _{1}^{*}\left( s\right) >0\Leftrightarrow u^{\prime }\left( \eta \underline{s}\right) <\lambda ^{*}\). To see this, suppose that \(\lambda ^{*}>u^{\prime }\left( \eta \underline{s}\right) \). Then, \(\lambda ^{*}>u^{\prime }\left( \eta s\right) \, \forall \ s\le x^{*}\). But \(\hat{R}^{*}\left( s\right) \le \left( 1-\eta \right) s \, \forall \ s\le x^{*}\), so we have that \(\lambda ^{*}>u^{\prime }\left( \eta \underline{s}\right) >u^{\prime }(\eta s)=u^{\prime }(s-\left( 1-\eta \right) s)\ge u^{\prime }[s-\hat{R}^{*}\left( s\right) ]\ \forall \ s\le x^{*}\). By (3), this means that \(0>\frac{\mu _{2}^{*}\left( s\right) -\mu _{1}^{*}\left( s\right) }{h\left( s\right) }\) which implies \(\mu _{1}^{*}\left( s\right) >0\). Then, (8) implies \(\hat{R}^{*}\left( s\right) =\left( 1-\eta \right) s \,\forall \ s\le x^{*}\). Limited liability then implies that \(\left( 1-\eta \right) x^{*}\ge \bar{R}^{*} \ge \hat{R}^{*}\left( x^{*}\right) =\left( 1-\eta \right) x^{*}\), and the SDC is optimal. To see that the converse is true, suppose that a SDC is optimal. Then, \(\hat{R}^{*}\left( s\right) =\left( 1-\eta \right) s \, \forall \ s\le x^{*}\) and \(\bar{R}^{*}=\left( 1-\eta \right) x^{*}\). By (8), this implies \(\mu _{1}^{*}\left( s\right) >0\) which clearly means that \(\mu _{2}^{*}\left( s\right) =0\). In turn, this implies, by (3), that \(\forall \ s\le x^{*}:\) \(0>u^{\prime }[s-\hat{R}^{*}\left( s\right) ]-\lambda ^{*}=u^{\prime }(\eta s)-\lambda ^{*}\) since the SDC is optimal. In turn, given \(u^{\prime }(\eta s)\le u^{\prime }(\eta \underline{s})\), this implies that \(\lambda ^{*}>u^{\prime }\left( \eta \underline{s}\right) \).

  2. (ii)

    Now suppose that \(\mu _{1}^{*}\left( s\right) =0\) for some \(s<x^{*}\). As long as \(\mu _{1}^{*}\left( x^{*}\right) >0\), the contract is continuous since \(\hat{R}^{*}\left( x^{*}\right) =\left( 1-\eta \right) x^{*}=\bar{R}^{*}\). Using again condition (3), we know \(\mu _{1}^{*}\left( s\right) =0\Leftrightarrow u^{\prime }\left( \eta s\right) \ge \lambda ^{*}\). There are two cases to consider. First suppose that \(\mu _{2}^{*}\left( s\right) =0\ \forall \ s\le x^{*}\) which holds iff \(\lambda ^{*}>u^{\prime }\left( s\right) \). Then, the optimal contract specifies that \(\hat{R}^{*}\left( s\right) >0 \, \forall \ s\), \(\hat{R}^{*}\left( s\right) =s-u^{\prime \left( -1\right) }\left( \lambda ^{*}\right) \) whenever \(u^{\prime }\left( \eta s\right) \ge \lambda ^{*}>u^{\prime }\left( s\right) \) and \(\hat{R}^{*}\left( s\right) =\left( 1-\eta \right) s\) when \(\lambda ^{*}>u^{\prime }\left( \eta s\right) \). Next suppose that \(\mu _{2}^{*}\left( s\right) >0\ \)for some \(s\) which implies that \(s<u^{\prime \left( -1\right) }\left( \lambda ^{*}\right) \). Then, the optimal contract specifies \(\hat{R}^{*}\left( s\right) =0\) whenever \(u^{\prime }\left( s\right) >\lambda ^{*}\), \(\hat{R}^{*}\left( s\right) =s-u^{\prime \left( -1\right) }\left( \lambda ^{*}\right) \) whenever \(u^{\prime }\left( \eta s\right) \ge \lambda ^{*}>u^{\prime }\left( s\right) \) and \(\hat{R}^{*}\left( s\right) =\left( 1-\eta \right) s\) as long as \(\lambda ^{*}>u^{\prime }\left( \eta s\right) \). That \(\hat{R}^{*\prime }\left( s\right) =1\) for some \(s\) follows immediately from \(\hat{R}^{*}\left( s\right) =s-u^{\prime \left( -1\right) }\left( \lambda ^{*}\right) \) since \(\lambda ^{*}\) is unique and independent of \(s\).

  3. (iii)

    Finally, suppose that \(\mu _{1}^{*}\left( s\right) =0\ \forall \ s\le x^{*}\) and \(\mu _{2}^{*}\left( s\right) =0\) for some \(s\le x^{*}\). Then, (3) implies that \(\left( 1-\eta \right) x^{*}>\hat{R}^{*}\left( x^{*}\right) =x^{*}-u^{\prime \left( -1\right) }\left( \lambda ^{*}\right) \). Thus, LLC and Lemma 2 imply that the optimal contract is discontinuous, i.e., \(\bar{R}^{*}>\hat{R}^{*}\left( x^{*}\right) \). To see this, suppose that it is continuous and find a contradiction. Continuity implies \(\hat{R}^{*}\left( x^{*}\right) =\bar{R}^{*}<\left( 1-\eta \right) x^{*}\) which in turn implies that \(\phi =0\). But then (5) implies that \(0=-\lambda \gamma \). This is true only if \(\lambda =0\) but then (4) requires \(\int _{x^{*}}^{\bar{s}}u^{\prime }\left[ s-\bar{R}^{*}\right] \hbox {d}H\left( s\right) =0\), a contradiction since \(\bar{R}^{*}<\left( 1-\eta \right) x^{*}\). \(\square \)

Lemma 5

The rank constraint qualification (RCQ) holds in problem (\(\mathcal {PP}.1\))–(\(\mathcal {PP}.3\)).

Proof

In an optimal contract, constraint (\(\mathcal {PP}.2\)) will bind. The remaining constraints may or may not bind depending on the form of the optimal contract. However, a key observation is that \(\left( 1-\eta \right) s\ge \hat{R}^{*}\left( s\right) \) and \(\hat{R}^{*}\left( s\right) \ge 0\) cannot both be binding at a given \(s\). That the RCQ is satisfied can now be seen by noticing that (a) at least one control is present in each of the binding constraints, and (b) the number of binding constraints at any given \(s\) is at most three, (weakly) less than the number of controls in the problem. In fact, if the solution to the problem (\(\mathcal {PP}.1\))–(\(\mathcal {PP}.3\)) belongs to family (i) of contracts in Proposition 1, \(\left( 1-\eta \right) x^{*}=\bar{R}^{*}\) and \(\left( 1-\eta \right) s=\hat{R}^{*}\left( s\right) \ \forall \ s\). The Jacobian matrix of all active constraints, \(\mathbf {J}_{i}(s,x^{*},\bar{R}^{*},\hat{R}^{*}\left( s\right) )\), in such case is therefore (omitting the arguments):

$$\begin{aligned} \mathbf {J}_{i}=\left[ \begin{array}{c@{\quad }c@{\quad }c} h\left( x^{*}\right) [\hat{R}^{*}\left( x^{*}\right) -\bar{R}^{*}-\gamma ] &{}\quad 1-H\left( x^{*}\right) &{}\quad h\left( s\right) \\ 1-\eta &{}\quad -1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad -1 \end{array} \right] \equiv \left[ \begin{array} [c]{c} J_{i}^{1}\\ J_{i}^{2}\\ J_{i}^{3} \end{array} \right] \end{aligned}$$

and one can verify that \(\mathbf {det}\left( \mathbf {J}_{i}\right) =-h\left( x^{*}\right) \gamma -\left( 1-\eta \right) \left[ 1-H\left( x^{*}\right) \right] \ne 0\) since \(\eta ,\gamma \in \left( 0,1\right) \) and \(h\left( x^{*}\right) >0\). Let \(\mathbf {J}_{ii}\) and \(\mathbf {J}_{iii}\) denote the Jacobian matrices of all active constraints when the optimal contract belongs, respectively, to families (ii) and (iii) in Theorem 1 (again, omitting the arguments). If the optimal contract belongs to family (ii), there are two cases to consider. First, if \(\hat{R}^{*}\left( s\right) >0 \, \forall \ s\), \(\left( 1-\eta \right) x^{*}=\bar{R}^{*}\) and \(\left( 1-\eta \right) s=\hat{R}^{*}\left( s\right) \) whenever \(s\in (s^{\lozenge },x^{*}]\) for some threshold \(s^{\lozenge }>\underline{s}\), then \(\mathbf {J}_{ii}=\mathbf {J}_{i}\) for \(s>s^{\lozenge }\) and \(\mathbf {J}_{ii}=[ \,\, J_{i}^{1} \,\, J_{i}^{2} \,\, ]^{\prime }\) for \(s\in \left[ \underline{s},s^{\lozenge }\right] ;\) in the latter case, \(\mathbf {rank}\left( \mathbf {J}_{ii}\right) =2\) since \(\hat{R}^{*}\left( x^{*}\right) \le \bar{R}^{*}\Rightarrow h\left( x^{*}\right) [\hat{R}^{*}\left( x^{*}\right) -\bar{R}^{*}-\gamma ]-\left( 1-\eta \right) \left[ 1-H\left( x^{*}\right) \right] \ne 0\), so that at least one of the \(2\times 2\) submatrices of \(\mathbf {J}_{ii}\) has nonzero determinant. Secondly, if \(\left( 1-\eta \right) x^{*} =\bar{R}^{*}\), \(\left( 1-\eta \right) s=\hat{R}^{*}\left( s\right) \) whenever \(s\in (s^{\Delta },x^{*}]\) for some threshold \(s^{\Delta }\le x^{*}\) and \(\hat{R}^{*}\left( s\right) =0\) whenever \(s\in [\underline{s},s^{\triangledown }]\) for some threshold \(\underline{s}<s^{\triangledown }<s^{\Delta }\), then \(\mathbf {J}_{ii}=\mathbf {J}_{i}\) for \(s>s^{\Delta }\), \(\mathbf {J}_{ii}=[ \,\, J_{i}^{1} \,\, J_{i}^{2} \,\, ]^{\prime }\) for \(s\in (s^{\triangledown },s^{\Delta }]\) and \(\mathbf {J}_{ii}=[ \,\, J_{i}^{1} \,\, J_{i}^{2} \,\, -J_{i}^{3} \,\, ]^{\prime }\) for \(s\in [\underline{s},s^{\triangledown }];\) in the latter case, \(\mathbf {det}\left( \mathbf {J}_{ii}\right) =h\left( x^{*}\right) \gamma +\left( 1-\eta \right) \left[ 1-H\left( x^{*}\right) \right] \ne 0\). The condition also holds if the optimal contract belongs to family (iii) in Theorem 1 since \(\mathbf {J}_{iii}=[ \,\, J_{i}^{1} \,\, -J_{i}^{3} \,\, ]^{\prime }\) and \(H\left( x^{*}\right) <1\) implies that at least one of the \(2\times 2\) submatrices of \(\mathbf {J}_{iii}\) has nonzero determinant. \(\square \)

Proof of propositions 16

Proof

(Proposition 1) First, notice that continuity of the contract implies that \(\hat{R}\left( x\right) =\bar{R}=\left( 1-\eta \right) x^{*}\), so that condition (\(\mathcal {PP}.2\)) can be written as follows:

$$\begin{aligned} \int _{\underline{s}}^{x^{*}}\hat{R}\left( s\right) \hbox {d}H\left( s\right) -\gamma H\left( x^{*}\right) +\left( 1-\eta \right) x^{*}\left[ 1-H\left( x^{*}\right) \right] =b \end{aligned}$$
(10)

Next, write \(x^{*}\left( \eta \right) \) to explicitly account for the dependence of \(x^{*}\) on \(\eta \) and totally differentiate (10) w.r.t. \(\eta \) while solving for \(\frac{dx^{*}\left( \eta \right) }{d\eta } \):

$$\begin{aligned} \frac{dx^{*}\left( \eta \right) }{d\eta }=\frac{x^{*}\left( \eta \right) \left[ 1-H\left( x^{*}\left( \eta \right) \right) \right] }{\left( 1-\eta \right) \left[ 1-H\left( x^{*}\left( \eta \right) \right) \right] -\gamma h\left( x^{*}\left( \eta \right) \right) } \end{aligned}$$
(11)

Now, \(\left( 1-\eta \right) \int _{x^{*}}^{\bar{s}}u^{\prime }\left[ s-\left( 1-\eta \right) x^{*}\right] \hbox {d}H\left( s\right) =\lambda ^{*}\left\{ \left( 1-\eta \right) \left[ 1-H\left( x^{*}\right) \right] -\gamma h\left( x^{*}\right) \right\} \) and \(\lambda ^{*}>0\) together imply that \(\frac{dx^{*}}{d\eta }>0\) and the first part of the proposition follows. Now write \(x^{*}\left( \gamma \right) \) and differentiate (10) w.r.t. \(\gamma \) while solving for \(\frac{dx^{*}\left( \gamma \right) }{d\gamma }\):

$$\begin{aligned} \frac{dx^{*}\left( \gamma \right) }{d\gamma }=\frac{H\left( x^{*}\left( \gamma \right) \right) }{\left( 1-\eta \right) \left[ 1-H\left( x^{*}\left( \gamma \right) \right) \right] -\gamma h\left( x^{*}\left( \gamma \right) \right) }>0 \end{aligned}$$

\(\square \)

Proof

(Proposition 2) Recall that a SDC is optimal iff \(\lambda ^{*}>u^{\prime }\left( \eta \underline{s}\right) \) or, equivalently, iff:

$$\begin{aligned} C\left( \eta \right) \equiv \frac{\int _{x^{*}}^{\bar{s}}u^{\prime }\left[ s-\left( 1-\eta \right) x^{*}\right] \hbox {d}H\left( s\right) }{\left[ 1-H\left( x^{*}\right) \right] }-u^{\prime }\left( \eta \underline{s}\right) +u^{\prime }\left( \eta \underline{s}\right) \frac{\gamma }{1-\eta }\varPsi \left( x^{*}\right) >0 \end{aligned}$$

We will therefore show that \(C\left( \eta \right) >0\) if and only if \(\eta >\eta ^{*}\) for some \(\eta ^{*}>0\). Notice that \(\lim _{\eta \mapsto 1}C\left( \eta \right) >0\) while Assumption 4 ensures that \(\lim _{\eta \mapsto 0}C\left( \eta \right) <0\). By the continuity of \(\eta \mapsto C\left( \eta \right) \), it suffices to show that \(C\left( \eta \right) >0\Rightarrow C^{\prime }\left( \eta \right) >0\). To show this, suppose that \(C\left( \eta \right) >0\) and differentiate:

$$\begin{aligned} C^{\prime }\left( \eta \right)&=\frac{\int _{x^{*}} ^{\bar{s}}u^{\prime \prime }\left[ s-\left( 1-\eta \right) x^{*}\right] \hbox {d}H\left( s\right) }{1-H\left( x^{*}\right) }\left\{ x^{*}-\left( 1-\eta \right) \frac{dx^{*}}{d\eta }\right\} \nonumber \\&\quad -u^{\prime \prime }\left( \eta \underline{s}\right) \underline{s}\left\{ 1-\frac{\varPsi \left( x^{*}\right) \gamma }{1-\eta }\right\} +u^{\prime }\left( \eta \underline{s}\right) \frac{\varPsi \left( x^{*}\right) \gamma }{\left( 1-\eta \right) ^{2}} \nonumber \\&\quad +\frac{dx^{*}}{d\eta }\Bigg \{ \frac{\varPsi \left( x^{*}\right) \int _{x^{*}}^{\bar{s}}u^{\prime }\left[ s-\left( 1-\eta \right) x^{*}\right] \hbox {d}H\left( s\right) }{1-H\left( x^{*}\right) }+\frac{\varPsi ^{\prime }\left( x^{*}\right) u^{\prime }\left( \eta \underline{s}\right) \gamma }{1-\eta } \nonumber \\&\quad -u^{\prime }\left( \eta x^{*}\right) \varPsi \left( x^{*}\right) \Bigg \} \end{aligned}$$
(12)

The expression inside braces in the first term is negative. To see this, notice that SDC optimal implies:

$$\begin{aligned} \left( 1-\eta \right) \frac{\hbox {d}x^{*}}{\hbox {d}\eta }=\frac{\left( 1-\eta \right) \left\{ \int _{\underline{s}}^{x^{*}}s\hbox {d}H\left( s\right) +x^{*}\left[ 1-H\left( x^{*}\right) \right] \right\} }{\left( 1-\eta \right) \left[ 1-H\left( x^{*}\right) \right] -\gamma h\left( x^{*}\right) }>\frac{x^{*}}{1-\frac{\varPsi \left( x^{*}\right) \gamma }{1-\eta }} \end{aligned}$$

and \(\lambda ^{*}>0\) requires \(1-\eta >\varPsi \left( x^{*}\right) \gamma \), which in turn means that \(x^{*}-\left( 1-\eta \right) \hbox {d}x^{*}/\hbox {d}\eta <0\). Next, \(1-\eta >\varPsi \left( x^{*}\right) \gamma \) also implies that the second term in (12) is positive. Lastly, notice that the expression inside braces in the third line of (12) is also positive. This is true by construction when Assumption 5-B holds. To see that this is also true if Assumption 5-A holds instead, rewrite as:

$$\begin{aligned}&\varPsi \left( x^{*}\right) \left\{ \frac{\int _{x^{*}}^{\bar{s}} u^{\prime }\left[ s-\left( 1-\eta \right) x^{*}\right] \hbox {d}H\left( s\right) }{1-H\left( x^{*}\right) }+\frac{u^{\prime }\left( \eta \underline{s}\right) \gamma }{1-\eta }\varPsi \left( x^{*}\right) -u^{\prime }\left( \eta x^{*}\right) \right\} \\&\quad +\,\,\frac{u^{\prime }\left( \eta \underline{s}\right) \gamma }{1-\eta }\frac{h^{\prime }\left( x^{*}\right) }{1-H\left( x^{*}\right) } \end{aligned}$$

Clearly, \(C\left( \eta \right) >0\) and \(h^{\prime }\left( x^{*}\right) \ge 0\) guarantee that this last expression is positive. Hence, we conclude that \(C\left( \eta \right) >0\Rightarrow C^{\prime }\left( \eta \right) >0\) and the statement of the proposition follows. \(\square \)

Proof

(Corollary 3) Let \(\rho _{e}\) denote the entrepreneur’s degree of risk aversion and suppose that for \((\eta ,\rho _{e}^{\prime })\) a SDC is optimal. Notice that since a SDC is optimal, (\(\mathcal {PP}.2\)) becomes (\(\mathcal {P}.2\)) and since verification costs are deadweight losses, \(x^{*}\) is the smallest \(x\) such that (\(\mathcal {P}.2\)) binds. Moreover, (\(\mathcal {P}.2\)) does not depend on \(\rho _{e}\), which implies \(dx^{*}(\rho _{e}^{\prime })/d\rho _{e}=0\). Next, recall that for any \(\rho _{e}\), \(\lambda ^{*}\left( \eta \right) >u^{\prime }\left( \eta \underline{s}\right) \Leftrightarrow \eta >\eta ^{*}\) and notice that (1), \(dx^{*}(\rho _{e}^{\prime })/d\rho _{e}=0\) together imply that \(D(\eta ,\rho _{e}^{\prime })=\lambda ^{*}\left( \eta ,\rho _{e}^{\prime }\right) /u^{\prime }\left( \eta \underline{s}\right) \) is decreasing in \(\rho _{e}\) for each \(\eta \). Hence, we may write the threshold found in Proposition 2 as \(\eta ^{*}(\rho _{e}^{\prime })\), so that \(\eta >\eta ^{*}(\rho _{e}^{\prime })\Rightarrow \lambda ^{*}\left( \eta ,\rho _{e}^{\prime }\right) >u^{\prime }\left( \eta \underline{s}\right) \). To complete the proof, choose \(\eta _{\varepsilon }=\eta ^{*}(\rho _{e}^{\prime })+\varepsilon \) for \(\varepsilon >0\) arbitrarily small. Then, \(\rho _{e}^{\prime \prime }>\rho _{e}^{\prime }\Rightarrow D(\eta _{\varepsilon },\rho _{e}^{\prime })>0\ge D(\eta _{\varepsilon },\rho _{e}^{\prime \prime })\). That is, a SDCs is optimal for \((\eta _{\varepsilon },\rho _{e}^{\prime })\) but not for \((\eta _{\varepsilon } ,\rho _{e}^{\prime \prime })\). \(\square \)

Proof

(Proposition 3) Let \(\theta _{-\eta }=\left\{ b,\gamma ,\rho _{e},\psi \right\} \). Applying the Envelope theorem to (\(\mathcal {P}.1\))–(\(\mathcal {P}.2\)):

$$\begin{aligned} \frac{\hbox {d}v\left( \theta _{-\eta };\eta \right) }{\hbox {d}\eta }&= \int \limits _{\underline{s}}^{x^{*}}\left[ u^{\prime }\left( \eta s\right) -\lambda ^{*}\right] s\hbox {d}H\left( s\right) \\&\quad +\,\,x^{*}\left\{ \int \limits _{x^{*}}^{\bar{s}}u^{\prime }\left[ s-\left( 1-\eta \right) x^{*}\right] \hbox {d}H\left( s\right) -\lambda ^{*}\left[ 1-H\left( x^{*}\right) \right] \right\} \end{aligned}$$

using the expression for \(\lambda ^{*}\) and after a minor algebraic manipulation, we get:

$$\begin{aligned} \frac{\hbox {d}v\left( \theta _{-\eta };\eta \right) }{d\eta }&= \int _{\underline{s}}^{x^{*}}\left[ u^{\prime }\left( \eta s\right) -\lambda ^{*}\right] s\hbox {d}H\left( s\right) \\&\quad -\,\,x^{*}\frac{\gamma h\left( x^{*}\right) \int _{x^{*}}^{\bar{s}}u^{\prime }\left[ s-\left( 1-\eta \right) x^{*}\right] \hbox {d}H\left( s\right) }{\left[ 1-H\left( x^{*}\right) \right] \left( 1-\eta \right) -\gamma h\left( x^{*}\right) } \end{aligned}$$

now, the last term of this expression is clearly negative since \(\lambda ^{*}>0\Rightarrow \left[ 1-H\left( x^{*}\right) \right] \left( 1-\eta \right) -\gamma h\left( x^{*}\right) >0\). Moreover, the first term is also negative since the optimality of SDCs implies that \(\forall \ s\le x^{*}, \,\, u^{\prime }\left( \eta s\right) <\lambda \). Thus, we conclude that \(\hbox {d}v\left( \theta _{-\eta };\eta \right) /\hbox {d}\eta <0\).

Proof

(Proposition 4) It suffices to show that: \(\int _{\underline{s}}^{x^{*}}\left[ u^{\prime }\left( \eta s\right) -\lambda \right] s\hbox {d}H\left( s\right) >x^{*}\frac{\gamma h\left( x^{*}\right) \lambda ^{*}}{\left( 1-\eta \right) }\) for some parametrization that satisfies Assumptions 13 and violates condition (1). Consider the case used in Corollary 2, that is, suppose that \(\lim _{c\rightarrow 0}u^{\prime }\left( c\right) =\infty \) and \(\eta =0\). Then, obviously \(\lambda ^{*}<u^{\prime }\left( \eta \underline{s}\right) \) and assumptions 1–3 are satisfied but \(\int _{\underline{s}}^{x^{*}}\left[ u^{\prime }\left( \eta s\right) \right] s\hbox {d}H\left( s\right) >\lambda ^{*}\left[ H\left( x^{*}\right) +x^{*}\gamma h\left( x^{*}\right) \right] \). \(\square \)

Proof

(Lemma 4) The complete system of necessary and sufficient conditions is given by (7)–(9) and:

$$\begin{aligned} -\mu _{1}\left( s\right)&=\left\{ u^{\prime }\left[ s-\hat{R}\left( s\right) \right] -\lambda W^{\prime }\left[ \hat{R}\left( s\right) \right] \right\} h\left( s\right) -\mu _{2}\left( s\right) \ \ \ \forall \ s\le x \end{aligned}$$
(13)
$$\begin{aligned} -\phi&=\int _{x}^{\bar{s}}u^{\prime }\left[ s-\bar{R}\right] \hbox {d}H\left( s\right) -\lambda W^{\prime }\left( \bar{R}\right) \left[ 1-H\left( x\right) \right] \end{aligned}$$
(14)
$$\begin{aligned} \frac{-\phi \left( 1-\eta \right) }{h\left( x\right) }&=u\left[ x-\hat{R}\left( x\right) \right] -u\left[ x-\bar{R}\right] +\lambda \left\{ W\left[ \hat{R}\left( x\right) \right] -W\left( \bar{R}\right) -\gamma \right\} \end{aligned}$$
(15)
$$\begin{aligned} 0&=\lambda \left\{ \int _{\underline{s}}^{x}W\left[ \hat{R}\left( s\right) \right] \hbox {d}H\left( s\right) +W\left( \bar{R}\right) \left[ 1-H\left( x\right) \right] -\gamma H\left( x\right) -W\left( b\right) \right\} \end{aligned}$$
(16)

An argument identical to that found in the proof of Theorem 1-(i) shows that a SDC is optimal iff:

$$\begin{aligned} \lambda ^{*}&= \frac{\left( 1-\eta \right) \int _{x^{*}}^{\bar{s}} u^{\prime }\left[ s-\left( 1-\eta \right) x^{*}\right] \hbox {d}H\left( s\right) }{\left( 1-\eta \right) W^{\prime }[\left( 1-\eta \right) x^{*}]\left[ 1-H\left( x^{*}\right) \right] -\gamma h\left( x^{*}\right) }\\&> u^{\prime }\left( \eta s\right) /W^{\prime }[\left( 1-\eta \right) s]\ \forall \ s\le x^{*} \end{aligned}$$

The proof is completed by noticing that \(\rho _{e}>\rho _{i}\Rightarrow u^{\prime }\left( \eta \underline{s}\right) /W^{\prime }[\left( 1-\eta \right) \underline{s}]\ge u^{\prime }\left( \eta s\right) /W^{\prime }[\left( 1-\eta \right) s]\) for all \(s\le x^{*}\) while \(\rho _{e}<\rho _{i}\Rightarrow u^{\prime }\left( \eta x^{*}\right) /W^{\prime }[\left( 1-\eta \right) x^{*}]\ge u^{\prime }\left( \eta s\right) /W^{\prime }[\left( 1-\eta \right) s]\ \forall \ s\le x^{*}\). \(\square \)

Proof

(Proposition 5) Differentiate: \(\int _{\underline{s}} ^{x^{*}}W\left[ \hat{R}\left( s\right) \right] \hbox {d}H\left( s\right) -\gamma H\left( x^{*}\right) +W[ \left( 1-\eta \right) x^{*}] [ 1-H\left( x^{*}\right) ] \) w.r.t. \(\gamma \) and \(\eta \) to obtain

$$\begin{aligned} \frac{dx^{*}\left( \eta \right) }{d\eta }&= \frac{x^{*}\left( \eta \right) \left[ 1-H\left( x^{*}\left( \eta \right) \right) \right] W^{\prime }\left[ \left( 1-\eta \right) x^{*}\right] }{\left( 1-\eta \right) W^{\prime }\left[ \left( 1-\eta \right) x^{*}\right] \left[ 1-H\left( x^{*}\left( \eta \right) \right) \right] -\gamma h\left( x^{*}\left( \eta \right) \right) } \\ \frac{dx^{*}\left( \gamma \right) }{d\gamma }&= \frac{H\left( x^{*}\left( \gamma \right) \right) }{\left( 1-\eta \right) W^{\prime }\left[ \left( 1-\eta \right) x^{*}\right] \left[ 1-H\left( x^{*}\left( \eta \right) \right) \right] -\gamma h\left( x^{*}\left( \eta \right) \right) } \end{aligned}$$

and, as before, \(\lambda ^{*}>0\) implies that \(\frac{dx^{*}\left( \eta \right) }{d\eta },\frac{dx^{*}\left( \gamma \right) }{d\gamma }>0\). The last statement of the proposition trivially follows by replacing \(W^{\prime }\left[ \left( 1-\eta \right) x^{*}\right] \lessgtr 1\) and comparing the results with the proof of Proposition 1. \(\square \)

Proof

(Proposition 6) It suffices to show that \(x^{*}\left( \rho _{i}\right) <x^{*}\left( 0\right) \) for \(\rho _{i}>0\). To do so, first define \(\mathbb {E}\left( C_{x}\right) =\int _{\underline{s}}^{x}\left( 1-\eta \right) s\hbox {d}H\left( s\right) +\left( 1-\eta \right) x\left[ 1-H\left( x\right) \right] \). Next, notice that \(W\left( \cdot \right) \) concave implies that the function \(\Omega \left( \gamma \right) =W\left[ \mathbb {E}\left( C_{x}\right) \right] -\gamma H\left( x\right) -W\left[ \mathbb {E}\left( C_{x}\right) -\gamma H\left( x\right) \right] \) is nonnegative, continuous and increasing for each \(x\). In fact, \(\Omega \left( 0\right) =0\) and \(\lim _{\gamma \rightarrow \infty }\Omega \left( \gamma \right) =\infty \). Hence, for each \(\rho _{i},\ \exists \ \hat{\gamma }\left( \rho _{i}\right) \) such that \(\gamma \ge \hat{\gamma }\left( \rho _{i}\right) \) \(\Rightarrow \Omega \left( \gamma \right) >W\left[ \mathbb {E}\left( C_{x}\right) \right] -\mathbb {E}\left[ W\left( C_{x}\right) \right] \ \forall \ x\). Thus, for a given \(\rho _{i}>0\) and \(\gamma \ge \hat{\gamma }\left( \rho _{i}\right) \), we have: \(\mathbb {E}\left[ W\left( C_{x^{*}\left( 0\right) }\right) \right] -\gamma H\left( x^{*}\left( 0\right) \right) >W\left[ \mathbb {E}\left( C_{x^{*}\left( 0\right) }\right) -\gamma H\left( x^{*}\left( 0\right) \right) \right] =W\left( b\right) \), where the last inequality follows from the observation that \(x^{*}\left( 0\right) \) must satisfy \(\mathbb {E}\left( C_{x^{*}\left( 0\right) }\right) -\gamma H\left( x^{*}\left( 0\right) \right) =b\). Therefore, at \(x^{*}\left( 0\right) \), \(F\left( x^{*}\left( 0\right) \right) \equiv \mathbb {E}\left[ W\left( C_{x^{*}\left( 0\right) }\right) \right] -\gamma H\left( x^{*}\left( 0\right) \right) -W\left( b\right) >0\) and the IR constraint is slack. Next, notice that, in general, \(x^{*}=\min \left\{ x\ |\ F\left( x\right) =0\right\} \) which in turn means that \(x<x^{*}\Rightarrow F\left( x\right) <0\). Therefore, it must be that \(x^{*}\left( 0\right) >x^{*}\left( \rho _{i}\right) \). \(\square \)

The dynamic contracting problem and Propositions 78

In this section, we spell out the details of the dynamic extension of the contracting problem. As in the static case, we assume that the entrepreneur has access to a technology but lacks enough funds to operate it, so she must enter a (long term) contract with an investor. We also maintain the assumption of two-sided commitment. In addition to the modifications mentioned in the first two paragraphs of Sect. 3.2, we allow for the transfers to the investor to be positive or negative and assume that he has instant access to a credit market. Thus, the lower LLC is relaxed to a negative number, \(-s_{0}\) with \(s_{0}>0\).

At time \(t=0\), the investor makes a take-it-or-leave-it offer to the entrepreneur with the terms of a financial contract. Each \(t\) is associated with a history of events \(h^{t}=\left\{ h_{1},h_{2},\ldots ,h_{t}\right\} \in \mathcal {H}^{t}\) where \(\mathcal {H}^{t}\) is the set of all possible such histories and without loss of generality \(h^{t}=\emptyset \ \forall \ t\le 0\). Under CSV, histories include all past announcements by the entrepreneur and the list of previous periods in which verification took place. That is, the typical component of a particular history is a pair \(h_{t}=\left\{ \hat{s}_{t},q_{t}\right\} \) where \(\hat{s}_{t}\) is what the entrepreneur reports as the state and \(q_{t}=1\), if monitoring occurred in period \(t\), and \(q_{t}=0\) otherwise.Footnote 21 Moreover, under CSV, the contract also includes \(B\left( h^{t-1}\right) \), a set of states in which the lender verifies after observing history \(h^{t-1}\). In the symmetric information case \((\gamma =0), \, h^{t}=s^{t}=\left\{ s_{1,}s_{2},\ldots ,s_{t}\right\} \). In order to proceed to the formulation of the problem, we need some definitions:

Definition 2

A reporting strategy for the entrepreneur, \(\mathfrak {\hat{Z},}\) is a sequence of functions that maps histories up to \(t\) into reports of the state, i.e., \(\mathfrak {\hat{Z}= \,}\{\hat{s}_{t}\left( h^{t}\right) \}_{t=1}^{\infty }=\{\hat{s}_{t}\left( h^{t-1},s_{t}\right) \}_{t=1}^{\infty }\).

Let \(\mathbf {S}\) denote the set of all possible reporting strategies. As in the static problem, the entrepreneur will not misrepresent in the verification region, so that \(q_{t}=1\Rightarrow h_{t}=\left\{ s_{t},1\right\} \) and \(q_{t}=0\Rightarrow h_{t}=\left\{ \hat{s}_{t},0 \right\} \). Next, let \(\mathfrak {Z}\) be the strategy in which the agent always reports truthfully.

Definition 3

A verification strategy for the lender is a sequence of set-valued mappings \(\{B_{t}(h^{t-1})\}_{t=1}^{\infty }\) assigning to each history \(h^{t-1}\) a verification region, i.e., a set of states for which verification occurs.

The set \(B_{t}\) is the natural time-varying extension of the set \(B\) in Sect. 2.3. Next, define:

Definition 4

A dynamic contract under CSV is a sequence of mappings \(\varvec{\Phi }= \{B_{t}(h^{t-1}),R_{t}(h^{t})\}_{t=1}^{\infty }\), assigning current period verification strategies and repayments to each history.

Notice that \(B_{t}(h^{t-1})\subseteq \tilde{\varSigma }\) depends upon the history of events up to \(t-1\), as verification decisions are independent of the current period realization of the state. On the other hand, \(R_{t}(h^{t})\) is contingent on the current realization of the state and therefore depends upon the history up to \(t\). Let \(\mu (B_{t}(h^{t-1}))\) be the probability with which verification takes place after history \(h^{t-1}\) when the principal uses strategy \(B_{t}\). That is, \(\mu (B_{t}(h^{t-1}))=\sum _{i\ |\ s_{i}\in B_{t}(h^{t-1})}\pi _{i}\). Next, we define the lender’s expected discounted payoff from the strategies in the subgame starting after \(h^{t}\) given a contract \(\varvec{\Phi }\) and a reporting strategy \(\mathfrak {\hat{Z}}\):

$$\begin{aligned} \mathcal {Q}_{t}\left( h^{t},\mathfrak {\hat{Z}},\varvec{\Phi }\right) =\mathbb {E}_{t}\sum _{\tau =t+1}^{\infty }\beta ^{\tau -t-1}\!\left\{ \! R_{\tau }\left( h^{\tau -1},\hat{s}_{\tau }\!\left( \! h^{\tau -1},s_{\tau }\!\right) ,q_{\tau } \right) -\mu \!\left( \!B_{\tau }\left( h^{\tau -1}\right) \right) \! \gamma \right\} \end{aligned}$$

where the expectation is conditional on \(h^{t}\) taken with respect to the probability measure that the reporting strategy and verification policy implicitly induce on \(\mathcal {H}^{t}\). On the other hand, the entrepreneur’s payoff is:

$$\begin{aligned} V_{t}\left( h^{t},\mathfrak {\hat{Z}},\varvec{\Phi }\right) =\mathbb {E} _{t}\sum _{\tau =t+1}^{\infty }\beta ^{\tau -t-1}u\left[ s_{\tau }-R_{\tau } \left( h^{\tau -1},\hat{s}_{\tau }\left( h^{\tau -1},s_{\tau }\right) ,q_{\tau }\right) \right] \end{aligned}$$

Let \(\mathcal {Q}_{1}(h^{0},\mathfrak {\hat{Z}},\varvec{\Phi }),V_{1} (h^{0},\mathfrak {\hat{Z}},\varvec{\Phi )}\) be the parties’ payoffs at the beginning of time 1. We can now define incentive compatible and feasible contracts:

Definition 5

A contract \(\varvec{\Phi }\) is incentive compatible if \(\forall h^{t}\) and \(\forall \ \mathfrak {\hat{Z}}\in \mathbf {S}\)

$$\begin{aligned} V_{t}\left( h^{t},\mathfrak {Z},\varvec{\Phi }\right)&= \mathbb {E}_{t} \sum _{\tau =t+1}^{\infty }\beta ^{\tau -t-1}u\left[ s_{\tau }-R_{\tau }(h^{\tau -1},s_{\tau },q_{\tau })\right] \nonumber \\&\ge \mathbb {E}_{t}\sum _{\tau =t+1}^{\infty }\beta ^{\tau -t-1}u\!\left[ s_{\tau }-R_{\tau }\left( h^{\tau -1},\hat{s}_{\tau }\!\left( h^{\tau -1},s_{\tau }\!\right) \! ,q_{\tau }\right) \right] =V_{t}(h^{t},\mathfrak {\hat{Z},}\varvec{\Phi )}\nonumber \\ \end{aligned}$$
(17)

Definition 6

A contract is feasible if, for any \(h^{t}\),

$$\begin{aligned} R_{t}(h^{t})\le \left( 1-\eta \right) \hat{s}_{t}\;\text {if}\; q_{t}=0,\; R_{t}(h^{t})\le \left( 1-\eta \right) s_{t} \;\text {if}\; q_{t}=1,\;\text {and}\; R_{t}(h^{t})\ge -s_{0} \end{aligned}$$
(18)

Notice that conditions (18) imply that \(\forall h^{t},V_{t}(h^{t},\mathfrak {Z},\varvec{\Phi )\ge }\frac{1}{1-\beta }\sum _{i=1}^{N} \pi _{i}u\left( \eta s_{i}\right) =\underline{v}\). Now let \(\mathcal {V}\) be the set of all entrepreneur’s expected discounted payoffs, \(v\), that can be generated by a contract satisfying (17)–(18) and \(V_{1}\left( h^{0},\mathfrak {Z},\varvec{\Phi }\right) =v\). Let \(\bar{v}=\sup u\left( c\right) /\left( 1-\beta \right) =\frac{1}{1-\beta }\sum _{i=1} ^{N}\pi _{i}u\left[ s_{0}+s_{i}\right] \) be the maximum attainable expected payoff for the entrepreneur. Statement (i) of Lemma 2 in Wang (2005) (pp. 902) established that \(\mathcal {V=\,}[\underline{v},\bar{v}]\) when \(\eta =0\). This result can be trivially extended for the case of \(\eta >0\). Next, for each \(v\in [\underline{v},\bar{v}]\), an optimal contract maximizes the value obtained by the investor among all incentive compatible and feasible contracts that deliver an initial value \(v\) to the entrepreneur. This defines a frontier of values \(\mathcal {J}\left( v\right) \mathcal {=}\max \left\{ \mathcal {J\ } |\ \exists \ \varvec{\Phi \ }\text {such that }V_{1}(h^{0},\mathfrak {Z} ,\varvec{\Phi )}=v\text { and }\mathcal {Q}_{1}(h^{0},\mathfrak {Z},\varvec{\Phi })=\mathcal {J}\right\} \).

Under mild condition shown to hold below, the extensive form contract has an equivalent recursive representation \(\{B(v),R\left( v,s_{i}\right) ,w\left( v,s_{i}\right) ,s_{i}\in \tilde{\varSigma },v\in \left[ \underline{v},\bar{v}\right] \}\) where \(R\left( v,s_{i}\right) \) is the repayment schedule and \(w\left( v,s_{i}\right) \) is next period “promised” utility when the current expected payoff for the borrower is \(v\) and the state of nature is \(s_{i}\). Hence, the optimal contract is found by solving:

$$\begin{aligned} \mathcal {J}\left( v\right)&= \max _{R\left( v,\cdot \right) ,w\left( v,\cdot \right) ,B(v)}\left\{ \sum _{i=1}^{N}\pi _{i}\left[ R\left( v,s_{i}\right) +\beta \mathcal {J}\left( w\left( v,s_{i}\right) \right) \right] -\sum _{i|s_{i}\in B\left( v\right) }\pi _{i}\gamma \right\} \nonumber \\&\text {subject to}: \end{aligned}$$
(19)
$$\begin{aligned}&u(s_{i}-R\left( v,s_{i}\right) )+\beta w\left( v,s_{i}\right) \ge u\left( s_{i}-R\left( v,s_{j}\right) \right) \nonumber \\&\quad +\beta w\left( v,s_{j}\right) \ \forall \ s\in \tilde{\varSigma },\quad \forall \ s_{j}\notin B(v) \end{aligned}$$
(20)
$$\begin{aligned}&w\left( v,s_{i}\right) \in \left[ \underline{v},\bar{v}\right] \quad \forall \ s_{i}\in \tilde{\varSigma } \end{aligned}$$
(21)
$$\begin{aligned} v&= \sum _{i=1}^{N}\pi _{i}\left[ u(s_{i}-R\left( v,s_{i}\right) )+\beta w\left( v,s_{i}\right) \right] \end{aligned}$$
(22)
$$\begin{aligned}&-\,s_{0}\le R\left( v,s_{i}\right) \le \left( 1-\eta \right) s_{i} \quad \forall \ s_{i}\in \tilde{\varSigma } \end{aligned}$$
(23)
$$\begin{aligned}&B\left( v\right) \subseteq \tilde{\varSigma } \end{aligned}$$
(24)

where (20) require temporary incentive compatibility (t.i.c.), (23) are the modified LLCs and (22) is the so-called promise-keeping constraint (PKC). \(\{ R\left( v,\cdot \right) ,w\left( v,\cdot \right) ,B(v)\} \) should be though of as policy rules in the sense that they are invariant functions of the state variable \(v\).

There are two main requirements for (19)–(24) to be an equivalent formulation to the sequential problem. The first and most obvious one is that, if the entrepreneur’s continuation utility is to be considered a candidate for summarizing history, her preferences over continuation contracts must be common knowledge after any history. This condition is satisfied in the current problem given our assumption that the entrepreneur cannot save or engage in side trades.

The second condition is that continuation payoffs lie in a bounded set.Footnote 22 This condition is satisfied in the current problem since \(\frac{1}{1-\beta }u\left( \eta s_{1}\right) =\underline{v}>-\infty \) while \(\frac{1}{1-\beta }\sum _{i=1}^{N}\pi _{i}u\left[ s_{0}-(1-\eta )s_{i}\right] =\bar{v}<\infty \). Notice that since we are not assuming \(\lim _{c\rightarrow 0}u\left( c\right) >-\infty \), it is essential that \(\eta \in \left( 0,1\right) \) and \(s_{1}>0\). Under this boundedness condition, one can show that, if for a given \(v_{0}\), an allocation \(\{B_{t}(h^{t-1}),R_{t}(h^{t})\}_{t=1}^{\infty }\) is generated recursively by the policy rules, then the allocation delivers the promised utility \(v_{0}\). To see this, define \(c_{t}\left( h^{t}\right) =s_{t} -R_{t}(h^{t})\) and iterate on the PKC (22) to get:

$$\begin{aligned} v_{0}=\sum _{t=0}^{T}\beta ^{T}\mathbb {E}_{0}u(c_{t}(h^{t}))+\beta ^{T} \mathbb {E}_{0}w_{T}(h^{T}),\ \ \forall \ T \end{aligned}$$

so that \(v_{0}=\sum _{t=0}^{\infty }\beta ^{T}\mathbb {E}_{0}u(c_{t}(h^{t} ))+\lim _{T\rightarrow \infty }\beta ^{T}\mathbb {E}_{0}w_{T}(h^{T})=\sum _{t=0}^{\infty }\beta ^{T}\mathbb {E}_{0}u(c_{t}(h^{t}))\), where the last equality follows since \(w_{T}\in [\underline{v},\bar{v}]\Rightarrow \lim _{T\rightarrow \infty }\beta ^{T}\mathbb {E}_{0}w_{T}(h^{T})=0\). Under these conditions, Lemma 2 in Green (1987) implies that t.i.c. is equivalent to incentive compatibility in the sense of (17). This justifies the use of a recursive formulation, from which it is easy to obtain Propositions 7 and 8.

Proof

(Proof of Proposition 7) See Wang (2005), page 915. \(\square \)

Proof

(Proof of Proposition 8) Suppose that the borrower has been promised \(\underline{v}\). It is enough to show that if \(s_{N-1}\notin B^{*}\left( \underline{v}\right) \) the contract fails to be optimal. Suppose that \(s_{N-1}\notin B^{*}\left( \underline{v}\right) \) and suppose that the borrower uses a strategy that calls her to report \(s_{N-1}\) when she observes \(s_{N}\). Then, the PKC is necessarily violated for the minimum expected utility that the borrower can obtain is \(\sum _{i=1}^{N-2}\pi _{i}u \left( \eta s_{i}\right) +\sum _{i=N-1}^{N}\pi _{i}u \left[ s_{i} -R^{*}\left( \underline{v},s_{N-1}\right) \right] +\beta w^{*}\left( \underline{v},s_{N-1}\right) \). But, since we know that \(w^{*}\left( \underline{v},s_{N-1}\right) \ge \underline{v}\) and \(R^{*}\left( \underline{v},s_{N-1}\right) \le \left( 1-\eta \right) s_{N-1}\) [by (23)], this lower bound cannot be less than:

$$\begin{aligned} \sum _{i=1}^{N-1}\pi _{i}u\left( \eta s_{i}\right) +\pi _{N}u\left[ s_{N}-\left( 1-\eta \right) s_{N-1}\right] +\beta \underline{v}>\sum _{i=1}^{N}\pi _{i}u\left( \eta s_{i}\right) +\beta \underline{v}=\underline{v} \end{aligned}$$

So define the left-hand side of the first inequality to be the threshold \(\hat{v}\left( \eta \right) \). Clearly then, for any \(v\le \hat{v}\left( \eta \right) ,\left\{ s_{1},s_{2},\ldots ,s_{N-1}\right\} \nsubseteq B^{*}\left( v\right) \) is not incentive compatible. It is now easy to see that \(\eta \mapsto \hat{v}\left( \eta \right) \) is increasing, so that the statement of the proposition follows. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tamayo, C.E. Investor protection and optimal contracts under risk aversion and costly state verification. Econ Theory 59, 547–577 (2015). https://doi.org/10.1007/s00199-014-0856-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00199-014-0856-5

Keywords

  • Investor protection
  • Risk aversion
  • Financial contracts
  • Standard debt

JEL Classification

  • D86
  • E61
  • G10
  • G18
  • G38
  • K40