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Collective reputation with stochastic production and unknown willingness to pay for quality

Abstract

In many cases, consumers cannot observe a single firm’s investment in environmental quality or safety, but only the average quality of the industry. The outcome of the investment is stochastic, since firms cannot control perfectly the technology or external factors that may affect production. In addition, firms do not know consumers’ valuation of quality. We characterize the solution of the firms’ investment game and show that the value of stopping investments when firms are already investing in quality can be negative when the free-riding incentives dominate. The existence of systematic uncertainty on the outcome of investment slows down investment in quality, compared to a situation without uncertainty. The uncertainty on consumers’ willingness to pay for quality can speed up or slow down investment. We also obtain the counterintuitive result that information acquisition may decrease the overall level of quality.

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Fig. 1

Notes

  1. 1.

    This can be the case of experience goods, for which the consumer cannot assess quality ex-ante, but it can also be the case for credence goods (Darby and Karni 1973), i.e., goods for which quality characteristics depend on production processes that are difficult to verify.

  2. 2.

    Levin (2009) introduced a stochastic version of the seminal Tirole (1996) model of collective reputation, where the workers’ cost of effort evolves randomly and there are noisy signals of a worker’s type. In these models, the workers’ types are imperfectly observed and collective reputation arises through beliefs, and can be expressed as the fraction of workers with good reputation. By comparison, there are no individual reputations in our model and collective reputation is a pure public good.

  3. 3.

    Static analyses by Fleckinger (2007), McQuade et al. (2016) and Rouvière and Soubeyran (2011) focus on the impact of market structure and of regulation, such as minimum quality standards.

  4. 4.

    We thus deliberatively abstract from firms’ other strategic interactions, such as the quantity choice, to focus on the provision of quality as a collective good problem.

  5. 5.

    For instance, we could assume, as in Claude and Zaccour (2009), a simple linear inverse demand curve:

    $$\begin{aligned} p=A+R-C(nq), \end{aligned}$$

    where p is the output price per unit, nq is the total output, A is an exogenous shock to demand, and C is a non-negative constant representing the slope of the linear demand function. Setting \(a=A-C(nq)\), we would obtain the expression (1) in the text

  6. 6.

    The support of \(\theta \) can be \([\underset{-}{\theta },\bar{\theta }]\) without affecting the results. However as seen from Eq. (2), as long as \(Q_{t}>0\), \(R_{t}\ge 0\) if and only if \(\underset{-}{\theta }\) \(\ge -1\).

  7. 7.

    Equation (3) implies that, starting from Q, the random position of the actual stock \(Q_{t}\) has a lognormal distribution, with mean \(\ln Q+(\frac{ \alpha }{n}\sum _{i=1}^{n}k_{it}-\frac{1}{2}\sigma ^{2})t\), and variance \( \sigma ^{2}t\). Since the process is Markovian, at any point in time the value \(Q_{t}\) observed by the firms is the best predictor of the future stock of quality.

  8. 8.

    Since \(p_{t}\), \(a_{t}\), and \(Q_{t}\) are cointegrated, a sequential learning process could be set in the following way: defining \( X_{t}\equiv \frac{p_{t}-a_{t}}{Q_{t}},\) each firm observes over time a sequence of realizations of a random variable \(X_{t}=(1+\theta )+\varepsilon _{t}\), where \(\varepsilon _{t}\) is a sequence of “noise” terms. If \(\varepsilon _{t}\) are independent and N(0, 1),  by the strong law of large numbers, the value of \( \theta \) would eventually be revealed, i.e., almost surely lim\( _{t\rightarrow \infty }\sum X_{t}/t\rightarrow 1+\theta \). This implies that asymptotically, a learning process would reveal the true value of \(\theta \). However, we are not interested in this paper to evaluate the time asymptotically or studying how firms could estimate \( \theta \) in finite time. We simply assume that by paying \(\phi \) firms can acquire in finite time, a knowledge of \(\theta \) that all of them can fully rely on.

  9. 9.

    To make the model more realistic, we could also introduce quality depreciation, for instance, by assuming that quality lifetime follows a Poisson process. This would mean that, over any short period \(\mathrm{d}t\), there would be a given probability that \(Q_{t}\) got reduced or even completely cancelled. Since the effect of depreciation would make irreversibility weaker, none of the results of this paper would be affected by such an assumption.

  10. 10.

    It is worth to note that, even if it is not possible to exclude a priori the existence of time-dependent Nash equilibria, stationary strategies are the ‘natural’ choice in this context, where firms are symmetric and invest a constant amount over time (Dangl and Wirl 2004).

  11. 11.

    In our framework, the absolute integrability condition \(E[ \int _{0}^{\infty }e^{-rt}\mid p(Q_{t},a_{t};\theta )\mid \mathrm{d}t] <\infty \) is satisfied by the assumption that \(r>\alpha \) (the proof is available from the authors upon request).

  12. 12.

    From now onward, we drop the time index for notational convenience, whenever it is not explicitly needed.

  13. 13.

    Interestingly enough, such a result is similar to the one obtained by Wirl (2008) in his study of a dynamic game among firms contributing to the same aggregate externality (Wirl 2008, Proposition 6, p. 108).

  14. 14.

    This result agrees with Bernanke’s (1983) Bad News Principle, for which relative uncertainty asymmetrically influences firms’ decision processes, since unfavourable events have a higher impact on investment decisions than favourable ones.

  15. 15.

    We thank an anonymous reviewer for pointing out this.

  16. 16.

    Originally introduced in 1996 in the fruit and vegetable sector (Council Regulation 2200/96), the EU Regulation 1308/2013 extended the use of Producer Organisations to all agricultural sectors.

  17. 17.

    We do not allow firms to know ex-ante that the consortium has the possibility to release such information—in other words, no signalling game is allowed between the consortium and the firms. We also assume that once the information is acquired, it is immediately visible to all firms.

  18. 18.

    For coherence with the preceding section, we consider only the case of \(Q< \hat{Q}\). Proposition 4 holds also for \(Q>\hat{Q}\) as proved in Fontini et al. (2013).

  19. 19.

    We thank an anonymous reviewer for pointing this out.

  20. 20.

    See (Dixit and Pindyck 1994, chapters 6 and 7) for a thorough discussion.

  21. 21.

    This condition follows from the maximization of Eq. (7). Each firm i will invest \(k=1\) if

    $$\begin{aligned} \frac{1}{2}\sigma ^{2}Q^{2}V^{\prime \prime }+\frac{\alpha }{n}\left( 1+\sum _{j=1, j\ne i}^{n}k_{j}\right) QV^{\prime }+p(Q;0)-c\ge \frac{1}{2}\sigma ^{2}Q^{2}V^{\prime \prime }+\frac{\alpha }{n}\left( \sum _{j=1, j\ne i}^{n}k_{j}\right) QV^{\prime }+p(Q;0) \end{aligned}$$

    which at \(\hat{Q}\) reduces to

    $$\begin{aligned} \frac{\alpha }{n}\left( 1+\sum _{j=1, j\ne i}^{n}k_{j}\right) \hat{Q}V^{\prime }-c&=\frac{\alpha }{n}\left( \sum _{j=1, j\ne i}^{n}k_{j}\right) \hat{Q}V^{\prime } \\ \hat{Q}V^{\prime }&= \frac{c}{\displaystyle \frac{\alpha }{n}\left( 1+\sum _{j=1, j\ne i}^{n}k_{j}\right) -\frac{\alpha }{n}\left( \sum _{j=1, j\ne i}^{n}k_{j}\right) }. \end{aligned}$$

    Then, by symmetry, we obtain

    $$\begin{aligned} \hat{Q}V^{\prime }=\frac{c}{\alpha - \alpha \frac{n-1}{n}}. \end{aligned}$$

    The marginal gain of investing one more unit in quality should be equal to the marginal cost.

  22. 22.

    To be precise, since \(\frac{\mathrm{d}\tilde{Q}(\theta )}{\mathrm{d}\theta }<0\) for all \( \theta \in [-1,+1],\) by inverting (16) we get

    $$\begin{aligned} \tilde{\theta }(Q)=\left\{ \begin{array}{ll} -1 &\quad \hbox{when } {\tilde{Q}}(-1)=\infty \\ \frac{\hat{Q}}{Q}-1 &\quad {\hbox {for }} \frac{{\hat{Q}}}{2}<Q<\infty \\ +1 &\quad Q\le \tilde{Q}(+1)=\ \frac{\hat{Q}}{2}. \end{array} \ \right. \end{aligned}$$

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Acknowledgements

Katrin Millock and Fulvio Fontini warmly thank the Department of Economics and Management at the University of Padova and the Centre d’Économie de la Sorbonne at the University of Paris 1 Panthéon-Sorbonne for hospitality during their respective research stays. Michele Moretto gratefully acknowledges financial support by the University of Padova (Research Grant BIRD173594/17).

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Correspondence to Katrin Millock.

Appendices

Technical Appendix 1

The general solution to the differential equations (8) and (9) takes the form:Footnote 20

$$\begin{aligned} V_{0}(Q;0)=A_{0}Q^{\beta _{1}}+B_{0}Q^{\beta _{2}}+\frac{a+Q}{r}\quad {\text{for }}\,\,Q<{\hat{Q}} \end{aligned}$$

and

$$\begin{aligned} V_{1}(Q;0)=A_{1}Q^{\gamma 1}+B_{1}Q^{\gamma _{2}}+\frac{a}{r}+\frac{Q}{ r-\alpha }-\frac{c}{r}\quad {\text{for }}\,\,Q>{\hat{Q}}, \end{aligned}$$

where \(\beta _{1},\gamma _{1}>1\ \)and \(\beta _{2},\gamma _{2}<0\) are the roots of the characteristic equations \(\Phi (\beta )=\frac{1}{2}\sigma ^{2}\beta (\beta -1)-r=0\) and \(\Phi (\gamma )=\frac{1}{2}\sigma ^{2}\gamma (\gamma -1)+\alpha \gamma -r=0\) respectively. \(A_{0}\), \(B_{0},\) \(A_{1}\), \(B_{1}\) are four constants to be determined. Note that under \(Q<\hat{Q}\) the first and second terms stand for the value of the option to switch to investment. However, since the value of the option vanishes as \(Q\rightarrow 0\), we set \( B_{0}=0\). Similarly, under \(Q>\hat{Q}\), the option to suspend investment is valueless as \(Q\rightarrow \infty \)  and then, we set \(A_{1}=0\). To find the constants \(A_{0},\) \(B_{1}\), and the optimal trigger \(\hat{Q}\), we impose a matching value condition and smooth pasting at \(\hat{Q}\):

$$\begin{aligned} A_{0}\hat{Q}^{\beta _{1}}+\frac{a+\hat{Q}}{r}=B_{1}\hat{Q}^{\gamma _{2}}+ \frac{a}{r}+\frac{\hat{Q}}{r-\alpha }-\frac{c}{r} \end{aligned}$$
(23)
$$\begin{aligned} A_{0}\beta _{1}\hat{Q}^{\beta _{1}-1}+\frac{1}{r}=B_{1}\gamma _{2}\hat{Q} ^{\gamma _{2}-1}+\frac{1}{r-\alpha } \end{aligned}$$
(24)

and the incentive constraintFootnote 21

$$\begin{aligned} A_{0}\beta _{1}\hat{Q}^{\beta _{1}}+\frac{\hat{Q}}{r}=\frac{c}{\alpha -\alpha \frac{n-1}{n}} \end{aligned}$$
(25)

Solving the system [23-25] yields the following:

$$\begin{aligned} \hat{Q}=\frac{(\beta _{1}-\gamma _{2})\frac{rn}{\alpha }-\gamma _{2}\beta _{1}}{(\beta _{1}-\gamma _{2})-\beta _{1}(\gamma _{2}-1)\frac{\alpha }{ r-\alpha }}c>0 \end{aligned}$$
(26)

and

$$\begin{aligned} \hat{A}_{0}=\left[ \frac{nc}{\alpha }-\frac{\hat{Q}}{r}\right] \frac{\hat{Q} ^{-\beta _{1}}}{\beta _{1}} \end{aligned}$$
$$\begin{aligned} \hat{B}_{1}=\left[ \frac{nc}{\alpha }-\frac{\hat{Q}}{r-\alpha }\right] \frac{ \hat{Q}^{-\gamma _{2}}}{\gamma _{2}}. \end{aligned}$$

Then, the condition for \(A_{0}>0\) is \(\hat{Q}<n\frac{r}{ \alpha }c\), or \(\frac{(\beta _{1}-\gamma _{2})\frac{rn}{\alpha } -\gamma _{2}\beta _{1}}{(\beta _{1}-\gamma _{2})-\beta _{1}(\gamma _{2}-1) \frac{\alpha }{r-\alpha }}<n\frac{r}{\alpha }\), which can be simplified into \(-\gamma _{2}<-(\gamma _{2}-1)\frac{nr}{r-\alpha }\), which is always satisfied. On the other hand, the condition for \( B_{1}>0 \) is \(\hat{Q}>n\frac{r-\alpha }{\alpha }c\), or \( \frac{(\beta _{1}-\gamma _{2})\frac{rn}{\alpha }-\gamma _{2}\beta _{1}}{ (\beta _{1}-\gamma _{2})-\beta _{1}(\gamma _{2}-1)\frac{\alpha }{r-\alpha }} >n\frac{r-\alpha }{\alpha }\), which can be simplified into \(\gamma _{2}\alpha (\beta _{1}(n-1)-n)>0\); this is satisfied for \(\beta _{1}<n/(n-1)\). \(\square \)

Technical Appendix 2

Let us calculate \(E_{\theta }\left[ V(Q;\theta )\right] -V(Q;0).\) In particular, since (16) is monotonic, we write

$$\begin{aligned} E_{\theta }\left[ V(Q;\theta )\right] -V(Q;0)= \left\{ \begin{array}{rl} \int _{-1}^{\tilde{\theta }(Q)}V_{0}(Q;\theta )f(\theta )\mathrm{d}\theta +\int _{ \tilde{\theta }(Q)}^{+1}V_{1}(Q;\theta )f(\theta )\mathrm{d}\theta -V_{0}(Q;0) &\quad \frac{ \hat{Q}}{2}<Q<\hat{Q} \\ \int _{-1}^{+1}V_{0}(Q;\theta )f(\theta )\mathrm{d}\theta -V_{0}(Q;0) &\quad \text{for } Q< \frac{{\hat{Q}}}{2}, \end{array} \right. \end{aligned}$$
(27)

where \(\tilde{\theta }(Q)=\frac{\hat{Q}}{Q}-1\) Footnote 22. The first line of Eq. (27) shows that, for any given stock Q,  the ex-ante value \(E_{\theta }\left[ V(Q;\theta )\right] \) is formed by two terms. The first integral indicates the firm’s value when the revealed value of \(\theta \) is so low that it is not optimal to invest in quality, while the second integral reflects the case, where \(\theta \) is found sufficiently high to induce the firm to invest in quality.

Assuming that there exists a value \(Q_{I}\in [Q,\hat{Q}]\) beyond which each firm decides to coordinate spending on acquiring information on \( \theta \), the solution of (20) must solve the following Bellman equation:

$$\begin{aligned} rI=\frac{1}{2}\sigma ^{2}Q^{2}I^{\prime \prime }\quad {\text{for }} Q<Q_{I}< {\hat{Q}}, \end{aligned}$$
(28)

where \(I^{\prime }\) and \(I^{\prime \prime }\) stand for the first and second derivatives with respect to Q, respectively. The general solution for I takes the form:

$$\begin{aligned} I(Q)=M_{0}Q^{\beta _{1}}+M_{1}Q^{\beta _{2}}\quad \text{for }Q<Q_{I}, \end{aligned}$$

where \(\beta _{1}>1\ \)and \(\beta _{2}<0\) are the roots of the characteristic equations \(\Phi (\beta )=\frac{1}{2}\sigma ^{2}\beta (\beta -1)-r=0.\) However, since the value of the option to acquire the information vanishes as \(Q\rightarrow 0\), we set \(M_{1}=0\).

Assume \(\phi \ge 0.\) To find the constant \(M_{0}\) and the optimal trigger \( Q_{I}\), we impose a matching value condition and smooth pasting at \(Q_{I}\). Using (27), we have

$$\begin{aligned} M_{0}Q_{I}^{\beta _{1}}&= \int _{-1}^{\tilde{\theta }(Q_{I})}V_{0}(Q_{I}; \theta )f(\theta )\mathrm{d}\theta \nonumber \\&\quad+\int _{\tilde{\theta }(Q_{I})}^{+1}V_{1}(Q_{I};\theta )f(\theta )\mathrm{d}\theta -V_{0}(Q_{I};0)-\phi \end{aligned}$$
(29)

and

$$\begin{aligned} M_{0}\beta _{1}Q_{I}^{\beta _{1}-1}&= \frac{\mathrm{d}\tilde{\theta }(Q_{I})}{\mathrm{d}Q_{I}} \left[ V_{0}(Q_{I};\tilde{\theta }(Q_{I}))f(\tilde{\theta }(Q_{I}))\right] \\&\quad+\int _{-1}^{\tilde{\theta }(Q_{I})}V_{0}^{\prime }(Q_{I};\theta )f(\theta )\mathrm{d}\theta \nonumber \\&\quad-\frac{\mathrm{d}\tilde{\theta }(Q_{I})}{\mathrm{d}Q_{I}}\left[ V_{1}(Q_{I};\tilde{\theta } (Q_{I}))f(\tilde{\theta }(Q_{I}))\right] \nonumber \\&\quad+\int _{\tilde{\theta }(Q_{I})}^{+1}V_{1}^{\prime }(Q_{I};\theta )f(\theta )\mathrm{d}\theta -V_{0}^{\prime }(Q_{I};0) \nonumber . \end{aligned}$$
(30)

Since by (17) \(V_{0}(Q_{I};\tilde{\theta }(Q_{I}))=V_{1}(Q_{I};\tilde{ \theta }(Q_{I})),\) then (30) reduces to

$$\begin{aligned} M_{0}Q_{I}^{\beta _{1}}&= \int _{-1}^{\tilde{\theta }(Q_{I})}\frac{Q_{I}}{ \beta _{1}}V_{0}^{\prime }(Q_{I};\theta )f(\theta )\mathrm{d}\theta \\&\quad +\int _{\tilde{\theta }(Q_{I})}^{+1}\frac{Q_{I}}{\beta _{1}}V_{1}^{\prime }(Q_{I};\theta )f(\theta )\mathrm{d}\theta -\frac{Q_{I}}{\beta _{1}}V_{0}^{\prime }(Q_{I};0). \end{aligned}$$

Substituting in (29), we get

$$\begin{aligned}&\int _{-1}^{\tilde{\theta }(Q_{I})}V_{0}(Q_{I};\theta )f(\theta )\mathrm{d}\theta \\&\qquad +\int _{\tilde{\theta }(Q_{I})}^{+1}V_{1}(Q_{I};\theta )f(\theta )\mathrm{d}\theta -V_{0}(Q_{I};0)-\phi \\&\quad =\int _{-1}^{\tilde{\theta }(Q_{I})}\frac{Q_{I}}{\beta _{1}}V_{0}^{\prime }(Q_{I};\theta )f(\theta )\mathrm{d}\theta \\&\qquad +\int _{\tilde{\theta }(Q_{I})}^{+1}\frac{Q_{I}}{\beta _{1}}V_{1}^{\prime }(Q_{I};\theta )f(\theta )\mathrm{d}\theta -\frac{Q_{I}}{\beta _{1}}V_{0}^{\prime }(Q_{I};0). \end{aligned}$$

Substituting in for \(V_{0}(Q_{I};\theta )\) and \(V_{1}(Q_{I};\theta )\), and their derivatives gives

$$\begin{aligned}&\int _{-1}^{\tilde{\theta }(Q_{I})}\left[ \tilde{A}_{0}(\theta )Q_{I}^{\beta _{1}}+\frac{a+(1+\theta )Q_{I}}{r}\right] f(\theta )\mathrm{d}\theta \\&\qquad +\int _{\tilde{\theta }(Q_{I})}^{+1}\left[ \tilde{B}_{1}(\theta )Q_{I}^{\gamma _{2}}+\frac{a}{r}+\frac{(1+\theta )Q_{I}}{r-\alpha }-\frac{c}{ r}\right] f(\theta )\mathrm{d}\theta \\&\qquad -\left[ \hat{A}_{0}Q_{I}^{\beta _{1}}+\frac{a+Q_{I}}{r}\right] -\phi \\&\quad =\int _{-1}^{\tilde{\theta }(Q_{I})}\frac{Q_{I}}{\beta _{1}}\left[ \tilde{A} _{0}(\theta )\beta _{1}Q_{I}^{\beta _{1}-1}+\frac{(1+\theta )}{r}\right] f(\theta )\mathrm{d}\theta \\&\qquad +\int _{\tilde{\theta }(Q_{I})}^{+1}\frac{Q_{I}}{\beta _{1}}\left[ \tilde{B} _{1}(\theta )\gamma _{2}Q_{I}^{\gamma _{2}-1}+\frac{(1+\theta )}{r-\alpha } \right] f(\theta )\mathrm{d}\theta \\&\qquad -\frac{Q_{I}}{\beta _{1}}\left[ \hat{A}_{0}\beta _{1}Q_{I}^{\beta _{1}-1}+ \frac{1}{r}\right] \end{aligned}$$

or

$$\begin{aligned} \frac{\alpha Q_{I}}{r(r-\alpha )}\int _{\tilde{\theta }(Q_{I})}^{+1}(1+\theta )f(\theta )\mathrm{d}\theta&= \frac{\beta _{1}}{\beta _{1}-1}\left[ \phi +\frac{c}{r} \left[ 1-F(\tilde{\theta }(Q_{I}))\right] \right] \\&\quad -\frac{\beta _{1}-\gamma _{2}}{\beta _{1}-1}\left\{ \int _{\tilde{\theta } (Q_{I})}^{+1}\tilde{B}_{1}(\theta )Q_{I}^{\gamma _{2}}f(\theta )\mathrm{d}\theta \right\} \nonumber , \end{aligned}$$
(31)

Recall that the expression above should hold for \(Q_{I}<\hat{Q}\). We can show that if \(Q_{I}\) existed, it would belong to the interval \((\frac{\hat{Q}}{2},\hat{Q})\). This can be seen by noting that if we look for \(Q_{I}\) that tends to \(\frac{\hat{Q}}{2}\) from above, i.e., \(\tilde{\theta }(Q_{I})\simeq +1,\) the condition (31) is never satisfied. This means that, since the initial condition \(\frac{\hat{Q}}{2}<Q<\hat{Q}\) must be true whatever is the value of \(Q_{I}\), it is never optimal to acquire information about the true value of \(\theta \), i.e., \(Q<Q_{I}\) and thus \(Q_{I}\in (\frac{\hat{Q}}{2},\hat{Q})\). Note, however, the exception when the initial condition is \(Q=\frac{\hat{Q}}{2}\) and \(\phi =0\) in which case \(Q_{I}\) would be undetermined.

Consider now the case when \(Q\le \frac{\hat{Q}}{2}.\) Equation (27) becomes

$$\begin{aligned} E_{\theta }\left[ V(Q;\theta )\right] -V(Q;0)&= \int _{-1}^{+1}V_{0}(Q;\theta )f(\theta )\mathrm{d}\theta -V_{0}(Q;0) \\&= \int _{-1}^{+1}[V_{0}(Q;\theta )-V_{0}(Q;0)]f(\theta )\mathrm{d}\theta . \end{aligned}$$

Recalling that \(V_{0}(Q;\theta )=\tilde{A}_{0}Q^{\beta _{1}}+\frac{ a+(1+\theta )Q}{r}\) and \(V_{0}(Q;0)=A_{0}Q^{\beta _{1}}+\frac{a+Q}{r},\) we can write \(E_{\theta }[V(Q;\theta )] -V(Q;0)=S(\theta )Q^{\beta _{1}},\) where \(S(\theta )=[ \int _{-1}^{+1}\Delta A(\theta )f(\theta )\mathrm{d}\theta ]\) and \(\Delta A=\tilde{A}_{0}-A_{0}\) is given by

$$\begin{aligned} \Delta A(\theta )=\tilde{A}_{0}-A_{0}=\left[ \frac{nc}{\alpha }-\frac{\tilde{ Q}}{r}\right] \frac{\tilde{Q}^{-\beta _{1}}}{\beta _{1}}-\left[ \frac{nc}{ \alpha }-\frac{\hat{Q}}{r}\right] \frac{\hat{Q}^{-\beta _{1}}}{\beta _{1}} \end{aligned}$$

and \(\tilde{Q}=\frac{\hat{Q}}{(1+\theta )}\). To find the constant and the optimal trigger \(Q_{I}\), we impose a matching value condition and smooth pasting at \(Q_{I}\):

$$\begin{aligned} M_{0}Q_{I}^{\beta _{1}}=S(\theta )Q_{I}^{\beta _{1}}-\phi \end{aligned}$$

and

$$\begin{aligned} M_{0}\beta _{1}Q_{I}^{\beta _{1}-1}=S(\theta )\beta _{1}Q_{I}^{\beta _{1}-1}. \end{aligned}$$

Defining \(Y(Q)=\left[ M_{0}-S(\theta )\right] Q^{\beta _{1}}\) we should distinguish two cases:

  1. 1.

    \(S(\theta )>0\). In this case, there always exists a positive constant \( M_{0}\) such that \(Y(Q)>0,\) then \(\max Y(Q)\rightarrow Q_{I}=\frac{\hat{Q}}{2} \).

  2. 2.

    \(S(\theta )<0\). In this case, it is never optimal to invest.

\(\square \)

Technical Appendix 3

Proof of Proposition 4

Assume \(Q < \hat{Q}\). Recall that when Q is between 0 and \(\frac{\hat{Q}}{2}\), it is never optimal to invest in quality. Next, consider the case when \( \frac{\hat{Q}}{2}\le Q<\hat{Q}.\) Recall that Q, defined in Eq. (3), is our stochastic process. We calculate the probability that a level \(\hat{Q}\) is hit starting from a generic starting value \( Q_{t}\). Since

$$\begin{aligned} \mathrm{d}\ln [Q_{t}]=\mu \mathrm{d}t+\sigma \mathrm{d}z_{t}, \end{aligned}$$

where \(\mu =(\alpha -\frac{1}{2}\sigma ^{2})\), the probability \(\Pr (Q_{\tau }=\hat{Q}\mid Q_{t})\) is given by (Cox and Miller 1965, p. 212; Harrison 1985, p. 43):

$$\begin{aligned} \Pr \left( \hat{Q},Q\right) =\left\{ \begin{array}{ll} 1 &{}\quad \text{if } 2\frac{\alpha }{n}\sum _{i=1}^{n}k_{it}/\sigma ^{2}\ge 1 \\ \left( \dfrac{\hat{Q}}{Q}\right) ^{(2\frac{\alpha }{n}\sum _{i=1}^{n}k_{it}/ \sigma ^{2})-1} &{}\quad \text{if } 2\frac{\alpha }{n}\sum _{i=1}^{n}k_{it}/\sigma ^{2}<1. \end{array} \right. \end{aligned}$$
(32)

Starting at Q in the interior of the range \((0,\hat{Q}]\), after a “sufficient” long interval of time, the process is sure to hit the trigger \(\hat{Q}\) if the trend is positive and sufficiently large with respect to the uncertainty. However, if \(\frac{\alpha }{n} \sum _{i=1}^{n}k_{it}\) is positive but low with respect to the uncertainty or it is negative, the process may drift away and never hit \(\hat{Q}\). Taking into account that the firms never invest below \(\hat{Q}\), applying the expression in (32) to Eq. (22) with \(\frac{\alpha }{n} \sum _{i=1}^{n}k_{it}=0\), we obtain

$$\begin{aligned} \Delta (P(Q))&= \int _{-1}^{\tilde{\theta }(Q)}\left( \frac{Q}{\tilde{Q} (\theta )}\right) f(\theta )\mathrm{d}\theta +\int _{\tilde{\theta }(Q)}^{+1}1f(\theta )\mathrm{d}\theta -\left( \frac{Q}{\hat{Q}}\right) \\&= \left[ 1-F(\tilde{\theta }(Q))\right] \left[ 1-\left( \frac{Q}{\hat{Q}} \right) \right] +\left( \frac{Q}{\hat{Q}}\right) \int _{-1}^{\tilde{\theta } (Q)}\theta f(\theta )\mathrm{d}\theta \nonumber . \end{aligned}$$
(33)

We see that \(\Delta (P(\frac{\hat{Q}}{2}))=0\) and

$$\begin{aligned} \frac{\partial \Delta (P(Q))}{\partial Q}=\left( \frac{1}{\hat{Q}}\right) \left[ \int _{-1}^{\tilde{\theta }(Q)} \theta f(\theta )\mathrm{d}\theta -\left[ 1-F(\tilde{\theta }(Q)) \right] \right]< 0,\text{ for all }\frac{\hat{Q}}{2}< Q<\hat{Q}. \end{aligned}$$

Then \(Q=\frac{\hat{Q}}{2}\) is the single root of Eq. (33). \(\square \)

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Fontini, F., Millock, K. & Moretto, M. Collective reputation with stochastic production and unknown willingness to pay for quality. Environ Econ Policy Stud 20, 387–410 (2018). https://doi.org/10.1007/s10018-017-0198-1

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Keywords

  • Collective reputation
  • Dynamic game
  • Real options
  • Stochastic quality

JEL Classification

  • C73
  • D92
  • Q52