A dynamic model of certification and reputation

Abstract

Markets typically have many ways of learning about quality, with two of the most important being reputational forces and certification, and these types of learning often interact with and influence each other. This paper is the first to consider markets where learning occurs through these different sources simultaneously, which allows us to investigate the rich interplay and dynamics that can arise. Our work offers four main insights: (1) Without certification, market learning through reputation alone can get “stuck” at inefficient levels and high-quality agents may get forced out of the market. (2) Certification “frees” the reputation of agents, allowing good agents to keep working even after an unfortunate string of bad signals. (3) Certification can be both beneficial and harmful from a social perspective, so a social planner must choose the certification scheme carefully. In particular, the market will tend to demand more certification than socially optimal because the market does not bear the certification costs. (4) Certification and reputational learning can act as complementary forces so that the social welfare produced by certification can be increased by faster information revelation.

Appendix

1.1 Proof of Theorem 1

First note that if the price is less than $c$, the agent would refuse to accept any offers, and so the social welfare is equal to $0$ regardless of the information process. Thus, we only need to consider prices $p\ge c$.

Note that under any blind process, the agent’s expected quality is never updated, and so since we assume that $\mu _o >p$ the agent will never stop working. The ex ante social surplus can thus be calculated as $\frac{\mu _0 -c}{\rho }$. For a general information process the market continues to hire any agent until its expected quality drops below $p$, and for an admissible process this happens at the first time that $\mu _t =p$. We can write out the ex ante expected social welfare for any information process as

$$\begin{aligned} W\left( {p,\mathcal{L}} \right)&= {\int }_{-\infty }^\infty \left[ {{\int }_0^\infty e^{-\rho t}\left( {q-c} \right) dt} \right] f_0 \left( q \right) dq-E_{q,t^{*}} \left[ {{\int }_{t^{*}}^\infty e^{-\rho t}\left( {q-c} \right) dt} \right] \\&= {\!\int }_{-\infty }^\infty \left[ {{\int }_0^\infty e^{-\rho t}\!\left( {q\!-\!c}\right) \!dt} \right] f_0 \left( q \right) dq\!-\!E_{t^{*}} \left[ {{\int }_{t^{*}}^\infty E_q \left[ {e^{-\rho t}\!\left( {q\!-\!c}\right) \!|t^{*}} \right] dt}\! \right] \\&= \frac{\mu _0 -c}{\rho }-\frac{p-c}{\rho }E_{t^{*}} \left[ {e^{-\rho t^{*}}} \right] \le \frac{\mu _0 -c}{\rho } \end{aligned}$$

$\square$

1.2 Proof of Theorem 2

Suppose the market believes that agents are following some (not necessarily type independent) certification strategy $\tilde{\tau } (q)$. Consider any time $t^{\prime }$ and history $\mathcal{H}_{t^{{\prime }}}$, and let the expected quality for an agent that has not certified by $t^{\prime }$ be given by $\mu _{t^{{\prime }}}^N \equiv E\left[ {q|\mathcal{H}_{t^{{\prime }}} ,\theta _{t^{{\prime }}} =\phi } \right]$. Note that for any market strategy beliefs, the value of $\mu _{t^{{\prime }}}^N$ depends only on the work history $\mathcal{H}_{t^{{\prime }}}$and certification status $\theta _{t^{{\prime }}}$, and not the agent’s true quality. Thus, fixing the work history and certification status, all agents will have the same $\mu _{t^{{\prime }}}^N$ regardless of their true quality. There are two possible cases: $\mu _{t^{{\prime }}}^N >p$ or $\mu _{t^{{\prime }}}^N \le p$, and we show that for both cases, either all types of agents with qualities $q\ge \underline{q}$ will choose to certify at $t^{\prime }$ or no types of agents will choose to certify at $t^{\prime }$.

First consider the case where $\mu _{t^{{\prime }}}^N >p$. Then agents can still work even without certifying. The payoff of an agent that chooses to certify is given by $\frac{p-c}{\rho }-k$. This payoff is identical for all agents with quality $q\ge \underline{q}$ because of our assumption that $\underline{q}\ge p$, and so certified agents will never stop working. Now consider the alternate strategy of waiting until the time $\hat{t} \equiv \inf \left\{ {t|\mu _t^N \le p} \right\}$ and then certifying. This strategy gives a payoff of

$$\begin{aligned} {\int }_0^{\hat{t}} e^{-\rho t}\left( {p-c} \right) dt-\left( {e^{-\rho \hat{t}}} \right) \left( \frac{p-c}{\rho }-k\right) \end{aligned}$$

This alternate strategy gives a payoff higher than certifying immediately by $(1-e^{-\rho \hat{t}})*k$. So certifying at time $t^{{\prime }}$ is not optimal, and with this $\mu _{t^{{\prime }}}^N$ all types of agents would choose not to certify.

Next consider the case where $\mu _{t^{{\prime }}}^N \le p$. An agent that does not certify will not be able to work at time $t^{\prime }$, and so receives a maximum payoff of

$$\begin{aligned} \left( {1-\rho dt} \right) \left( {\frac{p-c}{\rho }-k} \right) \end{aligned}$$

This is the payoff the agent would receive if it certified at time $t^{{\prime }}+dt$. Since no observations are made, the mean at a later time cannot be greater than $p$ unless the agent were to certify at some time $t^{{\prime }}+dt$. If the agent instead chooses to certify immediately at time $t^{\prime }$, it would get a payoff of

$$\begin{aligned} \frac{p-c}{\rho }-k>\left( {1-\rho dt} \right) \left( {\frac{p-c}{\rho }-k} \right) \end{aligned}$$

Thus, certifying in the current time step would increase the payoff by $\rho dt\left( {\frac{p\!-\!c}{\rho }\!-\!k} \right)$. Therefore, all types of agents would choose to certify given this value of $\mu {t^{{\prime }}}\,\,N$.

Since all agents with quality $q\ge \underline{q}$ would choose the same certification decision for any value of $\mu _{t^{{\prime }}}^N$, every equilibrium must feature all agents with quality $q\ge \underline{q}$ utilizing the same certification strategy $\tau _\varepsilon ^*$. $\square$

1.3 Proof of Proposition 1

Consider any equilibrium that requires the agent to certify at a time $t^{{\prime }}$ where $\mu _{t^{{\prime }}} >p$. Now let us compare the payoffs to the agent against the equilibrium where the agent certifies at the first time $t$ such that $\mu _t \le p$. In the second equilibrium, at time $t^{{\prime }}$ the agent would instead delay certification until the time $t^{*}=\inf \left\{ {t|\mu _t =p} \right\}$. Since the payoff from certification is the same regardless, the agent would be able to improve its payoff by the amount $(e^{-\rho t^{{\prime }}}-e^{-\rho t^{*}})k$. Thus, no equilibrium that requires the agent to certify at a $\mu _t >p$ can be agent optimal. $\square$

1.4 Proof of Proposition 2

We fix an arbitrary equilibrium and compute the social welfare flow payoff difference at the equilibrium certification time $\tau ^{*}$ between certification and no certification. Without certification, the social welfare flow payoff is given by

$$\begin{aligned} E_{\tau ^{*}} \left[ {q-c} \right] ={\int }_{-\infty }^\infty \left( {q-c} \right) f_{\tau ^{*}} (q)dq \end{aligned}$$

With certification this would become

$$\begin{aligned} E_{\tau ^{*}} \left[ {q\!-\!c} \right] =\left( {1\!-\!F_{\tau ^{*}}^- \left( {\underline{q}} \right) } \right) \int _{\underline{q}}^\infty \left( {q\!-\!c} \right) f_{\tau ^{*}} (q|q\ge \underline{q})dq=\int _{\underline{q}}^\infty \left( {q-c} \right) f_{\tau ^{*}} (q)dq \end{aligned}$$

Thus, the difference in the two expectations is given by

$$\begin{aligned} \mathop {\lim }\limits _{q^{{\prime }}\nearrow \underline{q} } \int _{-\infty }^{q^{{\prime }}} \left( {q-c} \right) f_{\tau ^{*}} (q)dq \end{aligned}$$

Certification has a higher flow payoff if and only if the difference in expectations plus the flow cost of certification is less than zero:

$$\begin{aligned} \mathop {\lim }\limits _{q^{{\prime }}\nearrow \underline{q}} \int _{-\infty }^{q^{{\prime }}} \left( {q-c} \right) f_{\tau ^{*}} (q)dq+\rho k\left( {1-F_{\tau ^{*}}^- \left( {\underline{q}} \right) } \right) \le 0 \end{aligned}$$

Or equivalently $\left( {\mu _{\tau ^{*}}^{NC} -c-\rho k} \right) F_{\tau ^{*}}^- (\underline{q})\le -\rho k$, which leads to (1).

Now suppose for the sake of contradiction that (1) does not hold at the certification time $\tau ^{*}$. We will consider the following strategy that can be an equilibrium, and we show that it provides a higher social welfare than certifying at time $\tau ^{*}$ if (1) does not hold at $\tau ^{*}$. Suppose instead that the agent keeps working until 1: the first $t>\tau ^{*}$ such that (1) holds, or until 2: the first $t>\tau ^{*}$ such that $\mu _t \le p$, and then the agent certifies.

Note that the agent keeps working without certifying as long as the social welfare flow payoff from not certifying is greater than the flow payoff from certifying at $\tau ^{*}$. Once the agent certifies, the flow payoffs are the same as with certifying at $\tau ^{*}$. Thus, under this alternate strategy the flow payoffs can never be less than under certifying at $\tau ^{*}$. Therefore, the total social welfare generated must be higher as well. $\square$

1.5 Proof of Proposition 3

The short run time $t$ principal’s utility is given by $\mu _t -p=E_t [q-p]$, so the principal prefers certification if and only if this expectation with certification is higher than the expectation without certification. Without certification, we have

$$\begin{aligned} E_t \left[ {q-p} \right] =\int _{-\infty }^\infty \left( {q-p} \right) f_t (q)dq \end{aligned}$$

With certification this would become

$$\begin{aligned} E_t \left[ {q-p} \right] =\left( {1-F_t^- \left( {\underline{q}} \right) } \right) \int _{\underline{q}}^\infty \left( {q-p} \right) f_t (q|q\ge \underline{q})dq=\int _{\underline{q}}^\infty \left( {q-p} \right) f_t (q)dq \end{aligned}$$

Thus, the difference in the two expectations is given by

$$\begin{aligned} \mathop {\lim }\limits _{q^{{\prime }}\nearrow \underline{q}} \int _{-\infty }^{q^{{\prime }}} \left( {q-p} \right) f_t (q)dq \end{aligned}$$

Certification is preferred if and only if the above term is less than zero.

$$\begin{aligned} \int _{-\infty }^p \left( {q-p} \right) f_t (q)dq+\mathop {\lim }\limits _{q^{{\prime }}\nearrow \underline{q}} \int _p^{q^{{\prime }}} \left( {q-p} \right) f_t (q)dq\le 0 \end{aligned}$$

This means that the benefit of removing bad agents (qualities below $p)$ from the market outweighs the costs of removing the good agents (qualities above $p)$. Or equivalently:

$$\begin{aligned} \left( {\mu _t^{NC} -p} \right) F_t^- (\underline{q})\le 0\\ \mu _t^{NC} \le p \end{aligned}$$

This results in Eq. (2) in the Proposition. Since at each time $t$, that time $t$ principal wishes for the agent to certify if and only if this equation holds, the resulting certification strategy will feature the agent certifying at the first moment that this equation holds. By the corollary to theorem 2, we know that such a certification strategy can be an equilibrium. $\square$

1.6 Proof of Theorem 4

First note that if $p<c+\rho k$ there can be no certification in equilibrium. The optimal way to implement no certification is to set $p=c,\underline{q}=\infty$ from Theorem 1. This results in a social welfare of $\frac{\mu _0 -c}{\rho }$ for any admissible information process. Next, suppose that we wish to allow certification in equilibrium. Thus, we need to set $p\ge c+\rho k$. We show that for any $p\ge c+\rho k$ and any $\underline{q}\ge p$, we can achieve at least as high of a welfare by setting $\underline{q}=p=c+\rho k$. Given the first set of parameters, denote the socially optimal certification stopping time by $\tau _s^*$. But under the second set of parameters, $\tau _s^*$ can also be implemented because $\mu _{\tau _s^*}^{NC} \le p$ will also hold at any $\tau _s^*$ (recall that $\mu _t^{NC} \le p$ for all $t$ if $\underline{q}=p)$, and $\mu _t >p$ for all $t<\tau _s^*$. In addition, once implemented the social welfare provided by certification will be at least as high, because all types of agents with qualities $\underline{q}>c+\rho k$ contribute positively to social welfare by certifying instead of exiting. Thus, the social welfare with $\underline{q}=p=c+\rho k$ must be at least as high as with any other standard. $\square$

1.7 Proof of Theorem 5

This proof will proceed in several steps. Note that certification can be broken up into three possible types: immediate certification at $t=0$, delayed certification that takes place at some $t>0$, and no certification for all times. Which type of certification results will depend on the specific values of $\underline{q}$ and $p$. We prove that under pessimistic principal beliefs, delayed certification is never optimal. Then, we characterize the social welfare generated by immediate and no certification. We prove that immediate certification and no certification can both be optimal depending on how high the certification cost is.

First we show that implementing delayed certification is never socially optimal. If the price is set lower than $c+\rho k$, then certification can never occur because agents would prefer to exit than certify. Thus, for any type of certification to be implemented, we must have $p\ge c+\rho k$. Next, note that under pessimistic principal beliefs, the certification standard $\underline{q}$ must be set high enough such that $\mu _0^{NC} >p$ because otherwise agents would be expected and thus forced to certify immediately in a principal optimal equilibrium by Proposition 3. In particular, this requires that $\underline{q}>p$.

Now we analyze the social welfare generated by any delayed certification scheme, and we show that the welfare is strictly less than under no certification. Let $\hat{t} =\inf \left\{ {t|\mu _t^{NC} \le p} \right\}$. In a principal optimal equilibrium, $\hat{t}$ is the time at which certification would occur. Note that admissibility implies that at $\mathop t\limits$, the truncated expected mean $\mu _{\hat{t}}^{NC} =p$. For any $p$ and $\underline{q}$ that satisfy the above conditions, we can compute the resulting social welfare as:

$$\begin{aligned} W\left( p \right)&= \int _{-\infty }^\infty \left[ {\int _0^\infty e^{-\rho t}\left( {q-c} \right) dt} \right] f_0 \left( q \right) dq-E_{\hat{t}} \left[ ke^{-\rho \hat{t}}\left( {1-F_{\hat{t}}^- \left( {\underline{q}} \right) } \right) \right. \\&\left. +F_{\hat{t}}^- \left( {\underline{q}} \right) \int _{\hat{t}}^\infty e^{-\rho t}\left( {\mu _{\hat{t}}^{NC} -c} \right) dt \right] \\&= \frac{\mu _0 -c}{\rho }-\left( {kE_{\hat{t}} \left[ {e^{-\rho \hat{t}}\left( {1-F_{\hat{ t}}^- \left( {\underline{q}} \right) } \right) } \right] +\frac{p-c}{\rho }E_{\hat{t}} \left[ {e^{-\rho \hat{ t} }F_{\hat{t}}^- \left( {\underline{q}} \right) } \right] } \right) \\&\le \frac{\mu _0 -c}{\rho } \end{aligned}$$

This proves that the social welfare of any delayed certification scheme is bounded above by setting $p=c$ and $\underline{q}=\infty$, which results in no certification. From the blind boundedness theorem, we know that the payoff of such a scheme is exactly $\frac{\mu _0 -c}{\rho }$.

Now, fixing a $\underline{q}$, the welfare provided by a de facto license is given by the expression $\int _{\underline{q}}^\infty (\frac{q-c}{\rho }-k)f_0 \left( q \right) dq$. Given a $k$, the optimal certification standard is $\underline{q}=c+\rho k$. The reason is that any agent that certifies will give a social welfare of $\frac{q-c}{\rho }-k$, and this is the quality where this expression is equal to zero. Any agent with a quality higher than this amount contributes positively to welfare by certifying. Since we require that $p\le \underline{q}$ and we need $p\ge c+\rho k$ for certification to occur, this implies that we need to set $p=c+\rho k$ in order to implement immediate certification. Thus, the highest ex ante surplus generated by any immediate certification scheme is $\int _{c+\rho k}^\infty (\frac{q-c}{\rho }-k)f_0 \left( q \right) dq=\left( {\frac{\mu _0^C -c}{\rho }-k} \right) \left( {1-F_0^- \left( {c+\rho k} \right) } \right)$.

Thus, to see whether immediate certification is better, or whether no certification is better, we need to see which of the two surpluses is higher. This depends on the value of $k$, and specifically the cutoff value will be given by $k^{*}=\frac{\mu _0^C -c}{\rho }-\frac{\mu _0 -c}{\rho \left( {1-F_0^- \left( {c+\rho k} \right) } \right) }$ . $\square$

1.8 Proof of Theorem 6

First note that if either no certification or a de facto license is implemented, the social welfare will always be bounded away from the social optimal. With no certification, the welfare always equals the benchmark welfare for any $k$, and with a de facto license, the price and standard have to be set to at least $\mu _0$. The social welfare of the de facto license is thus equal to $\int _{\mu _0 }^\infty (\frac{q-c}{\rho }-k)f_0 \left( q \right) dq$, which is bounded away from the first best welfare, $\int _c^\infty (\frac{q-c}{\rho })f_0 \left( q \right) dq$, for any value of $k$.

Thus, in order to get asymptotic efficiency as $k\rightarrow 0$, we need to do it through a delayed certification scheme. We now show that the social welfare of any delayed certification scheme will also be bounded away from first best. First note that if the standard $\underline{q}{\nrightarrow }\, c$, then first best cannot be achieved. The reason is that before certification occurs, we are losing welfare from letting bad agents work, and after certification occurs we also lose welfare since not all good agents are working. Then assume that $p,\underline{q}\rightarrow c$. Fix any path of the expected mean for the agent. Let $t_c^*=\hbox {inf}\{t|\mu _t \le p;p=c\}$ be the stopping time of this path in the limit as the price approaches $c$. In order to achieve first best as $k\rightarrow 0$, we must have $t_c^*=0$ or else bad agents will be working for some stretch of time. But for any admissible information process this is impossible since $\mu _0 >p$. Thus, $t_c^*$ is strictly above $0$, and so delayed certification cannot achieve first best. $\square$

1.9 Proof of Proposition 4

From the proof of Theorem 6, we see that immediate certification and no certification cannot asymptotically achieve first best as the information speed increases, because the speed of the reputational mechanism does not affect social welfare in either case. So we must show that the social welfare of a delayed certification scheme approaches first best. We wish to show that as the speed becomes very high, $t_c^*\left( q \right) \rightarrow 0\,\forall q<c$, because doing so means that all agents who have socially inefficient qualities will be kicked out extremely quickly, and the good agents will be able to stay in forever (perhaps paying the certification cost that asymptotically approaches 0). Since the process is fully revealing, almost surely $t_c^*(q)<\infty$ for agents with quality $q<c$. Then as $n$ gets large, agents will be kicked out at time $\frac{t_c^*\left( q \right) }{n}$ instead, which approaches $0$ for all finite $t_c^*(q)$. Thus, $t_c^*\left( q \right) \rightarrow 0\,\forall q<c$ and so delayed certification asymptotically achieves the first best outcome. $\square$