Silviase/jgov_v1.4_full_ann huggingface dataset:
in the following example, this subsidy policy seems to be fairly effective as a pump-priming mechanism for production.

The next proposition specifically shows that without condition (RC) not only $ s \equiv 0 $ but also another unnatural solution is allowed in the case where y and c are linear.

Proposition 3: Suppose that y and c are linear functions such that $ 0 < \frac{c'}{y'} < \frac{5}{24} $ and $ \inf_{\theta \in (0,1)} \frac{2c(\theta)}{y(\theta)} < \varphi(-2 + \sqrt{9 - 24c'/y'}) $ , where $ \varphi(x) = \frac{3}{4} - \frac{1}{2}x - \frac{1}{4}x^2 $ . Then, there exist $ k_1, k_2 $ ( $ 0 < k_1 < k_2 < 1 $ ) such that $ s = 1_{(k_1, k_2)} $ is an optimal strategy for a sufficiently small $ \sigma > 0 $ .

The following numerical example indicates that $ k(\sigma) $ is much larger than $ \theta $ , which means that even the productive equilibrium is inefficient. This is because the region of fundamentals, where a positive payoff to production is possible but all individuals choose not to produce, is not small, even in the productive equilibrium. Further, we can see how effective the subsidy in Proposition 2 is as a pump-priming policy.

Example1: For $ y(\theta)=4\theta, c(\theta)=\frac{1}{2}\theta+\frac{1}{4}, \theta=\frac{1}{14}\approx0.07, \sigma_{0}=\frac{1}{28}\approx0.04 $ . Then, by taking $ \sigma=\frac{1}{32}\approx0.03 $ , we have $ k(\sigma)=\frac{11}{72}\approx0.15 $ and $ \overline{W}\approx1.49 $ for Proposition1.

Denoting the amount of total subsidies by $ \bar{S} $ , we have $ \bar{S} \approx 0.02 $ and $ \frac{\bar{W}}{\bar{S}} \approx 64.27 $ for $ \sigma = \frac{1}{32} \approx 0.03 $ and $ \lambda = 0 $ for Proposition 2.

Further, it holds for Proposition 3 that

$$ \inf_{\theta\in(0,1)}\frac{2c(\theta)}{y(\theta)}=\frac{3}{8}\approx0.34\quad,\quad\varphi(-2+\sqrt{6})=-\frac{3}{4}+\frac{1}{2}\sqrt{6}\approx0.47\quad,\quad k_{1}=\frac{1}{4\sqrt{6}-8}\approx0.56 $$
and $ k_{2} - k_{1} = (1 - A^{*})\sigma = (3 - \sqrt{6})\sigma \approx 0.02 $ .

## 4 Game Under Imperfect Information with

## Unsophisticated Players

In this section we extend the model in two ways. One extension is to generalize the distribution of $ \theta $ and $ \varepsilon $ , which enables us to fit our model to a broader range of situations. In particular, by changing the distribution of $ \theta $ we can represent various situations in which the priors for the state of fundamentals, i.e., the prevailing sensations for the fundamentals, are different. It is useful to analyze the impact of the prior on the equilibrium of the game.

The other is to introduce two types of individuals who choose whether to produce without considering the choice of other individuals. It is up to the reader if she calls these players more "realistic". In the basic model all individuals are identical and optimize the choice in a Bayesian manner, in which they compute the expected payoff to production based on private signals, taking the choices of others into account. We call these individuals type B (Bayesian). As an extension of the basic model, first we consider the situation in which a proportion of individuals make decisions simply based on the prevailing sensation for the state of fundamentals. This characterization implies that the individuals are not analytic, but naive, under the strong influence of the atmosphere prevailing in the economy. We call that type of individual type S (Sensation).

Next, we consider the case in which a different kind of unsophisticated individual, who we call type N (Noise trader), coexists with type B individuals. Type N people falsely and always take the signals at their face values and make decisions based only on what the signals just told them. They never take account the strategic interactions of players like type S, and they neglect any prior unlike type S because they completely believe that
they know the true value of $ \theta $ based on the signals, somewhat like the noise traders in De Long et al. (1990).

In the previous section, we have shown that if all individuals are Bayesian, generally multiple equilibria may emerge, and productive equilibrium as well as unproductive equilibrium may appear. In this section we show that in the case in which type S-individuals are introduced, if the prevailing atmosphere for fundamentals is optimistic, only productive equilibrium appears, as long as the proportion of type S is not so small. Further, in the case in which type N individuals are introduced, the appearance of productive equilibrium is guaranteed independently of the atmosphere as long as the proportion of type N stays within some range.

In any case, the players who play the Bayesian equilibrium game know what unsophisticated players do. It is interesting to note that the existence of unsophisticated players does not necessarily make the situation more difficult. In our cases, unsophisticated players facilitate the fulfillment of condition (RC) for rational players. Then this fulfillment of the local condition at the extreme value of fundamentals brings about only the productive equilibrium of production in a large region. Thus bounded rationality, if not ignorance, may well serve the stability of an economic system with coordination problems.

### 4.1 Structure and Assumption

Now we suppose that the state $ \theta $ is a random variable on $ [0,1] $ , of which distribution function is denoted by G, which has a density g that is strictly positive and continuously differentiable. We denote by $ \theta_{m}^{G} $ the expectation value of $ \theta $ by G. The noise term $ \varepsilon $ is distributed on $ [-1,1] $ with distribution function F, which also has a continuous density f and of which support is assumed to be not contained in $ [-1,0] $ . As in the previous section, $ \varepsilon $ is independent of $ \theta $ and independent across individuals. Further, there is some critical value $ \theta_{T} \in (0,1) $ such that the state of fundamentals is regarded as productive (unproductive) when the true value
of $ \theta $ exceeds (resp. does not exceed) $ \theta_{T} $ . The prevailing sensation for economic conditions is reflected on the prior G, and we say that the atmosphere is optimistic (pessimistic) when $ \theta_{m}^{G} $ exceeds (resp. does not exceed) $ \theta_{T} $ .

We consider two cases in which sophisticated and unsophisticated individuals coexist in the economy: Case 1 assumes that type B (Bayesian) and type S (Sensational) coexist, while Case 2 assumes that type B and type N (Noise trader) coexist. In both cases we assume that the ratio of type B is $ \rho $ and that of type S and type N is $ 1-\rho $ for $ \rho\in(0,1] $ .

In Case 1, individuals of type S decide whether to produce depending only on the prevailing atmosphere, i.e., they choose to produce if the atmosphere is optimistic, and otherwise choose not to produce. Meanwhile, in Case 2, individuals of type N do it based only on what their signals just told them, i.e., those who observe the signal x, falsely believe that the true value of $ \theta $ is x, and they choose to produce if $ x > \theta_{T} $ and otherwise choose not to produce. On the other hand, individuals of type B choose to produce based on the Bayesian computation of the expected payoff, knowing how unsophisticated individuals make decisions and taking into account the strategic interactions of players. In the above, individuals of type B know that $ G, F, \theta_{T} $ and $ \rho $ , but individuals of type S and type N know only G and $ \theta_{T} $ .

In the extended model the game proceeds according to the following steps:

1. Nature selects a state $ \theta $ according to G.

2. Each individual is given a private signal x.

3. In Case 1, an individual of type S whose signal is x chooses to produce if the atmosphere is optimistic and otherwise chooses not to produce, irrespective of the signal x. In Case 2, an individual of type N whose signal is x choose to produce if $ x > \theta_{T} $ and otherwise chooses not to produce. In the both cases an individual of type B whose signal is x decides whether to produce based on the expected payoff conditional on x, knowing how
individuals of type S and N make decisions and considering the strategic interactions of players.

4. Each individual carries out his decision and obtains the payoff.

The choice of individuals of type S (type N) are exogenously determined by G and $ \theta_{T} $ (resp. $ F, G $ and $ \theta_{T} $ ), and only individuals of type B are the Bayesian players of the game. Knowing this, individuals of type B choose whether to produce based on the signal x. The aggregated strategy of type B is represented by a [0,1]-valued measurable function s as in the basic model, and an equilibrium of the game in the extended model is also defined as in the basic model.

Definition 3': An unproductive equilibrium is a profile in which each individual of type B always chooses not to produce.

Definition 4: A productive equilibrium is a profile that represents an optimal strategy, in which each individual of type B chooses to produce if and only if his private signal is above some $ k \in (0,1) $ .

Production function $ y(\theta) $ and cost function $ c(\theta) $ are continuously differentiable and satisfy the following conditions, which are also met by $ y(\theta) = 4\theta $ , $ c(\theta) = \frac{1}{4} (2\theta + 1) $ if F is uniform. As will be seen in Proposition 4.

and Proposition 5, $ \rho $ gives the lower bound of the proportion of type B for

the existence of the productive equilibrium to be ensured in the case where individuals of type S or type N choose not to produce. Meanwhile, $ 1-\bar{\rho} $ gives that of type S and type N for the productive equilibrium to appear as a unique equilibrium in the case where individuals of type S or type N choose to produce.

$$ A1.\ c(0)>y(0)>0. $$

$$ A2．y(1)>\frac{2c(1)}{(1-F(0))}>0． $$

$$ \mathrm{A}3^{\prime}.y^{\prime}(\theta)>2c^{\prime}(\theta)>0,\forall\theta\in[0,1]. $$
$$ A4^{\prime \prime}.\underline{\rho}<\overline{\rho},where\underline{\rho}=(\max_{0\leq\theta\leq1}\frac{2c^{\prime}(\theta)}{y^{\prime}(\theta)})\vee(\frac{2c(1)}{y(1)(1-F(0))})\ and\overline{\rho}=1-\frac{c(1)}{y(1)}. $$

### 4.2 Results

Here we present results for Cases 1 and 2. In both cases, multiple equilibria can be eliminated for productive equilibrium since unsophisticated players (individuals of type S and type N) facilitate the fulfillment of local conditions (RC) for rational players (individuals of type B). We start with the result for Case 1, where individuals of type B and type S coexist.

Proposition 4: In the case where individuals of type B and type S coexist, when the atmosphere is optimistic (pessimistic), for any $ \rho\in(0,1] $ (resp. $ \rho\in(\rho,1] $ ) and any sufficiently small $ \sigma>0 $ , there exists

$ k(1,\sigma,\rho) $ (resp. $ k(0,\sigma,\rho) $ ) such that $ s = 1_{(k(1,\sigma,\rho),1]} $ (resp. $ s = 1_{(k(0,\sigma,\rho),1]} $ ) gives the productive equilibrium of the game.

In particular, $ s = 1_{(k(1, \sigma, \rho), 1]} $ gives a unique equilibrium if and only if the atmosphere is optimistic and $ 0 < \rho < \overline{\rho} $ . While s = 0 gives also an equilibrium as well as $ s = 1_{(k(0, \sigma, \rho), 1]} $ if the atmosphere is pessimistic and $ \rho < \rho \leq 1 $ .

Remarks: Proposition 4 implies the following. Under both pessimistic and optimistic atmospheres, if the proportion of type B is sufficiently large, productive equilibrium—where all individuals of type B whose signal is above some threshold value choose to produce—may exist.

However, when the elimination of multiple equilibria for the productive equilibrium is guaranteed, it is restricted to the case in which the atmosphere is optimistic and the proportion of type N is above $ 1-\bar{\rho} $ . Therefore, this seems more likely to happen, for instance in the situation in
which individuals of type B become type S and choose to produce without considering the choices of other individuals in an optimistic atmosphere.

For obtaining only productive equilibrium in other cases, some policy treatment is needed like the production subsidy explained in the previous section.

The next corollary is proved as Corollary 1 and useful for the computation of $ \lim_{\sigma\to0}k(1,\sigma,\rho) $ and $ \lim_{\sigma\to0}k(0,\sigma,\rho) $ .

Corollary 2: $ \lim_{\sigma\to0}k(1,\sigma,\rho)\left(\lim_{\sigma\to0}k(0,\sigma,\rho)\right) $ is given by the solution of $ (1-\frac{\rho}{2})y(x)-c(x)=0 $ (resp. $ \frac{\rho}{2}y(x)-c(x)=0 $ ).

Next, we turn to the result for Case 2, in which individuals of type B and type S coexist. In this case an individual of type B knows that all individuals of type N choose to produce when his signal x is more than $ \theta_{T} + 2\sigma $ and choose not to produce when his signal x is less than $ \theta_{T} - 2\sigma $ . Based on this fact, Proposition 5 is also proved by making use of lemmas ready for Proposition 4 and stated in terms of $ k(1, \sigma, \rho) $ and $ k(0, \sigma, \rho) $ in Proposition 4.

Proposition 5: In the case where individuals of type B and type N coexist, for any sufficiently small $ \sigma > 0 $ and any $ (\rho, \theta_{T}) \in (\underline{\rho}, 1] \times (0, 1) $ such that either $ \theta_{T} < \lim_{\sigma \to +0} k(1, \sigma, \rho) $ or $ \lim_{\sigma \to +0} k(0, \sigma, \rho) < \theta_{T} $ holds, there exists $ k(\sigma, \rho) $ with $ k(1, \sigma, \rho) < k(\sigma, \rho) < k(0, \sigma, \rho) $ such that $ s = 1_{(k(\sigma, \rho), 1]} $ gives the productive equilibrium. In particular, $ s = 1_{(k(\sigma, \rho), 1]} $ gives a unique equilibrium if such $ \rho $ is contained in $ (\rho, \overline{\rho}) $ .
Remarks: It seems that $ \lim_{\sigma\to0}k(0,\sigma,\rho)<\theta_{T} $ usually holds, as will be indicated in the following example. Therefore, roughly speaking, the proposition says that the elimination of multiple equilibria for productive equilibrium is guaranteed only by the condition that the ratio of type N stays within $ (1-\overline{\rho},1-\underline{\rho}) $ . An optimistic atmosphere is no longer needed, in contrast to Case 1. Because when an individual of type B observes the signal equal to unity, he knows that all individuals of type N surely choose to produce, which implies condition (RC).

Therefore, it is easy to see for the same reason that Proposition 5 holds even for the case where individuals of type N make decisions based on the posterior distribution of $ \theta $ conditional on the signals as long as they neglect the strategic interactions of individuals, although in the present case individuals of type N simply always take the signals always at their face values.

Finally, we give an example for Cases 1 and 2. The example indicates that $ (\rho,\bar{\rho}) $ , the period of the proportion of type B where only the productive

equilibrium can appear in Case 2, is rather restricted. On the other hand, in Case 1 the atmosphere must be optimistic for obtaining only productive equilibrium. Therefore, the elimination of multiple equilibria for productive equilibrium does not seem so easy in general despite the presence of type S and N.

Example 2: For $ y(\theta) = 4\theta, c(\theta) = \frac{1}{4}(2\theta + 1), dF(z) = \frac{1}{2}dz, \theta_T = \frac{1}{2} $ , it holds that

$ \rho = \frac{1}{4} \vee \frac{3}{4} = \frac{3}{4} $ and $ \bar{\rho} = \frac{13}{16} \approx 0.81 $ . By taking $ \rho = \frac{25}{32} $ and $ \sigma \downarrow 0 $ , we have $ \lim_{\sigma \to 0^+} k(0, \sigma, \rho) = \frac{4}{17} \approx 0.24, \lim_{\sigma \to 0^+} k(1, \sigma, \rho) = \frac{4}{31} \approx 0.13 $ , which shows
$$ \lim_{\sigma \to 0^+} k(0, \sigma, \rho) < \theta_T $$

Hence by Proposition 4, for any G such that

$ \theta_{m}^{G}>\frac{1}{2} $ as $ g(\theta)=\frac{1}{K}\exp(-\frac{(\theta-3/5)^{2}}{200}) $ with $ K=\int_{0}^{1}\exp(-\frac{(\theta-3/5)^{2}}{200})d\theta $ , the limit of the productive equilibrium s is given by $ s=1_{(\frac{4}{31},1]} $ , which gives the limit of a unique equilibrium for G and $ \rho=\frac{25}{32} $ . But, if G is such that $ \theta_{m}^{G}\leq\frac{1}{2} $ as $ g(\theta)=1 $ , the set of the limit of the equilibria contains the unproductive equilibrium as well as the productive one given by $ s=1_{(\frac{4}{17},1]} $ .

Further, by Proposition 5, for any sufficiently small $ \sigma > 0 $ , there exists $ k(\sigma, \rho) $ with $ k(1, \sigma, \rho) < k(\sigma, \rho) < k(0, \sigma, \rho) $ such that $ s = 1_{(k(\sigma, \rho), 1]} $ gives a productive and unique equilibrium for Case 2.

## 5 Concluding Remarks

In this paper, we have asked the question how multiple equilibria can be eliminated in a simplified barter economy model with imperfect information. We have found that in general multiple equilibria cannot be eliminated. A sufficient condition for the elimination of multiple equilibria and realization of the equilibrium with production is that each individual, who observes the signal about the state of the highest value of fundamentals, never fails to produce. Furthermore, we have shown that multiple equilibria can be eliminated for productive equilibrium in the presence of unsophisticated individuals. Curiously, it seems to indicate the possibility that bounded rationality may well serve rather than disrupt the stability of an economic system that possesses coordination problems.
Our results have the following implication to the nature of multiple equilibria in a macroeconomy. Equilibria are represented as response functions of individuals to the noisy signal about the state of fundamentals. Productive equilibrium implies that all rational individuals produce in the upper region exceeding a critical level of the states of fundamentals. This feature is brought about by the application of the method of Morris and Shin (1998, 2001).

These results in a standard model are based on two fundamental assumptions of the models. One is the assumption of imperfect information, i.e., all individuals do not know the true value of the state of fundamentals and are given the noisy private signals about the state. The other is that all the rational players behave in the Bayesian manner, i.e., they optimize their choices conditional on the signal, while unsophisticated individuals make decisions in irrational ways. These two assumptions make the response to one signal interdependent with that to another signal, and equilibria are obtained as response functions. This mechanism never disappears however infinitesimal the amount of noise is, nor if one introduces, not strictly the Bayesian, unsophisticated players into the model. The coexistence of unsophisticated players, however, may mitigate some difficulties.

## A Proofs

### A.1 Proofs for Section 3

The proof proceeds in the same way as in Morris and Shin (1998). For a measurable set $ B \subset [0,1] $ , we denote by $ I_{s} $ the indicator function of B. Then we have the following lemmas.

Lemma 1: $ u(k,I_{(k,1)}) $ is continuous on $ (0,1) $ and has a unique zero point
$ k(\sigma) $ in (0,1) for any $ \sigma\in(0,\sigma_{0}) $ , where $ \sigma_{0}=[\sup\{\sigma>0\}|y(1-2\sigma)>c(1)\}]\wedge\frac{\theta}{2} $

Proof of lemma 1. By the definition of signal x, $ u(k, I_{(k,1)}) $ is obviously continuous on $ (0,1) $ . For the proof we show that $ u(k, I_{(k,1)}) $ is strictly increasing on $ (\sigma, 1-\sigma) $ and that $ u(k, I_{(k,1)}) < 0 $ for $ k \in [0, \sigma] $ and $ u(k, I_{(k,1)}) > 0 $ for $ k \in [1-\sigma, 1) $ .

First, by using the concavity and convexity of y, c, for $ k \in (\sigma, 1 - \sigma) $ we have

$$ \begin{align*}\frac{d}{dk}u(k,I_{(k,1]})&=\frac{1}{2\sigma}[y(k+\sigma)-\frac{1}{2\sigma}\int_{k-\sigma}^{k+\sigma}y(\theta)d\theta-(c(k+\sigma)-c(k-\sigma))]\\&\geq\frac{1}{2\sigma}[y(k+\sigma)-y(k)-2(c(k+\sigma)-c(k))]\\&=\frac{1}{2\sigma}\int_{k}^{k+\sigma}[y^{\prime}(\theta)-2c^{\prime}(\theta)]d\theta>0,\end{align*} $$

which shows $ u(k,I_{k}) $ is strictly increasing on $ (\sigma,1-\sigma) $ .

Next, since $ \pi(l,\theta)<0 $ for all $ (l,\theta)\in[0,1]\times[0,\theta) $ by the assumption, clearly $ u(k,I_{(k,1)})<0 $ for $ k\in[0,\sigma] $ . As for $ k\in[1-\sigma,1) $

$$ \begin{align*}u(k,k)=&\int_{k-\sigma}^{1}\pi\left(1-F(\frac{k-\theta}{\sigma}),\theta\right)\frac{d\theta}{1-k+\sigma}=\int_{\frac{k-1}{\sigma}}^{1}\pi\left(1-F(z),k-\sigma z\right)\frac{\sigma dz}{1-k+\sigma}\\ >&y(k-\sigma)\int_{\frac{k-1}{\sigma}}^{1}(1-F(z))\frac{\sigma dz}{1-k+\sigma}-c(1).\end{align*} $$

Changing variables to $ \frac{k-1}{\sigma}=\eta(-1\leq\eta<0) $ , we have

$$ \int_{\frac{k-1}{\sigma}}^{1}(1-F(z))\frac{\sigma dz}{1-k+\sigma}=\int_{\eta}^{1}(1-F(z))\frac{\sigma dz}{\sigma\left(1-\eta\right)}=\frac{1}{1-\eta}\int_{\eta}^{1}\frac{1-z}{2}dz=\frac{1-\eta}{4}>\frac{1}{4}. $$
Thus, for $ 1 - \sigma \leq k < 1 $ , we have

$$ \begin{align*}u(k,k)>\frac{y(k-\sigma)}{4}-c(1)\geq\frac{y(1-2\sigma)}{4}-c(1)\geq0\\ if0<\sigma\leq\sigma_{0}where\sigma_{0}=\sup\{\sigma>0\mid y(1-2\sigma)>c(1)\}.\end{align*} $$

Lemma 2: Under condition (RC), $ s = I_{(k(\sigma),1]} $ is a unique optimal strategy for individuals.

Proof of lemma 2. For any optimal strategy s for individuals, set $ \bar{x}=\sup\{00,\end{align*} $$

which implies $ \bar{x} < 1 $ .

When $ s(x)<1 $ , there are some individuals who do not produce, which means $ u(x,s)\leq0 $ . From this we have $ \lim_{x\to\bar{x}}u(x,s)\leq0 $ .

Clearly $ \lim_{x\to\bar{x}}u(x,s) $ is increasing in s, i.e., $ \lim_{x\to\bar{x}}u(x,s_1)\geq\lim_{x\to\bar{x}}u(x,s_2) $ for any pair of measurable functions such that $ s_{1}(\theta)\geq s_{2}(\theta),\forall\theta\in[0,1] $ . So, by $ I_{(\bar{x},1]}(\theta)\leq s(\theta),\forall\theta\in[0,1] $ we have $ \lim_{x\to\bar{x}}u(x,I_{(\bar{x},1]})\leq\lim_{x\to\bar{x}}u(x,s)\leq0 $ , which shows $ \bar{x}\leq k(\sigma) $ .
By the symmetric argument, we have $ x \geq k(\sigma) $ . Therefore,

$ \bar{x}=x=k(\sigma) $ and $ s=1_{(k(\sigma),1]} $ .

Proof of proposition 1. Since $ l(\theta, I_{(k(\sigma),1]}) = \frac{1}{2\sigma} \int_{\theta - \sigma}^{\theta + \sigma} I_{(k(\sigma),1]}(\theta) d\theta $ and $ \pi(l, \theta) = l y(\theta) - c(\theta) $ , (i) and (iii) immediately follow. By assumption, there exist $ \beta_{1}, \beta_{2} \in (0,1) $ such that $ \beta_{1} \leq \frac{c(\theta)}{y(\theta)} \leq \beta_{2}, k(\sigma) - \sigma \leq \theta \leq k(\sigma) + \sigma $ . So by the continuity of $ l(\theta, I_{(k(\sigma),1]}) $ and $ \frac{c(\theta)}{y(\theta)} $ , $ l(\theta, I_{(k(\sigma),1]}) $ crosses $ \frac{c(\theta)}{y(\theta)} $ from below only once at $ \theta^{*} \in (k(\sigma) - \sigma, k(\sigma) + \sigma) $ , which implies (ii).

Proof of proposition 3. For $ k_{1}
Thus, $ u(k_{2}, I_{(k_{1}, k_{2})}) - u(k_{1}, I_{(k_{1}, k_{2})}) = 0 $ has solution $ A^{*} = -2 + \sqrt{9 - 24c'(1)} / y'(1) $ in (0,1) when $ \frac{c'}{y'} < \frac{5}{24} $ .

Next, we observe that $ u(k_{1}, I_{(k_{1}, k_{2})}) $ converges to

$$ \begin{align*}&\frac{1}{2}\int_{A}^{1}{\ d}z[\frac{1-z}{2}y(k_{1})-c(k_{1})]+\frac{1}{2}\int_{A}^{1}{\ d}z[\frac{1-A}{2}y(k_{1})-c(k_{1})]\\=&\frac{1}{2}[\varphi(A)y(k_{1})-2c(k_{1})],\end{align*} $$

uniformly in $ k_{1} $ and A as $ \sigma $ goes to zero. Under assumption $ \inf_{\theta\in(0,1)}\frac{2c(\theta)}{y(\theta)}<\varphi(-2+\sqrt{9-24c'/y'}) $ , clearly $ \varphi(A)y(k_{1})-2c(k_{1})=0 $ has a solution $ k_{1}^{*}\in(\theta,1) $ by the continuity of y,c. Therefore, for a sufficiently small $ \sigma>0 $ , there exist $ k_{1},k_{2} $ ( $ 00,\quad\lim_{x\to k_{2}-0}\frac{d}{dx}u(x,I_{(k_{1},k_{2})})<0 $ and $ \frac{d^{2}}{dx^{2}}u(x,I_{(k_{1},k_{2})})<0,k_{1} 0 $ for $ k_{1} < x < k $ and the proof is completed. ☐

### A.2 Proofs for Section 4

The optimal strategy for type B in Case 1 is obtained in the same way as in section 2. We denote by $ u(x,s,\alpha,\sigma) $ the expected payoff to production of an individual of type B observing signal x when the strategy for type B is s and the proportion of type-S individuals who choose to produce is $ \alpha $ ( $ \alpha = 0,1 $ ).

Lemma 3: When $ \alpha=0 $ ( $ \alpha=1 $ ), for any $ \delta>0 $ and $ \rho_{1}\in\left(\max_{0\leq\theta\leq1}\frac{2c'(\theta)}{y'(\theta)},1\right] $ (resp.