# Regulating a firm with bilateral unknown demands

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## Abstract

We consider the impact of bilateral unknown information about the market capacity on the optimal regulatory policies in a regulation problem. We first analyze how to solve such problem when the market capacity is full information, a case which allows us to obtain most of key insights from regulatory models. We then extend the analysis to the bilateral unknown information case. We do this by assuming that neither the regulator nor the firm knows exactly the true market capacity, but they can make respective estimates about it, the same or different. The results show that when the regulator and the firm make the same estimate about the true market capacity, the optimal price is distorted downwards from that under full information, and the transfer payment is distorted upwards from that under full information, but bilateral unknown information does not necessarily result in the distortion of the firm’s output. When the regulator and the firm make the different estimates about the true market capacity, and if the firm’s estimate is more optimistic than the regulator’s, the optimal price and output are more than those under the case that they have the same estimate. Contrary to this case, if the firm’s estimate is more pessimistic than the regulator’s, the optimal price and output are less than those under the case that they have the same estimate.

## Keywords

Uncertainty modeling Pricing Regulation Bilateral unknown information## 1 Introduction

In order to keep the regulated firm from abusing her monopoly power, the price regulation problem arises. If the regulator knows the products’ demand, the optimal regulatory policy is that the firm should follow marginal cost pricing and be subsidized for the fixed cost. However, in reality, the market capacity about the products, which plays a heavy impact on the demand, is usually unknown to the market participants (Yang and Ng 2014). That is to say, the products’ demand is bilateral unknown information for the regulator and the firm. For example, with respect to a R&D project, the demand of the new developed products is uncertain and subject to many uncertainties, e.g., the consumer’s preference for the new products, substitute products’ impacts and so on. Therefore, neither the regulator nor the firm can exactly know the true market capacity. Nevertheless, the regulator and the firm can only make their respective estimates about the market capacity depending on their own experience. Consequently, considering the regulation problem under bilateral unknown information with different estimates about the market capacity is necessary and challenged.

Despite this kind of situations are widespread in reality, the literature has devoted much less attention. Our paper intends to fill this gap by exploring bilateral unknown information’s impact on the optimal regulatory policies, and meanwhile, investigating the influence of different estimates about the market capacity from the regulator and the firm on the optimal regulatory policies.

To do so, we develop a model in which the regulator (he) is a policy designer, who proposes a policy menu consisting of unit price of products, firm’s output, and transfer payment for the firm (she). To investigate the influence of the market capacity on the optimal regulatory policies under different information structures, we limit our attention to the following two cases: one is that the products’ market capacity is full information; The other is that the products’ market capacity is bilateral unknown information, that is to say, neither the regulator nor the firm knows the true market capacity, but they have their respective estimates about it.

Particularly, our analysis begins with the case that the market capacity is known to all, which we call the full information case. Under full information, the regulator seeks to maximize the social total surplus subject to gaining the firm’s participating. And then we extend the analysis to the bilateral unknown information case, assuming that the regulator and the firm make their respective estimates about the true market capacity, and also assuming that the firm will share her estimate to the regulator. Consequently, when the regulator designs the regulatory policy in order to maximize the social total surplus, he only needs to guarantee the firm’s participating. We firstly analyze the optimal regulatory policy when the regulator and the firm make the same estimate about the market capacity. After that, we further explore the optimal regulatory policy when the regulator and the firm make the different estimates about the market capacity. Moreover, we analyze how different estimates from the regulator and the firm affect the optimal regulatory policies by a key comparison between the optimal regulatory policy under the same estimate and that under different estimates. Via the derived results, we establish the following main findings.

In the first place, when the regulator and the firm make the same estimate about the true market capacity, the optimal price is distorted downwards from that under full information, and the transfer payment is distorted upwards from that under full information, but bilateral unknown information does not necessarily result in the distortion of the firm’s output. In particular, even though both the regulator and the firm do not know the true market capacity, the regulator can mandate the firm to produce the products quantity depending on that under full information. In the second place, when the regulator and the firm make the different estimates about the true market capacity depending on their own experience, their different estimates play an important role on the optimal regulatory policy design. Specially, if the firm’s estimate is more optimistic than the regulator’s, the optimal price and the optimal output are more than those under the case that they have the same estimate. Contrary to this case, if the firm’s estimate is more pessimistic than the regulator’s, the optimal price and the optimal output are less than those under the case that they have the same estimate.

Our paper is related to two streams of literature. The first stream focuses on the regulatory policy design under the case that the products’ demand is full information for the regulator and the firm. The second stream investigates the regulatory policy design when the regulator does not know exactly the products’ demand, that is, the products’ demand is the firm’s private information.

The first stream of literature focuses on the regulatory policy design when the products’ demand is full information for the regulator and the firm. Baron and Myerson (1982) study a regulation problem that the demand is common knowledge but the firm’s cost is private information. The optimal regulatory policy indicates that the marginal cost pricing is no longer optimal. In this case, the firm is able to extract information rents, which results in the optimal price distorted upwards for all except the most efficient firm, but the optimal output distorted downwards from the socially optimal level. The interested readers may refer to Sappington (1983), Baron and Besanko (1984), Laffont and Tirole (1986), Sappington and Sibley (1988), Laffont and Tirole (1993), Laffont and Rochet (1998), Feng et al. (2017) for further analysis. And the readers are referred to Armstrong and Sappington (2005) for a comprehensive review of this literature. As comprehensive as it is, these papers limit themselves to analyzing the effect of the firm’s private cost information on the regulator’s optimal regulation policy, do not address bilateral unknown information issues about the demand, which has been identified as a limitation. Therefore, we extend the current regulation literature by considering the environment where the demand of the products is unknown to both the regulator and the firm and highlighting the different impact of bilateral unknown information on the optimal regulatory policy, which leads to results which are quite different from those found in the literature under full information about the demand.

The second stream analyzes the regulatory policy design when the products’ demand is the firm’s private information. The results show that the optimal regulatory polices are different from those in the case of full information. For example, Lewis and Sappington (1988a) analyze that the firm has private information about the demand but the firm’s cost is full information. They show that the optimal regulatory policy is sensitive to the firm’s cost and analyze the nature of the demand uncertainty. Similarly, Riordan (1984), Aguirre (2004) also study the case that the demand is the firm’s private information and the firm’s marginal cost is constant. Riordan (1984) introduces an endogenous capacity level and constructs a mechanism that induces the firm to choose socially optimal price. Aguirre (2004) introduces a cost of public funds into the regulator’s objective function and finds that the costly public funds leads to similar qualitative results to those obtained under private cost information. Some research focus on the applications of regulation theory, such as Georges and David (2012), Muthuraman et al. (2008), Resende (2008), Thanassoulis (2000), Botero et al. (2004). Moreover, Babich et al. (2012), Li and Zhang (2015), Wu et al. (2017) address supply chain contract design when the products’ demand is the firm’s private information. Our paper is most related to Lewis and Sappington (1988a), whose model considers the impact of the firm’s private information about the demand on the optimal regulatory, but differs from Lewis and Sappington (1988a) along two essential dimensions: 1) By comparing the differences of the optimal regulatory policies between under full information and under bilateral unknown information, our paper highlights the impact of bilateral unknown information about the demand on the optimal regulatory. 2) More importantly, since the regulator and the firm might possess different estimates about the demand, we also analyze the effect of different estimates about the demand on the optimal regulatory policy.

The paper is organized as follows. Section 2 gives the problem formulation. Section 3 analyzes the problem and derives the optimal regulatory policy when the market capacity is full information. Section 4 deals with the case that the market capacity is bilateral unknown information and investigates the optimal regulatory policies under the cases that the regulator and the firm have the same estimate and different estimates about the true market capacity, respectively. Section 5 gives a numerical example. Finally, Section 6 summarizes the findings of the paper. The proofs of all the formal results are relegated to an A.

## 2 Problem formulation

Consider a regulation problem, in which there are three participants: a regulator (he), a firm (she) and the consumer (it), among them, the firm produces a kind of products and then sells them to the consumer, the consumer gives payment for the products at a given price and a transfer payment. Intended for maximizing the social total surplus, the regulator designs a regulatory policy, denoted by a vector (*p*,*q*,*t*), where *p* is the unit price for the products, *q* denotes the firm’s output, *t* is the transfer payment paid by the consumer to the firm.

Let *C*(*q*) denote the firm’s cost function, which is known to the regulator. Moreover, assume that *C*(*q*) is continuous, increasing and convex, which can be thought of as having increasingly more expensive production capacity or input (e.g., supplied materials). Such an assumption is commonly used in economics and marketing literatures (e.g., Kalai et al. 1992; Porteus 2002, Ha et al. 2011; Laffont and Mortimort 2002; Zeithammer and Thomadsen 2013). It is supported by empirical evidence in industries such as petroleum refining (Griffin 1972) and auto-making (Mollick 2004). The firm possessing cost function like this may be a natural monopoly, for example, the energy industries (water, electricity, coal etc.). Consumer demand follows a linear, downward sloping demand function, *Q*(*p*,*Y* ) = *Y* − *b**p*, where *Y* represents the market capacity, note that when *p* = 0, the demand equals to the market capacity. The parameter *b* measures a demand sensitivity in response to the price change. Moreover, assume that \(p\leq \frac {Y}{b}\), ensuring that the demand is nonnegative. This demand function is the most popular form used in the related literature, e.g., Ertek and Griffin (2002), Kurata et al. (2007), Yao et al. (2008) and Lau et al. (2006).

*V*(⋅) means that, the consumer’s final purchasing quantity is the minimum value between supply and demand. The transfer payment

*t*may be thought of as the fixed fee of a two-part tariff. If

*t*is positive, the firm would get a subsidy; If

*t*is negative, then it represents a tax on the firm.

*α*≤ 1, which means that the regulator pays more attention to the consumer surplus than the firm’s profit. Note that the social total surplus function (3) is commonly used in regulation literature (Baron and Myerson 1982 and Armstrong and Sappington 2004).

Since the regulatory policy is strongly influenced by the market capacity, the following two cases are considered as follows: one case is that the market capacity is known to both the regulator and the firm, which is called full information. The other case is neither the regulator and the firm knows the true market capacity, which is called bilateral unknown information, in which we consider two scenarios that the regulator and the firm make the same estimate and different estimates about the true market capacity, respectively.

## 3 Regulatory policy under full information

To begin with, we solve the problem under full information, and the first-best policy is served as a benchmark to the case with bilateral unknown information. Under full information, both the regulator and the firm know the true market capacity. For this case, assume that *Y* = *y*, representing the market capacity known. Note that so long as the optimal price is given, the optimal demand can be determined by *Q*(*p*,*y*) = *y* − *b**p*. After knowing the demand, in order to achieve the optimal social total surplus, the consumer demand must be satisfied, therefore, the regulator will mandate the firm to serve all products demand, i.e., *q* = *Q*(*p*,*y*). The regulator can ensure that the firm serves all demand at the regulated price. He does so by allowing the consumer to register grievances if it is refused to serve at the regulated price, and punishing the firm for such transgressions.

*p*,

*t*) to the firm, if the firm accepts the regulatory policy, then the firm produces the products to serve all demand. When the regulator designs the regulatory policy, he must take into account the firm’s participation constraint, that is to say, he must ensure the firm to obtain at least as great as her reservation profit. In this paper, we normalize her reservation profit as zero without loss of generality (Laffont and Rochet 1998; Özer and Raz 2011). Of course, this assumption is analytically convenient but not crucial to the analysis, we can establish similar results when the reservation profit is (strictly) positive. Therefore, to maximize the social total surplus, the regulator’s problem can be formulated as follows

### **Theorem 1**

*p*

^{∗},

*q*

^{∗},

*t*

^{∗}) satisfies

Theorem 1 shows that from the regulator’s perspective, the ideal outcome is to set the optimal price determined by the equation that marginal utility of the consumer is equal to marginal cost of the firm, and set the optimal transfer payment to compensate the firm’s cost, the above regulatory policy would leave the firm with her reservation profit. The reason why the regulator leaves the firm with reservation profit is that the regulator pays more attention to the consumer surplus. Therefore, under full information, the regulator allocates all surplus to the consumer.

## 4 Regulatory policy under bilateral unknown information

In this section, we consider the case that neither the regulator nor the firm knows the true market capacity, but they can make their respective estimates about it based on their own experience, which are represented by random variables *Y*_{1} and *Y*_{2}, respectively. Let *F*_{1}(*y*), *F*_{2}(*y*) and *f*_{1}(*y*), *f*_{2}(*y*) denote the probability distributions and the probability density functions of *Y*_{1} and *Y*_{2} on the support \([\underline {y},\overline {y}]\), respectively. And assume that the firm is willing to share her estimate to the regulator. After knowing the firm’s and his own estimates, the regulator opts to offer a regulatory policy (*p*,*q*,*t*). If the firm accepts the regulatory policy, then the firm produces the products and sells them to the consumer.

In order to reflect the impact of different estimates from the regulator and the firm on the optimal regulatory policy, we begin with investigating the case that the regulator and the firm have the same estimate about the true market capacity, and then focus on the case that the regulator and the firm make different estimates about it.

### 4.1 The same estimate

*t*until it binds, i.e.,

*E*[π(

*p*,

*q*,

*t*,

*Y*)] = 0. That is to say, under the same estimate, the firm’s expected profit is zero. Therefore,

*t*into the objective function of Model (11) yields

The following theorem establishes the optimal regulatory policy under bilateral unknown information when the regulator and the firm have the same estimate about the true market capacity.

### **Theorem 2**

According to Eq. 14, under bilateral unknown information, if the regulator and the firm have the same estimate about the true market capacity, then the optimal price policy only depends on the lowest market capacity \({\underline {y}}\), which specifies that the regulator and the firm have an implicit understatement of the market capacity. Such an understatement amounts to a claim that the regulator and the firm have a conservative attitude to the market capacity.

Equation 15 indicates that the regulator implements the optimal output policy resulting from the marginal utility of the consumer equal to the marginal cost of the firm. By comparing the optimal output under bilateral unknown information with the same estimate with that under full information, we can observe that the optimal output under bilateral unknown information with the same estimate is not distorted from that under full information. In other words, the regulator will implements the optimal output police under full information even if he has no exact information about the true market capacity. In addition, recall that, the optimal output under full information is equal to the consumer demand, but it is worth noting that the optimal output under full information still equals to the consumer demand under under bilateral unknown information with the same estimate.

- To interpret Eq. 16, we firstly observe Eq. 12, which implies, in the presence of bilateral unknown information, the transfer payment itself consists of two components: one component,
*C*(*q*) −*p**q*, represents the firm’s cost subsidy when the output can be sold totally; The other component, \(p{\int \limits }_{\underline {y}}^{q+bp}F(y){\mathrm {d}y}\), represents the residual products’ cost subsidy. This is always positive except for \({q}+{bp}={\underline {y}}\). Due to the presence of bilateral unknown information, the expected demand is less than the output which results in residual products, therefore, the firm must be compensated for addition residual products. However, note that Eq. 14 has shown that \(q_{1}^{*}+bp_{1}^{*}=\underline {y}\), as a result, the optimal transfer paymentThis means that there is no residual products under bilateral unknown information with the same estimate, which is realized by lowering price to stimulate the consumer demand, the reason for that is both the regulator and the firm have a conservative attitude to the market capacity.$$t_{1}^{*}=C(q_{1}^{*})-p_{1}^{*}q_{1}^{*}.$$

In the following proposition, we drive some important results by comparing the optimal regulatory policies under full information and bilateral unknown information with the same estimate.

### **Proposition 1**

*p*

^{∗},

*q*

^{∗},

*t*

^{∗}) and \((p_{1}^{*},q_{1}^{*},t_{1}^{*})\) be the optimal regulatory policies under full information and bilateral unknown information with the same estimate, respectively. Then we have

Equation 17 states that when neither the regulator nor the firm knows the true market capacity and they have the same estimate about it, the regulator’s optimal regulatory policy is to ask the firm to produce the products quantity depending on the optimal output *q*^{∗} under full information, which implies that bilateral unknown information with the same estimate has no impact on the optimal output.

Inequality (18) indicates that the optimal price under bilateral unknown information is lower than that under full information. The reason is that bilateral unknown information causes the products’ expected demand less than the firm’s output, in order to stimulate the demand such that the firm can have no residual products, the regulator should lower the price.

The intuition for Inequality (19) is as follows: the firm must be paid more transfer payment under bilateral unknown information with the same estimate than that under full information. To see why this may be optimal, observe that the outputs in two cases are identical, thus, the production costs are identical, but the price *p*^{∗} is above \(p_{1}^{*}\). Consequently, in order to ensure the firm’s participation in the trade, the firm should be given more transfer payment so that the firm’s production cost can be compensated, i.e., *t*_{1} ≥ *t*^{∗}.

### **Corollary 1**

*E*[

*W*

^{∗}(

*p*

^{∗},

*q*

^{∗},

*t*

^{∗})] and \(E[W_{1}^{*}(p_{1}^{*},q_{1}^{*},t_{1}^{*})]\) denote the optimal expected social total surpluses under full information and bilateral unknown information with the same estimate, respectively. We have

Corollary 1 indicates that bilateral unknown information with the same estimate gives rise to the decrease of the expected social total surplus by comparing with that under full information. To understand why it is, in the expression of expected social total surplus, firstly, notice that the firm’s expected profit is zero, so we only need to compare the expected consumer surplus. Secondly, the outputs in two cases are identical, furthermore, the same estimate causes the expected demand is equal to the optimal output, so that the consumer expected utilities under two cases are identical. Moreover, even though the price under bilateral unknown information is lower than that under full information, the transfer payment under bilateral unknown information is greater than that under full information. Furthermore, the decrease in consumer surplus resulted from the greater transfer payment is more than the increase in consumer surplus resulted from lower price. Consequently, the expected social total surplus under bilateral unknown information with the same estimate is distorted downwards from that under full information.

In a word, under bilateral unknown information, although the optimal output is not distorted from that under full information, but the presence of bilateral unknown information makes the regulator more conservative about the market capacity, leading to the optimal price distorted downwards from that under full information. Furthermore, in order to ensure the firm’s participation, the optimal transfer payment is distorted upwards from that under full information. Besides, bilateral unknown information gives rise to no impact on the firm’s profit, but makes the expected social total surplus distorted downwards from that under full information.

### 4.2 The different estimates

In this subsection, we consider the case that the regulator and the firm make different estimates about the true market capacity, which means their estimates with different probability distributions, i.e., *F*_{1}(*y*)≠*F*_{2}(*y*), \(\forall y\in [\underline {y},\overline {y}]\). To facilitate the exposition, we only consider regulatory policies under two scenarios that *F*_{1}(*y*) > *F*_{2}(*y*) and *F*_{1}(*y*) < *F*_{2}(*y*), \(\forall y\in [\underline {y}, \overline {y}]\), respectively. Note that *F*_{1}(*y*) > *F*_{2}(*y*) means that the expected market capacity of the regulator is less than that of the firm, which specifies the firm’s estimate is more optimistic than the regulator’s. In contrast, *F*_{1}(*y*) < *F*_{2}(*y*) means that the expected market capacity of the regulator is more than that of the firm, which specifies the firm’s estimate is more pessimistic than the regulator’s.

*E*[π(

*p*,

*q*,

*t*,

*Y*

_{2})] = 0 that

*t*into the objective function of Model (10) yields

### *Remark 1*

To analyze the impact of EVBA on the optimal regulatory policy, we can easily obtain the following results: EVBA is increasing with respect to both *p* and *q* when *F*_{1}(*y*) > *F*_{2}(*y*); EVBA is decreasing with respect to both *p* and *q* when *F*_{1}(*y*) < *F*_{2}(*y*).

Now we return to the optimal solutions problem in this case. But unfortunately, we cannot judge that the concavity of objective function, thus, we do not obtain the optimal solutions’ necessary and sufficient conditions for Model (10), but we can derive the optimal solutions’ necessary condition. The following proposition will reveal the optimal regulatory which varies with different estimates of the regulator and the firm.

### **Proposition 2**

*α*< 1, we have

- (i)
If

*F*_{1}(*y*) >*F*_{2}(*y*), then \(p_{2}^{*}>p_{1}^{*}\) and \(q_{2}^{*}>q_{1}^{*}\). - (ii)
If

*F*_{1}(*y*) <*F*_{2}(*y*), then \(p_{2}^{*}<p_{1}^{*}\) and \(q_{2}^{*}<q_{1}^{*}\).

- (1)
The different estimates of the regulator and the firm make the optimal price distorted from that under the same estimate. To get a more intuitive understanding of these results, it is necessary to return to the first-order condition of the regulator’s problem given by Eq. 32. Recall that in the case of the same estimate, from the regulator’s angle, the optimal price policy is to set the optimal price \(p_{1}^{*}\) determined by the equation that the left-hand side of Eq. 32 is equal to zero. However, in the case of the different estimates, the left-hand side of Eq. 32 may be either greater than or less than zero, resulting in either \(p_{2}^{*}>p_{1}^{*}\) or \(p_{2}^{*}<p_{1}^{*}\), respectively. In essence, it is EVBA that makes the optimal price distorted from \(p_{1}^{*}\). According to Remark 1, when

*F*_{1}(*y*) >*F*_{2}(*y*), EVBA is increasing with respect to the price, therefore, the regulator sets the price higher in order to increase EVBA such that the expected social total surplus is up to maximum; In contrast, when*F*_{1}(*y*) <*F*_{2}(*y*), the regulator sets the lower price in order to increase EVBA. - (2)
Similarly, with respect to the optimal output, let us see the first-order condition of the regulator’s objective function given by Eq. 34. Recall that in the case of the same estimate, the optimal output \(q_{1}^{*}\) follows that the left-hand side of Eq. 34 is equal to zero. However, in the case of the different estimates, the left-hand side of Eq. 34 may be either less than or greater than zero, resulting in either \(q_{2}^{*}>q_{1}^{*}\) or \(q_{2}^{*}<q_{1}^{*}\), respectively. Specifically, EVBA may be either increasing or decreasing with respect to the output, then the regulator sets the output higher or lower in order to improve EVBA.

- (3)
As we know, in face of bilateral unknown information, the regulator only needs to consider the firm’s participation. Although the optimal transfer payment \(t_{2}^{*}\) and \(t_{1}^{*}\) are to ensure the firm’s participation, there is no obvious relationship between them. The reason is that the optimal price and the optimal output under same estimate and different estimates are different, so we can not compare \(t_{2}^{*}\) with \(t_{1}^{*}\).

Proposition 2 has shown the important comparison results when 0 ≤ *α* < 1, particularly, if the regulator treats equally the consumer surplus and the firm’s profit, that is, *α* = 1, then it will give rise to the results of Proposition 3.

### **Proposition 3**

*F*

_{1}(

*y*)≠

*F*

_{2}(

*y*). When

*α*= 1, we have \(p_{2}^{*}= p_{1}^{*}\) and \(q_{2}^{*}= q_{1}^{*}\). Furthermore,

- (1)
If

*F*_{1}(*y*) >*F*_{2}(*y*), then \(t_{2}^{*}<t_{1}^{*}\), - (2)
If

*F*_{1}(*y*) <*F*_{2}(*y*), then \(t_{2}^{*}>t_{1}^{*}\).

Proposition 3 implies that the optimal price and the optimal output are not influenced by different estimates of the regulator and the firm when *α* = 1, that is to say, if the regulator values equally the consumer surplus and the firm’s profit, he would implement the optimal regulatory policy about the price and output identical to that under the same estimate. Nevertheless, there exists a deviation between \(t_{1}^{*}\) and \(t_{2}^{*}\). The reason is that although the regulator’s estimates are identical under two cases, since the firm’s estimates are different, and the optimal transfer payment paid to the firm is dependent on the firm’s estimate, and as a result, \(t_{2}^{*}\) is not equal to \(t_{1}^{*}\).

## 5 Numerical example

In this section, we take the applications in a kind of new products R&D project problem as a numerical example. In general, the market capacity of the new developed products, which is subject to many uncertainties, for instance, the consumer’s preference for the new products, substitute products’ impacts and so on, is often unknown to both the regulator and the firm, but they can make their respective estimates about the true market capacity. To demonstrate the impacts of the bilateral unknown information on the optimal regulatory policy, we will compare these optimal regulatory polices and the consumer surpluses under different information structures.

We assume the new products’ production cost function \(C(q)=\frac {1}{2}q^{2}+2q+8\), this assumption is analytically convenient. However, it is not crucial to the analysis. We can establish similar results if use other cost function. And assume \(\alpha =\frac {1}{4}\), *b* = 1. In the case of bilateral unknown information, when the regulator and the firm have the same estimate about the true market capacity, for simplicity, we assume that *Y* obeys uniform distribution on the support [8,10]. In the case of different estimates, there exist two cases: when the firm is more pessimistic than the regulator, *Y*_{1} and *Y*_{2} obey uniform distributions on the support [8,10] and [8,9], respectively, i.e., *F*_{1}(*y*) < *F*_{2}(*y*); While the firm is more optimistic than the regulator, *Y*_{1} and *Y*_{2} obey uniform distributions on the support [8,9] and [8,10], respectively, i.e., *F*_{1}(*y*) > *F*_{2}(*y*).

*p*

^{∗},

*q*

^{∗},

*t*

^{∗}), \((p_{1}^{*},q_{1}^{*},t_{1}^{*})\), \((p_{2}^{*},q_{2}^{*},t_{2}^{*})\) and \((p_{3}^{*},q_{3}^{*},t_{3}^{*})\) as the optimal regulatory polices under full information, bilateral unknown information with same estimate, different estimates (

*F*

_{1}(

*y*) >

*F*

_{2}(

*y*)) and (

*F*

_{1}(

*y*) <

*F*

_{2}(

*y*)), respectively. And we compare consumer surpluses under different cases in Fig. 4, still, denote

*S*

^{∗}, \(S_{1}^{*}\), \(S_{2}^{*}\) and \(S_{3}^{*}\) as consumer surpluses under full information, bilateral unknown information with same estimate, different estimates (

*F*

_{1}(

*y*) >

*F*

_{2}(

*y*)) and (

*F*

_{1}(

*y*) <

*F*

_{2}(

*y*)), respectively. Because the firm’s profits under different cases always are zero, the social total surplus equals to the consumer surplus, therefore, Fig. 4 may regard as the comparison results of the social total surpluses under different cases.

From Fig. 1, first, we find that the optimal price under full information is greater than that under bilateral unknown information except for the case that the firm has an optimistic attitude about the market capacity, which implies that the presence of bilateral unknown information can make the regulator reduce price. The reason accounting for this result is that under bilateral unknown information with the same estimate, the regulator underestimates the true market capacity, inducing the optimal price to become low. However, when the regulator knows that the firm has an optimistic attitude about the market capacity, he will design a higher price than that under full information based on the firm’s estimate when the market capacity is less. Second and more interestingly, we can observe from Fig. 1 that under full information, the optimal price is increasing with respect to the market capacity *y*, however, the optimal prices are decreasing with respect to the market capacity *y* under bilateral unknown information, which bilateral unknown information alters the monotonicity of the optimal price with respect to the market capacity. In addition, when \({y}={\underline {y}}\), note that the optimal price under full information equals to that under bilateral unknown information with the same estimate.

Figure 2 shows the optimal output *q*^{∗} under full information is the same as \(q_{1}^{*}\) under bilateral unknown information with the same estimate. This result specifies that in the case of bilateral unknown information, the regulator can only ask the firm to produce the products quantity depending on the the optimal output *q*^{∗} under full information although they does not know exactly the true market capacity. Note that when the regulator and the firm have different estimates about the true market capacity, the optimal output \(q_{2}^{*}\) is higher than *q*^{∗} under full information, while \(q_{3}^{*}\) is lower than *q*^{∗}, which demonstrates that the optimal output heavily depends on the firm’s estimate. In particular, when the firm possesses an optimistic attitude about the true market capacity, the regulator will design a higher output; Conversely, when the firm possesses an pessimistic attitude about the true market capacity, the regulator will design a lower output. Additionally, the optimal outputs under different cases are increasing with respect to the market capacity *y*, that is to say, the larger the market capacity is, the higher the output is, regardless of the information structure about the market capacity, which is a reasonable result.

Figure 3 shows that the optimal transfer payments under bilateral unknown information are bigger than that under full information. The reason for this is that the presence of bilateral unknown information results in loss due to overproduction, which is paid via transfer payment from the consumer to the firm. Meanwhile, notice that the optimal transfer payments under bilateral unknown information are increasing in *y*, but the optimal transfer payment *t*^{∗} under full information is decreasing with respect to *y*. This result specifies that bilateral unknown information changes the monotonicity of the optimal transfer payment with respect to the market capacity. When \(y=\overline {y}\), *t*^{∗} = 0, this result implies that the firm’s production cost is equal to the firm’s gross profit when the market capacity is up to maximum.

Figure 4 shows that if the regulator knows the true market capacity, that is, under full information, the consumer surplus can be up to maximum. It’s worth noting that the consumer surplus under full information and under bilateral unknown information are identical when \({y}={\underline {y}}\), which implies that the presence of bilateral unknown information arises no impact on the consumer surplus only when \({y}={\underline {y}}\). In addition, more interestingly, under bilateral unknown information, the same estimate can result in more consumer surplus than that under the different estimates. The reason for this is that the same estimate makes no residual products, inducing transfer payment reduced, which makes the consumer surplus increased. From this, we can obtain the conclusion that the regulator may prefer the regulatory policy when the regulator and the firm make the same estimate about the true market capacity. And notice that the consumer surplus under full information is increasing with respect to the market capacity, while the consumer surplus under bilateral unknown information is decreasing with respect to the market capacity, which specifies that bilateral unknown information changes the monotonicity of consumer surplus with respect to the market capacity.

## 6 Conclusions

First, when the market capacity is full information, the optimal price follows that the marginal utility of the consumer is equal to marginal cost of the firm, and the optimal output is determined by the optimal price. The optimal transfer payment compensates the firm’s cost such that the firm only obtains her reservation profit, i.e., zero profit. Thereby, the regulator allocates all surplus to the consumer under full information.

Second, when the market capacity is unknown to both the regulator and the firm, the different estimates will result in obvious effect on the optimal regulatory policy. In other words, the optimal regulatory policy focuses on whether the firm’s estimate about the market capacity is the same as the regulator’s, or not. If the estimates of the regulator and the firm about the true market capacity are identical, the optimal output is the same as that under full information, but the optimal price is distorted downwards from that under full information, and the transfer payment is distorted upwards from that under full information; If their estimates are different, this leads to different results: when the firm’s estimate is more optimistic than the regulator’s, the optimal price and the optimal output are more than those under the case that they have the same estimate. Contrary to this case, when the firm’s estimate is more pessimistic than the regulator’s, the optimal price and the optimal output are less than those under the case that they have the same estimate. In addition, it is worth noting that the firm still obtains her reservation profit under bilateral unknown information regardless of the same estimate or different estimates, such that the regulator allocates all surplus to the consumer even if neither the firm nor he knows the true market capacity, which implies that the presence of bilateral unknown information only affects the consumer surplus, but not the firm’s profit.

Finally, we highlight two important directions for future research. Firstly, we may consider a regulation issue when some cost information is unknown to the regulator and the firm, analyzing the effect of bilateral unknown cost information on the optimal regulatory policy. Secondly, despite assuming the firm will share her estimate about the true market capacity in this paper, she may have a motivation to misreport her estimate, therefore, the regulator must design an incentive regulatory policy so as to induce the firm reporting truthfully her estimate, consequently, leading the issue to become complicated.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 71771166, Tianjin Natural Science Foundation under Grant No. 18JCQNJC04200, and partly by a S\(\hat {e}\)r Cymru II COFUND Fellowship, UK.

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