Production planning for static Cournot duopoly competition under random yield

Abstract

The duopoly model with a Cournot form under random yield has received little attention. This article revisits the classic Cournot duopoly competition model by considering manufacturer yield uncertainty. We analyze the model under symmetric, asymmetric and monopoly situations. The manufacturers’ best production policy and expected market share are compared with each other in each situation. The expected market outputs of the three models are also compared under certain conditions. In the asymmetric case, the asymmetric model is extended by considering a powerful oligarch who can exert influence on her rival’s production decision by releasing her cost-type information. We derive that the maximization of the stronger manufacturer’s expected profit is highly correlated with consumer surplus. Numerical examples are presented to illustrate the results.

Introduction

Research on static and dynamic oligopolies is one of the richest areas in the economics literature. Since the by now famous duopoly theory was first explained by Cournot in 1838, many scholars have examined oligopoly models and their extensions. A comprehensive review of results about the single-product model was provided by Okuguchi, ¹ and the extensions of multiple products extensions with applications were discussed by Okuguchi and Szldarovszky. ² Long and Soubeyran ³ provided a new proof of the existence and uniqueness of a Cournot equilibrium characterized in terms of marginal costs. Most of the models discussed in the literature have assumed that the firms have full knowledge of all price functions as well as are able to respond instantly to the rival firm’s outputs and price decisions. However, there is a basic underlying presumption that the quantity a firm intends to produce equals the final output production quantity. In fact, almost every manufacturing firm suffers from yield uncertainty, namely, that the final output does not always equal the firm’s target production quantities. Thus, it would be worthwhile to reconsider the classic Cournot competition model under yield randomness.

In many industries, the final output quantity of a production process is not deterministic and is influenced by many factors. For example, weather is one of the most important factors that affect agriculture-related industries; yet, when planting decisions are made, it is almost impossible to precisely forecast the weather in the future. Chiu et al. ⁴ considered the algebraic derivation for the production lot-sizing problem with backlogging, random defective rate and rework. They extend the classic economic production quantity (EPQ) model to a modified model by taking backlogging, random defective rate and rework into consideration. Hsu and Hsu ⁵ developed an integrated inventory model for the coordination of a two-echelon supply chain under an imperfect vendor production process. They assume that the proportion of defective items in each production lot is stochastic and follows a known probability density function. Dada et al. ⁶ considered the procurement problem of a newsvendor that is served by multiple suppliers, where the supplier is defined to be either perfectly reliable or unreliable. There have been some attempts at modeling the production and procurement decisions in supply chain management involving risk supply. Xu ⁷ tried to manage production and procurement in a decentralized supply chain with random supplier yield and stochastic manufacturer demand through option contracts. Babich et al. ⁸ studied the effects of disruption risk in a supply chain where one retailer deals with competing risky suppliers who may default during their production lead times. The phenomenon of yield uncertainty can also be found in remanufacturing companies. Bakal and Akcali ⁹ investigated the pricing decisions in a remanufacturing supply chain with random recovery yield rate. Mukhopadhyay and Ma ¹⁰ studied the production and procurement problem of an integrated manufacturing and remanufacturing system by considering yield randomness in recycling.

The studies above mainly discuss the ordering and production policies in a single supplier–vendor supply chain. Vendor with multiple suppliers should also be of concern. Deo and Corbett ¹¹ formulated a two-stage model of Cournot competition with N-symmetric firms and analyzed the effect of yield uncertainty on firms’ entry and production decisions. Tang and Kouvelis ¹² presented supplier diversification strategies by considering yield uncertainty and buyer competition, but the production decision strategies between competing suppliers with random yield in a market when there is no retailer has been insufficiently studied. Xanthopoulos et al. ¹³ proposed a single-period inventory model for capturing the trade-off between inventory policies and disruption risks in a dual-sourcing supply chain. Yan et al. ¹⁴ studied supplier diversification in a multiple supplier competition model with random yield under symmetric environment by considering a case in which the suppliers’ random yields are correlated. Yu et al. ¹⁵ focused on evaluating the impacts of supply disruption risks on the choice between single- and dual-sourcing methods in a two-stage supply chain with nonstationary and price-sensitive demand. The above work mainly focused on the ordering policy of the vendor from a supply chain perspective while neglecting the production strategies of the manufacturers.

However, competition among manufacturers does change the structure of the market. Deo and Corbett ¹¹ and Yan et al. ¹⁴ studied the random yield problem in a market with manufacturers who have symmetric information and the same costs. Therefore, it is worthwhile to study how manufacturers with different costs compete under asymmetric or monopoly situations. The duopoly model with a Cournot form under random yield has received little attention. Yet, few researchers have worked on this problem with yield randomness. The motivation for exploring the topic is to try to answer the following questions: (1) How do duopoly manufacturers compete or cooperate under random yield in a noncooperative or monopoly situation? (2) What is the impact of yield uncertainty on the duopoly manufacturers’ production policy and the market output? and (3) Which manufacturer shares a larger market share under random yield?

In this article, we remodel the classic Cournot production competition model by using the stochastic proportional yield model under noncooperative (symmetric and asymmetric) and monopoly conditions, respectively, to try to derive the best target production quantities for the duopoly manufacturers. Then, our model is extended by considering a powerful oligarch who can exert influence on her rival’s production decision not only by her own production decision but also by releasing her cost-type information.

The objective of this article is to reconsider the classic Cournot duopoly competition model under yield uncertainty. We aim to derive the equilibrium target production policies for the duopoly and analyze the impact of production yield uncertainty on the equilibrium outcome and the social optimum for the classic Cournot model of duopoly competition and monopoly under yield uncertainty. The results of this article can be applied in particular to Cournot-characteristic industries, which are industries that supply necessities of life: oil, plant production, tap water and electricity, for instance. Likely users of this research are manufacturers in these industries, who can use the model in setting their best target production quantities under yield uncertainty.

The rest of this article is organized as follows: In section “Model formulation,” we present the basic model and related literatures. Sections “Symmetric Cournot duopoly with random yield” and “Asymmetric Cournot duopoly with random yield” study the Cournot competition model under random yield and outline the competitive equilibrium under a symmetric and asymmetric environment, respectively. Section “The case of Cartel-monopoly” focuses on the situation in which the duopolists cooperate to monopolize the market in order to reach maximum revenue. Section “An extension: ρ is ‘made’ and released by M₂” extends the asymmetric model to consider a stronger oligarch who can exert influence on her rival’s production decision by releasing her cost-type information. Numerical examples are presented to illustrate the results and study the effect of yield uncertainty on the market.

Model formulation

Consider a single-period model in a local market consisting of two manufacturers denoted by M_i (i = 1,2). The duopoly manufacturers make the same final products and compete in their production quantities at the same time. Their aims are to decide their best target production quantities to maximize their profit. M_i competes in output Q_i (i = 1,2). q_i (i = 1,2) denotes their planed production quantities and equals their final production quantity when there is no production risk. However, by considering random yields, their actual realized production quantities are Q_i = α_iq_i, where α_i (i = 1,2) is a random variable attached to the planned production quantity q_i representing the yield risk of manufacturer i. Because it is impossible to avoid yield uncertainty, the two manufacturers can only decide their target production quantities to maximize their expected profit. Yano and Lee ¹⁶ presented a thorough review of single-stage single-period models that dealt with lot-sizing models with stochastically proportional yield models, especially when the yield losses in the production process and the final production quantities are not predictable in manufacturing systems. Furthermore, we assume that α_i is independent but with an identical probability distribution. For example, there are two farms in the same area planting the same crop with the same seed and planting techniques. They suffer the same yield risk, which is mainly determined by the weather, and their production processes are independent. Thus, it is feasible to assume “α” as an independent and identically distributed variable. The mean and variance of α_i are denoted by µ = E(α_i) and σ ² = Var(α_i) (0 < µ < 1, σ > 0), respectively. This form of random yield has been used by many researchers, such as Yan et al., ¹⁴ Agrawal and Nahmias, ¹⁷ Li and Zheng ¹⁸ and Keren. ¹⁹ Note that, as a special case, our formulation covers the model in the studies by Creane ²⁰ and Serel, ²¹ where the yield risk α_i is a Bernoulli distributed variable. We assume that the manufacturers’ final production quantities are realized with a nonzero probability.

The manufacturers’ retail price (market price) of the product is given by the linear reverse demand curve p = k−bQ_t (k, b > 0), where k is the reservation price, b is the elasticity coefficient and Q_t stands for the total output of M₁ and M₂. There are two marginal costs for the manufacturers: one is the unit-preparing production cost c_i1 and the other is the unit-realized production quantity cost c_i2. This type of production cost has been used by many researchers, such as Deo and Corbett ¹¹ and Kazaz. ²² Then the manufacturer’s sales revenue is ${\pi }_i(Q_i)=[k-b\sum Q_i]·Q_i$ , and the manufacturers’ operational profit is

$${\pi }_{M_i}\left(q_i\right)=[k-b\sum Q_i]·Q_i-c_{i1}{q}_i-c_{\text{i}2}{Q}_i$$

(1)

Without taking into consideration of random yield (Q_i = q_i), the solution of equation (1) will be the classical Cournot production competition model. However, because of disruptions such as human error, technical error, weather and so on, yield uncertainty exists almost in every production system, especially in agriculture production. Thus, the final output or profit cannot be effectively predicted, and the only thing they can do is to choose a best target production quantity to maximize their expected profit when α_i is stochastically distributed. Section “Symmetric Cournot duopoly with random yield” will focus on the manufacturers’ optimal decisions under symmetric condition with random yield.

Symmetric Cournot duopoly with random yield

This section examines the scenario in which two manufacturers compete with full information about their mutual costs. Each manufacturer sets her target production quantity q_i, and after realizing the production risk α_i, we have the total production quantity as $Q_t=\sum {\alpha }_i{q}_i={\alpha }_1{q}_1+{\alpha }_2{q}_2$ . Based on equation (1), in view of the fact that ${\pi }_{M_i}(q_i)$ is a function of a random variable, we maximize the expected profit function of M₁ and M₂ in the following expressions

$$\begin{array}{c}\multicolumn{1}{c}{\underset{q_1}{\max }E\left({\pi }_{M_1}\left(q_1\right)\right)=E_{{\alpha }_i}([k-b(Q_1+Q_2)]·Q_1-c_{11}{q}_1-c_{12}{Q}_1)}\end{array}$$

(2)

$$\underset{q_2}{\max }E\left({\pi }_{M_2}\left(q_2\right)\right)=E_{{\alpha }_i}([k-b(Q_1+Q_2)]·Q_2-c_{21}{q}_2-c_{22}{Q}_2)$$

(3)

By substituting Q_t = α₁q₁+α₂q₂ into the above two equations, equations (2) and (3) can be rewritten as

$$\underset{q_1}{\max }E\left({\pi }_{M_1}\left(q_1\right)\right)=\mathrm{kE}\left({\alpha }_1\right)q_1-b(E\left({\alpha }_1^2\right)q_1^2+E\left({\alpha }_1{\alpha }_2\right)q_1{q}_2)-(c_{11}+c_{12}E\left({\alpha }_1\right))q_1$$

(4)

$$\underset{q_2}{\max }E\left({\pi }_{M_2}\left(q_2\right)\right)=\mathrm{kE}\left({\alpha }_2\right)q_2-b(E\left({\alpha }_2^2\right)q_2^2+E\left({\alpha }_2{\alpha }_1\right)q_1{q}_2)-(c_{21}+c_{22}E\left({\alpha }_2\right))q_2$$

(5)

Because M₁ and M₂ decide their target production quantities at the same time, the Nash equilibrium production quantity $(q_1^{*},q_2^{*})$ will satisfy

$$\frac{\partial E\left({\pi }_{M_1}\left(q_1\right)\right)}{\partial q_1}=k\mu -2b({\mu }^2+{\sigma }^2)q_1-b{\mu }^2{q}_2-(c_{11}+\mu c_{12})=0$$

(6)

$$\frac{\partial E\left({\pi }_{M_2}\left(q_2\right)\right)}{\partial q_2}=k\mu -2b({\mu }^2+{\sigma }^2)q_2-b{\mu }^2{q}_1-(c_{21}+\mu c_{22})=0$$

(7)

Let $E({\alpha }_i)c_i^{\circ }=(c_{i1}+E({\alpha }_i)c_{i2})$ , namely, $\mu c_i^{\circ }=(c_{i1}+\mu c_{i2})$ to transform equations (6) and (7). Then, we have

$$q_1\left(q_2\right)=-\frac{{\mu }^2}{2({\mu }^2+{\sigma }^2)}·q_2+\frac{\mu (k-c_1^{\circ })}{2b({\mu }^2+{\sigma }^2)}$$

(8)

$$q_2\left(q_1\right)=-\frac{{\mu }^2}{2({\mu }^2+{\sigma }^2)}·q_1+\frac{\mu (k-c_2^{\circ })}{2b({\mu }^2+{\sigma }^2)}$$

(9)

Equations (8) and (9), respectively, denote the production policy reaction function for the duopoly manufacturers under random yield. To ensure that $q_1^{*}$ and $q_2^{*}$ are positive, there should exist $(k-c_i^{\circ })>0$ , that is, $c_{i1}<\mu (k-c_{i2})$ . To resolve by combining equations (8) and (9), we have Theorem 1.

• Theorem 1. Under random yield, there exists a unique Nash best target production quantity equilibrium for the duopoly manufacturers in a single-period symmetric environment

$$\{\begin{array}{c}\hfill q_1^{*}=\frac{2\mu ({\mu }^2+{\sigma }^2)(k-c_1^{\circ })-{\mu }^3(k-c_2^{\circ })}{b(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4)}\hfill \\ \multicolumn{1}{c}{q_2^{*}=\frac{2\mu ({\mu }^2+{\sigma }^2)(k-c_2^{\circ })-{\mu }^3(k-c_1^{\circ })}{b(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4)}}\end{array}$$

(10)

$(q_1^{*},q_2^{*})$ is the best target production decision for the duopoly manufacturers in a complete information Cournot model under random yield. Comparing $q_1^{*}$ and $q_2^{*}$ , we have Proposition 1.

• Proposition 1. In the symmetric case, a manufacturer with lower cost has a larger (expected) market share, that is, (i) if $c_1^{\circ }\le c_2^{\circ }$ , there is $q_1^{*}\ge q_2^{*}$ , $E{M}_{1,s}\ge E{M}_{2,s}$ and (ii) if $c_1^{\circ }>c_2^{\circ }$ , there is $q_1^{*}<q_2^{*}$ , $E{M}_{1,s}<E{M}_{2,s}$ .

• Proof. $q_1^{*}-q_2^{*}=(3{\mu }^3+2\mu {\sigma }^2)(c_2^{\circ }-c_1^{\circ })/b(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4)$ , therefore, it is easy to show that $q_1^{*}<q_2^{*}$ when $c_1^{\circ }>c_2^{\circ }$ , then $E{M}_{1,s}=\mu q_1^{*}/(\mu q_1^{*}+\mu q_2^{*})<\mu q_2^{*}/(\mu q_1^{*}+\mu q_2^{*})=E{M}_{2,s}$ . Thus, Proposition 1 is proved. Following Deo and Corbett, ¹¹ we use $\delta =\sigma /\mu$ , $(\delta >0)$ by keeping µ constant and only analyzing the effect of changes in σ to analyze the effect of yield uncertainty on the market. Results expressed in terms of δ rather than σ facilitate the exposition. From equation (10), we have Lemma 1.

• Lemma 1. In the symmetric case, the expected market output decreases with respect to δ.

• Proof. $E{q}_{t,s}=\mu (q_1^{*}+q_2^{*})=({\mu }^4+2{\mu }^2{\sigma }^2)(2k-c_1^{\circ }-c_2^{\circ })/b(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4)=(1+2{\delta }^2)(2k-c_1^{\circ }-c_2^{\circ })/b(3+8{\delta }^2+4{\delta }^4)$ , then, to derive the first-order condition of $E{q}_{t,s}$ with respect to $\delta$ , we have

$$\frac{\partial E{q}_{t,s}}{\partial \delta }=\frac{-4\delta (4{\delta }^4+3{\delta }^2+1)(2k-c_1^{\circ }-c_2^{\circ })}{b{(3+8{\delta }^2+4{\delta }^4)}^2}<0$$

The introduction of the δ facilitates the analysis of the effect of yield randomness on the expected market output. Lemma 1 indicates that, if µ is fixed, Eq_t,s decreases in σ. Furthermore, if we fix µ = 1, σ = 0 (δ = 0, the deterministic yield condition), as Eq_t,s is decreasing in δ, then Eq_t,s < q_t,s (δ = 0). Thus, the expected market output under yield uncertainty is smaller compared with the deterministic yield condition. From equation (10), we can also derive $\partial q_i^{*}/\partial c_i^{\circ }<0$ , which means that a larger $c_1^{\circ }$ leads to a smaller target production quantity.

Figure 1 illustrates the effect of random yield on the production reaction function of the duopoly manufacturers. Solid lines l₁ and l₂, respectively, denote M₁ and M₂’s production reaction curve under deterministic yield. The curve can be obtained by setting µ = 1, σ = 0 in equations (8) and (9). The shadow between the parallel dash lines, lying on both sides of the solid line, stands for the fluctuation effect on M₁’s target production reaction under random yield.

Figure 1.

The impact of random yield on the production reaction of the duopoly.

The slope coefficient of the reaction curve of M₁ under yield uncertainty can be derived from equations (8) and (9). $\left|{\eta }_{u1}\right|$ and $\left|{\eta }_{d1}\right|$ , respectively, denote the slope coefficient’s absolute value of the reaction curve of M₁ under yield uncertainty and deterministic environment. By comparison, we have

$$\begin{array}{c}\multicolumn{1}{c}{\left|{\eta }_{u1}\right|=\frac{\partial q_1}{\partial q_2}=\frac{{\mu }^2}{2({\mu }^2+{\sigma }^2)}=\frac{1}{2(1+{\left(\sigma /\sigma \mu \mu \right)}^2)}}\\ <\frac{1}{2}=\frac{\partial q_{d1}}{\partial q_{d2}}=\left|{\eta }_{d1}\right|\hfill \end{array}$$

(11)

It can be observed in Figure 1 that the slope coefficient of the reaction curve of M₁ under yield uncertainty is smaller than that of the corresponding solid line, which also exists for M₂. An explanation for equation (11) could be that yield uncertainty decreases the reaction on their mutual target production quantity. The actual location of the reaction curve under random yield (dash line) depends on the comparison of $\mu (k-c_1^{\circ })/2b({\sigma }^2+{\mu }^2)$ with $(k-c_1^{\circ })/2b$ and so on. The rhombus ABCD in Figure 1 stands for the solution set of duopoly manufacturers’ Nash target production quantity equilibrium. The actual solution depends on the realization of the random yield risk α_i.

Asymmetric Cournot duopoly with random yield

Information asymmetry exists widely in competitive markets, especially in oligopoly competition. Vives ²³ analyzed an incomplete information setting where firms receive signals about uncertain demand, and they proved that the Bertrand Bayesian–Nash quantity is higher than the Cournot Bayesian–Nash quantity. Lofaro ²⁴ studied the efficiency of Cournot and Bertrand competition under incomplete information environment. However, their results would be changed if yield uncertainty was considered. In this section, a Cournot duopoly production competition model with incomplete cost information under random yield is analyzed. Here, we assume that M₁ produces at a constant unit production cost $c_1^{\circ }$ (contains c₁₁ and c₁₂), which is known by M₂. However, M₁ is not sure whether M₂ is high ( $c_2^h>c_1^{\circ }$ , $c_2^h$ contains $c_{21}^h$ and $c_{22}^h$ ) or low ( $c_1^{\circ }>c_2^l$ , $c_2^l$ contains $c_{21}^l$ and $c_{22}^l$ ) cost type. Yet, M₁ knows that M₂ is high cost type with a probability ρ (0 < ρ < 1) and low cost type with a probability 1 −ρ. ρ is common knowledge, known by both M₁ and M₂. Surely, we can assume that M₂ knows about her own cost type. The static Cournot production competition occurs when the two manufacturers make target production decisions at the same time.

Assuming that M₂ has accurate knowledge about M₁’s cost, M₂ should make her production decision (q₂) to maximize her profit function as

$${\pi }_{M_2}\left(q_2\right)=Q_2·[k-b(Q_1+Q_2)]-c_{21}{q}_2-c_{22}{Q}_2$$

(12)

By considering the random yield risk α_i, M₂ will choose the best target production quantity $q_2^{*}$ to maximize her expected profit function as

$$\begin{array}{c}\multicolumn{1}{c}{\underset{q_2}{\max }E\left({\pi }_{M_2}\left(q_2\right)\right)=E_{{\alpha }_i}({\alpha }_2{q}_2[k-b({\alpha }_1{q}_1+{\alpha }_2{q}_2)]-(c_{21}+c_{22}{\alpha }_1)q_2)}\end{array}$$

(13)

By using a similar transformation as in section “Symmetric Cournot duopoly with random yield,” here we set $\mu c_2^{\circ }=(c_{21}+\mu c_{22})$ to transform equation (13). To derive the first-order condition of $E{\pi }_{M_2}(q_2)$ with respect to q₂, we have the reactive production function of M₂ upon M₁ as

$$q_2\left(q_1\right)=\frac{-{\mu }^2}{2({\sigma }^2+{\mu }^2)}{q}_1+\frac{\mu (k-c_2^{\circ })}{2b({\sigma }^2+{\mu }^2)}$$

(14)

Equation (14) explains that M₂’s best production target quantity depends not only on the target production quantity of M₁ but also depends on her own unit production cost. To set, $q_2^h$ stands for the target production quantity of M₂ when $c_2^{\circ }=c_2^h$ , and $q_2^l$ stands for the target production quantity when $c_2^{\circ }=c_2^l$ . The best target production quantity of M₂ is given by

$$\begin{array}{cc}\multicolumn{1}{c}{q_2^h\left(q_1\right)}& =\frac{-{\mu }^2}{2({\sigma }^2+{\mu }^2)}{q}_1+\frac{\mu (k-c_2^h)}{2b({\sigma }^2+{\mu }^2)},\hfill \\ \multicolumn{1}{c}{q_2^l\left(q_1\right)}& =\frac{-{\mu }^2}{2({\sigma }^2+{\mu }^2)}{q}_1+\frac{\mu (k-c_2^l)}{2b({\sigma }^2+{\mu }^2)}\hfill \end{array}$$

(15)

By observing the equations above, it is easy to find that 2b(σ ² +µ ² ), the denominator of q₂, is a positive constant. To ensure q₂ is positive, the numerator must also be positive, such that $k>b\mu q_1+c_2^{\circ }>c_2^{\circ }$ . In the asymmetric condition, M₁ is not sure about whether M₂’s real cost is high or low cost type, such that M₂’s reaction function could be $q_2^h$ or $q_2^l$ for M₁. Thus, M₁’s expected profit function could be maximized by considering the probability of M₂’s cost type. The expected profit of M₁ is given by

$$\begin{array}{c}{\max }_{q_1}E\left({\pi }_{M_1}\left(q_1\right)\right)\hfill \\ =\rho ·[E\left({\alpha }_1\right)k{q}_1-b(E\left({\alpha }_1^2\right)q_1^2+E\left({\alpha }_1{\alpha }_2\right)q_1{q}_2^h)]\hfill \\ -c_1^{\circ }E\left({\alpha }_1\right)q_1+(1-\rho )\hfill \\ ·[E\left({\alpha }_1\right)k{q}_1-b(E\left({\alpha }_1^2\right)q_1^2+E\left({\alpha }_1{\alpha }_2\right)q_1{q}_2^l)]\hfill \\ =\rho ·[\mu k{q}_1-b(({\mu }^2+{\sigma }^2)q_1^2+{\mu }^2{q}_1{q}_2^h)]\hfill \\ -\mu c_1^{\circ }{q}_1+(1-\rho )·[\mu k{q}_1-b(({\mu }^2+{\sigma }^2)q_1^2+{\mu }^2{q}_1{q}_2^l)]\hfill \end{array}$$

(16)

To solve the first-order condition of equation (16) with respect to q₁, the reactive production function of M₁ upon M₂ should satisfy

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial E\left({\pi }_{M_1}\left(q_1\right)\right)}{\partial q_1}=\mu (k-c_1^{\circ })-2b({\mu }^2+{\sigma }^2)q_1}\\ -b\rho {\mu }^2{q}_2^h-(1-\rho )b{\mu }^2{q}_2^l=0\hfill \end{array}$$

(17)

Because M₁ is not sure about M₂’s cost type, it is hard to get the exact reaction curve of M₁ as in Figure 1. By substituting equation (15) into equation (17), and using simple algebraic manipulation, we have Theorem 2:

• Theorem 2. Under random yield, there exists a unique Bayes–Nash equilibrium best target production quantity for the duopoly manufacturers in a single-period asymmetric environment

$$\{\begin{array}{c}{q}_1^{*}=\frac{\mu (k-c_1^{\circ })-b{\mu }^2[\rho H+(1-\rho )L]}{b(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)}\hfill \\ q_2^{h*}=\frac{b\nu {\mu }^2[\rho H+(1-\rho )L]+\mathrm{bH}[2({\mu }^2+{\sigma }^2)-\nu {\mu }^2]-\mu \nu (k-c_1^{\circ })}{b(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)}\hfill \\ q_2^{l*}=\frac{b\nu {\mu }^2[\rho H+(1-\rho )L]+\mathrm{bL}[2({\mu }^2+{\sigma }^2)-\nu {\mu }^2]-\mu \nu (k-c_1^{\circ })}{b(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)}\hfill \end{array}$$

(18)

where $H=\mu (k-c_2^h)/2b({\mu }^2+{\sigma }^2)>0$ , $L=\mu (k-c_2^l)/2b({\mu }^2+{\sigma }^2)>0$ , $\nu ={\mu }^2/2({\mu }^2+{\sigma }^2)$ , that is, $q_2^{h*}=H-\nu q_1^{*}$ , $q_2^{l*}=L-\nu q_1^{*}$ , $0<H<L$ . $(q_1^{*},q_2^{h*}(q_2^{l*}))$ is the best target production decision for the duopoly manufacturers in an asymmetric information Cournot model under random yield. By simplifying equations (15) and (18), we have the following results

• Lemma 2. M₂’s best target production quantity $q_2^{*}$ under asymmetric information decreased with her production cost $c_2^{\circ }$ , and there is $q_2^{h*}<q_2^{l*}$ .

• Proof. To derive the first-order condition of $q_2^h$ with respect to $c_2^h$ and $q_2^l$ with respect to $c_2^l$ , then

$$\begin{array}{l}\frac{\partial q_2^{h\ast }}{\partial c_2^h}=-\frac{\mu }{2b\left({\sigma }^2+{\mu }^2\right)}\\ -\frac{\rho {\mu }^5}{2b\left({\sigma }^2+{\mu }^2\right)\left(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4\right)}<0\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial q_2^{l*}}{\partial c_2^l}=-\frac{\mu }{2b({\sigma }^2+{\mu }^2)}}\\ -\frac{(1-\rho ){\mu }^5}{2b({\sigma }^2+{\mu }^2)(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4)}<0\hfill \end{array}$$

Thus, M₂’s best production quantity q₂ decreases with her cost $c_2^{\circ }$ . $q_2^h<q_2^l$ also indicates that M₂ would have a larger expected market share when she is low cost type. By using the same method, Lemma 2 could also be applied to q₁ on $c_1^{\circ }$ . From Lemma 2, we can conclude that a lower unit production cost would let M₂ set a larger target production quantity, and we derive that regardless of the random yield risk and the competitor’s production decision, the unit production cost always plays a negative role in the manufacturers’ target production decisions.

• Lemma 3. (i) M₁’s best target production quantity $q_1^{*}$ is proportional to the probability ρ that she believes M₂ is high cost type and (ii) M₂’s best target production quantity $q_2^{*}$ is inversely proportional to the probability ρ, and the difference of q₂ and q₁ is a decreasing function of ρ.

• Proof of (i). Derive the first-order condition of $q_1^{*}$ with respect to $\rho$ . Note that $c_2^l<c_1^{\circ }<c_2^h$ and $\nu ={\mu }^2/2({\mu }^2+{\sigma }^2)=1/2({\mu }^2+1)<1/2$ , thus $2({\mu }^2+{\sigma }^2)-\nu {\mu }^2>0$ , then we obtain $\partial q_1^{*}/\partial \rho ={\mu }^2(L-H)/2({\mu }^2+{\sigma }^2)-\nu {\mu }^2>0$ , which proves (i). (i) explains that if M₁ considers M₂ as a high cost type, she would set her target production quantity based on her own cost advantage.

• Proof of (ii). From equation (18), we have

$$\begin{array}{l}\frac{\partial q_2^{h\ast }}{\partial \rho }=\frac{{\mu }^4\left(H-L\right)}{2\left({\sigma }^2+{\mu }^2\right)\left(2\left({\mu }^2+{\sigma }^2\right)-\nu {\mu }^2\right)}<0;\\ \frac{\partial \left(q_2^{h\ast }-q_1^{\ast }\right)}{\partial \rho }\\ =\frac{{\mu }^2\left(3{\mu }^2+2{\sigma }^2\right)\left(H-L\right)}{2\left({\mu }^2+{\sigma }^2\right)\left(2\left({\mu }^2+{\sigma }^2\right)-\nu {\mu }^2\right)}<0\end{array}$$

The case of the low cost–type M₂ can also be proved. Recall that M₁’s best target production quantity q₁ is proportional to ρ and M₂’s best target production quantity q₂ decreases when ρ increases. Hence, the difference of q₂ and q₁ decreases with the increase of ρ.

• Lemma 4. (i) The expected market output in the asymmetric case increases with the probability ρ that M₁ believes M₂ is high cost type and (ii) the expected market output is larger than the symmetric case when M₂ is low cost type and is smaller when M₂ is high cost type.

• Proof of (i). Take high cost–type M₂, for example, $E{q}_{t,\mathrm{as}}=\mu (q_1^{*}+q_2^{h*})$ . Then, to derive the first-order condition of $E{q}_{t,\mathrm{as}}$ with respect to ρ, for $\nu -1<0$ , $H-L<0$ and ${\mu }^2+{\sigma }^2-\nu {\mu }^2>0$ , we have

$$\frac{\partial E{q}_{t,\mathrm{as}}}{\partial \rho }=\frac{{\mu }^2(\nu -1)(H-L)}{2({\mu }^2+{\sigma }^2)-\nu {\mu }^2}>0$$

which proves (i).

• Proof of (ii). This is obtained by solving the difference of $E{q}_{t,\mathrm{as}}$ with $E{q}_{t,s}$

$$\begin{array}{l}E{q}_{t,as}-E{q}_{t,s}=\frac{\left({\mu }^6-2{\mu }^4\left({\mu }^2+{\sigma }^2\right)\right)\left[\rho \left(k-c_2^h\right)+\left(1-\rho \right)\left(k-c_2^l\right)\right]+\mu \left(k-c_2^j\right)\left(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4\right)}{2b\left({\mu }^2+{\sigma }^2\right)\left(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4\right)}\\ -\frac{{\mu }^2\left({\mu }^2+2{\sigma }^2\right)\left(k-c_2^j\right)}{b\left(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4\right)},j=h,l\end{array}$$

To derive the first-order condition of the difference with respect to ρ, we have

$$\frac{\partial (E{q}_{t,\mathrm{as}}-E{q}_{t,s})}{\partial \rho }=\frac{{\mu }^6-2{\mu }^4({\mu }^2+{\sigma }^2)(c_2^l-c_2^h)}{2b({\mu }^2+{\sigma }^2)(3{\mu }^4+8{\mu }^2{\sigma }^2+4{\sigma }^4)}>0$$

When M₂ is high cost type, if ρ = 1, we have $E{q}_{t,\mathrm{as}}=E{q}_{t,s}$ , that is, $E{q}_{t,\mathrm{as}}-E{q}_{t,s}=0$ , as $(\partial (E{q}_{t,\mathrm{as}}-E{q}_{t,s})/\partial \rho )>0$ , thus $E{q}_{t,\mathrm{as}}-E{q}_{t,s}<0$ when 0 < ρ < 1. When M₂ is low cost type, if ρ = 0, $E{q}_{t,\mathrm{as}}=E{q}_{t,s}$ , that is, $E{q}_{t,\mathrm{as}}-E{q}_{t,s}=0$ , as $\partial (E{q}_{t,\mathrm{as}}-E{q}_{t,s})/\partial \rho >0$ , thus $E{q}_{t,\mathrm{as}}-E{q}_{t,s}>0$ when 0 < ρ < 1. (ii) is proved.

(i) indicates that information asymmetry has a positive effect in enlarging the market scale. (ii) shows that whether information asymmetry magnifies the expected market output compared with the symmetric case depends on M₂’s cost type. By observing equations (10) and (18), it is interesting to find that the solution of the asymmetric case is a special condition of the symmetric case for ρ = 1, when M₂ is high cost type, for ρ = 0, when M₂ is low cost type. By comparing the optimal solutions in equation (18), we have

Proposition 2. When the mean value and variance of yield risk are fixed and known to the duopoly manufacturers, the value of ρ decides the relationship of the double oligarchies’ target production quantities.

• If ${\Gamma }_l<1$ , and ${\Gamma }_l<\rho <1$ , then $q_2^h<q_2^l<q_1$ , and M₂’s expected market share is less than M₁’s expected market share regardless of her cost type. If ${\Gamma }_l>1$ , then there does not exist ρ, to have $q_2^h<q_2^l<q_1$ ;

• If ${\Gamma }_h>0$ and $0<\rho <{\Gamma }_h$ , then $q_1<q_2^h<q_2^l$ , and M₂’s expected market share is larger than M₁’s expected market share regardless of her cost type. If ${\Gamma }_h>1$ , for $\forall \rho$ , then $q_1<q_2^h<q_2^l$ ;

• If ${\Gamma }_h<1$ , ${\Gamma }_l>0$ and ${\Gamma }_h<\rho <{\Gamma }_l$ , then $q_2^h<q_1<q_2^l$ , M₂’s expected market share is larger than M₁’s expected market share when M₂ is low cost type and M₂’s expected market share is less than M₁’s expected market share when M₂ is high cost type.

where

$$\begin{array}{cc}\multicolumn{1}{c}{{\Gamma }_l}& =\frac{b\mu L-k+c_1^{\circ }}{b\mu (L-H)}+\frac{(k-c_2^l)((2-\nu ){\mu }^2+2{\sigma }^2)}{b\mu (L-H)(3{\mu }^2+2{\sigma }^2)},\hfill \\ \multicolumn{1}{c}{{\Gamma }_h}& =\frac{b\mu L-k+c_1^{\circ }}{b\mu (L-H)}+\frac{(k-c_2^h)((2-\nu ){\mu }^2+{\sigma }^2)}{b\mu (L-H)(3{\mu }^2+2{\sigma }^2)}\hfill \end{array}$$

• Proof of (i). From Lemma 2, we have $q_2^h<q_2^l$ . There is $q_2^h<q_2^l<q_1$ when $q_1-q_2^l>0$ is proved

$$\begin{array}{l}{q}_2^l-q_1=\frac{\mu \left(k-c_2^l\right)-b{\mu }^2{q}_1}{2b\left({\sigma }^2+{\mu }^2\right)}\\ -\frac{\mu \left(k-c_1\text{}\text{}°\right)-b{\mu }^2\left(\rho H+\left(1-\rho \right)L\right)}{b\left(2\left({\mu }^2+{\sigma }^2\right)-\nu {\mu }^2\right)}\\ =\frac{\mu \left(k-c_2^l\right)}{2b\left({\mu }^2+{\sigma }^2\right)}-\frac{3{\mu }^2+2{\sigma }^2}{2\left({\mu }^2+{\sigma }^2\right)}\\ ·\frac{\mu \left(k-c_1\text{}\text{}°\right)-b{\mu }^2\left(\rho H+\left(1-\rho \right)L\right)}{b\left(2\left({\mu }^2+{\sigma }^2\right)-\nu {\mu }^2\right)}\\ =\frac{\text{1}}{2b\left({\mu }^2+{\sigma }^2\right)}\\ ·(\mu \left(k-c_2^l\right)-\left(3{\mu }^2+2{\sigma }^2\right)\\ ·\frac{\mu \left(k-c_1\text{}\text{}°\right)-b{\mu }^2\left(\rho H+\left(1-\rho \right)L\right)}{2\left({\mu }^2+{\sigma }^2\right)-\nu {\mu }^2})\end{array}$$

Note that 2b (σ ² +µ ² ) > 0, L − H > 0. By simplifying in order to reach $q_1-q_2^l>0$ , the proposition can be easily proved with these expressions. If ${\Gamma }_l>1$ , then ${\Gamma }_l>1>\rho$ , thus, there does not exist a ρ to have $q_2^h<q_2^l<q_1$ .

• Proof of (ii). Similar to the proof of Lemma 4.

• Proof of (iii). Since 0 < H < L, it is easy to get Γ _h < Γ _l , and (iii) can be derived by combining the proof of (i) and (ii). As $k-c_2^l>k-c_1^{\circ }$ ^° and ν < 1/2, to simplify Γ _l , we have

$${\Gamma }_l=\frac{(1+\frac{(2\delta -1)\nu +2}{2\delta +3})(k-c_2^l)-(k-c_1^{\circ })}{b\mu (L-H)}>0$$

where it is uncertain whether Γ _h is positive or negative. Combining these, we find that Proposition 2 leads to the following results. Table 1 illustrates Proposition 2 more directly by taking account of the value range of ρ (0 < ρ < 1).

Table 1.

Γ _h , Γ _l and ρ with respect to q₁, $q_2^h$ , $q_2^l$ ’s relationship.

${\Gamma }_h$ and ${\Gamma }_l$	ρ’s range size	$q_1$ , $q_2^h$ and $q_2^l$	Expected market share
$0<{\Gamma }_h<{\Gamma }_l<1$	${\Gamma }_l<\rho <1$	$q_2^h<q_2^l<q_1$	$E{M}_{1,\mathrm{as}}>E{M}_{2,\mathrm{as}}$
	$0<\rho <{\Gamma }_h$	$q_1<q_2^h<q_2^l$	$E{M}_{1,\mathrm{as}}<E{M}_{2,\mathrm{as}}$
	${\Gamma }_h<\rho <{\Gamma }_l$	$q_2^h<q_1<q_2^l$	${\mathrm{EM}}_{2,\mathrm{as}}^h<E{M}_{1,\mathrm{as}}<{\mathrm{EM}}_{2,\mathrm{as}}^l$
${\Gamma }_h\le 0<{\Gamma }_l<1$	${\Gamma }_l<\rho <1$	$q_2^h<q_2^l<q_1$	$E{M}_{1,\mathrm{as}}>E{M}_{2,\mathrm{as}}$
	No $\rho$ makes	$q_1<q_2^h<q_2^l$	$E{M}_{1,\mathrm{as}}<E{M}_{2,\mathrm{as}}$
	$0<\rho <{\Gamma }_l$	$q_2^h<q_1<q_2^l$	${\mathrm{EM}}_{2,\mathrm{as}}^h<E{M}_{1,\mathrm{as}}<{\mathrm{EM}}_{2,\mathrm{as}}^l$
${\Gamma }_l>{\Gamma }_h\ge 1$	$\forall \rho$ , $0<\rho <1$	$q_1<q_2^h<q_2^l$	$E{M}_{1,\mathrm{as}}<E{M}_{2,\mathrm{as}}$
	No $\rho$ makes	$q_2^h<q_1<q_2^l$	${\mathrm{EM}}_{2,\mathrm{as}}^h<E{M}_{1,\mathrm{as}}<{\mathrm{EM}}_{2,\mathrm{as}}^l$
		$q_2^h<q_2^l<q_1$	$E{M}_{1,\mathrm{as}}>E{M}_{2,\mathrm{as}}$
${\Gamma }_l\ge 1>{\Gamma }_h$	$0<\rho <{\Gamma }_h$	$q_1<q_2^h<q_2^l$	$E{M}_{1,\mathrm{as}}<E{M}_{2,\mathrm{as}}$
	${\Gamma }_h<\rho <1$	$q_2^h<q_1<q_2^l$	${\mathrm{EM}}_{2,\mathrm{as}}^h<E{M}_{1,\mathrm{as}}<{\mathrm{EM}}_{2,\mathrm{as}}^l$
	No $\rho$ makes	$q_2^h<q_2^l<q_1$	$E{M}_{1,\mathrm{as}}>E{M}_{2,\mathrm{as}}$

The comparison of M₁ and M₂’s target production quantity suggests that a larger production quantity means a bigger expected market share in a direct relationship. After determining the value of Γ _h and Γ _l , the value of ρ also determines the relationship of $q_1$ , $q_2^h$ and $q_2^l$ . Traditionally, researchers assume that ρ is common knowledge which is fixed and cannot be changed. However, in the real world, this may not be true because the competitors’ ability to format the competition information could be different according to their market capability. As denoted in section “Asymmetric Cournot duopoly with random yield,” the unit production cost of M₁ is common knowledge, but the cost type of M₂, $c_2^j(j=h,l)$ is asymmetric to M₁. It also implies that M₂ is more competitive than M₁ in capturing her rival’s cost-type information than M_1. In section “An extension: ρ is ‘made’ and released by M₂,” we study further the assumption that M₂ has a greater capability in controlling the market.

• Lemma 5. (i) Expected market output of the asymmetric case decreases with yield uncertainty δ when M₂ is low cost type and (ii) when M₂ is high cost type, expected market output decreases with δ if

$$\rho >1-\frac{2k-c_1^{\circ }-c_2^h}{\nu (c_2^h-c_2^l)}$$

• Proof of (i). This is similar to the proof of Lemma 1.

• Proof of (ii). Simplifying $E{q}_{t,\mathrm{as}}^h$ , we have

$$\begin{array}{c}{\mathrm{Eq}}_{t,\mathrm{as}}^h=\mu (q_1^{*}+q_2^{h*})\hfill \\ =\frac{{\mu }^2((1-\nu )(k-c_1^{\circ })+(1-{\nu }^2)(k-c_2^h)+\rho (\nu -{\nu }^2)(c_2^h-c_2^l)-(\nu -{\nu }^2)(k-c_2^l))}{2b(1-{\nu }^2)({\mu }^2+{\sigma }^2)}\hfill \\ =\frac{{\mu }^2((k-c_1^{\circ })+(1+\nu )(k-c_2^h)+\rho \nu (c_2^h-c_2^l)-\nu (k-c_2^l))}{3b{\mu }^2+2b{\sigma }^2}\hfill \\ =\frac{(k-c_1^{\circ })+(1+\nu )(k-c_2^h)+\rho \nu (c_2^h-c_2^l)-\nu (k-c_2^l)}{3b+2{\delta }^2}\hfill \end{array}$$

As $\delta >0$ , by solving $\partial {\mathrm{Eq}}_{t,\mathrm{as}}^h/\partial \delta <0$ , then (ii) is proved. Lemma 5 shows whether the expected market output decrease with yield uncertainty δ depends on M₂’s cost type. In the following section, we mainly focus on the situation when the duopolist choose to monopolize the market.

The case of cartel-monopoly

Duopoly manufacturers tend to form a combination of business organizations to control the market. For example, members of Organization of Petroleum Exporting Countries (OPEC), the well-known oil monopoly organization with cartel characteristics, aim to maximize their earnings by regulating their output. Prokop ²⁵ considered the process of cartel formation in the industries characterized by collusive price leadership. Andaluz ²⁶ analyzed the influence of the degree of differentiation on cartel sustainability, under both price and quantity competition in the context of a vertically differentiated duopoly. The literatures above mainly focused on the existence of cartel stability. In this section, we attempt to derive the best monopoly policy for the duopoly manufacturers and study the effect of random yield on market output. The duopoly firms would join up to maximize their total expected profit as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\pi }_{\mathrm{mon}}=Q_t(k-b{Q}_t)}\\ -(c_{11}{q}_1+{\alpha }_1{c}_{12}{q}_1+c_{21}{q}_2+{\alpha }_2{c}_{22}{q}_2)\hfill \\ =({\alpha }_1{q}_1+{\alpha }_2{q}_2)(k-b({\alpha }_1{q}_1+{\alpha }_2{q}_2))\hfill \\ -(c_{11}{q}_1+{\alpha }_1{c}_{12}{q}_1+c_{21}{q}_2+{\alpha }_2{c}_{22}{q}_2)\hfill \end{array}$$

(19)

Their expected total profit can be maximized by deciding their target production set (q₁,q₂)

$\begin{array}{l}\underset{\left(q_1,q_2\right)}{\max }E\left({\pi }_{mon}\right)=E_{{\alpha }_i}[\left({\alpha }_1{q}_1+{\alpha }_2{q}_2\right)\left(k-b\left({\alpha }_1{q}_1+{\alpha }_2{q}_2\right)\right)\\ -\left(c_{11}{q}_1+{\alpha }_1{c}_{12}{q}_1+c_{21}{q}_2+{\alpha }_2{c}_{22}{q}_2\right)\\ =k\mu \left(q_1+q_2\right)-b\left({\sigma }^2+{\mu }^2\right)\left(q_1^2+q_2^2\right)\\ -2b{\mu }^2{q}_1{q}_2-\mu \left(c_1\text{}\text{}°q_1+c_2\text{}\text{}°q_2\right)\\ =\left(k-c_1\text{}\text{}°\right)\mu q_1+\left(k-c_2\text{}\text{}°\right)\mu q_2\\ -b\left(\left({\sigma }^2+{\mu }^2\right)\left(q_1^2+q_2^2\right)+2{\mu }^2{q}_1{q}_2\right)\end{array}$

(20)

• Theorem 3. Under random yield, there exists an optimal target production policy $(q_1^{*},q_2^{*})$ for the duopoly firms when they monopolize together.

• Proof. To derive the first-order condition of Eπ_mon with respect to q₁ and q₂, we have

$$\begin{array}{c}\frac{\partial E{\pi }_{\mathrm{mon}}}{\partial q_1}=(k-c_1^{\circ })\mu -b(2({\sigma }^2+{\mu }^2)q_1+2{\mu }^2{q}_2);\hfill \\ \frac{\partial E{\pi }_{\mathrm{mon}}}{\partial q_2}=(k-c_2^{\circ })\mu -b(2({\sigma }^2+{\mu }^2)q_2+2{\mu }^2{q}_1)\hfill \end{array}$$

To derive the second order condition (soc) of Eπ_mon on q₁ and q₂, we have

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_1^2}}& =\frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_2^2}=-2b({\sigma }^2+{\mu }^2);\hfill \\ \multicolumn{1}{c}{\frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_1{q}_2}}& =\frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_2{q}_1}=-2b{\mu }^2\hfill \end{array}$$

The Hessian matrix is written as

$$\begin{array}{c}\multicolumn{1}{c}{H_m=\left|\begin{array}{cc}\hfill \frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_1^2}\hfill & \hfill \frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_1{q}_2}\hfill \\ \multicolumn{1}{c}{\frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_2{q}_1}}& \hfill \frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_2^2}\hfill \end{array}\right|=\frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_1^2}·\frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_2^2}}\\ \multicolumn{1}{c}{-{\left(\frac{{\partial }^{2}E{\pi }_{\mathrm{mon}}}{\partial q_1{q}_2}\right)}^2=4b^2(2{\sigma }^2{\mu }^2+{\sigma }^4)>0}\end{array}$$

Because $H_m>0$ and ${\partial }^{2}E{\pi }_{\mathrm{mon}}/\partial q_1^2<0$ , there exists an optimal production decision $(q_1^{*},q_2^{*})$ to maximize $E{\pi }_{\mathrm{mon}}$ . $(q_1^{*},q_2^{*})$ can be solved by setting $\partial E{\pi }_{\mathrm{mon}}/\partial q_1=0$ and $\partial E{\pi }_{\mathrm{mon}}/\partial q_2=0$ . Furthermore, we have

$$\{\begin{array}{c}{q}_1^{*}=\frac{(k-c_1^{\circ })({\mu }^5+2{\sigma }^2{\mu }^3+{\sigma }^4\mu )-(k-c_2^{\circ })({\mu }^5+{\sigma }^2{\mu }^3)}{2b{\sigma }^2({\mu }^2+{\sigma }^2)(2{\mu }^2+{\sigma }^2)}\hfill \\ q_2^{*}=\frac{(k-c_2^{\circ })({\mu }^3+{\sigma }^2\mu )-(k-c_1^{\circ }){\mu }^3}{2b({\sigma }^4+2{\sigma }^2{\mu }^2)}\hfill \end{array}$$

(21)

$(q_1^{*},q_2^{*})$ is the best target production quantity for the duopoly manufacturers under the monopoly situation. Comparing $q_1^{*}$ and $q_2^{*}$ in equation (21), we have Proposition 3:

• Proposition 3. In the monopoly case, manufacturer with lower cost owns a larger (expected) market share, that is, (i) if $c_1^{\circ }\le c_2^{\circ }$ , there is $q_1^{*}\ge q_2^{*}$ , $E{M}_{1,\mathrm{mon}}\ge E{M}_{2,\mathrm{mon}}$ and (ii) if $c_1^{\circ }>c_2^{\circ }$ , there is $q_1^{*}<q_2^{*}$ , $E{M}_{1,\mathrm{mon}}<E{M}_{2,\mathrm{mon}}$ .

• Proof. Because $q_1^{*}-q_2^{*}=(c_2^{\circ }-c_1^{\circ })\mu /2b{\sigma }^2$ , thus, $c_1^{\circ }$ and $c_2^{\circ }$ determine the relationship of $q_1^{*}$ and $q_2^{*}$ . The proof of the comparison of expected market share is similar to the proof of Proposition 1. From equation (21), we can also find that, in the monopoly case, q_i also decreases with $c_i^{\circ }$ . Proposition 2 indicates that, under random yield, the manufacturer with cost advantage (low cost) shares a larger (expected) market and expected profit. This is commonly seen in the real world as the low-cost manufacturer having a competitive advantage in production.

• Lemma 6. (i) In the monopoly case, the expected market output decreases with δ and (ii) the expected market output under the monopoly case is smaller than the symmetric case, that is, $E{q}_{t,\mathrm{mon}}<E{q}_{t,s}$ .

• Proof of (i). This is similar to the proof of Lemma 1.

• Proof of (ii). This is given by

$$\begin{array}{cc}\multicolumn{1}{c}{E{q}_{t,\mathrm{mon}}}& =\mu (q_1^{*}+q_2^{*})=\frac{{\mu }^2{\sigma }^2(2k-c_1^{\circ }-c_2^{\circ })}{2b(2{\mu }^2{\sigma }^2+{\sigma }^4)}\hfill \\ =\frac{2k-c_1^{\circ }-c_2^{\circ }}{2b(2+{\delta }^2)}\hfill \end{array}$$

As $k-c_i^{\circ }>0$ , then, $2k-c_1^{\circ }-c_2^{\circ }>0$ . To solve $E{q}_{t,\mathrm{mon}}-E{q}_{t,s}$ , we have

$$\begin{array}{l}E{q}_{t,mon}-E{q}_{t,s}=\frac{2k-c_1\text{}\text{}°-c_2\text{}\text{}°}{2b\left(2+{\delta }^2\right)}-\frac{\left(1+2{\delta }^2\right)\left(2k-c_1\text{}\text{}°-c_2\text{}\text{}°\right)}{b\left(3+8{\delta }^2+4{\delta }^4\right)}\\ =-\frac{\left(2{\delta }^2+1\right)\left(2k-c_1\text{}\text{}°-c_2\text{}\text{}°\right)}{2b\left(2+{\delta }^2\right)\left(3+8{\delta }^2+4{\delta }^4\right)}<0\end{array}$$

which proves (ii). Comparing the results with the asymmetric case and combining Lemma 6, we have

• Lemma 7. (i) If M₂ in the asymmetric case is low cost type, then there is $E{q}_{t,\mathrm{mon}}<E{q}_{t,s}<{\mathrm{Eq}}_{t,\mathrm{as}}^l$ and (ii) if M₂ in the asymmetric case is high cost type, then there is ${\mathrm{Eq}}_{t,\mathrm{as}}^h<E{q}_{t,\mathrm{mon}}<E{q}_{t,s}$ when $\rho$ satisfies

$$\begin{array}{cc}\multicolumn{1}{c}{1>\rho }& >\frac{(3+8{\delta }^2+4{\delta }^4)(k-c_2^h)+(1+3{\delta }^2+2{\delta }^4)(k-c_1^{\circ })}{(2+5{\delta }^2+2{\delta }^4)(c_2^l-c_2^h)}\hfill \\ -\frac{k-c_2^l}{c_2^l-c_2^h}\hfill \end{array}$$

• Proof of (i). Because $E{q}_{t,s}<{\mathrm{Eq}}_{t,\mathrm{as}}^l$ has already been proved in Lemma 4, combining the results of Lemma 6, then we have (i).

• Proof of (ii). This is derived by solving the difference of $E{q}_{t,\mathrm{mon}}$ and ${\mathrm{Eq}}_{t,\mathrm{as}}^h$ . Lemma 6 indicates that in the monopoly case, consumers suffer from a larger expected market price $(\mathrm{Ep}=k-\mathrm{bE}{q}_t)$ compared with the symmetric case. Lemma 7 shows that under random yield, the expected market output in the asymmetric case is smaller compared with the monopoly case when there exists a large enough ρ.

An extension: ρ is “made” and released by M₂

M₂’s expected profit

In some activities of commercial and social politics, certain participants have the ability to release some information to confuse their competitor(s). Stronger participants may not only have superiority in getting access to important intelligence but also perform better in creating valuable information which can change their opponent’s decision-making. In a monopolistic market, the existing monopoly enterprise would like to pretend to be an extreme low cost type which will hold back the potential entry by threatening to launch a price war. Because cost c₂ is not known to M₁, M₁’s production decision depends on the estimation of the probability of M₂’s cost type. In many competitive industries, stronger firms tend to take actions and release some information to induce or confuse their opponent firm’s appraisal of her cost type or financial condition. We further assume that the probability that M₁ considers M₂’s cost type is gained by evaluating several pieces of marketing information which are formatted and released by the stronger M₂. That is, ρ is “made” by M₂. M₂ could affect M₁’s production decision by covertly controlling M₁’s evaluation of ρ. Because M₁ is not aware that ρ is “made” by M₂, M₂ could choose a best ρ by calculating M₁’s best reaction function to maximize M₂’s own expected profit function. To solve for the optimal condition of the high cost–type M₂, for example, M₂’s expected profit (from equation (13)) can be turned into

$$E_{{\alpha }_i}\left({\pi }_{M_2}\right)=\mu (k-c_2^h)q_2^h-b({\mu }^2(q_1+q_2^h)+{\sigma }^2{q}_2^h)q_2^h$$

From equation (13), it could be found that the realization of $q_1$ and $q_2^h$ is correlated with ρ. Thus, we could take M₂’s expected profit function as a function of ρ. Combining equations (13), (15) and (18), we have Proposition 4.

• Proposition 4. The expected profit function of M₂ is a convex function on the probability ρ.

• Proof. Because M₁ is not aware that her measurement of ρ is “made” by M₂, the process of obtaining the optimal target production quantity equilibrium $(q_1^{*},q_2^{*})$ does not affect the optimization of ρ. To derive the first-order condition of M₂’s expected profit function on ρ, we have

$$\begin{array}{c}\frac{\partial E_{{\alpha }_i}\left({\pi }_{M_2}\left({\rho }_h\right)\right)}{\partial {\rho }_h}=(H-L)\hfill \\ (\frac{\nu {\mu }^3(k-c_2^h)}{2({\mu }^2+{\sigma }^2)-\nu {\mu }^2}-\frac{(2b\nu {\mu }^2({\mu }^2+{\sigma }^2)(\mathrm{bH}(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)+b\nu {\mu }^{2}L-\nu \mu (k-c_1^{\circ }))+b\nu {\mu }^5((k-c_1^{\circ })-2b\mu L)-\frac{1}{2}{b}^2\nu {\mu }^6(H-L)\rho )}{b{(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)}^2})\hfill \end{array}$$

(22)

To derive the second-order condition on ρ, we obtain

$$\frac{{\partial }^2{E}_{{\alpha }_i}\left({\pi }_{M_2}\left({\rho }_h\right)\right)}{\partial {\rho }^2}=\frac{b{\mu }^6{(H-L)}^2}{2({\mu }^2+{\sigma }^2){(2{\sigma }^2+(2-\nu ){\mu }^2)}^2}>0$$

(23)

A same result can be reached when M₂ is a low cost type. Thus, we can derive that there exists a unique ρ to minimize the expected profit function of M₂. Recall in this section that ρ is decided by M₂. For M₂, the final realization of ρ depends on her market capability which she needs to monitor in order to make sure that M₁ believes it. Because of that consideration, an extreme value of ρ (such as 0 or 1) cannot be believed by M₁. Generally, we use [ρ₁, ρ₂] (0 < ρ₁ < ρ₂ < 1) to denote the probability interval that can be controlled by M₂, and then, we have Proposition 5.

Proposition 5. To maximize the expected profit function, M₂ should make her optimal decision as follows:

• If ${\rho }_h^{\circ }<1/2({\rho }_1+{\rho }_2)$ , then ${\rho }^{*}={\rho }_2$ , and $\max E_{{\alpha }_i}({\pi }_{M_2})=E_{{\alpha }_i}({\pi }_{M_2}({\rho }_2))$ ;

• If ${\rho }_h\text{}\text{}°>1/2({\rho }_1+{\rho }_2)$ , then ${\rho }^{*}={\rho }_1$ , and $\max E_{{\alpha }_i}({\pi }_{M_2})=E_{{\alpha }_i}({\pi }_{M_2}({\rho }_1))$ .

where ρ_h^° stands for the extreme point of E(π₂(ρ)). ρ^* maximizes M₂’s expected profit. By setting equation (14) equal to 0, and the “worst”ρ for M₂ (high cost type) is given as follows

$${\rho }_h^{\circ }=\frac{2\left(\left(1-b\right)\mu \left(k-c_2^h\right)\left(2\left({\mu }^2+{\sigma }^2\right)-\nu {\mu }^2\right)-{\mu }^3\left(\left(k-c_1^{\circ }\right)-2b\mu L\right)-2\left({\mu }^2+{\sigma }^2\right)\left(b\nu {\mu }^{2}L-\nu \mu \left(k-c_1^{\circ }\right)\right)\right)}{b{\mu }^4\left(L-H\right)}$$

(24)

• Proof of (i). If ${\rho }_h\text{}\text{}°<({\rho }_1+{\rho }_2)/2$ , then we have ${\rho }_h^{\circ }-{\rho }_1<{\rho }_2-{\rho }_h^{\circ }$ . Because the function $E{\alpha }_i({\pi }_{M_2}(\rho ))$ is monotonically decreasing on ρ when $\rho <{\rho }_h^{\circ }$ , and monotonically increasing on ρ when $\rho \ge {\rho }_h^{\circ }$ . Because $E{\alpha }_i({\pi }_{M_2}(\rho ))$ is axially symmetrical on ${\rho }_h^{\circ }$ , therefore, $E{\alpha }_i({\pi }_{M_2}({\rho }_1))-E{\alpha }_i({\pi }_{M_2}({\rho }_h^{\circ }))<E{\alpha }_i({\pi }_{M_2}({\rho }_2))-E{\alpha }_i({\pi }_{M_2}({\rho }_h^{\circ }))$ , thus, we have (i): ρ^* = ρ₂.

• Proof of (ii). This can be obtained similarly. For some examples of this phenomenon, see section “Numerical illustrations.”

Consumer surplus

Formally, when the production quantity is denoted as Q, then the expected consumer utility can be expressed as

$$E\left[\mathrm{CU}\left(Q\right)\right]=E\left\{\underset{0}{\overset{Q}{\int }}(k-\mathrm{bx})\mathrm{dx}\right\}$$

(25)

As the expected payment by the consumer is denoted as E[(k−bQ)Q], then the expected consumer surplus under random yield is given by

$$\begin{array}{c}{E}_{{\alpha }_i}\left[\mathrm{CS}\left(Q_U\right)\right]\hfill \\ =E_{{\alpha }_i}\{\underset{0}{\overset{\sum _{i=1}^2{\alpha }_i{q}_i^{*}}{\int }}(k-\mathrm{bx})\mathrm{dx}-(k-b\sum _{i=1}^2{\alpha }_i{q}_i^{*})·\sum _{i=1}^2{\alpha }_i{q}_i^{*}\}\hfill \\ =\frac{b}{2}(2{\mu }^2{q}_1^{*}{q}_2^{*}+({\mu }^2+{\sigma }^2)(q_1^{*2}+q_2^{*2}))(26)\hfill \end{array}$$

(26)

• Proposition 6. The expected consumer surplus under asymmetric information is convex on ρ.

• Proof. To derive the first-order condition of equation (26) on ρ, we have

$$\begin{array}{c}\frac{\partial E_{{\alpha }_i}\left[C{S}_{\mathrm{as}}\left(Q_U\right)\right]}{\partial \rho }\hfill \\ =\frac{\begin{array}{c}\hfill \mu ({\mu }^2+{\sigma }^2)\left(\begin{array}{c}\hfill 2b{\mu }^3(1+{\nu }^2)(\rho H^2-(1-\rho )L^2+(1-2\rho )\mathrm{HL})\hfill \\ \multicolumn{1}{c}{+2\nu H(2b\mu -(k-c_1^{\circ }))(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)(H-L)}\\ \multicolumn{1}{c}{-2(1+{\nu }^2){\mu }^2(k-c_1^{\circ })(H-L)}\end{array}\right)\hfill \\ \multicolumn{1}{c}{+2{\mu }^4\left(\begin{array}{c}\hfill -2b\nu {\mu }^2(\rho H^2-(1-\rho )L^2+(1-2\rho )\mathrm{HL})\hfill \\ \multicolumn{1}{c}{+2(\nu \mu (k-c_1^{\circ })-H(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2))(H-L)}\end{array}\right)}\end{array}}{8{(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)}^2}\hfill \end{array}$$

(27)

To derive the second-order condition on ρ, we obtain

$$\begin{array}{c}\frac{{\partial }^2{E}_{{\alpha }_i}\left[C{S}_{\mathrm{as}}\left(Q_U\right)\right]}{\partial {\rho }^2}\hfill \\ =\frac{b{\mu }^4({\mu }^2+{\sigma }^2)(1+{\nu }^2){(H-L)}^2-2b\nu {\mu }^6{(H-L)}^2}{4{(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)}^2}\hfill \\ =b{\mu }^4\frac{{\sigma }^2(1+{\nu }^2)+{\mu }^2{(1-\nu )}^2}{4{(2({\mu }^2+{\sigma }^2)-\nu {\mu }^2)}^2}{(H-L)}^2>0\hfill \end{array}$$

(28)

Equation (28) indicates that there exists a unique ${\rho }_c^{*}$ which could minimize the expected consumer surplus. ${\rho }_c^{*}$ can be obtained by setting equation (27) to 0

$${\rho }_c^{\ast }=\frac{\left({\mu }^2+{\sigma }^2\right)\left(2b{\mu }^3\left(1+{\nu }^2\right)L-2\left(1+{\nu }^2\right){\mu }^2\left(k-c_1^{\circ }\right)+2\nu H\left(2b\mu -\left(k-c_1^{\circ }\right)\right)\left(2\left({\mu }^2+{\sigma }^2\right)-\nu {\mu }^2\right)\right)+4{\mu }^3\left(\left(\nu \mu \left(k-c_1^{\circ }\right)-H\left(2\left({\mu }^2+{\sigma }^2\right)-\nu {\mu }^2\right)\right)-b\nu {\mu }^{2}L\right)}{2b{\mu }^3\left(2\nu {\mu }^2-\left(1+{\nu }^2\right)\right)\left(H-L\right)}$$

(29)

Based on Propositions 5 and 6, we derive that the optimization of M₂’s expected profit is highly related to consumer surplus:

Proposition 7. The optimization of M₂’s expected profit is highly related to the maximization of consumer surplus under asymmetric information:

• If ${\rho }_h^{\circ }<1/2({\rho }_1+{\rho }_2)$ , and ${\rho }_2<{\rho }_c^{*}$ , expected consumer surplus reaches its minimum value $E_{{\alpha }_i}[C{S}_{\mathrm{as}}({\rho }_2)]$ (as M₂ will choose ${\rho }_2$ to maximize her expected profit);

• If ${\rho }_h^{\circ }<1/2({\rho }_1+{\rho }_2)$ , and ${\rho }_c^{*}<1/2({\rho }_1+{\rho }_2)$ , expected consumer surplus reaches its maximum value $E_{{\alpha }_i}[C{S}_{\mathrm{as}}({\rho }_2)]$ (as M₂ will choose ${\rho }_2$ to maximize her expected profit);

• If ${\rho }_h^{\circ }>1/2({\rho }_1+{\rho }_2)$ , and ${\rho }_1>{\rho }_c^{*}$ , expected consumer surplus reaches its minimum value $E_{{\alpha }_i}[C{S}_{\mathrm{as}}({\rho }_2)]$ (as M₂ will choose ${\rho }_1$ to maximize her expected profit);

• If ${\rho }_h^{\circ }>1/2({\rho }_1+{\rho }_2)$ , and ${\rho }_c^{*}>1/2({\rho }_1+{\rho }_2)$ , expected consumer surplus reaches its maximum value $E_{{\alpha }_i}[C{S}_{\mathrm{as}}({\rho }_2)]$ (as M₂ will choose ${\rho }_1$ to maximize her expected profit).

Proof of (i) and (ii). If ${\rho }_h^{\circ }<({\rho }_1+{\rho }_2)/2$ , from Proposition 5, M₂ will choose ρ^* = ρ₂ to maximize her profit, when ${\rho }_1<{\rho }_2<{\rho }_c^{*}$ , then in the interval (ρ₁, ρ₂), which is the probability interval that can be controlled by M₂, Eα_i[CS_as(ρ)] is decreasing in ρ. Thus, Eα_i[CS_as(ρ)] reaches its minimum value as Eα_i[CS_as(ρ₂)], when ${\rho }_c^{*}<({\rho }_1+{\rho }_2)/2$ , Eα_i[CS_as(ρ)] is increasing in ρ, thus, Eα_i[CS_as(ρ)] reaches its maximum value as Eα_i[CS_as(ρ₂)].

Proof of (iii) and (iv). This is similar to the proof of (i) and (ii).

The results above can be used as a reason for government policies to control monopolistic behavior. To gain more insights into the results given above and the impact of yield uncertainty upon Cournot duopoly competition model, numerical examples are presented in the next section.

Numerical illustrations

We take the values of the parameters involved in the model as b = 0.52, k = 800, c₁₁ = 0.93, c₁₂ = 0.2, $c_{21}^h=0.95$ , $c_{22}^h=0.6$ , $c_{21}^l=0.2$ , $c_{22}^l=0.15$ , ρ₁ = 0.35 and ρ₂ = 0.70. First, we fix µ = 0.83, for example (any µ, 0 < µ < 1 is suitable), and use δ to study the effect of yield uncertainty on the expected profits of manufacturers in three cases (symmetric, asymmetric and monopoly). Since $c_i^{\circ }=(c_{i1}+\mu c_{i2})/\mu$ , then $c_1^{\circ }=1.22$ , $c_2^h=1.74$ and $c_2^l=0.39$ .

In the following discussions, we assume that M₂ is high cost type. To study the effect of yield uncertainty on the market, we draw Figure 2 to study the effect of yield uncertainty on the expected market output.

Figure 2.

Comparison of expected market output in three cases when δ varies

From Figure 2, we can see that the expected market output of all three cases decreases when δ increases. The comparison shows that $E{q}_{t,\mathrm{mon}}<E{q}_{t,\mathrm{as}}<E{q}_{t,s}$ , which follows the results derived in Lemmas 4–6. Figure 2 indicates that consumers would suffer from the highest market price in the monopoly case. As δ increases, the output in all three situations all becomes very small. This can be explained by the idea that when yield uncertainty becomes too large, manufacturers have little willingness to produce, which could lead to a small market output. To further study the effect of ρ on output and profit in the asymmetric situation, we fix δ = 0.83 (any δ, δ > 0 is suitable).

Table 2 explores the optimal production decisions when ρ is controlled by M₂. From Table 2, it can be observed that q_as1, $(q_{\mathrm{as}1}-q_{\mathrm{as}2}^h)$ and the expected market output ${\mathrm{Eq}}_{t,\mathrm{as}}^h$ all increase when ρ increases, and that $q_{\mathrm{as}2}^h$ decreases when ρ increases. This also indicates that there is a larger increase in q_as1 compared with the decrease in $q_{\mathrm{as}2}^h$ . We also find that $E{\pi }_{t,M_1}^h$ increases with ρ. This can be explained that if M₁ evaluates M₂ as high cost type, she would produce more due to her cost advantage. Substituting the parameters into equation (24), we obtain ${\rho }_h^{\circ }=66.68$ . From Proposition 3, we have ${\rho }_h^{\circ }=66.68>0.525=({\rho }_1+{\rho }_2)/2$ . Thus, ρ^* = ρ₁ = 0.35. E(π₂(ρ)) reaches its maximum when ρ^* = 0.35, and reaches its minimum when ρ^* = 0.7. Substituting the parameters into equation (29), we have ρ_c = −1634.9. However, ρ is defined in [0,1], and ρ can only be reached in [0.35,0.7] for M₂. Because M₂ would choose ρ^* = ρ₁ = 0.35, the expected consumer surplus reaches its minimum as ECS_as = 1.5379 × 10⁶, which follows the results given in Proposition 7. Figure 3 shows the effect of ρ on the duopolist’s expected market share. From Figure 3, we can see that, as ρ increases, M₁’s expected market share increases and M₂’s decreases.

Table 2.

Optimal results for different values of ρ.

ρ	qas 1	$q_{\mathrm{as}2}^h$	$q_{\mathrm{as}1}-q_{\mathrm{as}2}^h$	${\mathrm{Eq}}_{t,\mathrm{as}}^h$	$E{\pi }_{\mathrm{as},M_1}$	$E{\pi }_{\mathrm{as},M_2}^h$	ECSas
0.35	60.4575	45.4492	15.0083	87.9026	1632.6	1340.6	1.5379 × 106
0.40	60.7044	45.3761	15.3282	88.0468	1634.2	1337.1	1.5455 × 106
0.45	60.9512	45.3031	15.6482	88.1911	1635.8	1333.6	1.5531 × 106
0.50	61.1981	45.2300	15.9681	88.3353	1637.4	1330.1	1.5607 × 106
0.55	61.4449	45.1569	16.2880	88.4795	1638.9	1326.6	1.5682 × 106
0.60	61.6918	45.0838	16.6080	88.6238	1640.4	1323.1	1.5757 × 106
0.65	61.9387	45.0107	16.9279	88.7680	1641.8	1319.6	1.5832 × 106
0.70	62.1855	44.9376	17.2479	88.9122	1643.2	1316.2	1.5907 × 106

Figure 3.

The effect of ρ on the duopolist’s expected market share.

Discussion and conclusion

The duopoly model with a Cournot form with yield randomness has received little attention. In this article, a Cournot duopoly production competition model was examined under random yield. The model was analyzed by giving the expectation value and variance of the random yield risk in a symmetric, asymmetric and monopoly environment, respectively. In a symmetric and asymmetric situation, we respectively derive that there exist a Nash and Bayes–Nash target production policies for the duopoly manufacturers. For the monopoly situation, an optimal production policy which could maximize their total expected profit was given.

The effect of yield uncertainty on the expected market output has been studied in each case. Regardless of the random yield and the competition environment, the unit production cost always plays a negative role in the manufacturers’ setting of a target production quantity, which is the same within the deterministic yield case. The comparison of the duopolist’s expected market share has been studied. We also compared the expected market output among the three situations. Then, we extended the asymmetric model by considering a powerful oligarch who can exert influence on her rival’s production decision by releasing her own cost-type information. We derive that the maximization of the stronger manufacturer’s expected profit is highly correlated with consumer surplus.

To the best of our knowledge, few researchers have studied the Cournot duopoly model under random yield. The model we analyzed is a static game. However, firms may compete in a dynamic way. Thus, it is necessary and worthwhile to extend our model in the future to study the case when the manufacturers play dynamically.