1 Introduction

The purpose of the present paper is to investigate a market with two competing producers of an identical commodity. We consider two stages: before globalization (separate markets) and after globalization (united market). Before globalization, each producer satisfies the separate demand of the market that it monopolizes. After globalization, both firms compete in a united market. This model is often said to have the structure of a pure (classic) duopoly where both companies satisfy the complete market demand.

One can find numerous studies on the effects of combining two or more markets in the literature. According to [1], there are two types of global markets: (a) the free trade market which allows the existence of n different markets with a separate supplier; and (b) a single integrated market in which all producers compete.

Since the 1980s, there has been a lot of research on the role of imperfect competition. This was pointed out in [2], which deals with global markets of type (a). In fact, there are several works which models correspond to these type of markets. Some examples are found in [3,4,5], too. On the other hand, [1] analyzes a globalized market of type (b), through a Nash-Cournot equilibrium model, whereas in [1], the authors examine cases where all producers’ profits are degraded in the same manner. For each producer, they use the ratio of the profit obtained after globalization to the profit before globalization to represent the degree of the profit degradation, and the largest of the ratios among the producers is a measure of coincident degradation. They found that under a complete symmetry, i.e. when the values of parameters of cost and demand functions are equal, all producers have profit degradation coincidently. For the model they use which boasts linear demand functions for the separated markets and the globalized market, as well as linear cost functions, under Nash-Cournot conjectures, the value of the measure of coincident degradation is the lowest (the worst) when the firms are identical.

The present paper also discusses the situation of type (b). As in [1], we use the ratio of the profit obtained after globalization to the profit before globalization to represent the degree of profit degradation or improvement. However, our purpose is to analyze the effects of globalization considering the diverse values that can take the parameters of the cost functions of the companies, which in our case are quadratic. We reveal that is possible that one producer loses while the other one gains; or both lose.

Next, at the stage of globalization, when competition takes place, we raise a kind of equilibrium with consistent conjectural variations (CCVE). Conjectural Variations Equilibria (CVE) were introduced by Bowley in 1924 [6] and Frisch in 1933 [7] as another possible solution concept in static games. According to this concept, agents behave as follows as was stated in [8]: each agent chooses her most favorable action taking into account that every rival’s strategy is a conjectured function of her own strategy. In [8, 9], we studied mixed oligopoly models with consistent conjectural variations (CCV), which correspond to the market price variations due to the change in the output level of a producer. Concepts such as exterior and interior equilibrium were introduced, and proofs of existence and uniqueness of equilibrium were presented in the above-mentioned papers. We apply these concepts in our present paper, too.

Since any conjectures inevitable bear doubts about whether they will be followed by all the players at all, this topic has a direct link to the area of solving problems with uncertainties. The conjectures accepted by each agent can be considered as attempts to evaluate the robustness of the model’s solution subject to the players’ preferences and market powers.

The paper is organized by follows. In Sect. 2, we describe the mathematical model and specify the assumptions to accept for each stage. This section also shows the optimal output levels produced by each firm before globalization. The consistent conjectural variations equilibrium price and production volumes are justified in this section as well. In Sect. 3, we define two types of agents: low-marginal and high-marginal cost firms (abbreviated as LMCF and HMCF, respectively). As we study a market with 2 agents, we have four feasible situations corresponding to the possible combination of types of firms. We define the profit ratio and compute it for each situation in terms of the parameters in order to analyze the effect of the cost parameters on this ratio. To do so we use the concept of technical advantage introduced by [10]. In that section, we also display an example showing that, unlike the Nash-Cournot case [1], a complete symmetry does not necessarily render the worst-case ratio under consistent conjectures. Finally, in Sect. 4, we present our conclusions and outline our future work. The list of references and acknowledgments finish the paper.

2 The Model Specification

We assume that before globalization, there exist two monopolistic markets. Each monopoly faces an active demand \( D_{i} , \, i = 0,1 \), which does not depend on market price, and current demand \( G_{i} = G_{i} \left( {p_{i} } \right), \, i = 0,1 \), whose argument \( p_{i} \) is the market clearing price. We will also assume that in every market, the price value \( p_{i} = \bar{P} \) is the cap price. This means that the demand functions have a discontinuity point (a break) and for prices higher than \( \bar{P} \) the demand is zero. Therefore, the company \( i \) output volume, \( q_{i} \ge 0 , \) will satisfy the following inequality if the market is “balanced”:

$$ g_{i} \left( {p_{i} } \right) + D_{i} \le q_{i} \le G_{i} \left( {p_{i} } \right) + D_{i} , \, i = 0,1. $$
(1)

Here, \( g_{i} \left( {p_{i} } \right) \) is the right limit of the function \( G_{i} , \, i = 0,1 \), and it may happen that \( g_{i} \left( {p_{i} } \right) < G_{i} \left( {p_{i} } \right) \) for some price \( p_{i} \), whereas the left limit of the current demand function at each point is assumed to coincide with its proper value.

After globalization, both firms compete in the common market. The consumers’ (current) demand is described by a demand function \( G = G\left( {p_{w} } \right) \), whose argument \( p_{w} \ge 0 \) is the (common) market clearing price. An active demand value \( D \) is nonnegative and does not depend on the market price. Here we take for granted that after globalization, the cap price will be the same as before globalization. Since the demand function has a point of discontinuity (a break at the cap price \( \bar{P} \)), the balance between the demand and supply for a given price \( p_{w} \ge 0 \) is described by the following (“balance”) inequality:

$$ g\left( {p_{w} } \right) + D \le Q \le G\left( {p_{w} } \right) + D. $$
(2)

Here again, \( g = g\left( {p_{w} } \right) \) is the right limit of the function \( G = G\left( {p_{w} } \right) \) at any point \( p_{w} \ge 0 \) while \( Q = q_{0} + q_{1} \).

2.1 The Model’s Assumptions

Accept the following assumptions about the demand and cost functions in order to study the effects of globalization.

2.1.1 Before Globalization

A1.1.

The inverse demand function for each firm \( i, i \in \left\{ {0, 1} \right\}, \) is defined as follows:

$$ p_{i} \left( {\theta_{i} } \right) = \left\{ {\begin{array}{*{20}l} {\bar{P},} \hfill & {{\text{if }}0 \le \theta_{i} \le {{\bar{Q}} \mathord{\left/ {\vphantom {{\bar{Q}} 2}} \right. \kern-0pt} 2};} \hfill \\ {c - d \cdot \theta_{i} ,} \hfill & {{\text{if }}{{\bar{Q}} \mathord{\left/ {\vphantom {{\bar{Q}} 2}} \right. \kern-0pt} 2} < \theta_{i} \le {c \mathord{\left/ {\vphantom {c d}} \right. \kern-0pt} d}.} \hfill \\ \end{array} } \right. $$
(3)

Here \( c \) and \( d \) are positive values, and \( \bar{P} = c - d\bar{Q}/2 \). The total quantity demanded in the market \( i \) at the price \( p_{i} \) is \( \theta_{i} \), which includes the passive and the active quantities demanded.

A1.2.

For each \( i \in \{ 0, 1\} \), the cost function \( f_{i} (q_{i} ) \) is quadratic, i.e., \( f_{i} \left( {q_{i} } \right) = {1 \mathord{\left/ {\vphantom {1 {2a_{i} q_{i}^{2} + b_{i} q_{i} }}} \right. \kern-0pt} {2a_{i} q_{i}^{2} + b_{i} q_{i} }}, \) where \( a_{i} > 0 \) and \( 0 \le b_{i} \le c, \, i = 0,1. \)

A1.3.

Also, assume that \( \bar{P} > \mathop {\hbox{max} }\limits_{i = 0,1} \left\{ {{{a_{i} \bar{Q}} \mathord{\left/ {\vphantom {{a_{i} \bar{Q}} 2}} \right. \kern-0pt} 2} + b_{i} } \right\}. \)

2.1.2 After Globalization

A2.1.

The market inverse demand function is defined as follows:

$$ p_{w} \left( \theta \right) = \left\{ {\begin{array}{*{20}l} {\bar{P},} \hfill & {{\text{if }}0 \le \theta \le \bar{Q};} \hfill \\ {c - d \cdot {\theta \mathord{\left/ {\vphantom {\theta 2}} \right. \kern-0pt} 2},} \hfill & {{\text{if }}\bar{Q} < \theta \le {{2c} \mathord{\left/ {\vphantom {{2c} d}} \right. \kern-0pt} d}.} \hfill \\ \end{array} } \right. $$
(4)

Here \( c,d, \) and \( \bar{P} \) are defined as in A1.1. The variable \( \theta \) is the total quantity demanded (including both the passive and active demands).

Assumption A1.2 about the cost function is also made; the cost structure won’t change after globalization. As a consequence of A1.3, if \( q_{0} + q_{1} < \, \bar{Q} \) then \( \bar{P} > a_{i} q_{i} + b_{i} \) for at least one \( i, i \in \{ 0, 1\} \).

2.2 Objective Functions of the Companies

2.2.1 Before Globalization

Recall that before globalization, there exists a single company in each market commercializing the commodity. Firm \( i, \left( { i \in \left\{ {0, 1} \right\}} \right) \) chooses its output volume so as to maximize its net profit function: \( \pi_{i} (q_{i} ) = p_{i} (q_{i} )q_{i} - f_{i} \left( {q_{i} } \right) \).

Note that assumption A1.3 implies that the output value that maximizes the benefits cannot be lower than \( \bar{Q}/2 \). Because of that, we can rewrite the maximization problem of any firm with sub-index \( i \) as follows:

$$ \mathop {\hbox{max} }\limits_{{q_{i} \ge {{\bar{Q}} \mathord{\left/ {\vphantom {{\bar{Q}} 2}} \right. \kern-0pt} 2}}} \pi_{i} \left( {q_{i} } \right) \equiv p_{i} \left( {q_{i} } \right) - f_{i} \left( {q_{i} } \right), $$
(5)

which can be easily replaced with Karush-Kuhn-Tucker (KKT) equations (cf., [13, p. 26]). The optimal output value \( \bar{q}_{i} \) for private firm \( i, i \in \{ 0, 1\} \) is found as:

$$ \bar{q}_{i} = \left\{ {\begin{array}{*{20}l} {\frac{{c - b_{i} }}{{2d + a_{i} }},} \hfill & {{\text{if }}\frac{{c - b_{i} }}{{2d + a_{i} }} > \frac{{\bar{Q}}}{2};} \hfill \\ {\frac{{\bar{Q}}}{2},} \hfill & {{\text{otherwise}} .} \hfill \\ \end{array} } \right. $$
(6)

2.2.2 After Globalization

After globalization, there is an integrated market where both companies compete in a classic duopoly. The price at this stage is determined in the global market, so it obeys the inverse demand function Eq. (4) cited in assumption A2.1.

The problem of each private company \( i \) is to maximize its net profit

$$ \xi_{i} \left( {q_{i} } \right) \equiv p_{w} \left( Q \right)q_{i} - f_{i} \left( {q_{i} } \right), \, i = 0,1. $$
(7)

The output level by each company under the assumptions made is found using the theory from [11]. As in [11], we also claim that the output volume chosen by a producer influences the market price. This can be described by a conjectured function of the variations of the price upon variations of the production volume. Then, the first order maximum condition to define the equilibrium would have the form for each \( i, i \in \{ 0, 1\} \):

$$ \frac{{\partial \xi_{i} }}{{\partial q_{i} }} \equiv p_{w} \left( Q \right) + \frac{{\partial p_{w} \left( Q \right)}}{{\partial q_{i} }} \cdot q_{i} - a_{i} q_{i} - b_{i} \left\{ {\begin{array}{*{20}l} { = 0,} \hfill & {{\text{if }}q_{i} > 0;} \hfill \\ { \le 0,} \hfill & {{\text{if }}q_{i} = 0.} \hfill \\ \end{array} } \right. $$
(8)

As in [11], the (negative of the) rate of the price function \( p_{w} \) variation implied by a possible variation of output conjectured by agent \( i{ (}i = 0,1 ) \) is denoted as \( v_{i} = - \partial p_{w} \left( Q \right)/\partial q_{i} \). In order to describe each agent’s behavior, we need to estimate \( v_{i} \). The conjectured dependence of \( p_{w} \) on \( q_{i} \) must account for the (local) concavity of the \( i \)-th agent’s objective function; otherwise, one cannot guarantee that the output volumes found via the first order optimality conditions Eq. (8) maximize (but not minimize) the profit functions. For instance, it suffices to assume that the coefficient \( v_{i} \) (from now on referred to as the \( i \)-th agent’s influence coefficient) is nonnegative and constant, for \( i, i \in \{ 0, 1\} \).

In [11, 12], we defined the concept of exterior equilibrium, i.e., conjectural variations equilibrium (CVE) with the influence coefficients fixed in an exogenous model. As the competition after globalization has been described with the model presented in [11], the equilibrium would be found exactly as before. Theorem 1 in [11] establishes the existence and uniqueness of the exterior equilibrium \( (p_{w} ; \tilde{q}_{0} , \tilde{q}_{1} ) \) under assumptions A1.2 and A2.1, and also provides the left and right derivatives of the equilibrium price \( p_{w} = p_{w} (D,v_{0} ,v_{1} ) \) with respect to \( D \). This theorem serves as a base for the concept of interior equilibrium, which was defined in [11] as the exterior equilibrium with consistent conjectures (influence coefficients). Under the above assumptions, according to Theorem 2 in [11], there exists interior equilibrium after globalization. Namely, we define the following concept.

2.2.3 Consistent Conjectural Variations Equilibrium (CCVE)

Let us define the following auxiliary parameter

$$ \tau = \left\{ {\begin{array}{*{20}l} { - \infty ,} \hfill & {{\text{if }}p_{w} = \bar{P};} \hfill \\ {{{ - 2} \mathord{\left/ {\vphantom {{ - 2} d}} \right. \kern-0pt} d},} \hfill & {{\text{if }}p_{w} < \bar{P}.} \hfill \\ \end{array} } \right. $$

Given the previous results obtained in [11], the consistent (justified) influence coefficient of agent \( i, i \in \{ 0, 1\} , \) after globalization is found by solving the following (nonlinear) equation system:

$$ v_{i} = \frac{1}{{\frac{1}{{v_{ - i} + a_{ - i} }} - \tau }}, \, i = 0,1, $$
(9)

where the symbol \( \left( { - i} \right) \) represents the competitor’s sub-index.

The CVE with the consistent conjectures (9) is called interior equilibrium. In Eq. (9), \( \tau \in \left[ { - \infty ,0} \right) \). When \( \tau = - \infty , \) system (9) has the unique solution \( v_{i} = 0, i \in \{ 0, 1\} \). The latter result corresponds to the perfect competition equilibrium (cf., [11]).

The following result was already derived and published as Theorem 3 in [11] and Theorem 4.3 in [12], including for the case of a mixed oligopoly (competition among a public firm and several private companies).

Theorem 2.1

([11, 12]). Under assumptions A1.2 and A2.1, for any \( \tau \ge 0 \), Eq. (9) has a unique solution \( v_{i} = v_{i} \left( \tau \right), \, i = 0,1 \), continuously depending upon \( \tau \). Furthermore, \( v_{i} \left( \tau \right) \to 0 \) when \( \tau \to - \infty , \) and strictly increases and tends up to \( v_{i} \left( 0 \right) > 0 \) as \( \tau \to 0, i = 0, 1 \).

In our case, the solution of the system formed by equations Eq. (9) for the firm i’s influence coefficient is:

$$ v_{i} = \left\{ {\begin{array}{*{20}l} { - \frac{{a_{i} }}{2} + \sqrt {\frac{{a_{i}^{2} }}{4} + \frac{\varGamma }{{K_{ - i} }},} } \hfill & {{\text{if }}\tau = - \frac{2}{d};} \hfill \\ {0,} \hfill & {{\text{if }}\tau = - \infty ,} \hfill \\ \end{array} } \right. \, i = 0,1, $$
(10)

where \( \varGamma = a_{i} + a_{ - i} + 2a_{i} a_{ - i} /d \) and \( {\rm K}_{ - i} = 2\left( {2 + \frac{2}{d}a_{ - i} } \right)/d \).

For the interior equilibrium price \( p_{w} \ge \bar{b} = \hbox{max} \left\{ {b_{1} ,b_{2} } \right\} \), Theorems 1 and 2 from [11] imply that relationship Eq. (8) defines uniquely the equilibrium production volumes \( \tilde{q}_{i} , \, i = 0,1 \) (taking into account that \( p_{w} = \bar{P} \) implies \( v_{i} = 0 \)):

$$ \tilde{q}_{i} = \left\{ {\begin{array}{*{20}l} {\frac{{p_{w} - b_{i} }}{{v_{i} + a_{i} }},} \hfill & {{\text{if }}p_{w} < \bar{P};} \hfill \\ {\frac{{p_{w} - b_{i} }}{{a_{i} }},} \hfill & {{\text{if }}p_{w} = \bar{P},} \hfill \\ \end{array} } \right.,i = 0,1. $$
(11)

In the particular case when \( b_{0} = b_{1} \) assumption A1.3 entails that the total output level given by Eq. (11) at \( p_{w} = \bar{P} \), is greater than \( \bar{Q} \). However, at this price the quantity demanded is at most \( \bar{Q} \), which means that the market is not balanced. Hence, in this particular case, the equilibrium can be reached only when \( p_{w} < \bar{P} \). From now onward, for simplicity, we restrict ourselves to this case with \( b_{0} = b_{1} = b \).

In the equilibrium when \( p_{w} < \bar{P} \), the total supply output equals the demand in the market. Then, from A2.1, \( p_{w} = c - dQ/2 \), where \( Q = \tilde{q}_{0} + \tilde{q}_{1} \). Plugging in this in the equilibrium outputs Eq. (11), it is easy to obtain both the total output and the equilibrium price \( p_{w} \).

3 The Effect of Globalization for the Profits

To find the effects of globalization on profits we look for the ratio of benefits. We determine conditions involving the parameters under which these ratios are greater or smaller than 1. If the profit ratio is greater than 1 for company \( i, i \in \{ 0, 1\} \), we say that globalization is beneficial for this firm, and it is not otherwise, that is, if the profit ratio is less than 1. In order to do that, we first introduce the properties of companies being low-marginal, or vice versa, high-marginal cost firms.

Definition 3.1.

We say that agent \( i \) is a low-marginal cost firm (LMCF) if the marginal cost \( f_{i}^{\prime } \left( {q_{i} } \right) \) evaluated at \( \bar{Q}/2 \) is less than the cap price minus a proportion \( d \) of the quantity \( \bar{Q}/2 \), that is, \( f^{\prime }_{i} \left( {\bar{Q}/2} \right) < \bar{P} - d\bar{Q}/2 \). Conversely, agent \( i \) is a high-marginal cost firm (HMCF) if \( f^{\prime }_{i} \left( {\bar{Q}/2} \right) \ge \bar{P} - d\bar{Q}/2 \).

Before globalization, the output level produced by firm i to supply to a separate market depends on the value of the corresponding parameters. On the one hand, if firm \( i \) is an LMCF, it produces \( \bar{q}_{i} = \frac{c - b}{{2d + a_{i} }} \). Finally, if it is an HMCF it supplies \( \bar{q}_{i} = \frac{{\bar{Q}}}{2} \). Because of that, before globalization, four situations in total are feasible depending on the characteristics of the firms of both markets. These situations are described in Table 1, which shows the optimal outputs and the profits for both firms.

Table 1. Possible situations before globalization.
Full size table

Let \( R_{i} \) denote the profit ratio of company \( i,i \in \left\{ {0,1} \right\} \) and be given by:

$$ R_{i} \equiv \frac{{\xi_{i} \left( {\tilde{q}_{i} } \right)}}{{\pi_{i} \left( {\bar{q}_{i} } \right)}}, \, i = 0,1. $$
(12)

Formula (12) would take different values according to the situation encountered.

3.1 Measure of Coincident Profit Degradation

Globalization may improve or degrade the profits of the companies. However, [1] study the cases when coincident profit degradation occurs, that is, both firms have smaller profits after globalization than before. In the above-mentioned work, the profit ratio of a producer after globalization to that before globalization is proposed as the degree of profit degradation for the producer due to globalization. They utilize the largest of the ratios of profit degradation among producers as a measure of coincident degradation.

According to [1], the reason is: a smaller value of the measure is supposed to indicate stronger coincident degradation. The situation where only one of the producers suffers profit degradation cannot be considered as a coincident producer profit degradation, as far as the other producer enjoys profit improvement. The measure of coincident profit degradation used in [1] for a duopoly is defined in the following terms:

$$ k_{R} = \hbox{max} \left\{ {R_{0} ,R_{1} } \right\} . $$
(13)

The main result obtained in [1] is that the worst-case ratio of coincident profit degradation for all producers due to globalization is reached by a market system if, and only if the system is in a complete symmetry. In the next subsection, we show how this result is not necessarily true in the case of the equilibrium with consistent conjectural variations (CCVE), at least for a system of two firms with quadratic cost functions. We also describe the effect of the cost parameters values, \( a_{i} \) and \( a_{ - i} \) on the profit ratios. In order to do so, we use the concept of technical advantage. According to the definition of technical advantage introduced in [10], a firm has a technical advantage over its rival if it can produce the same output that its rival produces at lower marginal and total costs than its rival. Therefore, we say that firm \( i \) has the technical advantage over the other firm \( \left( { - i} \right) \) if \( a_{i} \) < \( a_{ - i} \). The proofs of the propositions are too long and will be published elsewhere.

3.2 Situation 1

Situation 1 stated in Table 1 refers to the case when both agents are low-marginal cost firms (LMCF). Substitute the profits at the equilibrium and the optimal profits (Table 1) into formula (12), and after some algebraic manipulations obtain:

$$ R_{i} = \frac{{2d + a_{i} }}{{2v_{i} + a_{i} }} \cdot \left( {\frac{{2v_{i} }}{d}} \right)^{2} ,i = 0,1, $$
(14)

where \( v_{i} = - \frac{{a_{i} }}{2} + \sqrt {\frac{{a_{i}^{2} }}{4} + \frac{\varGamma }{{K_{ - i} }}} \), \( i = 0,1 \), according to Eq. (10).

Proposition 3.1.

There is a degradation of the profits of private firm i, \( i = 0,1 \), if and only if

$$ \lambda_{1} \left( {a_{i} ,a_{ - i} ,d} \right) + \lambda_{2} \left( {a_{i} ,a_{ - i} ,d} \right) > 1, $$
(15)

where \( \lambda_{1} \left( {a_{i} , a_{ - i} , d} \right),\lambda_{2} (a_{i} , a_{ - i} , d) \in (0,1) \) are defined as follows:

$$ \begin{aligned} & \lambda_{1} \left( {a_{i} ,a_{ - i} ,d} \right) = \left[ {{{2\sqrt {{{a_{i}^{2} } \mathord{\left/ {\vphantom {{a_{i}^{2} } 4}} \right. \kern-0pt} 4} + {\varGamma \mathord{\left/ {\vphantom {\varGamma {K_{ - i} }}} \right. \kern-0pt} {K_{ - i} }}} } \mathord{\left/ {\vphantom {{2\sqrt {{{a_{i}^{2} } \mathord{\left/ {\vphantom {{a_{i}^{2} } 4}} \right. \kern-0pt} 4} + {\varGamma \mathord{\left/ {\vphantom {\varGamma {K_{ - i} }}} \right. \kern-0pt} {K_{ - i} }}} } {\left( {2d + a_{i} } \right)}}} \right. \kern-0pt} {\left( {2d + a_{i} } \right)}}} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} , \, \\ & \lambda_{2} \left( {a_{i} ,a_{ - i} ,d} \right) = 1 - {{2\left( { - {{a_{i} } \mathord{\left/ {\vphantom {{a_{i} } 2}} \right. \kern-0pt} 2} + \sqrt {{{a_{i}^{2} } \mathord{\left/ {\vphantom {{a_{i}^{2} } 4}} \right. \kern-0pt} 4} + {\varGamma \mathord{\left/ {\vphantom {\varGamma {K_{ - i} }}} \right. \kern-0pt} {K_{ - i} }}} } \right)} \mathord{\left/ {\vphantom {{2\left( { - {{a_{i} } \mathord{\left/ {\vphantom {{a_{i} } 2}} \right. \kern-0pt} 2} + \sqrt {{{a_{i}^{2} } \mathord{\left/ {\vphantom {{a_{i}^{2} } 4}} \right. \kern-0pt} 4} + {\varGamma \mathord{\left/ {\vphantom {\varGamma {K_{ - i} }}} \right. \kern-0pt} {K_{ - i} }}} } \right)} d}} \right. \kern-0pt} d}, \, i = 0,1. \\ \end{aligned} $$
(16)

The simultaneous degradation of the benefits occurs when inequality (15) is valid for both \( i = 0, 1 \). The degradation or increase of company i’s profit depends not only on the cost parameters of the same company but also on the cost parameters of the other agent.

Proposition 3.2.

The profit ratio of competitor \( i \) increases if \( a_{ - i} \) grows.

Proposition 3.2 states that the larger the coefficient of the quadratic term of the rival’s cost function, the lower the profit degradation for producer \( i \) (or, its profits may even increase). The proof of Proposition 3.2 shows that the increase of the parameter \( a_{ - i} \) of the cost function of the rival affects positively the profit ratio of player \( i, \) as expected.

For the current situation, the profits of the weaker firm \( \left( { - i} \right) \) are degraded after globalization. Another important fact is that if a firm has the technical advantage over the other, the degradation of its own profit due to globalization (if the latter happens at all) is lower than the profit degradation of the other firm. Even more, the profits of firm \( i \) can increase. These results are summarized in Proposition 3.3.

Proposition 3.3.

If \( a_{i} < a_{ - i} \) , i.e., competitor \( i \) has technical advantage over \( \left( { - i} \right) \) , then

  1. (a)

    \( R_{ - i} < 1 \);

  2. (b)

    \( R_{ - i} < R_{i} \).

Note that if we consider the case where there is coincident degradation of the profits, the measure of the latter in this case would equal \( k^{R} = R_{i} , \) i.e., the profit degradation of the firm with the technical advantage.

Under the complete symmetry, both producers suffer coincident profit degradation. This result is the same as in [2] and is stated in Proposition 3.4:

Proposition 3.4.

If the firms are identical \( (a_{i} = a_{ - i} = a) \) the ratio of both firms is given by

$$ R = \frac{2d + a}{{2\sqrt {\frac{{a^{2} }}{4} + \frac{ad}{2}} }} \cdot \left( {\frac{a}{{\frac{a}{2} + \sqrt {\frac{{a^{2} }}{4} + \frac{ad}{2}} }}} \right)^{2} . $$
(17)

This resulting value is less than 1 for any positive values of \( a \) and \( d \) , which means that globalization degrades profits for each company when both firms face the same costs.

In contrast to [1], the latter is not necessarily the worst case under consistent conjectures. We introduce a numerical example to show it. In the following examples, we compute, together with the consistent conjectural variations equilibrium (CCVE), the equilibrium under Nash-Cournot conjectures considering the quadratic cost functions. In [1], the cost function is linear. Nevertheless, our examples with quadratic costs show that a complete symmetry implies the worst-case ratio under Nash-Cournot conjectures, too.

Example 1.

Consider a duopoly with c \( = 50, d = 10, \bar{P} = 30,\bar{Q} = 4, b = 1, \) and \( a_{0} = 0.1 \). Table 2 simulates Situation 1 for the above-given values of the parameters and different values of the parameter \( a_{1} \), starting with \( a_{1} = 0.06 \) and increasing with a mesh of 0.02. The above-mentioned table shows the influence coefficients in the case of CCVE, while the values of the influence coefficients at the Nash-Cournot equilibrium are always \( v_{0} = v_{1} = - d/2 \). For the consistent CVE, the minimal value of the degradation measure \( k_{R}^{CV} = 0.230 \) (among the values presented in Table 2) is achieved when \( a_{1} = 0.06 \), which means that the worst case is not the one where the firms are identical, unlike the Nash-Cournot case in which the worst case ratio is obtained when \( a_{1} = a_{0} = 0.10 \). That is, a complete symmetry does not necessarily entail the worst case ratio in for consistent CVEs.

Table 2. Example 1: \( c = 50, d = 10, \bar{P} = 30, \bar{Q} = 4, b = 1 \) and \( a_{0} = 0.1 \)
Full size table

3.3 Situations 2 and 3

In Situation 2 from Table 1, there is one low-marginal cost firm and the rival is a high-marginal cost producer. Plug in the equilibrium and optimal values into formula (12), and simple algebraic manipulations yield (in Situation 3, the results are similar):

$$ R_{0} = \frac{{2d + a_{0} }}{{2v_{0} + a_{0} }} \cdot \left( {\frac{{2v_{0} }}{d}} \right)^{2} ; \, R_{1} = \frac{{\left( {c - b} \right)^{2} \left( {{{2v_{1} } \mathord{\left/ {\vphantom {{2v_{1} } d}} \right. \kern-0pt} d}} \right)^{2} }}{{\bar{Q}\left( {2v_{1} + a_{1} } \right)\left[ {c - b - {{\bar{Q}\left( {2d + a_{1} } \right)} \mathord{\left/ {\vphantom {{\bar{Q}\left( {2d + a_{1} } \right)} 4}} \right. \kern-0pt} 4}} \right]}}. $$
(18)

Here, Propositions 3.1 and 3.2 are still valid for firm 0 because the formulas of the profit ratio for firm 0 from Eq. (18) are identical to the formulas (14). Firm 0 would face degradations of her profits if, and only if (15) holds. The profit ratio of firm 0 increases with respect to \( a_{1} \). The larger the value of \( a_{1} \) the higher is the profit ratio for firm 0. The latter means that if globalization damages the profits of firm 0, the degradation would not be too strong as it would be with a smaller value of \( a_{1} \). For firm \( 1 \), the higher values of the slope of the rival’s marginal cost would result in a better profit ratio as stated in Proposition 3.5.

Proposition 3.5.

Profit ratio of competitor \( 1 \) increases together with \( a_{0} \).

3.4 Situation 4

In Situation 4 from Table 1, both producers are high-marginal cost firms. Substituting the equilibrium and optimal values in formula (12), after simple algebraic manipulations one obtains for \( i = 0, 1 \):

$$ R_{i} = \frac{{\left( {c - b} \right)^{2} \left( {{{2v_{i} } \mathord{\left/ {\vphantom {{2v_{i} } d}} \right. \kern-0pt} d}} \right)^{2} }}{{\bar{Q}\left( {2v_{i} + a_{i} } \right)\left[ {c - b - {{\bar{Q}\left( {2d + a_{i} } \right)} \mathord{\left/ {\vphantom {{\bar{Q}\left( {2d + a_{i} } \right)} 4}} \right. \kern-0pt} 4}} \right]}} . $$
(19)

Proposition 3.6.

The profit of competitor i increases together with \( a_{ - i} . \)

Therefore, the effect of the increase of the quadratic cost coefficient \( a_{ - i} \) on the rival’s profit (player \( i) \) is positive.

4 Conclusions and Future Works

In this paper, we examine consistent conjectural variations equilibrium (CCVE) for a duopoly in a market of a homogeneous product. We study the effects of uniting two separate markets each monopolized by a producer: after globalization, both firms compete in one common market. Our model assumes an inverse demand function with a cap price and quadratic cost functions of both agents. Similar to previous studies, we investigate if the companies lose or gain due to globalization by evaluating their profit ratios, i.e., the ratios of their net profits after and before entering the common market. For the situations where both agents are low-marginal cost firms, we find that the company with a technical advantage over its rival has a better profit ratio. In addition, as the rival becomes weaker, this is, as the slope of the rival’s marginal cost function increases, the first agent’s profit ratio enhances, too. Moreover, when both agents are low-marginal cost firms, at least the weaker company suffers the degradation of its profits due to the globalization.

Unlike the previous study [1], which considers Nash-Cournot equilibrium, we show with an example that the complete symmetry of the agents does not always provide the worst case (the lowest profit ratio) in the case of consistent CVE. As a consequence, we demonstrate that under consistent conjectures it is important to analyze not only the case where firms are identical (although this leads one to deal with more complicated or even intractable problems).

In our forthcoming papers, we are going to analyze a system with a public firm whose maximized objective is distinct from its net profit.