The Open Monopolistic Competition Models: Market Equilibrium and Social Optimality

Abstract

We study the monopolistic competition model of Dixit–Stiglitz–Krugman with additive separable utility and transport costs of “iceberg types”. The production costs are not necessary linear. We study two concepts: market equilibrium and social optimality. There are well-known facts in the closed economy under monopolistic competition: (1) “in market equilibrium, the elasticity of revenue equals the elasticity of total costs” and (2) “in social optimality, the elasticity of utility equals the elasticity of total costs”. Moreover, earlier Prof. Sergey Kokovin (HSE University, Russia) generated the idea that (3) “the search for equilibrium is equivalent to the problem of optimization, but revenue, not utility”. For the case of several countries, it turns out that facts (1) and (2) and prediction (3) remain mostly true.

1. Introduction

A market situation where there are many independent sellers and many independent buyers, and competition is imperfect due to product differentiation and other factors, is called “monopolistic competition”. At the moment, monopolistic competition perhaps most adequately simulates real economic processes.

It is generally accepted that the basis of this theory was laid almost simultaneously in 1933 by the American economist E. H. Chamberlin (see [1], it was here that the fundamental concept of “product differentiation” was introduced) and a few months later by the British economist J. Robinson [2]. The “imperfection” of competition is confirmed by many empirical observations. However, for a long time, the development of the theory was hampered by its insufficient formalization, which did not allow the use of mathematical tools.

A real breakthrough should be considered the work of A. K. Dixit and J. E. Stiglitz [3], where a model for the additive-separable utility function is built and investigated. In the model of A. K. Dixit and J. E. Stiglitz, the sub-utility function is of CES type, i.e., “Constant Elasticity of Substitution”. Hereinafter, P. R. Krugman generalizes this model to the case of international trade [4] (see also [5,6]). Further, M. J. Melitz generalizes the model to the heterogeneous case [7] (see also [8]).

Unfortunately, the CES type of sub-utility function generates some inadequacy of the results obtained. For example, in equilibrium, an increase in market size is accompanied by an increase in competition (an increase in mass of firms), but it can also be accompanied by constant prices.

An important example of a model with a non-CES type of sub-utility function is the work of K. Behrens and Y. Murata [9].

However, it is always of interest to develop a technique that allows us to analyze general models and does not contain specific types of functions. In other words, it is interesting to abandon a specific functional type of function (utility functions, cost functions, etc.).

In 2012, E. Zhelobodko, S. Kokovin, M. Parenti and J.-F. Thisse published the work [10], where the mathematical apparatus is finally formalized, which makes it possible to study models of monopolistic competition of a sufficient general kind.

In 2012, C. Arkolakis, A. Costinot and A. Rodríguez-Clare published the work [11] (see also [12]), where doubts are expressed about the universal benefits of international trade (or rather, about the universal harm of the “fading” of international trade). This work prompts researchers to try to explain this phenomenon.

In [13], we compare the magnitude of social welfare at the point of free trade and at the point of autarky and show that, generally speaking, any situation is possible: both the benefits of free trade and the benefits of autarky. Moreover, in [14], we obtain the following “counter intuitive” result: with the additive-separable utility function, linear production costs and transport (trade) costs of the “iceberg type”, if autarky arises, then with an increase in transport costs “near” autarky, social welfare does not decrease, but, on the contrary, increases.

In the above works, the “market equilibrium” is investigated. However, the study of the behavior of social welfare in market equilibrium allows us to conclude that not only market equilibrium is of interest, but also social optimality. Indeed, the problem of maximizing social welfare can be seen as the problem of a “social planner” who maximizes “world welfare”. In the work of S. Dhingra and J. Morrow [15] (see also [16]), social optimality studies are described in great detail. In particular, the question of the coincidence of market equilibrium and social optimality is investigated.

It is important to note that the question of the relationship between market equilibrium and social optimality is discussed in the works of A. K. Dixit and J. E. Stiglitz and P. R. Krugman. Moreover, there are works devoted entirely to the study of this relationship, see, for example, the classic work of L. C. Corchón [17].

In [18], we study questions similar to those from [14], not for market equilibrium, but for social optimality. Therefore, there is a natural assumption that the concepts of market equilibrium and social optimality, being concepts of a fundamentally “different” nature, have something in common from a “mathematical” point of view. As far as we know, both of these concepts have not been considered from a single position before. Thus, it is of undoubted interest to try to find a mathematical framework within which the search for market equilibrium and the optimization of social welfare will be the problems of the same nature.

Our colleague Prof. Sergey Kokovin (HSE University, Russia) once suggested that in the case of a closed economy (the case of one country), the search for market equilibrium is equivalent to solving the problem of optimization, not of the public welfare, but of the revenue.

For the case of two countries, in [19], we obtain some results on a unified approach to the study of market equilibrium and social optimality. Moreover, generalizations are obtained for the case of two countries of the known consequences from the basic equations for a closed model of monopolistic competition. (In a closed homogeneous economy, in market equilibrium, the elasticity of revenue is equal to the elasticity of costs; in social optimality, the elasticity of utility is equal to the elasticity of costs.) However, the methodology developed in [19] seems to use the structure of the two-country model too well, which does not allow us to directly generalize the obtained results to the case of an arbitrary number of countries.

The purpose of the proposed work is to generalize the results of [19] to the case of an arbitrary (fixed and previously known) number of countries. Since direct generalization is impossible, it is necessary to develop a system of special functions such that the system of equations of market equilibrium and the system of equations of social optimality have the same structure, but the functions in these two systems are calculated at different points.

The article is organized as follows. Section 2 contains a description of the model and the definitions of symmetric equilibrium and symmetric optimality. Section 3 contains the main results and discussions. Section 4 contains the proofs. Section 5 concludes.

2. Open Economy Monopolistic Competition Model

Let us set the homogeneous international trade model with monopolistic competition of the producers, the model of Dixit–Stiglitz–Krugman.

Let $I=\{1,\dots ,K\},$ where K is the number of countries in the economy.

2.1. Monopolistic Competition Assumptions of Dixit–Stiglitz–Krugman

As it is usual in a homogeneous model of international trade under monopolistic competition, we assume (cf. [3,4,10])

• the identity of consumers;
• the identity of producers (firms);
• the only production factor is labor;
• each consumer has one unit of labor;
• utility function of each consumer is linear additive;
• all firms produce “varieties” of goods (“almost the same” goods);
• there is one-to-one correspondence between producers and varieties: each producer produces one variety, each variety is produced by one producer;
• the demand of each variety is influenced by other varieties;
• in each country, number of firms is big enough; thus, the influence of each firm on the whole industry can be ignored;
• the condition of free entry and exit holds, so the profits equal;
• in each country, there is labor balance;
• in each country, there is trade balance.

2.2. Consumers

Let for each country $k\in I,$

• $L_k$ be the number of consumers, it is an exogenous parameter;
• $N_k$ be the number (mass) of firms, it is an endogenous variable.

Due to assumptions of monopolistic competition, the number of firms is big enough. Therefore, instead of the standard “number of firms”, we consider the “mass of firms”. More precisely (cf. [10]), we consider the intervals $\left[0,N_k\right],k\in I,$ with uniformly distributed firms. (A popular interpretation is as follows: gas stations are uniformly located on the “long” road; we are not interested in the number of these stations, but the length of the road. In this case, $\left[0,N_k\right],k\in I,$ are called not “the number of firms”, but “the mass of firms”. This mass is determined endogenously and does not have to be an integer at all).

Now, we introduce the individual consumptions and corresponding prices. Let

• $x_{kj}^{\left(i\right)}=x_{kj}\left(i\right)$ be the (individual) consumption of a consumer in country $j\in I$ of the variety produced in country $k\in I$ by producer $i\in \left[0,N_k\right];$
• $p_{kj}^{\left(i\right)}=p_{kj}\left(i\right)$ be the price of corresponding variety, $k\in I,j\in I,i\in \left[0,N_k\right]$.

Further, let the wage of each consumer in country $k\in I$ be $w_k.$

As usual, we assume that sub-utility function $u\left(·\right)$ is at least twice differentiable and, moreover, is such that

• $u\left(0\right)=0,$ i.e., “zero utility if zero consumption”;
• $u^{\prime }\left(\xi \right)>0,$ i.e., “the rationality of consumers’ behavior”, $u\left(\xi \right)$ is strictly increasing;
• $u^{″}\left(\xi \right)<0,$ i.e., “decreasing of marginal utility”, $u\left(\xi \right)$ is strictly concave.

• The problem of the representative consumer in country $k\in I$ is

  $$\sum _{j\in I}{\int }_0^{N_j}u\left(x_{jk}^{\left(i\right)}\right)di\to max$$

  s.t.

  $$\sum _{j\in I}{\int }_0^{N_j}{p}_{jk}^{\left(i\right)}{x}_{jk}^{\left(i\right)}di\le w_k.$$

As usual, the inverse demand functions we obtain from First Order Conditions $\left(FOC\right)$:

$$p_{kj}^{\left(i\right)}=p_{kj}^{\left(i\right)}\left(x_{kj}^{\left(i\right)},{\lambda }_j\right)=\frac{u^{\prime }\left(x_{kj}^{\left(i\right)}\right)}{{\lambda }_j},k\in I,j\in I,i\in \left[0,N_k\right],$$

(1)

where ${\lambda }_j,j\in I,$ are Lagrange multipliers.

2.3. Producers

The transport costs of “iceberg type” are the parameters. (To sell in another country y units of the goods, the firm must produce $\tau ·y$ units. “During transportation, the product melts like an iceberg …”).

$${\tau }_{kj}\ge 1,{\tau }_{kk}=1,k\in I,j\in I.$$

Each firm in each country produces for consumers in each country. Thus, to sell in country $j\in I,$ the firm $i\in \left[0,N_k\right]$ in country $k\in I$ must produce the amount

$$q_{kj}^{\left(i\right)}={\tau }_{kj}{L}_j{x}_{kj}^{\left(i\right)},k\in I,j\in I,i\in \left[0,N_k\right].$$

(2)

Moreover, the total output (the size) of firm $i\in [0,N_k]$ in country $k\in I$ is

$$Q_k^{\left(i\right)}=\sum _{j\in I}{q}_{kj}^{\left(i\right)},k\in I,i\in \left[0,N_k\right].$$

(3)

Let V be the production costs (for each firm in each country). As usual, we assume that function V is increasing and twice differentiable.

Thus, the profits ${\pi }_k^{\left(i\right)},k\in I,i\in \left[0,N_k\right],$ of firm $i\in \left[0,N_k\right]$ in country $k\in I$ are

$${\pi }_k^{\left(i\right)}=\sum _{j\in I}{L}_j·p_{kj}^{\left(i\right)}·x_{kj}^{\left(i\right)}-w_k·V\left(Q_k^{\left(i\right)}\right),$$

i.e., substituting inverse demand functions (1),

$${\pi }_k^{\left(i\right)}=\sum _{j\in I}{L}_j·\frac{u^{\prime }\left(x_{kj}^{\left(i\right)}\right)·x_{kj}^{\left(i\right)}}{{\lambda }_j}-w_k·V\left(Q_k^{\left(i\right)}\right),k\in I,i\in \left[0,N_k\right].$$

(4)

Let us introduce the function

$$R_{\lambda }\left(\xi \right):=\frac{u^{\prime }\left(\xi \right)·\xi }{\lambda }.$$

(5)

Due to inverse demand (1), this function can be called “revenue per consumer”.

Using (5), we obtain

$${\pi }_k^{\left(i\right)}=\sum _{j\in I}{L}_j·R_{{\lambda }_j}\left(x_{kj}^{\left(i\right)}\right)-w_k·V\left(Q_k^{\left(i\right)}\right),k\in I,i\in \left[0,N_k\right].$$

(6)

The labor balance in country $k\in I$ is

$${\int }_0^{N_k}V\left(Q_k^{\left(i\right)}\right)di=L_k,k\in I.$$

(7)

2.4. Symmetric Case

Since we consider a homogeneous monopolistic competition model, we assume the identity of all consumers and the identity of all producers. Therefore, it is natural to consider the symmetric case. More precisely, we omit index i in (1)–(3), (6) and (7).

This way, (1)–(3), (6) and (7) have the form

$$p_{kj}=p_{kj}\left(x_{kj},{\lambda }_j\right)=\frac{u^{\prime }\left(x_{kj}\right)}{{\lambda }_j},k\in I,j\in I,$$

(8)

$$q_{kj}={\tau }_{kj}{L}_j{x}_{kj},k\in I,j\in I,$$

(9)

$$Q_k=\sum _{j\in I}{q}_{kj},k\in I,$$

(10)

$${\pi }_k=\sum _{j\in I}{L}_j·R_{{\lambda }_j}\left(x_{kj}\right)-w_k·V\left(Q_k\right),k\in I,$$

(11)

$$N_k·V\left(Q_k\right)=L_k,k\in I.$$

(12)

Moreover, the trade balances (“export equals import”) are

$$L_i·\sum _{j\in I}{N}_j·p_{ji}\left(x_{ji},{\lambda }_i\right)·x_{ji}=N_i·\sum _{j\in I}{L}_j·p_{ij}\left(x_{ij},{\lambda }_j\right)·x_{ij},i\in I,$$

i.e., using (5), (8) and (12),

$$\sum _{j\in I}\frac{L_j}{V\left(Q_j\right)}·R_{{\lambda }_i}\left(x_{ji}\right)=\frac{1}{V\left(Q_i\right)}·\sum _{j\in I}{L}_j·R_{{\lambda }_j}\left(x_{ij}\right),i\in I.$$

(13)

Moreover, the social welfare function is

$$U=\sum _{k\in I}{U}_k=\sum _{k\in I}{L}_k·\sum _{i\in I}{N}_i·u\left(x_{ik}\right)=\sum _{i\in I}{N}_i·\sum _{k\in I}{L}_k·u\left(x_{ik}\right),$$

i.e., substituting (12),

$$U=\sum _{i\in I}\frac{L_i}{V\left(Q_i\right)}·{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}\right).$$

(14)

2.5. Symmetric Equilibrium and Symmetric Optimality

In a market equilibrium situation, producers maximize profits (11).

The maximization means that First Order Conditions $\left(FOC\right)$

$$\frac{\partial {\pi }_k}{\partial x_{ki}}=0,k\in I,i\in I,$$

(15)

and Second Order Conditions $\left(SOC\right)$

$${\pi }_k^{″}<0,k\in I,$$

(16)

hold. ((16) means that if (15) holds, then matrices ${\pi }_k^{″},k\in I,$ are negatively defined).

How many firms are in the market? Usually,

• if the profit is positive, the firms enter into the market;
• if the profit is negative, the firms exit the market.

Therefore, the “free entry” and “free exit” condition means the zero-profit in equilibrium, i.e.,

$${\pi }_k=0,k\in I.$$

(17)

A bundle

$$\left({\left\{x_{ij}^{equ}\right\}}_{i\in I}^{j\in I},{\left\{p_{ij}^{equ}\right\}}_{i\in I}^{j\in I},{\left\{{\lambda }_i^{equ}\right\}}_{i\in I},{\left\{N_i^{equ}\right\}}_{i\in I},{\left\{w_i^{equ}\right\}}_{i\in I}\right)$$

is called the symmetric market equilibrium if it satisfies the following conditions:

• $FOC$ (8)—rationality in consumption;
• $FOC$ (15) and $SOC$ (16)—rationality in production;
• zero profit (17)—free entry and free exit;
• “total costs equal total labor” (12)—balance in labor;
• “export equals import” (13)—balance in trade.

In a social optimality situation, a “social planner” maximizes the social welfare (“total utility”) (14). Therefore, First Order Conditions $\left(FOC\right)$

$$\frac{\partial U}{\partial x_{ij}}=0,i\in I,j\in I,$$

(18)

and Second Order Conditions $\left(SOC\right)$

$$U^{″}<0,$$

(19)

hold. ((19) means that if (18) holds, then matrix $U^{″}$ is negatively defined).

A bundle

$$\left({\left\{x_{ij}^{opt}\right\}}_{i\in I}^{j\in I},{\left\{N_i^{opt}\right\}}_{i\in I}\right)$$

is called symmetric social optimality if it satisfies the following conditions:

• $FOC$ (18) and $SOC$ (19)—the rationality in welfare;
• “total costs equal total labor” (12)—balance in labor.

3. Results

Let us introduce the functions

$$S_{ij}\left(x_{ij},A\right):=L_j·A\left(x_{ij}\right),i\in I,j\in I,$$

(20)

where A is a real function.

Let us denote

$$q_{ij}^{equ}=L_j·{\tau }^{ij}·x_{ij}^{equ},i\in I,j\in I,$$

$$q_{ij}^{opt}=L_j·{\tau }^{ij}·x_{ij}^{opt},i\in I,j\in I.$$

Then

$$Q_i^{equ}=\sum _{j\in I}{q}_{ij}^{equ}=\sum _{j\in I}{L}_j·{\tau }^{ij}·x_{ij}^{equ},i\in I,$$

is the market equilibrium size of the firm in country $i\in I,$ while

$$Q_i^{opt}=\sum _{j\in I}{q}_{ij}^{opt}=\sum _{j\in I}{L}_j·{\tau }^{ij}·x_{ij}^{opt},i\in I,$$

is the social optimal size of the firm in country $i\in I.$

Let us denote

$$S_{ij}^{equ}:=S_{ij}\left(x_{ij}^{equ},R_{{\lambda }_j^{equ}}\right)\equiv L_j·\frac{u^{\prime }\left(x_{ij}^{equ}\right)·x_{ij}^{equ}}{{\lambda }_j^{equ}},i\in I,j\in I,$$

(21)

$$S_{ij}^{opt}:=S_{ij}\left(x_{ij}^{opt},u\right)\equiv L_j·u\left(x_{ij}^{opt}\right),i\in I,j\in I.$$

(22)

As usual, let

$${\displaystyle {\mathcal{E}}_g\left(\xi \right)=\frac{g^{\prime }\left(\xi \right)\xi }{g\left(\xi \right)}}$$

be the elasticity of function g. Note that

$${\mathcal{E}}_{R_{{\lambda }^H}}\left(\xi \right)={\mathcal{E}}_{R_{{\lambda }^F}}\left(\xi \right)={\mathcal{E}}_R\left(\xi \right),$$

where $R\left(\xi \right)=u^{\prime }\left(\xi \right)\xi$ is “normalized” revenue.

Before formulating the results, it is important to note the following. We assume that situations of market equilibrium and social optimality exist and, moreover, are unique. The questions of existence and uniqueness are mathematically very non-trivial, but these questions are not the subject of this article. Let us only note that the situations of market equilibrium and social optimality are “observable”, which is confirmed by many publications based on the results of empirical studies. Thus, the results below should be understood in the following context: “Suppose that situations of market equilibrium and/or social optimality exist and, moreover, are unique. Then …”

3.1. Main Result

The main result of the work is

Proposition 1.

1. 

In the situation of symmetric market equilibrium, the conditions

$$\frac{S_{ij}^{equ}}{{\displaystyle \sum _{k\in I}}{S}_{ik}^{equ}}·{\mathcal{E}}_R\left(x_{ij}^{equ}\right)=\frac{q_{ij}^{equ}}{Q_i^{equ}}·{\mathcal{E}}_V\left(Q_i^{equ}\right),i\in I,j\in I,$$

(23)

hold, where $S_{ij}^{equ},i\in I,j\in I,$ is (21).

2. 

In the situation of symmetric social optimality, the conditions

$$\frac{S_{ij}^{opt}}{{\displaystyle \sum _{k\in I}}{S}_{ik}^{opt}}·{\mathcal{E}}_u\left(x_{ij}^{opt}\right)=\frac{q_{ij}^{opt}}{Q_i^{opt}}·{\mathcal{E}}_V\left(Q_i^{opt}\right),i\in I,j\in I,$$

(24)

hold, where $S_{ij}^{opt},i\in I,j\in I,$ is (22).

Due to Proposition 1,

• Let the situation of symmetric market equilibrium hold. Then, the elasticities of normalized revenue in individual consumptions and the elasticities of production costs satisfy conditions (23);
• Let the situation of symmetric social optimality hold. Then, the elasticities of sub-utility of individual consumptions and the elasticities of production costs satisfy conditions (24).

Therefore, Proposition 1 generalizes the well-known facts in closed economy monopolistic competition:

• in equilibrium, the elasticity of revenue equals the elasticity of total costs;
• in optimality, the elasticity of utility equals the elasticity of total costs.

3.2. Corollary, Discussion

Another generalization mentioned at the end of Section 3.1 of the well-known facts in closed economy monopolistic competition gives the following.

Corollary 1.

1. 

In the situation of symmetric market equilibrium, the conditions

$$\sum _{j\in I}\frac{S_{ij}^{equ}}{{\displaystyle \sum _{k\in I}}{S}_{ik}^{equ}}·{\mathcal{E}}_R\left(x_{ij}^{equ}\right)={\mathcal{E}}_V\left(Q_i^{equ}\right),i\in I,$$

hold, i.e., the elasticity of production costs is a convex combination of the elasticities of normalized revenue in individual consumption.

2. 

In the situation of symmetric social optimality, the conditions

$${\displaystyle \sum _{j\in I}}\frac{S_{ij}^{opt}}{{\displaystyle \sum _{k\in I}}{S}_{ik}^{opt}}·{\mathcal{E}}_u\left(x_{ij}^{opt}\right)=\frac{q_{ij}^{opt}}{Q_i^{opt}}·{\mathcal{E}}_V\left(Q_i^{opt}\right),i\in I,$$

holds, i.e., the elasticity of production costs is a convex combination of the elasticities of sub-utility of individual consumption.

It is important to note that although the formulas in Proposition 1 and Corollary 1 seem “elegant” enough, the interpretation (for example, economic) of the coefficients (21) and (22) seems rather difficult. In this regard, it is of interest to obtain formulas devoid of such shortcomings.

It seems that for a situation of symmetrical market equilibrium, the following Proposition allows for a good economic interpretation.

Proposition 2.

In the situation of symmetric market equilibrium, the conditions

$$\sum _{j\in I}\frac{q_{ij}^{equ}}{{\mathcal{E}}_R\left(x_{ij}^{equ}\right)}=\frac{Q_i^{equ}}{{\mathcal{E}}_V\left(Q_i^{equ}\right)},i\in I,$$

(25)

hold.

Note that (25) can be written as

$$\sum _{j\in I}\frac{q_{ij}^{equ}}{Q_i^{equ}}·\frac{1}{{\mathcal{E}}_R\left(x_{ij}^{equ}\right)}=\frac{1}{{\mathcal{E}}_V\left(Q_i^{equ}\right)},i\in I.$$

(26)

Thus, the elasticities of normalized revenue in individual consumptions and the elasticities of production costs satisfy the following easily interpretable conditions: the “inverse” elasticities of production costs are the convex combination of the “inverse” elasticities of normalized revenue in individual consumptions.

Finally, what about the analogue of Proposition 2 for social optimality? Unfortunately, not direct analogue, but …

Proposition 3.

In the situation of symmetric social optimality, the conditions (let us recall that K is the number of countries)

$${\displaystyle \sum _{j\in I}}\frac{{\displaystyle \sum _{k\in I}}{\displaystyle L_k·u\left(x_{ik}^{opt}\right)}}{K·L_j·u\left(x_{ij}^{opt}\right)}·\frac{q_{ij}^{opt}}{{\mathcal{E}}_u\left(x_{ij}^{opt}\right)}=\frac{Q_i^{opt}}{{\mathcal{E}}_V\left(Q_i^{opt}\right)},i\in I,$$

(27)

hold.

Thus, in the situation of symmetric social optimality, the elasticities of sub-utility of individual consumptions and the elasticities of production costs satisfy (27). Of course, (27) is much less elegant than (25). Moreover, (27) can not be rewritten in the same “interpretable” manner as (26).

4. Proofs

4.1. Proof of Proposition 1

4.1.1. Market Equilibrium

For

$${\pi }_k=\sum _{i\in I}\frac{L_i}{{\lambda }_i}·R\left(x_{ki}\right)-w_k·V\left(Q_k\right),k\in I,$$

$FOC$ are

$$\frac{\partial {\pi }_k}{\partial x_{ki}}\equiv \left(\frac{R^{\prime }\left(x_{ki}\right)}{{\lambda }_i}-w_k·{\tau }_{ki}·V^{\prime }\left(Q_k\right)\right)·L_i=0,k\in I,i\in I,$$

i.e.,

$$\frac{R^{\prime }\left(x_{ki}\right)}{{\lambda }_i}=w_k·{\tau }_{ki}·V^{\prime }\left(Q_k\right),k\in I,i\in I,$$

i.e.,

$$L_k·\frac{R\left(x_{ki}\right)}{{\lambda }_i}·\frac{R^{\prime }\left(x_{ki}\right)·x_{ki}}{R\left(x_{ki}\right)}=w_k·V\left(Q_k\right)·\frac{L_k·{\tau }_{ki}·x_{ki}}{Q_k}·\frac{V^{\prime }\left(Q_k\right)·Q_k}{V\left(Q_k\right)},$$

$$k\in I,i\in I,$$

i.e.,

$$\frac{L_k·{\displaystyle \frac{R\left(x_{ki}\right)}{{\lambda }_i}}}{w_k·V\left(Q_k\right)}·{\mathcal{E}}_R\left(x_{ki}\right)=\frac{q_{ki}}{Q_k}·{\mathcal{E}}_V\left(Q_k\right),k\in I,i\in I,$$

i.e., due to free entry conditions, see (17),

$$\frac{L_k·{\displaystyle \frac{R\left(x_{ki}\right)}{{\lambda }_i}}}{{\displaystyle \sum _{j\in I}}{\displaystyle \frac{L_j}{{\lambda }_j}}·R\left(x_{kj}\right)}·{\mathcal{E}}_R\left(x_{ki}\right)=\frac{q_{ki}}{Q_k}·{\mathcal{E}}_V\left(Q_k\right),k\in I,i\in I.$$

(28)

4.1.2. Social Optimality

For

$$U=\sum _{i\in I}\frac{L_i}{V\left(Q_i\right)}·{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}\right),$$

$FOC$ are

$$\frac{\partial U}{\partial x_{ij}}\equiv L_i·L_j·\left(-{\displaystyle \sum _{k\in I}}{L}_k·\frac{u\left(x_{ik}\right)·V^{\prime }\left(Q_i\right)·{\tau }_{ij}}{{\left(V\left(Q_i\right)\right)}^2}+\frac{u^{\prime }\left(x_{ij}\right)}{V\left(Q_i\right)}\right)=0,i\in I,j\in I,$$

i.e.,

$$\frac{L_j·u\left(x_{ij}\right)}{{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}\right)}·\frac{u^{\prime }\left(x_{ij}\right)·x_{ij}}{u\left(x_{ij}\right)}=\frac{L_j·{\tau }_{ij}·x_{ij}}{Q_i}·\frac{V^{\prime }\left(Q_i\right)·Q_i}{V\left(Q_i\right)},i\in I,j\in I,$$

i.e.,

$$\frac{L_j·u\left(x_{ij}\right)}{{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}\right)}·{\mathcal{E}}_u\left(x_{ij}\right)=\frac{q_{ij}}{Q_i}·{\mathcal{E}}_V\left(Q_i\right),i\in I,j\in I.$$

(29)

Due to (21)–(29), Proposition 1 is proved.

4.2. Proof of Corollary 1

Due to (28),

$${\displaystyle \sum _{k\in I}}\frac{L_k·{\displaystyle \frac{R\left(x_{ki}\right)}{{\lambda }_i}}}{{\displaystyle \sum _{j\in I}}{\displaystyle \frac{L_j}{{\lambda }_j}}·R\left(x_{kj}\right)}·{\mathcal{E}}_R\left(x_{ki}\right)={\displaystyle \sum _{k\in I}}\frac{q_{ki}}{Q_k}·{\mathcal{E}}_V\left(Q_k\right),i\in I,$$

i.e.,

$${\displaystyle \sum _{k\in I}}\frac{L_k·{\displaystyle \frac{R\left(x_{ki}\right)}{{\lambda }_i}}}{{\displaystyle \sum _{j\in I}}{\displaystyle \frac{L_j}{{\lambda }_j}}·R\left(x_{kj}\right)}·{\mathcal{E}}_R\left(x_{ki}\right)={\mathcal{E}}_V\left(Q_k\right),i\in I.$$

(30)

Further, due to (29),

$${\displaystyle \sum _{j\in I}}\frac{L_j·u\left(x_{ij}\right)}{{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}\right)}·{\mathcal{E}}_u\left(x_{ij}\right)={\displaystyle \sum _{j\in I}}\frac{q_{ij}}{Q_i}·{\mathcal{E}}_V\left(Q_i\right),i\in I,$$

i.e.,

$${\displaystyle \sum _{j\in I}}\frac{L_j·u\left(x_{ij}\right)}{{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}\right)}·{\mathcal{E}}_u\left(x_{ij}\right)={\mathcal{E}}_V\left(Q_i\right),i\in I.$$

(31)

Due to (21), (22), (30) and (31), Corollary 1 is proved.

4.3. Proof of Proposition 2

For

$${\pi }_k=\sum _{i\in I}\frac{L_i}{{\lambda }_i}·R\left(x_{ki}\right)-w_k·V\left(Q_k\right),k\in I,$$

$FOC$ are

$$\frac{\partial {\pi }_k}{\partial x_{ki}}\equiv \left(\frac{R^{\prime }\left(x_{ki}\right)}{{\lambda }_i}-w_k·{\tau }_{ki}·V^{\prime }\left(Q_k\right)\right)·L_i=0,k\in I,i\in I,$$

i.e.,

$$\frac{1}{{\lambda }_i}=w_k·{\tau }_{ki}·\frac{V^{\prime }\left(Q_k\right)}{R^{\prime }\left(x_{ki}\right)},k\in I,i\in I.$$

(32)

Further, free entry conditions (17) are

$$\sum _{i\in I}\frac{L_i}{{\lambda }_i}·R\left(x_{ki}\right)=w_k·V\left(Q_k\right),k\in I.$$

(33)

Let us substitute (32) in (33):

$$\sum _{i\in I}{L}_i·w_k·{\tau }_{ki}·\frac{V^{\prime }\left(Q_k\right)}{R^{\prime }\left(x_{ki}\right)}·R\left(x_{ki}\right)=w_k·V\left(Q_k\right),k\in I,$$

i.e.,

$$\sum _{i\in I}{L}_i·{\tau }_{ki}·\frac{R\left(x_{ki}\right)}{R^{\prime }\left(x_{ki}\right)}=\frac{V\left(Q_k\right)}{V^{\prime }\left(Q_k\right)},k\in I,$$

i.e.,

$$\sum _{i\in I}{L}_i·{\tau }_{ki}·x_{ki}·\frac{1}{{\displaystyle \frac{R^{\prime }\left(x_{ki}\right)·x_{ki}}{R\left(x_{ki}\right)}}}=\frac{Q_k}{{\displaystyle \frac{V^{\prime }\left(Q_k\right)·Q_k}{V\left(Q_k\right)}}},k\in I,$$

i.e.,

$$\sum _{i\in I}\frac{q_{ki}}{{\mathcal{E}}_R\left(x_{ki}\right)}=\frac{Q_k}{{\mathcal{E}}_V\left(Q_k\right)},k\in I.$$

Proposition 2 is proved.

4.4. Proof of Proposition 3

For

$$U=\sum _{i\in I}\frac{L_i}{V\left(Q_i\right)}·{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}\right),$$

$FOC$ are

$${\displaystyle \sum _{k\in I}}{L}_k·\frac{u\left(x_{ik}^{opt}\right)·V^{\prime }\left(Q_i^{opt}\right)·{\tau }_{ij}}{V\left(Q_i^{opt}\right)}=u^{\prime }\left(x_{ij}^{opt}\right),i\in I,j\in I,$$

i.e.,

$${\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}^{opt}\right)·\frac{1}{u\left(x_{ij}^{opt}\right)}·\frac{{\tau }_{ij}·x_{ij}^{opt}}{{\mathcal{E}}_u\left(x_{ij}^{opt}\right)}=\frac{Q_i^{opt}}{{\mathcal{E}}_V\left(Q_i^{opt}\right)},i\in I,j\in I,$$

i.e.,

$$\frac{{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}^{opt}\right)}{L_j·u\left(x_{ij}^{opt}\right)}·\frac{q_{ij}^{opt}}{{\mathcal{E}}_u\left(x_{ij}^{opt}\right)}=\frac{Q_i^{opt}}{{\mathcal{E}}_V\left(Q_i^{opt}\right)},i\in I,j\in I.$$

Hence,

$${\displaystyle \sum _{j\in I}}\frac{{\displaystyle \sum _{k\in I}}{L}_k·u\left(x_{ik}^{opt}\right)}{L_j·u\left(x_{ij}^{opt}\right)}·\frac{q_{ij}^{opt}}{{\mathcal{E}}_u\left(x_{ij}^{opt}\right)}={\displaystyle \sum _{j\in I}}\frac{Q_i^{opt}}{{\mathcal{E}}_V\left(Q_i^{opt}\right)},i\in I,j\in I,$$

i.e.,

$${\displaystyle \sum _{j\in I}}\frac{{\displaystyle \sum _{k\in I}}{\displaystyle L_k·u\left(x_{ik}^{opt}\right)}}{K·L_j·u\left(x_{ij}^{opt}\right)}·\frac{q_{ij}^{opt}}{{\mathcal{E}}_u\left(x_{ij}^{opt}\right)}=\frac{Q_i^{opt}}{{\mathcal{E}}_V\left(Q_i^{opt}\right)},i\in I.$$

(Let us recall that K is the number of countries.) Proposition 3 is proved.

5. Conclusions

The paper investigates the structure of monopolistic competition in a homogeneous model of international trade of the type Dixit–Stiglitz–Krugman with additively separable utility functions. The only production factor is the labor.

Main attention is paid to the following two concepts.

• Market equilibrium—optimization of producer behavior: firms maximize profits by using inverse demand functions (provided to them by representative consumers), free entry conditions (firms enter the market until their profits are positive), labor balances (in each country, the total costs are equal to total labor) and trade balances (in each country, exports equal imports).
• Social optimality—optimization of the behavior of the state in the interests of consumers: the state (more precisely, all countries together) maximizes the social welfare function under the only constraint—the labor balances in each country.

The concepts of “market equilibrium” and “social optimality” seem completely fundamentally different and even contradictory concepts: perhaps the interests of firms and the interests of societies as a whole strongly disagree.

As far as we know, both of these concepts have not been considered from a single position before.

The article proposes a unified approach to the study of these concepts.

In particular, the main result (see Proposition 1) shows that the structure of the basic formulas for situations of symmetrical market equilibrium and symmetrical social optimality is the same: it depends on function (20) calculating at different points.

This allows clarifying the nature of the concepts of market equilibrium and social optimality.

Moreover, the results can help to generalize the results of [14] on the behavior of social welfare with respect to transport costs near “free trade” and near “autarky” not only to the case of market equilibrium, but also to the case of social optimality.

In addition, it is of interest to see how the results obtained can be transferred to the case of models with retail, see [20,21], etc.

Finally, it is interesting to investigate models in which utility is not additively separable from the same perspective, see [22], etc.

In conclusion, let us draw attention to the limitations of the research. We consider the classical (homogeneous) model of Dixit–Stiglitz–Krugman. Transport costs are “iceberg type”. The utility function is additively separable, but the sub-utility function is not necessarily a CES function. The requirements for the sub-utility function are standard: twice differentiable, $u\left(0\right)=0$, strictly increasing, strictly concave. The only production factor is labor. The cost function has a rather general form: it is only required to be twice differentiable and strictly increasing. Thus, we can conclude that the limitations of the research are as follows:

• homogeneity of the model (cf. with heterogeneous models of Melitz [7]);
• additive separability of utility;
• the transport costs are “iceberg type’’.

Of course, we do not consider the question of the existence (and uniqueness) of market equilibrium and social optimality: our results should be understood as follows: “Let market equilibrium and social optimality exist and be unique. Then …”.