We study the homogeneous model of international trade under the monopolistic competition of producers. The utility function assumes additive separable. The transport costs are of “iceberg types”. We consider both market equilibrium and social optimality and study the idea of Sergey Kokovin (NRU HSE): “the search for equilibrium is equivalent to the problem of optimization, but revenue, not utility”. For the case of two countries, we show that (i) in symmetric market equilibrium, the elasticity of production costs is a convex combination of the elasticities of normalized revenue in individual consumption; while (ii) in symmetric social optimality, the elasticity of production costs is a convex combination of the elasticities of sub-utility of individual consumption. These generalize the well-known facts in closed economy under monopolistic competition: “in equilibrium, the elasticity of revenue equals the elasticity of total costs” and “in optimality, the elasticity of revenue equals the elasticity of utility”. Moreover, we find that, in symmetric market equilibrium, the “inverse” elasticities of production costs is a convex combination of the “inverse” elasticities of normalized revenue in individual consumption. It turns out that the last result can be generalized in the case of international trade of several countries.
market equilibrium, the elasticities of production costs and the elasticities of normalized revenue in individual consumptions; (ii) in symmetric social optimality, the elasticities of production costs and the elasticities of sub-utility of individual consumptions. It turns out that these conditions have the identical form. Therefore, the search for equilibrium is equivalent to the problem of optimization, but revenue, not utility1. Corollary 1 shows that (i) in symmetric market equilibrium, the elasticity of production costs is a convex combination of the elasticities of normalized revenue in individual consumption; while (ii) in symmetric social optimality, the elasticity of production costs is a convex combination of the elasticities of sub-utility of individual consumption2. Let us note that the obvious disadvantage of the formulas in Proposition 1 and Corollary 1 is the poor interpretability of the coeficients
,
,
, turns out that it is necessary to compare not elasticities (of costs, revenues, utility), but their “inverse” values, (1/elasticities). Due to Proposition 2, in symmetric market equilibrium, the “inverse” elasticities of production costs is a convex combination of the “inverse” elasticities of normalized revenue in individual consumption. Moreover, the coeficients of this convex combination have a clear meaning: they are the ratio of total domestic consumption to the size of the firm. Moreover, Proposition 2 can be generalized to the case of several countries, see Section 4, Proposition 3. Section 5 concludes.
, see (27)-(30). We hope that Proposition 2 does not have this disadvantage. It
In this section we set the basic monopolistic competition model for open economy (international trade case case). Let be two countries, (“big”) and (“small”).
As it is usual in monopolistic competition, we assume that (cf. [1, 2, 3]) • consumers are identical, each endowed with one unit of labor; • labor is the only production factor; consumption, output, prices etc. are measured in labor; • firms are identical, but produce “varieties” (“almost the same”) of good; • each firm produces one variety as a price-maker, but its demand is influenced by other varieties; • each variety is produced by one firm that produces a single variety; • each demand function results from additive utility function; • number (mass) of firms is big enough to ignore firm’s influence on the whole industry/economy; • free entry drives all profits to zero; • labor supply/demand in each country is balance; • trade in each country is balance.
1As far as the author knows, this idea was first formulated several years ago by Sergey Kokovin, Center for Market Studies and Spatial Economics, National Research University Higher School of Economics.
2Thus Corollary 1 generalizes the well-known facts in closed economy monopolistic competition: “in equilibrium, the elasticity of revenue equals the elasticity of total costs” and and “in optimality, the elasticity of revenue equals the elasticity of utility”.
Let
consumer in country , by a consumer in country .
Note that and are parameters (the known constants) while and are the variables determined endogenously. Moreover, let us recall that, in monopolistic competition models, number of ifrms is big enough. Therefore, instead of standard “number of firms is intervals [0, ] and [0,
] with uniformly distributed firms.
Now we introduce four kinds of the individual consumption and prices. Let for every , ∈ { , }3, • be the amount of the variety produced in country by firm ∈ [0, ] and consumed by a (or )” we consider the • be the price of the unit of the variety produced in country by firm ∈ [0, ] and consumed the conditions Let (⋅) be a sub-utility function. As usual, we assume that (⋅) is twice diferentiable and satisfies
In country , the problem of representative consumer is while the wage rate in country be normalizing to one, = 1. i.e., it is strictly increasing and strictly concave. Further, let = be the wage rate in country s.t. s.t. while, in country , the problem of representative consumer is ∫ 0 ∫ 0 ∫ 0
∫ 0
0 0 0 0
( ) + ∫
≤ ,
≤ 1.
Using First Order Conditions, we get the inverse demand functions ′ ( )
( , )=
3Hereinafter, due to the tradition of monopolistic competition, we use the notation for the function ( ), etc. ifrm in country are “revenue per consumer” Let us substitute (1) in (4) and (5). Using (6), we get
Let the production costs be determined for each firm in each country by the increasing twice differentiable function . Then the profits , ∈ [0, ], of firm in country and , ∈ [0, ], of =
= + + − ( ), ∈ [0, ],
− ( ), ∈ [0, ].
Of course, the firms choose inverse demand functions (1) as the prices. Let us introduce so-called
country. Thus, To introduce the production amount of the firms (the “size” of the firm), let us introduce the parameter ≥ 1 as transport costs of “iceberg type”4. Each firm in each country produces for consumers in each ( )
= , ( ) = .
The Labor Balances in countries and are, correspondingly, melts like an iceberg ...”
4To sell in another country units of the goods, the firm must produce ⋅ units. “During transportation, the product
Let us recall that the consumers are assume identical, the producers are assumed identical. Thus, as usual, we consider the symmetric case. More precisely, we omit index in consumption, inverse demand functions (1), sizes of the firms (2), (3), profits (7), (8) and Labor Balances (9), (10). This way (1)-(3) and (7)-(10) are ( , )=
, , ∈ { , },
The following Corollary generalizes the well-known facts in closed economy monopolistic competition: “in equilibrium, the elasticity of revenue equals the elasticity of total costs” and “in optimality, the elasticity of revenue equals the elasticity of utility”.
Corollary 1.
1. In symmetric market equilibrium, the elasticity of production costs is a convex combination of the elasticities of normalized revenue in individual consumption, i.e., 2. In symmetric social optimality, the elasticity of production costs is a convex combination of the elasticities of sub-utility of individual consumption, i.e.,
Let us note that the obvious disadvantage of the formulas in Proposition 1 and Corollary 1 is the poor interpretability of the coeficients (27)-(30). We hope that the proposition below does not have this disadvantage.
Proposition 2. In symmetric market equilibrium, the elasticities of normalized revenue in individual consumptions and the elasticities of production costs satisfy the conditions
Moreover, Proposition 2 can be generalized to the case of several countries.
• be the amount of variety produced in country by firm ∈ [0, ] and consumed in country by a consumer, • be the corresponding prices, • be the number of consumer in country , • = ≥ 1,
= 1, • be the wage in country .
be the total output for firm ∈ [0, ] in country for selling in country , The problem of representative consumer in country ∈ is s.t.
For symmetric case, FOC is where is the corresponding Lagrange multiplier.
For a producer in country ∈ , total output is Let us substitute the inverse demand function in profits. Then the profit of a producer in country (Let us recall that ( ) = ′
To define symmetric equilibrium, as usual, we write first and second order conditions, free entry conditions, labor and trade balances.
It turns out that first second order conditions and free entry conditions allow to generalize Proposition 2.
Let be equilibrium consumption and
enue in individual consumptions and the elasticities of production costs satisfy the condition Proposition 3. For country ∈ , in symmetric market equilibrium, the elasticities of normalized rev
In this paper, we study, in the monopolistic competition framework, the homogeneous model of international trade with additively separable utility function for each consumer.
One of the most interesting topic in these studies is the so-called “comparative statics”, i.e., the influence of the models’ parameters (market size, transport costs, etc.) on the equilibrium and optimal variables: consumption, firm sizes, market sizes, social welfare, etc., see, e.g. [17, 18, 19, 20, 21, 22, 23].
Instead, we study a unified approach to both market equilibrium and social optimality. The following results are obtained.
• For the case of international trade between two countries, – in symmetric market equilibrium, the elasticities of production costs and the elasticities of normalized revenue in individual consumptions; moreover, the elasticity of production costs is a convex combination of the elasticities of normalized revenue in individual consumption; – in symmetric social optimality, the elasticities of production costs and the elasticities of sub-utility of individual consumptions; moreover, the elasticity of production costs is a convex combination of the elasticities of sub-utility of individual consumption; – in symmetric market equilibrium, the “inverse” elasticities of production costs is a convex combination of the “inverse” elasticities of normalized revenue in individual consumption; moreover, the coeficients of this convex combination have a clear meaning: they are the ratio of total domestic consumption to the size of the firm.
• The last result generalizes to the case of international trade between several countries.
Therefore, we generalize the well-known facts in closed economy monopolistic competition: “in equilibrium, the elasticity of revenue equals the elasticity of total costs” and and “in optimality, the elasticity of revenue equals the elasticity of utility”. It can allow to clarify the nature of these concepts.
Finally, it would be also glad to know whether the best choice for the two economies can gives the best choice for each economy separately. This can be the topic of future research.
The author is very grateful to the anonymous reviewers for very useful suggestions for improving the text. The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0018). The work was supported in part by the Russian Foundation for Basic Research, projects 18-010-00728 and 19-010-00910.