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2019 | OriginalPaper | Chapter

Social Optimality in International Trade Under Monopolistic Competition

Author : Igor Bykadorov

Published in: Mathematical Optimization Theory and Operations Research

Publisher: Springer International Publishing

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Abstract

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”. It is known that, in the situation of market equilibrium, under linear production costs, the social welfare, as a function of transport costs, decreases near free trade while (counter-intuitively!) increases near total autarky. Instead, we study the situation of social optimality. We show that total welfare decreases. We restrict our study by the case of two countries, “big” and “small”. Moreover, we study two important “limited” situations: near free trade and near total autarky. We show that near free trade, the welfare in the small country decreases; as to the big country, we find examples when (1) the welfare decreases and (2) the welfare (counter-intuitively!) increases. Besides, in the autarky case, we describe the situations of decreasing/increasing of welfare in each country.

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previous chapter The Interaction of Consumers and Load Serving Entity to Manage Electricity Consumption

next chapter Hamilton-Jacobi-Bellman Equations for Non-cooperative Differential Games with Continuous Updating

In welfare economics [25, 32], a social planner is a decision-maker who attempts to achieve the best result for all parties involved. Usually this means or the maximization of a social welfare function (in neo-classical welfare economics), or Pareto optimality (in modern welfare economics).

A popular interpretation is as follows: gas stations are equally spaced on the “long” road; we are not interested in the number of these stations, but the length of the road. In this case, N and n 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.

Hereinafter, due to the tradition of monopolistic competition, we use the notation $X_i$ for the function X(i), etc.

To sell in another country y units of the goods, the firm must produce $\tau \cdot y$ units. “During transportation, the product melts like an iceberg ...”.

Usually function $u\left( \cdot \right) $ is assumed to be increasing (not necessarily strictly increasing). For example, in the case of quadratic sub-utility

$$ u\left( \xi \right) ={\left\{ \begin{array}{ll} A\xi -\frac{B}{2}\cdot \xi ^{2}, &{} \xi \in \left[ 0,\frac{A}{B}\right] ;\\ \frac{A^{2}}{2B}, &{} \xi \ge \frac{A}{B}; \end{array}\right. } $$

with

$$ u^{\prime }\left( \xi \right) ={\left\{ \begin{array}{ll} A-B\cdot \xi , &{} \xi \in \left[ 0,\frac{A}{B}\right] ;\\ 0, &{} \xi \ge \frac{A}{B}. \end{array}\right. } $$

We assume strictly increase only for convenient.

It may seem more appropriate to maximize W and w separately. But here comes the problem of reconciling the resulting solutions, because of

$$ \frac{\partial W}{\partial Z}=0\Longleftrightarrow X=0,\quad \frac{\partial W}{\partial x}=0\Longleftrightarrow z=0, $$

$$ \frac{\partial w}{\partial X}=0\Longleftrightarrow Z=0,\quad \frac{\partial w}{\partial z}=0\Longleftrightarrow x=0. $$

Hence, we maximize the sum of welfares as a special case of scalarization in multiple-objective optimization. Of course, another linear combinations of the items can be considered and lead to the similar results.

Usually in this paper we omit the proofs, they are rather technical.

Note that concavity of sub-utility u restricts its elasticity as $\mathcal {E}_{u}\left( \xi \right) <1\;\forall \xi >0.$ Indeed, for every $\forall \xi >0$,

$$ \mathcal {E}_{u}\left( \xi \right)<1\Longleftrightarrow u^{\prime }\left( \xi \right) \cdot \xi -u\left( \xi \right) <0\quad \forall \xi >0. $$

Consider the function $g\left( \xi \right) =u^{\prime }\left( \xi \right) \cdot \xi -u\left( \xi \right) $. One has $g^{\prime }\left( \xi \right) \equiv u^{\prime \prime }\left( \xi \right) \cdot \xi <0\;\forall \xi >0$ due to strictly concavity of u. But $g\left( 0\right) =u\left( 0\right) =0.$ Hence $g\left( \xi \right) <0\;\forall \xi >0,$ i.e., $u^{\prime }\left( \xi \right) \cdot \xi -u\left( \xi \right) <0\;\forall \xi >0.$

In equilibrium situation (see, e.g., [15]), due to trade balance, the moving from the trade to the total autarky is under the unique transport cost. Instead, in social optimality situation, several kinds of partial autarky can be before the total autarky.

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Title: Social Optimality in International Trade Under Monopolistic Competition
Author: Igor Bykadorov
Publisher: Springer International Publishing
Book: Mathematical Optimization Theory and Operations Research
Print ISBN: 978-3-030-33393-5

Electronic ISBN: 978-3-030-33394-2

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-33394-2_13

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