Oligopoly Model and Its Applications in International Trade

Abstract

Each firm in the oligopoly plays off of each other in order to receive the greatest utility, expressed in the largest profits, for their firm. When analyzing the market, decision makers develop sets of strategies to respond the possible actions of competitive firms. In international stage, firms are competitive and they have different business strategies, their interaction becomes essential because the number of competitors is increased. This paper will provide an examination in international trade balance and public policy under Cournot’s framework. The model shows how the oligopolistic firm can decide the business strategy to maximize its profit given others’ choice, and how the public maker can find out the optimal tariff policy to maximize its social welfare. The discussion in this paper can be significant for both producers in deciding their quantities needed to be sold in not only domestic market but also international stage in order to maximize their profits and governments in deciding the tariff rate on imported goods to maximize their social welfare.

Appendices

A Appendix (Proof of Proposition 1)

From the equation system, we have the FOCs of the welfare functions equal to zero:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{c} \dfrac{dW_{d}}{dt_{d}}=0\\ \dfrac{dW_{f}}{dt_{f}}=0 \end{array} &{} \Leftrightarrow {\left\{ \begin{array}{ll} \begin{array}{c} Q_{d}^{*}\dfrac{dQ_{d}^{*}}{dt_{d}}+y_{d}^{*}+t_{d}\dfrac{dy_{d}^{*}}{dt_{d}}+\dfrac{d\pi _{d}^{*}}{dt_{d}}=0\\ Q_{f}^{*}\dfrac{dQ_{f}^{*}}{dt_{d}}+x_{f}^{*}+t_{f}\dfrac{dx_{f}^{*}}{dt_{f}}+\dfrac{d\pi _{f}^{*}}{dt_{f}}=0 \end{array}\end{array}\right. }\end{array}\right. } \end{aligned}$$

From the optimal quantity produced by firms, equilibrium prices and quantity sold in markets, we can derive:

$$\begin{aligned}&Q_{d}^{*}+\dfrac{1}{3}t_{d}=Q_{f}^{*}+\dfrac{1}{3}t_{f}\\&Z_{d}^{*}-\dfrac{1}{2k+1}t_{d}=Z_{f}^{*}-\dfrac{1}{2k+1}t_{f} \text { and }p_{d}^{*}-\dfrac{1}{3}t_{d}=p_{f}^{*}-\dfrac{1}{3}t_{f} \end{aligned}$$

Thus, there is a unique solution of this equation system: $t_{d}^{*}=t_{f}^{*}=t^{*}$.

The second-order conditions of the welfare functions:

$$\begin{aligned}&\dfrac{d^{2}W_{d}}{dt_{d}^{2}}=\dfrac{d^{2}W_{f}}{dt_{f}^{2}}=\left( \dfrac{dQ_{d}}{dt_{d}}\right) ^{2}+2\dfrac{dy_{d}}{dt_{d}}+\dfrac{d^{2}\pi _{d}}{dt_{d}^{2}}\\&=\left( -\dfrac{k+3}{3(2k+3)}\right) ^{2}-\dfrac{2(4k+3)(k+2)}{3(2k+1)(2k+3)}+\dfrac{2}{9}-\dfrac{k}{(2k+1)^{2}(2k+3)^{2}}<0 \end{aligned}$$

Therefore, in Nash equilibrium, both countries will impose the same tariff rates to imported goods: $t_{d}^{*}=t_{f}^{*}=t^{*}$.

B Appendix

The first-order conditions to maximize firms’ profits show:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{lll} \dfrac{d\pi _{i}}{dx_{ij}}&{}=p_{j}(Q_{j})+x_{ij}\dfrac{dp_{j}}{dx_{ij}}-\dfrac{dC_{i}(Z_{i})}{dx_{ij}}-t_{j}&{}=0 (j=\overline{1,n},j\ne i)\\ \dfrac{d\pi _{i}}{dx_{ii}}&{}=p_{i}(Q_{j})+x_{ii}\dfrac{dp_{i}}{dx_{ii}}-\dfrac{dC_{i}(Z_{i})}{dx_{ii}}&{}=0 \end{array}\end{array}\right. } \end{aligned}$$

Suppose that: $p_{j}(Q_{j})=a-Q_{j}$     $\forall j\in {1,2,3,\ldots ,n}$ and $C_{i}(Z_{i})=f_{i}+\dfrac{1}{2}kZ_{i}^{2}$

We have $n^{2}$ equation and $n^{2}$ variables:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} \dfrac{d\pi _{i}}{dx_{ij}}&{}=p_{j}(Q_{j})+x_{ij}\dfrac{dp_{j}}{dx_{ij}}-\dfrac{dC_{i}(Z_{i})}{dx_{ij}}-t_{j}=0\left( j=\overline{1,n},j\ne i\right) \\ \dfrac{d\pi _{i}}{dx_{ii}}&{}=p_{j}(Q_{j})+x_{ii}\dfrac{dp_{i}}{dx_{ii}}-\dfrac{dC_{i}(Z_{i})}{dx_{ii}}=0 \end{array} \end{array}\right. } \end{aligned}$$

$$\begin{aligned} \Leftrightarrow&{\left\{ \begin{array}{ll} a-Q_{j}-x_{ij}-kZ_{i}-t_{j}&{}=0 \\ a-Q_{i}-x_{i}i-kZ_{i}&{}=0 \end{array}\right. } \end{aligned}$$

(4)

From (4), we have: $\forall i=\overline{1,n}$

$$\begin{aligned} Z_{i}^{*}=\displaystyle \sum _{j=1}^{n}x_{ij}^{*}=\frac{na}{nk+n+1}+\frac{t_{i}}{nk+1}-\frac{(nk+2)}{(nk+1)(nk+n+1)}\displaystyle \sum _{j=1}^{n}t_{j}\\ Q_{i}^{*}=\displaystyle \sum _{j=1}^{n}x_{ji}^{*}=\frac{na}{nk+n+1}-\frac{n-1}{n+1}t_{i}+\frac{k(n-1)}{(n+1)(nk+n+1)}\displaystyle \sum _{j=1}^{n}t_{j} \end{aligned}$$

The equilibrium price in country i will be:

$$\begin{aligned} p_{i}^{*}&=a-Q_{i}^{*}\\&=\dfrac{a(nk+1)}{nk+n+1}+\dfrac{n-1}{n+1}t_{i}-\dfrac{k(n-1)}{(n+1)(nk+n+1)}\displaystyle \sum _{j=1}^{n}t_{j} \end{aligned}$$

From (4), the optimal quantity each firm i produce in domestic market i will be:

$$\begin{aligned} x_{ii}^{*}=p_{i}^{*}-kZ_{i}^{*}. \end{aligned}$$

The optimal quantity each firm i produce in foreign market j is:

$$\begin{aligned} x_{ij}^{*}=p_{j}^{*}-kZ_{i}^{*}-t_{j}. \end{aligned}$$

C Appendix (Proof of Proposition 2)

The optimal quantity each firm i produce in domestic market i will be:

$$x_{ii}^{*}=p_{i}^{*}-kZ_{i}^{*}$$

The optimal quantity each firm i produce in foreign market j will be:

$$x_{ij}^{*}=p_{j}^{*}-kZ_{i}^{*}-t_{j}$$

We have, the total welfare of country i will be determined as:

$$W_{i}=\dfrac{1}{2}Q_{i}{}^{2}+t_{i}(2Q{}_{i}+kZ_{i}-a)+\pi _{i}$$

Where:

$$\begin{aligned} \pi _{i}=\displaystyle \sum _{j=1}^{n}x_{ij}p_{j}(Q_{j})-C_{i}(Z_{i})-\displaystyle \sum _{j=1,j\ne i}^{n}t_{j}x_{ij} \end{aligned}$$

(5)

Using the first-order-conditions of $W_{i}$:

$$\begin{aligned} \dfrac{dW_{i}}{dt_{i}}=0 \Leftrightarrow Q_{i}\dfrac{dQ_{i}}{dt_{i}}+(2Q{}_{i}+kZ_{i}-a)+t_{i}(2\dfrac{dQ_{i}}{dt_{i}}+k\dfrac{dZ_{i}}{dt_{i}})+\dfrac{d\pi _{i}}{dt_{i}}=0 \end{aligned}$$

From (5):

$$\begin{aligned} \dfrac{d\pi _{i}}{dt_{i}}&=\displaystyle \sum _{j=1}^{n}\left( \dfrac{dx_{ij}}{dt_{i}}p_{j}(Q_{j})+x_{ij}\dfrac{dp_{j}(Q_{j})}{dt_{i}}\right) -\dfrac{dC_{i}(Z_{i})}{dt_{i}}-\displaystyle \sum _{j=1,j\ne i}^{n}t_{j}\dfrac{dx_{ij}}{dt_{i}}\\&=2\displaystyle \sum _{j=1}^{n}\dfrac{dp_{j}}{dt_{i}}x_{ij}-kZ_{i}\dfrac{dZ_{i}}{dt_{i}} (\text {By using }x_{ii}=p_{i}-kZ_{i} \text { and }x_{ij}=p_{j}-kZ_{i}-t_{j}) \end{aligned}$$

Thus, the first-order-conditions of $W_{i}$ $(i=1,2,3,\ldots ,n)$ can be expressed as:

$$\begin{aligned}&\dfrac{dW_{i}}{dt_{i}}=Q_{i}\dfrac{dQ_{i}}{dt_{i}}+(2Q_{i}+kZ_{i}-a)+t_{i}\left( 2\dfrac{dQ_{i}}{dt_{i}}+k\dfrac{dZ_{i}}{dt_{i}}\right) +\dfrac{d\pi _{i}}{dt_{i}}=0\\&\,\, \Leftrightarrow Q_{i}\dfrac{dQ_{i}}{dt_{i}}+(2Q_{i}+kZ_{i}-a)+t_{i}\left( 2\dfrac{dQ_{i}}{dt_{i}}+k\dfrac{dZ_{i}}{dt_{i}}\right) +2\displaystyle \sum _{j=1}^{n}\dfrac{dp_{j}}{dt_{i}}x_{ij}-kZ_{i}\dfrac{dZ_{i}}{dt_{i}}=0 \end{aligned}$$

From these above conditions, we have the best-response functions of $t_{i}$:

$$\begin{aligned} t_{i}=-\dfrac{Q_{i}\frac{dQ_{i}}{dt_{i}}+(2Q_{i}+kZ_{i}-a)+2\displaystyle \sum _{j=1}^{n}\dfrac{dp_{j}}{dt_{i}}x_{ij}-kZ_{i}\dfrac{dZ_{i}}{dt_{i}}}{2\dfrac{dQ_{i}}{dt_{i}}+k\dfrac{dZ_{i}}{dt_{i}}} ~~~~ (i=1,2,3,\ldots ,n) \end{aligned}$$

In order to find out the Cournot-Nash equilibrium, we need to solve the equation system of n variables $(t_{1},t_{2},t_{3},\ldots ,t_{n})$. From the optimal quantity produced by firms, equilibrium prices and quantity sold in markets in Appendix B, we can derive:

$$Q_{i}+\dfrac{n-1}{n+1}t_{i},\,Z_{i}-\dfrac{1}{nk+1}t_{i}~ \text {and}~p_{i}-\dfrac{n-1}{n+1}t_{i}~ \text {are constant with}~ i=1,2,3,\ldots ,n.$$

Thus, there is a unique Nash equilibrium: $t_{1}=t_{2}=t_{3}=\ldots .=t_{n}=t^{*}$.