2.1 Third stage: firms compete à la Cournot

During the third stage of the game, each firm believes that the other will hold its output fixed while the output level of that firm changes⁸. At this point, each firm has already considered that both markets are segmented. Profit functions for each of the firms take the following form:

For the domestic firm, and:

For the foreign firm⁹.

Where Q = x 1 + x 2 , represents the domestic market demand, whereas Z = y 1 + y 2 , represents the foreign country’s market demand, so that P ( Q ) and P ( Z ) are the prices for the domestic and the foreign markets, respectively, where the Cournot competition takes place.

Likewise, F stands for the fixed cost of entry, and t 1 is the specific trade tax ”administered” by the domestic government, while t 2 is the specific trade tax "administered" by the foreign government. Finally, I consider a constant marginal cost of production c for the domestic and foreign production levels and τ to export between countries, considering symmetric cost structures between countries.

Naturally, for the domestic market, reaction functions are obtained directly from ∂ π 1 ∂ x 1 = π x 1 = 0 = π x 2 = ∂ π 2 ∂ x 2 . The foreign market reaction functions are similarly obtained from: ∂ π 1 ∂ y 1 = π y 1 = 0 = π y 2 = ∂ π 2 ∂ y 2 . Therefore, for the domestic market:

With second order conditions ∂ ∂ x i [ ∂ π i ∂ x i ] = π x i x i < 0 , for any i = 1,2 , and ∂ ∂ x i [ ∂ π i ∂ x j ] = π x i x j < 0 , for any i ≠ j , so that I have for the domestic market:

At this point, the question would be about the effect of the trade tax t 1 on the production levels realized for the domestic market. Totally differentiating eqs. (3) and (4) to get the answers for the following expressions: d x 1 d t 1 and d x 2 d t 1 , I obtain:

Since π x 1 t 1 = 0 and π x 2 t 1 = − 1 , the equation system can be solved simultaneously to obtain:

Where Δ = π x 1 x 1 π x 2 x 2 − π x 2 x 1 π x 1 x 2 > 0 ¹⁰.

Provided the Routh-Hurwitz condition for reaction function stability takes place, eqs. (9) and (10) clearly show that an increase in the trade tax t 1 , leads to a domestic output increase, while decreasing simultaneously the foreign output’s share in the domestic market.

The analysis in the foreign country is quite similar. I will take the second subset of reaction functions. In the foreign market these are: ∂ π 1 ∂ y 1 = π y 1 = 0 = π y 2 = ∂ π 2 ∂ y 2 . Then I obtain:

With second order conditions ∂ ∂ y i [ ∂ π i ∂ y i ] = π y i y i < 0 , for any i = 1,2 , and ∂ ∂ y i [ ∂ π i ∂ y j ] = π y i y j < 0 , for any i ≠ j , i.e., for the foreign market:

Totally differentiating eqs. (11) and (12) to obtain the effects of t 2 on the production levels y 2 and y 1 , I obtain symmetric results as those shown for the domestic market from the following equation system for the foreign country:

Since π y 1 t 2 = − 1 and π y 2 t 2 = 0 , this equation system can be solved simultaneously to obtain:

Where Δ = π y 1 y 1 π y 2 y 2 − π y 2 y 1 π y 1 y 2 > 0 .

As I expected, the effect of the trade tax on the domestic exports’ market share in the foreign market is negative, whereas the effect of the trade tax on the foreign production market share is positive. In both markets, the application of trade taxes, shifts to the left the counterpart’s reaction function, reducing its counterpart’s market share.

Proposition 1: An increase in the trade tax committed by the domestic government:

1. Increases the domestic price of the good,
2. Increases the domestic market share,
3. Increases domestic profits, and
4. Reduces foreign profits.

Proof:

a) dP ( Q ) dt 1 = P ' ( Q ) Δ [ π x 1 x 1 − π x 1 x 2 ] > 0

b) d ( x 1 Q ) dt 1 = − ( x 2 π x 1 x 2 Δ + x 1 π x 1 x 1 Δ ) Q 2 > 0

c) d π 1 dt 1 =     x 1 P ' ( Q ) π x 1 x 1 Δ > 0

d) d π 2 dt 1 =     − x 2 < 0

2.2 Second stage: the domestic government determines the optimal trade tax t ₁ :

Under freely managed trade and with the aim to determine the optimal trade tax for the domestic economy, I proceed to examine the consumer and producer surpluses and trade revenues. These three elements allow me to have an expression for the domestic welfare function:

First, let us obtain the sign of the optimal trade tax by obtaining dW ( t 1 ) dt 1 = 0 . Therefore, on the consumer surplus the effect of the trade tax is as anticipated:

Provided:

On the producer surplus, the general expression is:

From the first order conditions for the domestic firm, I know that: ∂ π 1 ∂ x 1 = 0 = ∂ π 1 ∂ y 1 . On the other hand, from eq. (1) I know that ∂ π 1 ∂ t 1 = 0 . As for the remaining of the first order conditions for the domestic firm, although mixed between firms and countries, choices on y 2 -taken by the foreign firm-, and on x 1 -taken by the domestic firm-, can be thought of separately for firms 1 and 2, respectively¹¹. Finally, for the producer surplus the effect of the trade tax is unambiguously positive, as it was expected:

Regarding the tariff revenue, the following expression reflects the negative effect the specific trade tax has on x 2 , but considering that the specific trade tax is not prohibitive, this effect should be non-negative in a context where the trade tax plays the role of a countervailing duty¹².

Naturally, there is a negative effect coming from the consumer surplus, but as long as the trade tax is not prohibitive, that effect can be more than compensated by the effect of the trade tax on the producer surplus and the tariff revenues, plus the gain in the consumer surplus as a result of the practice of reciprocal dumping, so that the optimal trade tax would be no negative:

Accordingly, the optimal t 1 would be a function of the price elasticity of the demand in the domestic country, and the market share of the foreign exports in the domestic country¹³, i.e., t 1 o p t = f ( α , η ) ( + ) ( + ) ≥ 0 . Where: α = x 2 x 1 + x 2 represents the market share of the foreign output in the domestic market demand, and η = d Q d P P Q , the price elasticity of the demand in the domestic country. The result shown in eq. (24) lead us to the following proposition:

Proposition 2: Even though the domestic government has a limited commitment, it has an incentive to independently provide a trade tax to the domestic firm because dumping is better than autarky for producers and consumers.

At this stage of the game, even though the government has a limited commitment, the sequence of the game -with the trade tax offered before the Cournot competition takes place-, influences firms’ behavior, so that the non-cooperative equilibrium of the domestic government favors the domestic firm.

The intuition behind this result is as if the trade tax applied by the domestic government in stage two, with the aim to protect the domestic firm, played the role of a crossed subsidy on exports of the foreign firm. In this context, there is a profit-shifting motive for intervention¹⁴.

2.3 First stage: entry choice

At this stage of the game, the choice to enter the market for the domestic firm will reflect the possibility to obtain positive net benefits from having administered protection in a context of reciprocal dumping, compared to a situation of autarky that may cover or not, the fixed cost of entry F 1 . In other words, the domestic firm will entry if:

However, this means that the private incentive to enter the market may not be sufficient from the social welfare point of view. To analyze this, I examine through the private domestic welfare.

Let us define B 1 = π 1 ( t 1 o p t ) − π 1 ( t 1 = 0 ) as the domestic firm’s incentive to dump in the foreign country, once substituting t 1 o p t using the general linear inverse demand function, I obtain the following expression for the domestic private welfare:

In the absence of managed trade protection, the autarky equilibrium for the domestic firm implies that the domestic firm will have the following profit²⁰:

And if both countries are reciprocally dumping at each other’s market I obtain:

To finally get an explicit expression for the domestic private welfare in eq. (42):

If I additionally assume that both firms have symmetric cost structures with c x = c = c y , where c , τ ≫ 0 , and c ∈ ( 0,1 ) with τ ∈ ( 0, 1 4 ) , the domestic private welfare is positive B 1 ≥ 0 and an increasing function in the transport τ and the marginal ccosts²¹.

However, from the private welfare point of view and for entry costs F 1 relatively high, there might be not enough private incentives to enter the market and to dump in the counterpart’s market.

3.1 Domestic welfare

To explore how private and social incentives to dump interact each other, I will compare the change in the domestic social welfare that is obtained from the practice of dumping in the context of this model i.e., Δ W 1 to the private incentive net of the investment cost, i.e., B 1 − F 1 . The abovementioned change can be examined in the following equation:

The first term in eq. (58) represents the change in the consumer surplus, i.e., Δ C S , which is unambiguously positive even though trade protection rises the domestic price to a level higher than that observed under the potential practice of dumping without the managed trade regime. The second term -the change in the producer surplus net of the investment cost F 1 -, is clearly negative as long as the equilibrium price under the managed trade regime is lower than the price obtained under autarky conditions²³. The third term reflects the change in trade revenues, i.e., Δ T R , and is clearly positive when changing from autarky to trade under this specific trade regime, as long as the trade tax be non-prohibitive. At this point, the main analysis is performed to examine:

Where the change in the consumer surplus is the following:

Which is clearly positive for strictly positive values of the marginal cost and the transport cost c, τ ≫ 0 , and c ∈ ( 0,1 ) with τ ∈ ( 0, 1 4 ) . As for the change in trade revenues, I obtain the next expression:

Therefore,

is clearly positive for all relevant values of c and τ .

Proposition 5: The managed trade regime provides both social and private incentives to dump in the counterpart’s market, but from the social perspective, these incentives are not enough when the sunk costs to enter the market are in the range F 1 ∈ ( B 1 − F 1 ) .

The social benefits of dumping in the counterpart’s market while being reciprocally dumped in its own market -net of private benefits-, are positive for all relevant values of marginal production and transport costs. Naturally, sunk costs represented by F 1 can be potentially high, but if F 1 falls in the range ( B 1 − F 1 ) , there are no private incentives that compensate social welfare loses to reciprocally dump: adding the sunk cost F 1 to transport and marginal production costs, i.e., τ and c , would prevent from the practice of reciprocally dumping itself.

3.2 Global welfare

This section’s main concern is about the effects of the managed trade regime on global welfare. On the one hand, it has been pointed out that there are clear incentives for the domestic and foreign firms to dump each other as long as: i) the perception of the segmented market leads to generate extra revenues from dumping in the counterpart’s market, and ii) there is a credible commitment of the government to provide administered protection preventing the price from falling below perfect-competition levels once the practice of dumping has taken place. However, for sufficiently high levels of F 1 and F 2 , although privately desirable, the practice of dumping might not take place and therefore, trade taxes’ influence on firm’s behavior would be null.

To obtain the expression of the world welfare we will add up the expressions for the domestic and foreign welfare in equations (56) and (57) in the following way:

Then substituting t 1 o p t = 1 − τ − c x 3 and t 2 o p t = 1 − τ − c y 3 , allowing the practice of dumping to exist, and comparing with a situation where the practice of dumping does not take place, i.e., t 1 = 0 = t 2 , this expression becomes:

Finally,

This expression is positive for all relevant values of τ and c, i.e., c, τ ≫ 0 , where c ∈ ( 0,1 ) while τ ∈ ( 0, 1 4 ) .

Proposition 6: Global welfare is higher under reciprocal dumping in a managed trade regime. Any bilateral trade liberalization can never increase global welfare.

Like in the domestic welfare analysis, any trade liberalization from the optimal trade tax, being even and unbound tariff, will only deprive domestic and foreign firms with sufficient incentives to start trading each other.