Vertical industry and strategic trade policy

This paper investigates whether and how vertical industries have the characteristics to be targeted with the subsidy policy instrument. Under both price and quantity competition, the results of the established literature are reversed when exporters are in vertical relations. In particular, the conventional result that the emerging subsidising equilibrium is Pareto inefficient (the well-known prisoner's dilemma outcome) vanishes in the following cases: (1) under quantity competition, when products are sufficiently differentiated, and (2) under price competition, when products are neither too differentiated nor too substitutes. In such cases, the policy intervention leads to welfare improvement with respect to the free trade regime in both producing countries. Moreover, under price competition, export subsidization becomes the rule instead of taxation. Our results suggest that, depending on the degree of competition, vertical industries can be efficiently targeted through the subsidy policy.


Introduction
The subsidy (tax) policy for exporting firms is a cornerstone of the public intervention in the productive sectors which compete in terms of quantity as well as price. While its effectiveness in gaining an advantage is undeniable when other rival countries do nothing, in the case of quantity competition, if the other country also uses a subsidy policy, the well-known outcome is that both exporter countries are better off if neither uses the policy, and conversely in the case of price competition. 1 A demanding policy task consists of identifying the types and characteristics of industries to be targeted with the subsidy policy instrument, in that, as noted by Spencer (1986, 70-71), "identification of these characteristics is a preliminary step toward translating theory into practical policy proposals." This feature of contemporary economics is timely and significant for investigation. In fact, renewed debate on whether free trade or trade policies are suitable to improve the national welfares of those countries which are involved in international trade is high on the political agendas. As a clarifying example, consider the case of the United States. In recent years, the Obama administration has been supporting bilateral free-trade agreements with countries such as Colombia, Panama, and South Korea; it has also been conducting and advancing negotiations on two vast regional multilateral agreements such as the Asia-centred Trans-Pacific Partnership and the Transatlantic Trade and Investment Partnership with the European Union. However, the subsequent Trump administration chose to re-evaluate those advancements with the opportunity to introduce some trade policies (see Scott and Glass 2016;Scott 2016).
To process one step further in this direction, we focus on an industry characterised by a vertical structure. In fact, as noted by Horn and Wolinski (1988, 408), "some important industrial inputs are acquired not in conventional markets, but rather through relations between suppliers and buyers that are often characterized by elements of bilateral monopoly, which may result from many sources." The strategic trade policy literature has addressed both cases of quantity and price competition, producing mixed results. On the one hand, Brander and Spencer (1985) show that, when firms engage themselves in quantity competition in a third country's market, it is always convenient to subsidize unilaterally exports; however, such policy interventions by both countries are welfare inferior if compared to the case of free trade. On the other hand, Eaton and Grossman (1986) found that, in the case of price competition in the third market, each exporting country levies taxes (instead of subsidizing) on its exports and obtains a level of social welfare greater than that under a free-trade regime.
Subsequently, many other studies have enriched the literature on strategic trade policy theory, showing how the above-mentioned basic results may hold or not under different scenarios of imperfect competition (see, inter alias, Spencer and Brander 1983;Dixit 1986;Richardson 1986;Brander and Spencer 1988;Gosh and Pal 2014;Toshimitsu 2017;Tsai et al. 2018;Cho et al. 2019;Fanti and Buccella 2017, 2020, 2021aLivanis and Geringer 2021).
Although the relations between suppliers and buyers characterized by elements of bilateral monopoly seem to be rather common business phenomena, to the best of our knowledge, so far, the effects of the interaction between them and the strategic trade policies have been scantly addressed, and this paper attempts to fill this gap.
Remarkable exceptions are Spencer andJones (1991, 1992), Ziss (1997), Bernhofen (1997), Ishikawa and Lee (1997), and Ishikawa and Spencer (1999) and the more recent contributions of Kawabata (2010), Chou (2011), Chang and Ryu (2016), Choi et al. (2017), and Choi and Lim (2019). Bernhofen (1997) is perhaps the work most closely related to ours. Bernhofen studies strategic export policies in a vertical industry framework in which downstream Cournot competitors, subject to governments' export policies, must buy an intermediate product which is supplied to all firms by a unique foreign upstream monopolist to produce a final product for consumption. The main result is that the government's optimal export policy strongly depends on the vertical industry relationships, particularly the ability of the upstream supplier to price discriminate between the downstream producers of the final consumption goods. However, in our model, the monopolist supplier is domestic, and thus it is included in the domestic welfare when the government decides whether to intervene.
In a third-country market model, Kawabata (2010) studies the strategic export policies when vertically related industries are characterized by Bertrand duopoly with product differentiation. The work shows that, when the input markets are segmented, the optimal export policy towards the final good is a tax; with integrated markets, the optimal policy for final goods is a subsidy. Bilateral strategic trade policy can be welfare improving compared with free trade depending on the degree of product differentiation among final goods. Chou (2011) analyses the optimal export policy in a framework of a vertically related industry with differentiated products and focuses on the impact that the degree of product substitutability and market structure has on the design of such a policy. However, in this model, the upstream supplier is from a different country from those of the downstream firms, and it sells its input to all companies. In such a context, the policy is sensitive to the degree of product substitutability. Chang and Ryu (2016) assume, in contrast to our model, that the domestic supplier is a public firm and aim to show that the privatization of such a supplier is never optimal and that the privatization policy can be used as a pseudo trade policy.
In a third-country model with strategic trade policies, Choi et al. (2017) consider first-and second-mover advantages in a vertically related industry in which national upstream firms offer a two-part contract (input price plus a fixed fee) to downstream exporters of final goods. The authors show that the upstream firms' and governments' preferences over sequential versus simultaneous moves, and over free trade versus public intervention contrast the common wisdom. In fact, the endogenous market structure is: (1) with quantity strategies, regardless of the nature of goods, the potential leader chooses sequential moves in a leading position, and the equilibrium trade regime is unilateral subsidy; (2) with price strategies, the potential leader chooses simultaneous moves, and the equilibrium trade regime is bilateral taxes (free trade) when goods are substitutes (complements).
Also, Choi and Lim (2019) develop a third-market model with a vertically related market, first in a free-trade framework and then in an export subsidy regime. However, they concentrated on the choice of strategic variables (quantity or price) in line with the analysis of Singh and Vives (1984), differently from the present model, whose focus is on the unexpected welfare effects of the strategic policies with respect to the free trade regime under both given (quantity and price) modes of competition.
Thus, it is natural to ask whether and how a vertical industry might be suited to be targeted with the subsidy (tax) policy instrument. In this regard, we analyse a duopoly in which firms acquire inputs through bilateral monopoly relations with suppliers. We suppose a vertical industrial structure as simple as possible; that is, a monopolist in each country, which can be a labour union, sells an intermediate good (labour) to each firm, which uses it to produce two symmetrically differentiated final goods. The single national input supplier monopolistically sets the factor price (wage). Given their national operativity, there is no direct competition among suppliers (unions). Note that this is the polar case in which, when the input price is determined in bargaining between the firm and its supplier, the latter has full bargaining power; thus, the results of this paper qualitatively tend to hold also under firm-supplier bargaining when the supplier has sufficiently high power.
With this in mind, the study focuses on the design and implementation of the strategic trade policy only in the downstream sectors. In other words, we assume the presence of vertical industries in all countries, which have national bilateral monopoly relations in place. This simplified frame is very common in the industrial organization literature and permits a clearer interpretation of the results.
The timing of moving in this industry takes place in two stages as follows. First, each supplier monopolistically fixes the price of the input, then the input price is known, and the firms interact in the product market. During the product market interaction, the firms alternatively decide, according to the two standard different modes of competition: (1) on the quantities they will produce (i.e., Cournot) and (2) on the prices they fix (i.e., Bertrand) and as a consequence on their input purchases. As argued by Horn and Wolinski (1988, 410), this two-stage structure "makes the model an appropriate description of a situation in which input prices are determined for a relatively long period while production decisions are made for a relatively shorter period." To the two-stage structure discussed above, we add a first stage in which the governments determine the specific subsidies/taxes, 2 while in the last stage, firms compete in the third market.
The paper analyzes which policy instrument (subsidy or tax) will be chosen as well as the trade policy intervention results that are truly detrimental to the national social welfare in comparison to the free trade regime, as argued by the traditional literature.
The main findings of this paper challenge this common wisdom in both cases of Cournot and Bertrand competition, depending on the fierceness of the product competition in the final market. The study shows the cases in which policy intervention leads to welfare improvement with respect to the free trade regime in both producing countries. Therefore, our results suggest that vertical industries are efficient candidates to be targeted with the subsidy policy instrument.
However, the policy suggestions are differentiated between modes of competition; in fact, trade policies are efficient under quantity competition when products are sufficiently differentiated and under price competition when products are neither too differentiated nor too substitutes between them. Moreover, we show that, under price competition, the policy instrument traditionally used should be reversed; the optimal tool is a subsidy instead of a tax.
The rest of the paper is organised as follows. "The model" develops the model of duopolistic vertical industry and the equilibrium outcomes of the Cournot and Bertrand cases are separately presented under both policy intervention and freetrade regimes. "Trade policy and free-trade regimes" focuses on the examination of the relative efficiency properties of both regimes under both modes of competition. "Conclusion" concludes the paper.

Technologies and demand
We consider a single industry consisting of two firms, 1 and 2, located in two exporting countries, which produce differentiated goods sold to a third country, an importing country. We assume, as usual, a constant return to scale technology so that the two firms face the constant marginal cost given by the price per unit of input, z i (which is assumed to capture all short-run marginal costs). The present model builds on Brander-Spencer (1985) and Eaton-Grossman (1986). Firms compete between them in terms of either quantity or price (i.e., a duopolistic Cournot or Bertrand market). The governments of Country 1 and 2 provide their producers with specific export subsidies, s i . Thus, firm i 's linear cost function is: Therefore, profits of firm i can be written as where q i represents output. Regarding the determination of the product market demand, following an established branch of the literature (see, e.g., Dixit 1979;Singh and Vives 1984;Qiu 1997;Hackner 2000), we assume that preferences of the representative consumer over q are: such that they generate a system of linear demand functions. The parameter a > 0 captures the size of the market demand, and −1 < < 1 represents the degree of horizontal product differentiation: 0 < < 1 captures the case of imperfect substitutability between goods; a negative value of implies that goods are complements. In our analysis, we restrict the attention to the case of 0 < < 1 , i.e. to imperfect substitute goods. From the maximisation of the representative consumer utility in Eq.
(3), subject to the budget constraint p 1 q 1 + p 2 q 2 + y = M (where y is the numeraire good 3 and M denotes the consumer's exogenously given income), the inverse demand functions of goods 1 and 2 are:

Cournot competition
At the last stage, from (2), (4.1), and (4.2), under the maximization problems of the firms, the best response functions are given by with q i q j < 0 , that is, the quantity game is in strategic substitutes. From (5) and (6), we obtain equilibrium output (respectively, by firm i , given z i , z j , s i and s j ): In stage 2, a firm-specific national supplier (union) (that, without loss of generality, produces at zero cost) fixes input price (wage) by maximising its profit (utility), Ω , given by: where q i (that is, the firm's output for any given level of prices) is given by Eqs. (7) and (8), respectively. If we interpret the supplier as a labour union, Eq. (9) informs us that the union's objective function is to maximize the wage bill. Making use of Eqs. (7) and (8), maximisation of (9) by both suppliers leads to the input prices reaction functions. One can easily check (results available upon request) that z i z j > 0 ; that is, the input price game among suppliers (unions) is in strategic complements. Then, solving the system of the input price reaction functions, one obtains the following sub-game perfect equilibrium input prices as a function of the trade policy instrument After substitution of (10) and (11) in (7) and (8), the sub-game perfect equilibrium quantities are: By substituting Eqs. (10)-(13) backwards, also profits are obtained as functions of only subsidy/tax rates. The standard definition of the social welfare (that is, the sum of the outcomes of the economic agents) of country's i with a vertical industry exporting in a third market is where the latter term denotes the cost (revenue) of export subsidization (taxation). To economize on space, the expressions for i (s i , s j ) , Ω i (s i , s j ) and SW i (s i , s j ) are here omitted (available upon request). The process of maximisation of (14) with respect to subsidy/tax rates by each government (for a given subsidy/tax rate of the other government) leads to the following reaction function in subsidy/tax rates of government 1 (and its counterpart for government 2) with s i s j < 0 for ∈ (0, 1) ; that is, the strategic trade policy game among governments is in strategic substitutes. The rationale for this result is straightforward: if one government increases its own subsidy, the respective exporter expands output, q 2 = 2 8(a + s 2 ) − 2 (a + s 2 ) − 2 (a + s 1 ) and this exerts a downward pressure on prices. As a consequence, the rival government reduces its subsidy to avoid extra production from its exporter, which would lead to an excessive price decrease. From (15) and its counterpart for government 2, we get the sub-perfect equilibrium subsidy/tax rates: where Γ = (16 + 8 − 4 2 − 3 ) , and the apex C denotes Cournot competition.
By exploiting (16), after the usual algebra, the equilibrium values of input prices, output, downstream firms' profits, upstream firms' profits, and social welfare are easily derived, as reported in Table 1, first row, in which I = 8 − 2 .

Bertrand competition
In this section, we assume that the product market game is characterized by firms' price-setting behaviour. From (4.1) and (4.2), we can derive the product demand firm i faces It follows that the profits of firm i are: From (18), the first-order condition for profit-maximization gives, at the third stage, the choice of price of firm i in the function of the price chosen by firm j: and hence, p i p j > 0 for > 0 , the Bertrand product market game is played in strategic complements. Equilibrium prices as a function of input prices and subsidy/tax rates are: Cournot, free trade Bertrand, strategic trade policy Bertrand, free trade Hence, substituting Eqs. (21) and (22) in (17), the sub-game perfect equilibrium output as a function of input prices and subsidy/tax rates is obtained Also in this case, at the second stage, each independent supplier maximises its profit (utility), Ω , as in Eq. (9), where q i is now given by Eqs. (23) and (24), respectively. From this maximisation, one obtains the sub-game perfect best-reply function in the input prices of supplier-firm pair i (and of its counterpart, firm j ) under the non-cooperative Bertrand-Nash equilibrium in the product market with z i z j > 0 , that is, the input price game among suppliers (unions) is again in strategic complements. Then, solving the system of the input price reaction functions, one gets the input prices as a function of subsidy/tax rates: Substituting Eqs. (25) and (26) in (23) and (24), the sub-game perfect equilibrium quantities, as only function of subsidy/tax rates, are: Then, by substituting Eqs. (27) and (28) backwards, profits and social welfares at the second stage as only a function of subsidy/tax rates are obtained, which are here omitted to economize on space (available upon request).

Free-trade regime
To complete the picture and for comparison purposes, one needs the equilibrium outcomes when governments do not intervene; that is, the free-trade (FT) regime. Those outcomes can be easily obtained substituting s 1 = s 2 = 0 in all equilibrium expressions under both Cournot and Bertrand competition. Those outcomes are reported in Table 1, second row, for Cournot competition, where K = (4 − )(2 + ) , and fourth row, for Bertrand competition, where Λ = 4 − − 2 2 .

Trade policy and free-trade regimes
To discuss the results under trade policy and free trade regimes, we first present this Lemma.
Lemma 1 In accord with (resp. in contrast with) the conventional wisdom under quantity competition (resp. price competition), governments always fix an export subsidy (except for in extremely high substitutability, i.e., for > ).
Lemma 1 tells us that, while the result from Brander and Spencer (1985) is always preserved, the result from Eaton and Grossman (1986) is reversed for most part of the degrees of product differentiation; that is, if the price competing industry is vertical, then a subsidy instead of a tax is the rule. Now we are positioned to investigate whether the policy intervention leads to an inefficient outcome (i.e., a prisoner's dilemma game structure). Indeed, the analysis of the game between governments reveals that the conventional results indicating that intervention is always the sub-perfect Nash equilibrium of the game holds true under vertical industries as well. For brevity, the analysis is not reported here (available upon request). We define the following welfare differentials: Therefore, and Since the welfare differentials are not elegant enough to examine their properties analytically, a numerical analysis has been conducted, whose results are however exhaustive.
Result 1. Under quantity competition, the strategic trade policy is welfare improving with respect to the free-trade regime when < • = 0.6; that is, when products are sufficiently differentiated.
A graphical representation of Result 1 is provided in Fig. 2. Result 2. Under price competition, the strategic trade policy is welfare-improving with respect to the free-trade regime when ≤ 1 ∧ ≥ 2 ; that is, when products are either sufficiently differentiated or close substitutes between them. P r o o f : Δ B ≥ 0 ⇔ ≤ 1 = 0.495 and ≥ 2 = 0.966; Δ B < 0 ⇔ 1 < ≤ 2 (see also Fig. 2, below). The conventional result that the trade intervention is always welfare improving under Bertrand competition no longer holds true when the industry is characterised by vertical relations. The economic intuition behind the emergence of this unconventional result is as follows.
Without upstream suppliers, the mechanism that leads to the selection of a tax is clear: price competition tends to lower the prices of the final goods and expand production. Therefore, by taxing output (exports), firms shrink production, and taxation sustains prices. Consequently, profits and social welfare increase as well.
With upstream suppliers, the provision of a subsidy to the downstream firm has a positive impact on the upstream profitability (union utility): in fact, one can verify that 47 , which holds always true if s i ≥ 0(analytical details are available upon request). A subsidy leads the downstream firm to expand output. Consequently, the downstream firm increases the demand of the intermediate goods (labour) that the upstream firm (union) produces (furnishes). Increased demand allows the upstream supplier (union) to charge a higher price (wages) for the intermediate product (labour). By anticipating this move of the supplier, the government provides a subsidy that precisely reduces the downstream firm's increased costs of production to sustain its competitiveness.

Conclusions
This paper has analysed the effects of strategic trade policies on the social welfare of exporting countries in comparison to the free trade regime in a simple vertical industrial structure, in which upstream firm-specific, national monopolists, that can be interpreted as unions, sell intermediate goods (labour) to their respective national, downstream firms which use it to produce two horizontally differentiated The common wisdom is that, under quantity competition (resp. price competition) policy intervention reduces (resp. improves) social welfare in comparison to the free-trade regime. Moreover, in the former (resp. latter) case, the suited policy instrument results in a subsidy (resp. a tax). These textbook results are shown by Brander and Spencer (1985) and Eaton and Grossman (1986), respectively, and they abstract from vertical relationships between firms/unions.
We show that, when exporters are involved in vertical relations, the results of the established literature are reversed, both under price and quantity competition. In particular, the conventional result that the emerging equilibrium choice of subsidising is Pareto inefficient (i.e., the well-known prisoner's dilemma outcome) vanishes (1) under quantity competition, when products are sufficiently differentiated, and (2) under price competition, when products are neither too differentiated nor too substitutes. Therefore, in such cases, the policy intervention leads to a welfare improvement with respect to the free trade regime in both producing countries. Moreover, we show that, under price competition, subsidization instead of taxation of export becomes the rule.
The policy suggestions in the case of vertical industry are straightforward. When firms compete on prices, (1) an export subsidy (instead of an export tax, as in the conventional result) should be generally implemented, and (2) policy intervention is efficient only if the degree of product competition is neither too high nor too low (instead of for any degree of product competition as in the conventional result). In particular, we have shown the novel result that, when price competition is very fierce, governments at equilibrium will tax exports, in line with the results from Eaton and Grossman; however, in contrast to those authors, this equilibrium is inefficient. In other words, under a sufficiently high price competition, the vertical link of the industry implies that governments are trapped in a prisoner's dilemma game.
When firms compete on quantities, the policy intervention is efficient, provided that the product competition is sufficiently low (instead of always inefficient, regardless of the intensity of product competition as in the conventional result). Therefore, the generalization of the previous famous models of subsidy policy games under quantity and price competition (i.e., Brander andSpencer 1985, andEaton andGrossman 1986, respectively) to a vertical industry suggest very rich trade policies.
Author contributions Both authors have equally contributed to the model design and editing of the article.
Funding This study was not funded by any Institution.
Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of interest
The authors declare that they are Guest Editors of the article collection "The Industrial Organization of Vertically-Related Industries".
Ethical approval This article does not contain any studies with human participants performed by any of the authors.
Informed consent This article does not contain any studies with human participants performed by any of the authors.
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