Trade Policies, Firm Heterogeneity, and Variable Markups

Transcription

1 Trade Policies, Firm Heterogeneity, and Variable arkups Svetlana Demidova caster University ay 29, 204 Abstract In this paper we explore the effects of trade policy in form of wasteful and non-wasteful import tariffs in the model with firm heterogeneity and variable markups. First, we develop a graphical analysis similar to the one in Demidova and Rodríguez-Clare 203) and show that, as in the models with CES preferences that imply constant markups, in the model with variable markups, the welfare effect of a wasteful import tariff depends on the use of an outside good assumption: unilateral trade liberalization reduces welfare in the presence of the outside good, and raises it otherwise. Second, we show how to model a small one-sector economy with variable markups. We use this model to demonstrate that the type of an import tariff matters: while full trade liberalization is optimal in the case of a wasteful tariff, this is no longer true in the case of a non-wasteful tariff. We derive the formula for the optimal value of the non-wasteful tariff and show that a strictly positive tariff maximizes welfare of liberalizing economy. Introduction Recently, there has been a surge in international trade models with imperfect competition and heterogenous firms. In addition to the expansion in product variety studied by Krugman 980), these models offer new channels, through which trade affects welfare. A number of papers, headed by the seminal work of elitz 2003), highlight the mechanism of self-selection of more effi cient firms into exporting, through which trade liberalization can lead to the reallocation of resources towards more effi cient firms, raising average productivity in liberalizing countries. ost of these papers rely on the assumptions of monopolistic competition and constant elasticity of substitution CES) preferences, implying constant mark-ups charged by firms. Not only such prediction is inconsistent with the data, but it also ignores potential welfare gains that arise due to variable markups. These gains are emphasized in a number of recent papers that allow for variable markups by deviating from the assumption of monopolistic competition and/or incorporating non-ces preferences. 2 These Department of Economics, caster University, Hamilton, ON, Canada L8S demidov@mcmaster.ca. Phone: 905) , ext Fax: 905) See, for example,... 2 The examples of the settings with variable markups include, amongst many others, the model of monopolistic competition in elitz and Ottaviano 2008), the Cournot competition model in Atkeson and Burstein 2008) and

2 models give rise to what is called pro-competitive gains from trade. At the firm level, trade liberalization intensifies foreign competition, reducing markups charged by local producers. At the industry level, trade liberalization may affect the distribution of markups, reducing their dispersion. The latter effect can be welfare-improving, since it reduces misallocation of resources across firms that is caused by variable markups. The detailed discussion of this distortion can be found in Nocco, Ottaviano and Salto 203), who use the elitz and Ottaviano 2008) setting, 3 where more effi cient firms charge higher markups. They show that endogenous markups result in additional within-sector misallocation: more productive firms do not pass on their entire cost advantage to consumers by absorbing part of it in the markup and end up selling too small quantities compared to the optimal levels. The opposite happens with high cost producers, whose varieties end up being oversupplied. Hence, while variable markups allow us to study pro-competitive gains from trade, in the models with heterogeneous firms such markups introduce an additional distortion. Given the discussion above, the important question is what the new trade theory means for trade policy. In the presence of new mechanisms, is full trade liberalization optimal or is there need for some sort of protection? Does it matter what form trade liberalization takes, i.e., is there difference between a reduction in import tariffs and a fall in non-tariff trade barriers? What are the optimal trade policies in such models? Given the complexity of analytical derivations in the case of variable markups, it is not surprising that the answers to the aforementioned questions are given mostly for the models with the CES preferences. ainly, the results of trade liberalization in these models depend on whether one considers a one-sector or multiple-sector economy, and whether the import tariff is modeled as nonwasteful or wasteful i.e., whether tariff revenues are included into national income of a country or neglected in the analysis). 4 First, consider the case of two one-sector asymmetric economies. Demidova and Rodriguez-Clare 203) show that in such setting, a decline in wasteful import costs always benefits the liberalizing country, no matter whether this economy is large or small. This means that the optimal value of the wasteful import tariff is zero, i.e., full liberalization is optimal. However, if the import tariff is modeled as non-wasteful, its optimal value becomes strictly positive: Demidova and Rodríguez- Clare 2009) derive this value for the case of a small open economy, while Felbermayr, Jung and Larch 203) derive it in the setting with two large economies. Hence, in the case of non-wasteful import tariff, protection becomes an optimal policy. The intuition behind these results is that in addition to the terms of trade effect, imperfect competition together with firm heterogeneity gives rise to distortions, which prevent gains from trade happening. When it comes to the analysis of multiple-countries settings, it is worth mentioning the influential work by Arkolakis, Costinot and Rodríguez-Clare 202), who characterize gains from Edmond, idrigan and Xu 202), the Bertrand competition setting in Bernard, Eaton, Jensen and Kortum 2003) and Holmes, Hsu and Lee 203). 3 Variability of markups is achieved by using the demand system from Ottaviano, Tabuchi and Thisse 2002). 4 Here we discuss non-cooperative tariffs only. For the results of cooperative trade policies in the models with firm heterogeneity see, for example, Ossa 20), Cole and Davies 20) and Felbermayr, Jung and Larch 203). 2

3 trade in a general elitz-type setting with multiple countries. They provide a formula for welfare changes from trade liberalization that depends on the observed degree of endogenous openness of a country, which is measured by the expenditure share on domestically produced goods, and the trade elasticity measured by a standard gravity equation. However, since the authors consider only small changes in import tariffs, they neglect tariff revenues and their effect on income redistribution across countries. Hence, their approach cannot be used for the analysis of large changes in trade barriers and the derivation of the optimal tariff rates. Now, consider asymmetric economies with multiple sectors. In particular, let us focus on a very specific case of two sectors, where only one sector is modeled à la elitz 2003) and the other one is freely traded, produced one-to-one from labor in all countries, and used as numeraire. Usually the output of the latter sector is called an outside good. The use of the outside good is very popular in many extensions of elitz 2003) as well as in other works in trade literature. One well-known advantage of this assumption is significant simplification of analytical derivations, since wages across all countries become exogenous and can be normalized to. The second advantage is the opportunity to study cross-sectoral ineffi ciencies. However, such an assumption comes with a price. First, it excludes an important channel, through which trade affects welfare, namely, an income effect. Second, it adds an extra distortion into the model, as in the outside good sector there is no markup, while in the other sector producers charge prices above their marginal costs. As pointed out by Bhagwati 97), the presence of distortions can result in the breakdown of Pareto-optimality of laissez-faire. Hence, it is not surprising that the use of an outside good can potentially distort the welfare predictions. Consider, for example, unilateral trade liberalization UTL), when one of the countries unilaterally reduces trade barriers modeled as a wasteful tariff. Table summarizes some of the results from the previous literature. The main message of this table is that the assumption of the outside good reverses the effect of UTL on the welfare of liberalizing country: UTL is welfare-reducing in the presence of the outside good, and welfareincreasing otherwise. 5 Table. Unilateral Trade Liberalization: the Case of CES preferences Base odel Assumptions Results Krugman s odel 2 economies of any size) Graphical analysis from Demidova 2009): Extension of elitz 2003) with 2 large economies Felbermayr, Jung and Larch 203): Extension of elitz 2003) with 2 large economies Demidova and Rodríguez-Clare 203): Extension of elitz 2003) with 2 large economies and elitz 2003) with Home being a small economy Is there an outside good sector? Homogenous Firms Heterogeneous Firms and CES Preferences Productivity Distribution Welfare at Home No Identical Firms Welfare Yes wages ) General Form Welfare No Pareto Welfare No General Form Welfare 5 Note that in the case of two-sector economies it is assumed that before and after trade liberalization, there is no specialization in any of the countries, i.e., all countries produce both differentiated and outside goods. 3

4 Given the extensive analysis of the models with the CES utility functions and heterogeneous firms, what can we say about the models with variable markups and the impact trade liberalization has on welfare in the presence of potential pro-competitive gains from trade? While there is an extensive empirical evidence of changes in markups due to changes in openness of a country, the theoretical results are scarce and deal only with wasteful import tariffs or non-tariff barriers). For example, the well-known extension of elitz 2003) that incorporates endogenous markups by using the linear demand system with horizontal product differentiation developed by Ottaviano, Tabuchi and Thisse 2002) is the work by elitz and Ottaviano 2008). Its main prediction with regard to unilateral trade liberalization UTL) is that UTL is welfare-reducing for the liberalizing country. However, to simplify their derivations, the authors rely on the outside good assumption, which creates the same distortion as the one discussed above in the case of the models with CES preferences. This raises a natural question of whether the results derived by elitz and Ottaviano 2008) depend on the this specific assumption. oreover, the authors consider wasteful import tariffs. How do their results change if one considers a non-wasteful tariff instead? The main goal of our paper is to fill the gap in the literature and provide tractable analytical results for the optimal trade policy in the case of variable markups. To eliminate potential distortions, we extend the model of monopolistic competition and variable markups from elitz and Ottaviano 2008) by dropping the assumption of the outside good. Although it significantly complicates the derivations, we show that the analysis can still be carried out in a way similar to Demidova and Rodríguez-Clare 203), who deal with the case of CES preferences. In particular, we show that one can use the help of a simple figure that summarizes the key relationships in the model: similarly to Demidova and Rodríguez-Clare 203), the equilibrium results in two conditions that relate wage with the cost cutoff for domestic sellers in the Home economy. Using this figure, it is easy to show that, as in the case of CES preferences, in the absence of the outside good unilateral trade liberalization is welfare-increasing. Next, by using the same approach as in Demidova and Rodríguez-Clare 2009) to model a small economy, we show that UTL result for the case of 2 large economies also holds in the case of the small economy. Our results are summarized in Table 2. As can be seen from this table, a deviation from CES preferences and an introduction of pro-competitive gains from trade do not change the relationship between the results of UTL and the presence of the outside good: UTL is welfare-reducing in the presence of the outside good, and welfare-increasing otherwise, which confirms the conjecture of Ossa 20). Table 2. Unilateral Trade Liberalization: the Case of Variable arkups Base odel Assumptions Results elitz and Ottaviano 2008): 2 large economies Extension of elitz and Ottaviano 2008) with 2 large economies and with Home being a small economy Is there an outside good sector? Productivity Distribution Welfare at Home Yes wages ) Pareto Welfare No Pareto Welfare Finally, we use our small economy approach to study an optimal non-wasteful import tariff. 4

5 Although we do not get its value explicitly, we derive the equation for the optimal tariff rate, which can be studied further. We show that, as in the case of CES preferences, protection remains an optimal policy, i.e., a strictly positive import tariff maximizes welfare at Home. In other words, although the presence of variable markups creates an additional argument in favor of trade liberalization, since it can fix the misallocation distortion, this argument does not outweigh the effect of other distortions in the economy and does not shift policy choice from protection to full trade liberalization. Before we proceed with the details of our analysis, it is worth mentioning one of the caveats of our choice of the model. As in elitz and Ottaviano 2008), to deal with the complications caused by variable markups, we specify the cost distribution as Pareto. Unfortunately, it also means that the markup dispersion does not change with the changes in the level of openness see Feenstra 204) for the detailed discussion of this issue). Hence, our analysis excludes one of the important pro-competitive effects of trade from consideration. Nonetheless, trade liberalization still affects the markups at the firm level, reducing a revenue-weighted average markup. 6 The paper proceeds as follows. Section 2 explores the case of two large economies, while Section 3 looks at the case of the small economy. Section 4 offers concluding remarks. 2 The Case of 2 Large Economies First, we will modify the elitz and Ottaviano 2008) model by excluding the outside good assumption, i.e., wages everywhere will be determined endogenously. Then, after deriving the equilibrium conditions, we will show that in the modified model welfare of a liberalizing country rises as it unilaterally reduces trade barriers to foreign firms so that the result from elitz and Ottaviano 2008) is reversed). 2. Demand There are two countries, Home and Foreign, of size L and L, respectively. We will denote the Foreign country s parameters with asterisks. Wage in the Foreign country, w, is normalized to unity. In elitz and Ottaviano 2008) a household maximizes U = q0 c + α q c i) di 2 γ q c i) 2 di ) 2 2 η q c i) di s.t. p c 0q0 c + p i) q c i) di = w, where is the set of all differentiated good varieties available in the Home market, and q c 0 is a consumption of the numeraire good. By excluding the outside good from this model, we get the following household s maximization problem with a non-separable quadratic utility function α can be normalized to, but we keep it 6 Feenstra 204) argues that the average markup does not change, since a fall in domestic markups is balanced out by rising markups charged by foreign exporters. The reason that this is not the case in our paper is in the definition we use for the average markup: we follow Edmond, idrigan and Xu 202) and measure the average markup as a revenue-weighted mean of firm-level markups, while Feenstra 204) considers an unweighted mean. 5

6 in the formula to make the comparison with elitz and Ottaviano 2008) easy): U = α q c i) di 2 γ q c i)) 2 di 2 2 η q c i) di) s.t. Then from F.O.C., one can get λp i) = α γq c i) η p i) q c i) di = w. q c i) di, ) where λ is a Lagrangian multiplier. Denote qc i) di by Q. Then it can be shown that λ = αq γ qc i)) 2 di ηq 2. 2) w Using the same logic as in elitz and Ottaviano 2008), one can show that the set c of all varieties that are consumed q c i > 0) is the largest subset of that satisfies: ) p i γα + η p p max, η + γ λ where is the measure of consumed varieties in c and p max = /) i c p i di represents the choke price. Similarly, in the Foreign country λ p i) = α γq c, i) ηq, 3) λ = αq γ q c, i)) 2 di η Q ) 2, 4) where Q = q c i) di is the quantity of all varieties consumed by a Foreign individual. 2.2 Production and Firm Behavior Domestic arket. A firm with marginal cost C sells its variety i to L consumers. Hence, it sells q i) = Lq c i) and maximizes its profit, π = pq i) C i) q i), by choosing the appropriate level of p i). Given ), we get so that the F.O.C. results in q c i) = γ ) α λp i) η q c i) di, qi) = Lq c i) = L λ p Ci)). γ There is a continuum of domestic ) firms at Home that derive their unit labor cost from the k Pareto cost distribution G c) = c c, c [0, c ], k >, so that C c) = wc. Only firms with positive demand will sell domestically. We can define the cutoff such that q ) = 0 or p ) = C ), where, given wage w, p ) = w, 5) 6

7 so that only firms with c sell at Home. Using ), we have wλ = α ηq. 6) Similarly, we can define the cutoff for Foreign domestic sellers their initial cost distribution is G c) = c c ) k, c [0, c ]): p c D) = c D, λ c D = α ηq. 7) Given these cutoffs and the fact that λp i) = α γq c i) η qc i) di = wλ γ qi) L, where qi) = L γ λ p Ci)), we can rewrite the expressions for a domestic seller with a cost draw c as: p c) = 2 w + c) ; p c ) = 2 c D + c ), 8) q c) = L 2γ wλ c) ; q c ) = L 2γ λ c D c ), r c) = L 4γ w2 λ ) 2 c 2) ; r c ) = L 4γ λ c D) 2 c ) 2), π c) = L 4γ w2 λ c) 2 ; π c ) = L 4γ λ c D c ) 2. Exporting. The Foreign demand for the Home firm s variety is given by while for the Foreign firms selling at Home we have Thus, using the same logic as before, we get λ p X i) = α γq c X i) ηq, 9) λp X i) = α γq c, X i) ηq. 0) p X c) = 2 wτ c X + c) ; p X c ) = 2 τ c X + c ), ) q X c) = L 2γ wλ τ c X c) ; q X c ) = L 2γ λτ c X c ), r X c) = L 4γ w2 τ 2 λ c X ) 2 c 2) ; r X c ) = L 4γ λ τ ) 2 c X) 2 c ) 2), π X c) = L 4γ w2 τ 2 λ c X c) 2 ; π X c ) = L 4γ λ τ ) 2 c X c ) 2, where τ and τ are the iceberg transportation costs faced by exporters from the Home and Foreign countries, respectively, while c X and c X are the cost cutoffs for the exporters determined from q X c X ) = 0 and qx c X ) = 0, respectively, or c X = c D τw and c X = w τ. 2) 7

8 2.3 Equilibrium Conditions The free entry condition implies that the expected profits from entering the market should be equal to the entry cost. For Home firms this means that 0 π c) dg c) + c X 0 π X c) dg c) = wf e. Given the assumption of the Pareto cost distribution and 2), we can rewrite this condition as [ F E) : c ) k w Lλ ) k+2 + λ L w k 2 τ k c D) k+2] = Const, 3) where Const = 2γf e k + ) k + 2). 4) Similarly, the FE) condition for Foreign firms is F E) : c ) k [ L λ c D) k+2 + λlw k+2 τ ) k ) k+2] = Const. 5) Next, let us look at the mass of active firms in the Home economy. There is only one sector in the economy, thus, all labor is employed there. Due to free entry, total profits in the economy are zero, i.e., total revenues are equal to the labor payment: R = wl, where R = r D + G c ) X) G ) r X, and the expected revenues from domestic and export sales conditional on getting a cost draw below the corresponding cutoff, are r D = r X = so that given the FE) condition, Similarly, 0 c X 0 r c) r X c) r D + G c ) k X) G ) r c X = wf e k + ), and = = Then the masses of entrants in each economy are dg c) G ) = 2γ Lw2 λ ) 2 k + 2, 6) dg c) G c X ) = 2γ L w 2 λ τ 2 c X) 2 k + 2, 7) ) L k cd. f e k + ) c L c ) k D f e k + ) c. 8) L e = G ) = f e k + ), 9) e = G c D) = 8 L f e k + ) 20)

9 Now let us derive the trade balance condition. It is given by X r X = X r X, where X = G c X ) /G )) and X = G c X ) /G c D )), so that by using 2), 9), and 20), we get T B) : λ τ ) k c D) k+2 c ) k = w 2k+2 λτ k ) k+2 c ) k. 2) Finally, we need to derive the equations for Lagrangian multipliers λ and λ. Appendix A, and λ) : λ) : As shown in wλ = α k + 2 k + η ) 2, 22) λ = α c k + 2 D k + η ) c 2. 23) D Using T B) together with λ) and λ ), we get F E) : α k + 2 ) [ k + η ) k Lc k + L c ) k w k τ ) k] = Const, F E) : αc D k + 2 ) [ k + η c D) k L c ) k + Lc k w k τ k] = Const. oreover, the TB) condition itself can be re-written as so that from the new FE) and FE) conditions we have T B) : Lc k λ c D )k+2) τ c )k = τc ) k w 2k+ wλ ) k+2), ) k + L c ) k w k τ ) k = w 2k+ τc [ τ c L c ) k + Lc k w k τ k]. The TB) condition above is quite important: it implies that the wage in our model can be derived from the TB) condition alone, and then all the other variables in the equilibrium can be calculated as functions of wage. In other words, all values in the equilibrium depend only on the wage, and the equation for it is given by the TB) condition. To summarize, 3 unknown variables in the equilibrium w,, and c D ) can be found from 3 conditions, T B), F E), and F E). 2.4 Welfare As shown in Appendix A, welfare per capita at Home is: U = [ 2k + 3 α η k k + 2k + 3 ) 2 ]. Similarly, welfare in the Foreign country is: [ U = 2k + 3) α 2 k + ) c D ] η k + 2 ) 2k + 3 c 2. 24) D Hence, in each country welfare rises with a lower cutoff for its domestic sellers. 9

10 2.5 Unilateral Trade Liberalization Now we are ready to study unilateral trade liberalization by the Home country, i.e., the effect of falling τ on welfare at Home. Note that τ can be interpreted as a wasteful tariff, since it does not generate tariff revenues. Let us first look at the TB) condition. Using the FE) and FE) we can re-write it as w 2k+ α k + 2 ) ) k k + η ) k τc Const c τ c = )k L ) c k + Lc k w k τ. k As we will show in Section 3 see 37)), in the case of a small economy the RHS of the equation above becomes a constant, which will imply the negative relationship between w and. We can prove that this is also true in the case of 2 large economies. To see this, re-write the condition above as w k+ [ L c ) k w k + Lc k τ k] α k + 2 k + η ) ) k ) k τc τ c = Const c ) k. Hence, in the case of 2 large economies, from the TB) condition, w and are negatively related. Next, by using λ) and λ ), we can re-write the FE) condition given by 3) as c ) [L k α k + 2 ) k + η ) k + w k τ k L αc D k + 2 ) k + η c D) ] k = Const. ) In Section 3 we will show see 36)) that in the case of a small economy term L αc D k+2 k+ η becomes a constant, which will imply the positive relationship between w and. And again, we can prove the same for the case of 2 large economies, if we use the FE) condition and re-write the equation above as c ) [L k α k + 2 ) ] k + η ) k + w k τ k L Const c )k L ) c k = Const, + Lc k w k τ k or c ) [L k α k + 2 ) ] k + η ) k + L Const c )k τ k w k+ L ) c k = Const. + wlc k Hence, we proved the following result: Lemma The FE) condition implies a positive relationship between and w, while the TB) condition implies a negative relationship between and w. We can depict both relationships in Figure, where the TB and FE curves represent the FE) and TB) conditions, respectively. The intersection of two curves gives the unique equilibrium values of w and. oreover, it is straightforward to show that a reduction of inward variable trade barriers at Home, τ, affects only the TB curve by shifting it down, which immediately proves that both w and fall as τ falls. Finally, recall that welfare at Home falls with. Thus, we proved that: 0 c D )k

11 Figure : Unilateral Trade Liberalization by Home wage FE curve TB curve 0 Cutoff for domestic sellers Proposition In the case of two large economies, unilateral trade liberalization by Home a fall in a wasteful import tariff τ ) increases welfare there. This shows an important dependence of the effect of UTL on the outside good assumption used by elitz and Ottaviano 2008): only in the presence of the outside good, which results in the additional distortion in the model, UTL by Home is welfare-reducing. 7 3 The Case of a Small Economy In this Section we will consider the case of two countries, Home and Foreign, with Home being a small economy relative to the Foreign one. As in Demidova and Rodríguez-Clare 2009), this involves 3 assumptions: i) the Foreign demand for Home varieties depends only on the price of a variety, i.e., aggregate variables in the Foreign demand function are not affected by Home; ii) the mass of available Foreign varieties is fixed; and iii) the cost distribution of Foreign producers is fixed the last two assumptions mean that the mass of Foreign firms and Foreign wage are not affected by changes at Home). See Appendix B for justification of these three assumptions in our model. Derivations below are similar to those in the previous section. 7 Note that our Figure for the case of variable markups resembles the one in Demidova and Rodríguez-Clare 203) for CES preferences. This is not a coincidence. We relate w and the cost cutoff for domestic sellers,, while their graphical analysis relates w and the productivity cutoff for exporters, ϕ X. In elitz 2003) style models the productivity cutoff for exporters is negatively related to the productivity cutoff for domestic sellers, which, in turn, is an inverse of the cost cutoff. Hence, and ϕ X are positively correlated, which results in the similar relationships between w and in our Figure and and w and ϕ X in Figure in Demidova and Rodríguez-Clare 203).

12 3. Demand At Home a household maximizes U = α q c i) di 2 γ q c i)) 2 di 2 η q c i) di) 2 s.t. p i) q c i) di = w, where is the set of all varieties available in the Home market. Then one can get λp i) = α γq c i) η q c i) di, 25) where λ is Lagrangian multiplier. As before, we can show that given Q = qc i) di, wλ = αq γ q c i)) 2 di ηq 2. 26) 3.2 Production and Firm Behavior Domestic arket. There is a continuum of domestic firms at Home with the Pareto cost distribution G c) = c c ) k, c [0, c ], k >. As before, we define the cutoff such that q c ) = 0 or p ) = C ), so that Also, wλ = α ηq. 27) p c) = 2 w + c) ; q c) = L 2γ wλ c) ; r c) = L 4γ w2 λ ) 2 c 2) ; π c) = L 4γ w2 λ c) 2. Home Exporting. Our first small economy assumption is that the Foreign demand for the Home firm s variety is given by q X i) = A Bp X i), 28) where A and B are fixed constants. From the profit maximization problem, p X = A 2B + C X 2 and q X = 2 A B C X)), 29) where C X c) = wτc. Then, we can define the exporting cutoff from q X c X ) = 0 so that By using the same logic as before, we get c X = A B p X c) = 2 wτ c X + c) ; q X c) = Bwτ c X c) ; wτ. 30) r X c) = 4 Bw2 τ 2 c X ) 2 c 2) ; π X c) = 4 w2 τ 2 B c X c) 2. ) k Foreign Exporting. Active Foreign firms have the following cost distribution: G c) = c c, k >, c [0, c ].The next small economy assumption states that this distribution as well as the 2

13 mass of active Foreign firms is not affected by changes at Home, i.e., entry abroad is not affected. We normalize to. Note that not all active Foreign firms sell their goods in the Home market: only firms with c > c X become exporters, where as before, c X = w τ. 3) Then as before, we can derive the following expressions: p X c) = 2 τ c X + c), q X c) = L 2γ λτ c X c), r X c) = L 4γ λ τ ) 2 c X) 2 c) 2), π X c) = L 4γ λ τ ) 2 c X c) Equilibrium Conditions The new free entry condition for Home firms implies that ) ] F E) : c ) [Lwλ k ) k+2 A k+2 + γb τ k w k = Const, 32) B where as before, Const = 2γf e k + ) k + 2). Also, it is straightforward to show that as before, ) L k cd L = and e = f e k + ) f e k + ). 33) c Now let us derive the trade balance condition. It is given by X r X = r X, so that using c X = w/τ, c X = A/ Bwτ), and =, we get ) k = Const 2, 34) T B) : w 2k+2 λ ) k+2 τc τ c where ) A k+2 Const 2 = γb B f e k + ). 35) Finally, using the same logic as before, we again get λ) : wλ = α k + 2 k + η ) 2. Using this condition in FE) and TB), we get 2 equations for 2 unknown variables in the equilibrium, w and : F E) : c ) k [L T B) : w 2k+ α k + 2 k + η α k + 2 ) k + η ) k + γb ) ) ] A k+2 τ k w k = Const, 36) B ) k ) k τc τ c = Const 2, 37) where Const = 2γf e k + ) k + 2) and Const 2 = γb ) A k+2 B f. ek+) The formula for welfare per capita at Home is the same as before: U = 2k k + [ α η k + 2 2k + 3 ) 2 ]. 3

14 3.4 Unilateral Trade Liberalization Now we are ready to study unilateral trade liberalization by the Home country, i.e., the effect of falling τ on welfare at Home. As in the case of 2 large economies considered in Section 2, the FE) and TB) conditions give us a system of 2 equations in 2 unknowns. It can be shown straightforwardly that: Lemma 2 The FE) condition implies a positive relationship between and w, while the TB) condition implies a negative relationship between and w. Hence, we again can use Figure. The main difference between two figures the one for 2 large economies and the one for the small economy) is that the shapes of the curves can be different. Yet, the qualitative relationships between and w in these figures remain the same. The intersection of two curves gives the unique equilibrium values of w and. And again, as in the case of 2 large economies, a reduction of inward variable trade barriers at Home, τ, leads to a downward shift in the TB curve. As a result, both w and fall, leading to a fall in Home s welfare. We proved that Proposition 2 Under the assumption of Home being a small economy, unilateral trade liberalization by Home a fall in a wasteful import tariff τ ) increases welfare there. In other words, as in the case of 2 large economies, protection is not an optimal policy for the small economy. 3.5 Non-Wasteful Import Tariff In the last part of our analysis we will deal with a non-wasteful import tariff. In particular, assume that if a Foreign firm charges price p X, then consumers at Home pay tp X, so that the Home government collects tariff revenues of t ) p X per unit sold. The question we want to ask is whether charging t > i.e., a strictly positive tariff) is the optimal thing to do for the Home government. In other words, we want to see how unilateral trade liberalization affects welfare at Home, when an import tariff is modeled as a non-wasteful one. Before answering this question, let us discuss the formula for welfare at Home. As shown in Appendix A, per capita welfare can be written as a function of Q Q = qc i) di): U = [ 2 Q 2α ] α ηq) ηq = k + [ ] α 2k + 3) k k + 2) Q ηq, k + which means that the behavior of U with respect to Q depends on whether Q α 2k + 3) η 2k + 2). The restriction from the model is that α ηq > 0 so that prices in the equilibrium are nonnegative). Since in any equilibrium it has to be the case that Q < α/η < α2k+3) η2k+2), then, as Q rises, per capita welfare at Home rises as well. 4

15 We leave derivations of the equilibrium conditions as well as all the proofs to Appendix C. Here we provide the main results only. First, in the case of a non-wasteful tariff, there are 3 unknown variables, w,, and λ, 8 and 3 corresponding equilibrium conditions: 9 F E) : c ) k [Lwλ ) k+2 + γb w 2k+2 ) k T B) : t k+2 λ ) k+2 τc τ c = Const 2, α η k+2 k+ λ) : wλ = ) k. + η k+2 t k+ τ c 2γk+2)t k+2 wcd ) ] A k+2 τ k w k = Const, B Next, we will derive the optimal value of the non-wasteful tariff. All formal proofs are provided in Appendix C.) Proposition 3 There exists a unique non-wasteful tariff that maximizes welfare at Home. value can be found as a solution of the equation below k + t opt = + k + 2 where ) k + 2k+ k+ + r X rd r X = G c X) r X = c ) k f e k + ) w t opt )) k r D G ) r D c τ ) k L t opt ) k+2 Its k 2 + 2k+ k+ + r X rd, 38) is the ratio of the expected revenues from export and domestic sales, and w t) is the unique solution of the following equation: c τ ) ) k L t k+2 c τ f e k + ) w 2k+ + τ k w k+ = 2ck f e k + ) k + 2) B A/B) k+2. This result is in stark contrast to Proposition 2: full trade liberalization is no longer an optimal policy for the Home economy. Figure 2 provides the graphical representation of the equation 38), where the left-hand and right-hand sides of the equations are functions of t. Interestingly, the optimal value of t is less than 2, i.e., the optimal non-wasteful import tariff does not exceed 00%. Also, note that r X r D + = X r X D r D + = R X R D + = R X R D + = R R D λ jj, where we borrow the notation from ACDR: λ jj denotes the share of expenditures on domestic varieties at Home. Then the formula for the optimal non-wasteful tariff becomes ) k + k + t opt k+ = + + λ jj k + 2 k 2 + k+ +. λ jj 8 In the case of the small economy we no longer need to calculate λ and c D. 9 Note that if t =, these conditions become the ones we derived earlier for the case of a wasteful import tariff. 5

16 Figure 2: The LHS and RHS of the formula for the optimal non-wasteful import tariff. 2 LHS RHS 0 2 t opt tariff rate t The intuition is that the higher is the share of expenditures on domestic varieties, the smaller is the need for protection from foreign competition, and hence, the smaller is the optimal non-wasteful tariff. 4 Conclusion... References [] Arkolakis, C., A. Costinot, D. Donaldson and A. Rodríguez-Clare 202). The Elusive Pro- Competitive Effects of Trade, mimeo. [2] Arkolakis, C., A. Costinot and A. Rodríguez-Clare 202), New Trade odels, Same Old Gains?, American Economic Review 02): [3] Bhagwati, J. 97). The Generalized Theory of Distortions and Welfare, in J. Bhagwati, R. Jones, R. undell, and J. Vanek eds), Trade, Balance of Payments and Growth. Amsterdam: North-Holland Publishing Company. [4] De Loecker, J., P.K. Goldberg, A.K. Khandelwal and N. Pavcnik 202). Prices, arkups and Trade Reform, Princeton University working paper. [5] Demidova, S. 2008). Productivity Improvements and Falling Trade Costs: Boon or Bane?, International Economic Review 494):

17 [6] Demidova, S., and A. Rodrígues-Clare 2009). Trade Policy under Firm-Level Heterogeneity in a Small Economy, Journal of International Economics 78):00-2. [7] Demidova, S., and A. Rodrígues-Clare 203). The Simple Analytics of the elitz odel in a Small Economy, Journal of International Economics 902): [8] Edmond, C., V. idrigan and D. Xu 202). Competition, arkups, the Gains from International Trade, NBER Working Paper 804. [9] Felbermayr, G., and B. Jung 202). Unilateral Trade Liberalization in the elitz odel: A Note, Economics Bulletin 322): [0] Felbermayr, G., B. Jung and. Larch 203). Optimal Tariffs, Retaliation and the Welfare Loss from Tariff Wars in the elitz odel, Journal of International Economics 89 ):3 25. [] Gros, D. 987). A Note on the Optimal Tariff, Retaliation and the Welfare Loss from Tariff Wars in a Framework with Intra-Industry Trade, Journal of International Economics 23 3): [2] Krugman, P.R. 980). Scale Economies, Product Differentiation, and the Pattern of Trade, American Economic Review 70: [3] elitz,.j. 2003). The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity, Econometrica 7: [4] elitz,.j., and G.I.P. Ottaviano 2008). arket Size, Trade, and Productivity, Review of Economic Studies 75: [5] elitz,.j., and S.J. Redding 204). Heterogeneous Firms and Trade, Handbook of International Economics, Volume 4, Elsevier: North Holland, forthcoming. [6] Nocco, A., G.I.P. Ottaviano and. Salto 203), onopolistic Competition and Optimum Product Selection: Why and how heterogeneity matters, CEPR Discussion Paper No [7] Ossa, R. 20). A New Trade Theory of GATT/WTO Negotiations, Journal of Political Economy 9): [8] Ottaviano, G.I.P., T. Tabuchi and J.-F. Thisse 2002). Agglomeration and trade revisited, International Economic Review 43:

18 Appendix Appendix A: Equilibrium Conditions in Case of 2 Large Economies Condition for Lagrangian ultipliers From 6), λpq c = α ηq γq c ) q c = λw γq c ) q c. By integrating both parts over all varieties sold at Home, we get so that λ p i) q c i) di = wλ = λw Q γ q c i)) 2 di, where from 6) and 7) we get note that q c c) = qc) and so that Q = = 2γ = 2γ 0 q c c) q c i)) 2 di = λ = γ qc i)) 2 di w Q ), 39) dg c) G ) + X c X 0 q c, X L, qc, X c) dg c) G ) c X c) = q X c) L ) [ k + λ e w c ) k ) k+ + e c ) k τ c X ) k+] k + λ ) k+ [ e w c ) k + e c ) k τ ) k w k+], Thus, 39) can be re-written as Then from 2), wλ = αq γ which we can solve for wλ : 0 q c)) 2 dg c) G ) + X = ) k+2 wλ 2 2γ 2 k + ) k + 2) = γ wλc D k + 2 Q. c X 0 q c, X c)) 2 dg c) G ) c X [ e w c ) k + e c ) k τ ) k w k+] λ = γ γ wλ k+2 Q w Q ), Q = k ) k + q c i)) 2 di ηq 2 = α k + 2 wλ k + λ) : k + 2 k + η k + ) 2, wλ = α k + 2 k + η ) 2. 4) 8

19 Similarly, Q = k + 2 k + c D and λ ) : λ = α c k + 2 D k + η ) c 2. 42) D Welfare From 40), per capita welfare at Home is given by U = αq 2 γ q c i)) 2 di 2 ηq2 = αq 2 wλc D k + 2 Q 2 ηq2 43) = [ 2 Q 2α ] α ηq) ηq = [ 2k + 3 α η k + 2 ] k k + 2k + 3 ) 2. Welfare in the Foreign country can be derived in a similar manner. Appendix B: Justification of Small Economy Assumption Let us look at the system of the equilibrium equations for the model of two countries of size L and L considered in Section 2: or F E) : w [ Lλ ) k+2 + λ L w k 2 τ k c D) k+2] = Const, F E) : L λ c D) k+2 + λlw k+2 τ ) k ) k+2 = Const, λ) : wλ = α k + 2 k + η ) 2, λ) : λ = α c k + 2 D k + η ) c 2, D T B) : λ τ c ) k c D) k+2 = w 2k+2 λ τc ) k ) k+2. When L/L goes to 0, i.e., when Home becomes small relative to the Foreign country, we have F E) : λ c D) k+2 + λ L L wk+2 τ ) k ) k+2 = Const L, λ c D) k+2 Const L. In other words, λ c D )k+2 is no longer affected by changes in Home s variables. Then from λ ), c D and t are not affected as well, i.e., as we assumed in Section 3, the mass of Foreign active firms as well as their cost distribution is not affected by changes at Home. We are left with 3 unknowns now, w,, and λ, and 3 equations, F E) : wlλ ) k+2 + λ L c D) k+2) w k τ k = Const, λ) : wλ = α k + 2 k + η T B) : w 2k+2 τc λ τ c ) 2, ) k ) k+2 = Const L. 9

20 Note that the demand for Home variety abroad is given by q X i) = L x X i) = L γ [α ηx λ p X i)] = L γ [λ c D λ p X i)], where L γ λ c L D and γ λ are now constants. Let us denote them by A and B, respectively, so that q X i) = A Bp X i), as we assumed in Section 3. Then we can rewrite FE) as or F E) : wlλ ) k+2 + λ L w k τ k c D) k+2 = Const, F E) : Lwλ ) k+2 + γb ) A k+2 τ k w k = Const, B which is exactly what we have as the FE) condition under the small economy assumption in Section 3. Thus, we proved that our small economy assumptions follow directly from the case of two economies, when one of them becomes small relative to the other one. Appendix C: Non-Wasteful Import tariff Equilibrium Conditions If the import tariff is non-wasteful, then the national income becomes I = wl + T, where T is tariff revenues. Hence, income per capita becomes I/L = w + T/L) instead of w. This means that all our derivations for Home consumers remain the same except for the new formula instead of 26): I/L) λ = αq γ q c i)) 2 di ηq 2. 44) Also, all derivations for Home producers do not change, which gives us the same FE) condition as before: F E) : c ) k [Lwλ ) k+2 + γb ) ] A k+2 τ k w k = Const. B The main change for the Foreign exporters is that while their goods are sold at tp X in the Home market, i.e., the demand for their goods is defined by λ tp X ) = α γq X ηq, the exporters collect only p X so that their profits are π X = p X C X ) q X. This gives the following cost cutoff for the marginal Foreign exporter: c X = w tτ, so that from F.O.C., the price set by the Foreign export with the cost draw c is p X c) = 2t w + tτ c) = 2 τ w tτ + c ) = 2 τ c X + c). 20

21 Thus, we get the same equations as before: p X c) = 2 τ c X + c), q X c) = L 2γ λτ c X c), r X c) = L 4γ λ τ ) 2 c X) 2 c) 2), π X c) = L 4γ λ τ ) 2 c X c) 2. It is worth emphasizing here that rx c) is the revenue received by the Foreign exporter, while its total sales in the Home market are trx c). Next, let us look at the total expenditures at Home that are spent on domestic and imported varieties: wl + T = t x r x) + r d, or wl + t ) x r x) = t x r x) + r d. The TB) condition implies that x r x = x r x, so that the above equation can be re-written as wl = x r x + r d = x r x + r d = R, 45) i.e., total revenues earned by Home firms are equal to labor payments at Home. This implies that our derivations for the masses of firms at Home do not change, and as before, = ) L k cd and e = f e k + ) c L f e k + ). 46) Now let us look at the TB) condition. Given the new formula for c X, it can be written as T B) : w 2k+2 ) k t k+2 λ ) k+2 τc τ c = Const 2. Finally, we need to re-derive the λ) condition. Now we have see Appendix A for details): where, as before, λ = γ qc i)) 2 di w Q I/L, q c i)) 2 di = γ wλc D k + 2 Q. By using the expression above in the equation for λ, we get Q = I k + 2, wl k + where I/wL is no longer equal to. This means that wλ = α ηq = α η I k ) wl k + 2

22 Lemma 3 From the definition of I, we get so that where Proof. Recall that I wl I wl = + αy + η k+2 k+ y, λ) : wλ = α η k+2 k+ + η k+2 k+ y, y = t wcd ) k. 2γ k + 2) t k+2 τ c t ) = + wl x r x t ) = + wl [ L 2γ λ τ ) 2 k + 2 ] c X )k+2 ) c k ) t ) k wcd = + 2γ k + 2) t k+2 c τ wλ ) = + y α ηq) = + yα yη I k + 2 wl k +. By solving the equation above, we get the expression for I/wL, and, in turn, for wλ. QED Note that if t =, y = 0, so we get the same equations as in the case of a wasteful tariff in Section 3. Optimal Non-Wasteful Import Tariff As discussed in the main body of the paper, welfare per capita at Home increases with Q. Note that Q = α wλ ) /η. Hence, to see what happens with Q and welfare per capita, we can study the behavior of wλ. Denote it by z, i.e., z wλ. Let us re-write all the equilibrium conditions as functions of w,, and z. In other words, instead of keeping track of λ, we will look at z. The notation below seems to be a little bit messy, but it will help us to make derivations easier.) First, we get where C 4 = c τ c τ conditions to derive T B) : w2k+ t k+2 ) k+ z = C 4, 48) ) k γb A/B) k+2 / f e k + )). We can use this equation in the FE) and λ) F E) : C 4 t k+2 w 2k+ + C 5 w k+ = C 3, 49) z) : z + t ) ψ w k+ + η k + 2 = α, k + 50) 22

23 where C 5 = γb A/B) k+2 τ k /L, C 3 = 2γc k f e k + ) k + 2) /L, and ψ = C 4 η c τ ) k / 2γ k + 2)). Note that the FE) condition becomes very convenient, since it includes only one equilibrium unknown w. We can use this condition to look at the behavior of w with respect to t. From the implicit function theorem, we get: w w = t Then it is easy to see that w > 0 and k + 2 2k + Next, from the TB) condition, we get c D = k + 2 k + t + k+ 2k+ w w < k + 2 t 2k +. z z ). C5 w k C 4 t k+2 ) w 2k + ). w Finally, from applying the implicit function theorem to the z) condition, we get z c D t ) ψ w k+ + η k + 2 ) w k + ) t ) ψ k + w k+ w + ψ = 0. wk+ Using the expression for c D /, we can re-write it as z = ψ w w k+ w t ) k2 + k+ z k+ + k+2 k+ t ) ψ ) t t + η t t 2k+ k+2 ) w w ). + η k+2 w k+ k+ What can we say about the sign of z? Let us look at the denominator first. From z): Denominator = + α z) = + ηq > 0. k + z k + z Hence, the sigh of z depends on the sign of its nominator. Note that given the properties of w /w, the second term in the nominator is always positive: η t 2k + w ) > η t 2k + ) k + 2 = 0. t k + 2 w t k + 2 t 2k + However, the first term in the nominator can be negative. In fact, ψ w w k+ w t ) k 2 k + + k + 2 t ) = ψ k + t w k+ < 0. So what can we say about the sign of z? Lemma 4 In the case of a non-wasteful import tariff, dz dt < 0, = dq t= dt > 0. t= t= 23

24 Proof. Since we have already proved that the denominator of z is always positive, we need to look at its nominator. First, let us introduce the following notation: r D = G ) r D = 0 r c) dg c), r X = G c X ) r X = c X 0 r X c) dg c), 5) i.e., r D and r X are the expected revenues of the Home firms in local and Foreign markets. so that Next, the FE) condition can be re-written as Then from the FE) condition given by 49), w w = t F E) : r D + r X = wf e k + ). k + 2 2k + r X = C 5w k r D C 4 t k+2, + k+ 2k+ and the second term in the nominator of z is η t 2k + t k + 2 we get ) C5 w k C 4 t k+2 = w ) = η w t t r D 2k+ k+ r, D + r X r X 2k+ k+ r D + r X oreover, by using I/wL = + [t ) RX /wl] in 47) and then applying the result to 50), ψ w k+ = η k + 2 RX k + wl = η k + 2 R X k + R = η k + 2 R X k + R D + R X = η k + 2 e G c X ) r x = η k + 2 k + e G ) r D + e G c X ) r x k + r X r D + r X. Then the nominator of z becomes ψ w w k+ w t ) k 2 k + + k + 2 ) t + η t 2k + w ) k + t t k + 2 w ) = η k + 2 r X r D t ) k 2 k + r D + r 2k+ X k+ r D + r X t k + + k + 2 t + η r X k + t tc 2k+ D k+ r D + r X [ ) = η k + 2 r X r D k 2 t k + r D + r 2k+ X k+ r t ) D + r X k + + k + 2 t ) t) + k + k + k + 2 [ ) )] = η k + 2 r X t ) r D r D t k + r D + r X k + k + 2. This means that sign z ) = sign 2k+ k+ r D + r X k 2 + t ) k + k k + 2k+ k+ + r X rd k 2 + 2k+ k+ + r X rd 2k+ k+ r D + r X k + ). r D + r X 2k+ k+ r D + r X ]

25 The right-hand side of the expression above evaluated at t = i.e., no tariff is applied) is negative. But what about the comparison of for t >? t + k + k + 2 k + 2k+ k+ + r X rd k 2 + 2k+ k+ + r X rd 52) Lemma 5 The LHS of 52) rises with t, while the RHS of 52) falls with t. Proof. To see that the RHS of 52) falls, recall that r X / r D = C 5 w k /C 4 t k+2, where ) ] [ ] d w k dt t k+2 = [ wk t k+3 k + 2) + kt w = tk+ r D w w k k + 2) k 2k+ k+ r < 0, D + r X so that which proves the Lemma. QED. d k + 2k+ k+ + r ) X rd dt k 2 + 2k+ k+ + r < 0, X rd Figure 2 depicts both sides of 52) as functions of t. From Lemma 5, there exists a unique intersection at t = t opt, which means that z < 0 for all t < t opt and z > 0 for all t > t opt. Hence, the minimal value of z, and in turn, the highest level of Q and welfare at Home, is reached at t = t opt. To summarize, the optimal value of the non-wasteful tariff can be found as a solution of the equation below where t opt = k + k + 2 k + 2k+ k+ + r X rd k 2 + 2k+ k+ + r, X rd r X = c ) k f e k + ) w opt ) k r D c τ ) k L t opt ) k+2, and w opt = w t opt), where w t) is the function of t that is implicitly given by QED. ) [ A k+2 c B τ ) ] k L t k+2 τ k + B c τ f e k + ) w2k+ w k+ = 2c k f e k + ) k + 2) 25