# The Simple Analytics of the Melitz Model in a Small Economy

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4 2.1 Model Consider two countries indexed by i = 1, 2 and populated by L i identical households, each of which has a unit of labor supplied inelastically and earns wage w i. There is a continuum of goods indexed by ω Ω. The representative consumer has Dixit-Stiglit preferences in each country with elasticity of substitution σ > 1. Each country has an endogenous) measure Mi e of monopolistically competitive firms that pay a fixed cost w i F i to enter the market and draw their random productivity from the cumulative distribution function G i ). Given, a firm from country i faces a cost w i / of producing one unit and decides whether to sell in the domestic market and/or export. Firms from i have to use f ij units of labor in country i to export any quantity to country j this entails a fixed cost w i f ij. Iceberg trade costs are τ ij > 1 so that for a firm in i with productivity the cost of producing and selling one unit in j is w i τ ij /. We assume that τ ii = 1 for i = 1, Characteriation of the Equilibrium Since profits are monotonically increasing in productivity,, there is a productivity cut-off ij such that, among country i firms, only those with a productivity of at least ij decide to sell in market j. Letting ρ 1 1/σ, these cut-offs are defined implicitly by where P j is the price index in country j given by w j L j Pj wi τ ij /ρij ) 1 σ = σwi f ij, 1) P 1 σ j = 2 i=1 M e i ij ) wi τ 1 σ ij dg i ). 5 2) ρ The free entry condition for firms in country i equalies the expected profits of entering the market to the entry costs. Following Melit 2003), we let J i a) [ ] a a) 1 dg i ). Note that from the definition of cut-offs ij ) the expected profits for country i firms in country j are w i f ij J i ij ). Then the free entry condition in country i is 2 ) f ij J i ij = Fi. 3) j=1 Next, let us look at the labor market clearing condition that equalies total labor demand given by M e i F i + 2 j=1 L ij to labor supply in country i, L i, where L ij is variable and fixed) labor what happens with the wage, which is the only unknown variable in this case. The proof can be found in the online Appendix available at the authors web-sites.) 5 In establishing these conditions for the cut-offs, we have used four standard results. First, firms set prices equal to unit cost multiplied by the mark-up 1/ρ. Second, firms variable profits are revenues divided by σ. Third, revenues in market j given a price p are R jp j p 1 σ, where R j are total expenditures in j. And fourth, R j = w jl j, since due to free entry the only source of national income is labor payments. 4

5 employed by firms in i to sell to market j. Using 1), 3), and the definition of J i ij ), the labor market clearing condition can be written as M e i σ 2 [ ) )] f ij Ji ij + 1 Gi ij = Li. 4) j=1 Total sales by firms from i in j are X ij Mi eσw if ij /ij) ij dgi ), hence X ij = M e i σw i f ij [ Ji ij) + 1 G i ij) ]. Trade balance X ij = X ji ) can then be written as M e i w i f ij [ Ji ij) + 1 G i ij) ] = M e j w j f ji [ Jj ji) + 1 G j ji) ]. 5) To summarie, there are 10 unknown equilibrium variables: M e i, ii, ij, P i, and w i for i, j = 1, 2. We have 9 equilibrium conditions: two free entry conditions, four cut-off conditions, two price index equations, and trade balance. Setting labor in one of the countries as numeraire, we can then use the equilibrium conditions to solve for all the unknown variables. 6 For future reference, we note here that, as in Melit 2003), the effect of trade on welfare is completely determined by the behavior of the productivity cut-off for domestic sellers. Free entry implies that there are no profits, so the real wage, w i /P i, measures welfare per capita in our simple economy. But 1) directly implies that ) 1 w i Li = ρ P i σf ii. ii Hence, to know what happens to welfare as a result of trade liberaliation, we just need to see what happens to the domestic productivity cut-off, ii. 2.3 Graphical Analysis First, let us normalie wage in country 2 to unity, w 2 1. We now show how to use the equilibrium conditions to re-write 2 equations, the ero profit condition for exporters from country 1 and the trade balance condition as the functions of only 2 unknowns, w 1 and 12. But from 1) we get and ) 1 12 = h 12 w 1, 22) f12 τ 12 w1 ) 1 ρ f 22, 6) 22 ) 1 21 = h 21 w 1, 11) f21 τ 21 w1 ) 1 ρ f 11. 7) 11 Furthermore, 3) implies that 22 can be expressed as a function of 21, and 11 can be expressed as a function of 12. With a slight abuse of notation, we write these two functions as ) and 6 As is standard in the literature, we assume that iceberg trade and fixed marketing costs are such that ii < ij for all i, j = 1, 2. 5

7 Proposition 1 Welfare increases for a country that unilaterally reduces importing trade barriers. Proof. See the Appendix. This result stands in sharp contrast to that in Demidova 2008) for the setting with CES preferences and Melit and Ottaviano 2008) for the setting with linear demand, where lowering import trade barriers reduces welfare. The difference arises from the presence of an outside good in these papers and the absence of such a good in the current paper. The presence of an outside good allows for a Home Market Effect HME) on specialiation patterns. Specifically, liberaliation in Home makes Foreign a better export base, which results in the additional entry of firms there and a decline in entry of firms in Home. As a result, Foreign specialies in the differentiated good sector while Home specialies in the homogenous good sector. As shown in Venables 1987), this results in a welfare loss in Home. Proposition 1 shows that this result no longer holds when there is no outside good pinning down wages in both countries. Without this good, HME is no longer operative on specialiation patterns. Instead, trade liberaliation in Home leads to a decline in its relative wage, but this is smaller than the decline in the price index, hence, welfare rises. Nevertheless, our approach proves to be a useful tool even in models in which an outside good pins down the wage as long as there is no complete specialiation. 7 In this case the trade balance curve becomes a horiontal line, with the wage determined by productivity levels in the outside sector in the two countries. A reduction in importing barriers by Home lower τ 21 and/or f 21 ) shifts down the competitiveness curve while not affecting the horiontal TB curve, resulting in a higher 12, and hence in lower welfare at Home. Similarly, lower exporting barriers lower τ 12 and/or f 12 ) shift the competitiveness curve up and increase welfare. In the next Section we will show how the assumption that the Home country is a small economy used in Demidova and Rodrígue-Clare 2009) helps to significantly simplify the analysis. 3 Case of a Small Economy Here we assume Home i.e., country 1) can be treated as a small economy. Compared to Section 2, the small economy assumption requires two changes. First, we assume that foreign demand for a domestic variety is given by Ap σ. The term A includes both the national income and the price index in Foreign i.e., country 2). In line with the small economy assumption, A is not affected by changes at Home, i.e., A is exogenous in our small-country setting. Second, the measure M2 e of monopolistically competitive firms in Foreign is exogenous. However, since f 21 > 0, not all foreign firms sell at Home, so the measure of foreign varieties available at Home is endogenous. In the Appendix we show that our small economy model can be obtained from the model of two large countries as a limit case, where the share of labor in Home, n L 1 / L 1 + L 2 ), goes to ero. Formally, we show that if two large countries are symmetric in everything except for sie, and if the 7 It can be easily shown that if Home becomes small enough relative to the Foreign ecomomy, it completely specialies in the production of the outside good. In that case, our approach, which is based on the existence of the differentiated good sector at Home, is no longer valid. 7

8 productivity distribution in both countries is Pareto, then in the limit as n 0) we obtain the three key assumptions of the small economy model, namely: 1) the domestic productivity cut-off for firms in Foreign is not affected by changes in Home; 2) the mass of firms in Foreign is not affected by changes in Home; and 3) the demand in Foreign for Home goods exported at the price p can be expressed as Ap σ, where A is a constant not affected by changes in Home. 3.1 Characteriation of the Equilibrium As before, productivity cut-offs 11 and 21 are determined by 1), but 22 is now taken as exogenous, while 12 is determined by A w 1 τ 12/ρ12) 1 σ = σw 1 f 12. 8) In turn, the free entry, labor market clearing, and trade balance conditions at Home remain the same. To summarie, in the case of a small economy, there are 5 unknown variables in the equilibrium, M e 1, 11, 12, 21, and w 1, defined implicitly by 5 equilibrium equations: equations 3) and 4) for i = 1, and equations 5), 7), and 8). 3.2 Graphical Analysis Next, we show how to reduce the system of 5 equilibrium conditions with 5 unknowns to 2 equations with 2 unknowns, w 1 and 12. The first equation is obtained from 8), 12 = τ 12 f 1/) 12 w 1/ρ 1 σ/a) 1/) /ρ. CC) Note that this no longer depends on τ 21 or f 21. The reason is that these conditions no longer affect country 2 Foreign) if country 1 Home) is small. This will simplify the comparative statics below. The second equation is the trade balance condition. It is the same as in the case of two large economies except that M e 2 is now exogenous, M e 1 12)w 1 f 12 [J 1 12) + 1 G 1 12)] = M e 2 f 21 [J 2 21w 1, 12)) + 1 G 2 21w 1, 12))]. TB) Conditions CC and TB form a system of 2 equations in w 1 and 12. Again, it can be shown that Condition CC implies a positive relationship between w 1 and 12, while Condition TB implies a negative relationship between w 1 and 12. With the same intuition as before, Conditions CC and TB give us two curves, the competitiveness curve and the trade balance curve, as in Figure Unilateral Trade Liberaliation Consider again a reduction of variable and/or fixed trade barriers for foreign exporters. Unlike the case with a large Home economy, the competitiveness curve is not affected, and as shown in Figure 1a), only the trade balance curve moves down, implying an unambiguous decline in the equilibrium levels of w 1 and 12. As before, the decline in 12 implies an increase in 11 and, hence, an increase in the real wage in Home. The reason that the graphical analysis is now suffi cient to establish the result is that the CC curve does not depend on τ 21 or f 21. In turn, this is because 8

10 References [1] Baldwin, R. E., Forslid, R., Trade Liberaliation with Heterogeneous Firms. Review of Development Economics 142), [2] Baldwin, R. E., Okubo, T., Tax Reform, Delocation, and Heterogeneous Firms. Scandinavian Journal of Economics 1114), [3] Chor, D., Subsidies for FDI: Implications from a Model with Heterogenous Firms. Journal of International Economics 78, [4] Demidova, S., Productivity Improvements and Falling Trade Costs: Boon or Bane? International Economic Review 494), [5] Demidova, S., Rodrígue-Clare, A., Trade Policy under Firm-Level Heterogeneity in a Small Economy. Journal of International Economics 781), [6] Flam, H., Helpman, E., Industrial Policy Under Monopolistic Competition. Journal of International Economics 22, [7] Gros, D., 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, [8] Grossman, G. M., Helpman, E., Seidl, A., Optimal Integration Strategies for the Multinational Firm. Journal of International Economics 701), [9] Melit, M. J., The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity. Econometrica 71, [10] Melit, M. J., Ottaviano, G. I. P., Market Sie, Trade, and Productivity. Review of Economic Studies 75, [11] Venables, A., Trade and Trade Policy with Differentiated Products: A Chamberlinian- Ricardian Model. The Economic Journal 97,

11 5 Appendix 5.1 Proof of Lemma 1 ) 1 First, let us look at CC: 12 τ f12 12 f 22 w 1 ) 1 ρ 22 = 0. We need to show that dw 1 d 12 = LHS/ 12 LHS/ w 1 > 0, ) 1 where LHS/ 12 = 1 τ f12 12 f 22 w 1 ) 1 ρ d 22 d d21 21 d11 to derive d21 /d 11, we get where J i a) = 1 σ a LHS/ 12 = 1 a f 12 f 21 f 11 f 22 τ 12 τ 21 d11 d12. By using 3) to derive dii /d ij, and 7) f12 f 21 f 11 f 22 ϕ ) a dgi ϕ). Using CC and 7), we get ) 1 ) 2 J 1 12 ) J 2 21 ) J 1 11 ) J 2 22 ), LHS/ 12 = 1 τ 12 τ 21 ) 21 σ) ϕ dg 12 1 ϕ) ϕ dg 21 2 ϕ) ϕ dg 1 ϕ) ϕ dg 22 2 ϕ) > 0, since τ 12 τ 21 > 1, 1 σ < 0, 11 < 12, and 22 < 21. Next, note that LHS/ w 1 = 12 ρw 1 τ f12 f 22 ) 1 w1 ) 1 d ρ 22 d21 d21 < 0, dw 1 since from 3) and 7), d 22 /d 21 < 0 and d 21 /dw 1 = 21 /ρw 1 < 0. Hence, from CC, dw 1 /d 12 > 0. Now let us turn to the TB condition. Using 4) to solve for Mi e, we can rewrite TB as f22 [J 2 22 L ) + 1 G 2 22 )] ) 1 2 f 21 [J 2 21 ) + 1 G 2 21 )] + 1 f11 [J 1 11 = w 1 L ) + 1 G 1 11 )] ) 1 1 f 12 [J 1 12 ) + 1 G 1 12 )] + 1, Letting ψ i f ii ii ) ϕ/ii ) dg i ϕ) / f ij ij ) ϕ/ij) dgi ϕ) and solving for w 1 from 7) yields τ 21 f 21 /f 11 ) 1 ) ρ 21 ) ρ ψ 2 + 1) = 11) ρ ψ 1 + 1). 9) Equation 3) implies that an increase in ii leads to a decline in ij for j i), hence, given the definition of ψ i, we see that ψ i is decreasing in ii. Again, by using 3) for country 1, the RHS of 9) can be written as an increasing function of 12, while by using 3) for country 2, the LHS of 9) can be written as an increasing function of 21. It then follows that if 12 rises, then 21 must rise as well. From 3) for country 1, 11 must fall with rising 12. Hence, we conclude from 7) that w 1 should fall when 12 increases. 11

12 5.2 Proof of Proposition 1 Shift in the curves. First, let us show that for any given 12, a decrease in τ 21 and/or f 21 shifts down the competitiveness curve. To see this, note that if 12 is fixed, then from 3), 11 is fixed as well. But since from CC and 7), = τ 12τ 21 f 12 f 21 /f 11 f 22 ) , 22 must rise and 21 must fall from 3) they move in the opposite directions). Hence, from 6) w 1 falls for any fixed 12. Now we need to show that for any fixed w 1, a decrease in τ 21 and/or f 21 shifts the trade balance curve to the left, i.e., 12 falls for any given w 1. We will make use of the following two equations, which are just a reformulation of 7) and 9): ) 1 f21 τ f = w 1 ) 1 ρ, 10) ψ 1 + 1) = w 1 ψ 2 + 1). 11) We need slightly different argument for the case of a decline in τ 21 than for the case of a decline in f 21. Consider first a decline in τ 21. We proceed in two steps: first, we show that 21 decreases and then we use this to establish that 12 also decreases. For the first step, we proceed by contradiction, assuming that 21 increases given a constant w 1. From 10), 11 must rise as well. However, if 21 rises, then from 3), 22 falls, resulting in increasing ψ 2, and from 11)) increasing ψ 1, which from 3) implies that 11 falls, a contradiction. Hence, 21 falls with a fall in τ 21. Having proved that a decline in τ 21 results in a fall in 21, we now proceed to the second step and prove that 12 decreases. We proceed as follows. First, by 3) the fall in 21 leads to an increase in 22. This leads to a decline in ψ 2, which by 11) must be accompanied by a decline in ψ 1, implying an increase in 11 and again by 3)) a decline in 12. Now consider a decline in f 21. The previous logic does not work here because f 21 enters the free entry condition 3) for country 2. Consider then a decline in f 21 and assume by contradiction that 12 rises. Then from 3) for i = 1, 11 falls and, in turn, ψ 1 rises so that from 11), ψ 2 must rise as well. Moreover, from 10) we see that f 21 ) 1 / 21 must rise with falling 11, hence, since f 21 falls, we need to have 21 decrease. But from the definition of ψ 2 we have ) ψ 2 f 21 ) 1 / 21 ϕ dg 2 ϕ) = f 22 ϕ/22) dg 2 ϕ). 12) 21 Since both ψ 2 and f 21 ) 1 / 21 increase, while 21 falls, then the LHS of 12) must increase, implying that the RHS must also increase, hence, 22 must fall. Then from 3), f 21J 2 21 ) should fall as well. But note that [ ) ] f 21 J 2 21) = f 21 ) 1 / 21 ϕ dg 2 ϕ) 21) 1 G 21)). 21 We established above that f 21 ) 1 / 21 increases and 21 falls, hence, the RHS of this expression would increase, implying that f 21 J 2 21 ) would increase, a contradiction. Thus, we proved that for any given w 1, 12 falls with a fall in f

13 Welfare change. We know from Figure 1 that if both curves shift down, w 1 falls with a fall in τ 21 and/or f 21. We now prove that 12 must decrease. We proceed by contradiction. If 12 increases then then from 3), 11 falls. This implies that ψ 1 increases, and from 11) combined with the decline in w 1 we see that ψ 2 must increase. At this point, we need slightly different arguments for the case of a decline in τ 21 and the case of a decline in f 21. Consider first the case of a decline in τ 21. The increase in ψ 2 implies a decrease in 22, which combined with the decline in τ 12 and w 1 implies that the RHS of 6) falls, whereas by the assumption that 12 increases the LHS of 6) increases, a contradiction. Now consider the case of a decline in f 21. Using 6) and 7), we can rewrite ψ 2 as ψ 2 = τ 12τ 21 ) f )1 σ ϕ dg 22 2 ϕ) f )1 σ. 13) ϕ dg 21 2 ϕ) Assume by contradiction that 12 increases. From 3) for country 1 then 11 falls. Since w 1 falls, then 6) implies that 22 must increase. We know that ψ 2 increases, so the LHS of 13) must also increase. But since 12 and 22 both increase and 11 falls, the only way that the LHS of 13) can increase is for 21 to increase. However, 22 and 21 cannot rise at the same time, since in this case given a fall in f 21, the LHS of 3) for country 2, f 22 J 2 22 ) + f 21J 2 21 ), falls recall J i.) is decreasing), while its RHS remains constant, leading to a contradiction. Hence, as in the case of a decline in τ 21, 12 falls with a decline in f 21. Therefore, from 3), 11 rises, raising welfare at Home. To simplify the exposition, in the proofs above we have disregarded the possibility that the competitiveness curve does not shift or that 12 stays constant given a change in τ 21 or f 21. It is straightforward to extend the previous proofs by contradiction to show that this is indeed the case that is, that neither the competitiveness curve, nor 12 can stay unchanged. This establishes the strict monotonicity for the result in Proposition Justification of Small Economy Assumptions Here we will show that the assumptions we use to treat Home as a small economy can be obtained from the model of two large countries, Home and Foreign, with Home becoming small relative to the Foreign one the limit case). In particular, if two countries are endowed with n and 1 n) shares of the world s labor, L, L 1 = nl, L 2 = 1 n) L, n [0, 1], then the limit case we want to explore is the one when n 0. To simplify our analysis, we assume that 2 countries are symmetric in everything except for their sies, i.e., f 11 = f 22 = f, f 12 = f 21 = f x, F 1 = F 2 = F e, τ 12 = τ 21 = τ. Also, we assume that the productivity distribution in both countries is now specified as Pareto: G ) = 1 ) b β for b. Then, the free entry condition in country i can be written as θ 1) b β [ f ii) β + f x ij ) β ] = F e, FE) 13

14 where θ = β/ β σ 1)). Moreover, from 6) and 7), ij = τ ) 1 fx w i f w j ) σ jj B wi w j ) σ jj, where B τ fx f ) 1 > 1. Note that by using FE in the definition of M e i, we get M e i = θ 1) bβ L i /σf e. Hence, if we denote w 1 w 2 by w, then we get the new TB condition: [ ] n 1 n = σ w2β 1 β TB) To summarie, for given n, the equilibrium in the model with 2 countries can be described by 2 free entry and 1 trade balance conditions with 3 unknown variables, 11, 22, and w. What happens in the model described above when n 0? Solving FE for 11 and 22 gives [ ] β 11 = 1 fx 1 fx 22 so that the TB condition can be rewritten as As n 0, the LHS of 14) goes to 0. fx n 1 n = 1 σ w2β 1 1 fx f B β w β σ f B β w β σ f B β w β σ f B β w β σ,. 14) Moreover, the RHS of 14) rises with w here we use the fact that fx f B β < 1). Hence, as n falls, w falls as well, and when n 0, the RHS of 14) goes to 0. Note that if n < 1/2, then w < 1. If w > 1, then from FE, 11 < 22. But then in 14), the LHS<1, while the RHS>1, resulting in contradiction.). Thus, the denominator in the RHS of 14) is always positive and bigger than 1 fx f B β. Hence, as n 0, we must have ) [ ] w 2β σ 1 1 fx f B β w β σ 0. Can w be below fx f B β βσ for some n 0, 1/2)? The answer is no, since in this case the RHS of 14) would become negative, while n/ 1 n) > 0. Thus, [ ] as n 0, then w falls to fx f B β βσ. Moreover, from FE, if n falls, then 22 falls and 11 rises. Note that due to the Pareto distribution assumption, 22 cannot fall below b, the minimum value for φ, but from the solution of FE, it seems that 22 0 as n 0. How to explain this? The reason is that as n continues to fall, 22 reaches its minimum so that all foreign firms survive. As n continues to fall, 22 remains at level b, and the ero profit condition for country 2 is violated, so that FE is no longer true for country 2. 9 This also means that we proved assumption 1): productivity cutoff 22 is not affected by changes at Home, when n is small enough. Now let us derive the new FE conditions for n small enough so that 22 = b and π ) > 0. While for Home we have the same FE condition as before, for the Foreign country, 1 σ L 2P 2 ρ θb f + f x θ 1) b β 21) β = F e, 9 Note that this logic also applies to the other types of the productivity distributions. 14

15 which from the ero profit condition for exporters from Home can be rewritten as ) wτ wf x 12 θb f + f x θ 1) b β 21) β = F e. New FE) By using the new FE conditions for small enough n, we get M e 1 = θ 1) bβ nl σf e, M e 2 = which allows us to rewrite the TB condition as 1 n) L σ F e + f + b β f x 21 ) β), n 1 n = F e 12 / 21 )β θ 1) b β w F e + f + b β f x 21 ) β). As n 0, the LHS falls to 0 as well. Since the minimum value for F e + f + b β f x 21 ) β cannot be smaller than F e + f, then for the RHS 0, we need 12 / 21 )β /w 0 as n 0. Using this property in the new FE condition for country 2, which we can rewrite as ) wτ [ ] f x 12 θb 12) β + f x θ 1) b β 12/ 21) β /w = F e + f) 12 )β w, implies that we can ignore the second term in the LHS above, i.e., for small enough n, ) wτ f x 12 θb 12) β F e + f) 12 )β w, or wσ 12) 1 σ const. However, from the ero profit condition for exporters from Home, R 2 P2 w σ 12 )1 σ. Hence, we proved assumption 3): at some low level of n, we can treat R 2 P2 as a constant, i.e., the foreign demand for Home goods exported at the price p can be expressed as Ap σ. This also means that since for small n, P2 1 σ = M2 eθρ b β + M1 eθbβ ρ/τw) 12 ) β+) M2 eθρ b β as L 1 is very small) and R 2 L, then treating R 2 P2 as a constant implies treating M2 e as a constant, i.e., we proved assumption 2). 15

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