Trade Liberalization and Pollution in Melitz-Ottaviano Model

Transcription

1 Trade Liberalization and Pollution in Melitz-Ottaviano Model Zhou Mohan November 30, 20 Abstract I incorporate Copeland-Talor tpe pollution technolog into Melitz and Ottaviano 2008) s work horse model. I find in this model the pollution level of firms against productivit takes an inverse- U-shape. Also, trade liberalization does not necessaril increase total pollution. Under fixed pollution tax rate, this will depend on the tax rate charged b the government, the pollution level increases onl when the initial level of tax rate is high enough. However, when government s polic is responsive to trade, a numeric example shows that the aggregate pollution level nearl alwas increase. Kewords: Pollution; Trade; Heterogenous firms. JEL Classification Numbers: 00, 00, 00.

2 Introduction Empirical work on the relationship between trade liberalization, productivit and pollution emission at the firm level is scarce. Cole et al. 2005) examine the the relationship between firm-level characteristics and environmental regulation on industr-level emissions for manufacturing plants in the UK. The find emissions to be negativel related to firm size and productivit. Hollada 200) is a recent work tring to investigate the impact of trade on firm level emission. Using a panel of establishment-level data over the ears , he finds that exporters generate significantl less pollution emissions than non-exporters and that those emissions are less toxic than other establishments in the same industr. According to the heterogeneous firm literature, opening up to trade intensifies competition, driving the least efficient firms out of the market either through higher factor price or through small market demand. Intuitivel, this might become a mechanism through which trade liberalization ma reduce aggregate pollution level. Recentl, there are two papers tring to formalize this idea. Yokoo 2009) claimed that he is the first to formall incorporate the Copeland-Talor technolog into a heterogeneous firm framework a la Melitz 2003). He finds that more productive firms generate larger amounts of pollution. And since trade alwas increase consumption in the world econom, pollution will increase after trade. Bajona and Missios 200) also emplo a Melitz 2003) framework. But instead of assuming emission is proportional to the input used in production, the assume emission is proportional to the potential output. The result of this assumption is that more productive firms abate more intensivel and thus have lower pollution intensit than less productive ones, but still the level of pollution is increasing in firm productivit. Opening up to trade reduces the average pollution intensit, though has no impact on aggregate pollution level. In both of these two papers, constant markup is assumed following Melitz2003), which plas a crucial role driving the results. I uses Melitz and Ottaviano 2008) s work horse model. A result of appling this model is that though more productive firms produce more, their pollution level need not necessaril 2

3 be higher than a less efficient firm. In fact, the pollution level against productivit takes an inverse- U-shape. Also, the impact of trade liberalization on aggregate pollution level is not necessaril to increase total pollution. Under fixed pollution tax rate, it depends on the tax rate charged b the government, the pollution level increases onl when the initial level of tax rate is high enough. However, when government s polic is responsive to trade, a numeric result shows that the aggregate pollution level nearl alwas increase. 2 The model 2. Preference Following Melitz and Ottaviano2008), the consumer demand takes a quasi-linear form for the numeraire and all varieties in the differentiated sector: U = Q 0 + α q i di i Ω 2 β q i di) 2 i Ω 2 γ qi 2 di ) i Ω Assume there are L identical consumers in each countr. It can be easil derived that the market demand for varie i for all consumers is p i = α β q L j Ω jdj γ q L i. Invert this and we can get the demand for each variet: p i = A bq i 2) where A = αγ+βp βm+γ and b = γ L 2.2 Production I consider the short-run version of Melitz-Ottaviano Model. There is a continuum of incumbent firms with mass in each countr. No entr cost or fixed cost. At the beginning of the period, a productivit parameter ϕ is drawn from a Pareto distribution with c.d.f 3

4 Gϕ) = m ϕ ) k. Assume that m <, that is m is not too big. After that, all firms α decide simultaneousl if it would sta in market or exit, and in case it stas, how much to produce. The production and pollution technologies follows Copeland and Talor 2003). For simplicit, I assume the production process uses onl labor as input. Suppose a producer with productivit ϕ has a constant return to scale technolog q = ϕ θ)l 3) where θ is the proposition of potential production that is used for pollution abatement. I assume that the pollution tax charged b the government is high enough, so that each producer find it optimal to conduct a positive amount of abatement. Given θ and the labor input, the amount of pollution generated is z = θ) 2 l 4) I assume pollution is local throughout this paper. Combine this two equations and get ride of θ, the production technolog can be equivalentl viewed as using both labor and pollution as inputs q = ϕ z l 5) Suppose the unit pollution tax is τ and the wage rate is w. Then the dual unit cost function of a producer with productivit ϕ is ϕ, where = 2 wτ. Then each producer s optimal choice of price, quantit and the resulting mark-up are pϕ) = 2 A + ϕ ) 6) qϕ) = 2b A ϕ ) 7) pϕ) cϕ) = 2 A ϕ ) 8) 4

5 Notice that pϕ) cϕ) > 0 if and onl if A > 0, so onl firms with productivit ϕ > ϕ A will produce the set of which is called Ω ), while the rest of the firms exit the market. Assume that in equilibrium A > m, so there is a positive measure of firms exit in equilibrium. 3 Equilibrium under fixed pollution tax rate In this section, we assume that the pollution tax rate imposed b government is fixed, and does not change after trade liberalization. 3. Autark equilibrium Define = A. So a [ 0, m]. Then all equilibrium variables can be expressed as a function of. Aggregate price under autark is The mass of active firms is: P = p j dj = j Ω pϕ)dg ϕ) = 2k + 2k + 2 mk k+ 9) M = G ) = mk k 0) Using the above two equations and the definition A = αγ+βp βm+γ and = A, we get Φ) = 2k + 2 βmk k+ + γ αγ = 0 ) Obviousl, Φ ) is strictl increasing and Φ0) < 0. Since we have assumed m < α, so Φ m ) > 0. So, this function has a unique interior solution. Call it a [ 0, m]. The equilibrium quantit produced b a firm with productivit ϕ is: q a ϕ) = 2b a ϕ ) 2) 5

6 and the aggregate production is Q a = q j dj = j Ω qϕ)dg ϕ) = 2b a )dg ϕ) = ϕ 2k + 2 b mk a k+ 3) What about pollution b each firm? Since qϕ) = ϕ z l)and at optimum, τz = wl, qϕ) = ϕ z τz τ w ) = ϕz w zϕ) = = Kϕz 4) 2bK a ϕ ϕ 2 ) 5) so zϕ) has an inverse-u shape. z ϕ) > 0 if and onl if ϕ < 2 a. The pollution intensit of a tpe ϕ firm is: eϕ) = zϕ) qϕ) = Kϕ which is decreasing with productivi. Aggregate Pollution: 6) Z a = 2bK a ϕ mk )dg ϕ) = ϕ2 2bK k k + )k + 2) k+2 a 7) 3.2 Trade liberalization: 2 smmetric countries Now suppose that there are complete trade liberalization between the two countries, i.e. there is no trade frictions of an tpe. Two things change after trade liberalization. The first is that the mass of competing incumbent firms in the world market changed from to 2. The other is that the problem of each profit maximizing firm changed. Consider the smmetric equilibrium in which a tpical firm allocates the same amount of products to both home and foreign market export). Then a tpical firm s problem is max qi 0 2A b qi 2 ) qi 2 c i q i which is equivalent to max qi 0A b 2 q i) qi 2 c i q i, so each single firm is act as if it was facing a demand curve with slope b 2 instead of b. 6

7 Similar to the autark case, we can derive the aggregate price: P t = The mass of active firms: j Ω t p j dj = 2 pϕ)dg ϕ) = 2k + k + mk k+ 8) M = 2 G )) = 2mk k 9) Using the above two equations and A = αγ+βp βm+γ and = A,we can get Φ t ) = 2 2k + 2 βmk k+ + γ αγ = 0 20) the unique solution of which is called t. Comparing Φ t ) with Φ), we can immediatel find that t < a. More over, we can prove that Lemma t Lemma 2 t > Proof: a 2 k+ a 2 k+2, a ) if and onl if k + ) γ) k ) Obviousl t < a. Moreover, 2k+2 βmk 2 k+ t ) k+ + γ2 k+ t ) αγ > 2k+2 βmk 2 k+ t ) k+ + γ t αγ = 2 2k+2 βmk t k+ + γ t αγ = Φ t) = 0 But this is just Φ2 2k+ t ) > 0 = Φ a ) use the fact that Φ ) is strictl increasing: 2 2k+ t > a ) ) 2 k+ kk+2) + 2 kk+2) αm > β k 2) Using the fact that both Φ ) and Φ t ) are strictl increasing repeatedl: 2 k+2 t > a Φ2 k+2 t ) > Φ a ) = 0 = Φ t t ) 7

8 2k+2 βmk 2 k+2 t ) k+ + γ2 k+2 t ) αγ > 2 2k+2 βmk t k+ + γ t αγ γ2 k+2 ) > 2 2 k+ k+2 2k+2 t k < 2k + 2) γ ) 2 βm k 2k+2 )βmk k t k k+2 k+ k t < 2 k+2 k + ) γ βm k Φ t [2 k+2 k + ) γ ) βm k k ] > Φ t t ) = 0 ) k + ) γ) k 2 k+ kk+2) + 2 kk+2) > ) αm ) β k Q.E.D. 3.3 Individual pollution level How will the pollution that an individual firm emits change after trade liberalization? For those firms who are active both before and after trade liberalization: if and onl if but we know that t z t ϕ) z ϕ) = a 2 k+, a ) t bk ) ϕ ϕ 2 a ) = 2 2bK ϕ ϕ 2 ϕ > ϕ = t ϕ a ϕ ) > 2) 2 t a 22) so 2 t a < t or 2 t a > t. So there is alwas a positive measure of active firms whose pollution decrease, namel, the least productive firms ) with ϕ t, 2 t a while the more productive firms increases pollution. Notice in the current setting with no trade cost, all surviving firms are exporters. So the above discussion sas that that are some exporters whose pollution decrease the least productive ones). This is in contrast to the result of the two papers previousl cited, both of which predict that the pollution level of a firm will increase as it becomes an exporter. 8

9 3.4 Aggregate production and pollution The aggregate production is Q t = q j dj = j Ω The aggregate pollution: qϕ)dg ϕ) = 2 b t )dg ϕ) = ϕ k + b mk t k+ 23) 2 Z t = t bk a ϕ mk )dg ϕ) = ϕ2 bk k k + )k + 2) k+2 t 24) Comparing Q t, Z t with Q a, Z a using Lemma., we immediatel get: Proposition Q t > Q a alwas holds. Proposition 2 Z t > Z a if and onl if k + ) γ) k ) ) 2 k+ kk+2) + 2 kk+2) αm > β k Figure : Aggregate pollution under various β In other words, trade alwas increase production. However, trade does not necessaril increase aggregate pollution. Suppose the pollution tax is relativel high, then if this tax 9

10 rate is fixed, trade liberalization will increase aggregate pollution. However, if the initial tax rate is low, opening up trade will reduce the aggregate pollution. For a given tax rate, we can sa that when β which measures the substitutabilit between varieties) is high low), trade liberalization will reduce increase) aggregate pollution. The picture above plot the situation when k = L = γ = α = m = =. Red blue) line represent aggregate pollution level under autark liberalization). 4 Optimal pollution tax The indirect utilit associated with the utilit function is: and V t = I + 2 V a = I + 2 β + γ ) α P ) 2 + M M 2 ) β + γmt α P ) 2 t + M t 2 pϕ) P M ) 2 dg ϕ) Z 25) pϕ) P ) 2 t d 2 G ϕ)) Z t 26) M t In the two expressions above, is defined b Φ ) = 0 and Φ t ) = 0 respectivel. Assume the onl sourse of income of consumers is labor return. So in both equilibrium these are wl = const. Suppose that governments objective is to maximize consumer welfare and there is no strategic concern, i.e. the two governments coordinate to set a smetric optimal tax rate. The government chooses pollution tax τ to affect economic outcomes. Since in equilibrium, = 2 w τ = 2 τ Φ) = 2k+2 βmk k+ + γ αγ = 0 Φ t ) = 2 2k+2 βmk k+ + γ αγ = 0 so τ and or t are one to one, the government is in effect choosing or t according to: V = 0 a, and Vt = 0 t. 0

11 Proposition 3 Suppose there is a celling on the pollution tax rate that government can charge. Then as long as β is big enough, the optimal tax rate under free trade is larger than that under autark, τ t > τ a. Proof: For dispositional convinienc, I assume L = γ = α = m =. There is no loss of generalit as I assume both k and β can var. We call the first part of the indirect utilit gross utilit GU a = + U a + U 2a, where U a = 2 U 2a = 2 β + ) P ) 2 27) M M it can be calculated in equilibrium that: pϕ) P M ) 2 dg ϕ) 28) = k [ k ) 4k + 3) β k + 2k + 2 ] β k + 2k + 2) 3 29) U a U 2a = k k + 2 kk 2k + 2 β k ) β k + 2k + 2) 3 30) Similarl, = 2 k [ k ) 4k + 3) β k + k + ] β k + k + ) 3 3) U t Notice that U2t U 2a U 2t = 2 k k + 2 kk k + β k ) β k + k + ) 3 32) = 2 k + βk ) β k 2k + 2 β + 2k + 2 ) 3 k β k + k + ) 3 < = 2 33) U t U a = 2 β k + 2k + 2 ) 3 β k + k + ) 3 4k + 3) β k + k + 4k + 3) β k + 2k ) The function ξx) = 2 x+2k+2)3 x+k+) 3 4k+3)x+k+ 4k+3)x+2k+2 has the propert that it is single peaked in 0, + ) and that lim x + ξx) =. So for an given, there exist a minimum f k), 2

12 when β k > fk),or β > fk), we have k U t U a 2. If we impose a celling that the pollution tax τ could not be too big: τ τ c, then must also has a floor f, so [ f, ]. Then b continuit, there exists a uniform β k), when β > β k), conclude that under such circumstances, Ut U a 2 for an [ f, ]. So we GU t < 2 GU a, [f, ] 35) τ t However, Zt > τ a. Q.E.D = 2 Za. Which means, as long as β > β k), t < a, or equivalentl, 5 Pollution under optimal tax: a numerical example The following figure depict both optimal choice of which is inversel related to τ), under k =, 2, 4, L = γ = α = m = for various value of β. Red blue) represents autark trade liberalization). Figure 2: inverse of) optimal tax and pollution under various β 2

13 Notice that in this specific parameterized environment, for β close to 0, the optimal pollution tax is lower under trade. When β is high enough, the optimal pollution tax is higher under free trade. We can also plot the aggregate pollution level against various β in the picture above. We can see that the pollution under trade seems to be alwas higher than that under autark. Notice that we have previousl proved that, if the pollution tax does not respond to trade liberalization, then when β is big enough, trade can reduce pollution. Here, however, there is a rise in pollution tax, which undermines the condition under which pollution can be reduced via international trade. 6 Future job. Tr to identif conditions under which the pollution level under optimal tax is larger under trade than in autark. 2. Tr to identif pollution response under different polic regimes, e.g., non-tradable pollution quota, tradable pollution quota within countries, tradable pollution quota across globe. 3. Tr to identif pollution response under asmetric countr size. Reference Bajona and Missios 200). Trade and Environment with Heterogeneous Firms. Working Paper. Copeland, T. and M. S. Talor. 2003). Trade and the Environment: Theor and evidence. Princeton Universit Press, Princeton. Hollada, J. S. 200). Are Exporters Mother Natures Best Friends? Mimeo. Available at SSRN: 3

14 Yokoo, H. 2009). Heterogeneous Firms, the Porter Hpothesis and Trade. Institute of Economic Research Koto Universit, Japan. 4