Pareto-Efficient Climate and Trade Policies in the Presence of Non-Tradeable Goods

Transcription

1 Business School Department of Economics Pareto-Efficient Climate and Trade Policies in the Presence of Non-Tradeable Goods ECONOMISING, STRATEGISING AND THE VERTICAL BOUNDARIES OF THE FIRM Nikolaos Vlassis Discussion Paper in Economics No 15-8 October 2015 ISSN

2 Pareto-ecient climate and trade policies in the presence of non-tradeable goods Nikolaos Vlassis Department of Economics University of Aberdeen Business School Edward Wright Building Dunbar Street Aberdeen AB24 3QY Scotland, UK October 2015 Abstract The purpose of this paper is to extend the existing literature on the relationship between international trade and climate change by introducing non-tradeable goods and examining their role in the characterization of the Pareto-ecient environmental and trade policies It is argued that the presence of non-tradable goods does not impede the central planner from imposing the Pigovian carbon taxes Keywords: Environmental taxation; international trade; Pareto eciency; Non-tradeable goods; climate change JEL classication: H20; F18 Acknowledgments: Special thanks to Christos Kotsogiannis for introducing me to the issue and for his helpful comments and suggestions I also thank Pascalis Raimondos-Møller, Ben Zissimos and Catia Montagna for their comments and suggestions 1 Introduction Climate change, in economic terms, is a global externality: emitting countries ignore the damage they cause to others, thereby they are emitting more than it is 1

3 desirable from a global perspective But it is a particularly complex externality given the asymmetric impact of the stock of emissions on the geographical distribution and economic activity Given the global aspect of this challenge, the literature both theoretical and empirical on the relationship between international trade and climate change is fairly sizeable, 1 with considerable attention being paid to the characterization of Pareto-ecient environmental and trade policies 2 This literature, however, has neglected the role of non-tradeable goods, arguably a realistic feature of any economic environment The importance of non-tradeable goods to the characterization of implemented policiesdue to the substitution eect between tradeable and non-tradeable goods from the demand and supply side is well known in the trade literature 3 The obective of this paper is to extend the existing literature by examining the role of non-tradeable goods In particular, we ask whether the existence of non-tradeable goods alters the characterization of Pareto-ecient environmental and trade policies that are derived from a model with tradeable goods only We nd that that the answer to this question is: it does not Specically, we nd that, even with non-tradeable goods present, a Pareto-ecient environmental policy dictates that, in the presence of lump sum taxes, carbon taxes have a Pigouvian form and are uniform across all tradeable and non-tradeable sectors and countries while, in the absence of lump sum taxes, they are uniform across all sectors in a given country but not across countries which is consistent with the results of Keen and Kotsogiannis Description of the model The model is essentially that of Keen and Kotsogiannis 2014 appropriately modied to include non-tradeable goods There are J countries, indexed by the superscript, each of which produces M = T + N goods: T of these goods are tradeable and N are non-tradeable The T tradeable goods are traded at a T -vector of world prices given by p T 0 4 The price vector of non-tradeable goods in country is denoted by p N 0 International trade is subect to trade taxes or subsidies, the vector of which in country is denoted by τ The commodity price vector of the tradeable 1 Recent insightful surveys are by Copeland and Taylor 2004, Chen and Woodland 2013, Jones et al For contributions see, among others, Markusen 1975, Baumol and Oates 1988, Krutilla 1991 Hoel 1996,Copeland 1994, Lubema and Wooton 1994, Neary 2006, Hatzipanayotou et al 2008, Keen and Kotsogianis 2014 and Tsakiris et al See among others Dornbush 1974, Fukushima 1979, Clements 1982, Rivera-Batiz and Almansi The following convention is used: if x = x 1,, x N, then x 0 means x n > 0 for all n = 1,, N; x > 0 means x n 0 for all n = 1,, N and at least one x n 0; and x 0 means x n 0 for all n = 1,, N 2

4 goods in country is thus given by the T -vector p T = p T + τ The model is very general in allowing for all types of trade taxes and subsidies Within each country there is a perfectly competitive private production sector Producers in country use factor endowments, denoted by vector v, to produce the M-vector y of commodities The production of each commodity generates some pollutant such as carbon emissions, with the M-vector z denoting emissions produced by the M commodities in country Total emissions in country are thus given by ι z, where ι is the M-vector of 1s and a prime indicates transposition Each national government may impose carbon taxes on the emissions from each commodity sector, and the M-vector of carbon taxes is given by s Hence, we allow for pollution taxes to be sector-specic Global emissions are therefore given by k = ι z 1 =1 The production sector in country is perfectly competitive and characterized by a revenue function 5 r p, s, v = max y, z {p y s z : f y, z, v 0}, 2 where f is the implicit production possibility frontier in country Following from 2, and as an envelope property, the net output M vector, y, and the vector of emissions, z, are given by r pm p, s, v = y, 3 r sp, s, v = z 4 The consumption sector in country is characterized by the expenditure function e p, u, k = min x {p x : U x u }, 5 where U x is the utility attained by consuming vector x of commodities Notice that pollution k aects utility presumably negatively Shephard's lemma gives the M-vector of compensated demands, e pm p, u It is convenient to dene the net expenditure function in country as 6 S p, s, u, k e p, u, k r p, s, v 6 5 The revenue expenditure function has the standard properties of homogeneity, convexity concavity and dierentiability For the properties of the revenue and expenditure function see Dixit and Norman 1980 and Woodland The function S has the properties of the underlying expenditure and revenue functions 3

5 The net expenditure function has the useful derivative properties that the vector of compensated import functions, denoted by m T, is given by m T p, s, v, u, k Sp T p, s, v, u, k = e p T p, u, k rp T p, s, v, 7 where e p T denotes the T -vector of compensated demands for the tradeable goods Similarly, rp T denotes the net output T -vector of the tradeable goods The vector of emissions produced by country is given by z p, s, v, k Ssp, s = rsp, s 8 The vector of non-tradeable goods is given by m N p, s, v, u, k Sp N p, s, v, u, k = e p N p, u rp N p, s, v = 0 9 Without loss of generality, the rst commodity of the tradeable goods is taken as the numeraire, with unit world price, and it is assumed to be untaxed by all countries, so p 1 = 1 and τ 1 = 0 for all = 1,, J7 The equilibrium conditions for the world economy can be compactly expressed as 8 p T Sp T p, s, u, k + b = 0, = 1,, J, 12 p, s, u = 0, 13 =1 ι S q Ssp, s = k, 14 =1 b = 0, 15 =1 S p N p, s, u, k = 0, = 1,, J, 16 and S q denotes the net expenditure function for the T 1 tradeable goods in country It is the presence of equation 16 that is central to the analysis here 7 For notational convenience, the price and trade tax vectors will be partitioned accordingly to p T = 1, q ; p T = 1, q ; q = q + σ ; τ = 0, σ, 10 and so domestic prices are given by p M = p T, p N = 1, q, p N 11 8 The vector of endowments v, since it is xed, will be suppressed throughout 4

6 The J equations in 12 are the consumers budget constraints: They simply state that the balance of trade decit value of net imports at world prices plus any international transfers to from country must be equal to zero The T 1 equations in 13 are the world market equilibrium conditions for the nonnumeraire tradeable commodities Equation 14 species that global emissions are the sum of emissions produced by the J countries Condition 15 requires that the sum of international transfers be zero Equation 16 represents the market clearing conditions for the non-tradeable goods Given the tari vectors τ, = 1,, J, the carbon tax vectors s, = 1,, J and the vector b = b 1,, b J of international transfers satisfying 15, the national budget constraints 12, the market equilibrium conditions 13, the global emissions equation 14, and the market clearing for non tradeable goods 16 may be solved for the competitive equilibrium world price vector for tradeable commodities, p T, the equilibrium price vector in country of the non-tradeable goods p N, the level of global emissions, k, and the vector of national utility levels, u = u 1,, u J The existence of a competitive equilibrium solution with p 9 M 0 is assumed The dierential comparative static system is therefore Adu + Bdq + C T d q + C N dp N + Dds + Edk + F db = 0 17 To characterize Pareto optimality of the initial equilibrium it proves convenient to use Tucker's Theorem of the Alternative Mangasarian, 1969, p 34, which states that either the system in 17 has a solution with du > 0 where du > 0 is a semipositive vector with, that is, du 0 for all = 1,, J and du 0 for some perturbation dq, d q, dp N, ds, dk, dbso that the initial equilibrium is Pareto inecientor there is a vector y = y 1, y 2, y 3, y 4, y 5 R J+T 1+1+T +JN where y 1 = y1, 1, y1 J R J, y 2 = y2, 2, y2 T R T 1, y 4 = y4, 1, y4 T R T y 5 = y5, 1, y5 J R JN with y 5 y = 1 5,, yn 5 R N such that y B, C T, C N, D, E, F = 0, 18 y A 0, 19 in which case the initial equilibrium is Pareto ecient The analysis now proceeds to derive the Pareto ecient climate and trade policies 10 3 Pareto-ecient climate and trade policies This section considers the characterization of Pareto-ecient carbon tax and trade tax policies Starting from an initial equilibrium in which the world economy is characterised by Pareto-ecient policies, no government can alter carbon 9 The denitions of the matrixes A, B, C T, C N, D, E and F are given to the appendixa 10 The details of the derivations are available upon request 5

7 or trade taxes to make one country better o without making some other country worse o In the current framework, with both tradeable and non-tradeable goods, the following results establish the two key features of any Pareto ecient allocation Proposition 1 In the presence of lump sum transfers across countries and with both tradeable and non-tradeable goods, Pareto eciency requires that in every country = 1,, J : a carbon taxes are set at s = J i=1 Si k ι, ie they are uniform across production sectors within a country and also uniform across countries, b trade tax vectors are equal, σ = σ, = 1,, J, implying that domestic price vectors of the tradeable goods are all equal c the prices of non-tradeable goods are country-specic and satisfy p N = y 5 /y 4 According to Proposition 1 Pareto eciency requires thateven in the presence of non-tradeable goodseach country sets a Pigovian carbon tax in each of the M sector to equate the marginal cost of an extra unit of carbon emissions, s, to the marginal global damage that the extra unit of emissions causes through climate change, J i=1 Si kι The uniformity of carbon taxes within and across countries follows from the fact that carbon emissions from each sector and country contribute equally, at the margin, to the stock of carbon in the atmosphere and, hence, to climate change Part b of the Proposition 1, implies equality of domestic prices of the tradeable goods and the collinearity of the tari vectors across all countries The importance of this is in emphasizing thatin the presence of lump sum transfersproduction eciency for the tradeable goods is part of a Pareto ecient allocation To see this, recall that producer prices in country are p T = 1, q + 0, σ and so with σ = σ for = 1,, J, it is the case that p T = p T for all countries Parts c of the Proposition 1 states the Pareto ecient country-specic prices of the non-tradeable goods Notice that, if it happens that y 5 = y 5and so the shadow value of non-tradeable goods is the same across countries 11 then the price vectors for non-tradeable goods are collinear across countries with a degree of collinearity 1/y 4 Clearly, as Proposition 1 shows, the presence of the non-tradeable goods does not change the uniformity structure of carbon taxes within and across countries nor the collinearity of the trade tax vectors This is, perhaps, not surprising once seen aswith redistribution being taken care of by lump sum transfers and carbon taxes being uniform across all M sectorsnon-tradeable goods have no additional role to play, at a Pareto ecient allocation, in pollution policies Therefore, the presence of non-tradeable goods does not change the uniformity structure of carbon taxes within and across countries nor the collinearity of the trade tax vectors This is, perhaps, not surprising once seen aswith redistribution being taken care of by lump sum transfers and carbon taxes being 11 There is no reason, of course, to suppose that this will be the case 6

8 uniform across all M sectorsnon-tradeable goods have no additional role to play, at a Pareto ecient allocation, in pollution policies It is therefore interesting to consider the possibility that international lump sum transfers are unavailable In this case, we obtain: Proposition 2 In the absence of international lump sum transfers and assuming that the substitution matrix for each country has maximal rank, Pareto eciency requires that in every country = 1,, J: a carbon taxes are set such that y 1 si = J i=1 yi 1S i k ι, where y 1 is a scalar, and so carbon taxes are uniform across production sectors within a country but dierent across countries in the sense that for any countries and i they satisfy s = α i s i, where α i y1/y i 1, and b trade tax vectors are collinear across countries in the sense that σ = α i σ i, = 1,, J, implying that domestic price vectors are also collinear across countries c the prices of the non-tradeable goods are country-specic and satisfy p y 5 /y 1 N = Part a of the proposition simply states that in a Pareto ecient allocation carbon taxation reects the global changes in utility, taking into account the cross country income implications of this through the scalar multipliers y 1 Part b is more striking, implying that the social planner uses taris and so the prices of the tradeable goods as a redistribution device Consistently with Keen and Kotsogiannis 2014, there is generally global production ineciency for the tradeable goods in the allocations characterized by Proposition 2 Though increasing the net output of some good in some country without reducing the net output of any other good or increasing emissions requires that both producer prices and carbon taxes be equalized across countries, the proposition shows that Pareto eciency allows for trade taxes, and hence domestic prices, to dier internationally Turning now to non-tradeable goods and their prices, one notices that these prices also reect the redistribution motive of the central planner Clearly, in this case and even if the shadow prices of the non-tradeable goods where the same across countries in the sense that y 5 = y 5 non-tradeable price vector will not be collinear What Proposition 2 implies is intuitive: The central planner implicitly chooses goods prices either directly, in the case of non-tradable goods, or indirectly, in the case of tradable goods, through taris to correct any distributional misallocations The presence of non-tradable goods does not impede the central planner of imposing the Pigovian carbon taxes of part a of Proposition 2 7

9 4 Concluding remarks This paper has extend the existing literature of international trade and climate change examining the role of non-tradeable goods in the characterization of the Pareto-ecient environmental and trade policies It is argued that the presence of non-tradable goods does not impede the central planner of imposing the Pigovian carbon taxes with the Pareto-ecient carbon and trade taxes being consistent with the ndings of Keen and Kotsogiannis

10 References 1 Baumol W J and W Oates, 1988 The theory of environmental policy, 2nd edn, Cambridge University Press, Cambridge 2 Chen X and A Woodland, 2013 `International trade and climate change,' International Tax and Public Finance 203, pp Clements W K, 1982 `More on taris and non-traded goods,' Economics Letters 91, pp Copeland R B, 1994, `International trade and the environment: Policy reform in a polluted small open economy,' Journal of Environmental Economics and Management 261, pp Copeland B R and M S Taylor, 2004 `Trade, Growth, and the Environment,' Journal of Economic Literature 421, pp Dixit A and V Norman, 1980 Theory of International Trade, Cambridge University Press 7 Dornbusch R, 1974 `Taris and nontraded goods,' Journal of International Economics 42, pp Fukushima T, 1979 `Tari Structure, Nontraded Goods and Theory of Piecemeal Policy Recommendations,' International Economic Review 202, pp Hatzipanayotou P, S Lahiri and M Michael, 2008 `Cross-border pollution, terms of trade, and welfare,' Environmental & Resource Economics 413, pp Hoel M, 1996 `Should a carbon tax be dierentiated across sectors?,' Journal of Public Economics 59, pp Jones B, M Keen and J Strand, 2013 `Fiscal implications of climate change,' International Tax and Public Finance 201, pp Keen M and C Kotsogiannis, 2014 `Coordinating climate and trade policies: Pareto eciency and the role of border tax adustments,' Journal of International Economics 941, pp Krutila K, 1991 `Envirnmental regulation in an open economy,' Journal of Environmental Economics and Management 202, pp Ludema R D and I Wooton, 1994 `Cross-Border Externalities and Trade Liberalization: The Strategic Control of Pollution,' Canadian Journal of Economics, 274, pp , 15 Mangasarian O, 1969 Nonlinear Programming New York: McGraw-Hill 9

11 16 Markusen J R, 1975 `International externalities and optimal tax structures,' Journal of International Economics 5, pp Neary J, 2006 `International trade and the environment: Theoretical and policy linkages,' Environmental and Resource Economics 331, pp Rivera-Batiz L F and Almansi A A, 1983 `Non-traded goods and the general equilibrium eects of taris in the short run,' Economics Letters 13, pp Tsakiris N, M Michael and P Hatzipanayotou, 2014 `Asymmetric Tax Policy Responses in Large Economies With Cross-Border Pollution,' Environmental and Resource Economics 584, pp Woodland AD, 1982 International Trade and Resource Allocation, New York: North-Holland 10

12 A Appendix The matrices A, B, C T, C N, D, E and F are dened by p T S1 p T u p T S2 p T u 0 0 p T SJ p T u Squ 1 Squ 2 S J qu Adu Sp 1 N u Sp 2 N u 0 0 Sp J N u Bdq S 1 q S 2 q Sq J dq, du 1 du 2 du J, 1 C T d q p T S1 p T q p T S2 p T q p T SJ p T q Sqq 1 Sqq 2 Sqq J ι Ssq 1 ι Ssq 2 ι Ssq J Sp 1 N q Sp 2 N q 0 dq 1 dq 2 dq J, 0 0 S J p N q 1 Notice that the vector q is afollowing the normalization of pricest 1-vector 11

13 C N dp N p T S1 p T p N p T S2 p T p N p T SJ p T p N Sqp 1 N Sqp 2 N Sqp J N ι Ssp 1 N ι Ssp 2 N ι Ssp J N Sp 1 N p N Sp 2 N p N S J p N p N dp 1 N dp 2 N dp J N, p T S1 p T s p T S2 p T s 0 0 p T SJ p T s ds 1 Sqs 1 Sqs 2 Sqs J Dds ι Sss 1 ι Sss 2 ι Sss J ds , Sp 1 N s 0 0 ds J 0 Sp 2 N s Sp J N s p T S1 p T k p T S p T k p T SJ p T k S q T k =1 Edk dk, F db S p N k Sp 2 0 N k S J p N k db 1 db 2 db J 12

14 B Appendix Proof of Proposition 1: Given the dierentiability assumptions concerning the expenditure and revenue functions the systemre-written here again for convenience12, 13, 14, 15 and 16 p T Sp T p, s, u, k + b = 0, = 1,, J, B1 p, s, u = 0, B2 =1 ι S q Ssp, s = k, =1 b = 0, =1 S p N p, s, u, k = 0, = 1,, J B3 B4 B5 Equations 18 and 19 are necessary and sucient conditions for Pareto optimality in the present model The equations in 18 and 19 can be readily shown to be expressed as y A = y 1 p T Sp T u + y 2S qu + y 5 S p N u, = 1,, J 0, B6 y B = y 1 S = y 1S q = 0, B7 =1 q y C T = y 1 p T Sp T q + y 2S qq + y 3 ι Ssq + y 5 S p N q, = 1,, J = 0, B8 y C N = y 1 p T Sp T p N + y 2S qp + y 3 ι S N sp N + y 5 S p N p N, = 1,, J = 0, y D = y 1 p T Sp T s + y 2S qs + y 3 ι Sss + y 5 S p N s, = 1,, J = 0, y E = J y 1 p T S p T k + J y 2S qk y 3 + y F = =1 y 1 y 4, =1 =1 B9 B10 y 5 S p N k = 0, B11 = 1,, J = 0, B12 where we follow the convention to denote matrices with elements for each = 1,, J in the square brackets 13

15 Notice that combining B8 and B9 we have that y 1 p T where S pt q Sp T p N + y 2 Sqq S qp + y 3 ι S N sq Ssp N + y 5 S pn q Sp N p N = 0, B13 B13 can be re-written as y 1 p T S pt q S p T p N S qq S qpn S sq S sp N S pn q S p N p N T M 1, T 1 M 1 M M 1, N M 1 B14, B15 B16 B17 S pt q Sp T p N +0, y 2 Sp T q Sp T p +y 3 ι S N sq Ssp N +y 5 S pn q Sp N p N = 0, and so, after dening a 1 T vector, as ρ, y 5 ρ y 4 p T + 0, y 2, S p T q S p T p N S p N q S p N p N B18 + y 3 ι S sq S sp N = 0 B19 Equation B19 has used the fact that equation B12 implies that y 1 = y 4 for all = 1,, J and so y 1 = y 4 ι where ι is the unit vectorthe implication of this is that the marginal social utilities of incomegiven by y 1 for country are the same across all countries Following B6 we also have that S y A = ρ, y5 pm u, = 1,, J 0, B20 where Sp M u is an M 1-vector Similarly, following equation B10 we have that y D = y 1 p T Sp T s + y 2S qs + y 3 ι Sss + y 5 S p N s, = 1,, J = 0, B21 = ρ, y5 S pt s Sp + y 3 ι Sss, = 1,, J = 0 B22 N s 14

16 Equations B6-B11 may therefore be re-expressed as y A = ρ, y5 S pu, = 1,, J 0, B23 y B = y 4 S = 0, B24 y C = y D = y E = J =1 q ρ, y S pt q Sp T p N 5 Sp N q Sp + y 3 ι Ssq S sp N, = 1,, J = 0, B25 N p N ρ, y S pt s 5 + y 3 ι Sss, = 1,, J = 0, B26 =1 S p N s ρ, y 5 S p T s S p N s y 3 = 0 B27 Following from B25 and B26, equations y C = 0 and y D = 0 may be combined together as S ρ, y 5, y 3 ι pt q Sp T p N S pt Sp N q Sp s N p N Sp N s = 0, 0 B28 S sq Ssp N S ss Denoting S S pˆp pt q Sp T p N Sp N q Sp, N p N S sˆp Ssq S sp N, S Sps pt s, S p N s then B28 is equal to ρ, y 5, y 3 ι S pˆp Sps S sˆp Sss = 0, 0 B29 Homogeneity of the net expenditure function in price vector p, s implies that 2 p, s S pˆp Sps = 0, 0 B30 S sˆp holds as an identity 2 This follows from the fact that Sp p, s and Ss p, s are homogeneous of degree zero This implies that S ss S pp p + S ps s = 0 S sp p + S ss s = 0 15

17 Notice now that B29 can be written as y 4 p T + 0, y 2, y 5, y 3 ι S pˆp Sps S sˆp Sss = 0, 0, B31 whereas B30 can be written as after multiplying by y 4 y 4 p T + y 4 0, σ, y 4 p N, y 4s S pˆp Sps = 0, 0 B32 Subtracting one from the other we have that y4 σ y 2, y4 p N 5 y, This implies that S sˆp S ss y4 s y 3 ι S pˆp Sps S sˆp Sss σ = y 2/y 4, s = y 3 ι /y 4, = 0, 0 B33 B34 B35 p N = y 5 /y 4 B36 Assuming that the substitution matrix for each country has maximal rank, p, s is the only vector up to a factor of proportionality satisfying the equality in equation B30 Consequently, it must be the case that ρ, y 5, y 3 ι = p, s up to a factor of proportionality, implying that choosing the factor of proportionality to be y 4 0 p = ρ, y 5 /y 4, = 1,, J, B37 s = y 3 /y 4 ι, = 1,, J B38 Combining, we have that Spp Ssp Sps Sss p s 0 = 0, and so, upon transposing, p s Spp Sps Ssp Sss =

18 This shows that all domestic prices of the international traded goods must be equal proportional to one another the prices of the non-tradeable goods p T = ρ/y 4, = 1,, J, p N = y 5 /y 4, = 1,, J, and that carbon taxes are the same across countries and across sectors within each country For domestic prices of the international traded good to be equal, the specic tari vectors, σ = y 2/y 4, must also be equal across countries To complete the proof, we next need to characterize carbon taxes Following B11and upon using the fact that ρ/y 4 = p T we have that y 3 = J ρ, y5 S p T k =1 S = J ρ, y 5 S pk B39 p N k =1 We now know that, following from the homogeneity property of S p S pk S k, and so y 3 = J =1 ρ, y 5 S pk = y 4 = y 4 p S pk = y 4 =1 =1 =1 S k p T, p N S pk, B40 Substituting this expression for y 3 /y 4 into B38, one obtains that s = ι, B41 as required S k =1 17

19 C Appendix Proof of Proposition 2: The proof of this proposition makes use of the steps and the equations of the proof of Proposition 1 In the absence of international transfers, each country can only spend what it earns through production and net tax revenue and this constraint on national budgets complicates the outcomes of policy reforms In terms of the model given by and its dierential system 17, the international transfers are now set to zero and left unchanged Equation B12 no longer applies and so it is, therefore, no longer the case that the dual variables satisfy the condition y 1 = y 4, = 1,, J Accordingly, the equations y C = 0 and y D = 0 now become since ρ y 1 p T + 0, y 2, S y 1 p T + 0, y 2, y 5, y 3 ι pt q Sp T p N S pt Sp N q Sp s N p N Sp N s = 0, 0 S sq Ssp N S ss Equation C1 can be written as y 1 p T + 0, y 2, y 5, y 3 ι S pˆp Sps S sˆp Sss C1 = 0, 0, C2 whereas B30 can be written as after multiplying by y 1 y 1 p T + y 1 0, σ, y 1 p N, y 1 s S pˆp Sps = 0, 0 C3 S sˆp S ss Subtracting one from the other we have that y 1 σ y 2, y 1 p N y 5, This implies that y 1 s y 3 ι S pˆp S sˆp Sps Sss = 0, 0 C4 It then follows that y 3 = J =1 σ = y 2/y 1, s = y 3 ι /y 1, C5 C6 p N = y 5 /y 1 C7 ρ, y 5 S pk = J y 1 =1 = J y 1 p S pk = J y 1 S k =1 =1 p T, p N S pk, C8 18

20 Substituting this expression for C6, s = y 3 /y 1 ι, = 1,, J one obtains as required s = y1s i k i i=1 ι/y 1 C9 19