Effects of Unilateral Emission Permits Reduction on Terms of Trade, Capital Accumulation, and Welfare in a World Economy

Effects of Unilateral Emission Permits Reduction on Terms of Trade, Capital Accumulation, and Welfare in a World Economy Karl Farmer 1, Birgit Friedl 1,2,, Andreas Rainer 2 1 Department of Economics, University of Graz, Austria 2 Wegener Center for Climate and Global Change, University of Graz, Austria Corresponding author. Email: birgit.friedl@uni-graz.at, phone/fax: +43 316 380-7107/-9520 May 9, 2008 Abstract We develop a two country overlapping generations model where emissions arise from production. Both countries, denoted by Home and Foreign, implement a national emission permits system. If Home unilaterally reduces the emission permits level, Home s and Foreign s capital stocks fall while Home s terms of trade improve. Under dynamic efficiency and Home being a net debtor, Home s and Foreign s welfare is reduced and the welfare loss is larger in Home than in Foreign. Keywords: emission permits, trade, overlapping generations, welfare. JEL Codes: F11; Q56; D91. 1

1 Introduction The imposition of emission regulations increases production costs and deteriorates domestic firms ability to compete with foreign firms on international commodity markets (Copeland, 1994). Thus, unilateral environmental policy has undesirable consequences for a country s international competitiveness. Nonetheless, as the example of climate policy shows, unilateral policy approaches, like the envisioned strengthening of the European emissions trading system, are gaining prominence. Hence, the question is whether there is any economic rationale for the unilateral approach to climate policy beyond the effects on international competitiveness in an interdependent world economy. The aim of this paper is thus to develop a better understanding of the way in which unilateral environmental policy impacts on international competitiveness, capital accumulation and domestic and foreign welfare in an integrated world economy. We will disentangle the consequences for capital accumulation, the terms of trade and welfare of a unilateral environmental policy and show that the magnitude of the welfare effect depends on the country s net asset position. The issue of international competitiveness has been discussed intensively for unilateral fiscal policy and the necessity of international policy coordination (e.g., Feldstein, 1986; Frenkel and Razin, 1986). A key result in this literature is that the effect of unilateral fiscal policy on domestic terms of trade (as a measure of international competitiveness) depends on the net asset position of Home (Persson, 1985): if Home is a net creditor, a government debt expansion by Home leads to an improvement in its terms of trade while its terms of trade deteriorate if Home is a net debtor. However, the dependency of the terms of trade on the net asset position presupposes different production technologies across countries which contradicts the basic Heckscher Ohlin model of international trade. Thus, we will show that in our two country, two good model the terms of trade effect will be unambiguous and independent of the net asset position. Secondly, when proceeding from a quasi static, two period analysis to a full dynamic 2

analysis, unilateral fiscal expansion affects also capital accumulation (Lipton and Sachs, 1983). In a two good setting, Zee (1987) and Lin (1994) find that a higher level of government debt leads to lower steady state capital stocks in both countries while the steady state interest rates increase. The economic reason for this result is that government debt implies an increase in the tax burden which reduces both savings of younger households and the supply of loanable funds for private capital accumulation. We will show that a qualitatively similar result can be found for a more stringent unilateral environmental policy, in spite of a rising price of pollution. Thirdly, in the literature on unilateral fiscal policy the dynamic welfare impacts caused by unilateral policy have been decomposed into three effects, a terms-of-trade effect, a capital accumulation effect (caused by changes in the interest rate), and a wealth effect (direct income loss due to a shift in the tax burden), leading to an ambiguous net welfare effect for Home and for Foreign (Ono and Shibata, 2005). In contrast, in a static closed economy model the only effect of a more stringent environmental policy is a wealth effect (Hoel, 1991), while in a static open economy the only effect of a more stringent unilateral environmental policy is the terms of trade effect (Copeland and Taylor, 2005). In the dynamic context of our two country model, we analyze the lifetime welfare effects of unilateral environmental policy by considering the opposing forces of all three effects. In this way, we are able to unify the results of Hoel (1991) who finds that unilateral environmental policy is detrimental for the Home country and that of Copeland and Taylor (2005) who argue that a unilateral domestic environmental policy can cause a welfare loss for the foreign country and thus serve as a leverage for international policy coordination (i.e., environmental regulations become strategic complements). We will show that both cases can emerge, depending on the relative strength of the three effects. One increasingly popular instrument of environmental policy is the creation of a market for emission permits. Consequently, several authors have investigated the consequences of emission permits for capital accumulation and the environment in Overlapping Generations (OLG) models. For a closed economy where pollution arises from production, Ono (2002) finds that a permit reduction policy can decrease both capital accumulation and 3

environmental quality in the long run, even though pollution is reduced by the permits. Jouvet et al. (2005b) investigate the effects of emission permits in a closed economy, too, and find that auctioning permits is efficient, but not grandfathering. In a two country OLG model with a single commodity, Jouvet et al. (2005a) examine the equilibrium with an international market of tradable permits. However, their focus is on international permit trading and international policy coordination. We are interested in unilateral actions and we thus assume two nationally separated permit markets without trade. We investigate the effects of a unilateral reduction of emission permits in a world economy consisting of two large countries, interconnected through free trade in produced commodities and in bonds emitted by national governments. While commodities and governments bonds are internationally mobile, labor and real capital are not. We extend the leading example in Diamond s (1965) overlapping generations economy with productive capital to a two country setting with two tradeable goods with perfect specialization in each country. Following the setup of the dynamic Heckscher Ohlin models (see Chen, 1992; and more recently Ono and Shibata, 2005), we assume identical technologies and preferences across countries. As a prerequisite for international trade, countries differ in their levels of public debt per capita leading to diverging foreign net asset positions across countries. Regarding pollution and the permit market, we extend the single country model of Ono (2002) towards two economically interdependent economies with two separated national permit markets. In contrast to Ono (2002), however, we abstract from environmental maintenance investments (as in the first part of his paper). Unilateral environmental policy will be depicted as a marginal reduction in the number of permits issued in Home. This paper has six sections. The next section provides a description of the two country, two good model with nationally tradable emission permits. Section 3 investigates the existence and stability of the equilibria, defined by the three dynamic equations stated in the previous section. The long run and short term effects caused by a unilateral permits reduction on the terms of trade, and on domestic and foreign capital accumulation are analyzed in Section 4. We investigate the steady state welfare effects of such a unilateral permits reduction in Section 5. Section 6 summarizes our results and concludes. 4

2 The Basic Model We consider a two country two good OLG model and a permit market for greenhouse gas emissions, whereby the two country OLG model is an extension of Farmer and Zotti (2007). All economic variables of Foreign will be denoted by an asterisk ( ). Each country s economy is composed of perfectly competitive firms and finitely lived agents. Each country (Home and Foreign) produces a specific good, denoted by X for the Home good and Y for the Foreign good, which can be used for the purpose of consumption in both countries as well as for investment. 1 As a prerequisite for international trade, a necessary assumption is that countries differ in their levels of public debt per capita leading to diverging foreign net asset positions across countries. In order to simplify the equilibrium dynamics, we assume that the government runs a constant stock fiscal policy (b t = b t+1 = b, t for Home, and b t = b t+1 = b, t for Foreign) (as in Diamond, 1965). Without loss of generality, the rate of depreciation is set to be one, therefore the investment of the current period builds next period s capital stock. We further assume that there is no population growth and hence labor supply is exogenously set at L = 1. Since the greenhouse gas emissions of firms are considerably larger than those of households, the established literature focuses on producer emissions (Ono, 2002; Jouvet et al., 2005a,b). We will thus also focus on producer s carbon emissions. The government of each country implements a national permit trading system by setting a permit level. 2.1 Production and Pollution Production in each country is specified by a Cobb Douglas production function with constant returns to scale. Total output in Home X t is determined by three production 1 This assumption is a deviation of our model from the assumptions of the Heckscher Ohlin model. Our model can be regarded as an OLG analogue to Obstfeld s (1989) and Gosh s (1992) two country two good ILA models. 5

factors, namely capital services K t, labor services L t, and pollution flow P t : 2 X t = M (L t ) 1 α K α P (K t ) α K (P t ) α P, (1) where M denotes a productivity scalar. Defining inputs and output in per capita terms x t X t /L t, k t K t /L t and p t P t /L t, yields per capita output as: x t = M (k t ) α K (p t ) α P. (2) Total revenues of production (where the output price is set as numeraire) are spent on factor costs of labor w t L t and capital q t K t. Furthermore, following the specification of the permit market in Ono (2002), in each period emission quotas S are distributed free of charge to the firms. If the representative firm requires more allowances, it has to buy additional permits on the market for a price of e t each, while in the case of excess permits, the firm gains revenues from selling them on the market. Thus, e t (S P t ) depict the net payment on emission permits and the firm s profit maximizing problem reads as follows: max π t = M (L t ) 1 α K α P (K t ) α K (P t ) α P q t K t w t L t + e t (S P t ). (3) L t,k t,p t Assuming that labor supply L t is normalized to one, profits per capita are given by: π t = x t q t k t w t + e t (S p t.). (3 ) The first order conditions of the firm for an interior solution read as follows: q t = α K M (k t ) α K 1 (p t ) α P = α K x t k t, (4) w t = (1 α K α P )M (k t ) α K (p t ) α P = (1 α K α P ) x t, (5) e t = α P M (k t ) α K (p t ) α P 1 = α P x t p t. (6) Profit maximization implies that the firm s revenues net of the payments to production factors lead to a profit equal to the initial endowment of permits, e t S. This profit is collected by the government and reimbursed to the young households. 3 2 Ono (2002) shows that, by means of rescaling parameters, a production function which has constant returns to scale in using labor and capital as inputs and emission intensity as a scaling factor, can be transformed into a three factor constant returns to scale production function with labor, capital and pollution as inputs. 3 In essence, this particular modeling of the permit system guarantees that the subsidy is non distortionary and it is ensured that permits are not grandfathered. 6

Production in Foreign is specified by a similar constant returns to scale Cobb-Douglas production technology. Thus, total Foreign output, Y t, is determined by labor N t, capital K t and pollution P t. Output in per capita terms is then determined by y t = M(k t )α K (p t )α P. The first order conditions in Foreign are: qt = α K M (kt ) αk 1 (p t) α P yt = α K, (7) kt w t = (1 α K α P )M (k t )α K (p t )α P = (1 α K α P ) y t, (8) e t = α P M (kt ) α K (p t) α P 1 yt = α P. (9) p t 2.2 Intertemporal utility maximization and international asset allocation Household preferences are identical across and within periods and countries. As is standard in Diamond (1965) type OLG models, each generation lives for two periods, one working period and one retirement period. Lifetime utility depends on consumption during the working period, composed of the consumption goods of both countries, x 1 t and y 1 t, which are weighted by expenditure shares ζ and (1 ζ), and consumption during the retirement period, x 2 t+1 and y 2 t+1. The time preference factor is denoted by β, 0 < β 1. For the sake of analytical tractability, household s preferences are represented by a log linear utility function: U t = ζ ln x 1 t + (1 ζ) ln y1 t + β [ ζ ln x 2 t+1 + (1 ζ)ln y2 t+1]. (10) In maximizing intertemporal utility the young household is constrained by a budget constraint in each period of life. When young, wage income w t, net of a lump sum tax τ t imposed by the government, is spent on consumption of the Home good x 1 t and the Foreign good y 1 t, whereby the expenditures on the Foreign good in terms of the Home commodity are equal to y 1 t /h t with h t denoting the external terms of trade of Home (units of Foreign good per one unit of Home good). Furthermore, for transferring income to their retirement period, young households save in terms of capital k t+1 and in terms of bonds of Home b H t+1 and of Foreign b,h t+1 divided by h t. From saving, the old household gains interest 7

income, where i t+1 and i t+1 denote the interest rates in Home and Foreign. When old, the household spends interest income and capital on consumption, again for the Home and Foreign good (x 2 t+1 and y2 t+1, respectively). Thus, the first period budget constraint is given by: where savings are defined as x 1 t + 1 h t y 1 t + s t = w t τ t, (11) s t k t+1 + b H t+1 + (1/h t)b,h t+1. (12) After taking account of the no arbitrage condition for the capital market in Home (1+i t = q t, t), the second period budget constraint is given by: x 2 t+1 + 1 yt+1 2 = (1 + i t+1 ) [ ] ( ) k t+1 + b H t+1 + 1 + i 1 h t+1 b,h t+1 h t+1. (13) t+1 Since government bonds are perfectly mobile across Home and Foreign, the real interest parity condition holds across both countries ( 1 + i t+1 ) h t h t+1 = (1 + i t+1 ). (14) Taking account of (12) and (14), the first and second period budget constraint can be collapsed to the following intertemporal budget constraint for Home: x 1 t + 1 h t y 1 t + x2 t+1 (1 + i t+1 ) + 1 h t+1 y 2 t+1 (1 + i t+1 ) = w t τ t. (15) Since Home and Foreign consumers are specified as having identical preferences, the representative household in Foreign maximizes the following intertemporal utility function Ut = ζ ln x,1 t + (1 ζ) ln y,1 t + β [ ] ζ ln x,2 t+1 + (1 ζ)ln y,2 t+1 (16) given the two budget constraints h t x,1 t + y,1 t + s t = wt τt, (17) h t+1 x,2 t+1 + y,2 t+1 = ( ) ( ) 1 + i t+1 kt+1 + b,f t+1 + h t+1 (1 + i t+1 )b F t+1, (18) where savings are defined by s t k t+1 + b,f t+1 + h t b F t+1. (19) 8

Maximizing (10) and (16) gives the optimal consumption quantities as follows: x 1 t = ζ 1 + β (w t τ t ), x,1 t = ζ (wt τ t ), 1 + β h t (20) yt 1 = 1 ζ 1 + β (w t τ t ) h t, y,1 t = 1 ζ 1 + β (w t τ t ), (21) x 2 t+1 = βζ 1 + β (1 + i t+1)(w t τ t ), x,2 t+1 = βζ 1 + β y 2 t+1 = β(1 ζ) 1 + β (1 + i t+1)(w t τ t ) h t+1, y,2 t+1 = β(1 ζ) 1 + β ( ) 1 + i (wt τt ) t+1, h t+1 (22) ( ) 1 + i t+1 (w t τt ). (23) Reformulating the first period budget constraint for s t and substituting the optimal consumption quantities for x 1 t and y1 t gives s t = σ (w t τ t ), σ β (1 + β). (24) Denote total bonds issued in Home by b t and in Foreign by b t. Then market clearing for Home and Foreign bonds demands b t = b H t + b F t, b t = b,h t + b,f t. (25) To eliminate w t and τ t in (24), we first write down the government budget constraints: τ t + e t S = i t b t, τ t + e t S = i t b t. (26) The left hand side of (26) denotes the revenues from tax income and permit trading, while the right hand side gives the interest payments to the bond holders. Acknowledging the no arbitrage condition for the capital market in Home (1 + i t = q t, t), the market clearing for the permit market in Home (p t = S, t), and substituting for the firm s first order conditions (4) (6) yields an expression for s t which depends only on k t and exogenously given parameters: s t = σ [ (1 α K ) M (k t ) α K (S) α P ( b t αk M (k t ) αk 1 (S) α P 1 )], (27) and, similarly for Foreign, optimal savings s t are a function of k t parameters: and exogenously given s t = σ [ (1 α K )M (k t )α K (S ) α P b t ( αk M (k t )α K 1 (S ) α P 1 )]. (28) 9

2.3 Market Clearing and International Trade To complete the model, further market clearing conditions have to be specified. First, the national product markets have to be cleared. The market clearing condition requires output per capita in Home to be equal to the sum of consumption per capita of working and retired households in Home and in Foreign, and capital per capita in period t + 1 which is built by investments of period t: x t = x 1 t + x 2 t + k t+1 + x,1 t + x,2 t, t, (29) while the product market clearing condition in Foreign reads as follows: yt = y,1 t + y,2 t + kt+1 + y1 t + y2 t, t. (30) Clearing of the world financial capital market requires the supply of savings to be equal to the demand for savings (from (12), (19), and (25)): s t + 1 h t s t = k t+1 + b + 1 h t [ k t+1 + b ], t. (31) Defining the net asset positions of Home and Foreign as φ t+1 k t+1 + b s t, φ t+1 k t+1 + b s t, (32) gives the following relationship between Home s terms of trade and the net asset positions of Foreign and Home: h t = k t+1 + b s t k t+1 + b s t φ t+1 φ t+1, t. (33) Since h t > 0, either φ t+1 > 0 and consequently φ t+1 Foreign a net creditor, or vice versa if φ t+1 < 0 and φ t+1 > 0. < 0, i.e. Home is a net debtor and The balance of payment condition requires that the trade balance and the capital account match for all periods. To establish this result, we solve the budget constraints for young and old households in Home ((11) and (13)) for x 1 t and x 2 t and substitute these expressions into the product market clearing condition for Home (29). Due to constant returns of scale, firm s revenues match its factor payments, x t = q t k t + w t + e t S. Substituting 10

for the factor prices according to (4) (6) and the government budget constraint (26), and applying the international capital market clearing condition (31), the balance of payments condition results: 0 = tb t + φ t+1 (1 + i t )φ t, t, (34) where tb t x,1 t + x,2 t 1 h t (yt 1 + y2 t ) defines the trade balance and φ t+1 = b F t+1 1 h t b,h t+1 gives another expression for Home s net asset position. 2.4 Intertemporal Equilibrium Dynamics In this section, the dynamic equations describing the intertemporal equilibrium of our two country, two good model are presented. Considering the two national no arbitrage conditions of capital markets (1 + i t = q t, 1 + i t = qt, t) and the firms first order conditions (4) and (7) in the international interest parity condition (14), the equation of motion of the terms of trade follows ( ) ( ) 1 + i h t+1 = h t+1 k αk 1 t (1 + i t+1 ) = h t+1 (S ) α P t (k t+1 ) αk 1 (S) α. (35) P By inserting the optimal savings functions (27) and (28) into the international capital market clearing condition (31), we obtain the second equation of motion: h t k t+1 + k t+1 = h t [σ 0 (k t ) α K b (σ i t + 1)] + σ 0 (k t )α K b (σ i t + 1), (36) where σ 0 (1 α K )σms α P and σ 0 (1 α K ) σm (S ) α P. Multiplying the national product market clearing condition of Home (29) by h t and the one of Foreign (30) by ζ/(1 ζ), inserting optimal consumptions of households in Home and Foreign, and subtracting the second from the first, gives the combined product market clearing condition: h t k t+1 ζ (1 ζ) k t+1 = h t M (k t ) α K (S) α P ζ (1 ζ) M (k t ) α K (S ) α P. (37) The dynamic system for the terms of trade, h t, and for the capital stocks in Home and Foreign (k t+1 and kt+1 respectively) are thus described by Equations (35), (36), and (37). 11

To derive the law of motion for the Home net asset position, we solve (34) for φ t+1 : φ t+1 = (1 + i t )φ t tb t. (38) Substituting for the trade balance and using the optimal consumption quantities as determined by (20) (23), we get φ t+1 = (1 + i t )φ t ζ (wt τt ) + (1 + β) h t ζβ (1 + β) (1 ζ) (1 + β) (w t τ t ) ( w t 1 τt 1) h t (1 + i (1 ζ)β t ) + (1 + β) (w t 1 τ t 1 )(1 + i t ). After considering that according to (12) and (35) Home s savings in the previous period are β s t 1 (1 + β) (w t 1 τ t 1 ) = k t + b b F t + (1 + i t ) (1 + i t ) and analogously for s t 1 (see (19)), and taking account of φ t = b F t 1 of the net asset position simplify to b,h t h t h t 1 b,h t, the dynamics φ t+1 = (1 ζ)(w t τ t ) ζ (w t τ t ) +(1 ζ)(1 + i t )(k t + b) ζ (1 + i t )(k t + b ). (39) (1 + β) (1 + β)h t h t Thus, the net asset position in t + 1 becomes independent from its level in the previous period and is determined by the terms of trade h t and the factor prices, which, according to (4) (9), depend on the capital stocks k t and k t. 3 Existence and Stability of Equilibria 3.1 Existence A stationary state of the discrete dynamical system (35), (36) and (37) is defined by (h, k, k ) = (h t, k t, k t ) = ( h t+1, k t+1, k t+1). From (35) we thus get a relationship between k and k : k = Sk, S ( S S ) α P 1 α K. (40) 12

Inserting (40) into (37) at the steady state yields for the steady state terms of trade: h = S ζ 1 ζ. (41) Moreover, having a steady state for the net asset position φ t+1 = φ t = φ, t implies according to (38) that i φ = tb. (42) Thus, at the steady state the trade balance equals interest on net foreign assets. In general, three cases can be distinguished that fulfill (42): (i) in the Golden rule case (i = 0), the trade balance is zero independent of the sign of the net asset position φ, (ii) in the case of underaccumulation of capital in Home (i > 0), Home s trade balance is negative (positive) if and only if Home is a net creditor (debtor), and (iii) in the case of overaccumulation of capital Home s trade balance is negative (positive) if and only if Home is a net debtor (creditor). For finding the steady state value of k, we insert (40) and (41) into (36). Using the parameter ϑ ζb + (1 ζ) S 1 b, the following equation determines k: k + ϑ(1 σ) = MS α P k α K 1 [σ(1 α K )k ϑσα K ]. (43) To get information about the number and characteristics of the solutions of (43), let the parameter vector ω = (α K, α P, β, M, S, ϑ) be an element of the parameter space Ω = [0, 1] 3 R 3 +. Moreover, define k to be the solution of H(k) = MSα P k α K k = 0. Clearly, k = (MS α P ) 1/(1 αk). Finally, rearrange (43) such that it can be written as 0 = F(k, ϑ) with F(k, ϑ) MS α P k αk 1 [σ(1 α K )k ϑσα K ] ϑ(1 σ) k. Thus, the conditions for the existence and the resulting number of stationary states are summarized by the following Proposition: Proposition 1 For any ω Ω there exists ϑ R ++ such that 1. for ϑ < ϑ there are one trivial (k = 0) and two non trivial steady states k L and k H with 0 < k L < k H < k, 13

2. for ϑ = ϑ there are one trivial and one non trivial steady state, and 3. for ϑ > ϑ there is only the trivial steady state. Proof 1 see Appendix A.1. H(k) + k F(k) + k 45 k L κ k H k > k L,H Figure 1: Existence of distinct steady states Figure 1 illustrates the first case of Proposition 1. The graph illustrates the relationship 0 = F (k) as implicitly given by (43). Since this function cuts the 45 line twice, we have two non trivial steady states. Note that the existence condition is the two country analogy to Ono (2002, 82). 3.2 Stability In investigating the stability of the two equilibria, we calculate the eigenvalues λ i and the eigenvectors v i = (vi h, vk i, v i )T, (i = 1, 2, 3) of the Jacobian J (h, k, k ) at the economically interesting steady state (h, k, k ). 4 4 The elements of the Jacobian at the stationary state are given in Appendix A.2. 14

Linearizing our system by means of the Jacobian yields valuable information concerning the dynamical behavior near a steady state. Investigating the eigenvalues and eigenvectors for our model for the case ϑ < ϑ, we find that the lower steady state k L is saddle path unstable (therefore we do not explicitly calculate the eigenvectors for this case), while the higher steady state k H is saddle path stable, i.e. the three eigenvalues are λ 1 > 1, λ 2 (0, 1), and λ 3 (0, 1). 5 This result is summarized in Proposition 2. Proposition 2 For ϑ < ϑ, the eigenvalues λ i and eigenvectors v i = (v h i, vk i, v i )T (i = 1, 2, 3) of the Jacobian are given by and λ 1 = 1 + i > 1 (44) α K λ 2 = α K (0, 1) (45) ( λ 3 = σ(1 α K ) (1 + i) 1 + ϑ ) (0, 1) k = k H (46) k > 1 k = k L v 2 = v 3 = ( h ) T k, 1 + δ, α Sδ (47) K ( S) T 0, 1, (48) where δ = ζb(1 σ + σλ 1). k(α K λ 3 ) Proof 2 see Appendix A.3. To get more information about the analytical structure of the three dimensional equilibrium dynamics around the stable steady state solution, we approximate (35) (37) around (h, k, k ) = ( h H, k H, k,h) as follows: h t+1 k t+1 kt+1 = J (h, k, k ) h t k t kt + (I J (h, k, k )) h k k. (49) 5 If ϑ = ϑ holds, the dynamic system undergoes a saddle node bifurcation at a single steady state. The eigenvalues of the Jacobian evaluated at this steady state are as follows: the first is larger than one, the second eigenvalue equals α K and the third is equal to unity. 15

The general solution of the system of first order linear difference equations (49) takes the following form: h t = h + κ 2 υ2 h (λ 2 ) t + κ 3 υ3 h (λ 3 ) t, (50) k t = k + κ 2 υ2 k (λ 2) t + κ 3 υ3 k (λ 3) t, (51) kt = k + κ 2 υ2 (λ 2 ) t + κ 3 υ3 (λ 3 ) t, (52) where κ i, i = 2, 3 denote constants determined by initial conditions for capital intensities in Home and Foreign, while v i = (υi h, υk i, υ i )T, i = 2, 3 is the eigenvector associated with the eigenvalues λ i, i = 2, 3. From Proposition 2, we know that these eigenvalues are within the unit circle while the eigenvector associated with the eigenvalue larger than unity is excluded from (50) (52) by setting κ 1 = 0. Then, the linearized dynamics along the stable manifold can be derived: Corollary 1 A linear approximation of the three dimensional dynamics evaluated at the larger steady state k H takes the following form: { h t = h 1 + α [ K (k k t ) + (k t ]} k ), (53) k S k t+1 = k t + α K [ 1 αk α K δ(λ 3 α K ) ] (k k t ) + +(1 + α K δ)(λ 3 α K ) (k t k ), (54) [ ] S (1 kt+1 = kt λ3 ) + α K δ(λ 3 α K ) (k kt ) + α K δ α S(λ 3 α K )(k k t ), (55) K whereby k 0 and k 0 are exogenously given, and δ is defined according to Proposition 2. 4 The Effects of Home s Unilateral Permits Reduction Let us now turn to the primary focus of this paper what are the consequences of a unilateral permit reduction? To pursue this objective, we first assume that Home implements a more stringent permit policy (S ) while the permit policy of Foreign remains 16

unchanged at S. We further assume that the shock is unannounced and permanent such that the households and firms cannot act anticipatory prior to the shock (e.g., by adjusting their saving decision). 4.1 Comparative steady state effects of unilateral permits reduction To determine the effects of a marginal reduction of S on the three dynamic variables, we totally differentiate (40), (41), and (43), with respect to S. For k = k H, Proposition 3 states that a permanent decline in the permit volume of Home (S ), leads to an improvement in the equilibrium terms of trade (h ), and a decline in both the equilibrium values of k and k. This can also be seen from (41) where a reduction in S leads to an increase in S and therefore the terms of trade improve. On the other hand, for (43) to hold, the capital stock has to decline: since ϑ decreases along with a decline in S, k has to fall. By inspecting (40), we see that the foreign capital stock is influenced by the increase in S and the fall in k, with the net effect being negative according to Proposition 3 when γ > 0. Thus, capital accumulation is also impaired in Foreign, but the direct S effects are missing. When γ = 0, Foreign is insulated from the effects in Home in that case, the Foreign capital stock is not influenced by a reduction in the permit volume in Home. Proposition 3 An infinitesimal change of S leads to a shift of the equilibrium along the gradient given by where γ ζb(1 σ + σλ 1λ 2 ) k(1 λ 3 ) dh h dk = α P k(1 + γ) 1 α K dk k γ and γ > 0, for k = k H. ds S (56) Proof 3 see Appendix A.4. 17

What are the consequences for the remaining economic variables? A reduction in the capital stocks and in Home s permit volume leads to a reduction in Home s and Foreign s output (x, y ). Factor prices evolve, according to the firms first order conditions, as follows: the prices of permits increase in Home and the real interest rates increase in both Home and Foreign. Due to (14), interest rates are balanced across Home and Foreign: i = i, and the transfers to the households fall. Moreover, the wage rate declines in both countries. As a consequence, in Home both young household s net income and old household s wealth fall. However, since the terms of trade improve, the foreign good becomes relatively cheaper for Home s household. Thus, by evaluating (20) (21) at the steady state, we find that young household s consumption of domestic and foreign goods is reduced, but foreign consumption (y 1 ) falls by less than domestic consumption (x 1 ). Furthermore, from (22) (23) it follows that old household s consumption of the domestic good falls (but less severely than for the young because the rise in interest payments leads to a modest increase in wealth), while the consumption of the foreign good increases (y 2 ). By a similar reasoning for Foreign, young household s consumption declines for both goods (x,1, y,1 ), while old household s consumption of the foreign good increases (y,2 ) and of the domestic good declines (x,2 ). Finally, international trade is affected by a decline in Home s permit volume as follows: exports of Home fall (x,1 + x,2 ), but imports are ambiguous: the young household s consumption of the Foreign good falls, while that of the old households increases. In general, however, we find that imports ((y 1 + y 2 )/h) fall by more than exports and thus the trade balance improves (exports exceed imports). For Home s net asset position, this implies that φ increases while that of Foreign, φ, falls. 4.2 Short term effects of a shock in the permit volume S To investigate the short term effects of a reduction in Home s permit volume S in period t = t 0, recall that the shock is unannounced and permanent. While Home s and Foreign s capital stocks in t 0 remain at the initial steady state value, the terms of trade in the shock 18

period change according to (53) evaluated for the shock period t = t 0 : [ ] h t0 = ĥ k t0 1 + α K ˆk ( S 1 1), (57) where ĥ, ˆk denote the new stationary state values of h and k. Knowing that α K k t0 /ˆk is less than unity and ( S 1 1) is negative and since the new stationary state values ĥ is increased by a reduction in S, h t0 increases, too, but less than the new stationary state value ĥ. Thus, while jumping upwards, h t0 does not overshoot its new stationary state value. To provide an economic rationale for the upward jump of the terms of trade in the shock period, assume on the contrary that h t0 does not respond to the shock. The first order conditions for profit maximization (4) (6) imply that in Home q t0 (or, 1 + i t0 ) and w t0 decrease while e t0 increases. In addition, Home s output x t0 decreases while Foreign s output y t0 remains unchanged. Furthermore, the income of Home s young household, w t0 τ t0, falls. As a consequence, both young household s consumption (x 1 t0, y1 t0 ) and old household s consumption (x 2 t0, y 2 t0) of the domestic and the foreign good fall. By a similar reasoning, in Foreign young household s consumption of the domestic and the foreign good (x,1 t0, y,1 t0 ) are constant, but Foreign s old household consumes slightly more of the Foreign and the Home good: x,2 t0, y,2 t0. Thus, Home s exports (x,1 t0 +x,2 t0 ) increase slightly while its imports (y 1 t0 + y 2 t0) sharply decline and thus Home s trade balance improves. If Home is, e.g., a net creditor (φ t0 < 0), the current account (=trade balance plus interest on net foreign assets) in the shock period improves too, implying that the net asset position becomes more negative Home becomes an even stronger net creditor (see (34)). Since this trend cannot persist, Home s terms of trade have to improve and thus h t0 has to increase in the shock period. Let us now turn to the short term effects on capital accumulation. While the capital stocks are unaffected in the shock period, stocks in t 0 + 1 respond to the S shock. From the three dimensional dynamic system it is apparent, that also k t0+1 and kt0+1 change and that these effects depend mutually on each other and on the terms of trade. Solving (36) and (37) simultaneously for k t+1 and kt+1 and taking the total differentials of Home s 19

and Foreign s capital stocks in t 0 + 1 with respect to S yields: dk t0+1 ds = α PM (k t0 ) α K (S) α P 1 [ζ(1 α K )σ + (1 ζ)]+ + ζ (h t0 ) 2 {M (k t0) α K (S ) α P [1 (1 α K )σ] + b (σi t0 + 1)} dh t0 ds, (58) dk t0+1 ds = (1 ζ)α PM (k t0 ) α K (S) α P 1 [1 (1 α K )σ] (1 ζ) {M (k t0 ) α K (S) α P [1 (1 α K )σ] + b (σi t0 + 1)} dh t0 ds. (59) Investigating (58) at the initial steady state (h, k, k ), we find two opposing effects: the positive direct effect increasing the capital stock (holding h t0 fixed), and the negative one induced by the fall in the terms of trade (dh t0 /ds < 0). However, utilizing the linear approximation of the equilibrium dynamics (54), it is easy to see that the first effect dominates the second since (54) can be reduced to k t0+1 k t0 = (1 λ 3 )(k k t0 ), and (1 λ 3 ) > 0 and dk/ds > 0. Thus, Home s capital stock declines in the post shock period as a consequence of a fall in S. On the other hand, according to (59), in Foreign the direct effect on the capital stock is negative while the terms of trade effect is positive. Using the linear approximation (55), which can be reduced to kt0+1 kt0 = (1 λ 3 )(k kt0), it is apparent that the second effect dominates and thus a fall in S leads to a decline in Foreign s capital stock, too. 5 Welfare Effects of Unilateral Permits Reduction From the previous sections we know that a reduction in Home s permits volume has a positive effect on Home s terms of trade but negative consequences on capital accumulation. Hence, the question remains what the net effect of these two forces is on domestic welfare, and similarly for abroad. We start by analyzing the welfare effects for each country separately before proceeding to a comparison across countries. 20

5.1 Derivation of Home s welfare effects To be able to derive the welfare effect of a shock in S, we define the indirect intertemporal utility function as U(x 1, y 1, x 2, y 2 ) V (w τ, 1+ i, h). From the first order conditions of Home s utility maximization problem we know that U/ y 1 = U/ x 1 (h) 1, U/ x 2 = U/ x 1 (1+i) 1, U/ y 2 = U/ x 1 ((1+i)h) 1. Using these FOCs and collecting similar terms, the welfare effect is given by: dv ds = U {[ (w τ) dk x 1 k ds ] (w τ) + + σ w τ S 1 + i [ (1 + i) k dk ds (1 + i) + S + 1 ζ h ] + dh w τ ds h and a change in Home s intertemporal welfare can thus be decomposed into a change of the net wage rate, of the interest factor, and of the external terms of trade. By computing these derivatives we can sign these partial effects as follows: [ ] (w τ) dk (w τ) + = α P [(1 + i)(k + b)(1 + γ) + (w τ b)] > 0, k ds S S σ w τ [ ] (1 + i) dk (1 + i) + = σγ α P (w τ) < 0, 1 + i k ds S S 1 ζ w τ dh (1 ζ) α P = (w τ) < 0. h h ds (1 α K ) S In line with the results for the case of unilateral fiscal expansion by Ono and Shibata (2005), a country s unilateral permit reduction affects its lifetime utility through three channels: the negative wealth effect, the positive interest effect, and the positive terms of trade effect. The first effect, the wealth effect is caused by a decrease in Home s household lifetime net income (see the discussion in the previous section). The second effect is positive (the interest factor increases) and is caused by the reduced capital accumulation. Ono and Shibata (2005, 223) call this positive effect a foreign asset or intertemporal macroeconomic effect, since this effect cannot appear in static trade models. 6 The third effect is the positive terms of trade effect which is familiar from the comparative steady 6 The foreign asset effect does not play any role in our model since in contrast to Ono and Shibata (2005) in our model the terms of trade in Home are independent of the foreign net asset position of Home (see (41)). }, 21

state analysis above. The total welfare effect of a unilateral reduction in Home s permit volume S is then given by dv ds = α P(1 + β) S(w τ) { (1 + i)k γ [(1 + i)(k + b) σ(w τ)] + (w τ) (1 ζ) α K (1 α K ) }. (60) For the dynamically efficient case where Home is a net debtor (φ > 0), we are able to sign the net welfare effect as unambiguously positive, as summarized in Proposition 4: Proposition 4 Suppose that 1+i 1 (dynamic efficiency). Then, for φ > 0, dv/ds > 0. Proof 4 For φ > 0, i.e. k + b s = φ > 0 we get (1 + i)(k + b) > k + b > s = σ(w t). Therefore, using (60) and acknowledging that (w τ) = ((1 α K )/α K )k bi, dv ds α { } P(1 + β) (1 + i)k (w τ) (1 ζ) φ>0 S(w τ) α K (1 α K ) = α { } P(1 + β) (1 + i)k bi ζ + (1 ζ) > 0. qed S(w τ) α K (1 α K ) For the opposite case of Home being a net creditor, however, the expression in square brackets is generally negative and hence the net welfare effect can be signed unambiguously only for the case of the Golden Rule (denoted by superscript 0), as a special case of dynamic efficiency where (1 + i 0 ) = 1. At the Golden Rule, denoted by 0, the net wage is given by (w 0 τ 0 ) = ((1 α K )/α K )k 0. Then, dv /ds 1+i=1 simplifies to: dv ds = α } P (1 + β) {γ 0 φ 0 + ζ k0. (61) 1+i=1 S (w 0 τ 0 ) The following proposition states that, for Home being a net creditor and presupposing that the capital stock takes its Golden Rule value, dv/ds > 0 if α K is larger than σ 2 (1 α K ) 2. Thus, when the factor elasticity of capital takes a sufficiently large value, a reduction in the permit volume leads to a reduction in Home s intertemporal welfare. α K Proposition 5 Suppose that, in accordance with Proposition 1, 0 < ϑ < ϑ and that there exists a ϑ 0 (0, ϑ) such that (1+i 0 ) = α K M(k 0 ) (α K 1) S α P = 1 (Golden Rule). Then, for φ < 0 and σ 2 (1 α K ) 2 < α K < σ(1 α K ), dv ds = α P 1+i=1 S (1 + β) (1 α K ) ζ [ bα K φ 0 + (k 0 ) 2 (1 λ 0 3) 22 (k 0 ) 2 (1 λ 0 3) ] > 0. (62)

Proof 5 see Appendix A.5. Thus, for dynamic efficiency and Home being a net debtor, as well as for the Golden Rule and Home being a net creditor, a permit reduction in Home leads to a reduction in Home s welfare. For the remaining cases, however, the net welfare effect is determined by the relative strength of the interest effect, the wealth effect and the terms of trade effect. 5.2 Foreign s welfare effects For the consequences of Home s permit policy on Foreign s intertemporal welfare, we find the following effects: { dv ds = U (w τ ) dk y,1 k ds + τ ) σ(w (1 + i ) where (w τ ) dk k ds σ (w τ ) (1 + i ) dk (1 + i ) k ds ζ (w τ ) dh h ds = (1 + i ) dk k ds ζ (w τ ) dh h ds = α P S [(1 + i )(k + b )(1 + γ)] > 0, = σ(1 + γ)α P S (w τ ) < 0, ζ α P (1 α K ) S (w τ ) > 0. and, after acknowledging that i = i in the steady state: dv ds = α P(1 + β) S(w τ ) { γ [(1 + i)(k + b ) σ(w τ )] + ζ (w τ ) (1 α K ) }. }. (63) Since there is no direct effect of the shock in S on (w τ ) and (1+i ), both the wealth effect and the interest effect of Home s unilateral permit policy are smaller (in absolute terms) for Foreign. Moreover, in contrast to Home, if Home reduces S, Foreign s terms of trade, 1/h, deteriorate and thus Foreign s intertemporal welfare is reduced by this effect. However, while the interest effect and the wealth effect are simpler than for Home, the net effect remains again unclear. We are, however, able to sign the welfare effect for the case if Foreign is a net debtor (φ > 0): 23

Proposition 6 Suppose that 1 + i = 1 + i 1 (dynamic efficiency). Then, for φ = hφ > 0, dv /ds > 0. Proof 6 In analogy to the Proof to Proposition 4, for φ < 0, i.e. k + b s = φ > 0, we get (1 + i)(k + b ) > k + b > s = σ(w t ). Therefore, using (63), dv /ds > 0. qed In the opposite case, if Foreign is a net creditor (φ < 0), we have to restrict our result to the case of the Golden Rule again. As stated in Proposition 7, if moreover the factor elasticity of capital is sufficiently large, i.e. α K > σ 2 (1 α K ) 2, then Foreign s welfare effect of a permit reduction in Home is again unambiguously negative. Proposition 7 Suppose that, in accordance with Proposition 1, 0 < ϑ < ϑ and that there exists a ϑ 0 (0, ϑ) such that (1 + (i ) 0 ) = α K M((k ) 0 ) (α K 1) S α P = 1 (Golden Rule). Then, for φ > 0 and σ 2 (1 α K ) 2 < α K < σ(1 α K ), ] dv = α P (1 + β) [ bα K φ 0 + (k 0 ) 2 (1 λ 03 ) ds S (1 α K ) ζ (k 0 ) 2 > 0. (64) (1 λ 0 3) 1+i =1 Proof 7 At the Golden Rule we know that (1 + (i ) 0 ) = 1, and hence ((w ) 0 (τ ) 0 ) = ((1 α K )/α K )(k ) 0. Furthermore, (k ) 0 = hk 0. To derive (64), we substitute for γ 0 = ζb/(k 0 (1 λ 0 3 )) in (63). In analogy to the Proof to Proposition 5, dv /ds 1+i=1 > 0 if φ > 0 and σ 2 (1 α K ) 2 < α K < σ(1 α K ). qed Consequently, even if the foreign country does not implement a more stringent permit policy it looses in terms of economic welfare a result which contradicts the commonly used argument that a permit policy is detrimental only for domestic competitiveness. In an interdependent world economy, not acting and letting others act instead is thus not beneficial, but on the contrary, costly. 24

5.3 Comparing the welfare effects of Home and Foreign To compare the net welfare effect for Home (60) and for Foreign (63), we will investigate the expression separately for φ > 0 and φ < 0. We furthermore restrict the analysis to the case of dynamic efficiency, i.e. 1 + i 1, 1 + i 1. Proposition 8 states that the relative magnitude of the welfare effects in Home compared to Foreign depends on the net asset position of the countries. Proposition 8 Suppose that 1 + i 1 (dynamic efficiency). Then, where dv ds dv ds = α P (1 + i) (1 + β) S (w τ) (65) [ ] (w τ) i γφ 1 + h + b (w τ ) (1 + i) Depending on the sign of φ, two cases emerge: (i) For φ > 0 φ < 0, dv ds < dv ds. (ii) For φ < 0 φ > 0, and the Golden Rule case, where 1 + i = 1 + i = 1, dv ds > dv 1+i=1 ds. 1+i=1 Proof 8 From (33), φ = φ h and furthermore h > 0. To derive (65), we subtract (63) from (60) and utilize that [(1 + i)(k + b) σ(w τ)] = (1 + i)φ + σi(w τ), (w τ) = (1 α K )/α K (1+i)k bi, and analogously for Foreign. In accordance with Propositions 4 7 we distinguish two cases. Case i (φ > 0 φ < 0): Since φ > 0, > 0 and hence dv /ds < dv/ds. Case ii (φ < 0 φ > 0): Since φ < 0 and knowing that in the Golden Rule i = 0, < 0 and hence dv /ds 1+i =1 > dv/ds 1+i=1. qed 25

Consequently, if Home is a net debtor and Foreign a net creditor, the effect of Home s permit policy on Foreign s welfare is certainly smaller than on Home s welfare. On the contrary, if Foreign is a net debtor and for the Golden Rule case, the reverse case emerges with stronger welfare effects for Foreign than for Home. To facilitate the economic rationale for these results, let us investigate the partial effects again. Assume that we are in the Golden Rule and hence the interest rates remain unchanged by a shock in S. Then, the interest effect disappears in the utility differential and the wealth effect is determined solely by the impact of the S shock on the capital stocks. Moreover, in the initial steady state we have S = S and hence k = k. Then, if Home is a net debtor, i.e. φ > 0 and hence b > b, Home s capital demand on the international capital demand (k + b) exceeds that of Foreign s and hence Home s welfare effect exceeds Foreign s. On the other hand, if Foreign is a net debtor, i.e. φ > 0 and hence b < b, Foreign s international capital demand is larger than Home s and hence Foreign s welfare effect exceeds Home s. Moreover, investigating for the Golden Rule gives: ζb = 2 k 0 (1 λ 0 3) φ0. In other words, Home s higher (lower) capital demand is just compensated for by Foreign s lower (higher) capital demand. 6 Conclusion This paper investigates the effects of an unilateral reduction of emission permits on terms of trade, capital accumulation and welfare in a two country, two good OLG model. We first derive the intertemporal equilibrium dynamics of the terms of trade, Home s and Foreign s capital. In Proposition 1 we proof the existence of two distinct steady state solutions. The lower steady state turns out to be saddle path unstable, and the higher steady state is saddle path stable. Regarding a permits reduction in Home, we find that the terms of trade of the domestic 26

economy immediately jump upwards in response to an unilateral permits reduction and continue to increase along the stable manifold towards the new steady state. Alongside with rising terms of trade, capital intensities in both countries go down, but by more in Home than in Foreign. While these effects of unilateral emission permits reduction can be shown analytically and are independent of the net asset position of the countries, the effects on domestic household s welfare is more difficult to decide due to opposing effects. While the terms of trade improvement is welfare enhancing, a more stringent permit policy has welfare consequences via the factor prices, too: the wage rate declines, while the interest rate and the permit price increase. In total, for the dynamically efficient case, the welfare consequences caused by a permit reduction are negative for Home and this gives an economic explanation why climate policy has been implemented with large hesitation. How strongly Home s welfare is reduced, depends on her net asset position, with the decrease of Home s welfare being largest when Home is a net debtor to Foreign. For the special case of the Golden Rule, we are able to show that Home s welfare declines even when Home is a net creditor if the production elasticity of capital is sufficiently large. For Foreign, both the wealth effect and the terms of trade effect caused by Home s permit reduction are negative (since Foreign s terms of trade deteriorate). On the other hand, Foreign s interest effect is positive. The strength of Foreign s welfare effect depends again on her net asset position, with the decrease of Foreign s welfare being largest when Foreign is a net debtor to Home. As for Home, for the special case of the Golden Rule, Foreign s welfare declines even when Foreign is a net creditor and if the production elasticity of capital is not too small. Comparing the strength of the welfare decline caused by a reduction of Home s permits between Home and Foreign, we find that in the case of dynamic efficiency and Home as a net debtor (and consequently Foreign a net creditor), the welfare decline in Foreign to be smaller than in Home. For the opposite case when Home is a net creditor, we are able to show that Foreign s welfare declines by more than Home s welfare in the special case of the Golden Rule. As already claimed in the introduction, when taking account of both the international 27

trade link and of the dynamics of capital accumulation, Home s unilateral permit reduction can affect domestic and foreign welfare in different ways, depending on the net asset position of Home and Foreign. Our case of a stronger welfare effect in Home than in Foreign leads to the conclusion drawn previously by Hoel (1991): unilateral environmental policy harms the domestic economy and thus explains why climate policy has been implemented cautiously. On the other hand, our case of a stronger welfare effect in Foreign than in Home can be seen as a strong international spillover effect. Following the argument of Copeland and Taylor (2005), such a case can be regarded as one where environmental regulations become strategic complements across countries motivating a coordinated climate policy on the global level. By applying these results to the negotiating parties in the climate conferences, helps to better understand the large differences in standpoints across the groups of industrialized countries. There are several roots for improving our model. First, we assumed emission permit policy to be initially equal in both countries, which is an unrealistic simplification. Second, mitigation could be implemented in our model, but this comes at the cost of increasing the dimensionality of the dynamic system. Third, we could compare the welfare effects of the unilateral permit policy to an international permit solution (allowing for trade in emission permits across countries). References Chen, Zhiqi (1992). Long run equilibria in a dynamic heckscher ohlin model. Canandian Journal of Economics 25(4), 923 43. Copeland, Brian R. (1994). International trade and the environment: Policy reform in a polluted small open economy. Journal of Environmental Economics and Management 26, 44 65. Copeland, Brian R. and M. Scott Taylor (2005). Free trade and global warming: a trade 28