A Note on Correlated Uncertainty and Hybrid Environmental Policies

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1 This version: April 2014 A Note on Correlated Uncertainty and Hybrid Environmental Policies John K. Stranlnd* Department of Resorce Economics University of Massachsetts, Amherst Abstract: This note examines the effects of correlated ncertainty between abatement costs and polltion damage on hybrid emissions control policies. In particlar, I show how correlated ncertainty affects the strctre and performance of control policies that combine permit trading with price controls. Moreover, I show how correlated ncertainty affects the choice between a simple emissions tax verss a hybrid policy. Under correlated ncertainty, constant marginal damage is no longer necessary or sfficient for a tax to be an optimal policy. Keywords: Cap-and-trade, correlated ncertainty, emissions trading, emissions taxes, hybrid policies, price controls JEL Codes: H23, L51, Q58 *Correspondence to John K. Stranlnd, Department of Resorce Economics, 222 Stockbridge Hall, 80 Camps Center Way, University of Massachsetts-Amherst, Amherst, MA 01003, USA. Phone: (413) , stranlnd@resecon.mass.ed. This research has been spported by a grant from the U.S. Environmental Protection Agency s National Center for Environmental Research (NCER) Science to Achieve Reslts (STAR) program. Althogh the research described in the article has been fnded wholly or in part by the U.S. EPA throgh grant nmber RD , it has not been sbjected to the EPA s peer and policy review and therefore does not necessarily reflect the views of the Agency, and no official endorsement shold be inferred. Additional fnding for this research was provided by the Cooperative State Research Extension, Edcation Service, U. S. Department of Agricltre, Massachsetts Agricltral Experiment Station nder Project No. MAS

2 A Note on Correlated Uncertainty and Hybrid Environmental Policies 1. Introdction Analysts and policymakers have devoted mch attention in recent years to the design and evalation of hybrid emissions control policies, in particlar emissions permit trading with price controls. Shortly after Weitzman s (1974) seminal analysis of the reglatory choice between price and qantity instrments, Roberts and Spence (1976) showed that combining price and qantity instrments can be more efficient than either instrment alone. Under a pre price scheme, ncertainty abot abatement costs prodces ncertainty abot emissions, while pre emissions trading schemes prodce ncertainty abot emissions prices. A policy of emissions trading with price controls recognizes that it is often optimal to prodce a balance between emissions and price ncertainty. This note contribtes to the literatre on policy instrment choice nder ncertainty by considering how correlated abatement cost and polltion damage ncertainty affects the strctre and performance of hybrid environmental policies. Althogh environmental economists often ignore the possibility of correlated ncertainty in abatement costs and damages, sing a reslt from Weitzman (1974), Stavins (1996) showed how correlated ncertainty cold affect the choice of an emissions trading program vs. an emissions tax. A positive correlation between abatement cost and damage ncertainty tends to favor emission trading, while a negative correlation tends to favor an emissions tax. (Also see Shrestha 2001). Stavins provides several examples of possible positive correlation between abatement costs and damage driven by random weather shocks. For example, grond-level ozone if formed from nitrogen oxides and volatile organic componds in the presence of snlight. Ths, controlling concentrations of ozone is more costly when it is snny. Bt these are also the times when people prefer to be otside exerting themselves, so the damage from ozone exposre is also higher when it is snny. In the area of mitigating climate change, with a global integrated assessment model Pizer (1999) arges that the costs and benefits of CO2 control at this scale are more likely to be negatively correlated. In contrast, Qirion (2010) sggests that the costs 2

3 and benefits of controlling CO2 for a contry operating nder a global climate change treaty may be positively correlated. 1 In this note I show that correlated ncertainty affects the strctre of a hybrid tax/trading policy in intitive ways. The optimal hybrid policy incorporates a positive correlation between abatement costs and damage by increasing the likelihood that the permit spply binds and decreasing the likelihood that one of the price controls binds. On the other hand, negatively correlated ncertainty increases the likelihood that the price controls bind and redces the likelihood that the permit spply binds. Ths, an optimal hybrid policy reflects the relative preferences for emissions markets or taxes when abatement costs and damage are correlated. In all cases, however, hybrid policies prodce the optimal second-best emissions price and qantity; that is, these policies prodce the expected price and expected emissions that minimize expected social costs ex ante. 2 Interest in hybrid policies has grown rapidly in recent years becase of their proposed se in climate change reglation. 3 An interesting aspect of the literatre on hybrid policies in this context is that they may not be more efficient than a simple emissions tax. An important characteristic of the control of greenhose gases is that the expected marginal damage fnction in a relatively short compliance period (say one year) is estimated to be almost perfectly flat (e.g., Pizer 2002). This is mainly becase greenhose gases are stock polltants and their emissions in any one year do not 1 Qirion s (2010) argment rns as follows. Consider an international treaty that specifies contry-level emissions qotas like the Kyoto Protocol. A competitive international market for emissions permits exists, bt a contry may choose to implement a tax or tradable permits domestically. If a contry is a price-taker in the international permit market, its marginal benefit of complying with its qota is the international permit price. Moreover, this marginal benefit is constant, yet ncertain. Qirion sggests that since that contries abatement costs are determined by mch the same factors, if the marginal abatement cost in one contry is lower than expected, it is likely that many other contries marginal abatement costs are lower as well, which wold reslt in a lower global emissions price. In this way, a contry may find its marginal costs and benefits of complying with an international greenhose gas treaty to be positively correlated. 2 A first-best policy wold minimize aggregate abatement costs and damage ex post, that is, after ncertainty abot costs and damage is resolved. All the policies in this paper are determined before ncertainty is resolved, so they cannot reslt in the first-best otcome, except by accident. This is also tre of the vast majority of the literatre on instrment choice nder ncertainty. 3 Earlier proposals for controlling greenhose gas emissions only involved adding a price ceiling to trading programs, so-called safety valves (Jacoby and Ellerman 2004). However, a price floor along with a price ceiling, a so-called price collar, can be more efficient becase another instrment is available (Brtraw et al. 2010, Fell and Morgenstern 2010, Philibert 2008). Recent contribtions to this literatre also inclde Fell et al. (2012), Grll and Taschini (2011), and Weber and Nehoff (2011). For examples of price controls in actal and proposed greenhose gas markets see U.S. Congressional Bdget Office (2010) and Newell et al. (2013, pp ). 3

4 add mch to their accmlated stock. In this context, and with other common assmptions inclding ncorrelated ncertainty between abatement costs and damage, the optimal control policy is a pre emission tax adding a qantity restriction that has a non-zero chance of binding necessarily redces expected welfare (Webster et al. 2010). Given the importance of greenhose gas emissions control in motivating the literatre on hybrid control policies, I revisit the notion that a simple emissions tax is the optimal policy when the marginal damage fnction is flat. Correlated ncertainty between abatement costs and polltion damage modifies this reslt; in particlar, a flat marginal damage fnction is no longer necessary or sfficient for an emissions tax to dominate a hybrid policy. A hybrid policy is the optimal choice when the marginal damage fnction is flat and ncertainty in abatement costs and damage is positively correlated. Moreover, it is possible that an emissions tax is more efficient than a hybrid policy when marginal damage is an increasing fnction if abatement costs and damage are negatively correlated. 2. A hybrid policy: emissions trading with price controls The model of a hybrid emissions control policy in this paper is very close to the one proposed by Roberts and Spence (1976), with the addition of correlated damage and abatement cost ncertainty. Sppose that there are n firms that emit a niformly mixed polltant. 4 Firm i s emissions are q i and aggregate emissions are Q = n i=1 qi. Firm i s cost of controlling its emissions is given by the qadratic abatement cost fnction, c i (q i,) = a i 0 ( a i 1 + ) q i + ai 2 2 ( q i ) 2, (1) 4 The model is also close to Stranlnd and Moffitt (2014). In fact, the notation is the same and some of the reslts and proofs are similar. The difference is that Stranlnd and Moffitt focs on the role that enforcement costs play in determining hybrid policies and do not consider correlated ncertainty. This note sets aside the enforcement problem to highlight the role of correlated ncertainty in determining hybrid reglations. 4

5 where a i 0, ai 1 and ai 2 are positive constants. Random shocks that affect the abatement costs of all firms are captred by changes in. The emissions trading program with price controls has the following featres. L permits are distribted to the firms (free-of-charge) and firm i chooses to hold l i permits. Permits trade at the competitive price p. The government commits to selling additional permits beyond L at price t, and it commits to bying permits from firms at price s. For the permit market to clear it mst be tre that s p t. If p > t, then firms wold not trade permits, choosing instead to by additional permits from the government. If p < s, then the firms wold demand an nlimited nmber of permits in order to sell them back to the government. Note that t provides a price ceiling for permits, while s provides the price floor. The timing of events in the model is as follows. First, the government chooses and commits to all the elements of the policy, that is, the permit spply, and the price ceiling and floor. Uncertainty in abatement costs and polltion damage (to be specified shortly) is resolved after the policy is determined. The firms then choose their emissions and permit holdings, and the permit market clears. Roberts and Spence showed that this scheme retains the well-known cost-effectiveness property associated with emissions taxes and competitive emissions trading. That is, the distribtion of individal emissions minimizes the aggregate abatement costs of the aggregate level of emissions that reslts in eqilibrim. Moreover, the competitive eqilibrim price and aggregate emissions can be characterized by the eqality of the permit price and the minimm aggregate marginal abatement cost fnction. In sm: p = C Q (Q,), p [s,t], (2) where C(Q,) = min n {q i } n i=1 i=1 c i (q i,), sbject to n i=1 q i = Q. (3) As is common in the literatre on policy choices nder ncertainty, assme that (3) takes the 5

6 qadratic form, C(Q,) = c 0 (c 1 + )Q + Q 2 2, (4) where c 0, c 1, and, are all positive constants. See Yates (2012) for a demonstration of how an aggregate abatement cost fnction of this form is generated from qadratic individal abatement costs. Since the polltant is niformly mixed, polltion damage depends on aggregate emissions, not the geographical distribtion of emissions. Let the damage fnction take the qadratic form, D(Q,δ) = (d 1 + δ)q + d 2Q 2 2, (5) where d 1 and d 2 are positive constants and δ is a random variable. Let g(,δ) be the joint probability density fnction of and δ. The spport of is [,] and the spport of δ is. The nconditional expectation of both and δ is zero. Or analysis involves evalating conditional expectations of both random variables when the domain of is restricted at the top, at the bottom, and both top and bottom. Let E denote the expectation operator throghot the paper. Since E() = 0, then for some vale 0 <, E( 0 ) < 0. Likewise, for 0 >, E( 0 ) > 0. Moreover, limiting the domain of changes the expectation of δ if and δ are correlated. For example, if and δ are positively correlated then it is reasonable to assme that limiting the domain of at the top wold redce the conditional expectations of both and δ. That is, if and δ are positively correlated then E(δ 0 ) < 0 for 0 <, and E(δ 0 ) > 0 for 0 >. Likewise, if and δ are negatively correlated then E(δ 0 ) > 0 for 0 <, and E(δ 0 ) < 0 for 0 >. E(δ) = 0 regardless of restrictions on if and δ are independent. From (2) and (4), the eqilibrim permit price is p = c 1 + Q. (6) 6

7 The price controls limit the range of possible permit prices. Define s and t, sch that s is the vale of at which both the price floor and the permit spply bind and t is the vale of at which both the permit spply and the price ceiling bind. That is, s and t are defined by s = c 1 + s L and t = c 1 + t L, respectively, reslting in s = s + L c 1 (7) and t = t + L c 1. (8) For vales of < s the price floor binds and the permit price is eqal to s. For vales of between s and t, the permit spply binds and the permit price is determined by the aggregate marginal abatement cost fnction evalated at L, that is, p(l,) = c 1 + L. Vales of above t case the price ceiling to bind so the permit price is eqal to t. The following is the eqilibrim permit price schedle: p = t for [ t,] p(l,) = c 1 + L for [ s, t ] (9) s for [, s ]. Eqilibrim aggregate emissions are determined from (6) and (9): Q = Q(t,) = (c 1 + t)/ for [ t,] L for [ s, t ] Q(s,) = (c 1 + s)/ for [, s ]. (10) Assme that aggregate emissions are strictly greater than zero in all cases. The optimal hybrid policy is the triple (L,t, s) that minimizes the expected social costs of polltion and its control. Letting W(L,t, s) denote expected social costs, the reglatory objective 7

8 is to choose L, t, and s to minimize: W(L,t,s) = { t + + s s t [C(Q(t,),) + D(Q(t,),δ)]g(,δ)d [C(L,) + D(L,δ)]g(,δ)d [C(Q(s,),) + D(Q(s,),δ)]g(,δ)d } dδ, (11) sbject to (4), (5), (7), (8), (10), t s, s, and t. Assme that W(L,t,s) is strictly convex so that the standard first order conditions identify a niqe optimal policy. The model incldes policies with fewer instrments as special cases. For example, if the constraint t s binds, then the optimal policy is a pre tax. 5 As another possibility, if the constraints s and t bind together, then there is no chance that the price controls will bind, so the optimal policy is a pre trading scheme. In characterizing the optimal hybrid policy it is sefl to specify the second-best optimal vales of aggregate emissions and price as benchmarks. These vales minimize ex ante expected aggregate abatement costs and polltion damage. 6 That is, second-best emissions minimize E[C(Q,)+D(Q,δ)] and the second-best price minimizes E[C(Q(p,),)+D(Q(p,),δ)], given (4) and (5) and Q(p,) = (c 1 + p)/ from (6). The second-best emissions and price are, respectively, Q = (c 1 d 1 )/( + d 2 ), (12) p = (d 1 + c 1 d 2 )/( + d 2 ). (13) A pre emissions trading program optimally distribtes Q permits and their expected price is p, while the optimal emissions tax is p, which reslts in expected emissions Q. 5 Becase the model involves a fixed nmber of firms and tax receipts and sbsidy payments are transfers with no real effects, it cannot distingish between a pre sbsidy and a pre tax. However, becase a sbsidy wold be inferior to a tax in an extended model, I assme that if the optimal policy is a pre price scheme, it is implemented with a tax. 6 As noted in the introdction, these vales are second-best becase they are not generally optimal after the ncertainty has been resolved. 8

9 3. The strctre of a hybrid policy While the reglatory model can also prodce a pre tax, a pre trading scheme, and a trading scheme with a one-sided price control like a safety valve, in this section sppose that the optimal policy is one for which there are positive probabilities that each of the instrments will bind. (This reqires that each of the constraints on the reglatory problem, t s, s and t, do not bind). Given this assmption, the first proposition characterizes the optimal policy. The proof is in the appendix. Proposition 1: Under an optimal hybrid policy with non-zero probabilities that the permit spply, the price floor, and the price ceiling will bind: L = Q + E( s t ) E(δ s t ) + d 2 ; (14) t = p + d 2E( t ) + E(δ t ) + d 2 ; (15) s = p + d 2E( s ) + E(δ s ) + d 2. (16) In Proposition 1, s and t are evalated at (L,t,s ). (14) throgh (16) are not soltions for the optimal policy variables becase these variables appear on the right sides of these eqations. However, they are sefl for developing intition abot the strctre of the optimal hybrid policy, with the second-best emissions and permit price as benchmarks. Sppose that the marginal damage fnction is pward sloping (d 2 > 0). Since E() = 0, if there is a non-zero probability that the price ceiling binds (i.e., t < ), then the conditional expectation of given that the ceiling binds is strictly greater than zero (i.e., E( t ) > 0). Hence d 2 E( t ) > 0 serves to psh the price ceiling above the second-best price. Similarly, if there is a non-zero chance that the price floor binds (i.e., s > ), then the conditional expectation of given that the floor binds is strictly less than zero, and d 2 E( s ) < 0 serves to psh the price floor below the 9

10 second-best price. Correlated ncertainty between damage and abatement cost affects the strctre of the optimal hybrid policy in intitive ways. Recall that in the choice of pre emission trading or a pre emission tax, a positive correlation between abatement costs and damage tends to favor emission trading, while a negative correlation tends to favor an emissions tax. This insight has an analoge in the determination of hybrid policies. When and δ are positively correlated the conditional expectation of δ given that the ceiling binds is strictly greater than zero (E(δ t ) > 0), and the conditional expectation of δ given that the floor binds is strictly less than zero (E(δ s ) < 0). Conseqently, the price ceiling is higher and the price floor is lower. Pshing the price controls away from each other increases the likelihood that the permit spply binds and redces the likelihood that either of the price controls bind. This reflects the relative preference for qantity control when abatement costs and damage are positively correlated. When and δ are negatively correlated, then E(δ t ) < 0 and E(δ s ) > 0, and the price ceiling and price floor are pshed closer together, thereby increasing the likelihood that they bind instead of the permit spply. This reflects the relative preference for prices when abatement costs and damage are negatively correlated. Let s now examine how the expected emissions and permit price prodced by the optimal hybrid policy compare to their second-best vales. Expected aggregate emissions and the expected price nder (L,t,s ) are: E(Q) = { s E(p) = Q(s,)g(,δ)d + L t {s s s g(,δ)d + t g(,δ)d + p(l,)g(,δ)d +t s t Q(t,)g(,δ)d t g(,δ)d } } dδ; (17) dδ, (18) where again s and t are evalated at (L,t,s ), and p(l,) is defined by (9). The following proposition is proved in the appendix. Proposition 2: Under an optimal trading program with price controls, the expected aggregate emissions and permit price are eqal to their second-best optimal vales, Q and p. 10

11 Recall that an optimal pre tax and an optimal pre trading program also prodce the second best emissions and price. Proposition 2 indicates that this reslt extends to hybrid policies. Moreover, it is straightforward to show that this reslt also holds for trading programs with one-sided price controls. It is important to realize that a wide variety of optimally-chosen policies (i.e., pre taxes, pre trading, trading with one-sided price controls, and trading with two-sided price controls) will all prodce the same expected otcome nder a wide variety of settings (i.e., asymmetric and symmetric ncertainty, correlated and ncorrelated abatement costs and damage). 4. Correlated ncertainty and the optimality of a pre emissions tax To complete the analysis, let s examine how correlated ncertainty affects the choice between an emissions tax and a hybrid policy, paying particlar attention to the optimal policy when the marginal damage fnction is flat (i.e., d 2 = 0). As noted earlier, the literatre on price controls for permit trading programs has expanded in recent years, largely becase of debates abot how to control greenhose gas emissions. An important characteristic of greenhose gas control is that the marginal damage fnction in a particlar time period is essentially flat, and it is widely thoght that a flat marginal damage fnction calls for control with an emissions tax. In fact, a hybrid trading program with price controls in this sitation is less efficient (Webster et al. 2010). I now revisit this conclsion when abatement costs and damage may be correlated. Whether the optimal policy in a particlar sitation is a hybrid or a pre tax can be determined from (14) throgh (16). If the system implies t > s then the optimal policy is a hybrid, bt if the system implies t s then the optimal policy is a simple tax. Sbtracting (16) from (15) yields: sign(t s ) = sign ( d 2 { E( t ) E( s ) } + { E(δ t ) E(δ s ) }). (19) As already noted, E( t ) > 0 and E( s ) < 0; hence, the first term on the right side of (19), d 2 {E( t ) E( s )}, is greater than zero if d 2 > 0 and is eqal to zero if d 2 = 0. Therefore, in the absence of correlated ncertainty, the optimal policy is a hybrid if marginal 11

12 damage is increasing, while the optimal policy is a pre tax if marginal damage is constant. Now sppose that the ncertainty in damage and abatement cost is correlated. Recall that E(δ t ) > 0 and E(δ s ) < 0 when damage and abatement costs are positively correlated. Hence, given > 0 and and δ are positively correlated, {E(δ t ) E(δ s )} > 0, implying that the optimal policy is always a hybrid nder positively correlated damage and abatement cost ncertainty, even if marginal damage is constant. Under a negative correlation, E(δ t ) < 0 and E(δ s ) > 0, and {E(δ t ) E(δ s )} < 0. Hence, negatively correlated damage and abatement costs tends to favor a pre tax over a hybrid policy. In fact, there are circmstances nder which a tax is preferred to a hybrid even if marginal damage is increasing. The last proposition of the paper collects these insights into the choice of a hybrid emissions control policy verss an emissions tax nder correlated ncertainty. Proposition 3: When ncertainty abot abatement cost and damage is correlated, it is neither necessary nor sfficient that marginal damage be constant for an emissions tax to be the optimal policy. In fact, given a constant marginal damage fnction, a hybrid policy is more efficient than a simple emissions tax if damage and abatement costs are positively correlated. Moreover, when marginal damage is increasing, an emissions tax may be more efficient than a hybrid policy if damage and abatement costs are negatively correlated. Figres 1 and 2 illstrate the effects of emissions trading with price controls when marginal damage is constant. In both graphs the optimal tax τ is set eqal to the constant expected marginal damage, E(D Q ). The alternative hybrid policy consists of a permit spply L set eqal to the secondbest optimal level of emissions where E( C Q ) = E(D Q ). The price controls, s and t, are symmetric arond E(D Q ), althogh this featre is not necessary for this illstration. In both graphs, a random shock reslts in a higher than expected aggregate marginal abatement cost fnction, CQ 0. In Figre 1, ncertainty abot abatement cost and damage is negatively correlated so the shock also prodces a lower than expected marginal damage, D 0 Q. In Figre 2, ncertainty is positively correlated so D 0 Q is higher than expected. 12

13 Before considering the role of correlated ncertainty in choosing between a tax and a hybrid, first note the inefficiency of a hybrid policy when damage and abatement costs are ncorrelated. In Figre 1, if the shock that moves the aggregate marginal abatement cost fnction to C 0 Q does not move expected marginal damage becase they are ncorrelated, the optimal level of emissions is Q 0 τ, which is exactly what the firms will choose nder the tax τ. However, emissions nder the hybrid policy are limited to Q 0 h, reslting in a loss of area(a). The hybrid policy is less efficient becase it prevents the optimal adjstment of emissions to the change in aggregate marginal abatement costs. A negative correlation between damage and abatement costs exacerbates the inefficiency of a hybrid policy when expected marginal damage is flat. Again in Figre 1, if damage and abatement costs are negatively correlated, then the shock that increases marginal abatement costs to C 0 Q simltaneosly decreases marginal damage to D 0 Q. The optimal otcome is then Q0. Since the hybrid policy limits emissions to Q 0 h, the loss nder this policy is area(a+b+c). Note that this loss is greater than the loss from the hybrid when abatement costs and damage are ncorrelated. The emissions tax τ is more efficient than the hybrid when costs and damage are negatively correlated, becase the random shock cases firms to increase their emissions to Q 0 τ > Q 0 h, prodcing a smaller loss eqal to area(c). On the other hand, a positive correlation between damage and abatement costs makes a hybrid policy more efficient than a pre tax. In Figre 2, the random shock increases both marginal abatement cost and damage to C 0 Q and D0 Q, respectively, and the optimal otcome is Q0. However, nder the tax, firms respond to the shock to abatement costs by increasing their emissions to Q 0 τ. The loss, relative to Q 0, is eqal to area(d + e + f ). Since the hybrid policy limits the emissions increase to Q 0 h < Q0 τ, it limits the ex post loss to area(d + e), demonstrating that a hybrid policy is more efficient than a tax in this sitation. 13

14 Figre 1: Comparison of a hybrid policy and an emissions tax when expected marginal damage is constant and abatement costs and damage are negatively correlated Figre 2: Comparison of a hybrid policy and an emissions tax when expected marginal damage is constant and abatement costs and damage are positively correlated 14

15 5. Conclsion I have revisited the problem of how correlated damage and abatement cost ncertainty affects the optimal choice of policy instrments to control polltion. Previos work has focsed on the choice between a simple emissions tax and a simple emissions trading program, finding that a positive correlation favors emissions trading while a negative correlation favors an emissions tax. In this note I have extended the analysis to inclde hybrid policies, in particlar, emissions markets with price ceilings and floors. A positive correlation between damage and abatement costs tends to psh price controls apart, increasing the likelihood that the permit spply binds. A negative correlation pshes the price controls together, increasing the likelihood that one of them binds. Regardless of the strctre of correlated ncertainty between damage and abatement costs, an optimal hybrid policy prodces the second-best expected emissions and emissions price. Finally, a constant marginal damage fnction, which is so important in the design of policies to control of greenhose gas emissions, is neither necessary nor sfficient for an emissions tax to be the optimal control policy nder correlated damage and abatement cost ncertainty. It is well-known that a hybrid policy is less efficient than a pre tax when damage and abatement costs are ncorrelated. This inefficiency is exacerbated if there is a negative correlation between damage and abatement costs. However, if damage and abatement costs are positively correlated, an optimal hybrid policy will be more efficient than a simple emissions tax. 15

16 Appendix Proof of Proposition 1: To begin the proof, first note the following: Q(t, t ) = L from (10) and (8); Q(s, s ) = L from (10) and (7); p(l, t ) = t from (9) and (8); p(l, s ) = s from (9) and (7); C Q (Q(t,),) = t and C Q (Q(s,),) = s from (2) and (10). Using these facts, the first order conditions for (11), given t > s, t <, and s >, can be written as: W L (L,t,s) = W t (L,t,s) = W s (L,t,s) = t s To derive (14), first se (4), (5), and (12) to show s [C Q (L,) + D Q (L,δ)]g(,δ)ddδ = 0; (20) t [( t + D Q (Q(t,),δ))Q t ]g(,δ)ddδ = 0; (21) [( s + D Q (Q(s,),δ))Q s ]g(,δ)ddδ = 0. (22) C Q (L,) + D Q (L,δ) = ( + d 2 )(L Q) ( δ). (23) Sbstitte (23) into (20) to obtain t t W L (L,t,s) = ( + d 2 )(L Q) g(,δ)ddδ ( δ)g(,δ)ddδ = 0. (24) s s Note the following: E( s t ) = E(δ s t ) = t t t s g(,δ)ddδ s g(,δ)ddδ ; s δg(,δ)ddδ t s g(,δ)ddδ. (25) Use (25) to rearrange (24) to obtain (14). To derive (15) sbstitte Q(t,) = (c 1 + t)/ from (10) into D Q (Q(t,),δ) = d 1 + δ + 16

17 d 2 Q(t,) to obtain D Q (Q(t,),δ) = d 1 + δ + d 2(c 1 t) + d 2. (26) (13) and (26) yield t + D Q (Q(t,),δ) = + d 2 ( t + p) + d 2 + δ. (27) Sbstitte (27) and Q t = 1/ into (21) to obtain W t (L,t,s) = Note the following expectations: t [ c2 + d 2 ( ) 2 (t p) d 2 ( ) 2 δ ] g(,δ)ddδ = 0. (28) E( t ) = E(δ t ) = Rearrange (28) sing (29) to obtain (15). t g(,δ)ddδ t g(,δ)ddδ. t δg(,δ)ddδ t g(,δ)ddδ. (29) The derivation of (16) from (22) is very similar. Sbstitte Q(s,) = (c 1 + s)/ from (10) into D Q (Q(s,),δ) = d 1 + δ + d 2 Q(s,) to obtain D Q (Q(s,),δ) = d 1 + δ + d 2(c 1 s) + d 2. (30) Use (13) and (30)to show: s + D Q (Q(t,),δ) = + d 2 ( s + p) + d 2 + δ. (31) Sbstitte (31) and Q s = 1/ into (22) to obtain W s (L,t,s) = s [ c2 + d 2 ( ) 2 (s p) d 2 ( ) 2 δ ] g(,δ)ddδ = 0. (32) 17

18 With the following: rearrange (32) to obtain (16). E( s ) = E(δ s ) = s f ()d s f ()d ; s δ f ()d s f ()d, (33) Proof of Proposition 2: Begin the proof by sbstitting Q(t,) = (c 1 + t )/ and Q(s,) = (c 1 + s )/ from (10) into (17) and p(l,) = c 1 + L from (9) into (18) to obtain: { s ( c1 + s t ( E(Q) = )g(,δ)d + L c1 + t ) } g(,δ)d + g(,δ)d dδ s t t ( = L c1 s ) s ( ) 1 s g(,δ)ddδ + g(, δ)ddδ + f g(,δ)ddδ s ( c1 t ) ( ) 1 + g(,δ)ddδ + g(,δ)ddδ; (34) t t E(p) = s {s g(,δ)d + = (c 1 L ) +t t Define variables z and w sch that: t s g(,δ)ddδ + s (c 1 + L )g(,δ)d +t t s t g(,δ)ddδ + s g(,δ)d s } dδ g(,δ)ddδ t g(,δ)ddδ. (35) E(Q) Q z = 0; (36) E(p) p w = 0. (37) 18

19 In addition, define the following: A = G = J = s s s g(,δ)ddδ, B = g(,δ)ddδ, H = δg(,δ)ddδ, K = g(,δ)ddδ, F = s t g(,δ)ddδ, I = s t δg(,δ)ddδ, M = s t t t g(,δ)ddδ, g(,δ)ddδ, t δg(,δ)ddδ. (38) Note that A + B + F = 1 and G + H + I = 0, and J + K + M = 0. Combine (34), (35), and (38) to write (36) and (37) as: ( c s L ) B + A + ( c t ) F + ( G + I ) Q z = 0; (39) (c 1 L )B + H + s A +t F p z = 0, (40) respectively. Now se (38), (25), (29), and (33) to write (14), (15), and (16) as L = Q + (H/B) + (K/B) + d 2 ; (41) t = p + d 2(I/F) + (M/F) + d 2 ; (42) s = p + d 2(G/A) + (J/A) + d 2. (43) Treating all the terms in (38) as constants and noting that p and Q are constants, (39) throgh (43), sbject to A + B + F = 1, G + H + I = 0, J + K + M = 0, Q = (c 1 d 1 )/( + d 2 ), and p = (d 1 + c 1 d 2 )/( + d 2 ), form a system of five linear eqations in five variables, L, t, s, z, and w. The soltion to this system retrns (41), (42), and (43), as reqired. In addition, z = w = 0, which implies E(Q) = Q and E(p) = p. The proof is complete. 19

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