Appendix (For Online Publication) Community Development by Public Wealth Accumulation

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1 March 219 Appendix (For Online Publication) to Community Development by Public Wealth Accumulation Levon Barseghyan Department of Economics Cornell University Ithaca NY Stephen Coate Department of Economics Cornell University Ithaca NY

2 Appendix : Properties of H( ) and W() Recall from Section 5, that H( )wasdefined to be the largest housing level in the interval [ 1] that satisfies the equality (1 ) + ()+ (1 ) (1 ) = As noted, H( ) will not be well-defined if is so large that all types of households would get a positive benefit from living in the community or if is so small (i.e., sufficiently negative) that there is no population size at which residents would be willing to pay a price of housing to live in the community. To obtain the upper bound on,notefirstthatif (1) + (1 ) (1 ) all types would get a positive benefit from living in the community. Thus, the upper bound on is + (1 ) (1) 1 To obtain the lower bound on, consider the problem max ½ (1 ) (1 ) + ()+ ¾ (1 ) Let the solution be denoted ( )andlet be the wealth level such that (1 ( )) + (( )) + (1 ) ( ) (1 ) = The lower bound is. To characterize it in a more useful way, note that the first order condition for ( )is which implies that It follows that (1 ) ( ) + () (1 ) 2 = = ( )+( ) (( )) (1 ) (1 ( )) +(( ))+ (1 ) =(1 2( )) +(( ))+( ) (( )) (1 ) ( ) Thus,ifwedefine the housing level as the solution to: (1 ) + ()+ () (1 ) = + 1

3 then the lower bound can be written as = (1 ) + () (1 ) 1 On the interval [ ], it is straightforward to verify that H 1 ( )= (1 2H( )) + (H( )) + H( ) (H( )) (1 ) (A1) This is positive because the denominator must be positive given Assumption 1(i) and the fact that H( ).Moreover, H ( )= (1 ) 2 +2 (H( )) + H( ) (H( )) H ( ) (1 2H( )) + (H( )) + H( ) (H( )) (1 ) 2 which is negative by Assumption 1(i). Thus, H( ) is increasing and concave on the interval [ ]. Because H( ) is increasing on the interval [ ], it has an inverse, W(), definedonthe interval [H( ) H( )] = [ 1]. Note that the first inequality of Assumption 2 implies that which means that W() is well-definedontheentireinterval[ 1]. In light of the properties of H( ), W() will be increasing and convex. Note for future reference that W () = (1 2) + ()+ () (1 ) 1 (A2) 2

4 Appendix 1: Proof of Propositions 2, 3, and 4 Since Proposition 3 is a generalization of Proposition 2, it suffices to just prove Propositions 3 and 4. Both these results concern the nature of the solution to the problem faced by the period residents. It is helpful to begin with a clear statement of that problem. Statement of the problem The period residents problem is P n i h ³ io = () (1 ) h (1 ) ( +1 ) max { } = for all periods ³ (1 + ) = ³ =(1 +1 ) + +1 (1 ) ( +1) (=if +1 ) (A3) The first constraint is the budget constraint. equilibrium constraints. The remaining three constraints are the market The residents also need to respect the transversality condition that lim =, to prevent them operating a Ponzi scheme. Finally, the residents face initial conditions ( ). We assume that [ 1] and that W(1). Solving the problem Problem (A3) involves too many constraints to tackle head on by forming a Lagrangian. Rather, we must approach it through a process of simplification. Our first result concerns the objective function. Fact A.1.1. Suppose that the sequence of policies { } = satisfies in each period the market equilibrium conditions. Then, the period residents objective function is equal to + X () ( +1 1) + 1 (A4) = ProofofFactA.1.1. From the market equilibrium condition =(1 +1 ) + µ +1 (1 ) ( +1 ) + +1 (A5) 3

5 we have that µ +1 (1 ) ( +1 ) = +( +1 1) +1 + Thus, for any period, wehavethat = X = X = () ½(1 ) + () ½(1 ) + Expanding the right hand side, we have that X = () ½(1 ) + = (1 ) + ½ + (1 ) µ +1 (1 ) ( +1 ) ¾ + +( +1 1) +1 + ¾ + +( +1 1) +1 + ¾ 1 + +( +1 1) ( +2 1) +2 + ¾ 1 +() ½(1 2 ) ( +3 1) +3 + ¾ 1 + +() ½(1 ) + + +( +1 1) +1 + ¾ 1 Note that the +1 term in the first line cancels with that in the second line. Similarly, the +2 term in the second line cancels with the +2 term in the third line, etc, etc. Thus, we have X () ½(1 ) + + +( +1 1) +1 + ¾ 1 = X = + () ½(1 ) 1 + ( +1 1) + ¾ () = Given that the third market equilibrium condition implies that lim () =. Thus, = for all, wehavethat X () ½(1 ) + + +( +1 1) +1 + ¾ 1 = X = + () ( +1 1) + 1 This result tells us that housing prices have a direct impact on the objective function only in the initial period. Prices in all subsequent periods wash out. Moreover, all else equal, the initial residents prefer to have future housing levels as high as possible. This reflects that, when the size 4

6 of the community is determined by the market, higher housing levels correspond to greater surplus from living in the community. Our second result concerns the inter-temporal implications of the sequence of budget constraints. Fact A.1.2. Suppose that the sequence of policies { } = satisfies in each period the budget constraint µ +1 (1 + ) + 1 = (A6) and that lim =lim =.Then,{ } = satisfies the intertemporal budget constraint X = µ = (A7) ProofofFactA.1.2.Suppose that the sequence of policies { } = satisfies in each period the budget constraint (A6). From the period budget constraint, we have that = and from the period + 1 budget constraint, we have that +1 = µ µ Substituting the latter into the former yields µ = Similarly, the period + 2 budget constraint tells us that +2 = µ µ Substituting this into the period budget constraint yields µ µ = Iterating this logic, we find that for all periods X µ ( )= = Since lim ( ) =, this implies that X µ = = 5

7 which is (A7). The assumed properties of the public good benefit function imply that the public good level will be bounded above and hence that lim =. Moreover, the transversality condition requires that lim =. Thus, Fact A.1.2 suggests replacing the sequence of budget constraints (A6) in the period residents problem with the single intertemporal budget constraint (A7). Indeed, this is a standard procedure in models of optimal policy in which decision-makers face a sequence of budget constraints and have access to bonds. However, in our problem, we also have the market equilibrium constraints to worry about and these depend on the time path of taxes and hence local government debt. Specifically, while constraint (A7) is independent of the time path of debt and just depends on the present value of taxes, the market equilibrium conditions (A5) do depend on the time path of taxes. Thus, we need to verify that replacing the sequence of budget constraints with the single intertemporal budget constraint would not create problems in satisfying the market equilibrium conditions (A5). This is confirmed by our next result. Fact A.1.3. Suppose that the sequence of policies { } = satisfies in each period the market equilibrium condition (A5) and the budget constraint (A6) and that lim = lim =.Then,{ } = satisfies the constraints that X = µ µ (1 +1 ) + and that, for all periods, =(1 +1 ) + µ +1 (1 ) ( +1 ) (1 + ) + µ +1 (1 ) ( +1 ) + +1 = ³ Conversely, if { } = satisfies the constraints (A8) and, for all, (A9),then there exists { } = such that { } = satisfies for all the market equilibrium condition (A5) and the budget constraint (A6). Proof of Fact A.1.3. Let { } = be a sequence of policies satisfying in each period the market equilibrium condition (A5), the budget constraint (A6) and the requirement that lim = lim =. Then we know from Fact A.1.2 that X = (A9) µ = (A1) (A8) 6

8 Using (A5) to solve for and substituting this into (A1) yields (A8). That (A9) holds follows immediately from (A5) after solving (A6) for and substituting in for. For the converse, let { } = satisfy the constraints (A8) and, for all, (A9). For all, let =(1 +1 ) + µ +1 (1 ) ( +1 ) + +1 (A11) Simply rearranging this equation reveals that { } = satisfies (A5) for all. Solving (A11) for and using (A9) reveals that, for all periods, ³ = (1 + ) Rearranging this equation yields (A6). +1 Note that equation (A8) is obtained from the intertemporal budget constraint (A7) by substituting in the expression for the tax rate implied by the market equilibrium condition. Equation (A9) is obtained from the market equilibrium condition by substituting in the expression for the tax rate that is implied by the budget constraint. Thus, both equations reflect both the market equilibrium conditions and the budget constraint. Fact A.1.3 tells us that we can replace the per-period budget constraint and the market equilibrium condition with equations (A8) and (A9). It also allows us to eliminate from the set of choice variables. Thus, we can recast the initial residents problem as follows: + P = max () ( +1 1) + 1 { } = (8) and for all (9), +1, (=if +1 ) (A12) Our next result shows that there is no loss of generality in requiring the period residents choose policies so that the price of housing is equal to in all periods except period. Fact A.1.4. Let { } = be a sequence of policies satisfying the constraints (A8) and, for all, (A9). Then,thereexists{ e e } = such that e = for all +1, e =, and e =, with the property that { +1 e e } = satisfies the constraints (A8) and, for all, (A9). Proof of Fact A.1.4. Let { } = be a sequence of policies satisfying the constraints (A8) and, for all, (A9). Let{ e } = be such that e = for all +1 and 7

9 e =. Then we claim that { e } = satisfies the intertemporal budget constraint (A8). To prove this, it suffices to show that X = ³ ³ ³ i h (1 +1 ) + +1 (1 ) ( +1) + +1 = X =+1 h ³ ³ h ³ For this, we need to show that We have ³ +1 (1 ) (1 +1 ) + ³ +1 (1 +1 ) + X +1 ( +1 )= +1 ( )+ = i ( +1 ) + + ³ +1 (1 ) ( +1) (1 ) X =+1 +1 (1 ) X +1 ( +1 )= +1 ( +1 )+ +2 ( ) ( )+ = Let +1bethefirst time period in which. Then, = +1.Thus, i 1 ( )+ +1 ( +1 ) = 1 [ + ( +1 )] = 1 ( )+ +1 ( +1 ) Continuing this argument yields the result. and Now define the sequence of bond levels { e +1 } = inductively as follows: e +1 = µ (1 +1 ) + µ µ e =(1+) e (1 ) + Then we have that =(1 +1 ) + and, for all +1 =(1 +1 ) + µ +1 (1 ) ( +1 ) µ +1 (1 ) ( +1 ) (1 + ) + (1 + )e + µ +1 (1 ) ( +1 ) + µ (1 ) ( ) (1 ) ³ ³ e +1 + e

10 To understand this key result heuristically, imagine that, say, +1 were less than. Thenit must be the case that there is no new construction in period +1, so that +2 equals +1.The market equilibrium condition (A5) tells us that +1 will depend on the period +1 tax +1. Moreover, the government s budget constraint (A6) tells us that this tax will depend positively on the community s period + 1 debt +1. Now suppose that in period tax were increased to reduce +1 sufficiently to raise +1 to, with population held constant at +1.Thekeypoint to note is that this will not change the housing market equilibrium in period. In the market equilibrium condition (A5), the increase in taxes this period will be perfectly compensated by an increase in the value of housing next period. In this way, the time path of debt and taxes can be adjusted to make +1 equal to without impacting or the time path of housing. Given Fact A.1.1, this adjustment has no impact on the period residents payoff. Fact A.1.4 allows us to write the intertemporal budget constraint (A8) as X =+1 ³ ³ (1 +1 ) + h ³ ³ +1 (1 ) ( +1 ) + + ³ ³ i +1 (1 +1 ) + +1(1 ) ( +1 ) (1 ) The market equilibrium constraints (A9) can be written as ³ µ +1 (1 ) =(1 +1 ) + ( +1 ) (1 + ) and for all +1 =(1 +1 ) + µ +1 (1 ) ( +1 ) (1 + ) + +1 ³ = (A13) (A14) (A15) Finally, the constraint that (with equality if +1 ) need only be imposed for period. Next observe that given a choice of period + 1 debt, +1, and a sequence of public good and housing levels { } =, the market equilibrium constraints (A15) pin down the sequence of debt levels { +1 } =+1. Thus, these constraints can be eliminated and the debt levels { +1 } =+1 can be removed from the set of choice variables. This allows us to write the period residents 9

11 problem as: max { +1{ +1 +1} = } X =+1 + P = () ( +1 1) + ³ ³ +1 ³(1 +1 ) + +1 (1 ) 1 ( +1) + + ³ ³ +1 ³(1 +1 ) + +1 (1 ) ³ =(1 +1 ) + +1 (1 ) ( +1 ) = ( +1) (1 ) µ (1+) +( +1 1 ) (A16) +1 for all (=if +1 ) Our next result uses this formulation to tie down the public good levels. Fact A.1.5. In the period residents optimal plan, for all +1 =(1 ) ( +1 ) ProofofFactA.1.5. Inspecting problem (A16), it is clear that for all periods +1, +1 must equal (1 ) ( +1 ). The only place +1 enters the problem is in the intertemporal budget constraint and setting +1 equal to the level (1 ) ( +1 ) maximally relaxes this constraint. Given this, we can eliminate { +1 } =+1 from the choice variables and use the definition of W( +1 ) to write the intertemporal budget constraint as: ³ ³ X + =+1 (1 +1 ) + W( +1 )(1 ) = ³ +1 (1 ) ( +1 ) + In addition, recalling that = (1 + ),wecanwritetheperiod equilibrium market equilibrium constraint as =(1 +1 ) + µ Ã +1 (1 ) +1 ( +1 ) 1! Substituting this into the intertemporal budget constraint, we get X =+1 W( +1 )(1 ) = +1 µ

12 Thus, we can rewrite problem (A16) as max { { +1 } = } + P = () ( +1 1) + 1 X W( +1 )(1 ) = +1 µ =+1 =(1 +1 ) ³ +1 (1 ) ( +1) ³ (A17) +1 for all (=if +1 ) We can now prove that +1 must equal (1 ) ( +1 ). It is clear that this must be true if, since then the optimal +1 must maximize and hence µ Ã +1 (1 ) +1 ( +1 ) 1! Inthiscase,wehavethat µ +1 =(1 +1 ) + ( +1 ) Suppose then that = and that, contrary to our claim, +1 is not equal to (1 ) ( +1 ). Then, we know that µ Ã +1 (1 ) +1 = (1 +1 ) + ( +1 ) 1! µ +1 (1 +1 ) + ( +1 ) µ As an alternative to the policies ( ), consider the policies +1 (1 ) ( +1 ) where µ +1 =(1 +1 ) + ( +1 ) These policies do not change the price. However, since +1 +1, they relax the intertemporal budget constraint in the sense that X W( +1 )(1 ) = =+1 This permits the choice of a preferred sequence of housing levels { +1} =, which is a contradiction. 11

13 Fact A.1.5 allows us to eliminate { +1 } = from the choice variables. Moreover, substituting in the optimal public good levels and using the definitions of W( +1 )and,wecanwritethe period residents problem as max { +1 { +1 } = } + P = () ( +1 1) + X +1( ) 1 + = 1 W( +1 )= 1 = (1 )W(+1) ( +1) +1 (A18) +1 for all (=if +1 ) This formulation nicely illustrates how the community is constrained in the population it can attract by its initial wealth and that this constraint can be relaxed at the cost of reducing below. Reducing below corresponds to leaving the community with a higher wealth level in period + 1 (i.e., increasing +1 ). With this formulation, we are able to show that all new construction takes place in periods and +1. Fact A.1.6. In the initial residents optimal plan, for all +3 = +2 ProofofFactA.1.6. Suppose the Fact is not true. Let +3 be the first period which violates the claim; that is, 1 = = +2 Let be the multiplier on the intertemporal budget constraint in problem (A18). Given that we can raise 1 marginally without violating any of the constraints, we must have that benefits from so doing, which are () 2, are no greater than the costs, which are 2 W ( 1 ). This implies that 2 W ( 1 ) Given that we can lower marginally without violating any of the constraints, we must have that the benefits from so doing, which are 1 W ( ), are less than the costs, which are () 1.Thisimpliesthat W ( ) 1 12

14 Combining these two inequalities we have that W ( ) W ( 1 ), which contradicts the fact that W() isconvex. Fact A.1.6 allows us to eliminate the housing levels { +1 } =+2 from the choice variables and write the period residents problem as h i 1 ( +2 1) + 1 max { } +1( ) 1 + W( +1 )+ 1 W( +2) = 1 = (1 )W(+1) ( +1) (=if +1 ) (A19) Moreover, plugging the market equilibrium constraint into the intertemporal budget constraint reveals that W( +2) = 1 which immediately implies that +2 = H( +1 ). The period residents problem then reduces to max { +1 +1} (1 )W(+1) ( +1) +1 h i (H( +1) 1) + 1 H( +1 ) (1 )W( +1 )(=if +1 ) (A2) This problem involves just a choice of two variables - +1 and +1 -howmuchwealthto accumulate in period and how much new construction to undertake. The solution to this problem depends on whether is smaller or larger than where is defined in (18). The former situation is the case covered by Proposition 3 and the latter is covered by Proposition 4. We tackle the two cases separately. The solution when (Proposition 3) Fact A.1.7. Suppose that. Then, in the period residents optimal plan, there exist wealth levels ( ) and ( ), satisfying W( ) ( ) ( ),suchthat ( H ( )) if ( ) ( )= ( ( ) ) if ( ) 13

15 ProofofFactA.1.7. We know that ( ) solves problem (A2). There are two possibilities to consider: i) the period price constraint holds with equality at the optimal policies, and ii) the period price constraint holds with inequality at the optimal policies. We begin with the first possibility. Possibility i). If the period price constraint holds with equality, then (1 )W( +1 )= +1. It then follows that +1 = H(( +1 ) (1 )). The constraint that +1 implies that H(( +1 ) (1 )) or equivalently that (1 )W( ) +1 The constraint that H( +1 ) +1 implies that +1. It follows that the range of feasible +1 values is +1 [ (1 )W( ) ] For this interval to be non-empty it is necessary that W( ). Given the price constraint holds with equality, the optimal choice of period +1wealthmust solve the problem max { +1 } Thederivativeoftheobjectivefunctionis H( +1 1 )+ 1 H( +1) +1 [ (1 )W( ) ] 1 H ( +1 ) 1 H ( +1 ) 1 The concavity of the function H( ), implies that this derivative is negative for all +1. The optimal choice of period +1 wealth is therefore This in turn implies that +1 = H( ). We conclude that if the period price constraint holds with equality at the optimal policies, then the optimal policies are ( H ( )). A necessary condition for this to be the solution is that W( ). Note for future reference that the payoff from this candidate solution is + 1 (H( ) 1) + 1 (A21) Possibility ii). Iftheperiod price constraint holds as an inequality at the optimal policies, then (1 )W( +1 ) +1 and +1 =.Thismeansthat +1 (1 )W( ) 14

16 The constraint that H( +1 ) +1 requires that +1 W( ). Define the wealth level ( ) as the solution to the following problem max {+1} +1 ³ + 1 (H( +1) 1) +1 W( ) (A22) Then, the optimal policies must equal ( ( ) )anditmustbethecasethat ( ) (1 )W( ) (A23) The payoff from this candidate solution is µ (1 )W( ) ( ( )) + µ ( 1) + 1 (H( ( )) 1) + 1 We now provide some more information about the wealth level ( ). Claim A.1.1. ( ) W( ). (A24) Proof of Claim A.1.1. By definition, the wealth level ( ) solves problem (A22). The derivative of the objective function in problem (A22) is + 1 H ( +1 ) and the second derivative is 1 H ( +1 ) The objective function is therefore concave. To prove the claim, it suffices to show that if, then it is the case that From (A1), we have that + 1 H (W( )) It follows that H 1 (W( )) = (1 2 ) + ( )+ ( ) (1 ) = + 1 h + 1 H (W( )) 1 (1 )++ (1 ) ( ) ( ) i (A25) The right hand side of (A25) is positive if 2 (1 ) 1 1 (1 )+ + (1 ) ( ) ( ) 15 1

17 This inequality is equivalent to µ (1 ) + ( )+ ( ) (1 ) (1 ) 1 This follows from the fact that. Finally, note that it follows from the claim and the fact that the wealth level ( )solves problem (A22) that it must be the case that 1 H ( ( )) = 1 (A26) Which possibility arises? Having understood the two possibilities, we can now analyze which one arises. A necessary condition for possibility i) to be the solution is that W( )and condition (A23) implies that a necessary condition for possibility ii) to be the solution is that ( )+(1 )W( ). Given that ( ) W( ), we can conclude that the solution is ( ( ) )if W( )and( H ( )) if ( )+(1 )W( ). For values of in the interval [W( ) ( )+(1 )W( )) both possibilities are feasible. Thus, which possibility is optimal depends on a comparison of the payoffs (A21) and (A24). We can show: Claim A.1.2. There exists ( ) (W( ) ( )+(1 )W( )) such that the optimal policies are given by: ( H ( )) if ( ) ( )= ( ( ) ) if ( ) (A27) ProofofClaimA.1.2. When [W( ) ( )+(1 )W( )], the solution will be ( H ( )) if (A21) exceeds (A24) and ( ( ) ) if (A21) is less than (A24). Differencing (A21) and (A24) yields µ µ (1 1 H( )W( ) ( ( )) )+ + 1 H( ( )) µ µ (1 )W( ) ( ( )) = + 1 H( ( )) 1 1 H( ) Define the function ( ; )ontheinterval[w( ) ( )+(1 )W( )] to equal this difference. Note first that (W( ); )= ( ( ) W( )) µ 1 (H( ( )) H(W( ))) 16

18 By the Mean-Value Theorem, there exists (W( ) ( )) such that (W( ); )= 1 H ( ) 1 ( ( ) W( )) The concavity of the function H( ) then implies that (W( ); ) 1 H ( ( )) 1 ( ( ) W( )) = Ontheotherhand,wehavethat ( ( )+(1 )W( ); ) µ = + 1 H( ( )) 1 1 H( ( )+(1 )W( )) = 1 [H( ( )+(1 )W( )) ((1 ) + H( ( )))] 1 [H( ( )+(1 )W( )) ((1 ) H(W( )) + H( ( )))] where the first inequality follows from the fact that = H(W( )) H( ( )) and the second inequality follows from the concavity of H( ). Finally,wehavethatforall [W( ) ( )+(1 )W( )] ( ; ) = 1 µ H ( ) where the inequality follows from the concavity of H( )andthefactthat ( ). We conclude that there exists a unique ( ) (W( ) ( )+(1 )W( )) such that ( ; ) if ( )and( ; ) if ( ). Fact A.1.7 now follows from Claims A.1.1 and A.1.2. We have now completed the characterization of the solution to the period residents problem when and can verify it has the same form as described in Proposition 2. If ( ), then, given that +1 = H( ), in period the housing stock increases to H( ). Moreover, since +1 =(1 ) (H( )), the community invests in (H( )) units of the public good in period. Giventhat +1 =,itmustbethecasethat which implies that (1 ) (H( )) (1 + ) +1 = (1 + ) (1 + ) +1 (1 + ) = [(1 ) (H( )) ] 17

19 Thus, all but (H( )) of the cost of investment is financed with debt. Since = H( +1 ) and =(1 ) ( ) for all + 2, thereafter, the community maintains the public good at (H( )) and the market provides no more housing. From (A15) we have that (1 + ) = (1+) +1 for all +2, implying that debt remains constant. This means that the community s wealth remains at and taxes finance the maintenance of the public good and interest on the debt. The price of houses is in period andinallsubsequentperiods. If ( ), then, given that +1 =, no new construction takes place in period. Moreover, since +1 =(1 ) ( ), the community invests in ( ) units of the public good in period. Giventhat +1 = ( ),itmustbethecasethat which implies that (1 ) ( ) (1 + ) +1 = ( ) (1 + ) +1 (1 + ) = [(1 ) ( ) ] ( ( ) ) The price of houses in period is µ (1 )W( ) ( ( )) which is less than. Given that +2 = H( ( )), in period +1 the housing stock increases to H( ( )). Moreover, since +2 =(1 ) (H( ( ))), the community invests in (H( ( ))) (1 ) ( ) units of the public good in period. From (A15), we have that (1 + ) +2 (1 + ) +1 = [(1 ) (H( ( ))) (1 ) ( )] implying that all but (H( ( ))) of the cost of investment is financed with debt. Since = H( ( ))and =(1 ) ( ) for all + 3, thereafter, the community maintains the public good at (H( ( ))) and the market provides no more housing. From (A15) we have that (1+) =(1+) +2 for all +3, implying that debt remains constant. This means that the community s wealth remains at ( )andtaxesfinance the maintenance of the public good and interest on the debt. The price of houses is in period + 1 and in all subsequent periods. To complete the proof of Proposition 3, it only remains to establish that the functions ( ) and ( )defined in Fact A.1.7 have the properties claimed in Proposition 3. Fact A.1.8. The functions ( ) and ( ) defined in Fact A.1.7 are continuously differentiable and increasing on [ ), and satisfy the limit condition that lim % ( )= lim % ( )=W( ). 18

20 Proof of Fact A.1.8. We begin with the function ( ). From the proof of Fact A.1.7, we know that ( )satisfies (A26). It follows immediately from the concavity of H( )that ( )is increasing in.thedefinition of implies that 1 H (W( )) = 1 Along with (A26), this implies that lim % ( )=W( ). We now turn to the function ( ). We know that ( ) (W( ) ( )+(1 )W( )). We also claim that ( ) is increasing. To see this, note that ( )isimplicitly defined by the equation ( ( ); )=. Itfollowsthat ( ; ) = ( ; ) We have already established that ( ; ). Thus, to establish the claim we need to show that ( ; ). Differentiating and using the first order condition (A26), we have that ( ; ) = (1 )W ( ) [(1 )W( ) ( ( ))] ( ) 2 (1 )W ( ) where the inequality follows from the fact that [(1 )W( ) ( ( ))] ( ) 2 Using (A2) and the definition of W( ), we have that (1 )W ( ) = (1 )W( ) ( ) 2 + ( ) Thus, ( ; ) (1 )W( ) ( ) 2 + (1 ) ( ) It therefore suffices to show that (1 )W( ) Using the definition of W( ), we have that + (1 ) ( ) (1 )W( ) + (1 ) ( ) = (1 )+(1 )+ (1 ) ( ) ( ) 19

21 But since,wehavethat (1 ) ( ) ( ) µ (1 ) (1 ) 1 (1 )+ + (1 ) where the last inequality follows from the fact that 2 (1 ) 1 Given that ( ) (W( ) ( )+(1 )W( )) and that lim % ( ) = W( ), it must also be the case that lim % ( )=W( ). The solution when (Proposition 4) Fact A.1.9. Suppose that. Then, in the period residents optimal plan, ( H ( )) if W( ) ( )= (W( ) ) if W( ) ProofofFactA.1.9. We know that ( ) solves problem (A2). There are two possibilities to consider: i) the period price constraint holds with equality at the optimal policies, and ii) the period price constraint holds with inequality at the optimal policies. We begin with the first possibility. Possibility i). As argued in the proof of Fact A.1.7, in this case the optimal choice of period +1 wealth is which implies that +1 = H( ). Furthermore, a necessary condition for this to be the solution is that W( ). Possibility ii). As argued in the proof of Fact A.1.7, the optimal +1 must solve the problem ³ max (H( +1) 1) {+1 } (A28) +1 W( ) We claim that the solution is +1 = W( ). To show this, it is enough to show that 1 H (W( )) Following the steps in the proof of Fact A.1.7, this inequality is equivalent to µ (1 ) + ( )+ ( ) (1 ) (1 ) 1 2

22 This follows from the fact that. Given that +1 = W( ), for the price constraint to hold as an inequality it must be the case that W( ) (A29) Given that a necessary condition for possibility i) is that W( ) and a necessary condition for possibility ii) is that W( ), the result follows immediately. We have now completed the characterization of the solution to the period residents problem when and can verify it is as described in Proposition 4. If ( ), then, given that +1 = H( ), in period the housing stock increases to H( ). Moreover, since +1 = (1 ) (H( )), the community invests in (H( )) units of the public good in period. Given that +1 =,itmustbethecasethat (1 ) (H( )) (1 + ) +1 = (1 + ) which implies that (1 + ) +1 (1 + ) = [(1 ) (H( )) ] Thus, all but (H( )) of the cost of investment is financed with debt. Since = H( +1 ) and =(1 ) ( ) for all + 2, thereafter, the community maintains the public good at (H( )) and the market provides no more housing. From (A15) we have that (1 + ) = (1+) +1 for all +2, implying that debt remains constant. This means that the community s wealth remains at and taxes finance the maintenance of the public good and interest on the debt. The price of houses is in period andinallsubsequentperiods. If ( ), then, given that +1 =, no new construction takes place in period. Moreover, since +1 =(1 ) ( ), the community invests in ( ) units of the public good in period. Giventhat +1 = W( ),itmustbethecasethat (1 ) ( ) (1 + ) +1 = W( ) which implies that (1 + ) +1 (1 + ) = [(1 ) ( ) ] (W( ) ) The price of houses in period is µ W( ) 21

23 whichislessthan. Given that +2 = H(W( )) =, no new construction takes place in period +1. Moreover, since +2 =(1 ) (H(W( ))), the community maintains the public good level at ( )inperiod + 1. From (A15), we have that (1 + ) +2 (1 + ) +1 = so there is no change in debt. Since = H(W( ))and =(1 ) ( ) for all +3, thereafter, the community maintains the public good at ( ) and the market provides no more housing. From (A15) we have that (1 + ) =(1+) +2 for all + 3, implying that debt remains constant. This means that the community s wealth remains at W( )andtaxesfinance the maintenance of the public good and interest on the debt. The price of houses is in period + 1 and in all subsequent periods. It should be stressed here that increasing wealth from to W( ) is not the only way of financing the public good. It would be equally good to just keep wealth constant at.thekey point is that as long as the population remains constant at, which requires that wealth is does not exceed W( ), a form of Ricardian equivalence holds. Specifically, tax and debt finance are equivalent. Intuitively, if the residents increase tax finance and reduce debt finance, if they remain in the community, they will benefit from the lower taxes its higher wealth allows and if they leave, the higher wealth will be capitalized into the price of housing. Our method of solution picks out a financing patth in which wealth increases to W( ) because of Fact A.1.4. Recall, that this established that there is no loss of generality in assuming that future housing prices are equal to. This was then used to restrict attention to paths in which this was the case. 22

24 Appendix 2: Proof of Theorem Let Ψ and let ( ) be the associated candidate equilibrium. We need to show that ( ) is an equilibrium if satisfies the conditions of the Theorem. This requires showing that the policy rules and value function introduced in Section 5.3 satisfy the conditions for equilibrium described in Section 3.3. Recall that there are two such conditions. The first is that, for all states ( ), the policy rules solve the residents problem (8) when the continuation value is described by the value function and next period s price of housing is determined by the equilibrium price rule evaluated at next period s state variables. The second is that, for all states ( ), the value function satisfies the equality (9). It is helpful to write out problem (8) with all its constraints as follows: h i h ³ i (1 ) (1 ) ( ) + ( ) max ( ) (1 + ) + ³ 1 = + =(1 ) + ( (1 ) ( ) ) + ( ) (=if ). Solving the budget constraint for the tax, substituting this into the objective function and market equilibrium condition, and using the notation and to describe current and future wealth, we can remove the tax as a choice variable and write the problem as: h (1 ) + max ( ) + ( (1 ) ( ) ) 1 + i 1 + ( ( ) ) =(1 ) + ( (1 ) ( ) ) ( (1 ) ( ) ) + + ( ( ) ) 1 (A3) (=if ). While the wealth level replaces the debt level in the set of choice variables, the latter can be immediately recovered from the equation = ( ). This is the form of the residents problem we will work with. Since behavior in our candidate equilibrium differs in states in which the housing stock is 23

25 greater than and less than, we tackle the two situations separately. We begin with the former case, since it is simpler. This is not only because the policy rules and value function are simpler in this range of the state space but also because the durability of housing means that once the community is in this part of the state space it must remain in it. States ( ) such that The residents problem We need to show that the policy rules defined by (2), (22), (23), and (25) solve the residents problem (A3) given that the future price and continuation value are as described in (25) and (26) and (27). Our first observation is that we can assume that the residents policy choices are such that next period s wealth is at least as big as the threshold W( ). Fact A.2.1. Suppose that and let ( ) solve problem (A3) with future price and continuation value as described by (25) and (26) and (27). Then, we may assume without loss of generality that W( ). ProofofFactA.2.1. Suppose that W( ). We will show that increasing to W( ) will not violate the constraints or change the value of the objective function in problem (A3). Regarding the former, note that using the equilibrium price rule (25), the definition of P in (24), and the definition of W( )in(17),wehave = (1 ) + ( (1 ) ( ) ) = (1 ) + ( (1 ) ( ) ) = (1 ) + ( (1 ) ( ) ) = (1 ) + ( (1 ) ( ) ) ³ 1 ³ 1 ³ W( ) ³ 1 + W( ) + ( ( ) ) + P( W( ) ) + P( W( ) W( )) + Regarding the latter, note that using the equilibrium value function (26) and the definition of W( ) in (17), we have 24

26 ³ ( (1 ) 1 ( ) ) = ( (1 ) ( ) ) = ( (1 ) ( ) ) ³ 1 ³ ( ( ) ) + + W( ) + (W( )) + W( ) + (W( )) Fact A.2.1 simplifies matters considerably as it ties down the form of the value function and fixes the future housing price at. It allows us to rewrite problem (A3) as max ( ) h (1 ) + i ³ 1 + (1 ) ( ) =(1 ) + ( (1 ) ( ) ) 1 (=if ) + + ( ) (A31) W( ) Our next result shows that public good provision is efficient. Fact A.2.2. Suppose that and let ( ) solve problem (A31) with ( ) given by (27). Then, =(1 ) ( ). ProofofFactA.2.2.Note first that (1 ) ( ) = arg max ³ µ (1 ) ( ) 1 Thus, provided that such a choice does not make greater than, it will clearly be optimal to set equal to (1 ) ( ). Suppose then that such a choice does violate the price constraint; i.e., (1 ) + ( )+ + Then clearly it must be the case that the price constraint binds under the policies ( ), which means that =(1 ) + ( (1 ) ( ) ) ³

27 This implies that the payoff from the policies ( )is µ (1 ) ( 1) + ( ) Now choose c greater than to satisfy the price constraint with the efficient level of the public good; i.e., (1 ) + ( )+ c + = Consider the alternative policies ((1 ) ( ) c )whichinvolvetheefficient level of public good and a higher level of wealth passed to next period s residents. Clearly, these policies satisfy the constraints. Moreover, the payoff from them is µ (1 ) ( 1) + ( c ) This exceeds the payoff from the policies ( )since ( ) is increasing in wealth. Fact A.2.2 allows us to write the residents problem as h i i (1 ) h( )+ + ( ) max ( ) =(1 ) + ( )+ (=if ) W( ) + or, eliminating,as h i (1 ) (1 ) ( )+ + ( ) max ( ) (1 ) + ( )+ (1 ) (=if ) W( ) (A32) Our next result ties down the optimal ( ). Fact A.2.3. Suppose that and let ( ) solve problem (A32) with ( ) given by (27). Then, (W()) ( )= ( H ( )) if W() if (W() W(1)] 26

28 Proof of Fact A.2.3. There are two possibilities to consider: i) the price constraint holds with equality at the optimal policies, and ii) the price constraint holds with inequality at the optimal policies. We begin with the first possibility. Possibility i). If the price constraint holds with equality, then the definition of W( )implies that (1 )W( )=. It then follows that = H(( ) (1 )). The constraint that implies that H(( ) (1 )) or equivalently that (1 )W() The constraint that W( ) implies that H( ) = H(( ) (1 )) and hence that. It follows that the range of feasible values is [ (1 )W() ] For this interval to be non-empty it is necessary that W(). Using the fact that the price constraint holds with equality in problem (A32) together with (27), the optimal choice of wealth must solve the problem H( 1 )+ 1 max H( ) { } [ (1 )W() ] The derivative of the objective function in this problem is 1 H ( ) 1 H ( ) 1 The concavity of the function H( ), implies that this derivative is negative for all.the optimalchoiceofwealthistherefore This in turn implies that = H( ). We conclude that if the price constraint holds with equality at the optimal policies, then the optimal policies are ( H ( )). W(). A necessary condition for this to be the solution is that Possibility ii). If the price constraint holds as an inequality at the optimal policies, then (1 )W( ) and =. This means that (1 )W() The constraint that W( ) requires that W(). From (A32) and (27), the optimal 27

29 choice of wealth must solve the problem max { } ³ + 1 (H( ) 1) W() We claim that the solution is = W(). To show this, it is enough to show that 1 H (W()) Following the steps in the proof of Fact A.1.7, this inequality is equivalent to µ (1 ) + ()+ () (1 ) (1 ) 1 This follows from the fact that.giventhat = W(), for the price constraint to hold as an inequality it must be the case that which requires that W(). W() (1 )W() We conclude that if the price constraint holds as an inequality at the optimal policies, then the optimal policies are (W()). W(). A necessary condition for this to be the solution is that Which possibility arises? Having understood the two possibilities, we can now analyze which one arises. A necessary condition for possibility ii) to be the solution is that W(). Furthermore, a necessary condition for possibility i) to be the solution is that W(). Thus, we conclude that the optimal policies are as described in the statement of the fact. Using Facts A.2.2 and A.2.3, it is clear that the residents want to follow the equilibrium policy rules described in (2), (22), and (23). To verify the price rule, note that (1 ) + ()+ W() + if W() () = if (W() W(1)] Given (24), this is equal to (25). 28

30 Verifying form of value function It remains to verify that the value function as described by (26) and (27) satisfies (9). Suppose first that ( ) is such that (W() W(1)]. Then, we have that (1 ) ( )+ µ ( )(1 ) + 1 ( ) = (1 ) + (1 ) + (H( )) + + ( ) 1 H( ) = (1 ) + + +(1 ) + (H( ) 1) + ( ) 1 = (1 ) + + +(1 ) + (H( ) 1) + + = ( ) 1 ( )+ ( ( ) ( ) ( )) 1 + (H( ) 1) 1 as required. Now suppose that ( ) issuchthatw().then,wehavethat as required. (1 ) = (1 ) = (1 ) + = (1 ) + ( )+ + 1 P( W()) µ ( )(1 ) ( ) + ( )+ ( ( ) ( ) ( )) ()+ W() (1 )W() ()+ + (W()) + +(1 ) + ( 1) ( 1) 1 = (W()) + W() States ( ) such that + W() + (W()) + W() This case is more complicated both because the policy rules and value function are more complicated in this part of the state space and also because the residents choices could in principle take the community out of this range of the state space (i.e., by choosing ). Preliminaries The policy rules and value function depend on the functions ( )and () whicharedefined by (31) and (36). We begin by presenting two key results that establish some important properties of these functions. The first result concerns the function ( ). 29

31 Fact A.2.4 If [ () W( )) and,then ( ) is uniquely defined, belongs to the interval ( ), and is increasing in. Moreover, for any starting housing level, the sequence h i =1 defined inductively by 1 = ( ()) and = ( ( 1 )) converges monotonically to. Similarly, for any starting wealth level [ ( ) W( )), the sequence h i =1 defined inductively by 1 = ( ( )) and = ( ( 1 )) converges monotonically to W( ). Proof of Fact A.2.4 Using (24) and (31), ( )satisfies the equation P( ( ))=. Because [ () W( )) and Ψ we know from the assumed properties of () that P( ()) P( () ()) and that P( ( )) P( ( ) ( )) = Thus, by the Intermediate Value Theorem, theremustexistasolution ( )atwhich P( ( ))=. Moreover, at this solution, we must have that ( ). If not, then ( ). But then we have that = P( ( )) P( ( ) ( )) which is a contradiction. For uniqueness, it is sufficient that P( ()) at any solution of the equation P( ())=. Note from (24), that Moreover, at a solution P( ()) ( ()) = + () () ( ()) 2 = (1 ) + () (1 ) Thus, at a solution P( ()) = + () () (1 ) + () (1 ) + " # (1 )+ (1 2) () ()+ () = 3

32 Given that () is increasing, we have " P( ()) + (1 ) + ()+ () (1 ) # where the last inequality follows from Assumptions 1(i) and 2. A similar logic implies that ( )isincreasingin. Given that the solution satisfies = P( ( )), we have that ( ) ) P( ( ) ) P( = = 1 P( ( ) ) Given that P( ( )) is negative, the result follows. Now let and consider the sequence h i = defined inductively by 1 = ( ()) and = ( ( 1 )). We first show the sequence is increasing. To see this, recall that we showed that for any,if [ () W( )), then ( ) belongs to the interval ( ). Taking = 1 and = ( 1 ), this implies that ( ( 1 )) belongs to the interval ( 1 ). It remains to show that the sequence converges to. Since the sequence is bounded by and is increasing, it converges. Let denote this limit. We know that for all 1, we have that P( ( ) ( 1 )) =. Giventhat () is continuous, it follows that P( ( ) ( )) =. Since P( () ()) for all,thisimplies that must equal (recall that ( )=W( )andthatp( W( ) W( )) = ). Finally, let W( ) and consider the sequence h i =1 defined inductively by 1 = ( ( )) and = ( ( 1 )). Associated with this sequence of wealth levels, is a sequence of housing levels h i = defined inductively by = ( 1 ). This sequence satisfies the equation = ( ( 1 )) and hence converges monotonically to. Given that the function () isincreasingand ( )=W( ), it follows that the sequence h i =1 converges monotonically to W( ). We can use Fact A.2.4 to shed light on the nature of the function ( ) on the interval [ ( ) W( )) as defined in (35). For any [ ( ) W( )), we can construct a sequence h ( )i = that starts at (i.e., ( ) = ) and is defined inductively by ( )= ( ( 1 ( )))) as in Fact A.2.4. We can then write ( )=(1 ) + + ( ( ( ))) + ( ) 1 ( ) + ( 1 ( )) 1 ( ( )) Similarly, we can write ( 1 ( )) = (1 ) + + ( ( 1 ( ))) + 1( ) 2 ( ) + ( 2 ( )) 1 ( 1 ( )) 31

33 Iterating, we obtain X ( )= () µ(1 ) + + ( ( ( ))) + ( ) +1 ( ) 1 ( ( )) = We know from the definition of the function ( ) that for all (1 ( ( ))) + ( ( ( ))) + ( ) +1 ( ) ( ( )) (1 ) = This implies that ( ( )) = H( ( ) +1 ( ) 1 ) and allows us to write ( )= X = () H( ( ) +1 ( ) ) 1 1 (A33) Combining (27) and (A33), we may conclude that + 1 ( )= + P h i = () H( ( ) +1 ( ) 1 ) 1 if [ ( ) W( )) ³ (H( ) 1) if [W( ) W(1)] (A34) There are several points to note about the ( )functiondefined in (A34). First, note that it is continuous at = W( ), since for all lim ( ) +1 ( )=(1 )W( ) %W( ) Second, the function is increasing. This is obviously true on the interval [W( ) W(1)] given that H( ) is increasing. To see that it is true on the interval [ ( ) W( )), it is easiest to write the function as It follows that ( )= X () [ ( ( )) 1] ( ) = X = = () ( ( )) ( ) As shown in the proof of Fact A.2.4, ( ( )) is positive. We claim that for all, ( )is also positive. The proof is by induction. We know that ( )= and hence the result is true for =. Now suppose the result is true for all 1 and consider if it is true for By definition, we have that ( )= ( ( 1 ( ))). Thus, we have that ( )= ( ( 1 ( )))) ( 1 ( )) 1( ) Each term on the right hand side of this expression is positive, and hence ( ) is positive as required. 32

34 Third, the function defined in (A34) is likely to be kinked at = W( ). This is because ( ) = P = () H ( ) +1( ) ( 1 )( ( ) +1 ( ) ³ 1 1 )if [ ( ) W( )) H ( )if [W( ) W(1)] Under the assumption of the Theorem that ( ) is concave, it follows from the fact that that for all [ ( ) W( )) 44 (W( )) = 1 ( ) 1 Finally, note that for all [ ( ) W( )), the second derivative of the function ( )is 2 ( ) X 2 = () H ( ) +1( ) ( 1 )( ( ) +1 ( ) 1 ) 2 = +H ( ( ) +1 ( ) 1 )( ( ) +1 ( ) 1 ) The first part of the terms that is summed is clearly negative, but the second part is harder to sign. Thisiswhyitisdifficult to establish analytically that the ( ) function in this range is concave. The assumption of the Theorem that ( ) is concave amounts to assuming that the second part, if positive, does not fully offset the first part. The second fact concerns the function (). Fact A.2.5. If,then () (). Moreover,P( () ()). Proof of Fact A.2.5. We begin with the first claim. By definition (36), we know that P( () ()). Suppose, to the contrary, that () (). Then, since Ψ, wehave P( () ()) P( () ()) This is a contradiction. Turningtothesecondclaim,notefirstthatif () W( ), we have that P( () ()) P( W( ) ()) P( W( ) W( )) = 44 It can be shown analytically (see the proof of Fact A.4.1) that, if the value function is concave and kinked at W( ), then the left hand derivative of the value function at W( )mustbeequalto(1 )[(1 ) ]. Accordingly, a necessary condition for concavity is that exceed 1(1 + ). For realistic parameterizations, this condition will be always be satisfied. 33

35 and so the result is true. Thus, we can assume that () W( ). We will show in this case that if P( () ()) = it must be that the payoff from the policies ( ()) when the state is ( ()) is strictly lower than the payoff from the equilibrium policies ( ( ( ())) ( ())). But this is inconsistent with equality (37) being satisfied at, which is a contradiction. This will imply that it must be the case that P( () ()). Consider first the payoff from the policies ( ())whenthestateis( ()). By the first claim, we know that () () so that there will be new construction next period. It then follows that next period s price will be P( ( ( ())) ()) =. Moreover,the payoff from ( ())canbewrittenas (1 ) (1 ) ()+ () () + ( ()) 1 (A35) Since P( () ()) is equal to, wehavethat ()+ () () = (1 )+ (1 ) We can therefore write (A35) as µ 1 (1 )+ + ( 1) + ( ()) 1 Now note that = H( () () 1 ), so we can write this as (1 )+ µ H( () () ) ( ()) (A36) Next observe from (A34) that ( ()) = X = () = H( ( ()) +1 ( ()) ) 1 1 Substituting this into (A36), the payoff from ( ())canbewrittenas H( () () X ) 1 + () +1 H( ( ()) +1 ( ()) ) Note also that the payoff from the equilibrium policies ( ( ( ())) ( ())) in state ( ()) can be written in a similar way as H( () ( ( ())) 1 X + () +1 = ) 1 H( ( ( ( ()))) +1 ( ( ( ()))) )

36 Moreover, we know that since = H( () () 1 we must have that () ( ( ())). ) ( ()) = H( () ( ( ())) ) 1 Now define the function ( )ontheinterval[ ( ( ())) ()] as follows: ( )= H( () X ) 1 + () +1 H( ( ) +1 ( ) ) If we can show that this function is decreasing in, we will have established that ( ()) must yield a smaller payoff than ( ( ( ())) ( ())) in state ( ()), which contradicts equality (37). Consider then differentiating ( ). Ignoring multiplicative constants, the derivative is H ( () 1 ) + X = = () +1 H ( ( ) +1 ( ) )( 1 ( ) +1( )) Rearranging, and using the fact that ( )=1,wecanwritethisas H ( () ) H ( ( ) 1 ( ) ) 1 1 X () =1 H ( 1( ) ( ) 1 ) H ( ( ) +1 ( ) ) 1 ( ) We know that for all 1, ( ). In addition, we know () ( ) 1 ( ) and that for all 1, 1 ( ) ( ) ( ) +1 ( ). Since H is concave, this implies that the above expression is negative. We conclude that the derivative is decreasing in as required. We can use this result to get a sharper characterization of (). Given Fact A.2.5, we know from (36) that ½ () = arg max (1 ) P( ()) + + ()+ () ¾ + ( ) 1 Thederivativeoftheobjectivefunctionis µ ( ) 1 We know that for any W( )wehavethat ( ) (W( )) =

37 Accordingly, () W( ). Recall, however, that the value function may be kinked at W( ). It follows that () must either satisfy the first order condition ( ()) = 1 (A37) or equal W( )ifitisthecasethat The residents problem lim ( ) %W( ) 1 We are now ready to show that the policy rules defined by (2), (28), (3), (32), and (33) solve the residents problem (A3) when the continuation value is described by the appropriate value function and next period s price of housing is determined by the appropriate equilibrium price rule evaluated at next period s state variables. Given that the residents policy choices could in principle involve a housing level greater or less than, it is as well to be absolutely clear at the outset on what the appropriate value function and equilibrium price rules are. Combining the information in (29) and (34), when the appropriate value function is ( ) = ( ()) + () ( ) if () if ( () W(1)] (A38) where the function ( ) is as described in (A34). From (26), when the appropriate value function is ( ) = (W()) + W() ( ) if W() if (W() W(1)] (A39) where the function ( ) is again as described in (A34). Combining the information in (28) and (33), when the appropriate price rule is P( ()) ( ) = if () if [ () W(1)] (A4) From (25), when theappropriatepriceruleis P( W()) if W() ( ) = if (W() W(1)] (A41) 36

38 Our first observation is that the residents policy choices are such that next period s wealth is at least as big as the threshold level ( ) if their housing choice is less than and at least as big as W( )ifitisgreaterthan. Fact A.2.6. Suppose that and let ( ) solve problem (A3) with future price and continuation value as described in (A38), (A39), (A4), and (A41). Then, ( ) if and W( ) if. ProofofFactA.2.6. Suppose first, to the contrary, that ( )and. Then, from (A4) and (A38) and Thus, the current price is ( ( ) )=P( ( ) ) ( ( ) )= ( ( )) + ( ) =(1 ) + ( (1 ) ( ) ) and the payoff is (1 ) ( (1 ) ( ) ) ³ 1 + ³ P( ( ) ) µ + ( ( )) + ( ) Note that neither the price nor the payoff vary with respect to for any ( ). However, at = ( ), the price jumps up reflecting the fact that the future price jumps from P( ( ) ( )) (which is less than by Fact A.2.5) to. If it is the case, that ³ 1 (1 ) + ( (1 ) ( ) ) + ( ) + Then it is clear that we can replace ( )withapolicyinwhich = ( )andwe will have increased the value of the objective function. Since this is a contradiction, we can assume that ³ (1 ) + ( (1 ) 1 ( ) ) + ( ) + We can also write the payoff under ( )as ³ (1 ) + + ( (1 ) 1 1 ( ) ) + ( ) + ( ( )) (A42) 37

39 Now choose a wealth level c ( ) which keeps the current price equal to,butmakes the future price equal. Thispricesatisfies (1 ) + ( (1 ) ( ) ) ³ 1 + c + = (A43) The policies ( c ) yield a payoff (1 ) ( (1 ) ( ) ) ³ 1 + c + ( c ) (A44) Comparing (A42) and (A44), we see that choosing the alternative policies ( c ) will increase the objective function if ( c ) c ( ( )) ( ) Given that is strictly concave, this will be true if ( c ) 1 We know from (A37) that ( ( )) 1 thus it is sufficient to show that c is less than ( ). To establish this, note first from (A43) that (1 ) + ( (1 ) ( ) ) = (1 ) + ( (1 ) ( ) ) ³ 1 + ( ) ³ 1 + c Cancelling terms and dividing through by, this implies that Recall that, by definition, P( ( ) ( )) + c ( ) + + P( ( ) ( )) = P( ( ) ( )) (1 ) + ( )+ ( ) ( ) Thus, this implies that (1 ) + ( )+ c ( ) 38 + = +