Existence of a Radner equilibrium in a model with transaction costs
aa r X i v : . [ q -f i n . M F ] F e b Existence of a Radner equilibrium in a model with transaction costs
Kim Weston Rutgers UniversityDepartment of MathematicsPiscataway, NJ 08854, USASeptember 18, 2018
Abstract
We prove the existence of a Radner equilibrium in a model with proportional trans-action costs on an infinite time horizon and analyze the effect of transaction costs onthe endogenously determined interest rate. Two agents receive exogenous, unspannedincome and choose between consumption and investing into an annuity. After estab-lishing the existence of a discrete-time equilibrium, we show that the discrete-timeequilibrium converges to a continuous-time equilibrium model. The continuous-timeequilibrium provides an explicit formula for the equilibrium interest rate in terms ofthe transaction cost parameter. We analyze the impact of transaction costs on theequilibrium interest rate and welfare levels.
Keywords:
Transaction costs, Radner equilibrium, Shadow prices, Incompleteness
JEL Classification:
D52, G12, G11
Mathematics Subject Classification (2010):
We study an incomplete Radner equilibrium with proportional transaction costs. Trans-action costs influence asset prices, yet most asset pricing models with transaction coststake some or all asset prices as given. We seek to answer two questions: (1) Does a finite-agent equilibrium with transaction costs and an endogenously deter-mined interest rate exist?(2) If an equilibrium exists, what is the effect of transaction costs on the interest rateand welfare levels?
We devise a model to answer these questions and reveal an unexpected effect fromtransaction costs on the endogenous asset prices. Depending on agent risk preferencesand consumption smoothing over time, a proportional transaction cost can increase or The author would like to thank the anonymous referee and editor, Frank Riedel, as well as GordanˇZitkovi´c, Mihai Sˆırbu, Kasper Larsen, Johannes Muhle-Karbe, Frank Seifried, and Matteo Burzoni for helpfuldiscussions on this work. The author acknowledges support by the National Science Foundation under GrantNo. DMS-1606253. Any opinions, findings and conclusions or recommendations expressed in this materialare those of the author and do not necessarily reflect the views of the National Science Foundation (NSF). ecrease equilibrium interest rates. We provide an explicit formula for the continuous-time equilibrium interest rate in terms of the transaction costs and other input param-eters. We find that welfare decreases with increases in the transaction costs.Proving the existence of a general equilibrium is a challenging problem on its own,and frictions exacerbate the difficulties. Existing work on equilibria with transactioncosts lacks the ability to endogenously derive asset prices while providing rigorousjustification in a finite-agent Radner equilibrium. The works [22], [18], and [8] relyon an exogenously specified bank account, while [18] and [2] provide numerics butno existence result. Models with a continuum of agents are introduced in [23], [22],and [12]. An approximate equilibrium concept is introduced in [11]. In contrast to theexisting literature, we introduce transaction costs into a finite-agent Radner equilibriumwith unspanned income and consumption over an infinite time horizon. This set-up islongstanding without transaction costs; see, for instance, [3] and [24]. We prove theexistence of a proportional transaction cost Radner equilibrium and analyze its effectson endogenous asset prices.In our model, two exponential investors receive unspanned income, consume, andtrade in an annuity market on an infinite time horizon. The individual agent’s optimalinvestment problem in the presence of proportional transaction costs is well-studiedand dates back to [19] and [9]. The equilibrium setting in [23] is most similar to ours:In [23], a continuum of agents trade in two annuities in an overlapping generationsequilibrium while one of the annuities faces real proportional transaction costs fromtrading. However, [23] assume that the zero transaction cost economy supports anequilibrium in which the agents’ wealth first increases then decreases. We make nosuch assumption, and our model does not exhibit this behavior. We provide an exampleof an equilibrium with proportional transaction costs, which derives all traded assetprices endogenously and analytically describes the effect of transaction costs on theequilibrium interest rate.Understanding the effects of transaction costs analytically is often not possible. In[13], [20], and [16], the authors derive a Taylor expansion for the single-agent valuefunction and no-trade boundaries for small transaction costs. In equilibrium, we seekto understand the effect of transaction costs on the agents’ behavior and equilibriuminterest rate. The closeness of the agents’ input parameters determine explicitly whenthe agents are motivated to trade in equilibrium. We also derive an explicit formulafor the interest rate in terms of the transaction costs in continuous time. For agents i = 1 ,
2, we let α i > i ’s risk aversion, β i > µ i be agent i ’s income stream’s drift, and σ i be the volatility of the(unspanned) Brownian component in agent i ’s income stream. We let λ ∈ [0 ,
1) bethe proportional transaction cost parameter. Then in an equilibrium in which agent1 chooses to buy and agent 2 chooses to sell, the equilibrium interest rate is givenexplicitly by r ( λ ) = ˜ β /α + ˜ β /α α (1+ λ ) + α (1 − λ ) . The relationship between the agents’ risk aversions determine whether equilibriuminterest rates will increase or decrease when transaction costs are introduced. Ourmodel shows that typically the presence of transaction costs will decrease the equi-librium interest rate. However, when one agent seeks to trade aggressively in orderto ensure her future consumption while the other agent trades timidly and consumes ore readily, then it is possible for small transaction costs to increase the equilibriuminterest rate. In this case, the agents balance each other in their consumption timingand risk appetites so that a premium is placed on the annuity’s future consumptionstream when transaction costs are introduced. Our model is admittedly stylized, yet itcaptures the equilibrium interest rate and trading behavior with transaction costs in along-standing, classical setting.We employ a shadow price approach to establish an equilibrium in order to gaintractability of the single-agent problems. Shadow prices represent the traded assetprice in a least-favorable frictionless market completion, where the optimal investmentand consumption strategies align between the frictionless shadow market and the trans-action cost market. Shadow prices for proportional transaction costs were introducedby [14] and [6] and have since been established in increasingly greater generality; see,for example, [15] and [7]. Because least-favorability is investor specific, each economicagent will select her own frictionless shadow market to perform utility maximization.We link the investor-specific shadow markets using a “closeness” condition in equilib-rium. We show that a unique equilibrium asset price is only guaranteed when a tradeoccurs. Otherwise, agents’ shadow prices allow for a range of prices consistent withthe equilibrium.Several components of this equilibrium example are crucial for obtaining our results.We rely on the agents’ exponential preferences and income processes with independentincrements for tractability of the single-agent problem similar to [24], [4], and [17]. Ina frictionless model with deterministic interest rates, an annuity is spanned by a bankaccount, and vice versa. With transaction costs, we cannot freely move between anannuity and the bank account as the traded security. We work with the annuity as thetraded security similar to [12] and [23]. This choice yields trading strategies in whichthe agents choose to do the same thing at every time point: either buy, sell, or tradenothing. Theorem 5.2 proves that the constant interest rate equilibrium obtained bytrading in the annuity is not possible when the bank account is the traded security.This model is indeed highly stylized, yet it provides the first existence proof of a finite-agent Radner equilibrium with proportional transaction costs in which all traded assetprices are derived endogenously.The paper is organized as follows. Section 2 describes the discrete-time equilibriumand proves its existence in Theorem 2.4. Section 3 considers a continuous-time equi-librium model. The existence of an equilibrium is established in Theorem 3.4, and it isshown to be the limit of discrete-time equilibria. We analyze the impact of transactioncosts on interest rates and welfare in Section 4. Section 5 discusses a transaction costequilibrium with a traded bank account. The proofs are contained in Section 6. We consider a discrete-time infinite time horizon Radner equilibrium without a riskyasset. There is a single consumption good, which we take to be the numeraire. Timeis divided into intervals [ t n , t n +1 ), n ≥
0, where t n := n ∆ and ∆ >
0. An annuity,denoted by A , is in one-net supply and is available to trade with an exogenouslyspecified proportional transaction cost λ ∈ [0 , t n , t n +1 ). he risk-free rate r > A t n +1 − A t n = ( A t n r −
1) ∆ , A > . In this case, the annuity value will be the constant A t n = A = 1 /r . We choose to studya model with a traded annuity rather than a bank account because the annuity willprovide the mathematical structure needed to establish an equilibrium with transactioncosts. We discuss the implications of a traded bank account in Section 5.Each agent has the exogenous income stream Y i = ( Y it n ) n ≥ given by, Y it n +1 = Y it n + µ i ∆ + √ ∆ σ i Z it n +1 , Y i ∈ R , where µ i ∈ R , σ i >
0, and Z it n +1 ∼ N (0 ,
1) for i = 1 ,
2. The random variables ( Z it n ) n ≥ are independent, and Z · and Z · are possibly correlated. The agents are also endowedwith an initial allocation of annuity shares θ i ∈ R such that θ + θ = 1.The flow of information in this economy is given by F = ( F t n ) n ≥ , where F t n = σ ( Z it , · · · , Z it n : i = 1 , F , and allagents share the same filtration and probability P . All equalities are assumed to hold P -almost surely. Rather than deal directly with an optimization problem in a market with frictions,we cast each individual investor’s problem as a problem in her own frictionless shadowmarket. In equilibrium, the agents’ shadow markets will be related, and a unique (non-shadow) equilibrium price for the traded annuity will exist when a trade occurs. Yetthe individual optimization problems are treated in isolation as frictionless. Therefore,only in the next section (Section 2.2) will the parameter λ appear.We first consider the single-agent investment and consumption problem for agent i ∈ { , } . At time t n ≥
0, agent i chooses to consume c t n units of the consumptiongood and invest θ t n shares in the annuity beginning with an initial allocation of θ .We consider equilibria for which the value of the shadow annuity and shadow interestrate (to be determined endogenously in equilibrium) are constants A it n = A i = 1 /r i and r i >
0, respectively.For a given investment strategy θ , agent i ’s shadow wealth is defined by X it n := θ t n A it n , with the self-financing condition( θ t n +1 − θ t n ) A it n +1 = ( Y it n − c t n + θ t n ) ∆ , n ≥ . (2.1)For a given consumption and investment strategy ( c, θ ), the wealth evolves as X cit n +1 − X cit n = (cid:0) X cit n r i + Y it n − c t n (cid:1) ∆ , X i = θ A i . Given a consumption strategy c and an initial share allocation θ , the self-financingcondition dictates the investment strategy θ . e consider agents with exponential preferences over running consumption; that is,agent i ’s utility function is c
7→ − e − α i c for α i >
0. The agents prefer consumption nowto consumption later, which is measured by their time-preference parameters β i > i = 1 , t ≥
0, and c ∈ R , we define U i ( t, c ) := − e − β i t − α i c . Definition 2.1.
A consumption strategy c is called admissible for agent i if c satisfiesthe transversality requirement E (cid:2) exp (cid:0) − β i t n − α i r i X ct n − α i Y t n (cid:1)(cid:3) −→ n → ∞ .In this case, we write c ∈ A ∆ i .The value function is defined by V ∆ i ( x, y ) := sup c ∈A ∆ i ∞ X n =0 E [ U i ( t n , c t n )] , x, y ∈ R . (2.2)It is possible to consider non-constant shadow interest rates in the individual agentformulation, however it comes at the cost of not being able to explicitly describe theindividual agents’ transversality conditions. We refer the reader to Chapter 9 SectionD of [10] for further details. Our constant shadow interest rate formulation allowsus to explicitly characterize the optimal consumption and wealth processes, which wesummarize in Theorem 2.2. Theorem 2.2.
Agent i ’s optimal consumption and wealth process in (2.2) are givenby ˆ c it n = r i ˆ X it n + Y it n + 1 α i r i ∆ (cid:16) ˜ β i ∆ − log(1 + r i ∆) (cid:17) (2.3) and ˆ X it n = θ i r i + t n α i r i (cid:18)
1∆ log (1 + r i ∆) − ˜ β i (cid:19) , (2.4) where ˜ β i := β i + α i µ i − α i σ i . Moreoever, the value function can be expressed in theform V ∆ i ( x, y ) = J ∆ i ( x, y ) := − r i ∆ (1 + r i ∆) ri ∆ exp − α i r i x − α i y − ˜ β i r i ! . The definition of equilibrium must allow us to relate both agents’ willingness to trade,even when no trade occurs due to frictions. Shadow prices provide us with this mecha-nism and a way to compute the range of annuity values consistent with an equilibrium.Typically, shadow prices are used as a tool to establish properties of an original modelwith frictions; see, for example, [15] and [7]. Here, we work with shadow prices di-rectly, and we subsequently determine transaction cost models consistent with ouragents’ shadow markets.
Definition 2.3.
For the transaction cost parameter λ ∈ [0 , equilibrium withtransaction costs is given by a collection of processes ( A i , ˆ c i , ˆ θ i ) i =1 , such that i) Real and financial markets clear for each n ≥ X i =1 ˆ c it n ∆ = ∆ + X i =1 Y it n ∆ − λ (cid:12)(cid:12)(cid:12) ˆ θ t n +1 − ˆ θ t n (cid:12)(cid:12)(cid:12) A t n +1 and ˆ θ t n + ˆ θ t n = 1 , where in the event of a trade, we define A t n +1 := A itn +1 λ if agent i ∈ { , } purchases a positive number of annuity shares; that is, θ it n +1 − θ it n > i = 1 ,
2, the consumption and investment strategies, ˆ c i and ˆ θ i withˆ θ i = θ i , are optimal with the shadow annuity price A i : V ∆ i ( θ i A i , Y i ) = ∞ X n =0 E [ U i ( t n , ˆ c it n )] . (iii) The shadow markets remain “close enough” to the underlying transaction costmarket in the following sense: For each n ≥ A t n A t n ∈ (cid:20) − λ λ , λ − λ (cid:21) . Moreover, for n ≥
1, if ˆ θ t n − ˆ θ t n − > A t n = A t n · λ − λ . If ˆ θ t n − ˆ θ t n − < A t n = A t n · − λ λ . Remark . Let us consider a single-agent optimization prob-lem for a risky asset with frictions S and a shadow price ˜ S . The shadow price along withthe optimal trading strategy ˆ θ will satisfy ˜ S t n ∈ [(1 − λ ) S t n , (1+ λ ) S t n ], ˜ S t n = (1 − λ ) S t n when ˆ θ t n − ˆ θ t n − <
0, and ˜ S t n = (1 + λ ) S t n when ˆ θ t n − ˆ θ t n − >
0. Condition (iii) inDefinition 2.3 enforces this relationship between both agents’ shadow markets and theunderlying market. In the absence of condition (iii) and when trade does not occur,there is no connection between the shadow annuity markets and the underlying mar-ket. In this case, there are infinitely many no-trade equilibria, even in the frictionless( λ = 0) case.Since each agent optimizes in her own shadow market while maintaining the “close-ness” condition (iii), a unique market annuity rate is only guaranteed when tradeoccurs. When trade does not occur in a given period, there is a range of possible annu-ity values (and corresponding interest rates) consistent with equilibrium. In [18, 11],the authors have a single equilibrium price rather than two shadow prices, but theseworks need to allow for the agents to pay different transaction costs based on a total ex-ogenous cost. We are able to avoid this endogenous splitting of costs while maintainingtractability in part because we choose to work with shadow markets.The following is the main result of the section. The proof is in Section 6. Theorem 2.4.
Let ˜ β i := β i + α i µ i − α i σ i , and assume that ˜ β i is strictly positivefor i = 1 , . For λ ∈ [0 , , there exists an equilibrium with strictly positive constantshadow interest rates r , r and constant shadow annuity values A = 1 /r , A = 1 /r .The optimal consumption and wealth processes for investor i = 1 , , are given by (2.3) and (2.4) , respectively. Case 1:
A no-trade equilibrium occurs if e ˜ β ∆ − e ˜ β ∆ − ∈ (cid:20) − λ λ , λ − λ (cid:21) . (2.5) n this case, r = e ˜ β ∆ − and r = e ˜ β ∆ − . The range of possible constant, non-shadow interest rates that are consistent with thisequilibrium is given by r = ( r t n ) n ≥ with r t n ∈ (cid:20) − λ ∆ (cid:16) e max( ˜ β , ˜ β )∆ − (cid:17) , λ ∆ (cid:16) e min( ˜ β , ˜ β )∆ − (cid:17)(cid:21) = ∅ . Case 2:
There exists an equilibrium in which agent will purchase shares of theannuity in equilibrium at all times t n ≥ (while agent sells shares) if e ˜ β ∆ − e ˜ β ∆ − > λ − λ , (2.6) where the interest rate r > is uniquely determined by (cid:18) r ∆1 − λ (cid:19) α (cid:18) r ∆1 + λ (cid:19) α = e ˜ β α + ˜ β α , (2.7) and the shadow interest rates are given in terms of r = r (1 + λ ) = r (1 − λ ) . The parameters ˜ β and ˜ β represent the agents’ time preference parameters for con-sumption adjusted for risk and income. Strictly positive ˜ β i parameters correspond toan economy that allows for strictly positive equilibrium shadow interest rates. Strictlypositive shadow interest rates, in turn, ensure that the shadow annuity values arewell-defined. Allowing even one of the ˜ β i parameters to cross zero would cause the cor-responding shadow annuity to be infinitely valued. Financially, this case correspondsto the case when a (zero transaction cost) shadow annuity with a constant interest ratecannot be replicated by a bank account because of the bank account’s dwindling value.When the agents’ parameters ˜ β and ˜ β are sufficiently close, as in (2.5), thenthe agents are not motivated to trade because of the relatively high transaction costs.Trading occurs in equilibrium only when the parameters ˜ β and ˜ β are sufficiently farapart to overcome the transaction costs. In this case, the agents’ strategies are verysimple: either buy or sell the exact same amount at every time period. Remark . If the inequality (2.6) is flipped so that e ˜ β ∆ − e ˜ β ∆ − > λ − λ , then we can conclude an analogous result in which the roles of agent 1 and 2 areinterchanged. In our simple setting, the presence of only one traded security with constant dividendsallows for an optimal continuous-time trading strategy that is absolutely continuous ith respect to the Lebesgue measure ( dt ), even though Brownian noise enters theeconomy through the income streams. Since consumption occurs on the dt -time scale,the absolute continuity property will allow transaction costs to be paid on the same dt -time scale in the real goods market.In this section, we consider the continuous-time infinite time horizon Radner equi-librium. The only traded security is an annuity A , which is in one-net supply andavailable to trade with the proportional transaction cost rate λ ∈ [0 , r > A = 1 /r .Each of the two agents has an exogenous income stream given by Y i = ( Y it ) t , i = 1 ,
2, with dynamics dY it = µ i dt + σ i dB it , Y i ∈ R , where µ i ∈ R , σ i >
0, and B and B are possibly correlated Brownian motions. Theagents are also endowed with an initial allocation of shares in the annuity θ i ∈ R suchthat θ + θ = 1.The flow of information in the economy is given by F = ( F t ) t ≥ , where F t = σ ( B u , B u : 0 ≤ u ≤ t ). All process are assumed to be adapted to F , and all agentsshare the same filtration. We consider the single agent investment and consumption problem for agent i ’s shadowmarket, i ∈ { , } . We focus on models where the value of the shadow annuity andshadow interest rate (to be determined endogenously in equilibrium) are constants A it = A i = 1 /r i and r i >
0, respectively.For a given investment strategy θ , agent i ’s shadow wealth is defined by X it := θ t A it .For a measurable, adapted consumption process c = ( c t ) t for which R T | c t | dt < ∞ P -almost surely for all T >
0, the shadow wealth process associated with c evolves like dX cit = ( X cit r i − c t + Y it ) dt, X ci = θ i /r i ∈ R . As in the discrete-time case, we consider agents with exponential preferences overrunning consumption with risk aversion α i > β i > Definition 3.1.
Let i ∈ { , } . A consumption process c = ( c t ) t is called admissiblefor agent i if the transversality condition holds:lim t →∞ E h e − β i t − α i r i X ct − α i Y it i = 0 . In this case, we write c ∈ A i .For i ∈ { , } , agent i ’s value function is given by V i ( x, y ) := sup c ∈A i E Z ∞ U i ( t, c t ) dt, x, y ∈ R . e show in Theorem 3.2 (below) that V i = J i , where J i ( x, y ) = − r i exp − α i r i x − α i y + 1 − ˜ β i r i ! . (3.1)We note that V ∆ i ( x, y )∆ −→ J i ( x, y ) as ∆ →
0, where V ∆ i is the discrete-time valuefunction defined by (2.2).The following result establishes the individual agent optimal investment strategies.The proof is omitted, as it does not vary substantially from the discrete-time case. Theorem 3.2.
For i = 1 , , let ˜ β i := β i + α i µ i − α i σ i . The optimal consumption policyand wealth process for agent i are given by ˆ c it = r i ˆ X it + Y it + ˜ β i r i α i − α i , (3.2)ˆ X it = X ˆ c i t = θ i r i + 1 α i − ˜ β i r i ! t. (3.3) Moreoever, the value function coincides with (3.1) ; that is, V i = J i . In addition to establishing the existence of an equilibrium, we are interested in howthe equilibrium interest rate depends on λ . Definition 3.3.
For the transaction cost parameter λ ∈ [0 , equilibrium withtransaction costs is given by a collection of processes ( A i , ˆ c i , ˆ θ i ) i =1 , such that(i) For i = 1 ,
2, the optimal investment strategy ˆ θ i is differentiable in time withderivative ˆ θ ′ it .(ii) Real and financial markets clear for all t ≥ X i =1 ˆ c it = 1 + X i =1 Y it − λ (cid:12)(cid:12)(cid:12) ˆ θ ′ t (cid:12)(cid:12)(cid:12) A t and ˆ θ t + ˆ θ t = 1 , where in the event of a trade, we define A t := A it λ if agent i ∈ { , } purchases apositive number of annuity shares; i.e., ˆ θ ′ it > i = 1 ,
2, the consumption and investment strategies, ˆ c i and ˆ θ i withˆ θ i = θ i , are optimal with the annuity price A i : V i ( θ i A i , Y i ) = Z ∞ E [ U i ( t, ˆ c it )] dt. (iv) The shadow markets remain “close enough” to the underlying transaction costmarket in the following sense: For all t ≥ A t A t ∈ (cid:20) − λ λ , λ − λ (cid:21) . Moreover, if ˆ θ ′ t > A t = A t · λ − λ . If ˆ θ ′ t < A t = A t · − λ λ . emark . When trade occurs, we are able to define A t := A it λ , where agent i ∈ { , } purchases a positive number of shares, ˆ θ ′ it >
0. In this case, if we consider constantinterest rate equilibria, then the equilibrium interest rate is uniquely determined by r = 1 /A = 1 /A t .The following result establishes an equilibrium for the continuous-time model. Theproof is omitted, as it mirrors the proof of Theorem 2.4. Theorem 3.4.
Let ˜ β i := β i + α i µ i − α i σ i , and assume that ˜ β i is strictly positivefor i = 1 , . For λ ∈ [0 , , there exists an equilibrium with strictly positive constantshadow interest rates r , r and constant shadow annuity values A = 1 /r , A = 1 /r .The optimal consumption policies and wealth processes are given by (3.2) and (3.3) ,respectively. Case 1:
A no-trade equilibrium occurs if ˜ β ˜ β ∈ (cid:20) − λ λ , λ − λ (cid:21) . (3.4) In this case, r = ˜ β and r = ˜ β . The range of possible (non-shadow) interest rates thatare consistent with this equilibrium is r = ( r t ) t ≥ where r t ∈ [(1 − λ ) max( ˜ β , ˜ β ) , (1 + λ ) min( ˜ β , ˜ β )] = ∅ . Case 2:
There exists an equilibrium in which agent will purchase shares of theannuity in equilibrium at all times t ≥ (while agent sells shares) if ˜ β ˜ β > λ − λ . (3.5) In this case, the interest rate r > is determined by r = ˜ β /α + ˜ β /α α (1+ λ ) + α (1 − λ ) . (3.6) The shadow interest rates are given by r = (1 + λ ) r = (1 − λ ) r . The agents behave similarly in a continuous-time equilibrium as in discrete time.When the agents’ income-adjusted time preference parameters ˜ β and ˜ β are sufficientlyclose, as in (3.4), then the agents are not motivated to trade because of the relativelyhigh transaction costs. Their shadow interest rates reflect their individual frictionlessview of the market and do not differ significantly enough to encourage trade.Trading occurs in equilibrium only when the parameters ˜ β and ˜ β are sufficientlyfar apart to overcome the transaction costs. In (3.5), agent 1 values the annuity more(with a lower shadow interest rate) than agent 2, which encourages her to acquireshares in the annuity. The agents’ strategies are very simple: either buy or sell at thesame rate for all times.Theorem 3.5 proves that the discrete-time interest rate passes to the continuous-time equilibrium rate as the time step ∆ tends to zero. Theorem 3.5.
Suppose that ˜ β ˜ β > λ − λ . et r (∆) be the solution to (2.7) corresponding to the time step ∆ > , and let r (0) be the continuous-time equilibrium interest rate given by (3.6) . Then r (∆) > is theunique interest rate among constant interest rate equilibria for sufficiently small ∆ ,and r (∆) −→ r (0) as ∆ → .Remark . When ˜ β ˜ β > λ − λ , we can conclude analogous results to Theorem 3.4 Case2 and Theorem 3.5 in which the roles of agent 1 and 2 are interchanged. An analogousresult to Corollary 4.1 holds for ˜ β > ˜ β > In this section, we analyze the effects of transaction costs on the equilibrium interestrate and agent welfare. Our method of solving for an equilibrium is the same forzero and non-zero transaction costs, which allows us to easily compare the endogenousinterest rate as λ varies. The simplicity of the continuous-time limiting model lendsitself to further study as the transaction costs tend to zero. Even in this stylized model,the equilibrium interest rate is impacted by frictions in a non-trivial way and is notalways monotonic.Zero transaction costs and traded randomness from the stochastic income streamslead to complete markets and Pareto optimal equilibrium allocations. In our model,there is no market for the risk associated with the agents’ stochastic income streams.Consequently, the equilibrium allocation is non-Pareto optimal, even when transactioncosts are zero. In Section 4.3 below, we use certainty equivalents to compare the welfareloss due to transaction costs and unspanned income in equilibrium, similar to [8]. Bothtypes of incompleteness lead to a welfare loss. For i = 1 ,
2, we recall that the risk- and income-adjusted time preference parametersare given by ˜ β i := β i + α i µ i − α i σ i . When ˜ β and ˜ β differ and are strictly positive,then Theorem 3.4 establishes the existence of a continuous-time equilibrium in whichtrade will occur for sufficiently small transaction costs. We define a transaction costthreshold ˆ λ ∈ R by ˆ λ := (cid:0) √ α − √ α (cid:1) α − α . Corollary 4.1 describes the behavior of the equilibrium interest rate in the case whentrade occurs. The proof is contained in Section 6.
Corollary 4.1 (The Effects of Small Transaction Costs) . Suppose that ˜ β > ˜ β > . For λ ∈ [0 , ˜ β − ˜ β ˜ β + ˜ β ) , the equilibrium interest rate exists, is unique among constantinterest rate equilibria, and has the explicit form given by r = r ( λ ) = ˜ β /α + ˜ β /α α (1+ λ ) + α (1 − λ ) . The transaction cost threshold ˆ λ is strictly positive when α < α . In this case, r isstrictly increasing on (cid:16) , min (cid:16) ˜ β − ˜ β ˜ β + ˜ β , ˆ λ (cid:17)(cid:17) and strictly decreasing on (cid:16) min (cid:16) ˜ β − ˜ β ˜ β + ˜ β , ˆ λ (cid:17) , ˜ β − ˜ β ˜ β + ˜ β (cid:17) .The equilibrium interest rate r is strictly decreasing when α ≥ α , in which case ˆ λ ≤ . he impact of transaction costs on the equilibrium interest rate depends on theconfiguration of agent risk aversions α i and risk- and income-adjusted time preferenceparameters ˜ β i . An agent with a large ˜ β i is a planner: For small enough transactioncosts, she will choose to forgo consumption now in order to invest in shares of theannuity to ensure a future consumption stream. An agent with a small ˜ β i is a consumer:For small enough transaction costs, she will sell annuity shares in order to consumetoday, even though she will miss out on the future dividend stream.In addition to the adjusted time preference parameters, the agents’ risk aversionlevels determine the effect of transaction costs on the equilibrium interest rate andwelfare levels. Table 4.2 below outlines the four possible agent configurations thatassist in defining two equilibrium regimes. Table 4.2. small risk large riskaversion, α i > α i > Aggressive Planner Reserved Planner time preferences, invests in the annuity invests in the annuity˜ β i > Aggressive Consumer Reserved Consumer time preferences, sells the annuity sells the annuity˜ β i > β > ˜ β > Case 1: α ≥ α . The configuration ˜ β > ˜ β > α > α > Case 2: α < α . The configuration ˜ β > ˜ β > α > α > λ ) benefits the aggressive planner as she iscompensated for her desire to invest in the annuity to ensure her future consumption.Her modest appetite for risk is reflected by her risk aversion parameter α , which isdominated by α yet remains sufficiently small so that the agents are willing to tradewith ˜ β < ˜ β . λ in the Case 2 trading range [0 , ˜ β − ˜ β ˜ β + ˜ β ). The solid line plot represents Case 1 with a reservedplanner and aggressive consumer economy. The dashed line plots represent Case 2 with anaggressive planner and reserved consumer economy. When transaction costs become sufficiently large in that they go above the thresholdˆ λ , then the aggressive planner no longer receives an additional interest rate premiumfor her future planning. In this case, the interest rate decreases as transaction costs riseabove ˆ λ , and the reserved consumer begins to benefit from selling annuity shares in alower rate environment. In conclusion, for strictly positive but small transaction costs,a preference for future consumption and moderate amount of risk can be beneficial ifthe other agent is more risk averse and prefers immediate consumption. For λ > ˆ λ ,the interest rate premium declines.Figure 1 plots the equilibrium interest rate as a function of transaction costs λ in therange [0 , ˜ β − ˜ β ˜ β + ˜ β ) for three different input parameterizations and both cases describedabove. We assume that the agents in Figure 1 have identical income volatility σ i = . β i = . i = 1 ,
2; see [5]. Their risk aversions α i vary between 2 and 8, while their income drift µ i is between 0 .
02 and 0 . i = 1 , β > ˜ β >
0. Though the input parameters are in some sensereasonable and the expression for the interest rate is explicit, the effects of transactioncosts on the equilibrium interest rate vary significantly amongst the parameterizations,as illustrated in Figure 1.
Our model’s incompleteness stems from both the unspanned income streams and trans-action costs. When transaction costs are zero, it is possible to complete the marketby introducing additional financial securities. In order to measure the welfare loss dueto incompleteness, we compare our market to one with the same aggregate demand nd agent preferences, where the risks from trading are spanned in a dynamicallycomplete way by introducing risky financial assets. Since traded financial securitiesspan B and B , we denote the correlation between B and B by ρ ∈ [ − ,
1] so that d h B , B i t = ρdt .For i = 1 ,
2, we recall that U i ( t, x ) = − e − β i t − α i x . We consider a representativeagent with weight γ >
0, whose indirect utility for a given consumption stream c isgiven by the sup-convolution, U γ ( c ) := sup c ∈A ,c ∈A ∀ t, c t + c t ≤ c t E (cid:20)Z ∞ ( U ( t, c t ) + γU ( t, c t )) dt (cid:21) . We define the representative risk aversion parameter α r > β r by α r := 11 /α + 1 /α and β r := β /α + β /α /α + 1 /α . A corresponding representative utility function is given by U γ ( t, x ) = e − β r t − α r x . Forthe aggregate demand, c = Y + Y + 1, we have that U γ ( Y + Y + 1) = − E (cid:20)Z ∞ U γ ( t, Y t + Y t + 1) dt (cid:21) = − E "Z ∞ e − β r t − α r ( Y t + Y t +1) · α + α α (cid:18) α α γ (cid:19) − α α α dt = − α + α α (cid:16) α α γ (cid:17) − α α α e − α r ( y + y +1) β r + α r ( µ + µ ) − α r ( σ + σ + 2 ρσ σ ) , and the corresponding optimal individual consumptions c , c given in the sup-convolutionare c γ t = α α + α ( Y t + Y t + 1) − β − β α + α t + 1 α + α log (cid:18) α γα (cid:19) ,c γ t = α α + α ( Y t + Y t + 1) + β − β α + α t − α + α log (cid:18) α γα (cid:19) . The equilibrium state price density ξ = ( ξ t ) t is described by the first-order conditionfor the aggregate consumption by ξ t = U γc ( t,Y t + Y t +1) U γc (0 ,y + y +1) , where U γc denotes the derivativein the consumption variable of U γ . The dynamics of the state price density are givenby dξ t = − ξ t ( r compt dt + ν t dB t + ν t dB t ) , ξ = 1 , where r comp is the equilibrium interest rate, and ν and ν are the market prices ofrisk corresponding to the Brownian motions B and B , respectively. The equilibriuminterest rate in the complete market is computed using the first-order condition and ξ ’s dynamics. It is given by r comp = β r + α r ( µ + µ ) − α r σ + σ + 2 ρσ σ ) . e note that ξ , r comp , ν , and ν are independent of the weighting superscript γ . Sincethe agents’ preferences are described by exponential utility functions, these terms donot depend in equilibrium on γ .We are interested in the welfare level of a complete market economy and will usethe sum of the agents’ certainty equivalents as a proxy for welfare. Definition 4.3.
For i = 1 , γ >
0, a value CE compi is called the certainty equivalent for agent i if Z ∞ U i ( t, CE compi ) dt = − Z ∞ U i ( t, c γit ) dt, where c γi is agent i ’s optimal consumption stream. We write CE compi ( γ ) to emphasizethe dependence of the certainty equivalent on γ .The certainty equivalent represents a constant consumption stream level that anagent is willing to exchange for her optimal (stochastic) consumption stream. For i ∈ { , } and γ >
0, the certainty equivalent is given by CE compi ( γ ) = 1 α i log (cid:18) r comp β i (cid:19) + c γi . The sum of the certainty equivalents can be used as a welfare measure in the economy.In the complete market equilibrium, for initial wealth stream values y , y ∈ R andrepresentative agent weight γ >
0, we have CE comp ( γ ) + CE comp ( γ ) = 1 + y + y + 1 α log (cid:18) r comp β (cid:19) + 1 α log (cid:18) r comp β (cid:19) . We note that the sum of the certainty equivalents does not depend on the weight γ > In this section, we compare the agents’ welfare loss due to transaction costs. Similar tothe approach in [8], we use the sum of the agents’ certainty equivalents as our measureof welfare. We find that the introduction of transaction costs causes a strict loss ofwelfare.Our results contrast [8], which performs a related analysis in a one-period continuum-of-agents model with heterogenous beliefs on a risky asset, an exogenous riskless asset,and no income. [8] finds that under some parameter specifications, a strictly posi-tive transaction cost can provide a welfare gain. Our results from Section 4.1 showthat it may be possible for the interest rate to increase for a strictly positive level oftransaction costs, yet the welfare itself cannot be recouped.
Definition 4.4.
For i = 1 , λ ∈ [0 ,
1) and ( x, y ) ∈ R , a value CE i is called the certainty equivalent for agent i at transaction cost level λ , initial wealth x , and initialincome level y if Z ∞ U i ( t, CE i ) dt = V i ( x, y ) . We write CE i ( λ ) to emphasize the dependence of the certainty equivalent on λ . he certainty equivalent represents a constant consumption stream level that anagent is willing to exchange for her optimal (stochastic) consumption stream. For agiven shadow interest rate r i >
0, the certainty equivalent can be expressed as CE i = 1 α i log (cid:18) r i β i (cid:19) + ˆ c i , where ˆ c i is the optimal consumption level at time 0 and is given in (3.2).In equilibrium, we are interested in the sum of our agents’ certainty equivalents asa proxy for the welfare of the economy. For ˜ β > ˜ β > CE ( λ ) + CE ( λ ) = y + y + α (cid:16) log (cid:16) r ( λ ) β (1+ λ ) (cid:17) + ˜ β (1+ λ ) r ( λ ) − (cid:17) + α (cid:16) log (cid:16) r ( λ ) β (1 − λ ) (cid:17) + ˜ β (1 − λ ) r ( λ ) − (cid:17) , if λ < ˜ β − ˜ β ˜ β + ˜ β , y + y + α log (cid:16) ˜ β β (cid:17) + α log (cid:16) ˜ β β (cid:17) , if λ ≥ ˜ β − ˜ β ˜ β + ˜ β . The following result states that the economy’s welfare is decreasing in incompletenessdue to transaction costs and unspanned income. The proof of Proposition 4.5 is givenin Section 6.
Proposition 4.5.
Suppose that ˜ β > ˜ β > . CE ( λ ) + CE ( λ ) is strictly decreasingon [0 , ˜ β − ˜ β ˜ β + ˜ β ) and constant on [ ˜ β − ˜ β ˜ β + ˜ β , . Moreoever, complete market welfare levelsdominate incomplete market welfare levels in that for all transaction costs λ ∈ [0 , and weights γ > , we have CE ( λ ) + CE ( λ ) < CE comp ( γ ) + CE comp ( γ ) . Figure 2 plots the economy’s welfare change due to incompleteness as a function oftransaction costs λ for the three input parameterizations that were used in Figure 1.We assume that the income stream correlation is zero in that h B , B i t = 0 as in [5].The given parameter specifications allow for each agent to have dominant risk aversionwhile still ensuring that the income-adjusted time preference parameters are orderedby ˜ β > ˜ β >
0. Regardless of the input parameterizations, the sum of the certaintyequivalents, CE + CE , is decreasing in the transaction costs. When no trading occurs,the certainty equivalent sum is constant in transaction costs. The economy’s welfarealways decreases when moving from a complete to incomplete market. Though theinterest rate is possibly non-monotone in the transaction cost level as in Figure 1, thewelfare in the economy only decreases. Transaction costs in our model prevent us from trading freely between the annuity anda bank account. Using an annuity as our traded security allows for constant shadowinterest rates and trading strategies that are the same at every time point: either theagents buy, sell, or trade nothing. The simple structure of transaction cost equilibriawith a traded annuity is not possible when the bank account is traded instead.In this section, we consider a discrete-time equilibrium with transaction costs whenthe bank account is the traded security. Theorem 5.2 proves that the traded bankaccount model prevents a constant-interest rate transaction cost equilibrium.
In contrast to the annuity, the bank account is a financial asset in zero-net supply.For i = 1 ,
2, the shadow bank account B i has the associated interest rate process r i = ( r it n ) n ≥ and is given by B i = 1 and B it n = (1 + r i ∆) · . . . · (1 + r it n − ∆) , n ≥ . We focus on equilibria yielding constant shadow interest rates r it n = r i , as in the tradedannuity case.For a given investment strategy θ , agent i ’s shadow wealth is given by X it n := θ t n B it n . Since the bank account is in zero-net supply, the self-financing condition in(2.1) will be replaced by( θ t n +1 − θ t n ) B it n +1 = ( Y it n − c t n + θ t n ) ∆ , n ≥ . Thus, for a given consumption and investment strategy ( c, θ ), the shadow wealthevolves like X cit n +1 − X cit n = (cid:0) X cit n r i + Y it n − c t n (cid:1) ∆ , X i = θ B i = θ . The definitions of admissibility and the value function are unchanged from Defini-tion 2.1 and (2.2). As such, Theorem 2.2 holds for the frictionless shadow market witha bank account carrying a constant interest rate.
Definition 5.1.
For the transaction cost parameter λ ∈ [0 , transaction costequilibrium with a bank account is given by a collection of processes ( r i , ˆ c i , ˆ θ i ) i =1 , suchthat(i) Real and financial markets clear for each n ≥ X i =1 ˆ c it n ∆ = X i =1 Y it n ∆ − λ (cid:12)(cid:12)(cid:12) ˆ θ t n +1 − ˆ θ t n (cid:12)(cid:12)(cid:12) B t n +1 and ˆ θ t n + ˆ θ t n = 0 , here in the event of a trade, we define B t n +1 := B itn +1 λ if agent i ∈ { , } purchases a positive number of annuity shares; that is, θ it n +1 − θ it n > i = 1 ,
2, the consumption and investment strategies, ˆ c i and ˆ θ i withˆ θ i = θ i , are optimal with the shadow bank account value B i : V ∆ i ( θ i ) = − ∞ X n =0 E h e − β i t n e − α i ˆ c itn i . (iii) The shadow markets remain “close enough” to the underlying transaction costmarket in the following sense: For each n ≥ B t n B t n ∈ (cid:20) − λ λ , λ − λ (cid:21) . Moreover, if ˆ θ t n − ˆ θ t n − > B t n = B t n · λ − λ . If ˆ θ t n − ˆ θ t n − < B t n = B t n · − λ λ .Theorem 5.2 shows that aside from a stylized special case, any transaction costequilibrium with a bank account must have non-constant interest rates. The proof ispresented in Section 6. Theorem 5.2.
Let ˜ β i := β i + α i µ i − α i σ i , and suppose that ˜ β i and λ are strictlypositive for i = 1 , . Suppose that ( r i , ˆ c i , ˆ θ i ) i =1 , is a transaction cost equilibrium witha bank account. If r , r are strictly positive constants, then the following must hold:(1) The agents’ parameters satisfy ˜ β = ˜ β .(2) No trading occurs in equilibrium: ˆ θ it n − ˆ θ it n − = 0 for i = 1 , and n ≥ .(3) The shadow rates are identical and satisfy r = r = 1∆ (cid:16) e ˜ β ∆ − (cid:17) . Though it is possible to consider stochastic interest rates in a transaction costequilibrium with a bank account using a system of variational inequalities, it is notclear if such an equilibrium exists. The simple mathematical structure of the annuitycannot be obtained by studying a traded bank account in its place. The annuityprovides a constant dividend stream at all possible consumption times, which allowsagents to receive constant future dividends without trading or incurring transactioncosts.
We begin by proving Theorem 2.2 in the discrete-time case.
Proof.
We check that ˆ c i is admissible, by noting that E h exp (cid:16) − β i t n − α i r i X ˆ c i it n − α i Y it n (cid:17)i = E " exp − α i r i X i − n log(1 + r i ∆) − α i σ i n ∆ − α i n X k =1 √ ∆ σ i Z it k ! = (1 + r i ∆) − n exp ( − α i r i X i ) −→ n → ∞ . e have that c
7→ − e − α i c + e − β i ∆ E h J (cid:16) x (1 + r i ∆) + y − c, y + µ i ∆ + σ i √ ∆ Z (cid:17)i ismaximized for ˆ c = ˆ c ( x, y ) = r i x + y + ˜ β i α i r i − α i r i ∆ log(1 + r i ∆), where Z denotes astandard normal random variable. Thus, ( − n − X k =0 e − α i c tn + e − β i t n J ∆ i (cid:0) X cit n , Y it n (cid:1)) n ≥ is a supermartingale for all c ∈ A ∆ i and is a martingale for c = ˆ c i ∈ A ∆ i .Therefore, for ˆ c i , J ( x, y ) = − E " n X k =0 e − α i ˆ c itk + e − β i t n +1 E h J ∆ i (cid:16) X ˆ c i it n +1 , Y it n +1 (cid:17)i = − E " ∞ X k =0 e − α i ˆ c itk by the transversality condition , which implies that J ∆ i ≤ V ∆ i . Similarly, for any c ∈ A ∆ i , J ( x, y ) ≥ − E " n X k =0 e − α i c tk + e − β i t n +1 E h J ∆ i (cid:16) X cit n +1 , Y it n +1 (cid:17)i = − E " ∞ X k =0 e − α i c tk by the transversality condition . Thus, J ∆ i = V ∆ i , and ˆ c i ∈ A ∆ i is the optimal consumption policy. For initial wealth x = θ i A i = θ i /r i , the optimal wealth policy corresponding to ˆ c i is ˆ X i = X ˆ c i i , andˆ X it n = θ i r i + t n α i r i (cid:18)
1∆ log (1 + r i ∆) − ˜ β i (cid:19) . We now move towards the proof of Theorem 2.4. The self-financing condition (2.1)with the optimal policies (2.2) and (2.3) imply (cid:16) Y it n − ˆ c it n + ˆ θ it n (cid:17) ∆ = (cid:16) ˆ θ it n − ˆ θ it n − (cid:17) A it n = 1 α i r i (cid:16) log(1 + r i ∆) − ˜ β i ∆ (cid:17) . (6.1)For i = 1 ,
2, we define F i ( r ) := 1 α i r (cid:16) log (1 + r ∆) − ˜ β i ∆ (cid:17) , r > . (6.2)Using Definition 2.3 part (i), we seek solutions r , r > F ( r )+ F ( r ) = λ (cid:18) | F ( r ) | λ + | F ( r ) | − λ (cid:19) { F ( r ) ≥ } + λ (cid:18) | F ( r ) | − λ + | F ( r ) | λ (cid:19) { F ( r ) < } . By rewriting this equation and including Condition (iii) from Definition 2.3, we seek r , r > F ( r ) = ( − − λ λ F ( r ) , if F ( r ) ≥ , − λ − λ F ( r ) , if F ( r ) ≤ . (6.3) nd r r = λ − λ , if F ( r ) > , − λ λ , if F ( r ) < , ∈ h − λ λ , λ − λ i , if F ( r ) = 0 , (6.4) Proposition 6.1.
Let ˜ β i := β i + α i µ i − α i σ i / , and suppose that ˜ β i is strictly positivefor i = 1 , . There exists a unique strictly positive solution pair r , r to (6.3) and (6.4) . Case 1: If e ˜ β − e ˜ β − ∈ h − λ λ , λ − λ i , then r = e ˜ β ∆ − and r = e ˜ β ∆ − . (6.5) Case 2:
If we have e ˜ β − e ˜ β − > λ − λ , then the unique positive solutions satisfy r ∈ e ˜ β ∆ − , − λ λ e ˜ β ∆ − !! and r = 1 + λ − λ r . Proof.
We show the existence of the unique solution pair by examining both cases.Suppose that e ˜ β ∆ − e ˜ β ∆ − ∈ (cid:20) − λ λ , λ − λ (cid:21) . Then r and r as in (6.5) is the unique solution to (6.3) and (6.4) such that F ( r ) = F ( r ) = 0.To show uniqueness, we proceed by contradiction. Assume for the sake of contra-diction that there exist strictly positive solutions r , r such that F ( r ) >
0. We havethat F ( r ) > F ( r ) < r ∆ > e ˜ β ∆ −
1, and r ∆ < e ˜ β ∆ −
1. Thenby (6.4), e ˜ β ∆ − e ˜ β ∆ − < r r = 1 − λ λ ≤ e ˜ β ∆ − e ˜ β ∆ − , which is a contradiction. Here, we have used that ˜ β , ˜ β are strictly positive to ensurethat e ˜ β i ∆ − >
0. The same argument applies to rule out the case when F ( r ) < F ( r ) >
0. Therefore, we must have that F ( r ) = F ( r ) = 0, in which case r = e ˜ β − and r = e ˜ β − .We now consider the existence of a solution in Case 2. For F ( r ) >
0, (6.3) and(6.4) reduce to solving for r > (cid:18) r ∆ · λ − λ (cid:19) /α (1 + r ∆) /α = exp ˜ β α + ˜ β α ! ∆ ! , (6.6)while r = λ − λ · r . The assumption that ˜ β , ˜ β are strictly positive ensures that theright hand side of (6.6) is strictly bigger than 1. We note that x (cid:16) x · λ − λ (cid:17) /α (1 + x ) /α trictly increases from 1 to ∞ for x ∈ [0 , ∞ ). Thus, there exists a unique solution r > e ˜ β − e ˜ β − > λ − λ implies that r ∈ e ˜ β ∆ − , − λ λ e ˜ β ∆ − !! . We show uniqueness for Case 2 by contrapositive, which will rule out the possibilityof finding solutions for which F ( r ) ≤
0. Suppose that there exist strictly positivesolutions r , r such that F ( r ) ≤
0. Since F ( r ) ≤ F ( r ) ≥ r ∆ ≤ e ˜ β ∆ −
1, and r ∆ ≥ e ˜ β ∆ −
1, we have that1 + λ − λ ≥ r r ≥ e ˜ β ∆ − e ˜ β ∆ − , as desired. Proof of Theorem 2.4.
By Theorem 2.2 and Definition 2.3, we must solve (6.3) and(6.4) for the equilibrium shadow interest rates. Proposition 6.1 provides us with theexistence and uniqueness of positive shadow interest rates, as desired.The agents choose not to trade in Case 1, and the market interest rate cannotbe uniquely determined. The annuity values A consistent with this equilibrium mustsatisfy A ∈ (cid:20) A λ , A − λ (cid:21) ∩ (cid:20) A λ , A − λ (cid:21) = (cid:20) max( A , A )1 + λ , min( A , A )1 − λ (cid:21) . Since A i = ∆ e ˜ βi ∆ − for i = 1 ,
2, we can rewrite the above interval as A ∈ ∆(1 + λ ) (cid:16) e min( ˜ β , ˜ β )∆ − (cid:17) , ∆(1 − λ ) (cid:16) e max( ˜ β , ˜ β )∆ − (cid:17) . This interval is nonempty by (2.5). Since A = 1 /r , we have that r ∈ (cid:20) − λ ∆ (cid:16) e max( ˜ β , ˜ β )∆ − (cid:17) , λ ∆ (cid:16) e min( ˜ β , ˜ β )∆ − (cid:17)(cid:21) = ∅ . Trading occurs in Case 2, in which case we are able to determine a unique marketinterest rate. When e ˜ β ∆ − e ˜ β ∆ − > λ − λ , we have that r = λ − λ · r while r > /r = A = A (1 + λ ) = (1 + λ ) /r , which implies that r = (1 + λ ) r . Similarly, r = (1 − λ ) r .Therefore, the market interest rate r > (cid:18) r ∆1 − λ (cid:19) α (cid:18) r ∆1 + λ (cid:19) α = e (cid:16) ˜ β α + ˜ β α (cid:17) , and the shadow interest rates are given in terms of r = r (1 + λ ) = r (1 − λ ) . n continuous time, the proofs of Theorems 3.2 and 3.4 mirror their discrete-timecounterparts. The continuous-time analog of F i defined in (6.2) is given for i = 1 , F i ( r ) := 1 α i − ˜ β i r ! , r > . We now prove Theorem 3.5.
Proof of Theorem 3.5.
Since ˜ β / ˜ β > λ − λ , Theorem 3.4 shows that trade occurs for∆ = 0 and r (0) is given uniquely by (3.6). Moreover, e ˜ β ∆ − e ˜ β ∆ − −→ ˜ β ˜ β as ∆ → >
0. In thiscase, r (∆) > , r ) ∈ [0 , ∞ ) × (0 , ∞ ), we define G (∆ , r ) := (cid:16) r ∆1+ λ (cid:17) α (cid:16) r ∆1 − λ (cid:17) α , for ∆ > (cid:16) r (cid:16) α (1+ λ ) + α (1 − λ ) (cid:17)(cid:17) , for ∆ = 0.For sufficiently small ∆ > r (∆) is chosen such that G (∆ , r (∆)) =exp (cid:16) ˜ β α + ˜ β α (cid:17) . Since G is smooth on [0 , ∞ ) × (0 , ∞ ) and ∂G∂r (0 , r (0)) = 0 (a one-sidedderivative), the implicit function theorem implies that r (∆) −→ r (0) as ∆ → Proof.
By Theorem 3.4, an equilibrium with trade will occur for λ ∈ h , ˜ β − ˜ β ˜ β + ˜ β (cid:17) , inwhich agent 1 buys shares of the annuity, agent 2 sells shares of the annuity, and theequilibrium interest rate is given by r ( λ ) = ˜ β /α + ˜ β /α α (1+ λ ) + α (1 − λ ) . Differentiating in λ , we see that r has local extrema at λ − = (cid:0) √ α − √ α (cid:1) α − α and λ + = (cid:0) √ α + √ α (cid:1) α − α . When α > α , we have that both λ − , λ + <
0. In this case, r is strictly decreasing on h , ˜ β − ˜ β ˜ β + ˜ β (cid:17) . When α < α , we have that λ − ∈ (0 , λ + > r is strictly increasingon h , min (cid:16) λ − , ˜ β − ˜ β ˜ β + ˜ β (cid:17)(cid:17) , and r is strictly decreasing on (cid:16) min (cid:16) λ − , ˜ β − ˜ β ˜ β + ˜ β (cid:17) , ˜ β − ˜ β ˜ β + ˜ β (cid:17) .Recognizing that ˆ λ = λ − yields the desired result.We next prove Proposition 4.5. roof. The sum of the incomplete market certainty equivalents is differentiable on[0 , ˜ β − ˜ β ˜ β + ˜ β ). A calculation of the derivative yields CE ′ ( λ ) + CE ′ ( λ ) = −
2( ˜ β (1 − λ ) − ˜ β (1 + λ ))( α (1 + λ ) + α (1 − λ ) )( α (1 + λ ) + α (1 − λ ))(1 + λ ) (1 − λ ) ( ˜ β α + ˜ β α ) . Since ˜ β (1 − λ ) − ˜ β (1+ λ ) is strictly positive for λ ∈ [0 , ˜ β − ˜ β ˜ β + ˜ β ), we have that CE + CE is strictly decreasing. On [ ˜ β − ˜ β ˜ β + ˜ β , CE + CE is constant.We next verify that CE ( λ )+ CE ( λ ) < CE comp ( γ )+ CE comp ( γ ), and we recall thatthe right hand side does not depend on γ . Since λ CE ( λ ) + CE ( λ ) is decreasing,it suffices to check that the inequality holds for λ = 0.By algebra, we have that CE ( λ ) + CE ( λ ) < CE comp ( γ ) + CE comp ( γ ) if and onlyif r comp > r (0), which in turn holds if and only if α α σ + α α σ − ρσ σ > . When σ σ ≥
0, this inequality holds since α α σ + α α σ − ρσ σ ≥ α α σ + α α σ − σ σ = (cid:18)r α α σ − r α α σ (cid:19) > . The case when σ σ < Proof of Theorem 5.2.
Assume that r , r are strictly positive constants. By modifying(6.1) to account for a traded bank account, we arrive at the same form of F i as in (6.2).By Definition 5.1 (iii), r and r must satisfy (6.3) and (cid:18) r r (cid:19) n ∈ (cid:20) − λ λ , λ − λ (cid:21) , for n ≥ , while for each n ≥ B t n = B t n · λ − λ if ˆ θ t n − ˆ θ t n − >
0, and B t n = B t n · − λ λ if ˆ θ t n − ˆ θ t n − <
0. Since λ = 0, we must have that r = r and ˆ θ t n − ˆ θ t n − =ˆ θ t n − ˆ θ t n − = 0 for all n ≥
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