Existence and Uniqueness of Recursive Utility Models in L p
TThe Australian National University
Research School of Economics
Honours Thesis
Existence and Uniqueness of RecursiveUtility Models in L p Author
Flint
O’Neil
Supervisor
Prof. John
Stachurski
A Thesis Submitted in Partial Fulfilmentof the Requirements for the Degree ofBachelor of Economics (Honours) a r X i v : . [ ec on . T H ] M a y May 15, 2020
E C L A R AT I O NThis thesis contains no material which has been accepted for the award of any other degreeor diploma in any University, and, to the best of my knowledge and belief, contains nomaterial published or written by another person, except where due reference is made inthe thesis. Flint O’NeilMay 15, 2020 C K N O W L E D G E M E N T S*Redacted* B S T R A C TRecursive preferences, of the sort developed by Epstein and Zin [1989], play an integralrole in modern macroeconomics and asset pricing theory. Unfortunately, it is non-trivialto establish the unique existence of a solution to recursive utility models. We show thatthe tightest known existence and uniqueness conditions can be extended to (i) Schorfheideet al. [2018] recursive utilities and (ii) recursive utilities with ‘narrow framing’. Further,we sharpen the solution space of Borovička and Stachurski [2019] from L to L p so thatthe results apply to a broader class of modern asset pricing models. For example, using L Hilbert space theory, we find the class of parameters which generate a unique L solutionto the Bansal and Yaron [2004] and Schorfheide et al. [2018] models. O N T E N T SContents 4 Limitations to von Neumann-Morgenstern Preferences 9
Models of Recursive Preferences 10
Existence and Uniqueness 11 Setup and Intuition 13
Recursive Utility with Time Preference Shocks 15
Recursive Utility with Narrow Framing 17
Unbounding the State Space 20 Bansal-Yaron with Constant and Stochastic Volatility 23
Mehra-Prescott [1985] and Epstein-Zin [1990] 28
Schorfheide, Song and Yaron [2018] 29 a appendix a.1 General Mathematical Results 33 a.2
Proofs: Recursive Utility With Time Preference Shocks 40 a.3
Proofs: Recursive Utility with Narrow Framing 50 a.4
Proofs: Recursive Utility on an Unbounded State Space 52 a.5
Proofs from Chapter 4 57Bibliography 60I N T R O D U C T I O N “If the theory disagrees with the data, you throw out the data” - Rabee Tourky
Economic agents often make decisions under uncertainty. Whether it be firms calculatingoptimal inventories or hedge funds seeking to maximise a portfolio’s value, choice isinextricably linked to risk. Generally, economists analyse risk by assuming agents haveadditively separable preferences over states of nature. As a result, an agent’s risk aversioncannot be disentangled from their elasticity of substitution. This inability to isolate riskpreferences has consistently challenged economic models. In finance for example, Mehraand Prescott [1985], Weil [1990] and Bansal and Yaron [2004] note that standard economictheories of risk struggle to account for the 6% equity risk premium.This motivates our interest in the Epstein and Zin [1989] class of recursive utility functions.Whereas additively separable preferences combine risk aversion and intertemporal substi-tution, Epstein-Zin recursive preferences disaggregate these two forces. This distinctionmeans that Epstein-Zin recursive preferences are particularly powerful in asset pricing andlong-run risk models. Indeed, Roger Farmer notes that in the context of finance theory,“the dominant view [...] is that people maximize the [...] value of [...] preferences firstformalized by Epstein and Zin.” The dominance of Epstein-Zin recursive utilites in finance stems from their success inexplaining empirical asset pricing facts. For example, Bansal and Yaron [2004] puts forwardan economic mechanism of long-run risk which relies on an Epstein-Zin specification. Pohl These comments can be found on Roger Farmer’s personal blog at:http://rogerfarmerblog.blogspot.com/2015/07/behavioural-economics-and-exotic.html ntroduction et al. [2018] describes this to be the “foundation for a large literature on the ability oflong-run risk to solve empirical puzzles”. Simultaneously, Bansal and Yaron [2004] apply the Campbell and Shiller [1988] log-linearisation technique to analyse recursive utilities. But Pohl et al. [2018] show thatlong-run risk models exhibit economically significant non-linearities. Thus, log-linearisationintroduces large numerical errors. This highlights the necessity of understanding the fullEpstein-Zin recursive utility model, not just a linear approximation.Unfortunately, it is difficult to establish the existence of a unique solution to Epstein-Zin recursive utilities. This poses a severe limitation; without a unique solution, utilityrepresentations lack any informative content. Borovička and Stachurski [2019] provideexistence and uniqueness conditions which are both necessary and sufficient for classicEpstein-Zin recursive utilities. The conditions are “as tight as possible in a range ofempirically plausible settings” , though can be hard to evaluate analytically. This paperaims to build upon the results derived by Borovička and Stachurski [2019].This paper’s central results are theorems 3.2.2, 3.4.1, and proposition 3.3.1. These theoremsprovide conditions for existence and uniqueness of recursive utilities. The closest existingresult in the literature is Borovička and Stachurski [2019], theorem 3.1. The theorems inthis thesis extend the Borovička and Stachurski [2019] result in two significant ways.First, this paper’s results consider newer classes of recursive utility. Specifically, we considerrecursive utilities with (i) ‘time preference’ shocks and (ii) narrow framing. The latterconsideration has troubled economists for some time, with Guo and He [2019] havingonly been able to establish existence on a finite state space. Our results thus providemathematical foundation to recent asset pricing papers, including Albuquerque et al.[2016], Schorfheide et al. [2018] and Barberis et al. [2006].Second, our results generalise the solution space from L ( X ) to L p ( X ) . This generalisationis important in both an applied and theoretical sense. Regarding application, L solutionscan be of limited practical use due to infinite second moments. When one solves for theutility value of a consumption stream, the result is a ‘price’ over the stream. This price Comments found in Pohl et al. [2018] Comment found in Borovička and Stachurski [2019] ntroduction is typically a function of the current (Markovian) state. If the resulting price functionhas infinite variance, then it is of limited use to econometricians. By extending to higher L p , we guarantee finite moments and thus provide conditions which are more useful toempirical analysis.Regarding theory, Mandelbrot [1963] and others identify that asset pricing data is typicallyheavy tailed. This means that it is important to consider unbounded state spaces, asbounded approximations can be misinformative. By widening the solution space to L p ,our results apply to a larger set of unbounded asset pricing specifications. This is becausecompactness conditions, needed for regularity, are more readily met in higher L p . Toillustrate, this paper tackles (partially) unbounded specifications of the Bansal and Yaron[2004] and Schorfheide et al. [2018] models in L . These are major long-run risk papers,and existence of a unique solution has been an active question. Our results establishingexistence and uniqueness here are thus a major development.This paper is structured as follows. Chapter 2 canvasses recent developments in therecursive utility literature. Chapter 3 presents the main findings. Chapter 4 applies thesefindings to the major long-run risk models. Chapter 5 is dedicated to discussing results.The appendix contains the vast majority of mathematical proofs. Although this paper isprimarily theoretical, it is theoretical with a view towards results rather than technique.IL I T E R AT U R E R E V I E W “You only talk about Bansal and Yaron [2004]. Why don’t you ever ask me about my day?”- Laksshini Sundaramoorthy The Von Neumann and Morgenstern [1944] (vNM) expected utility representation is theworkhorse model of decision-making under uncertainty. Time is indexed discretely by t ∈ N and streams of risk-contingent consumption are ranked according to V t = E t (cid:20) ∞ X t = β t u ( c t ) (cid:21) (2.1.1)where β ∈ (
0, 1 ) , and u ( · ) is the one-period utility function.Under standard assumptions on u ( · ) (cid:62) V nt = n X t = β t u ( c t ) % ∞ X t = β t u ( c t ) = V t .Applying the monotone convergence theorem, we may rewrite (2.1.1) as V t = ∞ X t = E t (cid:20) β t u ( c t ) (cid:21) . models of recursive preferences This last expression is more useful as it can be written recursively as V t = u ( c t ) + β E ( V t + ) . (2.1.2)Although this utility representation is both simple and powerful, it is not without limitations.One significant drawback is that the vNM representation does not disentangle risk aversionfrom preferences over intertemporal substitution. Accordingly, vNM utilities have notfound total empirical success in financial economics and macroeconomics (e.g see Mehraand Prescott [1985] and Hansen and Singleton [1983]). Recursive preferences, pioneered by Kreps and Porteus [1978], Epstein and Zin [1989] andWeil [1990], generalise equation (2.1.2). In the (2.1.2) recursion, present value is a functionof consumption ‘today’ and value ‘tomorrow’. Drawing from this notion, Epstein and Zin[1989] define the class of stationary recursive preferences as V t = W [ c t , f ( V t + )] .This specification consists of two main components: a time aggregator representing timepreference, W , and a ‘Kreps-Porteus’ certainty equivalent capturing risk aversion, f . Todistinguish intertemporal substitution from risk-aversion, Epstein-Zin utilities allow forpreference over the timing of the resolution of uncertainty. Epstein-Zin utilities are integral to modern financial economics, particularly long-run riskmodels. Bansal and Yaron [2004] use Epstein-Zin to generate time-varying risk premia tojustify the ‘excess volatility’ of asset prices identified in Shiller [1981]. Subsequent long-runrisk contributions relying on Epstein-Zin utilities include Hansen et al. [2008], Bansal et al.[2012], Bansal et al. [2014] and Schorfheide et al. [2018] among others.Epstein-Zin utilities are also fundamental to macroeconomics. For instance, Tallarini [2000]and Dolmas [1998] examine the welfare effects of business cycles on agents with Epstein-Zin In this recursion an agent’s optimal decisions will be dynamically consistent. This is an important propertyfor most economic models. As noted by Backus et al. [2005], preferences derived this way are stationary and dynamically consistent. .3 existence and uniqueness utility. Both papers illustrate how models incorporating vNM utility underestimate thewelfare costs of macroeconomic volatility.Nevertheless, recursive utility is not without its flaws. As Campbell and Ammer [1993]and Cochrane [2011] note, variation in asset returns is overwhelmingly due to variationin discount factors. Although recursive utilities are able to isolate risk aversion, they donot consider time-varying discount factors. To address this shortcoming, Albuquerqueet al. [2016] add ‘time-preference’ discount shocks to the recursive utility valuation. Thisaugmentation has seen some success. For example, the long-run risk model of Schorfheideet al. [2018] uses ‘time preference’ shocks to estimate asset price persistence. In recursive utility models, a consumption stream’s value is found by solving a nonlinear,forward-looking difference equation. It is thus non-trivial to establish the existence of aunique solution. Originally, sufficient conditions were provided by Epstein and Zin [1989],and then built upon by Marinacci and Montrucchio [2010] and Pohl et al. [2019]. However,the proposed conditions require the asymptotic consumption rate, C t + C t , to be almost surely bounded. This in turn renders recursive utilities inapplicable to most asset pricing models.To achieve a tighter result, Borovička and Stachurski [2019] exploit a link between recursiveutilities and a Perron-Frobenius eigenvalue problem. The authors show that a uniquesolution can be found by considering the average “across all paths” . This condition ismuch weaker than requiring uniform bounds on the upper tail of the distribution.Borovička and Stachurski [2019] build upon Hansen and Scheinkman [2012]. AlthoughHansen and Scheinkman [2012] treat unbounded consumption paths, they only showthe existence of a solution for some preference parameters. By contrast, Borovička andStachurski [2019] allow for all parameters, while also establishing sufficient and necessaryconditions.This paper extends Borovička and Stachurski [2019], to make two contributions. First,we establish parallel results for recursive utilities with (i) time preference shocks and (ii) This comment is found in Borovička and Stachurski [2019] .3 existence and uniqueness narrow framing. The latter consideration in particular has confounded economists forsome time, with Guo and He [2019] having only been able to establish existence on afinite state space. Second, we generalise the solution space from L ( X ) to L p ( X ) . Asmentioned earlier, this allows for (i) solutions with greater empirical content and (ii)broader applications in an unbounded state space. Indeed, this thesis proves new resultsfor Bansal and Yaron [2004] and Schorfheide et al. [2018].IIR E C U R S I V E U T I L I T Y M O D E L S : E X I S T E N C E , U N I Q U E N E S SA N D S TA B I L I T Y “One day you’ll realise that theory should have empirical content.” - Tim Kam The Epstein and Zin [1989] model of recursive utility defines preferences by V t = (cid:20) ( − β ) C − / ψt + β {R t ( V t + ) } − / ψ (cid:21) / ( − / ψ ) (3.1.1)where β ∈ (
0, 1 ) is a time discount factor, { C t } is a consumption path and V t is the utilityvalue of the path extending from time t . The scalar ψ = R t is the Kreps-Porteus certainty equivalentdefined by R t ( V t + ) = ( E t V − γt + ) / ( − γ ) . (3.1.2)where γ = timing of the resolution of uncertainty (TRU). This breaks the link between risk aversion and IES.The function R t is a certainty equivalent of future utility because Epstein and Zin [1989]implicitly take u ( c t ) = c − / ψt . The certainty equivalent is over utilities , not consumption ,as a result of preference over the TRU. In the Epstein-Zin representation, the agent choosesto trade-off between utility ‘today’ and a certainty equivalent of value ‘tomorrow’. setup and intuition In Schorfheide et al. [2018], the Epstein-Zin recursion is modified to include a ‘timepreference shock’, λ t , so that lifetime value takes the form V t = (cid:20) ( − β ) λ t C − / ψt + β {R t ( V t + ) } − / ψ (cid:21) / ( − / ψ ) (3.1.3)The shocks { λ t } ∞ t = are a function of the state process. They are restricted to attain valuesin a compact set.In this paper, we surmise a general Markov environment on a state space X . This settinginvolves two mathematical assumptions. Assumption 3.1.1.
Consumption growth is specified according toln ( C t + / C t ) = κ ( X t , X t + , (cid:15) t + ) (3.1.4)where κ is continuous and { X t } ⊂ X is the exogenous Markov state process. The innovationprocess { (cid:15) t } is IID on R k , independent of { X t } .We let p ( x , · ) represent the stochastic transition kernel for X t + given that X t = x . As-sumption 3.1.1 is standard in the literature: see Hansen and Scheinkman [2012], Borovičkaand Stachurski [2019] and Guo and He [2019]. Assumption 3.1.2.
The transition kernel p is jointly continuous in its arguments. More-over, for some ‘ > p ‘ is everywhere positive. This ‘irreducibility’ assumption ensuresergodicity, and hence { X t } converges to a unique stationary distribution, which we denoteby π .Let F denote the standard Borel σ − algebra on X , and recall that π is the stationarydistribution of { X t } ⊂ X . For some p (cid:62)
1, we say that L p ( X , F , π ) is the space of(equivalence classes of) measurable functions f satisfying R | f ( x ) | p dπ < ∞ . We let L p ( X , F , π ) + denote the subset of these functions that are almost everywhere positive.We write L p ( X ) + for shorthand.The L p norm of a function f ∈ L p ( X ) is given by || f || p = R | f ( x ) | p dπ = E π | f | p . Foreach consumption process C = { C t } t ∈ N , consider the L p ( X , π ) norm of a long-run meanconsumption growth rate given bylim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( E x (cid:18) C n C (cid:19) − γ )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / np = lim n →∞ E π ( E x (cid:18) C n C (cid:19) − γ ) p ! / np = M − γC , p . (3.1.5) .2 recursive utility with time preference shocks where x ∈ X denotes some starting state, and π is the stationary distribution governingthe process { X t } . Proposition 3.2.1 shows existence of this expression. Let θ = − γ − / ψ .This paper’s results centre around a corresponding value Λ p = β M / θC , p .In the L ( X ) setting considered in Borovička and Stachurski [2019], p = n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( E x (cid:18) C n C (cid:19) − γ )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / n = lim n →∞ ( E π (cid:18) C n C (cid:19) − γ ) / n = M − γC ,1 . (3.1.6)Equation (3.1.6) is much simpler than the expression in (3.1.5), both in terms of numericalimplementation and economic intuition. Nevertheless, considering equation (3.1.5) willallow for a bigger set of applications and sharper solutions.We now consider the case where X is a compact metric space. In this section we find existence and uniqueness conditions for recursive utilities with timepreference shocks. Consistent with the approach taken in Hansen and Scheinkman [2012]and Borovička and Stachurski [2019], we seek a normalised solution to G t : = (cid:18) V t C t (cid:19) − γ (3.2.1)as it is easier to solve for G t than V t .We seek to express equation (3.1.3) in terms of G t to find a stationary Markov solution.By homogeneity of the aggregator W , we see V t C t = ( − β ) λ t + β ( R t (cid:18) V t + C t + C t + C t (cid:19)) − / ψ / ( − / ψ ) (3.2.2) .2 recursive utility with time preference shocks Using the consumption specification in assumption 3.1.1 and definition of R t yields V t C t = ( − β ) λ t + β " E t (cid:20) V t + C t + exp [ κ ( X t + , X t , (cid:15) t )] (cid:21) − γ − γ − / ψ / ( − / ψ ) (3.2.3)Taking this expression to the power of 1 − γ and rewriting in terms of G t gives the recursion G t = ( ( − β ) λ t + β (cid:18) E t (cid:20) G t + exp [( − γ ) κ ( X t , X t + , (cid:15) t + )] (cid:21)(cid:19) / θ ) θ (3.2.4)where θ = − γ − / ψ .We seek a stationary Markov solution of the form G t = g ( x ) where x ∈ X . This translatesinto the functional fixed point problem g ( x ) = ( ( − β ) λ ( x ) + β (cid:18)Z g ( x ) Z exp [( − γ ) κ ( x , y , (cid:15) )] ν ( d(cid:15) ) p ( x , y ) dy (cid:19) / θ ) θ (3.2.5)where ν is the distribution of (cid:15) t + , and λ ( x ) ∈ C ( X ) is continuous.It is convenient to express equation 3.2.5 in terms of an operator equation. Moreover, letthe preference shock ( − β ) λ ( x ) = ξ ( x ) . That is, Ag ( x ) = ( ξ ( x ) + β (cid:18)Z g ( x ) Z exp [( − γ ) κ ( x , y , (cid:15) )] ν ( d(cid:15) ) p ( x , y ) dy (cid:19) / θ ) θ . (3.2.6)In particular, the recursive utility representation has a unique solution if and only if A hasa fixed point. Drawing from Borovička and Stachurski [2019], we may further decomposethis problem into Ag ( x ) = ϕ ( x , Kg ( x )) where Kg ( x ) = Z g ( y ) Z exp [( − γ ) κ ( x , y , (cid:15) )] ν ( d(cid:15) ) q ( x , y ) dy , (3.2.7)and ϕ ( x , t ) is the scalar-valued function given by ϕ ( x , t ) = (cid:26) ξ ( x ) + βt / θ (cid:27) θ . (3.2.8)Note that K is a linear operator in g . Let ρ denote the spectral radius of a linear operator.This yields the following useful result. .3 recursive utility with narrow framing Proposition 3.2.1. Λ p is well defined and satisfies Λ p = β ρ ( K ) / θ .Proof. See appendix A.2. This is proven as proposition A.2.4.We take an auxiliary assumption for our main result.
Assumption 3.2.1.
The state space X is compact.We can now state a central theorem of this thesis. Theorem 3.2.2.
Let Λ p = β M / θC , p . Under assumptions 3.1.1, 3.1.2, 3.2.1, the followingstatements are equivalent:a) Λ p < .b) A has a fixed point in L p ( X ) + c) There exists a g ∈ L p ( X ) + such that { A n g } n (cid:62) converges to an element of L p ( X ) .d) A has a unique fixed point in L p ( X ) e) A has a unique fixed point, g ∗ ∈ L p ( X ) + , and A n g → g ∗ as n → ∞ for any g ∈ L p ( X ) + . Theorem 3.2.2 provides a necessary and sufficient condition for existence and uniquenessof recursive utilities in terms of the value Λ p . Part (e) also establishes global stability ofthe solution. In this section, we study the existence and uniqueness of a recursive utility model wherethe agent has ‘narrow framing’.The Barberis and Huang [2009] model of recursive utility with narrow framing can bewritten as U t = W ( C t , R t ( U t + ) + B t ) .3 recursive utility with narrow framing where W is the CES aggregator and B t is a function of the state X t which captures narrowframing.The intuition for this is as follows. As discussed by Guo [2017] and Guo and He [2019],narrow framing is conceptually equivalent to ‘utility for gains and losses’. Thus, thearguments in the aggregator W must change. Instead of trading off between consumptiontoday and value tomorrow, the agent chooses between consumption today and value plus some gain/loss tomorrow.Once again, we use homogeneity of the aggregator to seek a normalised solution to U t C t = (cid:26) ( − β ) + β (cid:20) R t (cid:18) U t + C t + C t + C t (cid:19) + B t C t (cid:21) − ψ (cid:27) − ψ .We then convert this problem into a functional fixed point equation, where the left handside is a Markovian function of the state. With regard to the framework presented earlier,this means our solution will be a fixed point to the operator defined by Bg ( x ) = (cid:26) ( − β ) + β (cid:18) Kg ( x ) + b ( x ) (cid:19) θ (cid:27) θ where K : L p ( X ) + → L p ( X ) + is a bounded, linear operator, β ∈ (
0, 1 ) , g ∈ L p ( X ) + and b ∈ C ( X ) is a strictly positive, continuous function on X .This mathematical framework allows us to establish sufficient conditions for the existenceand uniqueness of a non-trivial solution. Proposition 3.3.1.
Under assumptions 3.1.1, 3.1.2, 3.2.1, if Λ p < then B has a fixedpoint g ∗ ∈ L p ( X ) + . Moreover, B is globally stable in the sense that A n g → g ∗ as n → ∞ for any g ∈ L p ( X ) . For a proof of this proposition, see appendix A.3.Note that this proposition only ensures sufficiency. To see why necessity fails, considerthe following counter-example. Suppose that X = x is a singleton space. Then thedimension of the problem reduces from infinity to one. In this case, B : R + → R + , and K satisfies ρ ( K ) = | K | . If we choose 0 < K / β < θ <
0, then Λ p = βρ ( K ) / θ > .3 recursive utility with narrow framing Nevertheless, there may still be a non-trivial fixed point which occurs due to the narrowframing term. This can be seen in figure 1.Figure 1: Fixed points of the narrow framing recursive utility operator when Λ p > .4 unbounding the state space Figure 2: Without narrow framing, A has only the trivial fixed point. In practice, models using recursive utility will be implemented numerically. When this isthe case, the state space is discretized and hence finite. The state space will thus satisfythe compactness assumption in 3.2.1. In theoretical models however, the state space isoften unbounded. For example, Bansal and Yaron [2004] and Schorfheide et al. [2018]employ consumption processes which incorporate numerous unbounded shocks.Evidently, assumption 3.2.1 is restrictive. In the following analysis, we extend the un-boundedness results of Borovička and Stachurski [2019]. The analysis invokes a regularityassumption known as eventual compactness of the operator K . Definition 3.4.1.
An operator K is called compact if the closure of K ( B ) is compact forall bounded subsets B ⊂ L p ( X ) . Definition 3.4.2.
An operator K is called eventually compact if there exists an i ∈ N such that K i is compact. .4 unbounding the state space In the L ( X ) setting used in Borovička and Stachurski [2019], operator compactness ishard to determine. The Hilbert space structure of L ( X ) however allows more operatorsto satisfy compactness. This is a primary motivation for extending the solution spacefrom L ( X ) to L p ( X ) . Assumption 3.4.1.
The linear operator K is bounded (and hence continuous), andeventually compact. Moreover, let X be σ − finite.For the next theorem, recall that Λ p = β M / θC , p . We maintain the assumption that thefunction λ ( x ) is continuous takes values on a compact set. Theorem 3.4.1.
Let X be a (possibly unbounded) metric space, and A be the operator fromequation (3.2.6) . If assumptions 3.4.1, 3.1.1 and 3.1.2 hold, then the following statementsare equivalent:a) Λ p < .b) A has a fixed point in L p ( X ) + .c) There exists a g ∈ L p ( X ) + such that { A n g } n ∈ N converges to an element of L p ( X ) + . The proof of this theorem is located in the appendix A.4. Note, we lose stability of thesolution. Before moving on to applications, there is one more auxiliary result which canassist in establishing existence and uniqueness in L ( X , π ) . Proposition 3.4.2.
Let the (Schwartz) kernel be k ( x , y ) = Z exp [( − γ ) κ ( x , y , (cid:15) )] ν ( d(cid:15) ) q ( x , y ) . Let k ( x , y ) ∈ L ( X × X , π × π ) . Then the linear operator K , defined in equation (3.2.7) ,is compact in L ( X , π ) . We will see why this is true in proposition 3.4.2. This compactness result is unique to L , and does not hold in general for other L p spaces. This is because L is a Hilbert space, and the proof of compactness relies on an approximation argument relying on anorthonormal basis expansion. .4 unbounding the state space Proof.
Substituting the expression for k ( x , y ) into Kg ( x ) gives Kg ( x ) = Z g ( y ) k ( x , y ) dy . (3.4.1)If k ( x , y ) ∈ L ( X × X , π × π ) , then K is a Hilbert-Schmidt integral operator in L ( X , π ) .Hilbert-Schmidt operators are compact: see page 198 of Stein and Shakarchi [2005] for afull proof.VA P P L I C AT I O N S A N D S I M U L AT I O N S “I don’t give a damn about the applications - show me the equations.” - John Stachurski We now show existence and uniqueness of the recursive utility specification seen in sectionI.A of Bansal and Yaron [2004]. We also consider a truncated ‘stochastic volatility’ modelfrom section I.B. All results are new contributions which consider unbounded cases.Despite the broad success of Bansal and Yaron [2004] in explaining asset pricing puzzles,unique existence of a solution has not yet been resolved. The closest result is Pohl et al.[2019], which only proves the existence of a solution. Pohl et al. [2019] impose a stringentbounded condition on the parameters. By contrast, we establish uniqueness as well as existence. Further, the conditions we impose are less strict in that they are not onlysufficient, but necessary, for a unique solution.Bansal and Yaron [2004] represent preferences with the standard Epstein-Zin recursion V t = h ( − β ) C − / ψt + β {R t ( V t + ) } − / ψ i / ( − / ψ ) as seen in earlier sections. In this model, consumption grows according toln ( C t + / C t ) = µ c + z t + ση c , t + (4.1.1) Note that this is the degenerate form of the the Schorfheide et al. [2018] preference representation. Set λ ( x ) = bansal-yaron with constant and stochastic volatility where { η c , t + } are IID standard normal.In section I.A of their paper, Bansal and Yaron [2004] capture stochastic growth via theautoregressive process z t + = ρz t + ση z , t + (4.1.2)where the innovation process { η z , t + } is IID standard normal. We represent the stateprocess by X t = x . The central linear valuation operator can thus be written as: Kg ( x ) = Z g ( y ) Z exp (cid:20) ( − γ ) κ ( x , y , (cid:15) ) (cid:21) ν ( d(cid:15) ) q ( x , y ) dy (4.1.3) = Z g ( y ) Z exp (cid:20) ( − γ )( µ c + x + σ(cid:15) ) (cid:21) ν ( d(cid:15) ) q ( x , y ) dy . (4.1.4)Notably, the state space is given by X = R . Proposition 4.1.1.
The operator K , defined in equation (4.1.4) , is compact in L ( X , F , π ) . Proof.
Observe that k ( x , y ) = Z exp (cid:20) ( − γ )( µ c + x + σ(cid:15) ) (cid:21) ν ( d(cid:15) ) q ( x , y ) .First, note k ( x , y ) = Z exp [( − γ )( µ c + x + σ(cid:15) )] ν ( d(cid:15) ) q ( x , y )= exp (cid:20) ( − γ )( µ c + x ) + ( − γ ) σ / (cid:21) q ( x , y )= √ πσ exp (cid:20) ( − γ )( µ c + x ) + ( − γ ) σ / (cid:21) · exp (cid:20) − ( y − ρx ) / σ (cid:21) .1 bansal-yaron with constant and stochastic volatility Further, the stationary distribution of { z t } is given by π ( x ) = √ ( − ρ ) √ πσ exp [ − x ( − ρ ) / σ ] . Thus, using Fubini’s theorem over the product measure π = ( π × π )( x × y ) ,we obtain Z | k ( x , y ) | dπ = πσ Z e ( − γ )( µ c + x )+( − γ ) σ − ( y − ρx ) dπ = e ( − γ ) σ πσ Z e ( − γ )( µ c + x ) Z e − ( y − ρx ) / σ dπ ( y ) dπ ( x ) (cid:54) e ( − γ ) σ √ πσ Z e ( − γ )( µ c + x ) dπ ( x )= e ( − γ ) σ q ( − ρ ) πσ Z e ( − γ )( µ c + x ) e − x ( − ρ ) / σ dx = e ( − γ ) σ + ( − γ ) µ c q ( − ρ ) πσ Z e ( − γ ) x e − x ( − ρ ) / σ dx = e ( − γ ) σ + ( − γ ) µ c q − ρ √ πσ · e ( − γ ) σ ( − ρ ) < ∞ .That is, the Schwartz kernel k ( x , y ) is bounded in L ( R ) . Applying proposition 3.4.2shows that K is compact in L ( X , π ) .Proposition 4.1.1 and theorem 3.4.1 show that the constant volatility model of Bansal andYaron [2004] has a unique solution if and only if Λ <
1. Before moving on to see exampleparameterisations, we now consider the stochastic volatility case.In section I.B of Bansal and Yaron [2004], the authors employ the dual laws of motion z t + = ρz t + ϕ e σ t η z , t + (4.1.5) ¯ σ t + = ν ¯ σ t + d + ϕ σ η σ , t + (4.1.6)where the innovation process { η i , t } is IID standard normal for i ∈ { z , σ } . The state vector X t can be represented as X t = ( z t , σ t ) with x = ( x , x ) . In order to make σ t well-definedin R , we define it by σ t = ¯ σ t ¯ σ t (cid:62) − i ¯ σ t ¯ σ t < . .1 bansal-yaron with constant and stochastic volatility where i ∈ C is the imaginary unit.The operator K can thus be written as Kg ( x ) = Z g ( y ) Z exp (cid:20) ( − γ ) κ ( x , y , (cid:15) ) (cid:21) ν ( d(cid:15) ) q ( x , y ) dy (4.1.7) = Z g ( y ) Z exp (cid:20) ( − γ )( µ c + x + √ x (cid:15) ) (cid:21) ν ( d(cid:15) ) q ( x , y ) dy (4.1.8) = Z g ( y ) k ( x , y ) dy . (4.1.9)Here, the state space is given by X = R .The stationary distribution of { z t } t ∈ N is inconvenient; it is the product of two independentGaussian random variables. The corresponding probability density function is too heavytailed to establish existence. It decays asymptotically on the order of f ( X = x ) ∼ x − / e − hx . This means that bounding the kernel, k ( x , y ) , is generally intractable on anunbounded state-space.To overcome the heavy tail problem, we truncate the shocks to stochastic volatility butleave other shocks unbounded. That is, assume { η σ , t + } are bounded and that { σ t } thusconverges to some bounded distribution. The Bansal-Yaron dynamical system thus evolvesaccording to z t + = ρz t + ϕ e ( σ t + (cid:15) ) η z , t + (4.1.10) ¯ σ t + = ν ¯ σ t + d + ϕ σ η σ , t + (4.1.11)for some (cid:15) >
0. The inclusion of (cid:15) >
Proposition 4.1.2.
Let be { η σ , t + } be uniformly bounded with absolutely continuousdensity, such that x ∈ [ M ] for some M > . Then the linear operator K is compact inthe stochastic volatility Bansal-Yaron model.Proof. This proof builds upon the argument seen in the proof of proposition 4.1.1. Seeappendix A.5 for the full proof. This is demonstrated in Gaunt [2013]. As opposed to unbounded Gaussian as per the original model. .1 bansal-yaron with constant and stochastic volatility Proposition 4.1.1 implies that we may apply theorem 3.4.1 to the section I.A Bansal andYaron [2004] model. For constant volatility, this shows that the utility representationis well defined if and only if the parameters satisfy Λ <
1. For stochastic volatility,proposition 4.1.2 shows that the model with truncated ‘uncertainty’ is also well defined ifand only if Λ < µ c ρ ¯ σ ϕ e ν ϕ σ ψ γ · − β = Λ is different to that required inBorovička and Stachurski [2019]. In the latter paper, the authors need only test Λ becausethe state space is compact. In Borovička and Stachurski [2019], the authors show that Λ ≈ h ( x ) = E x (cid:18) C n C (cid:19) − γ (4.1.12) ≈ m m X j = (cid:18) C jn C j (cid:19) − γ (4.1.13)for 1000 draws of x ∼ U (
0, 100 ) . We then approximate (cid:18) Z h ( x ) dπ ( x ) (cid:19) / n .This process can also be implemented by choosing x from the stationary distribution, π ,and then estimating a new integral (cid:18) Z h ( x ) dπ ( x ) (cid:19) / n = (cid:18) Z h π ( x ) dx (cid:19) / n ≈ m m X i = h ( x i ) ! / n . .2 mehra-prescott [1985] and epstein-zin [1990] Choosing m = n = Λ ≈ ψ and µ c . Figure 3: Stability Map for the Bansal-Yaron Model
The seminal models of Mehra and Prescott [1985] and Epstein and Zin [1990] employ aconsumption specification with permanent innovations of the formln (cid:18) C t + C (cid:19) = t ln ( + g ) + ξ t + This result can be backed up by a heuristic mathematical argument. Note, that the process { z t } hasa stationary distribution in L ( X ) . Thus, the growth rate of C n for large n taken from this stationarydistribution is almost surely bounded. Averaging the draws of C n and taking to the power of 1 / n thenmakes this value arbitrarily close to 1 when the exponent 1 / n ‘dominates’ the growth rate. In this case,multiplying by the discount 0.998 means that the final integrated value will be almost surely arbitrarilyclose to 0.998. .3 schorfheide, song and yaron [2018] where ξ t + = ( − a ) + aξ t + u t + , for a ∈ (
0, 1 ] and u t + IID standard normal. The stateprocess here is determined by setting { X t } = { ξ t } .In terms of the state process, { X t } , we can write the consumption growth rate asln (cid:18) C t + C t (cid:19) = ln ( + g ) + ( − a ) + ( a − ) X t + u t + (4.2.1) Proposition 4.2.1.
The linear operator K , corresponding to equation (4.2.1) , is compact.Proof. Note that equation (4.2.1) is of a form similar to Bansal and Yaron with constantvolatility. Thus, the proof of compactness follows a similar logic.Applying proposition 4.2.1, we can see that Mehra and Prescott [1985] and Epstein andZin [1990] have an L solution precisely when Λ < Consumption in the Schorfheide et al. [2018] model is determined by the following statedynamics ln ( C t + / C t ) = µ c + z t + σ c , t η c , t + , (4.3.1) z t + = ρ z t + q − ρ σ z , t η z , t + , (4.3.2) σ i , t = φ i ¯ σ exp ( h i , t ) with h i , t + = ρ i h i + σ h i η h i , t + , i ∈ { c , z } . (4.3.3)Here, { η i , t } and { η h i , t } are iid and standard normal for i ∈ { c , z } . The associated statevector can be represented as X t = ( h c , t , h z , t , z t ) , where x = ( x , x , x ) . As discussedearlier, the Schorfheide-Song-Yaron model ranks consumption streams according to V t = (cid:20) ( − β ) λ t C − / ψt + β {R t ( V t + ) } − / ψ (cid:21) / ( − / ψ ) (4.3.4)There are two factors which complicate existence in this model. First, Schorfheide et al.[2018] augment the Epstein-Zin model of recursive utility with the time preference shock, .3 schorfheide, song and yaron [2018] λ t . This issue was solved by theorem 3.2.2. Second, the model of Schorfheide et al. [2018]involves an unbounded stochastic volatility process. Consequently, there are very fewdirect results regarding existence and uniqueness of this model.Theorem 3.2.2 shows that on a compact state space, the Schorfheide et al. [2018] modelstill has a unique, globally attracting solution in L p . We now consider a version of theunbounded case in L . In this model there are a number of unbounded shocks at play. Aswe now demonstrate, we only need to impose assumptions on the stochastic volatility toget uniqueness and existence. Proposition 4.3.1.
Let the shock processes { η z , t + } , { η c , t + } be uniformly bounded withabsolutely continuous density such that x , x ∈ [ − M , M ] for some M > . Then thelinear operator K is compact for the Schorfheide-Song-Yaron specification in L .Proof. See appendix A.5Using proposition 4.3.1, we may apply theorem 3.4.1 to the Schorfheide-Song-Yaron model.Consequently, a unique solution exists if and only if the model parameters satisfy Λ < Λ .3 schorfheide, song and yaron [2018] Figure 5: Values of Λ D I S C U S S I O N “Why would you microfound macro when there are real problems to work on?” - DamienEldridge
Borovička and Stachurski [2019] express recursive utility as the composition of an infinitedimensional linear operator and a real valued function. This allows the authors to combinePerron-Frobenius theory with monotone concave operator theory. In doing so, they obtainsharp results regarding the existence and uniqueness of Epstein-Zin utilities. This thesisextends the result of Borovička and Stachurski [2019]. Our contribution is twofold.First, this paper establishes existence and uniqueness conditions for recursive utilities with(i) time preference shocks and (ii) narrow framing. These results are particularly useful inmodern asset pricing, where theorists are increasingly relying upon more complex recursiveutility representations.Second, by altering the solution space from L to L p , we add greater flexibility to theexistence and uniqueness conditions. More specifically, if a solution can’t be established inone space, our results allow the practitioner to test another space. This is demonstrated inchapter 4, where we use L to tackle unbounded Bansal and Yaron [2004] and Schorfheideet al. [2018] environments.There are some shortcomings to this paper. The extension from L to L p makes numericalimplementation more costly. Moreover, all unbounded results rely on the eventual com-pactness of the operator K . But this assumption is not satisfied in heavy tailed models.As such, future research could try to relax this condition. This is non-trivial however. A P P E N D I X “If you drop out of theory, there is always applied micro.” - Sander Heinsalu a.1 general mathematical results
In this section, we set out the general fixed point and spectral radius results requiredfor this paper. Where results are taken from Borovička and Stachurski [2019], proofs aregeneralised from L ( X ) to either general Banach spaces or L p ( X ) spaces.Let E be a Banach space over R , and denote the zero element by ~ Definition A.1.1.
A nonempty, closed, convex set P ⊂ E is called a cone if1) x ∈ P , λ (cid:62) λx ∈ P x ∈ P and − x ∈ P implies that x = ~ normal if there exists a constant δ > || x + y || (cid:62) δ for all x , y ∈ P satisfying || x || = || y || = P ⊂ E induces a partial ordering (cid:54) , by defining u (cid:54) v ⇐⇒ v − u ∈ P . (A.1.1)We can take a strict ordering by also requiring that u − v P . A Banach space withthis cone-order structure may then be called a partially ordered Banach space. The cone general mathematical results generating the partial ordering is called a positive cone. Further, let D ⊂ E and definethe operator A : D → E . We say that A is an increasing (or isotone) operator if for all x , x ∈ D , where x (cid:54) x , we have Ax (cid:54) Ax .The next theorem is needed to establish basic fixed points results. Theorem A.1.1 (Du [1990] or Zhang [2013] Theorem 2.1.2) . Suppose that the cone P isnormal, u , v ∈ E and u < v . Let A : [ u , v ] → E denote an increasing operator. Ifone of the following holds:i) A is a concave operator, Au (cid:62) u + (cid:15) ( v − u ) , Av (cid:54) v where (cid:15) ∈ (
0, 1 ) is a constant;ii) A is a convex operator, Au (cid:62) u , Av (cid:54) v − (cid:15) ( v − u ) where (cid:15) ∈ (
0, 1 ) is a constant,then A has a unique fixed point x ∗ in [ u , v ] . Further, for any x ∈ [ u u , v ] , the iterativesequence { x n } given by x n = Ax n − satisfies || x n − x ∗ || (cid:54) M ( − (cid:15) ) n ( n =
1, 2, ... ) for some M ∈ R + independent of x .Proof. See Zhang [2013] theorem 2.1.2.Let X be a compact metric space, and µ be a measure. The set of Borel measurablefunctions g : X → R , such that || g || p = R | g | p dµ < ∞ , is canonically denoted by L p ( X , F , µ ) . We write L p ( X ) when it is clear which measure is being used. The dualspace of L p ( X ) is a Banach space identified with L q ( X ) when p , q satisfy p + q =
1. Notethat for g ∈ L ∞ ( X ) , we naturally define || g || ∞ = sup {| g ( x ) | : x ∈ X } . For g , h ∈ L p ( X ) ,we define g (cid:54) h to mean that g ( x ) (cid:54) h ( x ) for µ -almost every x ∈ X . Similarly, g (cid:28) h means that g ( x ) < h ( x ) µ -almost everywhere. These order-relations are induced by thepositive cone L p ( X , F , π ) + as per equation (A.1.1).Let K : L p ( X ) → L p ( X ) be a linear operator. Define the operator norm and spec-tral radius respectively by || K || = sup {|| Kg || p : g ∈ L p , || g || p = } and ρ ( K ) = sup { λ : λ satisfies Kg = λg for some g ∈ L p ( X ) } . Gelfand’s formula states that ρ ( K ) = lim n →∞ || K n || n . We say that K is positive if Kg (cid:62) g (cid:62)
0. It .1 general mathematical results is bounded if || K || is finite, and compact if the closure of K ( B ) is compact for all boundedsubsets B ⊂ L p ( X ) .By treating elements f , g ∈ L p ( X ) as points in a partially ordered Banach space, we getthe usual notions of convexity and concavity for an operator A : L p ( X ) → L p ( X ) .The next proposition generalises an argument found in Olver [2016]. Proposition A.1.2 (A general Neumann series result.) . Let K be a linear operator, λ ∈ R and h , f ∈ L p ( X ) . If ρ ( K ) < λ then the operator equation λh = Kh + f has a uniquesolution given by h = ( λI − K ) − f . In particular, this value exists if and only if thegeometric series expression h = P ∞ n = λ − ( n + ) K n f converges in norm.Proof. Write the equation λh = Kh + f as λIh = Kh + f . Rearranging gives h =( λI − K ) − f . We wish to see when this expression is well defined. As such, note ( λI − K ) − = λ (cid:18) I − Kλ (cid:19) − = λ (cid:18) I + Kλ + (cid:18) Kλ (cid:19) + ... (cid:19) (see Olver [2016] for details) = ∞ X n = λ − ( n + ) K n .Thus, h = P ∞ n = λ − ( n + ) K n f . Gelfand’s formula shows that this expression is well definedif ρ ( K ) < λ .The next lemma is stated without proof in Krasnosel’skii et al. [2012]. Lemma A.1.3.
Suppose K is a positive linear operator with Kh (cid:54) δh for some δ > ,and h ∈ P where P is a normal cone. If K is compact and h is a quasi-interior elementof P , we have ρ ( K ) (cid:54) δ .Proof. Observe that if K is a compact linear operator, then so too is the adjoint K ∗ .Hence, ρ ( K ) = ρ ( K ∗ ) must be attained by some f ∈ E and f ∗ ∈ E ∗ respectively. Thus .1 general mathematical results A ∗ ( f ∗ ) = ρ ( A ) f ∗ for some f ∗ ∈ E ∗ . By the assumption that h is quasi-interior, we have f ∗ ( h ) > f ∗ ∈ P and so ρ ( K ) = K ∗ f ∗ ( h ) f ∗ ( h ) is well defined. Consequently, ρ ( A ) = K ∗ ( f ∗ ( h )) f ∗ ( h )= f ∗ ( Kh ) f ∗ ( h ) (cid:54) f ∗ ( δh ) f ∗ ( h )= δ .This completes the proof. Definition A.1.2.
Let E be a Banach space, K : E → E a bounded linear operator and h ∈ E . We define ρ ( K , h ) = lim sup n →∞ || K n h || n as the local spectral radius of K at h .The next three lemmas are local spectral radius results. Lemma A.1.4.
Let h ∈ E . If K is a bounded linear operator, then (cid:54) ρ ( K , h ) (cid:54) ρ ( K ) . Proof.
The inequality 0 (cid:54) ρ ( K , h ) is immediate. Moreover, by Cauchy-Schwarz ρ ( K , h ) = lim sup n →∞ || K n h || (cid:54) lim sup n →∞ || K n || n || h || n (by Cauchy-Schwarz) = lim sup n →∞ || K n || n · lim n →∞ || h || n = lim sup n →∞ || K n || n = ρ ( K ) .This completes the lemma. .1 general mathematical results Lemma A.1.5.
The local spectral radius satisfies the following three properties.(1) ρ ( K , K n h ) = ρ ( K , h ) for all h ∈ E and m ∈ N (2) ρ ( aK , bh ) = | a | ρ ( K , h ) for all h ∈ L ( X , b = and a ∈ R (3) ρ ( K , f + h ) (cid:54) max (cid:26) ρ ( K , h ) , ρ ( K , f ) (cid:27) Proof.
Regarding (1), this follows immediately as lim sup n →∞ || K n + m h || n = lim sup n →∞ || K n h || n For (2), simply observe that ρ ( aK , bh ) = lim sup n →∞ || ( aK n ) bh || n (cid:54) lim sup n →∞ | a | | b | n || K n h || n = | a | ρ ( K , h ) For (3), take an arbitrary c ∈ R + . By the definition of local spectral radius, we maychoose an m c ∈ N such that for all n (cid:62) m c , || K n h || (cid:54) (cid:18) ρ ( K , h ) + c (cid:19) n and || K n f || (cid:54) (cid:18) ρ ( K , f ) + c (cid:19) n .From this, observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K n (cid:18) h + f (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:18) ρ ( K , h ) + c (cid:19) n + (cid:18) ρ ( K , f ) + c (cid:19) n (cid:54) (cid:18) max (cid:26) ρ ( K , h ) , ρ ( K , f ) (cid:27) + c (cid:19) n .Taking m c → ∞ , we can take c →
0. This completes the proof.The next lemma denotes the linear span of a set N by span { N } . Lemma A.1.6 (Daneš [1987]) . Let N ⊂ E . Then sup (cid:26) ρ ( K , h ) : h ∈ N (cid:27) = sup (cid:26) ρ ( K , h ) : h ∈ span ( N ) (cid:27) .1 general mathematical results Proof.
Since N ⊂ span ( N ) we get for free that sup (cid:26) ρ ( K , h ) : h ∈ N (cid:27) (cid:54) sup (cid:26) ρ ( K , h ) : h ∈ span ( N ) (cid:27) . By fixing an h ∈ span ( N ) , we may write h = P ni = t i h i , h i ∈ N for t i ∈ R .By (A.1.5) property (2) and (A.1.5) property (3), we see ρ ( K , h ) (cid:54) max i (cid:26) ρ ( K , t i h i ) (cid:27) (cid:54) max i (cid:26) ρ ( K , h i ) (cid:27) (cid:54) sup f ∈ N (cid:26) ρ ( K , f ) (cid:27) This completes the proof.This cohort of lemmas gives rise to the final corollary.
Corollary A.1.7 (Daneš [1987]) . Let N ⊂ E . Then sup (cid:26) ρ ( K , h ) : h ∈ N (cid:27) = sup (cid:26) ρ ( K , h ) : h ∈ M (cid:27) where M = span { K m h : h ∈ N , m (cid:62) } Proof.
By A.1.5 (1), we have ρ ( K , K n h ) = ρ ( K , h ) and so sup { ρ ( K , h ) : h ∈ N } = sup { ρ ( K , K n h ) : h ∈ N } . Then by A.1.6, we also have sup { ρ ( K , K n h ) : h ∈ N } = sup { ρ ( K , K n h ) : h ∈ span h ∈ N ( K n h ) } = sup { ρ ( K , h ) h ∈ span h ∈ N ( K n h ) } An L version of the following local spectral radius theorem is found in Borovička andStachurski [2019], theorem A.1. Theorem A.1.8 (Zabreiko–Krasnosel’skii–Stetsenko–Zima, Borovička and Stachurski[2019]) . Suppose h ∈ L p ( X ) . Let K be a positive compact linear operator. If h (cid:29) , then ρ ( K , h ) = lim n →∞ k K n h k / n = ρ ( K ) . (A.1.2) Proof.
In lemma A.1.4, we established that ρ ( K , h ) (cid:54) ρ ( K ) . Thus, it suffices to showthat ρ ( K , h ) (cid:62) ρ ( K ) . Let λ be a constant satisfying λ > ρ ( K , h ) and let h λ : = ∞ X n = K n hλ n + . (A.1.3) .1 general mathematical results The point h λ is a well-defined element of L p ( X ) + by lim sup n →∞ k K n h k / n < λ and theCauchy Root Test for convergence. It is also positive µ -almost everywhere since the sumin expression (A.1.3) includes h (cid:29) K is a positive operator. ApplyingA.1.2, the point h λ also has the representation h λ = ( λI − K ) − h , from which we obtain λh λ − Kh λ = h . Since h ∈ L p ( X ) + and h >
0, this implies that Kh λ (cid:54) λh λ . Accordingly,by the compactness of K , quasi-interiority of h λ and lemma A.1.3, we must have ρ ( K ) (cid:54) λ .Since this inequality was established for an arbitrary λ satisfying λ > ρ ( K , h ) , we concludethat ρ ( h , K ) (cid:62) ρ ( K ) . Hence ρ ( K , h ) = ρ ( K ) . Finally, since K is compact, corollaryA.1.7 implies that ρ ( K , h ) = lim n →∞ k K n h k / n , so equation (A.1.2) holds.The next result is an extension of theorem A.1.8. The L version is due to Borovička andStachurski [2019]. Theorem A.1.9.
Suppose h ∈ L p ( X ) and let K be a linear operator on L p ( X ) . If K i iscompact for some i ∈ N and Kf (cid:29) whenever f ∈ L p ( X ) + , then ρ ( K ) = lim n →∞ (cid:26) Z | K n h | p dµ (cid:27) pn . (A.1.4) for all h (cid:29) .Proof. Fix h ∈ L p ( X ) with h (cid:29) i ∈ N such that K i is a compact linearoperator on L p ( X ) . Fix j ∈ N with 0 (cid:54) j (cid:54) i −
1. By our assumptions on K , we knowthat K j h (cid:29)
0. Thus, theorem A.1.8 applied to K i with initial condition K j h yields (cid:26) Z ( K in K j h ) p dµ (cid:27) / pn = (cid:26) Z ( K in + j h ) p dµ (cid:27) / pn → ρ ( K i ) ( n → ∞ ) .But ρ ( K i ) = ρ ( K ) i , so taking both sides to the power of 1 / i yields (cid:26) Z ( K in + j h ) p dµ (cid:27) / ( ipn ) → ρ ( K ) ( n → ∞ ) .It follows that (cid:26) Z ( K in + j h ) p dµ (cid:27) / p ( in + j ) → ρ ( K ) ( n → ∞ ) .As j is an arbitrary integer satisfying 0 (cid:54) j (cid:54) i −
1, we conclude that (A.1.4) holds. .2 proofs: recursive utility with time preference shocks The next lemma is a fixed point result which holds when ( X , µ ) is a probability space. Lemma A.1.10.
Let { g n } be a positive, monotone increasing sequence in L p ( X ) .(1) If { g n } is bounded above by some h in L p ( X ) , then there exists a g in L p ( X ) such that R g pn dµ → R g p dµ .(2) Moreover, let g n = T n g for some continuous operator T mapping a subset of L p ( X , µ ) to itself. In this case, g must be a fixed point of T .Proof. Regarding the first claim, note that since { g n } ⊂ L p ( X ) , we have that { g pn } ⊂ L ( X ) . Thus, by Beppo Levi’s Monotone Convergence Theorem, R g pn dµ → R g p dµ forsome function g p ∈ L p ( X , µ ) . Then, since X is a finite measure space, applying Egorov’stheorem shows that R | g n − g | p dµ →
0. This g must be the limit. This establishes the firstpart of the lemma.To see that g is a fixed point of T , note that we have || g n − g || p → || T g n − T g || p →
0. But, by the definition of the sequence { g n } , we also have || T g n − g || p →
0. Hence
T g = g .Note that in this proof, we rely heavily on the measure µ being finite. This is obviouslysatisfied as µ is a probability measure. a.2 proofs: recursive utility with time preference shocks We now directly prove theorem 3.2.2. Our work draws heavily from the appendix ofBorovička and Stachurski [2019]. Recall that we write the recursive utility operator as Ag ( x ) = (cid:26) ξ ( x ) + β (cid:20) Kg ( x ) (cid:21) / θ (cid:27) θ where g ∈ L p ( X , B , µ ) , θ ∈ R , and ξ : X → R is continuous and strictly positive.Let p represent the transition density for the exogenous state process { X t } ⊂ X . By stan-dard Markov process results, we may write the i th iteration as p i ( x , y ) = R p ( x , z ) p i − ( z , y ) dz .2 proofs: recursive utility with time preference shocks for all x , y ∈ X . We assume that p is irreducible in the sense that p ( x , y ) > ∀ x , y ∈ X .We write k ( x , y ) = Z exp [( − γ ) κ ( x , y , (cid:15) )] ν ( d(cid:15) ) p ( x , y ) We define a linear operator K : L p ( X ) + → L p ( X ) + by Kg ( x ) = Z k ( x , y ) g ( y ) dy .We also define k i as the i th iterate of k such that k i ( x , y ) = R k ( x , z ) k i − ( z , y ) dz . Thus,for all x ∈ X and g ∈ L p ( X ) , we have K i g ( x ) = Z k i ( x , y ) g ( y ) dy .To see this, consider that K ( Kg ( x )) = Z k ( x , y ) Kg ( y ) dy = Z k ( x , y ) Z k ( y , z ) g ( z ) dz dy = µ ( X ) Z k ( x , y ) g ( y ) dy = Z k ( x , y ) g ( y ) dy .From this a simple induction shows K i g ( x ) = Z k i ( x , y ) g ( y ) dy .The next three lemmas are L p versions of results found in Borovička and Stachurski [2019]. Lemma A.2.1.
The density π is the unique stationary density for p ( x , · ) on X . Inaddition, π is everywhere positive and continuous on X .Proof. See Borovička and Stachurski [2019], lemma B.1.
Lemma A.2.2.
Regarding the operator K , the following statements are true:(a) K is a bounded linear operator on L p ( X , π ) that maps L p ( X , π ) + to itself.(b) Kg = whenever g ∈ L p ( X , π ) + and g = . .2 proofs: recursive utility with time preference shocks (c) Kg (cid:29) whenever g ∈ L p ( X , π ) and g (cid:29) .(d) For each g ∈ L p ( X ) + , Kg is a continuous L p function.Proof. Regarding claim (a), K is continuous and hence bounded by some constant M on X . Further, π is positive and continuous on a compact set, and hence bounded below bysome positive constant δ . This yields, for arbitrary f ∈ L p ( X , π ) , and sufficiently large N ∈ R | Kf ( x ) | p = (cid:12)(cid:12)(cid:12)(cid:12)Z k ( x , y ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) p (cid:54) M p (cid:18) Z | f ( y ) | π ( y ) π ( y ) dy (cid:19) p (cid:54) M p δ p (cid:18) Z | f ( y ) | π ( y ) dy (cid:19) p (cid:54) M p Nδ p (cid:18) Z | f ( y ) | p π ( y ) dy (cid:19) / p = M p Nδ p || f || p .The 2nd last inequality follows from the fact that as ( X , µ ) is a finite measure spacewe get L p ( X , µ ) ⊂ L ( X , µ ) . It follows directly that K is a bounded linear operator on L p ( X , π ) + . Moreover, note that since K is bounded, it must also be continuous.Regarding claim (b), suppose that, to the contrary, we have Kg = g ∈ L p ( X ) . Let B = { x : g ( x ) > } . Since g is nonzero, we have π ( B ) >
0. Since Kg = R B k ( x , y ) dy = x ∈ X . But then R B q ( x , y ) dy = x ∈ X . A simple induction argument shows that this extends to the n -step kernels, sothat, in particular, R B p ‘ ( x , y ) dy = x ∈ X . The last equality contradicts p ‘ > Kg ( x ) = R g ( y ) k ( x , y ) dy and the definition of k ( x , y ) .To see part (d), fix g ∈ L p ( X ) + , x ∈ X and x n → x . Note that we have k ( x n , y ) g ( y ) (cid:54) M g ( y ) π ( y ) π ( y ) (cid:54) Mδ g ( y ) π ( y ) (A.2.1) .2 proofs: recursive utility with time preference shocks Since g ∈ L p ( X ) + and X has finite measure, it must be the case that g is also in L ( X ) .Using (A.2.1), we can apply the dominated convergence theorem to obtainlim n →∞ Kg ( x n ) = Z lim n →∞ k ( x n , y ) g ( y ) dy = Kg ( x ) .This shows that Kg is continuous. Definition A.2.1.
A linear operator K : E → E is called irreducible if E and ~ Lemma A.2.3.
The operator K is irreducible and K is compact.Proof. To see that K is irreducible, see Borovička and Stachurski [2019], lemma B.3.Regarding compactness, we must alter the argument somewhat. The theory of compactoperators on an L-normed, Banach lattice implies that K will be compact whenever K isweakly compact, which requires that the image of the unit ball B in L p ( X , π ) under K is relatively compact in the weak topology. To prove this it suffices to to show that, given (cid:15) >
0, there exists a δ > R A ( K | f | ) p dπ < (cid:15) whenever f ∈ B and π ( A ) < δ .This is true because k is continuous and hence bounded on X , yielding Z A (cid:18) Z k ( x , y ) | f ( y ) | dy (cid:19) p π ( x ) dx (cid:54) Z A (cid:18) M p Nδ p || f || p (cid:19) π ( x ) dx (cid:54) M p Nδ p || f || p π ( A ) for constants M , N . By taking π ( A ) < (cid:15)δ p / ( M p N || f || p ) , we get the result. Proposition A.2.4. Λ p is well defined and satisfies Λ p = β ρ ( K ) / θ .Proof. Since K i is compact for some i and maps positive functions into positive func-tions (see lemmas A.2.3 and A.2.2), we can apply theorem A.1.9 to ≡ ρ ( K ) = lim n →∞ k K n k / n . An inductive argument based on the Borovička and To see this, I note that on an L-normed Banach lattice, if T : E → F is weakly compact, then T ( W ) isprecompact if W is weakly-compact. I thank participants on math.stackexchange who showed this to mehere:https://math.stackexchange.com/questions/3308257/product-of-two-weakly-compact-endomorphisms-is-compact .2 proofs: recursive utility with time preference shocks Stachurski [2019] consumption growth assumption shows that, for each n in N , we have K n ( x ) = E x ( C n / C ) − γ . Hence, k K n k / np = Z ( E x (cid:18) C n C (cid:19) − γ ) p π ( dx ) ! / np = E π ( E x (cid:18) C n C (cid:19) − γ ) p ! / np (A.2.2)Since ρ ( K ) = lim n →∞ k K n k / n , this yields ρ ( K ) = lim n →∞ E π ( E x (cid:18) C n C (cid:19) − γ ) p ! / np = M − γC , p .Because θ : = ( − γ ) / ( − / ψ ) , we now have βρ ( K ) / θ = β M − / ψC , p = Λ p . Theorem A.2.5.
The spectral radius ρ ( K ) of K is strictly positive. Moreover, thereexists an everywhere continuous eigenfunction e of K satisfying Ke = ρ ( K ) e and e (cid:29)
0. (A.2.3)
Proof.
The irreducibility and compactness properties of K obtained in lemma A.2.3 yieldpositivity of ρ ( K ) and existence of the positive eigenfunction in equation (A.2.3). This isby the De Pagter’s theorem, and the Krein-Rutman theorem respectively. Claim (d) oflemma A.2.2 implies that e is continuous, since e ∈ L p ( X ) and e = ( Ke ) / ρ ( K ) . Remark A.2.1.
Let ξ ( x ) ∈ C ( X ) . As ξ has compact support it is bounded. We alsodefine ξ >
0. As such there exist constants ξ , ξ ∈ R such that ξ < ξ ( x ) < ξ for all x ∈ X .From this define ϕ ( t , x ) = (cid:26) ξ ( x ) + βt θ (cid:27) θ , φ ( t ) = (cid:26) ξ + βt θ (cid:27) θ , φ ( t ) = (cid:26) ξ + βt θ (cid:27) θ . Lemma A.2.6.
Let A be defined as per the Schorfheide functional form. Let e be theKrein-Rutman eigenfunction of K . Let φ ∈ (cid:26) φ , φ (cid:27) refer to both φ and φ . Suppose ρ ( K ) is of the form that a ) lim t & φ ( t ) t ρ ( K ) > and b ) lim t %∞ φ ( t ) t ρ ( K ) <
1. (A.2.4) .2 proofs: recursive utility with time preference shocks Then we also have by boundedness and strict positivity of ξ ( x ) that a ) lim t & ϕ ( t , x ) t ρ ( K ) > and b ) lim t %∞ ϕ ( t , x ) t ρ ( K ) <
1. (A.2.5)
When this is the case, there exist positive constants c < c such that(1) If < c (cid:54) c and f = ce , then there exists a δ > such that Af (cid:62) δ f (2) If c (cid:54) c < ∞ and f = ce , then there exists a δ < such that Af (cid:54) δ f .As a remark, note that the intuition here can be conceived as ϕ being roughly ‘expansive’at first and then eventually ‘contractive’ along eigenfunction paths. This intuitively willsecure us a fixed point which is non-zero from any starting guess.Proof. We first show that if condition (A.2.4) holds for φ and φ then condition (A.2.5)will also hold for ϕ . First, note that if there exists (cid:15) > < t < (cid:15) , then by the joint continuity of ϕ ( t , x ) ϕ ( t , x ) t ρ ( K ) ∈ (cid:20) min (cid:26) φ ( t ) t ρ ( K ) , φ ( t ) t ρ ( K ) (cid:27) , max (cid:26) φ ( t ) t ρ ( K ) , φ ( t ) t ρ ( K ) (cid:27)(cid:21) .Since the whole interval is greater than 1, in view of expression (A.2.4) we get the result.Similarly, by part (b) of condition (A.2.4) there exists M ∈ R such that if t > M then ϕ ( t , x ) t ρ ( K ) ∈ (cid:20) min (cid:26) φ ( t ) t ρ ( K ) , φ ( t ) t ρ ( K ) (cid:27) , max (cid:26) φ ( t ) t ρ ( K ) , φ ( t ) t ρ ( K ) (cid:27)(cid:21) .By the assumption that (A.2.4) holds for φ and φ , the whole interval must now be lessthan one. This establishes (A.2.5).We now consider the first claim of the lemma. Let e be the Perron-Frobenius (Krein-Rutman) eigenfunction of K . Let e and e be the maximum and minimum values of e on X respectively. By (A.2.5) there exists a δ > (cid:15) > ϕ ( t , x ) t ρ ( K ) (cid:62) δ .2 proofs: recursive utility with time preference shocks for all x ∈ X and 0 < t < (cid:15) . Now, choosing c ∈ R such that 0 < c ρ ( K ) e < (cid:15) and c (cid:54) c ,we have cρ ( K ) e ( x ) < (cid:15) for all x ∈ X . Hence, Ace ( x ) = ϕ ( cKe ( x ) , x )= ϕ ( cρ ( K ) e ( x ) , x )= ϕ ( cρ ( K ) e ( x ) , x ) cρ ( K ) e ( x ) cρ ( K ) e ( x ) (cid:62) δ ce ( x ) For the second statement, by (A.2.5) we may choose constants δ < M < ∞ suchthat ϕ ( t , x ) t ρ ( K ) (cid:54) δ whenever t > M As such, choose c > max (cid:26) Mρ ( K ) e , c (cid:27) and c (cid:62) c . By definition of e , we take cρ ( K ) e ( x ) (cid:62) c ρ ( K ) e > M for all x ∈ X . Hence Ac e ( x ) = ϕ ( cρ ( K ) e ( x ) , x )= ϕ ( cρ ( K ) e ( x ) , xcρ ( K ) e ( x ) ρ ( K ) ce ( x ) (cid:54) δ ce ( x ) .By construction of 0 < c < c this completes the proof. Lemma A.2.7.
If the conditions from (A.2.5) hold and A has a fixed point g ∗ ∈ L p ( X ) + then there exist f , f ∈ L p ( X ) + such that f (cid:54) Ag , g ∗ (cid:54) f , Af (cid:62) f + (cid:15) ( f − f ) and Af (cid:54) f − (cid:15) ( f − f ) .Proof. Let g ∈ L p ( X ) + . Recall that ξ ( x ) > x ∈ X . Thus, since Ag is continuousand X is compact, Ag attains a finite maximum and strictly positive minimum. Similarly,the fixed point g ∗ = Ag ∗ and Krein-Rutman eigenfunction e ( x ) also attain finite maximumand strictly positive minimum. .2 proofs: recursive utility with time preference shocks From this, choose a , a > (cid:28) a e (cid:54) g ∗ , and Ag (cid:54) a e . If a is smallenough then lemma A.2.6 implies A ( a e ( x )) (cid:62) δ a e ( x ) for some δ >
1. If we thendefine f i : = a i e , we have Af (cid:62) δ a e . Since δ >
1, write Af (cid:62) a e + (cid:15) ( a − a ) forsmall enough (cid:15) >
0. From our definition of f and f we get the desired result that Af (cid:62) f + (cid:15) ( f − f ) .For the other inequality choose a large enough that f = a e and Af (cid:54) δ a e . Since δ < Af (cid:54) a e − (cid:15) ( a − a ) e = a e − (cid:15) ( f − f ) .Choosing (cid:15) = min { (cid:15) , (cid:15) } gives the overall result. Theorem A.2.8. If βρ ( K ) θ < , then A is globally stable on L p ( X ) + .Proof. We first show that if βρ ( K ) θ < ϕ ( t , x ) t = (cid:26) ξ ( x ) t θ + β (cid:27) θ (A.2.6)where ξ ( x ) ∈ [ M , M ] for some M , M ∈ R .Consider the case where θ < Λ p <
1. In this case we have β θ ρ ( K ) > β θ as t &
0. Thus the first inequality of (A.2.5)holds. The second inequality then holds because ϕ ( t , x ) / t → t → ∞ .For the case where θ > β θ ρ ( K ) < (cid:26) ξ ( x ) t θ + β (cid:27) % ∞ as t &
0. So the first inequality of (A.2.5) holds. The second inequality also holds because β θ ρ ( K ) < ϕ ( t , x ) / t → β θ as t → ∞ . Thus choosing t large enough will give theresult for the second inequality.This shows that the conditions in lemma A.2.6 hold. To conclude the proof, note that fora fixed x , ϕ ( t , x ) is either convex or concave in t , depending on θ . .2 proofs: recursive utility with time preference shocks Suppose that ϕ is concave in t . In this case, A is isotone and concave in g ∈ L p ( X ) as afunction from L p ( X ) to L p ( X ) . By lemma A.2.6 then choose c < c such that Ac e (cid:62) c e and Ac e (cid:54) c e .Applying A.1.1 implies that A has a fixed point g ∗ ∈ L p ( X ) + which satisfies c e (cid:54) g ∗ (cid:54) c e .Since e (cid:29) c > g ∗ (cid:29)
0. This gives us the uniqueness andexistence of a fixed point.To see global stability towards said fixed point, consider an arbitrary g ∈ L p ( X ) . Choose f , f as in lemma A.2.7. This gives f (cid:54) Ag (cid:54) f . Then by A.1.1 we have that everyelement of [ f , f ] converges to g ∗ under A . In particular, A n ( Ag ) → g ∗ in norm as n → ∞ by virtue of our definition of f i . But then A n g → g ∗ also holds and so A is stableon L p ( X ) .The convex case is largely the same. Proposition A.2.9 (Necessity) . If A has a nonzero fixed point in L p ( X ) + , then βρ ( K ) θ < .Proof. Recall that K is a linear operator on a Banach space. As such, let K ∗ be theadjoint operator. Since K is irreducible and K is compact note that by De Pagter’stheorem, ρ ( K ) >
0. Thus, by the Krein-Rutman and Riesz representation theorems, for q satisfying q + p = e ∗ ∈ L q ( X ) such that e ∗ (cid:29) K ∗ e ∗ = ρ ( K ) e ∗ .Before proceeding, it is helpful to note that if f ∗ ∈ L q ( X ) + and g ∈ L p ( X ) + then R f ∗ ( x ) g ( x ) dπ < ∞ is well defined. This can be seen by observing that if R | f ∗ ( x ) | q dπ < ∞ and R | g ( x ) | p dπ < ∞ then the embedding L p , L q ⊂ L on a finite measure space implies R f ∗ ( x ) dπ R g ( x ) dπ < ∞ . Hence, by Cauchy-Schwarz Z f ∗ ( x ) g ( x ) dπ (cid:54) Z f ∗ ( x ) dπ Z g ( x ) dπ < ∞ . .2 proofs: recursive utility with time preference shocks With this in mind, define g to be a nonzero fixed point of A in L p ( X ) + . We now provethe proposition for θ <
0. In this case we have ϕ ( t , x ) < β θ t whenever t > ξ (cid:29)
0. By assumption, we then also have that Kg (cid:29)
0, so g ( x ) = Ag ( x ) = ϕ ( Kg ( x )) < β θ Kg ( x ) . Since e ∗ (cid:29) R e ∗ ( x )( β θ Kg ( x ) − g ( x )) dπ > ρ ( K ) Z e ∗ ( x ) g ( x ) dπ = Z β θ K ∗ e ∗ ( x ) g ( x ) dπ = Z β θ e ∗ ( x ) Kg ( x ) dπ .Combining these two inequalities, it must be the case that β θ ρ ( K ) R e ∗ ( x ) g ( x ) dπ > R e ∗ ( x ) g ( x ) dπ . Since θ <
0, this shows that Λ p = βρ ( K ) / θ < θ >
0, note that ϕ ( t , x ) > β θ t whenever t >
0. As we again know that Kg (cid:29) g ( x ) = Ag ( x ) = ϕ ( Kg ( x )) > β θ Kg ( x ) . By a symmetricargument to above, β θ ρ ( K ) Z e ∗ ( x ) g ( x ) dπ = β θ Z K ∗ e ∗ ( x ) g ( x ) dπ = β θ Z e ∗ ( x ) Kg ( x ) dπ < Z e ∗ ( x ) g ( x ) dπ .Hence β θ ρ ( K ) <
1, and so βρ ( K ) θ < Proof of theorem 3.2.2.
We first note that that (e) = ⇒ (d). This is due K being abounded linear operator on L p ( X ) and ϕ being jointly continuous on R + . Hence, it followsthat A is continuous on L p ( X ) + , and so any limit of a sequence of iterates { A n g } n (cid:62) of A is a fixed point of A . As the limit is unique from any starting point, the fixed point isunique.Moreover, (d) = ⇒ (c) by taking g equal to the fixed point. Furthermore, (c) = ⇒ (b). This is again by continuity of A on L p ( X ) + , meaning that any limit of a sequence { A n g } n (cid:62) of A is a fixed point of A .The implication (b) = ⇒ (a) is due to proposition A.2.9. Finally (a) = ⇒ (e) by theoremA.2.8 and proposition A.2.4. .3 proofs: recursive utility with narrow framing a.3 proofs: recursive utility with narrow framing This section proves proposition 3.3.1. The proofs here have direct analogues from theprevious section.
Remark A.3.1.
Let b ( x ) ∈ C ( X ) . As b has compact support it is bounded. We alsodefine b >
0. As such there exist constants ξ , ξ ∈ R such that b < b ( x ) < b for all x ∈ X . Lemma A.3.1.
Let A be the recursive utility with narrow framing operator. Let e be the Krein-Rutman eigenfunction of K . Let φ ( t ) be defined as earlier and redefine ϕ ( t , x ) = (cid:26) − β + β (cid:18) t + b ( x ) (cid:19) θ (cid:27) θ . Suppose ρ ( K ) and φ ( t ) satisfy a ) lim t & φ ( t ) t ρ ( K ) > and b ) lim t %∞ φ ( t ) t ρ ( K ) <
1. (A.3.1)
Then we also have by boundedness and strict positivity of b ( x ) that ϕ ( t , x ) satisfies a ) lim t & ϕ ( t , x ) t ρ ( K ) > and b ) lim t %∞ ϕ ( t , x ) t ρ ( K ) <
1. (A.3.2)
When this is the case there exist positive constants c < c such that1) If < c (cid:54) c and f = ce , then there exists a δ > such that Af (cid:62) δ f
2) If c (cid:54) c < ∞ and f = ce , then there exists a δ < such that Af (cid:54) δ f .Proof. We first show that condition a) translates from (A.3.1) to (A.3.2). In this respectnote that for any value of θ ∈ R = , if t is small enough to satisfy a) , then1 < φ ( t ) t ρ ( K )= ρ ( K ) (cid:20) − βt θ + β (cid:18) tt (cid:19) θ (cid:21) θ (cid:54) ρ ( K ) (cid:20) − βt θ + β (cid:18) t + b ( x ) t (cid:19) θ (cid:21) θ = ϕ ( t , x ) t ρ ( K ) .3 proofs: recursive utility with narrow framing by positivity of b ( x ) .We now consider part b) of the translation. This is straightforward using the observationthat since b ( x ) is bounded for all x ∈ X we havelim t %∞ φ ( t ) t = lim t %∞ (cid:20) − βt θ + β (cid:18) tt (cid:19) θ (cid:21) θ = lim t %∞ (cid:20) − βt θ + β (cid:18) t + b ( x ) t (cid:19) θ (cid:21) θ = lim t %∞ ϕ ( t , x ) t .The rest of the proof then follows identically from A.2.6. Proposition A.3.2.
If the conditions from equation (A.3.2) in lemma A.3.1 hold and A has a fixed point g ∗ ∈ L p ( X ) then there exist f , f ∈ L p ( X ) such that f (cid:54) Ag , g ∗ (cid:54) f , Af (cid:62) f + (cid:15) ( f − f ) and Af (cid:54) f − (cid:15) ( f − f ) .Proof. This follows by a similar argument to lemma A.2.7 replacing ξ ( x ) with b ( x ) . Proposition A.3.3 (Sufficiency) . If βρ ( K ) θ < , then A is globally stable on L p ( X ) .Proof. This proof proceeds very similarly to A.2.8. We first show that if βρ ( K ) θ < ϕ ( t , x ) t = (cid:26) − βt θ + β (cid:18) t + b ( x ) t (cid:19) θ (cid:27) θ (A.3.3)where b ( x ) ∈ [ M , M ] for some M , M ∈ R .Consider the case where θ <
0. In this case we have β θ ρ ( K ) > t &
0. Thus the first inequality of (A.3.2) holds. The secondinequality then holds because ϕ ( t , x ) / t → t → ∞ . .4 proofs: recursive utility on an unbounded state space For the case where θ > β θ ρ ( K ) < (cid:26) − βt θ + β (cid:18) t + b ( x ) t (cid:19) θ (cid:27) % ∞ as t &
0. So the first inequality of (A.3.2) holds. The second inequality also holds because β θ ρ ( K ) < ϕ ( t , x ) / t → β θ as t → ∞ . Thus choosing t large enough will give theresult for the second inequality.This shows that the conditions in lemma A.3.1 hold. To conclude the proof, note that fora fixed x , ϕ ( t , x ) is either convex or concave in t , depending on θ . The rest of the prooffollows in exactly the same manner as A.2.8. a.4 proofs: recursive utility on an unbounded state space In this section we prove theorem 3.4.1. The following lemmas generalise results found inthe online appendix of Borovička and Stachurski [2019].
Lemma A.4.1.
Let { T n } and T be bounded linear operators on L p ( X , π ) such that (cid:54) T n (cid:54) T n + (cid:54) T for all n ∈ N . If R | T n f − T f | p dπ → as n → ∞ for each f in thepositive cone L p ( X ) + and T i is compact for some i ∈ N , then ρ ( T n ) % ρ ( T ) . Proof.
The proof follows identically to lemma 2.2 of the online appendix of Borovička andStachurski [2019], replacing the L norm with the L p norm. In particular, the spectralcontinuity result of Schep [1980] will apply to all L p spaces. Lemma A.4.2.
Let (E,d) be a metric space and let T and { T m } m ∈ N be operators on E with the property that T m u → T u in norm for all u ∈ E . Let ¯ u m be a fixed point of T m for each m and suppose that ¯ u m → ¯ u for some ¯ u ∈ E . If T is continuous on E and themaps { T m } are uniformly Lipschitz continuous, then ¯ u is a fixed point of T .Proof. See Borovička and Stachurski [2019] online appendix, lemma 2.3. .4 proofs: recursive utility on an unbounded state space We now turn to more direct results used in the proof of theorem 3.4.1. This proof relieson a limiting argument based on approximating X with compact sets. The proofs buildupon those seen in the online appendix of Borovička and Stachurski [2019].Let { F m } m ∈ N be an increasing sequence of compact sets such that F m ⊂ F m + for all m ∈ N . Since X is σ − finite, we set S m ∈ N F m = X . Let K m be the operator on L p ( X , π ) defined by K m g ( x ) = x ∈ F m Z F m k ( x , y ) g ( y ) dy Note that K m is also a positive linear operator and 0 (cid:54) K m (cid:54) K m + for all m ∈ N . Then K m is a bounded linear operator on L p ( X , π ) . Lemma A.4.3. If f ∈ L p ( X , π ) + , then || K m f − Kf || → as m → ∞ .Proof. Fix f ∈ L p ( X , π ) + . For any m ∈ N , we have || K m f − Kf || p (cid:54) Z (cid:18)Z k ( x , y ) ( − F m ( x ) F m ( y )) f ( y ) dy (cid:19) p dπ ( x ) .Since K is a bounded, linear operator, the integral on the right hand side is finite. Sincewe are on a finite measure space, Egorov’s theorem means that it suffices to show thatthe integrand converges pointwise to 0. This follows immediately from the definition of { F m } .Given g : F m → R , as per Borovička and Stachurski [2019], define its extension e m g to X as the function equal to g on F m and zero on F cm . Given g : X → R , its restriction c m g to F m is defined as the function c m g equal to g on F m . In addition, let ¯ K be the restrictionof K m to real functions on F m . That is, ¯ K m g ( x ) = Z F m k ( x , y ) g ( y ) dy .We regard ¯ K m as a mapping on L p ( F m , ¯ π ) , where ¯ π : = c m π . Note that A m = e m ¯ A m c m (A.4.1)on L p ( X ) + , where A m = ϕ ◦ K m and ¯ A m : = ϕ ◦ ¯ K m . The latter is a self-mapping on thepositive cone L p ( F m , ¯ π m ) + . .4 proofs: recursive utility on an unbounded state space Lemma A.4.4. If g ∈ L p ( F m , ¯ π m ) + is a fixed point of ¯ A m , then e m g is a fixed point of A m .Proof. For g ∈ L p ( F m , ¯ π ) + we have A m e m g = e m ¯ A m c m e m g = e m ¯ A m g = e m g . Lemma A.4.5.
For all m ∈ N , we have || ¯ K m || = || K m || .Proof. Fix f ∈ L p ( X , π ) with || f || (cid:54)
1. Let ¯ f be the restriction of f to F m . Note that, || ¯ f || p = Z | ¯ f | p ¯ π ( x ) dx (cid:54) || f || p (cid:54) || ¯ K ¯ f || p = Z F m (cid:12)(cid:12)(cid:12)(cid:12)Z F m k ( x , y ) f ( x ) dy (cid:12)(cid:12)(cid:12)(cid:12) p π ( x ) dx = Z | K m f ( x ) | p π ( x ) dx = || K m f || p .Thus, by the definition of the operator norm we have that || K m f || = || ¯ K m ¯ f || (cid:54) || ¯ K m || and then by taking the supremum over { K m f : || f || (cid:54) } on the left hand side we get || K m || (cid:54) || ¯ K m || .To see the reverse inequality holds, fix ¯ f ∈ L p ( F m , ¯ π ) with || ¯ f || (cid:54)
1. Let f ∈ L p ( X , π ) bedefined by f = ¯ f on F m and f = || f || p = Z | f | p π ( x ) dx = Z | ¯ f | p ¯ π ( x ) dx = || ¯ f || p (cid:54) || ¯ K ¯ f || = || K m f || . It follows that || ¯ K m ¯ f || (cid:54) || K m || , and taking the supremum on the left over all such ¯ f yields || ¯ K m || (cid:54) || K m || . Lemma A.4.6. If ρ ( K ) > / β θ , then there exists an M ∈ N such that ρ ( ¯ K m ) > / β θ whenever m (cid:62) M .Proof. In view of lemma A.4.5 and the definition of the spectral radius, it suffices to provethat ρ ( K m ) >
1, for sufficiently large m . This will be true if ρ ( K m ) → ρ ( K ) , which, bylemma A.4.1, will hold if (a) K i is compact for some i ∈ N , (b) 0 (cid:54) K m (cid:54) K m + (cid:54) K forall m and (c) K m f → Kf in norm for each f in L p ( , π ) + . We already have (a) by eventualcompactness and (b) is true by construction. Finally, (c) holds by lemma A.4.3. .4 proofs: recursive utility on an unbounded state space Lemma A.4.7.
Under the conditions of theorem 3.4.1, Λ p is well defined and satisfies Λ p = βρ ( K ) / θ . Proof.
Proof is identical to the compact state space case in proposition A.2.4.
Lemma A.4.8. If θ < and Λ p < , then there exists an M ∈ N such that, for all m (cid:62) M , the operator A m has a nonzero fixed point g m ∈ L p ( X , π ) + , and g m (cid:54) g m + forall such m ∈ N .Proof. If θ < Λ p <
1, by proposition A.4.7, we have ρ ( K ) > / β θ . Now, let M beas in lemma A.4.6 and take m (cid:62) M . Observe that ¯ A m has a unique nonzero fixed point ¯ g m in L p ( X , π ) , since F m is compact. It then follows from equation (A.4.1) that A m e m ¯ g m = e m ¯ A m c m e m ¯ g m = e m ¯ g m and so g m : = e m ¯ g m is a fixed point of A m . Since ¯ g m is nonzero on F m , the function g m is nonzero on X .It remains to prove that g m (cid:54) g m + for all m (cid:62) M . As such, choose some m (cid:62) M and notethat since K m (cid:54) K m + on L p ( X , π ) and ϕ is increasing, we have A m + g m (cid:62) A m g m = g m .Using isotonicity of A m + and iterating forward yields A nm + g m (cid:62) g m for all n ∈ N .Moreover, since g m is nonzero on F m and hence F m + , the convergence result of theorem3.2.2 applied to the compact set F m + implies that A nm + g m → g m + uniformly. Hence g m + (cid:62) g m , as was to be shown. Lemma A.4.9. If θ < , then the family { A n } is uniformly Lipschitz continuous on L p ( X , π ) + .Proof. When θ <
0, the scalar map ϕ is Lipschitz with Lipschitz constant 1. Hence, forarbitrary m ∈ N and f , g ∈ L p ( X , π ) + we have | A m f − A m g | (cid:54) | K m f − K m g | = | K m ( f − g ) | (cid:54) K m | f − g | (cid:54) K | f − g | .4 proofs: recursive utility on an unbounded state space from monotonicity of Lebesgue integration we then get Z | A m f − A m g | p dπ (cid:54) Z ( K | f − g | ) p dπ which in turn implies || A m f − A m g || (cid:54) || K || · || f − g || This brings us to the final proof of theorem 3.4.1.
Proof of Theorem 3.4.1.
We first show that (a) ⇐⇒ (b) under the assumptions claimed.We begin with (a) = ⇒ (b). When θ >
0, the proof follows directly from the proof of thecompact case, as the proof does not rely on compactness.Thus, suppose that θ <
0. Observe that by lemma A.4.3, for f ∈ L p ( X , π ) + we have K m f → Kf as m → ∞ . Since ϕ is Lipschitz continuous of order 1 when θ <
0, we getimmediately that A m f → Af as m → ∞ . By lemma A.4.8, there exists an M ∈ N suchthat for all m (cid:62) M , the operator A m has a nonzero fixed point g m ∈ L p ( X , π ) + , and g m (cid:54) g m + for all such m . Since ϕ is bounded above by ( − β ) θ when θ < g m (cid:54) g m + (cid:54) ( − β ) θ for all m .Note, any order bounded monotone sequence in L ( X , π ) converges to an element of thatset. Denote the limit by g . Then since R | g m ( x ) − g ( x ) | dπ →
0, observing that we are on afinite measure space and applying Egorov’s theorem shows that R | g m ( x ) − g ( x ) | p dπ → g is also the L p ( X , π ) limit. In view of lemma A.4.2, this g will be afixed point of A whenever A is continuous and { A m } is uniformly Lipschitz continuous.Continuity of A is immediate from the properties of K and ϕ , while uniform Lipschitzcontinuity of { A m } follows from lemma A.4.9. This shows that (a) = ⇒ (b).We now show (b) = ⇒ (a). In this case the exact same proof as used when X was compactcan be used. In proposition A.2.9 compactness was only used to ensure that K i wascompact on L p ( X , π ) for some i ∈ N . In the unbounded setting, this condition still holdsby eventual compactness. This shows that (b) = ⇒ (a). .5 proofs from chapter 4 Finally, the equivalence between (b) and (c) holds by standard arguments. The fact that(b) = ⇒ (c) follows by choosing g to be the fixed point in the statement of (c). To seethat (c) = ⇒ (b) note that g ∗ = lim n →∞ A n g is a fixed point of A by continuity of A . a.5 proofs from chapter 4 Proof of Theorem 4.1.2.
From the outset, we take x ∈ [ M ] .Now note that for some A σ > k ( x , y ) must satisfy k ( x , y ) = exp (cid:20) ( − γ )( µ c + x ) + ( − γ ) √ x + (cid:15) (cid:21) · q ( x , y ) (cid:54) A σ exp (cid:20) ( − γ )( µ c + x ) + ( − γ ) √ x + (cid:15) (cid:21)q π ( x + (cid:15) ) · exp (cid:20) − ( y − ρx ) ( x + (cid:15) ) (cid:21) .Since x is bounded, we can take sufficiently large B , C , D > B r>M ( ) = Ω ⊂ R × R , we have Z | k ( x , y ) | d ( π × π ) (cid:54) D Z exp [ Bx + C ] d ( π × π )= D (cid:18) Z Ω exp [ Bx + C ] d ( π × π ) + Z Ω c exp [ Bx + C ] d ( π × π ) (cid:19) (cid:54) D (cid:18) Q M + R N Z Ω c exp [ Bx + C ] exp (cid:20) − x M (cid:21) dx (cid:19) < ∞ for some Q M , R N >
0. In particular, the second last line follows from the fact that thetail density of x is weakly dominated by a normal distribution with variance M . Wetake a large ball around the origin so that we need only integrate around this tail densityin the latter term. .5 proofs from chapter 4 Proof of Theorem 4.3.1.
The Schwartz kernel is given by m ( x , y ) = Z exp [( − γ )( µ c + x + ( φ c ¯ σe x ) (cid:15) ] ν ( d(cid:15) ) q ( x , y )= q ( x , y ) exp (cid:20) ( − γ )( µ c + x ) (cid:21) · exp (cid:20) ( φ (cid:15) ¯ σe x ) (cid:21) where by assumption x is bounded.Using proposition 3.4.2, it suffices to verify that m ( x , y ) ∈ L ( R × R ) .Note that q = q (( x , x , x ) , ( y , y , y )) (cid:54) A q (( · , · , x ) , ( · , · , y )) for some A >
0. Thus q ( x , y ) (cid:54) AR exp (cid:20) − ( y − ρx ) Be x (cid:21) − M (cid:54) x (cid:54) M (cid:54) AR exp (cid:20) − ( y − ρx ) B · e M (cid:21) for B = q − ρ φ z ¯ σ , and R the constant of normalisation. Thus, for Q M > m ( x , y ) (cid:54) AR exp (cid:20) − ( y − ρx ) B · e M (cid:21) exp (cid:20) ( − γ )( µ c + x ) (cid:21) · exp (cid:20) ( φ (cid:15) ¯ σe x ) (cid:21) (cid:54) Q M · exp (cid:20) ( − γ ) x − ( y − ρx ) Be M (cid:21) (cid:54) Q M · exp (cid:20) ( − γ ) x (cid:21) Thus, note that for some sufficiently large ball around the origin B r> M ( ) = Ω ⊂ R × R ,and sufficiently large N , C > Z | m ( x , y ) | dπ (cid:54) Q M Z e ( − γ ) x dπ = Q M (cid:18) Z Ω e ( − γ ) x dπ + Z Ω c e ( − γ ) x dπ (cid:19) (cid:54) N + Q M Z Ω c e ( − γ ) x dπ (cid:54) N + Q M C Z Ω c e ( − γ ) x e − x ϕeM dx < ∞ That is, consider only the ‘slice’ density over x , and allow all other arguments to take any value. .5 proofs from chapter 4 where ϕ is a constant of normalisation. That is, m ( x , y ) ∈ L ( R × R ) . The second last line of the argument follows by noting that the tail distribution of x must be sub-Gaussian(see Lemma 2.1.1 of Vershynin [2018]), and that the density with respect to ( x , x , y , y , y ) decays tobe uniformly bounded by 1 for large values. I B L I O G R A P H YR. Albuquerque, M. Eichenbaum, V. X. Luo, and S. Rebelo. Valuation risk and assetpricing.
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