Discounted Utility and Present. Value A Close Relation


 Bethany Atkinson
 1 years ago
 Views:
Transcription
1 Discounted Utility and Present Value A Close Relation Han Bleichrodt a, Umut Keskin b, Kirsten I.M. Rohde b, Vitalie Spinu c, & Peter P. Wakker b a : As b ; corresponding author b : Erasmus School of Economics, Erasmus University Rotterdam, P.O. Box 1738, Rotterdam, 3000 DR, the Netherlands; +31(0) (F); c : The Anderson School at UCLA, 110 Westwood Plaza D511, Los Angeles, CA July, 2013 Abstract We introduce a new type of preference conditions for intertemporal choice, requiring independence of present values from various other variables. The new conditions are more concise and more transparent than traditional ones. They are natural, and directly related to applications, because present values are widely used in intertemporal choice. Our conditions give more general behavioral foundations, 1
2 which facilitates normative debates and empirical tests of time inconsistencies and related phenomena. Like other preference conditions, our conditions can be tested qualitatively. Unlike other preference conditions, however, our conditions can also be directly tested quantitatively, e.g. to verify the required independence of present values from predictors in regressions. We show how similar types of preference conditions, imposing independence conditions between directly observable quantities, can be developed for other decision contexts than intertemporal choice, and can simplify behavioral foundations there Keywords: present value, discounted utility, time inconsistency, preference foundation JELclassification: D90, D03, C60 2
3 4 1 Introduction Debates about appropriate models of intertemporal choice, both for making rational decisions and for describing and predicting human decisions, usually focus on the critical preference conditions that distinguish between those models. The two most discussed conditions are time consistency, distinguishing between constant and hyperbolic discounting, and intertemporal separability, pertaining to habit formation, addiction, and sequencing effects. 1 Both these conditions are assumed in the classical models but are usually violated empirically. Their normative status has also been questioned. 2 We introduce a new kind of preference conditions, stated directly in terms of present values (PVs). PVs are simple and tractable, and have been widely used in intertemporal choice throughout history (de Wit 1671; Fisher 1930). Hence preference conditions in terms of PVs are simple and natural too. In general, PVs can depend on many variables, such as the times of the receipt of future payoffs, the initial wealth levels at those times, and the wealth levels 1 See Attema (2012), Dolan & Kahneman (2008 p. 228), Epper, FehrDuda, & Bruhin (2011), Frederick, Loewenstein, & O Donoghue (2002), Gilboa (1989), Herrnstein (1961), Karni & Safra (1990), Loewenstein & Prelec (1993), Lucas & Stokey (1984 p. 169), Rozen (2010), Strzalecki (2013), Tsuchiya & Dolan (2005), Wakai (2008), and others. 2 See Broome (1991), Gold et al. (1996 p. 100), Parfit (1984 Ch. 14), Strotz (1956 p. 178), and others. 3
4 at other times. Our preference conditions will impose independence of PVs from part of those other variables, and we show that these independencies characterize various models of intertemporal choice. Like all preference conditions, our conditions can be tested qualitatively. Unlike other preference conditions, our conditions can also be directly tested quantitatively. We can carry out regressions with PV as the dependent variable, and the other relevant variables as predictors. 3 We can then test which of those other variables are significantly associated with PV, and whether the variables claimed to be independent in our conditions really are so. Such tests are more widely known and better understood than qualitative tests of preference conditions. They use econometric techniques based on wellestablished probabilistic error theories. To illustrate our new PV conditions clearly, we will apply them to wellknown models. A concise presentation of these models and their representations is in Table 1. Details of the table will be explained in the following sections. The table is presented here because it illustrates the organization of the first four sections. We provide the most concise preference foundations presently available in 3 Predictors are often called independent variables in regressions. The latter term independence does not refer to statistical independence, the concept relevant to our paper. We do not use it so as to avoid confusion. 4
5 the literature for: (a) constant discounted value as commonly used in financial economics (Hull 2013); (b) constant discounted utility (Samuelson 1937); (c) general discounted utility, which includes hyperbolic discounting. We also provide some results that are relevant to other than intertemporal contexts: (d) nobook and noarbitrage for uncertainty; (e) Gorman (1968)Debreu (1960) additive separability for general aggregations. All our preference conditions are given in the PV condition column in Table 1. Their statements take little space in this column, but yet give complete definitions as explained later. Further extensions to other decision contexts are a topic for future research, as are extensions to more general intertemporal models (Sections 5 and 7) Preferences and subjective PVs We consider preferences over outcome sequences x = (x 0,..., x T ) R T +1, yielding outcome x t in period t; t = 0 denotes the present. To avoid trivialities, we assume T 2; T is fixed. Besides indexed Roman letters x t that specify the period t of receipt, Greek letters α, β,... are used to designate outcomes (real numbers) when no period of receipt needs to be specified. By α t x we denote x with x t replaced by α. V represents if V : R T +1 R and x y V (x) V (y), for 5
6 Theory Functional Pref. conditions PV condition PV formula T t=0 x t Additive + π(ϕ) ϕ discounted value Symmetric Constant T t=0 λt xt Additive + π(ϕ, t) = τ(ϕ, t + 1) λ t ϕ discounted value Stationary Timedependent T t=0 λ txt Additive π(ϕ, t) λtϕ discounted value Non Non T t=0 U(x t) Separable + π(ϕ, x0, xt) U 1 (U(et + ϕ) discounted utility Symmetric U(et) + U(e0)) e0 Constant T t=0 λt U(xt) Separable + π(ϕ, t, e0, et) = U 1 (λ t U(et + ϕ) discounted utility Stationary τ(ϕ, t + 1, e 1 = e0, e t+1 = et) λ t U(et) + U(e0)) e0 Timedependent T t=0 U t(xt) Separable π(ϕ, t, x0, xt) U 0 1 (Ut(et + ϕ) discounted utility Ut(et) + U0(e0)) e0 Table 1: The column Functional gives the functional forms. Here: 0 < λ is a discount factor; 0 < λt is a perioddependent discount factor with λ0 = 1; U is a strictly increasing continuous utility function; Ut is a perioddependent strictly increasing continuous utility function. The column Pref. conditions gives preference conditions traditionally used in preference foundations of the functional forms, and defined in the Appendix. The column PV condition gives the PV condition (how π(ϕ, t, e) can be rewritten, with an extra equality imposed for constant discounting) used in our preference foundations. Note that these cells contain complete definitions. The column PV formula gives the formula of PV under each theory. 6
7 56 all x, y R T +1. If a representing V exists, then is a weak order; i.e., 57 is complete (x y or y x for all x, y) and transitive. We assume throughout that is a weak order. Strict preference, indifference, reversed preference, and strict reversed preference ( ) are as usual. We also assume monotonicity (strictly improving an outcome strictly improves the outcome sequence) and continuity of throughout. The following condition considers sums of outcomes x t = ϕ + e t. Here we call e t an initial endowment. We sometimes use the term payoff instead of outcome for ϕ to facilitate interpretations. Similarly, the distinction of the initial endowment only serves to facilitate interpretations and does not refer to any formal reference dependence. Formally, our analysis is entirely in terms of final wealth and is classical in this respect DEFINITION 1 π is the present value (PV ) of outcome ϕ received in period t with (prior) endowment e = (e 0,..., e T ) if (e 0 + π) 0 e (e t + ϕ) t e, i.e. (e 0 + π, e 1,..., e t,..., e T ) (e 0,..., e t + ϕ,..., e T ). (1) Eq. 1 means that, with e the current endowment, receiving an additional future payoff ϕ in period t is exactly offset by receiving an additional present outcome π. That is, the PV of a future payoff ϕ in period t is π. For 7
8 simplicity, we assume throughout that a PV always exists. By monotonicity, it is unique. The definition of PV (denoted π) can be naturally extended to an outcome sequence x with endowment e: (e 0 + π) 0 e e + x. However, we will not need this concept in this paper. The PV can in general depend on ϕ, t, and e, and is a function π(ϕ, t, e). For t = 0, it trivially follows that π(ϕ, 0, e) = ϕ. Note that π is subjective, depending on and, thus, reflecting the tastes and attitudes of the decision maker. The preference conditions presented in the following sections all amount to independence of PV from some of the variables (ϕ, t, e). We denote this independence by writing only the arguments that π depends on. For example, if π(ϕ, t, e) depends only on ϕ (i.e. is independent of t and e) then we write π = π(ϕ). (2) 86 Similarly, if π depends on e only through e 0 and e t, then we write π = π(ϕ, t, e 0, e t ). (3) These conditions, like all other PV conditions in this paper, are indeed preference conditions. Our conditions impose restrictions on indifferences. Preference conditions should be directly verifiable from preferences, the observable primitives in the revealed preference paradigm, without invoking theoretical constructs such as utilities. In Eq. 11 in the appendix we show that our PV 8
9 conditions can be written directly in terms of preferences. They, therefore, are real preference conditions, and our conditions can be tested in the same way as all qualitative preference conditions can be Linear Utility This section considers models with linear utility, as commonly used in financial markets. They can serve as approximations for subjective individual choices if the stakes are small (Epper, FehrDuda, & Bruhin 2011; Luce 2000 p. 86; Pigou 1920 p. 785). The first model in Table 1 maximizes the sum of outcomes: T Non Discounted Value: x t. (4) This model does not involve subjective parameters and, hence, is directly observable. It serves well as a first illustration of the nature of PV conditions. t=0 103 THEOREM 2 The following two statements are equivalent: 104 (i) Nondiscounted value holds. 105 (ii) π = π(ϕ). 106 Throughout this paper, Condition n.(ii) refers to the condition in State 107 ment (ii) of Theorem n. Theorem 2 shows that if π depends only on ϕ 108 (Condition 2.(ii)), then it must be the identity function (π(ϕ) = ϕ). This is 9
10 also displayed in the last two columns of the corresponding row of Table 1. It may be surprising that Condition 2.(ii) implies the addition operation of nondiscounted value. An informal proof in the appendix will clarify. Our next model involves a subjective parameter, the discount weight λ: Constant Discounted Value: T λ t x t for λ > 0. (5) t=0 113 Under the usual assumption that the decision maker is impatient, we have 114 λ 1. In PV calculations of cash flows, financial economists commonly use constant discounted value, setting λ = 1/(1 + r) with r the interest or discount rate. However, the discount factor λ is then no longer a subjective parameter reflecting the attitude of a decision maker, as in our approach, but it is a given constant, publicly known and determined by the market. The latter approach can be related to our approach if we interpret the market as a decision maker, setting rational PVs, with λ reflecting the market attitude. Condition 2.(ii) rationalizes this common evaluation system. If the parameter λ reflects the attitude of an individual decision maker, then it will probably be influenced by the market interest rate but need not be identical to it. As a preparation for the preference foundation, we define tomorrow s value as an analogue to PV: 126 DEFINITION 3 τ is tomorrow s value of payoff ϕ received in period t (t 1) 10
11 127 with endowment e, denoted τ = τ(ϕ, t, e), if (e 1 + τ) 1 e (e t + ϕ) t e, i.e., (e 0, e 1 + τ, e 2,..., e t,..., e T ) (e 0, e 1,..., e t + ϕ,..., e T ). (6) Now τ is the extra payoff in period 1 that exactly offsets the extra payoff ϕ in period t. It is tomorrow s PV. 131 THEOREM 4 The following two statements are equivalent: 132 (i) Constant discounted value holds. 133 (ii) π = π(ϕ, t) = τ(ϕ, t + 1) = τ Uniqueness of the parameter λ, and all other uniqueness results of this paper, are given in the proofs in the Appendix. Condition 4.(ii) entails that π and τ are independent of the endowments, and that tomorrow s perception of future income is the same as today s. The condition implies that PV depends only on stopwatch time (time differences) and not on calendar time (absolute time). 4 The notation here is short for: π = π(ϕ, t, e) = π(ϕ, t) = τ(ϕ, t + 1) = τ(ϕ, t + 1, e ). The two endowments e and e are immaterial, and are allowed to be different. Because of the shift by one period, the condition is imposed only for all t < T. Existence of π and the equality in Statement 4.(ii) imply that τ also exists. 11
12 Future present values are central tools in recursive intertemporal models (see Caplin & Leahy 2006; Maccheroni, Marinacci, & Rustichini 2006; Ju & Miao 2012 and their references). Stationarity, which requires these future values to be invariant over time, is a convenient special case. Statement 4(ii) formulates this condition in a simplified manner, with only one future payoff and one present value (today or tomorrow), rather than involving general preferences between general outcome sequences as do common versions of stationarity. Most tests of stationarity in the literature are in fact tests of our simplified version; see Takeuchi (2010) and his extensive survey. Many studies have shown that constant discounting is violated empirically. Hence the following model is of interest. TimeDependentDiscounted Value: T λ t x t (with λ 0 = 1). (7) t= This model allows for general discount weights whose dependence on time is unrestricted. Many special cases of such discount weights have been studied in the literature, the bestknown being hyperbolic weighting. The representation in Eq. 7 is not affected if all λ t s are multiplied by the same positive factor. The scaling λ 0 = 1 reflects the most common convention in the literature. 157 THEOREM 5 The following two statements are equivalent: 158 (i) Timedependent discounted value holds. 12
13 159 (ii) π = π(ϕ, t) An implication, displayed in the last two columns of Table 1, is that if π is any function of ϕ and t, then it must be the function λ t ϕ Nonlinear Utility The models presented in the preceding section take a sum, weighted or unweighted, of the outcomes. They assume constant marginal utility in the sense that an extra euro received in a particular period gives the same utility increment regardless of the endowment of that period. In individual choice this condition is often empirically violated and not normatively imperative. More realistic and more popular models allow for nonlinear utility. If marginal utility does depend on the endowment, then the models of the preceding section become: T NonDiscounted Utility: U(x t ); (8) Constant Discounted Utility: TimeDependent Discounted Utility: t=0 T λ t U(x t ); (9) t=0 T U t (x t ). (10) t= Eq. 9 is Samuelson s (1937) discounted utility, the most popular model for intertemporal choice. Each utilitymodel reduces to the corresponding value 13
14 model if utility is taken linear. In particular, in timedependent discounted utility, if the U t (α) are linear then they can be written as λ t α, where we can renormalize such that λ 0 = 1. Then timedependent discounted value results. The mathematics underlying the preference foundations for the utility models in Eqs is more advanced than it was for Theorems 2, 4, and 5. Whereas those theorems solved linear equalities, we now have to deal with general equalities, with nonlinear utilities intervening. Fortunately, the increased complexity of the underlying mathematics does not show up in the preference conditions and, consequently, in the empirical tests of the models. The relevant PV preference conditions are obtained directly from those defined in 3 by adding dependence on the endowment levels e 0, e t. This way we readily obtain Theorems 6, 7, and 8 from Theorems 2, 4, and 5, respectively. The transparancy of our preference conditions clarifies the logical relations between the different results. 188 THEOREM 6 The following two statements are equivalent: 189 (i) Nondiscounted utility holds. 190 (ii) π = π(ϕ, e 0, e t ). 5 5 Here t refers to the period where ϕ is added and e t specifies the endowment in that period. Note that there is no explicit dependence of π on t, which we denote by suppressing 14
15 An implication, displayed in the last two columns of Table 1, is that if π is any function of ϕ, x 0, and x t, then it must be of the form displayed there. Several authors have argued that any discounting, even if consistent over time, is irrational, and have thus recommended nondiscounted utility for intertemporal choice (Jevons 1871 pp ; Ramsey 1928; Rawls 1971; Sidgwick 1907). Condition 6.(ii) characterizes this proposal. Studies providing preference foundations for nondiscounted utility (sums of utilities) include Fishburn (1992), Kopylov (2010), Krantz et al. 1971), Marinacci (1998), Pivato (2013), and Wakker (1989). Blackorby, Davidson, & Donaldson (1977) provided results for lotteries with T + 1 equally likely lotteries, taking the representation as von NeumannMorgenstern expected utility. We now turn to discounting. 203 THEOREM 7 The following two statements are equivalent: 204 (i) Constant discounted utility holds. 205 (ii) π = π(ϕ, t, e 0, e t ) = τ(ϕ, t + 1, e 1 = e 0, e t+1 = e t ). 6 period t from the arguments of π(...). 6 The condition implies that the endowment levels e j, j 0, t of the PV endowment e, and similarly the endowment levels e j, j 1, t + 1 of the tomorrowvalue endowment e, are immaterial. Existence of π and the equality imply that τ also exists. The condition holds for all 0 t T 1, and whenever e 1 = e 0 and e t+1 = e t. We use the convenient argument matching notation popular in programming languages (R: see R Core Team 2013, 15
16 The first preference foundation of constant discounted utility was in Koop mans (1960, 1972), with generalizations in Harvey (1995), Bleichrodt, Ro hde, & Wakker (2008), and Kopylov (2010). Condition 7.(ii) requires that 209 the same tradeoffs are made tomorrow as today. Such requirements have 210 sometimes been taken as rationality requirements (see the introduction). 211 THEOREM 8 The following two statements are equivalent: 212 (i) Timedependent discounted utility holds. 213 (ii) π = π(ϕ, t, e 0, e t ) Statement 8.(ii) expresses that the tradeoffs between periods 0 and t are independent of what happens in the other periods, reflecting a kind of separa bility. Timedependent discounted utility is a general additive representation, 217 which has been axiomatized several times before. 7 It implies intertemporal separability, which is arguably the most questionable assumption of most intertemporal choice models. Python: see Python Software Foundation 2013, Scala: see École Polytechnique Fédérale de Lausanne 2013, among others) to express the latter two restrictions. Here the formal argument of a function (e 1 or e t+1) is assigned a value (e 0 or e t ). 7 Debreu (1960), Gorman (1968), Krantz et al. (1971); Wakker 1989). 16
17 Applications to contexts other than intertem poral choice The mathematical theorems provided in previous sections, and the preference conditions that we used, can be used in contexts other than intertemporal choice. For instance, Theorem 5 is of special interest for decision under un certainty, capturing nonarbitrage in finance. To see this point, we reinterpret the periods t as states of nature. Exactly one state obtains, but it is uncer tain which one (Savage 1954). Now x = (x 0,..., x T ) refers to an uncertain prospect yielding outcomes (payoffs) x t if state of nature t obtains. For sim plicity, we focus on a single time of payoff here, so that discounting plays no role. If we divide the discount weights λ t by their sum T t=0 λ t (relaxing the requirement of λ 0 = 1), then they sum to 1, and the representation becomes subjective expected value. Subjective expected value was first axiomatized by de Finetti (1937) using a nobook argument, called noarbitrage in finance. Then the representation is asif risk neutral, and the decision maker is the market which sets rational prices for statecontingent assets. For state j, a statecontingent asset x = (0,..., 0, 1, 0,..., 0) yields outcome 1 if j happens and nothing otherwise. In this interpretation, λ j becomes the market price of this statecontingent asset, 239 and PV is the offsetting quantity of state0contingent assets. Condition 17
18 (ii) provides the most concise formulation of the nobook and noarbitrage argument presently available in the literature. For decision under uncertainty, certainty equivalents are more natural 243 quantities than statecontingent prices. For risk, Chew (1983) explained basic techniques for applying functional equations to certainty equivalents to obtain preference foundations. Eq. 8 can be interpreted as von NeumannMorgenstern expected utility for equalprobability lotteries, which essentially covers all lotteries with rational probabilities (writing every probability i/j as i probabilities 1/j). Eq. 8 can also be interpreted as ambiguity under complete absence of information (Gravel, Marchant, & Sen 2012). Eq. 10 concerns Debreu s (1960) additively separable utility. Here again, our Statement 8.(ii) provides the most concise preference axiomatization presently available in the literature Discussion The primary purpose of preference foundations is not to provide surprising logical relations with deep and complex proofs, as it would be in abstract mathematics. Their primary purpose is to make decision models with theoretical constructs directly observable, by restating their existence (Statements (i) in our theorems) in terms of the preference relation (Statements 18
19 (ii) in our theorems). The simpler the preference conditions, the better they clarify the empirical meaning of the decision models. Similarity between the preference conditions and the functional is an advantage, because it helps to clarify the empirical meaning of the decision model. Hence the major motivation for this paper has been to introduce preference foundations that are as simple as possible and that reflect the corresponding decision models as well and transparently as possible. The conditions in our Statements (ii) use fewer words and characters than any conditions previously proposed in the literature, and thus are the most concise ones presently existing. The surprising aspect of our paper is that conditions for preference foundations of intertemporal decision models can be that simple. Further, our conditions are easy to understand and test because present values are familiar objects. Our present value conditions use indifferences rather than preferences. Conditions for indifferences are logically weaker, making their implications logically stronger. 8 Our conditions only concern the simplest tradeoffs possible involving the change of the present payoff and one future payoff, further enhancing their generality. Thus commonly used stationarity conditions are more restrictive than the conditions in our Statements 6(ii) and 7(ii). 8 They follow from preference conditions by applying the latter twice, first with preferences one way and then with preferences reversed. 19
20 An empirical advantage of our preference conditions is that they can be directly tested using analyses of variance and regression techniques. If we take PV as the dependent variable, then Eq. 3 predicts that ϕ, t, e 0, and 281 e 1 may be significant predictors, but not the e j s with j 0, t. We can test this prediction using common regression analyses, which allow us to use the probabilistic error theories underlying econometric regressions, which are easier to use than the more recently developed error theories for preferences (Wilcox 2008). The followup paper Keskin (2013) provides extensions of our results to some popular hyperbolic discount models, still maintaining intertemporal separability. Generalizations of the latter condition to situations with partial separability 9 are a topic for future research Conclusion We have introduced a new kind of preference conditions for intertemporal choice, requiring that (quantitative) PVs are independent of particular other variables. The resulting conditions can be used both in qualitative and in 9 Gilboa (1989)and Zuber & Asheim (2012) maintain rankdependent separability. Le Van, Cuong & Yiannis Vailakis (2005) maintain Bellman s equation. Noor (2009) gives up timeindependence of utility. 20
21 294 quantitative tests, and are more concise and transparent than preceding con 295 ditions proposed in the literature. The technique of obtaining preference conditions by taking a quantity with direct empirical meaning and then imposing independence conditions can be extended to other decision contexts such as decision under uncertainty. Then new, more concise characterizations result, which can easily be tested empirically. 21
22 Appendix. Proofs, uniqueness results, and clarification of the empirical status of PV conditions We present the proofs, with added the uniqueness results, of our theorems in reversed order, from most to least general, because this approach is most clarifying and most effi cient. The presentation of the proofs clarifies the relationship between our PV conditions and the wellknown preference conditions, showing that PV conditions indeed are preference conditions. In each proof, we start from our PV condition, which is always weaker than the conditions that are derived and that are commonly used in the literature. We thus show that our PV conditions give stronger results. Because each Statement (ii) is immediately implied by substitution of the functional, we throughout assume Statement (ii) and derive Statement (i) and the uniqueness results P T 8, The following uniqueness result holds for Statement 8.(i), which uses Eq. 10: A timedependent real constant µ t can be added to every U t, and all U t can be multiplied by a joint positive constant ν. This follows from well 22
23 known uniqueness results in the literature (Krantz et al Theorem 6.13; Wakker 1989 Observation III.6.6 ). We next derive Statement 8.(i). The equality π(ϕ, t, e) = π(ϕ, t, e 0, e t ) in Condition 8.(ii) means that π is independent of e j, j 0, t. This holds if and only if (e 0 + π, e 1,..., e t 1, e t, e t+1,... e T ) (e 0, e 1,..., e t 1, e t + ϕ, e t+1,... e T ) (e 0 + π, e 1,..., e t 1, e t, e t+1,... e T ) (e 0, e 1,..., e t 1, e t + ϕ, e t+1,... e T ).(11) 322 This holds if and only if the implication holds with twice preference instead 323 of indifference. 10 Eq. 11 with preference instead of indifference is known as separability of {0, t} (Gorman 1968). By repeated application of Gorman (1968), separability of every set {0, t} holds if and only if is separable; i.e., every subset of {0,..., T } is separable (preferences are independent of the levels where outcomes outside this subset are kept fixed, as with separability of {0, t}). This holds if and only if an additively decomposable representation holds 11, which we call timedependent discounted utility in the main text To wit, if the upper indifference is changed into a strict preference, then we find π < π to give indifference. The lower indifference follows with π instead of π. By monotonicity, replacing π by π leads to the lower strict preference. 11 See Debreu (1960), Gorman (1968), Krantz et al. (1971 Theorem 6.13), Wakker (1989 Theorem III.6.6). 23
24 P T 7, The following uniqueness result holds for Statement 7.(i), which uses Eq. 9: A realvalued timeindependent constant µ can be added to U, and U can be multiplied by a positive constant ν. λ is uniquely determined. These uniqueness results follow from those in Theorem 8, where, in terms of Eq. 10, λ is the proportion U t+1 /U t for any t. It is useful to note that the sum of weights, λ t, is the same for each outcome sequence evaluated, implying that there is no special role for utility value 0. We next derive Statement 7.(i). The equality π(ϕ, t, e 0, e t ) = τ(ϕ, t + 1, e 1 = e 0, e t+1 = e t ) in Condition 7.(ii) means that π is independent of e j, j 0, t, T (as in Theorem 8, but now only for t < T ), but also of whether it is measured in period 0 or period 1. This holds if and only if, writing α for e 0 = e 1 and β for e t = e t+1, (α + π, e 1,..., e t 1, β, e t+1,... e T ) (α, e 1,..., e t 1, β + ϕ, e t+1,... e T ) (e 0, α + π,..., e t 1, e t, β,... e T ) (e 0, α,..., e t 1, e t, β + ϕ,... e T ).(12) It implies separability of {0, t} for all t < T, as in Theorem 8, but now, instead of separability of {0, T } we have separability of all {1, t + 1} for all t < T. The latter separability can for instance be seen by replacing all primes in Eq. 12 by double primes, which should not affect the second indifference because of the maintained equivalence with the first indifference. By repeated 24
A Simpli ed Axiomatic Approach to Ambiguity Aversion
A Simpli ed Axiomatic Approach to Ambiguity Aversion William S. Neilson Department of Economics University of Tennessee Knoxville, TN 379960550 wneilson@utk.edu March 2009 Abstract This paper takes the
More informationRank dependent expected utility theory explains the St. Petersburg paradox
Rank dependent expected utility theory explains the St. Petersburg paradox Ali alnowaihi, University of Leicester Sanjit Dhami, University of Leicester Jia Zhu, University of Leicester Working Paper No.
More informationPDF hosted at the Radboud Repository of the Radboud University Nijmegen
PDF hosted at the Radboud Repository of the Radboud University Nijmegen The following full text is a publisher's version. For additional information about this publication click this link. http://hdl.handle.net/2066/29354
More information1. Probabilities lie between 0 and 1 inclusive. 2. Certainty implies unity probability. 3. Probabilities of exclusive events add. 4. Bayes s rule.
Probabilities as betting odds and the Dutch book Carlton M. Caves 2000 June 24 Substantially modified 2001 September 6 Minor modifications 2001 November 30 Material on the lotteryticket approach added
More informationInequality, Mobility and Income Distribution Comparisons
Fiscal Studies (1997) vol. 18, no. 3, pp. 93 30 Inequality, Mobility and Income Distribution Comparisons JOHN CREEDY * Abstract his paper examines the relationship between the crosssectional and lifetime
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationOptimal demand management policies with probability weighting
Optimal demand management policies with probability weighting Ana B. Ania February 2006 Preliminary draft. Please, do not circulate. Abstract We review the optimality of partial insurance contracts in
More informationSome Common Confusions about Hyperbolic Discounting
Some Common Confusions about Hyperbolic Discounting 25 March 2008 Eric Rasmusen Abstract There is much confusion over what hyperbolic discounting means. I argue that what matters is the use of relativistic
More informationDECISION MAKING UNDER UNCERTAINTY:
DECISION MAKING UNDER UNCERTAINTY: Models and Choices Charles A. Holloway Stanford University TECHNISCHE HOCHSCHULE DARMSTADT Fachbereich 1 Gesamtbibliothek Betrtebswirtscrtaftslehre tnventarnr. :...2>2&,...S'.?S7.
More informationThe Relation between Two Present Value Formulae
James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 6174. The Relation between Two Present Value Formulae James E. Ciecka,
More information1 Uncertainty and Preferences
In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package
More informationRegret and Rejoicing Effects on Mixed Insurance *
Regret and Rejoicing Effects on Mixed Insurance * Yoichiro Fujii, Osaka Sangyo University Mahito Okura, Doshisha Women s College of Liberal Arts Yusuke Osaki, Osaka Sangyo University + Abstract This papers
More informationOptimal Paternalism: Sin Taxes and Health Subsidies
Optimal Paternalism: Sin Taxes and Health Subsidies Thomas Aronsson and Linda Thunström Department of Economics, Umeå University SE  901 87 Umeå, Sweden April 2005 Abstract The starting point for this
More informationInfinitely Repeated Games with Discounting Ù
Infinitely Repeated Games with Discounting Page 1 Infinitely Repeated Games with Discounting Ù Introduction 1 Discounting the future 2 Interpreting the discount factor 3 The average discounted payoff 4
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More informationWhat is Behavioral Economics? 10/8/2013. Learner Outcomes
Making it Easier for Clients to Act Applying Behavioral Economic Concepts to Financial Planning Carol S. Craigie, MA, ChFC, CFP Professor College for Financial Planning Learner Outcomes Explain the critical
More informationIn recent years, Federal Reserve (Fed) policymakers have come to rely
LongTerm Interest Rates and Inflation: A Fisherian Approach Peter N. Ireland In recent years, Federal Reserve (Fed) policymakers have come to rely on longterm bond yields to measure the public s longterm
More informationBetting interpretations of probability
Betting interpretations of probability Glenn Shafer June 21, 2010 Third Workshop on GameTheoretic Probability and Related Topics Royal Holloway, University of London 1 Outline 1. Probability began with
More informationRed Herrings: Some Thoughts on the Meaning of ZeroProbability Events and Mathematical Modeling. Edi Karni*
Red Herrings: Some Thoughts on the Meaning of ZeroProbability Events and Mathematical Modeling By Edi Karni* Abstract: Kicking off the discussion following Savage's presentation at the 1952 Paris colloquium,
More informationRetirement Financial Planning: A State/Preference Approach. William F. Sharpe 1 February, 2006
Retirement Financial Planning: A State/Preference Approach William F. Sharpe 1 February, 2006 Introduction This paper provides a framework for analyzing a prototypical set of decisions concerning spending,
More informationMarket Value of Insurance Contracts with Profit Sharing 1
Market Value of Insurance Contracts with Profit Sharing 1 Pieter Bouwknegt NationaleNederlanden Actuarial Dept PO Box 796 3000 AT Rotterdam The Netherlands Tel: (31)10513 1326 Fax: (31)10513 0120 Email:
More informationUniversity of Pennsylvania The Wharton School. FNCE 911: Foundations for Financial Economics
University of Pennsylvania The Wharton School FNCE 911: Foundations for Financial Economics Prof. Jessica A. Wachter Fall 2010 Office: SHDH 2322 Classes: Mon./Wed. 1:303:00 Email: jwachter@wharton.upenn.edu
More informationA framing effect is usually said to occur when equivalent descriptions of a
FRAMING EFFECTS A framing effect is usually said to occur when equivalent descriptions of a decision problem lead to systematically different decisions. Framing has been a major topic of research in the
More informationAdditive decompositions for Fisher, Törnqvist and geometric mean indexes
Journal of Economic and Social Measurement 28 (2002) 51 61 51 IOS Press Additive decompositions for Fisher, Törnqvist and geometric mean indexes Marshall B. Reinsdorf a, W. Erwin Diewert b and Christian
More informationLOOKING FOR A GOOD TIME TO BET
LOOKING FOR A GOOD TIME TO BET LAURENT SERLET Abstract. Suppose that the cards of a well shuffled deck of cards are turned up one after another. At any timebut once only you may bet that the next card
More informationNoBetting Pareto Dominance
NoBetting Pareto Dominance Itzhak Gilboa, Larry Samuelson and David Schmeidler HEC Paris/Tel Aviv Yale Interdisciplinary Center Herzlyia/Tel Aviv/Ohio State May, 2014 I. Introduction I.1 Trade Suppose
More informationCPC/CPA Hybrid Bidding in a Second Price Auction
CPC/CPA Hybrid Bidding in a Second Price Auction Benjamin Edelman Hoan Soo Lee Working Paper 09074 Copyright 2008 by Benjamin Edelman and Hoan Soo Lee Working papers are in draft form. This working paper
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationShould we discount future generations welfare? A survey on the pure discount rate debate.
Should we discount future generations welfare? A survey on the pure discount rate debate. Gregory Ponthiere * ** Abstract In A Mathematical Theory of Saving (1928), Frank Ramsey not only laid the foundations
More informationEconomics Discussion Paper Series EDP1005. Agreeing to spin the subjective roulette wheel: Bargaining with subjective mixtures
Economics Discussion Paper Series EDP1005 Agreeing to spin the subjective roulette wheel: Bargaining with subjective mixtures Craig Webb March 2010 Economics School of Social Sciences The University of
More informationLecture 11 Uncertainty
Lecture 11 Uncertainty 1. Contingent Claims and the StatePreference Model 1) Contingent Commodities and Contingent Claims Using the simple twogood model we have developed throughout this course, think
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationDeterminants of Social Discount Rate, general case
Determinants of Social Discount Rate, general case The resulting equation r = ρ + θ g is known as the Ramsey equation after Frank Ramsey (928) The equation states tt that t in an optimal intertemporal
More informationA Simpli ed Axiomatic Approach to Ambiguity Aversion
A Simpli ed Axiomatic Approach to Ambiguity Aversion William S. Neilson Department of Economics University of Tennessee Knoxville, TN 379960550 wneilson@utk.edu May 2010 Abstract This paper takes the
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationThe MarketClearing Model
Chapter 5 The MarketClearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More information3 Introduction to Assessing Risk
3 Introduction to Assessing Risk Important Question. How do we assess the risk an investor faces when choosing among assets? In this discussion we examine how an investor would assess the risk associated
More informationSavages Subjective Expected Utility Model
Savages Subjective Expected Utility Model Edi Karni Johns Hopkins University November 9, 2005 1 Savage s subjective expected utility model. In his seminal book, The Foundations of Statistics, Savage (1954)
More informationTaxadjusted discount rates with investor taxes and risky debt
Taxadjusted discount rates with investor taxes and risky debt Ian A Cooper and Kjell G Nyborg October 2004 Abstract This paper derives taxadjusted discount rate formulas with MilesEzzell leverage policy,
More information.4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.
Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, $100, for one month, and is considering
More informationOptimal Consumption with Stochastic Income: Deviations from Certainty Equivalence
Optimal Consumption with Stochastic Income: Deviations from Certainty Equivalence Zeldes, QJE 1989 Background (Not in Paper) Income Uncertainty dates back to even earlier years, with the seminal work of
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationInflation. Chapter 8. 8.1 Money Supply and Demand
Chapter 8 Inflation This chapter examines the causes and consequences of inflation. Sections 8.1 and 8.2 relate inflation to money supply and demand. Although the presentation differs somewhat from that
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationCARLETON ECONOMIC PAPERS
CEP 1414 Employment Gains from MinimumWage Hikes under Perfect Competition: A Simple GeneralEquilibrium Analysis Richard A. Brecher and Till Gross Carleton University November 2014 CARLETON ECONOMIC
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationKeep It Simple: Easy Ways To Estimate Choice Models For Single Consumers
Keep It Simple: Easy Ways To Estimate Choice Models For Single Consumers Christine Ebling, University of Technology Sydney, christine.ebling@uts.edu.au Bart Frischknecht, University of Technology Sydney,
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationInvestment Decision Analysis
Lecture: IV 1 Investment Decision Analysis The investment decision process: Generate cash flow forecasts for the projects, Determine the appropriate opportunity cost of capital, Use the cash flows and
More informationCertainty Equivalent in Capital Markets
Certainty Equivalent in Capital Markets Lutz Kruschwitz Freie Universität Berlin and Andreas Löffler 1 Universität Hannover version from January 23, 2003 1 Corresponding author, Königsworther Platz 1,
More informationNew Axioms for Rigorous Bayesian Probability
Bayesian Analysis (2009) 4, Number 3, pp. 599 606 New Axioms for Rigorous Bayesian Probability Maurice J. Dupré, and Frank J. Tipler Abstract. By basing Bayesian probability theory on five axioms, we can
More informationDefinition. Hyperbolic discounting refers to the tendency for people to increasingly choose a
Hyperbolic Discounting Definition Hyperbolic discounting refers to the tendency for people to increasingly choose a smallersooner reward over a largerlater reward as the delay occurs sooner rather than
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More information6. Budget Deficits and Fiscal Policy
Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS 2012 6. Budget Deficits and Fiscal Policy Introduction Ricardian equivalence Distorting taxes Debt crises Introduction (1) Ricardian equivalence
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationUsing Discount Rate for Water Sector Planning in Bangladesh under Climatic Changes
Using Discount Rate for Water Sector Planning in Bangladesh under Climatic Changes Dr. Mohammad Ismail Khan Associate professor, Department of Agricultural Economics Bangabndhu Sheikh Mujibur Rahman Agricultural
More informationAnalyzing the Demand for Deductible Insurance
Journal of Risk and Uncertainty, 18:3 3 1999 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Analyzing the emand for eductible Insurance JACK MEYER epartment of Economics, Michigan State
More informationImplicit versus explicit ranking: On inferring ordinal preferences for health care programmes based on differences in willingnesstopay
Journal of Health Economics 24 (2005) 990 996 Implicit versus explicit ranking: On inferring ordinal preferences for health care programmes based on differences in willingnesstopay Jan Abel Olsen a,b,,
More informationChap 3 CAPM, Arbitrage, and Linear Factor Models
Chap 3 CAPM, Arbitrage, and Linear Factor Models 1 Asset Pricing Model a logical extension of portfolio selection theory is to consider the equilibrium asset pricing consequences of investors individually
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationThe Utility of Gambling Reconsidered
The Journal of Risk and Uncertainty, 29:3; 241 259, 2004 c 2004 Kluwer Academic Publishers. Manufactured in The Netherlands. The Utility of Gambling Reconsidered ENRICO DIECIDUE INSEAD, Decision Sciences,
More informationOperations Research and Financial Engineering. Courses
Operations Research and Financial Engineering Courses ORF 504/FIN 504 Financial Econometrics Professor Jianqing Fan This course covers econometric and statistical methods as applied to finance. Topics
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationInvestigación Operativa. The uniform rule in the division problem
Boletín de Estadística e Investigación Operativa Vol. 27, No. 2, Junio 2011, pp. 102112 Investigación Operativa The uniform rule in the division problem Gustavo Bergantiños Cid Dept. de Estadística e
More informationThe Equity Premium in India
The Equity Premium in India Rajnish Mehra University of California, Santa Barbara and National Bureau of Economic Research January 06 Prepared for the Oxford Companion to Economics in India edited by Kaushik
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationThe Values of Relative Risk Aversion Degrees
The Values of Relative Risk Aversion Degrees Johanna Etner CERSES CNRS and University of Paris Descartes 45 rue des SaintsPeres, F75006 Paris, France johanna.etner@parisdescartes.fr Abstract This article
More informationCHAPTER 7 GENERAL PROOF SYSTEMS
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationChapter ML:IV. IV. Statistical Learning. Probability Basics Bayes Classification Maximum aposteriori Hypotheses
Chapter ML:IV IV. Statistical Learning Probability Basics Bayes Classification Maximum aposteriori Hypotheses ML:IV1 Statistical Learning STEIN 20052015 Area Overview Mathematics Statistics...... Stochastics
More informationWhat Do Prediction Markets Predict?
What Do Prediction Markets Predict? by John Fountain and Glenn W. Harrison January 2010 ABSTRACT. We show that prediction markets cannot be relied on to always elicit any interesting statistic of aggregate
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationAppropriate discount rates for long term public projects
Appropriate discount rates for long term public projects Kåre P. Hagen, Professor Norwegian School of Economics Norway The 5th Concept Symposium on Project Governance Valuing the Future  Public Investments
More informationGambling Systems and MultiplicationInvariant Measures
Gambling Systems and MultiplicationInvariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously
More informationLecture 8 The Subjective Theory of Betting on Theories
Lecture 8 The Subjective Theory of Betting on Theories Patrick Maher Philosophy 517 Spring 2007 Introduction The subjective theory of probability holds that the laws of probability are laws that rational
More informationSubjective expected utility on the basis of ordered categories
Subjective expected utility on the basis of ordered categories Denis Bouyssou 1 Thierry Marchant 2 11 March 2010 Abstract This paper shows that subjective expected utility can be obtained using primitives
More informationAxioms for health care resource allocation
Axioms for health care resource allocation Lars Peter Østerdal University of Copenhagen July 2002 Abstract We examine welfare theoretic foundations for solutions to the health care resource allocation
More informationA Portfolio Model of Insurance Demand. April 2005. Kalamazoo, MI 49008 East Lansing, MI 48824
A Portfolio Model of Insurance Demand April 2005 Donald J. Meyer Jack Meyer Department of Economics Department of Economics Western Michigan University Michigan State University Kalamazoo, MI 49008 East
More informationChapter 1  Matrices & Determinants
Chapter 1  Matrices & Determinants Arthur Cayley (August 16, 1821  January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed
More informationDecision Theory. 36.1 Rational prospecting
36 Decision Theory Decision theory is trivial, apart from computational details (just like playing chess!). You have a choice of various actions, a. The world may be in one of many states x; which one
More informationLecture notes: singleagent dynamics 1
Lecture notes: singleagent dynamics 1 Singleagent dynamic optimization models In these lecture notes we consider specification and estimation of dynamic optimization models. Focus on singleagent models.
More informationUsefulness of expected values in liability valuation: the role of portfolio size
Abstract Usefulness of expected values in liability valuation: the role of portfolio size Gary Colbert University of Colorado Denver Dennis Murray University of Colorado Denver Robert Nieschwietz Seattle
More informationIntroduction to Share Buyback Valuation
By Magnus Erik Hvass Pedersen 1 Hvass Laboratories Report HL1301 First edition January 1, 2013 This revision August 13, 2014 2 Please ensure you have downloaded the latest revision of this paper from
More informationAnalyzing the Decision Criteria of Software Developers Based on Prospect Theory
Analyzing the Decision Criteria of Software Developers Based on Prospect Theory Kanako Kina, Masateru Tsunoda Department of Informatics Kindai University Higashiosaka, Japan tsunoda@info.kindai.ac.jp Hideaki
More informationThe value of tax shields IS equal to the present value of tax shields
The value of tax shields IS equal to the present value of tax shields Ian A. Cooper London Business School Kjell G. Nyborg UCLA Anderson and CEPR October 2004 Abstract In a recent paper, Fernandez (2004)
More informationChoice Under Uncertainty
Decision Making Under Uncertainty Choice Under Uncertainty Econ 422: Investment, Capital & Finance University of ashington Summer 2006 August 15, 2006 Course Chronology: 1. Intertemporal Choice: Exchange
More informationCurrent Accounts in Open Economies Obstfeld and Rogoff, Chapter 2
Current Accounts in Open Economies Obstfeld and Rogoff, Chapter 2 1 Consumption with many periods 1.1 Finite horizon of T Optimization problem maximize U t = u (c t ) + β (c t+1 ) + β 2 u (c t+2 ) +...
More informationSubjective Probability and Lower and Upper Prevision: a New Understanding
Subjective Probability and Lower and Upper Prevision: a New Understanding G. SHAFER Rutgers Business School Newark & New Brunswick, NJ, USA P. R. GILLETT Rutgers Business School Newark & New Brunswick,
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
More informationWellbeing and the value of health
Wellbeing and the value of health Happiness and Public Policy Conference Bangkok, Thailand 89 July 2007 Bernard van den Berg Department of Health Economics & Health Technology Assessment, Institute of
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationWald s Identity. by Jeffery Hein. Dartmouth College, Math 100
Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,
More informationDecision Making under Uncertainty
6.825 Techniques in Artificial Intelligence Decision Making under Uncertainty How to make one decision in the face of uncertainty Lecture 19 1 In the next two lectures, we ll look at the question of how
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More information