On Monotone Recursive Preferences

Abstract

We explore the set of preferences defined over temporal lotteries in an infinite horizon setting. We provide utility representations for all preferences that are both recursive and monotone. Our results indicate that the class of monotone recursive preferences includes Uzawa–Epstein preferences and risk‐sensitive preferences, but leaves aside several of the recursive models suggested by Epstein and Zin (1989) and Weil (1990). Our representation result is derived in great generality using Lundberg's (1982, 1985) work on functional equations.

Explicit Form of the Radiative and Collisional Branching Ratios in Polarized Radiation Transport with Coherent Scattering

“…With this straightforward exercise, we can verify that our second-order emissivity (13) and generalized redistribution function (14) agree with those derived by Bommier (2017) for the case of the two-term atom with non-coherent lower term (i.e., ρ ll ′ = δ ll ′ ρ ll ) and infinitely sharp lower levels (i.e., ǫ l , ǫ f → 0), once the redistribution function R(ω k , ω k ′ ) is replaced by the appropriate expression for this case (Casini et al 2014, equation (10)). In particular, depolarizing collisions can also be included analogously to Bommier (2017), taking into consideration the corresponding contributions to the first-order SE problem. In such case, a new level width contribution is added to the denominator of 1/(ǫ uu ′ + iω uu ′ ) in the second line of equation (14), as a consequence of this modification for the atomic density matrix solution of the SE problem (see Bommier 2017).…”

“…In this work we show how the two approaches lead to branching ratios that are in full agreement, when the generalized form of the emissivity of Casini, del Pino Alemán, & Manso Sainz (2017) is explicitly applied to the same atomic model considered by Bommier (2017).…”

“…In particular, depolarizing collisions can also be included analogously to Bommier (2017), taking into consideration the corresponding contributions to the first-order SE problem. In such case, a new level width contribution is added to the denominator of 1/(ǫ uu ′ + iω uu ′ ) in the second line of equation (14), as a consequence of this modification for the atomic density matrix solution of the SE problem (see Bommier 2017).…”

“…This has yielded the first numerical predictions of the polarimetric signature of magnetic fields on the emergent Stokes profiles of spectral lines that are formed under PRD conditions, for magnetic field strengths that extend down to the regime of the Hanle effect, and hence adequate for the investigation of the magnetism of the quiet Sun chromosphere (Trujillo Bueno,Štěpán, & Casini 2011;Sowmya et al 2014;del Pino Alemán, Casini, & Manso Sainz 2016;Alsina Ballester, Belluzzi, & Trujillo Bueno 2016) Among the various theoretical approaches to the problem of polarized radiative transfer (RT) with PRD, the two advanced by Bommier (1997aBommier ( ,b, 2016 and by Casini and collaborators (Casini et al 2014; have in particular come under some in-depth scrutiny. Both approaches rely on a derivation of the respective formalisms from the first principles of non-relativistic quantum electrodynamics, yet they have led to seemingly different interpretations of the driving physical processes of radiation scattering (see the discussions in Casini et al 2014;Bommier 2016Bommier , 2017.…”

“…Casini, del Pino Alemán, & Manso Sainz (2017) have derived a very general form of the scattering emissivity, which is broadly applicable to numerical problems of polarized radiation scattering in multi-term atoms of the Λ-type , see also Figure 1). Bommier (2017) has explicitly derived the expressions of the branching ratios for the CRD and PRD contributions to the emissivity redistribution function for the case of a two-term atom with non-coherent and unpolarized lower term.…”