Recent Research
NEW
Becker, Robert A. and RinconZapatero, Juan P, Distinct Solutions to Koopmans' Equation for Thompson Aggregators Represent Distinct Preference Orders (June 8, 2023). Available at SSRN: https://ssrn.com/abstract=4473832
Thompson aggregators generate recursive utility functions as solutions to Koopmans' functional equation. Multiple solutions may occur when the underlying commodity space is the positive cone of the space of bounded realvalued consumption sequences. Consider the case of extremal solutions: distinct least and greatest solutions exist. We show that these recursive utility functions represent distinct preference orders on the commodity space. We discuss the significance of this observation for uniqueness theory and optimal growth modeling with a Thompson aggregator based objective function.
NEW
Becker, Robert A. and RinconZapatero, Juan P, Recursive Utility for Thompson Aggregators: Least Fixed Point, Uniqueness, and Approximation Theories (May 22, 2023). Available at SSRN: https://ssrn.com/abstract=4456037
We reconsider the theory of Thompson aggregators proposed by Marinacci and
Montrucchio (J Econ Theory 2010). We demonstrate the Koopmans equation has a unique
utility function solution given a Thompson aggregator. Uniqueness holds only
on the interior of the commodity space's positive cone. We verify Du's
condition (Appl Anal 1990) holds. We show iteration of the Koopmans operator with
initial seed the zero function convergences uniformly to the operator's Least
Fixed Point on each order interval in the commodity space's norm interior.
This differs from Marinacci and Montrucchio's (J Econ Theory 2010, Math Oper Res 2019),
proofs as well as proofs given by MartinsdaRocha and Vailakis (Econometrica 2010).
Those papers rely on a form of the contraction mapping theorem where the space
of possibly utility functions is endowed with the topology induced by the
Thompson metric. Application of Du's theory works on the possible utility
function space with its norm topology. Our approach combines order and metric
structures to demonstrate uniqueness differently than in the existing literature.
Becker, Robert A. and RincónZapatero, J.P., Recursive Utility and Turnpike Theory for GMM Thompson Aggregators (January 17, 2020).
CAEPR WORKING PAPER SERIES #2020001. Available at SSRN: https://ssrn.com/abstract=3521498
The existence of a unique optimum, a unique optimal stationary program, and a turnpike theorem are demonstrated for a neoclassical one sector optimal growth model. The planner's allocation problem is formulated as a discrete time deterministic, infinite horizon programming model. The production sector is subject to diminishing marginal returns to capital. The planner's objective function is derived from a Generalized Marinacci and Montrucchio (GMM) Thompson aggregator preference. A given Thompson aggregator may be associated with many intertemporal utility functions (which may not be ordinally equivalent). The choice of one of these representations over another is shown to be a matter of mathematical tractability. There is an observational equivalence between those alternative objective functions: the qualitative features of the optimal solution do not depend on the particular utility function representation of the underlying Thompson aggregator preference structure.
Publications

Recursive utility for Thompson aggregators: Uniqueness via concave operator theory and iterative approximations
with Robert A. Becker
Communications in Optimization Theory, forthcoming.

(2024) Existence and uniqueness of solutions to the Bellman equation in stochastic dynamic programming
Theoretical Economics, Volume 19, Issue 3, Pages 12231260.
A previous draft, which contains some proofs not in the final version of the paper is
UC3M Working Paper. Series Economics 2207

(2024) Housing pricing and credit constraints in competitive search
with Antonia Díaz and Belén Jerez
The Economic Journal Vol. 134, Issue 657, Pages 220270.

(2021) Thompson aggregators, Scott Continuous Koopmans operators, and Least Fixed Point Theory
with Robert A. Becker
Mathematical Social Sciences, 112, 8497.
Available online 9 April 2021 https://doi.org/10.1016/j.mathsocsci.2021.03.015

(2020)
Differentiability of the value function and Euler equation in nonconcave discrete time stochastic dynamic programming
Economic Theory Bulletin 8, 7988.

(2019) Equilibrium strategies in a defined benefit pension plan game
with Ricardo JosaFombellida
European Journal of Operational Research 275(1), 374386.

(2019)
Certainty equivalence principle in stochastic differential games: an inverse problem approach
with Ricardo JosaFombellida
Optimal Control Applications and Methods 40(3), 545557.

(2018)
Stochastic differential games for which the openloop equilibrium is subgame perfect
with Ricardo JosaFombellida
Dynamic Games and Applications 8(2), 379400.

(2018) Portfolio optimization in a defined benefit pension plan where the risky assets are processes with constant elasticity of variance
with Ricardo JosaFombellida and Paula LópezCasado
Insurance: Mathematics and Economics 82(Sep), 7386.

(2018)
Envelope theorem in dynamic economic models with recursive utility
with Yanyun Zhao
Economics Letters 163(Feb), 1012.
See also the Appendix

(2015)
EulerLagrange equations of stochastic differential games: application to a game of a productive asset
with Ricardo JosaFombellida
Economic Theory 59(1), 61108.

(2012)
Differentiability of the value function in continuoustime economic models
with Manuel S Santos
Journal of Mathematical Analysis and Applications 394(1), 305323.

(2012) Stochastic pension funding when the benefit and the risky asset follow jump diffusion processes
with Ricardo JosaFombellida
European Journal of Operational Research 220(2) 404–413.

(2010) Optimal asset allocation for aggregated defined benefit pension funds with stochastic interest rates
with Ricardo JosaFombellida
European Journal of Operational Research 201(1), 211–221.

(2010)
On a PDE arising in onedimensional stochastic control problems
with Ricardo JosaFombellida
Journal of Optimization Theory and Applications 147(1), 126.

(2009) HopfLax formula for variational problems with nonconstant discount
Journal of Geometric Mechanics 1(3), 357368.

(2009)
Differentiability of the value function without interiority assumptions
with Manuel S Santos
Journal of Economic Theory 144(5), 19481964.

(2009)
On the impossibility of representing infinite utility streams
with Juan A Crespo and Carmelo Núñez
Economic Theory 40(1), 4756.

(2008) Meanvariance portfolio and contribution selection in stochastic pension funding
with Ricardo JosaFombellida
European Journal of Operational Research 187(1), 120137.

(2008) Funding and investment decisions in a stochastic defined benefit pension plan with several levels of laborincome earnings
with Ricardo JosaFombellida
Computers & Operations Research 35(1), 4765.

(2007)
Recursive utility with unbounded aggregators
with Carlos RodríguezPalmero
Economic Theory 33(2), 381391.

(2007)
New approach to stochastic optimal control
with Ricardo JosaFombellida
Journal of Optimization Theory and Applications135(1), 163177

(2006) Optimal investment decisions with a liability: The case of defined benefit pension plans
with Ricardo JosaFombellida
Insurance: Mathematics and Economics 39(1), 8198.

(2005)
Efficient Markov perfect Nash equilibria: theory and application to dynamic fishery games
with Guiomar MartínHerrán
Journal of Economic Dynamics and Control 29(6), 10731096.

(2004)
Characterization of Markovian equilibria in a class of differential games
Journal of Economic Dynamics and Control 28(7), 12431266.

(2004) Optimal risk management in definedbenefit stochastic pension funds
with Ricardo JosaFombellida
Insurance: Mathematics and Economics 34(3), 489503.

(2003)
Direct method comparing efficient and nonefficient payoffs in differential games
with Guiomar MartínHerrán
Journal of Optimization Theory and Applications 119(2), 695405.

(2003)
Existence and uniqueness of solutions to the Bellman equation in the unbounded case
with Carlos RodríguezPalmero
Econometrica 71(5), 15191555.
See also
Corrigendum to "Existence and Uniqueness of Solutions to the
Bellman Equation in the Unbounded Case"
with Carlos RodríguezPalmero
Econometrica 77, 317318 (2009).

(2001) Minimization of risks in pension funding by means of contributions and portfolio selection
with Ricardo JosaFombellida
Insurance: Mathematics and Economics 29(1), 3545.

(2000)
Identification of efficient subgameperfect Nash equilibria in a class of differential games
with Guiomar MartínHerrán and Julia Martínez
Journal of Optimization Theory and Applications 104(1) 235242.

(1998)
New method to characterize subgame perfect Nash equilibria in differential games
with Guiomar MartínHerrán and Julia Martínez
Journal of Optimization Theory and Applications 96(2), 377395.