On certain statistical properties of continued fractions with even and with odd partial quotients

We prove results concerning the joint limiting distribution of the renewal time of denominators and consecutive digits of random irrational numbers in the case of continued fractions with even partial quotients, with odd partial quotients, and for Nakada's \alpha-expansions.Comment: Final versio

Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface $\operatorname{PSL}(2,{\mathbb Z})\backslash {\mathbb H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $\Gamma\backslash {\mathbb H}$ and odd and grotesque continued fractions, where $\Gamma\cong {\Bbb Z}_3 \ast {\Bbb Z}_3$ is the index two subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by the order three elements $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix} \right)$, having an ideal quadrilateral as fundamental domain. A similar connection between geodesics on $\Theta\backslash {\mathbb H}$ and even continued fractions is discussed in our framework, where $\Theta$ denotes the Theta subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)$.Comment: 19 pages, minor typos corrected and clarifications added in the published versio

Endogeneous Household Interaction

Most econometric models of intrahousehold behavior assume that household decisionmaking is efficient, i.e., utility realizations lie on the Pareto frontier. In this paper we investigate this claim by adding a number of participation constraints to the household allocation problem. Short-run constraints ensure that each spouse obtains a utility level at least equal to what they would realize under (inefficient) Nash equilibrium. Long-run constraints ensure that each spouse obtains a utility level equal to a least what they would realize by cheating on the efficient allocation and receiving Nash equilibrium payoffs in all successive periods. Given household characteristics and the (common) discount factor of the spouses, not all households may be able to attain payoffs on the Pareto frontier. We estimate these models using a Method of Simulated Moments estimator and data from one wave of the Panel Study of Income Dynamics. We find that the model with long-run participation constraint fits the data best, and that 6 percent of sample households are not able to attain efficient outcomes. To meet the long-run participation constraint, over 90 percent of "efficient" households are required to modify the ex ante Pareto weight of 0.5 for each spouse assumed to apply to all households.Household Time Allocation, Grim Trigger Strategy, Household Production, Method of Simulated Moments

Visitations and Transfers in Non-Intact Households

Recent research reveals that divorce negatively impacts children's welfare as a consequence of the reduction in monetary and time contributions of the non-custodial parent. When the custody arrangement is sole custody, the variables that link the absent parent to the child are visitations and child support transfers. We explain visitations and child support transfers using a behavioral model of competitive equilibrium in which both variables are the results of competitive allocations realized in a decentralized non-cooperative manner. In our framework the mother has control over visitations and the father has control over child support. Estimates of the model are used to simulate the effects of alternative endowment levels on the proportion of time spent with the noncustodial parent and the ex-post parental income distribution. Our results show that a more equal allocation of time with the child, though beneficial to the children, may negatively affect the mother's welfare, increasing the income gap between ex-spouses.