Best Simultaneous Diophantine Approximations under a Constraint on the Denominator

We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals with smaller denominator. Finally, the approximation results are applied to the decomposition of integer vectors.Comment: 17 pages, corrected typo

Lattice Points in Large Borel Sets and Successive Minima

Let $B$ be a Borel set in $\mathbb E^{d}$ with volume $V(B)=\infty$. It is shown that almost all lattices $L$ in $\mathbb E^{d}$ contain infinitely many pairwise disjoint $d$-tuples, that is sets of $d$ linearly independent points in $B$. A consequence of this result is the following: let $S$ be a star body in $\mathbb E^{d}$ with $V(S)=\infty$. Then for almost all lattices $L$ in $\mathbb E^{d}$ the successive minima $\lambda_{1}(S,L),..., \lambda_{d}(S,L)$ of $S$ with respect to $L$ are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the semi-continuity of the successive minima.Comment: 8 page

An Optimal Lower Bound for the Frobenius Problem

Given $N$ positive integers $a_1, ..., a_N$ with $\gcd(a_1, ..., a_N)=1$, let $f_N$ denote the largest natural number which is not a positive integer combination of $a_1, ..., a_N$. This paper gives an optimal lower bound for $f_N$ in terms of the absolute inhomogeneous minimum of the standard $(N-1)$-simplex.Comment: 10 page

Knowledge-Capital Meets New Economic Geography

We incorporate the now standard knowledge-capital model of multinational firms in a new economic geography setting. The theoretical predictions of our model suggest that unskilled labor mobility leads to less concentration of production than skilled labor mobility does. This is in line with empirical evidence that agglomeration of production among European nations is less pronounced than among US regions. Our model shows that the different patterns in labor mobility can explain actual differences in the spreading of industries. According to our welfare analysis, trade liberalization is likely Pareto-improving for a larger (smaller) country with mobile unskilled (skilled) labor. In the supplement, we investigate the sensitivity of our results in several respects. In the first section, we provide the figures of real factor rewards for the trade liberalization scenarios discussed in and underlying Figures 7 and 8 of the paper. Second, in Figures 3(n) - 5(v) (6(n) - 6b(v)) we infer the existence, or non-existence, of each firm type separately in the τ - λ L-space (τ - λ S-space) for country i firms and all four scenarios of firm regimes. Third, we illustrate how changes in the parameters μ, ρ and σ affect the outcome. Finally, we analyze how the asymmetric endowment with the immobile factor influences the core-periphery patterns.knowledge-capital model, new economic geography, unskilled labor mobility, skilled labor mobility

Delays in Claiming Social Security Benefits

This paper focuses on Social Security benefit claiming behavior, a take-up decision that has been ignored in the previous literature. Using financial calculations and simulations based on an expected utility maximization model, we show that delaying benefit claim for a period of time after retirement is optimal in a wide variety of cases and that gains from delay may be significant. We find that approximately 10% of men retiring before their 62nd birthday delay claiming for at least one year after eligibility. We estimate hazard and probit models using data from the New Beneficiary Data System to test four cross-sectional predictions. While the data suggest that too few men delay, we find that the pattern of delays by early retirees is generally consistent with the hypotheses generated by our theoretical model.