Student Financial Aid:The Roles of Loans and Grants (Working Paper 37)

This paper addresses the roles of loans and grants as forms of student financial aid. It begins with a simple choice model where individuals decide to pursue post-secondary studies if i) the net benefits of doing so are positive and ii) no financing or liquidity constraints stand in their way. The effects of loans and grants on these two elements of the schooling decision are then discussed. It is argued that based on equity, efficiency, and fiscal considerations, loans are generally best suited for helping those who want to go but face financing constraints, whereas grants are more appropriate for increasing the incentives for individuals from disadvantaged backgrounds to further their studies. Loan subsidies, which make loans part-loan and part-grant, are also discussed, including how they might be used to address “debt aversion”. Given that subsidised loans have a grant (subsidy) element, while grants help overcome the credit constraints upon which loans are targeted, the paper then attempts to establish some general rules for providing loans, for subsidising the loans awarded, and for giving “pure” grants. It concludes with an application of these principles in the form of a recent proposal for reforming the student financial system in Canada. *

Degeneracy of velocity constraints in rigid body systems

The equations governing the dynamics of rigid body systems with velocity constraints are singular at degenerate configurations in the constraint distribution. In this report, we describe the causes of singularities in the constraint distribution of interconnected rigid body systems with smooth configuration manifolds. A convention of defining primary velocity constraints in terms of orthogonal complements of one-dimensional subspaces is introduced. Using this convention, linear maps are defined and used to describe the space of allowable velocities of a rigid body. Through the definition of these maps, we present a condition for non-degeneracy of velocity constraints in terms of the one dimensional subspaces defining the primary velocity constraints. A method for defining the constraint subspace and distribution in terms of linear maps is presented. Using these maps, the constraint distribution is shown to be singular at configuration where there is an increase in its dimension.