  We present functional forms allowing a broader range of analytic solutions to
common economic equilibrium problems. These can increase the realism of
pen-and-paper solutions or speed large-scale numerical solutions as
computational subroutines. We use the latter approach to build a tractable
heterogeneous firm model of international trade accommodating economies of
scale in export and diseconomies of scale in production, providing a natural,
unified solution to several puzzles concerning trade costs. We briefly
highlight applications in a range of other fields. Our method of generating
analytic solutions is a discrete approximation to a logarithmically modified
Laplace transform of equilibrium conditions.
