Until now, each of the matrices defining preferences, technologies, and information flows has been specified to be constant over time. This chapter relaxes this assumption and lets the matrices be strictly periodic functions of time. The aim is to apply and extend an idea of Denise Osborn (1988) and Richard Todd (1983, 1990) to arrive at a model of seasonality as a hidden periodicity. Seasonality can be characterized in terms of a spectral density. A variable is said to “have a seasonal” if its spectral density displays peaks at or in the vicinity of the frequencies commonly associated with the seasons of the year, for example, every 12 months for monthly data, every four quarters for quarterly data. Within a competitive equilibrium, it is possible to think of three ways of modeling seasonality. The first two ways can be represented within the time-invariant setup of our previous chapters, while the third way departs from the assumption that the matrices that define our economies are time invariant. The chapter focuses on a third model of seasonality following Todd. It specifies an economy in terms of matrices whose elements are periodic functions of time.
ON CERTAIN FOURIER SERIES EXPANSIONS OF DOUBLY PERIODIC FUNCTIONS OF THE THIRD KIND
