Paper Authors

Abstract

NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Session _____

THE SAVE-SPEND PROBLEM

Mohammad H. Alimi, and Howard B. Wilson North Dakota State University / University of Alabama

Introduction

Applied mathematics courses usually include topics from physics and engineering but seldom consider financial concepts. This omission is easy to remedy because the essential ideas of investment growth resulting from compounded investment earnings can be explained with a simple first-order differential equation. The current article analyzes what can be appropriately called the save-spend problem where funds earning interest are saved over one period and are consumed during a subsequent period. Special cases of this problem include mortgage financing as well as pension saving.

Mathematical Formulation

Investment capital Q growing due to a saving rate S (t ) while simultaneously earning a continuously compounded, after-tax, rate of investment return R satisfies the differential equation Q’(t ) = RQ (t ) + S (t ), Q(0) = q 0 . The general solution of this equation for constant R is t Q(t ) = e [q0 + ∫ e − Rt S (t )dt ] Rt

0 Inflation usually exists in real situations so it is desirable to think in terms of inflation adjusted capital defined by q (t ) = Q(t )e − It where I is the annual inflation rate. When I is zero, then q simply reduces to Q . The differential equation for q(t ) is q ’(t ) = ( R − I )q + S (t )e − It , q (0) = q 0 The form used here for S (t ) models a saving period followed by a payout period so that S (t ) = se At , 0 ≤ t ≤ t1 and S (t ) = − pe At , t > t1 where s, p and A are constants. Constants s and p are called the saving and payout rates. Parameter A , referred to as the saving growth constant, quantifies the rate at which S (t ) changes to account for inflation and wage growth. Common choices for A include A = 0 and A = I . The differential equation for q(t ) becomes q ’(t ) = rq(t ) + se at − ( s + p )e at (t > t1 )

1

• Citation

• Format
  • APA
  • APA - LaTeX bibitem
  • MLA
  • MLA - LaTeX bibitem
  • Bibtex
  • EndNote - RIS

Alimi, M. H., & Wilson, H. B. (2000, June), The Save Spend Problem Paper presented at 2000 Annual Conference, St. Louis, Missouri. 10.18260/1-2--8681