In this brief Article, using the elementary theory of differential equations as well as some basic economic theory, we will develop several estimates for national health expenditures for the United States: one using a linear model and three using non-linear models. We will derive the nonlinear models first and then compare them to the linear one in order to see if they differ significantly. While these estimates are for the United States, the methods used here, because they are robust, could be used for any country. Statistical information may be obtained from the World Bank databases which store health statistics by country [1].

What we will do here is estimate the total health costs as a percentage of gross domestic product (GDP) if no further copayments are required. In other words, we are seeking to estimate the total cost of health care as a percentage of GDP when all health care costs are covered by insurance and government subsidy. Several models will be discussed here since such estimates may be made using a variety of assumptions. There is no ‘best’ model, although such a decision is possible when comparing the estimates to actual data.