Health is getting better over time in some disembodied way and is also improved by purchases of health goods m. We assume that the effectiveness of health goods in improving health also changes over time through an efficiency parameter θ. Taking these together, we can write health status at time t as
where δt is the cumulated effects up to the beginning of t of the disembodied health progress, and θt is the efficiency of health goods and services m in producing health. Examples of δ would be improvements achieved through better childhood nutrition, lower pollution, or reductions in smoking. Combining (37) and (38), we can rewrite the budget constraint as
so that the disembodied technical progress δt acts like a gift of income (though because it works by reducing the need to purchase health care, its value is reduced the cheaper or more efficient health care is), and the “effective” price of health care is its quality-adjusted price pm/θt. In this set-up, the disembodied improvements in health status increase utility at any given set of prices and thus reduce the (unconditional) cost of living. Writing the budget constraint in the form of (39) allows us to see the consumer’s problem as a standard one; utility (36) is defined over q and h, and (39) gives their effective prices, p and pm/θt, as well as the effective budget available to fund them, x + pmδt/θt. Given this, we can immediately see that the unconditional cost function—the minimum cost of reaching u (including both health status and consumption) at prices pm and p can be written in the form
From (40) we see that (a) pm always appears deflated by the efficiency parameter θt, so that only the effective price matters, and (b) an increase in disembodied technical progress δt decreases the cost of living. The efficiency parameter reduces the price of health care, while the disembodied parameter effectively generates additional income.
Suppose that, in line with our discussion of the domain issue in the main text, we decide that the COLI price index should not fall in response to disembodied improvements in health status but should fall when new medical procedures or drugs mean that a given episode of illness can be treated at less cost. In the framework here, this decision can be implemented by including δt among the environmental variables, e, and holding it constant in cost-of-living comparisons while allowing θt to change in comparisons from 0 to t, so that we compare, not the prices and , but the quality-corrected prices, and /θt.The conditional cost function that we need to make this work is (40) with δt held constant,