# Wolfram Demonstrations Project

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An individual's happiness (or "utility") often depends on their wealth and their health. Spending on medical services, however, can often improve one's health. This Demonstration explores tradeoffs between wealth and health permitted by medical technology. You use a two-dimensional slider to select the parameters of a utility function from a family of transcendental logarithmic functions that depends on health and wealth. A sketch of the resulting utility surface appears to the left of the control. The cost of medical services to improve health from some baseline by an amount is modeled by the solution to the differential equation , subject to the constraint that . You choose the parameters and . You select baseline wealth and health using a locator. The Demonstration responds with a graphic showing the different utility contours and with a black locus showing the possible wealth-health states permitted by medical technology. A red point shows the optimal level of wealth and health given a baseline position and the cost of medical services.

Contributed by: Seth J. Chandler (March 2011)
Open content licensed under CC BY-NC-SA

## Details

Both health and wealth are constrained to line on the interval [0,1].

This Demonstration has implications for the proper operation of a torts and insurance system. A potential victim would likely want a system of compensation that optimizes expected utility over a distribution of wealth-health baselines given (a) the opportunity for medical services and (b) the relationship between health and the marginal utility of income.

The Demonstration assumes that the individual has only one opportunity to trade wealth for health. This assumption exists because the cost of improving health depends on a "baseline" of health. This assumption could be relaxed by letting the "baseline" evolve as the individual trades wealth for health. The code underlying this Demonstration readily permits such a relaxation, although in the interests of speed and simplicity, no controls exist in the Demonstration that permit the casual user directly to do so.

Snapshot 1: with medical services priced high, the individual prefers not to sacrifice much wealth to improve health above the baseline level

Snapshot 2: with medical services priced low, the individual prefers to attain a high level of health for a minimal sacrifice of wealth

Snapshot 3: with a utility function of , the individual is willing to sacrifice a great deal of wealth in order to improve health

Snapshot 4: with a utility function of , the individual is not willing to sacrifice wealth in order to improve health

Snapshot 5: with high baseline wealth, the individual is willing to sacrifice considerable wealth to improve health

## Permanent Citation

Seth J. Chandler

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