## The Utility Function, Indifference Curves, and Healthcare

By Brennan T. Beal

### Impetus For The Post

When I first learned about utility functions and their associated indifference curves, I was shown an intimidating figure that looked a bit like the image below. If you were lucky, you were shown a computer generated image. The less fortunate had a professor furiously scribbling them onto a board.

https://opentextbc.ca/principlesofeconomics/back-matter/appendix-b-indifference-curves/

A few things were immediately of concern: why are there multiple indifference curves for one function if it only represents one consumer? Why are the curves moving? And… who is Natasha? So, while answering my own questions, I thought sharing the knowledge would be helpful. This post will hopefully provide a better description than maybe most of us have heard and by the end you will understand:

1. What indifference curves are and what they represent
2. How a budget constraint relates to these indifference curves and the overall utility function
3. How to optimize utility within these constraints (if you’re brave)

For the scope of this post, I’ll assume you have some fundamental understanding of utility theory.

## Expected Loss Curves

By Brennan T. Beal, PharmD

The Second Panel on Cost-Effectiveness in Health and Medicine recommends model uncertainty be reflected by displaying the cost-effectiveness acceptability curve (CEAC) with the cost-effectiveness acceptability frontier (CEAF) overlaid (more on this can be seen here). However, on top of being relatively difficult to interpret, these graphical representations may miss a crucial part of decision-making: risk.

A risk-neutral approach to decision-making would mean choosing a strategy that is most likely to be cost-effective despite what one stands to lose economically when the strategy is not cost-effective. Though, we know that decision-makers are often not risk-neutral. With this in mind, selecting a strategy based solely on the probability of being cost-effective could expose a decision-maker to unnecessary risks. It is not always the case that the most likely to be cost-effective is truly the optimal decision; notably, the optimal decision should be thought of as the strategy with the lowest expected loss.

Consider the following example:

Let us suppose that you want to compare two strategies (Strategy A and Strategy B) to see which will be optimal for your company. Your head statistician informs you that Strategy A will be cost-effective 70% of the time and in the 70 times out of 100 that it is cost-effective, you stand to gain \$5 dollars each time (i.e., you lose \$0 each of those 70 times). She then proceeds to tell you that for every time you are wrong (30% of the time) you stand to lose \$100. Your expected loss would be \$30 (30% of the time losing \$100). With that in mind, you also calculate the expected loss for Strategy B. Turns out it is only \$7! (\$7 is arbitrary for the sake of example).

In this example, Strategy B would be favored on the CEAF given that it has the lowest expected loss but the CEAC would have shown it to be less likely. So, having the CEAF at least informs us what strategy is optimal, but we are still left with a relatively confusing picture of cost-effectiveness.

Below are three hypothetical distributions of the incremental net benefit (INB) of Strategy B when compared to Strategy A. Simply stated, the INB curves can be thought of as the probabilities of monetary outcomes when comparing strategies.

This example is described in greater detail in my recent blog entry on the topic of expected loss. For each distribution above, Drug B is considered optimal as it has the lowest expected loss in each scenario. However, in situations where the mean and median have opposite signs (such as in the case of the right skewed blue curve above, mean INB of \$90 vs. a median of -\$75), only considering the most likely to be cost-effective will not provide a decision-maker with the optimal decision. For the blue curve above, Drug B has a lesser chance of being cost-effective (46%), but an expected loss of \$271 vs. \$361 for Drug A.

Expected loss curves (ECLs) account for the probability that a strategy is not cost-effective and how drastic the consequences are in those scenarios. The ELC represents the optimal strategy at each willingness-to-pay threshold and provides a much clearer picture of risk for more informed decision-making.

In my full blog entry, I cover:

1. An in-depth explanation of ELCs;
2. A working example of the mathematics and associated R code;
3. and an interactive example at the end so you can see for yourself