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# Marginal Analysis

### Health Economics: 6 - Marginal Analysis

The principle of the margin is described in section 1 and an application of marginal analysis in health care is discussed in section 8. The examples that are given in section 1 show how examining marginal quantities, rather than average quantities, is valuable in decision making. However, there is much more to marginal analysis than that, and in fact this is one of the most important ideas in microeconomics. The reason why economics uses the terms ‘margin’ and ‘marginal’ for this concept are historical and the way that these are used here reflect their most common usage in modern economics.

As section 1 explained, a marginal value can be calculated when there is a causal relationship between two variables, for example - to be optimistic - between the amount of health care that is consumed and the level of health. At least up to some point, the more health care is consumed, the higher the level of health. However, it is likely that if someone has poor levels of health, health care has more impact on health than when they are healthy. To quantify this, let us assume that health care consumption is measured by current health spending and that levels of health are measured in the number of expected future QALYs over the next 5 years. If health spending rises, the number of expected QALYs rises.

Next, suppose that someone has a particular level of health spending of £300 and 3 expected QALYs. Health spending comes in packages that cost £100, so they currently have 3 packages. If they spend £400, they will have 3.5 expected QALYs in total; if they spend £500, they will have 3.75 expected QALYs.

The average number of QALYs that they obtain per package spent on health care can be calculated. Making the unrealistic assumption that they will have no expected QALYs if they spend nothing on health care, the average QALY per package at their current spending of £300 is 1. If they raise it to £400, that is 4 packages, it will be 0.875; if they raise it to £500, that is 5 packages, it will be 0.75. The total number of QALYs rises if spending rises, but the average QALY per package falls.

If the person was deciding whether or not to buy extra packages, they would be misled if they looked at the current average QALY per package, which would suggest to them that they will gain 1 QALY for each package. Instead, they should look at the marginal QALY per package. If they buy one more package, they will gain 0.5 of a QALY. If, having bought the first package, they buy another one, they will gain another 0.25. Both of these are well below 1, so if the person was expecting to obtain 1 QALY for each package, they would be disappointed.

This example, like those that are in section 1, simply shows how marginal values are more informative than average values in some circumstances. But how does this help? First, let us assume that the person likes health, but not above everything else. They are willing to pay £200 to obtain a QALY, but no more. If they are deciding whether to increase their spending from £300, they might observe that they would be willing to pay up to £700 to get 3.5 QALYs and up to £800 to get 4 QALYs, so to get them for £400 and £500 respectively is a bargain. But if we look at the marginal values, if they spend £100 more, they are getting 0.5 QALY, which means the price per QALY is £200, which is at the limit of what they are willing to pay. If they spend another £100, that would in effect cost them £400 per QALY, which is above what they want to spend.

So, the conclusion is that the person should buy one more package of health care, and this can only be deduced directly by looking at marginal values, not totals or averages. The same principle holds in more realistic cases, and is applied to many economic variables, such as costs, levels of output and levels of spending.

There is even more to marginal analysis. Mathematically, marginal values are the same as rates of change. If the relationship between spending and QALYs was a smooth one, with both measured as continuous variables, and these were plotted together on a graph as a curve, the marginal value at any point on the curve would be the slope of the curve at that point. Rates of change are very important in mathematical optimisation, so marginal values are important in determining optimal levels of economic variables - for example, maximum benefits or minimum costs.

The example above suggested a rule that only if the marginal cost of buying a QALY was at or above the person’s willingness to pay for one, then they should buy it, but if it is below then they should not. In fact, if they are already spending £500, they should decide to spend less next time they must decide how much to buy.

But suppose their willingness to pay for QALYs was also not constant. They are willing to pay more in total for more QALYs, but when they have few QALYs they are willing to pay more for an extra QALY than when they already have more. What is relevant in their decision making is this marginal willingness to pay, not their total or average. So, a more general rule than the one that was suggested earlier is that to optimise the person’s buying of health care, they should set their marginal willingness to pay equal to the marginal QALY gain per pound.

This general rule - equalising two marginal values to optimise - is used very widely in economics. For example, to maximise profits, firms should set their marginal costs equal to their marginal revenues. To maximise their consumer satisfaction, consumers should set the marginal benefit per pound spent on one good equal to the marginal benefit per pound spent on every other good. Health services should set the marginal health gain per pound spent on each health intervention that it funds equal to that of every other intervention - although that rule is rather more complicated with a limited health budget.

To be less extreme, however, the suggestion is not that the health service ought to measure such marginal values and literally attempt to achieve a perfect distribution of resources. Instead, it is that marginal values are a guide to what will lead to a better distribution of resources.

Finally, there is another advantage to examining marginal values, in that the marginal values that are of current interest are close to where we are at present. Concentrating on looking at the impact of a small change is likely to be closer to current experience and therefore more accurate than looking at the impact of a large change.