Marginal Rate of Substitution vs Risk Aversion
Marginal Rate of Substitution and Risk Aversion are two Microeconomic Theory concepts in AP Economics that students often mix up. The marginal rate of substitution is the rate at which a consumer will give up one good to get more of another while staying equally satisfied. Risk aversion is a preference for a certain outcome over a gamble with the same expected value, shown by diminishing marginal utility of wealth. Here is how they compare side by side.
It equals the slope of the indifference curve and diminishes as you move along it, the more you have of a good, the less of the other you'll sacrifice for it. At the optimal bundle, the MRS equals the ratio of the goods' prices.
A risk-averse person turns down a fair gamble, meaning one whose expected value equals its cost, and takes the sure amount instead. The reason is diminishing marginal utility of wealth: the extra utility from gaining $500 is smaller than the utility lost from dropping $500, so the downside outweighs the upside even at even odds. On a graph, a risk-averse person's utility of wealth curve is concave, bending toward the horizontal axis. It follows that their certainty equivalent, the sure amount that feels as good as the gamble, is less than the gamble's expected value, and the gap between the two is the risk premium they will pay to avoid the risk. That premium is why insurance can be priced above the expected payout and still be worth buying.
Marginal Rate of Substitution vs Risk Aversion: Trading Between Goods and Trading Between Outcomes
| Marginal Rate of Substitution | Risk Aversion | |
|---|---|---|
| What is being traded off | One good against another, both received for certain | The same money across states of the world, only one of which happens |
| What the curvature means | A diminishing rate of substitution, so mixtures beat extremes | Diminishing marginal utility of wealth, so a sure amount beats a fair gamble |
| Effect of relabeling the utility numbers | None; the rate survives any increasing transformation | Everything; one ranking can be risk averse or risk neutral |
| Kind of utility required | Ordinal, where only the order of the labels carries meaning | Cardinal in the expected-utility sense, where the numbers themselves matter |
| The number it produces | A rate, in units of one good per unit of the other | A money amount: the certainty equivalent and the risk premium |
| Where it shows up on the exam | Consumer choice, and the tangency with the budget line | Insurance, expected utility, and why a lower sure return gets accepted |
A convex indifference curve tells you nothing about whether someone will buy insurance
Both ideas involve a curve bending the same way, and the coincidence causes real trouble. Take preferences over apples and bananas represented by the product of the two quantities. The rate of substitution is bananas divided by apples, so along the curve where that product is thirty-six it falls from 9 at two apples, to 1 at six apples, to a ninth at eighteen apples. Now square the utility numbers, so every bundle carries the square of its old label. Both marginal utilities change, and yet the rate of substitution is still bananas divided by apples at every point. Each tangency, each demand curve and each prediction survives the relabeling untouched. Try the same trick on risk. Someone whose utility of wealth is the square root of wealth faces an even chance of ending with 16 or 64. Expected wealth is 40, expected utility is the average of 4 and 8, which is 6, and the sure amount worth 6 is 36. She would swap the gamble for 36, giving up 4 of expected wealth to be rid of the risk. Square that same utility function and it becomes wealth itself, which ranks sure amounts in exactly the order the square root did. Expected utility is now 40, the certainty equivalent is 40, and the gamble no longer bothers her. One ordering of certain outcomes, two opposite attitudes to risk. See /glossary/certainty-equivalent.
Risk aversion is convexity too, once the axes stop being goods and become states of the world
Redraw the diagram with wealth if the bad thing happens on one axis and wealth if it does not on the other. A point is now a gamble rather than a basket, and an indifference curve joins gambles the person ranks equally. Anywhere on the line where wealth is the same either way, the rate of substitution between states reduces to the odds, so a one in four chance of the bad state gives a rate of a third. That fact drives the standard insurance result. Fair insurance trades wealth across states at a third as well, since a premium of a quarter of the coverage costs one unit in the good state for every three units it delivers in the bad one, so the tangency lands exactly on the certainty line and a risk-averse buyer covers the whole loss. Put numbers on it. Wealth is 100, a quarter chance of a loss of 64 leaves 36, and utility is the square root of wealth. Expected utility is three quarters of 10 plus a quarter of 6, which is 9, so the certainty equivalent is 81 while expected wealth is 84. Full cover at fair odds costs 16 and the most she would pay is 19, leaving room of 3 for the insurer's own costs. That 3 is the risk premium wearing a different hat. See /glossary/expected-utility and /glossary/marginal-utility.
Frequently asked questions
Does a convex indifference curve mean the consumer is risk averse?
No. Convexity of ordinary indifference curves says that a mixture of two goods is at least as good as either extreme, which is a claim about variety among goods received for certain. Risk aversion concerns one quantity arriving in different amounts depending on chance. The cleanest test is relabeling: squaring a utility function leaves every indifference curve and every purchase exactly where it was, and yet it can turn a risk-averse decision maker into a risk-neutral one.
What is the marginal rate of substitution between wealth in two states of the world?
Drawing wealth in each state on its own axis turns a gamble into a point, and the slope of the indifference curve through that point equals the probability of the first state times the marginal utility of wealth there, divided by the same product for the second state. Along the line where wealth is equal in both states the marginal utilities cancel, and the rate collapses to the odds. That is why an actuarially fair policy leads a risk-averse buyer to insure the whole loss.
Why can utility numbers be relabeled for consumer choice but not for gambles?
Consumer choice uses only the order of the labels, since the buyer compares bundles she will definitely receive, so any increasing relabeling leaves every comparison intact. A gamble is scored by averaging utility numbers weighted by probabilities, and averaging is arithmetic on the numbers themselves rather than on their order. Stretch the labels unevenly and the average shifts even though the ranking of sure outcomes does not. Expected utility therefore needs a cardinal scale, pinned down only up to a positive linear transformation.
Related comparisons
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