Risk and Expected Utility

1. Risk and Expected Utility

2. Total and Marginal Utility

3. Here we show a generic example with a risk avoider. Two monetary values of interest are, say, X1 and X2 and those values have utility U(X1) and U(X2), respectively Utility U(X2) U(X1) $ X1 X2

4. Say the outcome of a risky decision is to have X1 occur p% of the time and X2 occur (1 – p)% . Then the EMV is p(X1) + (1 – p)(X2). The expected utility of the risky decision is found in a similar way and without proof I tell you the expected utility is Utility U(X2) U(X1) EU $ EMV X1 X2 along the straight line connecting the points on the curve directly above the EMV for the decision. We have the expected utility as EU = pU(X1) + (1 – p)U(X2)

5. In general we say people have one of three attitudes toward risk. People can be risk avoiders, risk seekers (or risk lover) , or indifferent toward risk (risk neutral). Utility Value Risk neutral Risk avoider Risk lover Monetary Value Utility values are assigned to monetary values and the general shape for each type of person is shown above. Note that for equal increments in dollar value the utility either rises at a decreasing rate (avoider), constant rate or increasing rate.

6. Diamond-Water Paradox • Paradox? • Water is essential to life, yet cheap • Diamonds are (almost) pragmaticallyuseless, yet expensive • What’s wrong? • Unfair comparison between MU of diamonds and MU of water. We consume a large amount of water, so MUwater is smaller. • Recall that MU is captured in the law of demand, and therefore by the price • However, TOTAL utility from water consumption is much greater than TU of diamond consumption!

7. Diamond-Water Paradox • Uses of water • Cooking, drinking, bathing  high value • Watering lawns, washing cars  low value • Prices • In most places, water is relatively abundant, and therefore cheap • Diamonds are relatively scarce and more expensive • With the high price of diamonds, you will only consume if you expect to get a high amount of marginal utility from the diamond

8. Water and Diamonds, Graphically

9. There are some goods in which we only purchase one of that good. Thus, diminishing MU may not apply. However, we still try to maximize our utility. Discrete choice model—we buy the one “best” choice out of many alternatives Airline ticket House College education Spouse! Discrete Choice Models

10. Conclusion • Money doesn’t make people happier, but it can allow them to buy more goods and services • Due to diminishing marginal utility, the amount of happiness gained from additional consumption will get smaller and smaller • When maximizing utility, consumers face a budget constraint and must consider income, prices, and marginal utility • Exogenous price changes will affect the optimal consumption bundle chosen by individuals

11. Practice What You Know Which of the following would most likely illustrate an example of negative marginal utility? Studying for another hour Sleeping in late on a weekend Eating too much food at an all-you-can-eat buffet Drinking a second glass of juice with a meal

12. Practice What You Know If I consume more of good Z, what happens? The price of good Z will fall The price of good Z will rise My marginal utility of good Z will fall My marginal utility of good Z will rise

13. Practice What You Know Suppose that To optimize utility, the consumer should: Buy more X Buy less X Buy more X and less Y Buy more Y and less X

14. Practice What You Know If the price of a good rises, consumers tend to purchase less of that good and instead purchase more of another good. This illustrates: The real-income effect The substitution effect Diminishing marginal utility The disadvantage of using “utils” to measure consumption