# Risk and Expected Utility

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Risk and Expected Utility

Total and Marginal Utility
Image: Animated Figure 16.1 Lecture notes: The relationship between total utility and marginal utility can be seen by observing the dashed line that connects (a) and (b). Since the marginal utility becomes negative after five brownies are consumed, the total utility eventually falls. To the left of the dashed line, the marginal utility is positive in panel (b) and the total utility is rising in panel (a). Conversely, to the right of the dashed line the marginal utility is negative, and total utility is falling. Calculus-minded students will note that MU is the derivative of TU.

Here we show a generic example with a risk avoider
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) \$ X X2

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 X 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)

In general we say people have one of three attitudes toward risk
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.

Water is essential to life, yet cheap Diamonds are (almost) pragmatically useless, 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! Lecture notes: Total utility is determined by the amount of consumer surplus enjoyed from a transaction. We learned that consumer surplus is the area under the demand curve and above the price, or the gains from trade a consumer enjoys.

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 Lecture notes: Therefore, if the price of a diamond rises, consumers enjoy less surplus from buying them.

Water and Diamonds, Graphically
Image: Animated Figure 16.2 Lecture notes: The diamond-water paradox exists because people fail to recognize that demand and supply are equally important in determining the value a good creates in society. The demand for water is large while the demand for diamonds is small. If we look at the amount of consumer surplus, we observe that the light green area (TUw) is much larger than the dark green area (TUd), because water is essential for life. As a result, water creates significantly more total utility than diamonds. However, since water is abundant in most places, the price, Pwater, is low. On the other hand, diamonds are rare and the price is high, Pdiamond.

Discrete Choice Models
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! “Beyond the Book” Slide Airline ticket Look for the best departure and arrival times, price, number of layovers House Look for price, location, distance from work, color, size, number of bedrooms College education Best program, location, scholarships, friends Spouse Looks, intelligence, personality, earning power, maybe race or religion. Note that you can choose your spouse you don’t really “buy” one.

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

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 Clicker Question Correct answer: C Negative MU means we’ve consumed up to a point where further consumption makes us worse off. Maybe it makes us feel sick or even injures us. If MU < 0, then TU decreases. At the buffet, it is easy and tempting to overeat since the marginal cost of more food is \$0.

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 Clicker Question Correct answer: C There is an inverse relationship between consumption of a good and MU of that good. Prices won’t change with consumption—one individual consumer won’t affect the price of good Z.

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 Clicker Question Correct answer: D The equation says you are getting more “bang for your buck” with good Y. Thus, you should buy more Y. However, if you are already spending all of your income, you must buy less X in order to be able to afford more Y.

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 Clicker Question Correct answer: B The substitution effect illustrates that in a two-good model, the two goods are at least somewhat substitutable in consumption.