## Problems for Section 34

Casino Destroyer

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1. a. The average and variability of return on money balances are both zero. That is, if he puts all of his \$100 in them he is at the point All Money in Figure A3.1; if he puts all of it in bonds, on the other hand, he is at a point such as All Bonds High Interest. He can pick any point on the line between the two and will pick Best High Interest. The proportion he puts in money can be measured along the Variability axis as M/(M + B) (or, for that matter, by a similar construct, along the Average axis: the measure is the same because both Average and Variability for the combination he chooses move in proportion to the amount put in bonds).

b. If the average return (interest rate) falls he has a new budget line, that is, All Money-All Bonds Low Interest, and chooses a new point, Best Low Interest. In the case portrayed in Figure A3.1, the proportion of money balances in his \$100 rises (look at the heavy arrow). That is to say, a fall in the interest rate has induced him to hold more money. The demand for money is sometimes called the demand for liquidity (or liquidity preference). What the diagram says is that because holding bonds is risky, a fall in the interest rate on bonds increases liquidity preference. A relationship between the interest rate and the amount of money demanded plays a large role in theories of inflation and unemployment: risk aversion is one justification for such a relationship. The justification was discovered by lames Tobin, in his "Liquidity Preference as Behavior Towards Risk," Review of Economic Studies 25 (February 1958): 65-86, reprinted in many places, for example, M. C. Mueller, ed., Readings in

Figure A3.1

### Liquidity Preference as Behavior Toward Risk

The price of average rate of return relative to safety (safety being the opposite of variability) Average rises as the interest rate falls. A rise in the interest rate will, by the law of demand, increase

The price of average rate of return relative to safety (safety being the opposite of variability) Average rises as the interest rate falls. A rise in the interest rate will, by the law of demand, increase

Macroeconomics, 2nd ed. (New York: Holt, Rinehart and Winston, 1971). The intuition behind the diagram is simple. Money is safe (ignoring the hazards of inflation) and a bond is risky. Safety is a good, risk a bad. But to get the safety of money one must give up another good, average return. The average return is the opportunity cost of more safety, that is, the price of holding money. By the Law of Demand, as the price goes down more is demanded.

COMMENT With other indifference curves the point Best Low Interest might be to the right of Best High Interest instead of to the left, in which case the theory would break down. As Tobin said, "The ambiguity is a familiar one in the theory of choice, and reflects the ubiquitous conflict between income and substitution [price] effects" [Mueller, ed., p. 185]. A fall of interest rates does reduce the price of safety facing the investors, but it also reduces his income (if bonds yield 1% rather than 10%, investors are obviously made poorer). If safety is a normal good he will demand less of it with a lower income. And the income effect, which leads him to demand less safe money balances, may offset the price effect, which leads him to demand more. Bear in mind this example of the significance of price and income effects: it will come up in various guises throughout the book (in, for example, the next chapter).

2. False. Draw the hint for David Bender in Figure A3.2. If Bender's house does not burn down he receives the income at Lucky, if it does burn down he receives the income at Unlucky, on average he receives Average House, which has a utility of only ¿/(Average House), because it is risky. If the insurance company offers him unfair insurance, that is, the riskless income Y Insured (look at the income axis), he will take it, for its utility is higher. Now he is at the point Insured on his utility of income curve. Suppose that he is offered an unfair gamble between Lose and Win, with an average payoff of Average Gamble. If the utility of Average Gamble is higher than that of Insured, he will take the gamble, too.

Figure A3.2

Gambling and Insurance Can Coexist

COMMENT Until 1948 many economists believed that the answer was "true": that is, they believed that simultaneous gambling and insuring were impossible for a man who did not get utility from the acts of gambling or insuring themselves. Still other observations of how people behave in the face of risk can be accommodated by adding further regions of convexity and concavity.