endowment in Figure 12.1?
12.1. We need a way to reduce consumption in the bad state and increase consumption in the good state. To do this you would have to sell insurance against the loss rather than buy it.
2. Which of the following utility functions have the expected utility property? (a) u(c1,c2,π1,π2)=a(π1c1 + π2c2), (b) u(c1,c2,π1,π2)=π1c1 + π2c2 2, (c)u(c1,c2,π1,π2)=π1 lnc1 + π2 lnc2 + 17.
12.2. Functions (a) and (c) have the expected utility property (they are a?ne transformations of the functions discussed in the chapter), while (b) does not.
3. A risk-averse individual is o?ered a choice between a gamble that pays $1000 with a probability of 25% and $100 with a probability of 75%, or a payment of $325. Which would he choose?
12.3. Since he is risk-averse, he prefers the expected value of the gamble, $325, to the gamble itself, and therefore he would take the payment.
4. What if the payment was $320?
12.4. If the payment is $320 the decision will depend on the form of the utility function; we can’t say anything in general.
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5. Draw a utility function that exhibits risk-loving behavior for small gambles and risk-averse behavior for larger gambles.
12.5. Your picture should show a function that is initially convex, but then becomes concave.
6. Why might a neighborhood group have a harder time self insuring for ?ood damage versus ?re damage?
12.6. In order to self-insure, the risks must be independent. However, this does not hold in the case of ?ood damage. If one house in the neighborhood is damaged by a ?ood it is likely that all of the houses will be damaged.
13 Risky Assets
1. If the risk-free rate of return is 6%, and if a risky asset is available with a return of 9% and a standard deviation of 3%, what is the maximum rate of return you can achieve if you are willing to accept a standard deviation of 2%? What percentage of your wealth would have to be invested in the risky asset?
13.1. To achieve a standard deviation of 2% you will need to invest x = σx/σm =2 /3 of your wealth in the risky asset. This will result in a rate of return equal to (2/3)0.09 + (1?2/3)0.06 = 8%.
2. What is the price of risk in the above exercise?
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13.2. The price of risk is equal to (rm ? rf)/σm = (9? 6)/3 = 1. That is, for every additional percent of standard deviation you can gain 1% of return.
3. If a stock has a β of 1.5, the return on the market is 10%, and the risk- free rate of return is 5%, what expected rate of return should this stock o?er according to the Capital Asset Pricing Model? If the expected value of the stock is $100, what price should the stock be selling for today?
13.3. According to the CAPM pricing equation, the stock should o?er an expected rate of return of rf + β(rm ?rf)=0.05 + 1.5(0.10?0.05) =0 .125 or 12.5%. The stock should be selling for its expected present value, which is equal to 100/1.125 = $88.89.
14 Consumer’s Surplus
1. A good can be produced in a competitive industry at a cost of $10 per unit. There are 100 consumers are each willing to pay $12 each to consume a single unit of the good (additional units have no value to them.) What is the equilibrium price and quantity sold? The government imposes a tax of $1 on the good. What is the deadweight loss of this tax?
14.1. The equilibrium price is $10 and the quantity sold is 100 units. If the tax is imposed, the price rises to $11, but 100 units of the good will still be sold, so there is no deadweight loss.
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2. Suppose that the demand curve is given by D(p) = 10?p. What is the gross bene?t from consuming 6 units of the good?
14.2. We want to compute the area under the demand curve to the left of the quantity 6. Break this up into the area of a triangle with a base of 6 and a height of 6 and a rectangle with base 6 and height 4. Applying the formulas from high school geometry, the triangle has area 18 and the rectangle has area 24. Thus gross bene?t is 42.
3. In the above example, if the price changes from 4 to 6, what is the change in consumer’s surplus?
14.3. When the price is 4, the consumer’s surplus is given by the area of a triangle with a base of 6 and a height of 6; i.e., the consumer’s surplus is 18. When the price is 6, the triangle has a base of 4 and a height of 4, giving an area of 8. Thus the price change has reduced consumer’s surplus by $10.
4. Suppose that a consumer is consuming 10 units of a discrete good and the price increases from $5 per unit to $6. However, after the price change the consumer continues to consume 10 units of the discrete good. What is the loss in the consumer’s surplus from this price change?
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14.4. Ten dollars. Since the demand for the discrete good hasn’t changed, all that has happened is that the consumer has had to reduce his expenditure on other goods by ten dollars.
15 Market Demand
1. If the market demand curve is D(p) = 100 ? .5p, what is the inverse demand curve?
15.1. The inverse demand curve is P(q) = 200?2q.
2. An addict’s demand function for a drug may be very inelastic, but the market demand function might be quite elastic. How can this be? 15.2. The decision about whether to consume the drug at all could well be price sensitive, so the adjustment of market demand on the extensive margin would contribute to the elasticity of the market demand.
3. If D(p) = 12?2p, what price will maximize revenue? 15.3. Revenue is R(p) = 12 p?2p2, which is maximized at p = 3.
4. Suppose that the demand curve for a good is given by D(p) = 100/p. What price will maximize revenue?
15.4. Revenue is pD(p) = 100, regardless of the price, so all prices maximize revenue.
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