Questiond are given below

2. "(From McAfee 1996) In May 1990, IBM announced the introduction of the LaserPrinter E, a lower cost alternative to its popular LaserPrinter. The LaserPrinter E was identical to the original Laser Printer except that it printed at 5 ppm instead of 10 ppm. According to Jones(1990), the engine and parts of the printer were virtually identical to the faster printer except that the controller for the slower printer had firmwear that inserted wait states to slow the print speed of the printer. This problem is designed to see how such damaging of a product may make everyone better off. (a) Assume first that you have a single high quality product that you are selling. The product has constant marginal cost to produce of $1. There are two types of consumers who are willing to buy a single unit of your good. 25% of the population is type 1 and are willing to buy your product if the cost of producing it is less than or equal to $11. 75% of the population is type 2 and are willing to buy your product 2 if it is priced at $3.00. Suppose that you must sell your product at a constant price (you can not screen your consumers) - show that you will only sell to the high types. (b) Now assume that you can produce a broken version of your product. Let s be a measure of how broken your product is. The high types get utility Up(x,,s) for consuming one unit of type s good. Thus Un(1, 0) is the value to the high type of buying one unit of the original quality good and Up (1, s) would be the utility of the high type for buying a good with quality s. Similarly, the low type get Uz(1,0) for the original good and Up (1, s) for consuming a low good of value s. Assume that a 0 (it is costly to produce an inferior good). i. Again assume that there is no screening other than offering a low quality good. Set up the monopolists problem. ii. Suppose that Un (1, a) = 11 -26, UL(1, a) =3-.250, and c(s) = 1. Find the optimal quality of the lower good and the amount charged for the two goods. ili. Suppose by law the monopolist can only reduce quality down to s = 3. Show that in this case, everyone is at least as well off due to the creation of the damaged good. 3. Suppose that a hospital carries K doses of an opium based medicine used to help patients suffering from micro fibralgia. N 2 K patients come seeking the drug at the same time. 6 of these actually suffer from the disease while N-6 of them are drug addicts looking for a fix. (a) Suppose that the hospital does not care about the utility of the drug addicts. The utility of a micro fiberalgia agent receiving the drug is 10. The utility of a micro fibralgia agent who does not receive the drug is zero. Calculate the total expected utility the hospital can provide to its micro fibralgia patients if it has no way to screen between drug addicts and true patients. (b) Suppose that the Hospital can force agents to waste time before being treated. The utility of a micro fibralgia user who must wait and receives the medication with probability p is: UMicro(w, p) = 10p - 2w Drug addicts have a utility function of UDA(w,P) =4-w The addicts are not rational - as long as they stay in the hospital they believe that the probability of receiving treatment is 1. Both agents outside option is zero, that is UMiers(0,0) = UpA (0,0) =0.1. Consider the monopolist from the last problem set who is serving two types of consumers with demand functions. Let c(y) = 0: Di(p.k) = 2-p ks (-p) Otherwise Du(p.k) = 4-p ks(-' Otherwise Recall from the last problem set that an agents utility from consuming good 1 is given by: UL(P.k) = (2-P) _ UN(P.k) = -P_ _ (a) Suppose that the low market all have student IDs that allow them to be differentiated from the high types. For an agent to participate it must be that: UL(p. k) 2 0 Un(p.k) 2 0 These are known collectively as individual rationality (/ R) constraints or participation constraints (PC). Draw graphically the following cases. You do not have to do the math for this problem: i. Suppose legislation requires that & = 0 and that pz = pu. Draw the optimal price and shade the producer surplus. ii. Suppose legislation requires that & = 0 but pr and py can be different. Draw the optimal prices and shade the producer sur- pluses ifi. Suppose legislation requires pr = pp and kt = ky. Draw the optimal price and shade the producer surpluses. Will the agent serve both markets? iv. Suppose both PL. PH. kL,and ky-can all be different Draw the optimal price and shade the producer surpluses. Will the agent serve both markets? (b) Now suppose that the agent can not tell one type of consumer from the other. In this case we require that each agent type gets more utility from choosing the bundle meant for them than the one meant for the other type of agent: UL(PL.KL) 2 UL(PH. KM) Un(PH, KH) 2 Un(PL, KL) These conditions are called incentive compatability constraints. They bascially say that a high type who pretends to be a low type must get less utility than he would acting as a high type. In standard screening problems it is typically the IIt constraint of the low type and the IC constraint of the high type that binds. That is: UL(p. k) 2 0 Un(PH, KH) 2 Un(PL.KL) To think about these concepts draw graphically the following cases. You do not have to do math for this problem. If you haven't already done it, assume that the low market is large enough that it isn't shut out.: i Suppose legislation requires pr = p# and ky = ky. Draw the optimal price and shade the producer surpluses. Will the agent serve both markets? Is this different from the problem with full signals above? ii. Suppose both PL, Pu. kz,and ky-can all be different Draw the optimal price and shade the producer surpluses. iii. Suppose instead of only being able to restrict p and & the mo- nopolist can also restrict quantity qr and qu. Draw the optimal price and shade the producer surpluses.4. A monopolist has a cost function of c(y) = cy so that its marginal costs are constant at $c per unit. The monopolist is operating at an output level where (= = 3. The government imposes a quantity tax of $6 per unit of output. (a) If the monopolist is facing linear demand curve, how much does the price rise? (b) If the monopolist has constant elasticity, how much doe the price rise? 5. *An economy has two kinds of consumers and two goods. Type A consumers have utility function UA(:1, 12) = 4x1 -(7)+ x2 and Type B consumers have utility function Up(21, 12) = 221 -(7) +12. Consumers can only consume nonnegative quantities. The price of good 2 is 1 and all consumers have incomes of 100. There are N type A consumers and N type B consumers. (a) Suppose that a monopolist can produce good 1 at a constant unit cost of c per unit and cannot engage in any kind of price discrimination. Find its optimal choice of price and quantity. For what values of c will it be true that it chooses to sell to both types of consumers? (b) Suppose that the monopolist uses a "two-part tariff" where a consumer must pay a lump sum k in order to be able to buy anything at all. A person who has paid the lump sum & can buy as much as he likes at a price p