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c) Determine the optimal combination of packages (of, 254'), (mi, 23') offered by the health agency graphically in the diagram of part a). Calculate the packages (qA', 2,0 and mg, 25'). 20 marks Asymmetric information: Now consider the case of asymmetric information, i.e. assume that the health agency does not know the type 1': A, B of individual suppliers, but does know that 50% of the suppliers areI of type A (and have low costs) and that the remaining 50% of the suppliers are of type B (and have high costs). That is, the health agency knows that the 'low' cost function can} = 2 q2 applies for 50% of the suppliers and that the 'high' cost function cr-(q) = 5 :32 applies for the other 50% of the suppliers. So we have cdq} = cam} and cH(q) = cam}, with c.4(q) and cq) as before. Incentive compatibility: d) Which constraints must an incentive compatible pair of packages (cu, 2H), (Qt. 2L) satisfy? 10 marks Optimal contracts: It can be shown that for the optimal packages (qr-r, 2H) and (qt, 2L): 1) the participation constraint of the suppliers with high costs ('type H') is binding (i.e. holds with equality} 2) the lncentfve constraint of the suppliers with low costs ('type L') is binding (i.e. holds with equality), 3) if 1) and 2) are satised, then the participation constraint of the suppliers with low costs ('type L') and the incentive constraint of the suppliers with high costs ('type H') are also satised. e) Under these assumptions 1) to 3), what does the health agency's mathematical opljmization problem reduce to? 10 marks Continued... 1) Calculate the optimal packages (qr-1. Zn) and (qt'. 2;) offered by the health agency for the suppliers in the industry to choose from. 15 marks Interpretation: 9) Compare the optimal packages the health agency offers under symmetric information. as in part c). with those under asymmetric information. as in part 1). Discuss. 10 marks h) Suppose the public health agency nds itself in the situation of asymmetric information and offers the packages calculated in part 1). What would you expect the reactions of different lobby groups and of nancial watch dogs to be? Discuss. 10 marks 9mm Consider a continuum of rms seeking developing a Corona virus antibody test. The rms have different prospects for their test's effectiveness. The prospective effectiveness of a nn's test is denoted by 6. with 0 S 9 s 1. The poorest prospective effectiveness is denoted by D. the best by 1. We refer to individual rms by their unique value of 9. The rms' 9 are uniformly (i.e. 'evenly') distributed over the interval [0, 1]. The actual prospective effectiveness of a n-n's test is only known to the rm. An-n can either sell the rights to its test before it is fully developed to one of many (possibly international) health agencies at a price of P, or market the test themselves after it is fully developed and its effectiveness is known. Marketing a test of prospective effectiveness 9 when it is fully developed gives the rm an expected revenue of mei=1 +59. There are as many identical health agencies potentially willing to buy the rights to a test as there are rms, so the health agencies do not derive economic rents from buying tests early. The health agencies have access to unlimited nancial resources. A health agency can only buy the rights to one single test before it is developed. If the nn's test is of prospective effectiveness 6 and the price is P. the health agency obtains a utility of ulQP):=2+9P. Not buying the rights to any test gives the health agency a utility of D. For any price P each health agency knows the average prospective effectiveness F of the not fully developed tests offered in the rights market at that price. But only after the sale of the rights to a test has been completed. the actual prospective effectiveness of the test becomes known to the rm. Sellers (Firms): a) Denote by WP) the rms who are willing to sell the rights to their not fully developed test at a price P. For P = 3 determine V(P) graphically in a diagram. 10 marks b) Denote by 9MP) the 'critical' or 'iuatershed' firm which at price P is indifferent between selling the rights to its test before it is fully developed and waiting to market its test after it has been fully developed. Calculate 9\"\"(P). Determine the supply function S(P) of rights to not fully developed tests. Depict the supply function S(P) in a diagram. 1 5 marks Continued... G) Let F(P) denote the average prospective effectiveness of not fully developed tests offered to the rights market at price P. Determine F(P). Depict HP) in a diagram. 10 marks Buyers (Health agencies): d) At which combinations (F, P} of average prospective effectiveness F of not fully developed test offered to the rights market. and of price P, will health agendas be willing to buy the rights to not fully developed tests? At which combinations (F. P) will they refuse to buy? When will they be indifferent between buying and not buying? Explain. 10 marks d) At which combinations (F. P) of average prospective effectiveness F of not fully developed test offered to the rights market. and of price P, will health agencies be willing to buy the rights to not fully developed tests? At which combinations (F. P) will they refuse to buy? When will they be indifferent between buying and not buying? Explain. 1 It marks e) Let 00'. F) denote the demand for the rights to not fully developed tests at price P, given that the average prospective effectiveness of the tests offered to the rights market is F. Depict the demand D(P: F) for F = 0.6 in the diagram of the supply function 8(P) in part b). At which price does demand equal supply for F = 0.6? Explain. 1 0 marks 1) Calculate DiP: F). At which combinations for P and F will the market for rights to notfully developed tests clear? Depict these combinations in the diagram of part c). 1 0 marks Equilibrium: 9) What is the firms' 'supply side' equilibrium condition on the combinations of F and P? What is the health agencies\" 'demand side' equilibrium condition on the combinations of F and P? 1 0 marks h) Determine the equilibrium combination (F. P) graphically in the diagram of part c). Calculate the equilibrium combination (F. P). 1 0 marks Continued... Interpretation: A53ume that health agencies start to worry that there may be a shortage of test kits at a later stage, and therefore become more eager to secure the ghts to a test before it is fully developed. Suppose that as a result their utility function changes from u(6, P) := 2 + 9 P to u(9, P) := 2 +1.256P. i) Adapting the above graphical analysis in a new diagram. how do you expect this change to affect the equilibrium price P' and the average prospective effectiveness F' in the market for rights to not fully developed test? How do you expect this change to affect the conditions in the market for test kits after the all tests have been developed? Discuss. 1 5 marks Continued... mm A public health agency would like to purchase PPE (Personal Protection Equipment) for its staff. It would like to do so in a cost-effective way. There are two types of suppliers offering PPEs. They appear in the industry in equal proportions. One type of suppliers face high production costs and the other type of supplier face low costs of production. Let q to denote the amount of PPEs supplied. Depending on the context as specied below. the health agency either knows the cost function of the individual suppliers. or it only knows the fractions of the two types of suppliers in the industry and does not know the individual cost functions. For suppliers with a low production costs. the cost function is cam) = 2 qnz. for suppliers with a high production cost it is cams) = 5 qaz. The health agency makes offers to purchase amounts q of PPEs for lump-sum payments 2. Suppose it offers (q;.. 2') to suppliers of type A and (cs. 23) to suppliers of type B. where qr denotes the amount of PPEs and z; the lump-sum payment for supplier i = A. B. For suppliers of type A the profit from the package (flat. 2..) is: mica. 2n) := 2a - click) = 24 - 2 142 and for suppliers of type B the profit from the package (in. 23) is name. 23) := 23 - ca(qa) = 25 - 5 qaz. The health agency's utility function regarding the total amount 0 of PPEs purchased and overall payments Z is: UlQ.Z):=Q - 0.52 where O = 0.5 in + 0.5 as is the total amount of PPEs purchased and Z = 0.5 Zn. + 0.5 23 the overall payment. Symmetric Information For the time being. assume that the health agency knows the cost function of each supplier. For each of the two types of suppliers. 1' = A. B. the health agency offers a specic take-it-or-leave-it package (at. It). A supplier of type t= A. B. will accept the package whenever the supplier's prot resulting from the package is non-negative. If the offer is declined. both qiand 2: equal zero and the supplier has a prot of zero. a) Depict the indifference curve of the health agency to a utility level of 0 in a (O. Zl-diagram. Both for suppliers of type A and of type B. depict the zero- protit curve in the same diagram with the axis representing all and Zn for supplier A and as and 23 for supplier B. 1 5 marks Continued... b) Formulate the health agency's decision problem mathematically. 10 marks c) Determine the optimal combination of packages (0;. 24'). (mi. 23') offered by the health agency graphically in the diagram of part a). Calculate the packages (QA'. 2.4-) and (Qa'. 23')- 20 marks Asymmetric information: Now consider the case of asymmetric information. i.e. assume that the health agency does not know mmrs. but does know that F...\
