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ECON20512/30512 – Intermediate Microeconomic Theory 2 Final Exam Paper 2022/2023

发布时间：2024-05-24

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ECON20512/30512 – Intermediate Microeconomic Theory 2

Final Exam Paper 2022/2023

Question 1 (15 marks; word limit = 100) Agent i is considering to buy one of the fol- lowing two ﬁnancial assets: (a) asset 1 (A1), that pays £103 with certainty, or (b) asset 2 (A2), that pays £110 with probability 0.95, and £0 otherwise. Each asset costs £100, and the initial wealth is wi ≥ 100.

(a) Assume agent i’s preferences over money x are represented by the utility function ui(x) = 10px. Determine the asset that agent i would buy (if any).

(b) Suppose ui(x) = 100(x2)0 , and let p be the probability that A2 pays £110. Find the

value of p such that agent i is indifferent between both assets. Illustrate in a graph for wi = 100.

Question 2 (27 marks; word limit = 300) Consider an agency problem in which a principal contracts an agent to do some effort. Let x be the monetary outcome of this principal-agent problem, and assume that x depends on both the agent’s effort (e) and the state of nature (θ). Speciﬁcally, let e 2 f4, 6g and θ 2 Θ = fθ1 , θ2 , θ3 g, and suppose

Table 1: Monetary outcomes

一

一一一 State

一

Effort 一一一

一

θ 1

θ2

θ3

e = 4	x = 30000	x = 60000	x = 30000
e = 6	x = 60000	x = 60000	x = 30000

Both parties believe that each state θ 2 Θ is equally likely. The payoff functions of the principal and the agent are give by B(x, w) = x - w and U(w, e) =pw - e2 , re- spectively, where w is the agent’s wage rate. Suppose that the agent has a reservation utility equal to U(-) = 114.

(a) Determine the equilibrium effort and the wage scheme when effort is observable. (b) Calculate the equilibrium contract when effort is not observable, and ﬁnd the effort that principal induces in equilibrium. Compare individual payoffs with the ﬁrst-best.

Question 3 (28 marks; word limit = 300) Consider a signalling model with two types θ 2 f10, 20g of workers’ abilities, which are equally likely. To signal its ability, each worker can choose one of three levels of education e 2 fe1 , e2 , e3 g, with 0 < e1 < e2 < e3. Workers’ individual utility function is U(w, e, θ) = w - c(e, θ), where the wage rate w 2 [10, 20]. The cost of education c(e, θ) is given by the discrete function below:

Table 2: Cost of education

一

一一一 Type

一

Education 一一一

一

θ = 10

θ = 20

c(e1 , 10) = 0 c(e2 , 10) = 7 c(e3 , 10) = 14

c(e1 , 20) = 0 c(e2 , 20) = 4 c(e3 , 20) = 8

(a) Describe the separating equilibria, determining the education level chosen by

each worker type and the wage function offers by the (risk neutral) employer.

(b) Describe the pooling equilibria, including the education levels and wage functions.

Question 4 (30 marks; word limit = 400) Consider a competitive market for health in- surance with 10000 potential costumers, which are divided into two groups. Assume that the number of people in group 1 (and, respectively, 2) is 9400 (and, respectively, 600). The probability of illness in group 1 isp1 = 0.01, whereas in group 2 it isp2 = 0.05. To treat an illness in either group costs £100000. All customers are risk averse, and the willingness to pay for insurance is £1200 in group 1, and £6000 in group 2.

(a) Describe the equilibrium assuming that the health insurance companies cannot differentiate between group 1 and group 2. Is the outcome efﬁcient?

(b) Describe the equilibrium if the health insurance companies devise a test to distin- guish between consumers of group 1 and 2. What can you say about efﬁciency?

(c) What happens if the government forbids the insurance companies to discriminate between individuals from the two groups? Consider introducing a subsidy to assist individuals of group 2; and show it is increasing in both, the total population and the expected cost of treatment.