ECO00001H Microeconomics 3 – SPECIMEN PAPER 2022
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ECO00001H
2022
Microeconomics 3 – SPECIMEN PAPER
SECTION A
1. Answer all of the following questions.
(a) Consider the following economy:
(i) 4 commodities: and are consumption goods, and are the production inputs labour and capital.
(ii) 2 consumers with utility functions 1(1, 1) and 2(2, 2).
(iii) Endowments: ̅1 = ̅2 = 1 = 2 = 0, 1 > 0, 2 > 0, 1 > 0,
and2 > 0 .
(iv) 2 producers with production functions: = ( , ) and =
( , ).
Suppose that the government subsidises commodity : The producer receives pounds for each unit of sold, whereas the consumers only pay − for each unit of consumed. Here, is the subsidy offered by the government. Derive the conditions for Pareto optimality, i.e., consumption, production and overall efficiency and investigate whether these are satisfied in the general competitive equilibrium outcome.
(b) Consider the same problem as in (a), but suppose that the
government also collects a tax so that the producer of only receives − for each unit of sold. Investigate whether the new general competitive equilibrium outcome satisfies production and overall efficiency.
(c) Consider a decision rule that, for every preference profile (consisting of rational preference relations), selects a social preference relation ≿ as follows. Let be the individual with the highest NHS number and let ≿ be individual ’s preference relation. The social preference relation ≿ reverses the preferences of . That is, for any alternatives and , ≿ if and only if ≿ . That is, if
prefers to, then society prefers to . Evaluate whether this decision rule satisfies each of the four conditions in Arrow’s impossibility theorem.
(d) Suppose there are six patient-donor pairs in a kidney exchange programme. The compatibility matrix and the priority order are given below:
0
1
=
(0
1
0
0
0
1
1
1
1
0
0
1
1
1
1
1
0
0
1
1 0
0 1
0 1
1 0)
(1) 2
(2) 6
(3) 3
, (4) 4
) 1
Furthermore, the patient-donor pairs have the following non- dichotomous preferences:
1 2 3 4 5 6
3
2
4
5
3
4
6
1
5
6
4
1
1
5
1
6
3
2
5
2
4
3
First use the TTC mechanism to find a matching, then draw the compatibility graph and find all priority matchings.
(e) We learned in class that priority matchings are Pareto efficient and
maximise the number of transplants. Using your answer in (d), show that this is not true in the particular scenario described in (d). Explain why this might be the case.
2. Answer all of the following questions, (a), (b) and (c).
(a) Consider the following economy:
(i) Two consumption goods and , and one production input (labour).
(ii) One consumer with utility function 1(1, 1, 1) = ln () , where
1, 1, 1 are the quantity of and consumed and the quantity of
labour supplied by the consumer each day.
2 1
(iii) Two producers with production functions = and = , where
and are, respectively, the quantity of labour used in the production of and each day.
(iv) Endowments: ̅1 = 1 = 1 = 0 .
The consumer chooses freely how much labour to supply each day. This labour is then used in the production of and in quantities and , respectively. Derive the condition corresponding to overall efficiency for Pareto optimality in the above economy. Show that the following condition must be satisfied in a Pareto optimal allocation:
1 2
31 = 41
Finally, investigate whether the condition above is satisfied in a general competitive equilibrium.
(b) Consider the same economy as in (a) but suppose instead that
1(1, 1, 1) = ln(1). Derive the general competitive equilibrium quantities of 1, 1 , 1, , and .
(c) Consider a society with 5000 individuals and three social alternatives = {, , }. All individuals have rational and strict preferences over social alternatives. Consider the four preference relations ≿1, ≿2, ≿3, ≿4 below, where ≻ indicates strict preference.
≿1:
≿2:
≿3:
≿4:
≻1 ≻1
≻2 ≻2
≻3 ≻3
≻4 ≻4
3000 individuals have preferences given by the preference relation ≿1 , 1000 individuals have preferences given by the preference relation ≿2 , 900 individuals have preferences given by the preference relation ≿3 , and 100 individuals have preferences given by the preference relation ≿4 .
Find the social preference relations selected by the majority rule and Borda count and determine whether they are transitive/quasitransitive. Finally, use the preferences above to demonstrate that Borda count violates Independence of Irrelevant Alternatives.
(d) Suppose there are four schools 1, 2, 3, 4 with capacities 1 =
2 = 3 = 4 = 1 and five children 1, 2, 3, 4, 5 . The priorities, preferences and capacities are given in the table below.
1 1 1 1 |
|
2 |
≿c4 |
2 3 5 4 1 |
3 2 1 4 |
Assume that all children report their true preferences. Use the Immediate Acceptance mechanism to find a matching. Explain why this matching is/isn’t Pareto efficient.
(e) Using the same priorities, preferences and capacities as in (d),
show that some child can benefit from reporting false preferences. Would the resulting matching still be Pareto efficient? Explain.
SECTION B
3.
(a)
(i) Explain the axiom of substitution in the context of expected utility representation.
(ii) Suppose now that y is a random variable with a continuous probability
distribution function f (y) over the closed interval [a,b] . Let f (y) be a concave function. Express a risk-averse decision-maker’s utility specification in terms of Jensen’s inequality.
(b) What is a risk premium? Suppose a decision-maker has a certain
income y . Show that her risk premium r can be expressed as
1 u//(y) 2
2 u/ (y)
Where 2 denotes the variance of a prospect. Explore the relationship between r and the individual’s degree of risk aversion. Clearly explain and illustrate how you can show the certainty equivalent income and the magnitude of risk premium in an indifference curve diagram for all risk categories.
(c) In the Rothschild-Stiglitz insurance market model with adverse selection where there are two types of individuals: Type A and Type B, with the following data:
Initial wealth y = 10 (e.g. £10K, same for both types)
Loss z if an accident happens, z = 5 (same for both types)
Accident probabilities: A 0.5, B 0.9
The proportion of Type A individuals in the population is = 0.6
Utility functions:
o Type A: u(x) ln x, > 0
o Type B: v(x) ln x, > 0 .
(i) Show that the utility functions u(x) and v(x) can be considered as linear transformations of one another. Show that the indifference
curves of the two types intersect one another on the certainty line. Find which type has the steeper indifference curve.
(ii) Suppose that the insurance firm can fully ascertain the individual’s
type. Determine the set of equilibrium insurance contracts, specifying the amount of cover and the premium rates, offered to each type. Show your result in a diagram clearly labelling all relevant values.
(iii) If the insurance company could not ascertain the individual’s risk
category, explain the consequence of offering the above full information contracts.
(iv) Consider a separating equilibrium.
Write down the incentive constraints that must be satisfied if the insurance firms want to induce separating equilibrium.
Specify a possible set of (equilibrium) separating contracts. Clearly show your results in a diagram.
(v) Now consider a pooling equilibrium.
Find the slope of the pooled budget line.
Explain why the pooled budget line can at most be tangent to the Type A agent’s indifference curve passing through their separating contract for a competitive equilibrium to exist.
4.
(a) Consider Lori who faces the problem of buying an insurance cover, q,
at an actuarially fair premium rate p, 0 < p < 1, in an incomplete market where there are two sources of income risks: insurable and non-insurable. Let L denote the loss associated with the insurable risk, and D denote the loss associated with the uninsurable risk. There are four possible states that Lori can face (given below), with the probability
4
of occurrence of each state beingi , i 1,.., 4; i 1 .
i1
- State s1 : the no accident state (1 )
- State s2 : where only the uninsurable loss D occurs (2 )
- State s3 : where only the insurable loss L occurs (3 )
- State s4 : where both types of losses L and D can occur (4 )
Let xi , i 1, 2,3, 4 denote Lori’s income in state si with her initial endowment income being x. Lori’s utility function is:
u(x) ln a x a 0
Show that, when risks are perfectly negatively correlated, Lori buys positive cover if and only if L > D whereas she buys zero cover if D L .
(b) Consider a standard model of an insurance contract in a competitive
market with moral hazard (as we have discussed in the lecture). Jill’s probability of accident happening is a continuous function of her effort level e with (e)/ 0 and (e)// 0 . Jill has the following utility function,
separable in income and effort: u(y, e) v(y) g(e); with v (y) 0,v/ // (y) 0; g (e) 0,/ and g// 0
where y denotes Jill’s income. Let us use the following notations:
- L: the magnitude of loss (if an accident happens)
- q: the amount of cover bought by her.
- p: the premium rate, 1 > p > 0.
2022-05-11