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Main Examination Period 2022

ECN211

Microeconomics II

Question 1

Consider an economy with two goods t = 1, 2. Whenever an individual consumes x1  units of good 1 and x2  units of good 2, their utility is given by

u(x1 , x2 )  =  ln x1 + 6 ln x2 ,

where 6 is a parameter taking values 0 < 6 < 1.

(i)  How does the parameter 6 affect the marginal rate of substitution between goods 1 and 2? Explain intuitively how does the relative preference for goods 1 and 2 change as the parameter 6 increases.

[3 marks]

Consider an exchange economy with two consumers, A and B , whose utilities are given as above. How- ever, the parameter 6 is lower for consumer A than for consumer B , i.e., we have 6A  ≤ 6B . Consumer A is endowed with 2 units of good 1, but has no endowment of good 2. Consumer B has no endowment of good 1, but has 2 units of good 2. In the remainder of this question, let p denote the price of good 1 and the price of good 2 be normalised to 1.

(ii)  Find the competitive equilibrium price p and the equilibrium allocation expressed as a function of

the parameters 6A and 6B . Who is consuming more of good 1 and who consumes more of good 2? Interpret your results by referring to your answer in part (i).

[7 marks]

(iii)  Plot your results using the Edgeworth box. Make sure to include the 45line.

[6 marks]

The example above can be interpreted as a dynamic, two-period economy with a perishable consump- tion good, i.e., one that can not be saved, stored, or transferred across time (e.g., fresh fruit). In this case, good 1 can be interpreted as a consumption good available in period 1 and good 2 is a consumption good available in the second period only.

(iii)  What is the interpretation of the parameters 6A, 6B  in this dynamic economy? Argue that the price p can be interpreted as the inter-temporal interest rate.  Discuss the equilibrium allocation from part (ii) in light of this interpretation.

[3 marks]

(v)  The government thinks that in equilibrium each consumer should consume exactly one unit of consumption in each period. Under what conditions (if any) is it possible to reach this objective by making transfers of the initial endowment between the two consumers?

[6 marks]

Question 2

(i)  What is the tragedy of the commons?

[3 marks]

A particular road is used by N drivers each day.  Each driver obtains a benefit v from the journey and incurs a cost c(N) that increases with the number of drivers on the road due to congestion. Suppose that the cost incurred is given by

c(N)  =  a + bN,

for some positive numbers a, b. If the agent decides not to use the road, they receive 0. You may assume throughout that the number of drivers N can be any real number (rather than an integer).  Each driver uses the road at most once.

(ii)  Show that the number of drivers that join the road is

Ne   =  v a

b    .

[5 marks]

(iii)  Suppose that we identify social welfare with the sum of benefits of all drivers minus the sum of

their individual costs.  Find the number of drivers N*  at which the social welfare is maximised. What is the value of social welfare when N = Ne ? Comment on your answers.

[7 marks]

(iv)  Suppose that the government charges a toll t (a congestion charge) for each journey.  Find the

optimum toll. Interpret your answer in terms of the externality that each driver imposes on others.

[5 marks]

By spending amount X on road improvements, the government reduces the cost per trip to:

c(N, X)  =  a + bN gX.

Assume there is no congestion charge.

(v)  Write down the new expression for the social surplus when there are N drivers (including the cost X incurred by the government to finance the improvements). Find the new equilibrium number of drivers and the equilibrium social surplus. What is the social value of road improvements?

[5 marks]

Question 3

Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will be equal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their income will be y — L. The chance of a safe return is 50%.

(i)  Draw and carefully label the merchant’s endowment point, their expected income, and their cer- tainty equivalent income in a 2-dimensional state-contingent consumption space.

[5 marks]

(ii)  Use the diagram to illustrate and explain how the merchant would benefit from buying insurance

in a competitive insurance market. At which point a risk-neutral insurance firm would maximise their profits by offering the merchant full insurance?

[7 marks]

(iii)  Suppose that the merchant is risk-seeking. Describe and illustrate in a state-contingent diagram an

agreement that would make the merchant better off, and that the firm would be willing to accept.

[6 marks]

Now suppose that there are two identical merchants, A and B , who are both risk averse expected utility maximisers with utility of income given by u(y) = ln y. The income of each merchant will be 8 if their own ship returns and 2 if it sinks.  As previously, the probability of a safe return is 50% for each ship. However, with probability p ≤ 1/2 both ships will return safely. With the same probability p both will sink. Finally, with the remaining probability, only one ship will return safely.

(iv)  Compute the increase in the utility of each merchant that they could achieve from pooling their

incomes (as a function of p). How does the benefit of pooling depend on the probability p? Explain intuitively why this is the case.

[7 marks]

Question 4

Three qualities of second-hand bicycles are available in equal numbers: high, medium, and low. There are many buyers and sellers, who value each quality of the bike differently. The value that each agent assigns to each quality of the bike is given below.

Buyers value    Sellers value

100

65

30

75

60

45

(i)  What is the efficient outcome in this market?  Equivalently, which types of bikes should change their owners to maximise the social welfare?

[3 marks]

Suppose that the buyers do not know the quality of any particular bicycle for sale, but the sellers do know the quality of the bike they sell. The price at which a bike is traded is determined by demand and supply. Each buyer wants at most one bicycle.

(ii)  Assuming that each buyer purchases a bike only if its expected quality is higher than the price, and each seller is willing to sell their bike only if the price exceeds their valuation, what is the equilibrium outcome in this market?

[7 marks]

Now assume that the sellers can offer a reimbursement of 50 to the buyer, payable if the bicycle breaks down. Bikes of high quality never fail; those of low quality always fail; those of medium quality fail 50% of the time.

(iii)  Suppose that all owners of high and low quality bikes offer reimbursement but owners of low qual-

ity bikes do not. Moreover, assume this to be common knowledge. Show that this is an equilibrium if the price satisfies 85 < p < 95. That is, show that none of the sellers is willing to change their behaviour, given actions of others. Is the resulting outcome socially optimal?

[6 marks]

(iv)  Are there any other (possibly inefficient) equilibria in which trade occurs?

[9 marks]