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ECON2003W1

SEMESTER 1 SUPPLEMENTARY ASSESSMENT 2020-21

ECON2003 Microeconomics of Markets

Section A

A1 (Equilibrium, 25 Points)

Suppose there are two goods and two consumers, Andy (A) and Beth (B). Andy’s utility function is denoted by uA (xA(1), xA(2)) = ′xA(1)′xA(2) where xA(1) is the amount Andy consumes of good 1 and xA(2) is the amount Andy consumes of good 2. His endowment consists of 20 units of good 1 and 40 units of good 2. Beth’s utility function is denoted by uB (xB(1), x B(2)) = ln(xB(1)) + ln(xB(2)) where x B(1) is the amount Beth consumes of good 1 and x B(2) is the amount Beth consumes of good 2. Her endowment consists of 30 units of good 1 and 60 units of good 2.

(a) Compute the marginal rate of substitution for both Andy and

Beth.

(b) Find the set of Pareto eﬃcient allocations.

(d) Consider the allocation where Andy consumes 25 units of good 1 and 30 units of good 2. Is it possible to redistribute the original endowment of Andy and Beth such that this bundle arises as Andy’s demand in a Walrasian equilibrium?

(e) Now, suppose there is a third consumer, Claire (C). Her utility function is denoted by uC (xC(1), x C(2)) = ′xC(1) +′xC(2) where x C(1) is the amount Claire consumes of good 1 and x C(2) is the amount Claire consumes of good 2. Her endowment consists of 81 units of good 1 and 144 units of good 2. Will her presence on the market change the Walrasian price and the allocation for Andy and Beth? [6]

A2 (Uncertainty, 25 Points)

Emily has bought a new car. She is not a good driver and likely to have an accident. She reckons that the probability that she will have an accident in 2021 is 25%. If undamaged, the car will have a value of £10,000 next year, but the value of the car decreases by £4,400 in case of an accident. Additionally she has £4,400 in cash. Her utility over wealth is given by u(z) = ′z .

(a) Write down the lottery, compute the expected value, and compute Emily’s expected utility.

(b) Compute the certainty equivalent of not being insured. Compare to the expected value computed in (a). Which one is larger? Why is this the case?

(c) Now suppose an insurance company oﬀers her, for an unconditional payment of £1200, to cover the full cost of a damage caused by an accident. Will she buy this insurance (if it is the only one available)? What is the highest price she would be willing to pay for this insurance?

(d) Now suppose the insurance company oﬀers another contract with a copay of £400 at a price of £1000, i.e. the insurance only pays £4000 in case of an accident. Will Emily buy this insurance (if it is the only one available)? What would be the maximal sum a risk neutral decision maker is willing to pay for an insurance with £400 copay? Discuss.

Section B

B1 (Endowments, 25 Points)

Sarah has preferences over consumption and recreation time. She does not have any money endowments or any endowments of the consumption good, but she can work at an hourly wage of w =£20 in order to buy consumption goods at a price of p =£1 per unit of consumption. Sarah can, therefore, allocate her time endowment of 24 hours per day between labor supply and recreation. C denotes the quantity of consumption and R denotes the hours spent on recreation. Sarah’s preferences are represented by the utility function

u(R, C) = ln(R) + ln(C).

(a) Suppose Sarah works 6 hours and consumes 120 units of the consumption good. Compute the marginal rate of substitution at this time-consumption bundle. Interpret the number you have computed.

(b) Find Sarah’s demand for recreation and consumption. How many hours will she work?

(c) How does the demand for recreation change in response to a change in Sarah’s wage? What is the substitution eﬀect for this

price

change?

(d) Suppose that Sarah’s hourly wage is raised to £25. Compute the new demand and identify the substitution eﬀect, the ordinary income eﬀect, and the endowment income eﬀect.

(e) Argue why the (overall) income eﬀect of a wage change on the demand for recreation is always positive here.

B2 (Equilibrium with production, 25 Points)

Consider the following economy with production: Robinson consumes recreation time and coconuts. He has an endowment of 24 hours of

time and 6 coconuts. Robinson can choose to use his time either for recreation or to gather coconuts. He manages to collect 2 ′L coconuts where L denotes the hours of time spent on gathering coconuts. His preferences over recreation R (measured in hours) and coconuts C are represented by the utility function

u(R, C) = ln(R) + ln(C)

(a) What are the feasibility and market clearing conditions for the coconut market?

(b) Which consumption of R and C is best for Robinson? How long does he work?

(c) From now, suppose there is a market at work on the island. The production of coconuts is organised by Crusoe INC, which uses labor as input at wage w = 1 (numeraire) and produces coconuts as output with the technology above (2 ′L) and supplies coconuts at price pc per coconut. What are the returns to scale of this production function? What is the supply function of Crusoe INC? What is the factor demand function?

(d) Which prices constitute a Walrasian equilibrium in this economy with production? What will be the associated quantities? What are the proﬁts of Crusoe INC?