ECON30001 Problem Set 2

Semester 1 2024

Question 1. Consider the following competitive labour market. There are many identical firms that can hire workers. Each firm produces the same output using a constant returns to scale technology and the price of output is set to one. The market determines a wage level w for labour. Let θ denote the number of units of output (equal to revenue) that a worker can produce.  There is asymmetric information.  Worker productivity levels are privately known, and are uniformly distributed between 24,000 and 40,000.  Let r(θ) denote the reservation wage of an employee with productivity θ .

(a) Explain fully what is meant by a competitive equilibrium in this market.

(b) Suppose that r(θ) = 0:9θ .

(i) Derive and graphically illustrate the firm’s conditional expected revenue as a function of the market wage, conditioning on available labour choosing to accept employment.

(ii) Find the competitive equilibrium and calculate the percentage of the population that will be unemployed.

(c) Suppose that r(θ) = 0:75θ .

(i) Derive and graphically illustrate the firm’s conditional expected revenue as a function of the market wage, conditioning on available labour choosing to accept employment.

(ii) Find the competitive equilibrium and calculate the percentage of the population that will be unemployed.

Question 2. Consider the following competitive market for insurance.

There are two states of the world: good G and bad B. Consumers have wealth 20, but if the bad state occurs their wealth is 10. Consumers are risk averse, expected utility maximisers, with common utility function u(x) = ln(x).  There are two types of consumers:  high risk types H and low risk types L . The probabilities that H and L types find themselves in the bad state are pH = 0.5 and pL = 0.4. There is probability β = 0.6 that a consumer picked at random is type L.

Firms are risk-neutral expected profit maximisers.   The firm offers consumers a state- contingent contracts c = (cG, cB) in exchange for their endowment e = (eG, eB).

(a) Explain what is meant by a competitive equilibrium of this market.

(b) Explain and derive the equilibrium insurance contracts under the assumption that there is perfect information.

(c) Explain and derive the equilibrium insurance contracts under the assumption that there is void information.

(d) Explain and derive the equilibrium insurance contracts under the assumption that there is asymmetric information, and assuming that equilibrium exists.