HE4001: A Decision-maker has Income x0 in Period 0 and Income Varying: Advanced Microeconomics Assignment, NTU, Singapore

Question 1

Consider the following savings problem. A decision-maker has income x0 in period 0 and income varying randomly between x1−ε and x1+ ε in period 1 (ε>0); the outcomes are equally likely. Now, the individual can save an amount s in period 0 returning (non‐randomly) rs, with r > 0, 0 ≤ s ≤ x0, in period 1. The individual has expected utility preferences exhibiting risk aversion; preferences are time additive. That is

u(s) = u(x0 − s) + 1/2 u(x1−ε+rs) + 1/2 u(x1+ε+rs)

• Analyze how s depends on x0 and x1 (i.e. derive comparative‐statics results and interpret them in terms of properties of the utility function).
• Note that ε measures the riskiness of future income. Show that for a prudent decision-maker (whose utility has a positive third-order derivative) the saving is an increasing function of ε.
• Briefly discuss how would the analysis in (a) change if the decision-maker were instead a risk lover?

Question 2

Consider a two-consumer, two-good exchange economy. Utility functions and endowments are

U^A(x1,x2) = (x1x2)² and e^A = (18, 4)

U^B(x1,x2) = ln(x1) + 2 ln(x2) and e^B = (3, 6)

• Characterize the set of Parato-efficient allocations as completely as possible.
• Find a Walrasian equilibrium and compute the WEA and verify that the WEA you found in part is in the core.
• A social planner wants to reallocate goods even among A and B and then lets the consumers transact through perfectly competitive markets. Find the Walrasian equilibrium.
• If instead, the social planner wants to allocate goods to maximize consumer B’s utility while holding consumer A’s utility at u^A = 5184. Find the assignment of goods to consumers that solves the planner’s problem and show that the solution is Pareto efficient.

Question 3

Two firms produce a homogeneous product. Let p denote the product’s price. The output level of firm i is denoted by qi, i=1, 2. The aggregate industry demand curve for this product is given by p = A – (q1+q2). Assume that the cost function of firm i is Ci(qi)=ci q²i, with A > c2 > c1 > 0. Solve the equilibriums for the following cases (Make sure you solve for the output level of each firm and calculate the market price as well as the profit for each firm).

• Solve for a Cournot equilibrium.
• Suppose firm 1 sets its output level first and firm 2 will react with the Cournot best-response. Solve for a Stackelberg equilibrium. Is there any first-mover advantage for firm 1?
• If firm 2 behaves as a price-taker and firm 1 knows the exact output of firm 2. How will firm 1 determine its output? Is there any first-mover advantage for firm 1?
• If firm 2 behaves as a price-taker and firm 1 knows the exact decision rule of firm 2. How will firm 1 determine its output? Is there any first-mover advantage for firm 1 now?

Question 4

• Professor Huang hires a teaching assistant, Miss Wong, for teaching x hours with a total wage s. For this arrangement, Professor Huang’s payoff is x-s while Miss Wong’s utility is s – c(x), where c(x) = x²/2. Assume Miss Wong’s reservation utility is zero.

1. If Professor Huang chooses x and s to maximize his utility subject to the constraint that Miss Wong is willing to work for her, what is the optimal x and s?
2. Suppose that Professor Huang sets a wage schedule of the form s(x) = ax + b and lets Miss Wong chooses the number of hours that she wants to work. What values of a and b should Professor Huang choose so as to maximize his payoff function? Could Professor Huang achieve a higher payoff if he were able to use a wage schedule of a more general functional form?

• Consider the standard hidden-action problem with outcomes x₁ < x₂ <… < xn resulting with probabilities π1b, π2b, …, πnb if the agent takes the desired action, b, and with probabilities π1a, π2a,…, πna if the agent takes the other action, a. The principal is risk-neutral and the agent is risk-averse with utility function u(s).

1. Show how the principal’s profit-maximization problem produces the relationship 1/u'(si) = λ+μ[1−(πia/πib)] where si is the payment given outcome xi, and λ and μ are Lagrange multipliers.
2. Explain the implications for the solution if πib is small for some I and the implications for the solution if μ is large.

Question 5

• Suppose that the government is considering building a public project with a cost K. There are I individuals indexed by i. Individual i’s privately known benefit from the project is bi. The government’s objective is to maximize the expected value of the aggregate surplus. Will the Groves-Clarke tax mechanism be applied in this project? Is it possible that the government balances its budget on average (overall realizations of the bi’s)?
• By your opinion, what can be done to improve Singapore’s COE bidding system? Please justify your argument based on what you learn from Auction Theory.