ECON 501-Microeconomics-MAIN 2022
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MSc/MEcon/PhD (integrated) in Economics
ECONOMICS
ECON 501-Microeconomics-MAIN
Part A: Answer all questions.
1. Consider a two good economy (x1 , x2 ) and an individual i whose utility function is given by
ui (x1 , x2 ) = x1 + αx2 -
The individual has wealth wi > 0 and the prices are given by p1 > 0 and p2 > 0. Parameter α is equal to the letters in your first name (e.g., α = 7 for Orestis).
(a) First substitute “your” value of α in the utility function. Solve for the consumer’s
Walrasian demand function. Show that it is appropriately homogeneous.
(b) Argue whether the indirect utility function is of the Gorman form. What does this
imply for the properties of the aggregate demand function?
[25 marks]
2. Two students (Elisa and Olivia) work on a group project. The effort they exert is denoted by xi e [0, 1] and is observable. The individual cost of effort is C(xi ) = 50xi(2) + 25xi for i = E, O . The outcome of the project (their final mark) is given by F (xi , x一i ) = 50(xi x一i + xi ). Each student’s utility function is ui = F (xi , x一i ) _ C(xi ).
(a) Write down the best-response functions if Elisa and Olivia simultaneously choose how
much effort to exert.
(b) Graph Elisa’s and Olivia’s best-response functions. In your graph, place xE on the
horizontal axis and xO on the vertical axis. Label the intercepts.
(c) What is the (pure strategy) Nash equilibrium of this game. Compare with the level of effort these students would supply if they were maximizing their joint payoff, uE + uO . Comment on your results.
(d) Now assume that the students work on the project sequentially. Elisa exerts some effort level xE . Olivia observes this effort level and then chooses her own level of effort, xO . Find the subgame perfect Nash equilibrium and compare with your results in part (c).
[25 marks]
3. The tax authority employs an inspector to audit tax returns. The dollar amount of tax evasion x revealed by the audit is either x1 = 100 or x2 = 50. It depends on the inspector’s effort level, e, and the random complexity of the tax return. The probability that x = xi conditional on effort e, πi (e) > 0, is given in the following table:
Effort |
Probabilities π 1 (e) π2 (e) |
|
Low (e = 0) High (e = 1) |
pL pH |
1 _ pL 1 _ pH |
With 0 < pL < pH < 1. The tax authority offers the inspector a wage rate wi = w(xi ), contingent on the result achieved and obtains the expected benefit B = π 1 (e)[x1 _ w1] + π2 (e)[x2 _ w2]. The inspector’s utility function is U (w, e) = ^w _ e. His reservation utility is given by u = 3.
(a) Under full information, what is the wage paid to the inspector if the tax authority wants
to induce (i) a low effort (e = 0), and (ii) a high effort (e = 1)? What is the condition on pH and pL such that the tax authority will be willing to induce a high level of effort?
(b) Set up the tax authority’s maximisation problem in the event the inspector’s level of
effort is not observable and it wants to induce the inspector to take the high effort level.
(c) Solve for the optimal contract (w1(*), w2(*)). Comment on your results in terms of risk sharing.
[25 marks]
Part B: Answer one of the two questions.
1. A firm has two plants. One plant produces output according to the production function x1(α)x2(1) 一α . The other plant produces output according to the production function x1(b)x2(1) 一b . Denoting by w1 and w2 the prices of inputs x1 and x2 , what is the cost function for this firm?
[25 marks]
2. Consider the following entry game. Firm E (for entrant) is considering entering a market that currently has a single incumbent (firm I). In order to prevent entry, the incumbent firm is considering building an additional production plant. The building costs can be low (CL = 0.5) or high (CH = 2). The building costs are known by the incumbent firm only. The probability that the building costs are high is denoted by p e (0, 1). This probability is known by the entrant. The entrant and the incumbent take their decisions simultaneously. Their payoffs are given in the table below:
|
Entrant |
||
E |
NE |
||
Incumbent |
B |
2 _ C ; _1 |
4 _ C ; 0 |
NB |
2 ; 1 |
3 ; 0 |
where C e {CL ; CH } represents the building costs. “E” and “NE” stand for Enter and Do Not Enter, respectively. “B” and “NB” stand for Build a new plant and Do Not Build a new plant, respectively. Find the pure strategy Bayesian Nash equilibria of the game. Comment on your results in terms of p.
[25 marks]
2022-08-18