Applied Microeconomics

Applied Microeconomics
Assessment 2: Problem Set

  1. Consider the following exchange economy. There are two goods (1 and
    2) and two consumers (A and B). Preferences and endowments are as
    follows:
    u
    A

x
A
1
; xA
2

= 2x
A
1 + x
A
2 x
A =

1
2
;
1
2

u
B

x
B
1
; xB
2

= x
B
1 + 2x
B
2 x
B =

1
2
;
1
2

(a) Depict the economy in an Edgeworth Box diagram. Your diagram
should include the autarkic allocation and some indi§erence curves
for each consumer.
[10 marks]
(b) Identify the set of Pareto e¢ cient allocations.
[Hint: No calculus is required to answer this question! Just use
your Edgeworth Box diagram and explain your reasoning.]
[15 marks]
(c) There is a market for each good and a Walrasian auctioneer who
announces a price vector p = (p1; p2).
i. Suppose the auctioneer announces prices p1 = 3 and p2 = 4.
Find each consumerís optimal consumption plan and verify
that these are not equilibrium prices.
[Hint: Use a picture rather than calculus to think about the
consumersíoptimal consumption plans.]
[10 marks]
ii. Suppose the auctioneer announces prices such that
p1
p2
= 1.
Show that markets will clear at such prices. What is the
WEA?
[10 marks]

  1. Recall the discussion of tax evasion in Week 3. The IRD has two policy
    parameters with which to deter evasion: the audit probability ( ) and
    the penalty rate. Denote the latter by q. The current IRD penalty rate
    for evasion is q =
    3
    2
    . That is, an evader who is caught must repay the
    evaded tax plus a penalty equal to 1.5 times the amount of evaded tax.
    (a) In class we showed that if all taxpayers are risk averse, must
    be at least 0:4 to deter all tax evasion (given q =
    3
    2
    ). This means
    auditing 40% of taxpayers, which is quite expensive! Suppose
    the IRD decides it can only a§ord to audit 10% of taxpayers, so
    = 0:1. To achieve its objective of zero tax evasion it will need
    to increase q. What is the smallest value of q consistent with
    achieving its objective? (You may assume that all taxpayers are
    risk averse.) [20 marks]
    (b) Suppose = 0:1 but the penalty rate remains at q =
    3
    2
    . This
    means that every risk-averse taxpayer will try to evade some tax.
    Recall Ngaire from Week 3. Ngaire is risk averse with vNM utility
    function v (x) = p
    x. She has pre-tax wealth W = $45; 000 and
    annual taxable income Y = $36; 000. Her tax rate is t =
    1
    2
    so if she
    declares all her income she pays $18; 000 in tax, leaving her with
    post-tax wealth of $27; 000. Ngaire can choose not to declare all
    income and risk being caught. Let z be her undeclared income,
    so she declares $ (36; 000 z) to the IRD. Ngaire can choose any
    (real number) value for z between $0 and $36; 000.
    i. Let state 1 be the state in which Ngaire is audited and state
    2 the state in which she is not. Thus, x1 is Ngaireís wealth,
    after deduction of tax and penalties, in state 1 and x2 is her
    post-tax wealth in state 2. Draw Ngaireís ìbudget lineî in
    a suitable HY diagram. [Hint: Recall our analysis of the
    Exercise on Slide 32 for Week 3 and note that we did not use
    any information about to construct the ìbudget lineî.]
    [10 marks]
    ii. Which point (x1; x2) on her ìbudget lineîmaximises her expected utility?
    [20 marks]
    iii. Based on your answer to (ii), how much income will Ngaire
    declare? [If you didnít manage to solve (ii), explain how you
    would have calculated the level of declared income if you did
    have a solution to (ii).] [5 marks
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