Assessment 2: Problem Set
- Consider the following exchange economy. There are two goods (1 and
2) and two consumers (A and B). Preferences and endowments are as
1 + x
1 + 2x
(a) Depict the economy in an Edgeworth Box diagram. Your diagram
should include the autarkic allocation and some indi§erence curves
for each consumer.
(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.]
(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.]
ii. Suppose the auctioneer announces prices such that
Show that markets will clear at such prices. What is the
- 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 =
. 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 =
). 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 =
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 =
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.]
ii. Which point (x1; x2) on her ìbudget lineîmaximises her expected utility?
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