## Applied Microeconomics

Applied Microeconomics

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

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]

- 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