1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 8.11
MICROECONOMICS
Principles and Analysis
Frank Cowell
November 2006

2. Ex 8.11(1): Question
purpose: to develop an analysis of individual tax evasion as an application of a risk-taking model
method: formulate consumption in each state of the world. Maximise expected utility. Determine effect of parameter changes by differentiating the FOC.

3. Ex 8.11(1): Consumption Case 1: concealed income is 0, so penalty is 0
so total required payment is just ty
therefore disposable income is y – ty
Case 2:
concealed income is y – x
penalty is st [y – x ]
so total required payment is ty + st [y – x]
therefore disposable income is
y – ty – st [y – x ]
= [1 – t – st]y + stx

4. Ex 8.11(2): Question Method Substitute in the values for c
Use standard maximisation technique

5. Ex 8.11(2a): Maximising utility
Maximand found from substituting in c:
Differentiate w.r.t. x:
For interior solution set this equal to 0
On rearrangement, this yields:
Can be interpreted as E(ruc(c)) = 0
where r is the rate of return to evasion

6. Ex 8.11(2b): Behaviour at x = y
Take the differential w.r.t. x:
Evaluate at x = y:
On rearrangement, this yields:
This is negative if 1 – p – sp > 0
So utility would increase if x were reduced

7. Ex 8.11(3): Question
Method
Take x* as a function of the parameters y, p, s, t
This function satisfies the FOC
So to get impact of parameter:
Differentiate the FOC w.r.t. the parameter
Rearrange to get x* /  parameter

8. Ex 8.11(3a): Effect of s Take the first-order condition:
Differentiate with respect to s:
We need to rearrange this expression…
…to get explicit formula for x* / s

9. Ex 8.11(3a): Effect of s (cont)
Define the second order term:
Then we get:
We know that uc > 0, ucc < 0 and x* < y
Therefore the above expression is positive
Reporting increases with the penalty rate
Underreported income decreases

10. Ex 8.11(3b): Effect of y Again take the first-order condition:
Differentiate with respect to y:
Rearranging we find:

11. Ex 8.11(3b): Effect of y (cont)
Effect on underreported income:
Rearranging:
Bottom line is negative (because ucc < 0 )
Top line is negative if we assume DARA
Under DARA underreporting rises with income

12. Ex 8.11: Points to remember
Identify the components of the optimisation problem
payoffs in each state of the world
return to evasion activity
Set up the maximand
Derive FOC
Check for interior solution
Get comparative static effects from FOCs