1. EC202: Worked Example #3.19 Frank Cowell April 2004
This presentation covers exactly the material set out in the file WorkedExamples.pdf, but with the addition of a few graphics and comments
To start the presentation select Slideshow\View Show or click on icon below left.
Mouse click or [Enter] to advance through slide show

2. WX3.19: Setting

3. WX3.19: Part 1 Case 1: obvious Case 2: 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. WX3.19: Part 2 Substitute in the values for c.
Use classical maximisation technique

5. WX3.19: Maximising utility
Maximand found from substituting in c:
Differentiate w.r.t. x:
For interior solution set this equal to 0:

6. WX3.19: Property of FOC Take the differential w.r.t. x:
Evaluate at x = y:
This is negative if 1 – p – sp > 0.
So utility would increase if x were reduced

7. WX3.19: Part 3 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. WX3.19: effect of s Take the FOC: Differentiate with respect to s:
Need to rearrange this to get explicit formula for x* / s

9. WX3.19: 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. WX3.19: effect of y Again take the FOC:
Differentiate with respect to y:

11. WX3.19: effect of y (cont) Effect on underreported income:
Bottom line is negative (because ucc < 0 )
Top line is negative if we assume DARA
Under DARA underreporting rises with income

12. WX3.19: Points to remember
Identify the components of the optimisation problem
Set up the maximand
Derive FOC
Check for interior solution
Get comparative static effects from FOCs