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

WX3.19: Setting

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

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

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

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

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

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

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

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

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

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