Frank Cowell: Microeconomics Exercise 8.11 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

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

Frank Cowell: Microeconomics Ex 8.11(1): Consumption Case 1: 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: 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 = [1 – t – st]y + stx

Frank Cowell: Microeconomics Ex 8.11(2): Question Method Substitute in the values for c Substitute in the values for c Use standard maximisation technique Use standard maximisation technique

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

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

Frank Cowell: Microeconomics Ex 8.11(3): Question Method Take x* as a function of the parameters y,  s,  Take x* as a function of the parameters y,  s,  This function satisfies the FOC This function satisfies the FOC So to get impact of parameter: So to get impact of parameter:  Differentiate the FOC w.r.t. the parameter  Rearrange to get  x* /  parameter

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

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

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

Frank Cowell: Microeconomics Ex 8.11(3b): Effect of y (cont) Effect on underreported income: Effect on underreported income: Rearranging: Rearranging: Bottom line is negative (because u cc < 0 ) Bottom line is negative (because u cc < 0 ) Top line is negative if we assume DARA Top line is negative if we assume DARA Under DARA underreporting rises with income Under DARA underreporting rises with income

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