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Exercise 8.11

Exercise 8.11. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006. Ex 8.11(1): Question. purpose : to develop an analysis of individual tax evasion as an application of a risk-taking model

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Exercise 8.11

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  1. 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

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