Frank Cowell: Microeconomics
November 2006
Exercise 8.11
MICROECONOMICS
Principles and Analysis
Frank Cowell
Ex 8.11(1): Question
Frank Cowell: Microeconomics
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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.
Ex 8.11(1): Consumption
Frank Cowell: Microeconomics
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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
Ex 8.11(2): Question
Frank Cowell: Microeconomics
Method
 Substitute in the values for c
 Use standard maximisation technique
Ex 8.11(2a): Maximising utility
Frank Cowell: Microeconomics
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Maximand found from substituting in c:
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Differentiate w.r.t. x:
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For interior solution set this equal to 0
On rearrangement, this yields:
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Can be interpreted as E(ruc(c)) = 0
 where r is the rate of return to evasion
Ex 8.11(2b): Behaviour at x = y
Frank Cowell: Microeconomics
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Take the differential w.r.t. x:
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Evaluate at x = y:
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On rearrangement, this yields:
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This is negative if 1 – p – sp > 0
So utility would increase if x were reduced
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Ex 8.11(3): Question
Frank Cowell: Microeconomics
Method
 Take x* as a function of the parameters y, p, s, t
 This function satisfies the FOC
 So to get impact of parameter:
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Differentiate the FOC w.r.t. the parameter
Rearrange to get x* /  parameter
Ex 8.11(3a): Effect of s
Frank Cowell: Microeconomics
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Take the first-order condition:
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Differentiate with respect to s:
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We need to rearrange this expression…
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…to get explicit formula for x* / s
Ex 8.11(3a): Effect of s (cont)
Frank Cowell: Microeconomics
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Define the second order term:
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Then we get:
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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
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Ex 8.11(3b): Effect of y
Frank Cowell: Microeconomics
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Again take the first-order condition:
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Differentiate with respect to y:
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Rearranging we find:
Ex 8.11(3b): Effect of y (cont)
Frank Cowell: Microeconomics
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Effect on underreported income:
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Rearranging:
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Bottom line is negative (because ucc < 0 )
Top line is negative if we assume DARA
Under DARA underreporting rises with income
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Ex 8.11: Points to remember
Frank Cowell: Microeconomics
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Identify the components of the optimisation
problem
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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