3. Karen works 50 hours per week. She can work legally as a book keeper and earn

Question

3. Karen works 50 hours per week. She can work legally as a book keeper and earn S50 per hour Alternatively, she can work illegally by helping clients evade taxes for $150 per hour. Karen can only choose one occupation or the other. If she chooses illegal work, the probability of being caught is 0.3, and if caught, Karen will face a penalty of $150 per hour of illegal activity. (Note if Karen is caught, she retains the proceeds from her illegal activity, but must pay the fine Suppose Karen's utility function is given by u(y)y, where y is her weekly income (a) Will Karen choose the legal or illegal activity? Explain. (b) Now suppose Karen has an additional option. She can undertake part time employment as a legal book keeper for 30 hours per week and devote 20 hours per week to illegal activities. Doe:s Karen work only legally, only illegally, or both? Explain. (c) Suppose again that Karen can only choose to spend all her time legally or all her time illegally. If the police or justice system want to deter Karen from ever committing the illegal activity (given that her only choices are as given in earlier parts of this question), what is the minimum probability with which they must catch her committing the crime?

Explanation / Answer

Part (a)

Weekly earning under

i) legal activity: 50 x 50 = $2500

ii) illegal activity: 150 x 50 = 7500 with probability of 0.7 = 7500 x 0.7 = $5250

[i.e., probability of not being caught is 0.7]

Since 5250 >> 2500, option is: illegal activity: ANSWER

Part (b)

30 hours of legal activity and 20 hours of illegal activity would yield a weekly earning of:

(30 x 50) + (20 x 150 x 0.7)

= 1500 + 2100

= 3600

Since 5250 is greater than both 2500 and 3600, option is: illegal activity: ANSWER

Part (c)

Let the required probability be p. Then, the weekly earning from illegal activity would be;

150 x 50 x (1 - p) = 7500(1 - p).

This should be less than 2500 (the weekly earning from legal activity).

Solving 7500(1 - p) < 2500 or

(1 - p) < 2500/7500 = 1/3

=> p > 2/3.

Thus, the deterring probability is 2/3 ANSWER

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