Consider an individual with a current income of $40,000 that is considering committing a crime that has a payoff of $20,000 and a fine of $20,000 if the individual is caught. Based on the standard Becker criminal decision rule, which of the probabilities of getting caught listed below is the highest probability that would still give the individual incentive to commit the crime?

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We start by restating the scenario in our own terms: the crime offers a potential payoff of 20,000 if successful, but if caught the individual faces a 20,000 fine. The Becker crime model yields an expected payoff of the crime equal to (1 − p) × B − p × F, where p is the probability of being caught, B is the benefit from committing the crime (20,000), and F is the penalty (20,000).

Option 1: 25% (1/4). If p = 0.25, the expected payoff is (1 − 0.25) × 20,000 − 0.25 × 20,000 = 15,000 − 5,000 = 10,000 positive. So this probability would still incentivize crime, albeit with a positive expected gain.

Option 2: 50% (1/2). If p = 0.50, the expected payoff becomes (1 − 0.50) × 20,000 − 0.50 × 20,000 = 10,000 − 10,000 = 0. This yields zero expected payoff, meaning there is no incentive to commit the crime when exactly at 50% risk of capture.

Option 3: 66.67% (2/3). If p ≈ 0.6667, the expected payoff is (1 − 0.6667) × 20,000 − 0.6667 × 20,000 ≈ 6,667 − 13,333 ≈ −6,666, a negative expected payoff. This clearly disincentivizes crime.

Option 4: 33.33% (1/3). If p = 0.3333, the expected payoff is (1 − 0.3333) × 20,000 − 0.3333 × 20,000 ≈ 13,333 − 6,667 ≈ 6,666, which is positive. Therefore this probability still provides an incentive to commit the crime.

Now comparing the options with Becker’s threshold in mind: the highest p that still yields a positive expected payoff must be strictly less than 0.5, since at p = 0.5 the expected payoff is zero. Among the listed probabilities, 25% and 33.33% are below 50%, with 33.33% being the larger of the two. The 50% option hits the threshold and gives zero payoff, which does not provide an incentive. The 66.67% option is well above the threshold and produces a negative payoff, removing any incentive.

In summary, each option’s outcome under the Becker rule is evaluated by plugging into the EV formula: EV = (1 − p)×20,000 − p×20,000. Positive EVs occur when p < 0.5, zero at p = 0.5, and negative for p > 0.5. Consequently, the highest listed probability that still offers an incentive to commit the crime is 33.33% (1/3), since 50% would yield zero and 66.67% would yield a negative payoff.

SP25 G202 Kreft & Mafi-Kreft Self-Test Quiz 4

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