Let the Punishment Fit the Crime

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Let the Punishment Fit the Crime
Indro Chakraborty R. Preston McAfee
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Let the Punishment Fit the Crime

Alleged
Indro Chakraborty R. Preston McAfee
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Local Externalities

• Some externalities have nonlinear individual
  effects
• Automobile speed
• Alcohol consumed
• Perfume worn
• Noise
• Some emissions

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Piguouvian Tax

• Local externalities can be regulated with an
  individual Pigouvian tax
• Nonlinear fines for speeding
• Impose social cost as a tax and individual
  internalizes social cost

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Princeton, NJ Speeding Fines
MPH Over Speed Limit
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Piguouvian Tax

• Local externalities can be regulated with an
  individual Pigouvian tax
• Nonlinear fines for speeding
• Impose social cost as a tax and individual
  internalizes social cost
• But theory assumes externalities observable

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Radar Guns

• Headline
• Radar gun clocks palm tree going 86 MPH

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Radar Gun Accuracy
Redacted to preserve fair use
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Finnish Penalties

• HELSINKI, Finland (Reuters) -- One of Finland's
  richest men has been fined a record 170,000 euros
  (217,000) for speeding through the center of the
  capital, police said on Tuesday.
• Jussi Salonoja, 27, heir to his family's sausage
  business, was caught driving 80 km per hour (50
  miles per hour) in a 40 kph (25 mph) zone last
  Thursday, the police said.
• Finnish traffic fines vary according to the
  offender's income.

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Typos

• AUDIT FINDS ERRORS IN DRUG SEIZURES.
• The Los Angeles Police Department continues to
  have problems with discrepancies in the amount of
  drugs reported seized and the actual amount
  confiscated.
• LAPD officials attributed the problems to
  careless errors'' such as writing down the
  wrong information or calculation errors.
• We found no evidence that seized narcotics were
  mishandled or misappropriated,'' Tuttle said.
  We are concerned, however, that the variances
  appear to be so common.''

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Blood Alcohol Levels

• Breath readings vary at least 15 percent from
  actual blood-alcohol concentrations.
• Simpson, Accuracy and Precision of Breath-Alcohol
  Measurements for a Random Subject in the
  Postabsorptive State, 33(2) Clinical Chemistry
  261 (1987)
• In study 404 DUIs, for 2/3rds breathalyzer
  measures differed by at least 0.01 from blood
  alcohol.
• 32(4) Journal of Forensic Sciences 1235 (1987)

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Melvin Effects

• Many breath testing machines assume a 2,100-to-1
  ratio in converting alcohol in the breath to
  estimates of alcohol in the blood. However, this
  ratio varies from 1,900 to 2,400 among people and
  also within a person over time.
• Each one degree of body temperature above normal
  will cause a substantial elevation (about 8) in
  apparent BAC.

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Are Alcohol Effects Nonlinear?
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Are Alcohol Effects Nonlinear?
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Are Alcohol Effects Nonlinear?
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Are Alcohol Effects Nonlinear?
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Are Alcohol Effects Nonlinear?
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Basic Model

• Risk neutral agent has net benefit B(x)
• Choice x imposes social cost s(x)
• Signal y generated given x
• Penalty p(y) assessed
• Agent chooses x to
• max B(x) Ep(y)x

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Observational Error Example

• Signal y x e, Ee 0
• Penalty s(y) is not Pigouvian tax
• s convex, then too large
• Right tax satisfies
• Ep(y)xs(x)

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Closed Form Solution

• s(x)x2, Ee2?2
• Let

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Multiplicative Errors

• Choice x, Externality s(x)
• Signal y xe, Ee 1
• Assume Eei lt ? for all i.
• If s(x)x2,

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Multiplicative Errors Solution

• If s is analytic, express with Taylor series
• Now let

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General Model

• Risk neutral agent
• Externality function s is continuous on compact
  set
• Density f(yx) of signal
• Goal Find penalty function p so that

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Separating Probability Distns

• A density f separates probability distributions
  if for any densities g ? h,

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Separating Probability Distns

• A density f separates probability distributions
  if for any densities g ? h,
• Separation is necessary. If separation fails,
  there are distinct g, h with

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Necessity

• If separation fails,
• Not all s can be supported

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Theorem

• The density f separates probability distributions
  if and only if for every continuous s, there is a
  continuous p satisfying
• Corollary to Hahn Banach Theorem

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Approximations

• What does p look like for an arbitrary s?
• Consider a small additive error be
• p(x,0)s(x)

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Derivatives

• Thus
• And hence

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More Derivatives

• Thus

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Approximation

• Or b 1
• Thus, for small gambles, p is less than s if s is
  convex

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Approximation with Multiplicative Errors

• With multiplicative errors, approximation similar

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Risk Aversion

• Assume constant absolute risk aversion to
  eliminate wealth effects
• Parameter ?
• Problem

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Small Variance Approximation

• Using the same methods

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Conclusions

• Erroneous measurements common
• Radar guns
• Theft
• Emissions
• Proved existence of adjustments
• If and only if measurement distinguishes
  behavioral strategies

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Conclusions, Continued

• Offer approximation
• valid for small additive measurement error
• Exact for quadratic s.
• Not necessarily non-negative.

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Penalties for Drunk Boating
Redacted to preserve fair use