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Part D-II Part D-II

The Economics of Tort Law The Economics of Tort Law

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Objectives Objectives

The Social Cost of AccidentsThe Social Cost of Accidents

A Formal ModelA Formal Model

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The Social Cost of AccidentsThe Social Cost of Accidents

The model:The model:

pp is the probability of an accident (bad event) is the probability of an accident (bad event) occurringoccurring

xx is the amount of precaution takenis the amount of precaution taken

p = p(x) p = p(x) where where ΔΔp / p / ΔΔx x < 0 < 0

(p decreases as x increases - the more precaution taken, (p decreases as x increases - the more precaution taken, the lower the probability of an accident)the lower the probability of an accident)

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A picture of the relationship between the A picture of the relationship between the probability of an probability of an accidentaccident and the amount of and the amount of precautionprecaution - p(x)- p(x)

Probability of an accidentProbability of an accidentp(x)p(x)

xPrecautionPrecaution0

1.0

p(x)

Note that this is a probability and must be between 0 and 1 on the vertical axis

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AA is the monetary harm from an accident (medical costs, lost is the monetary harm from an accident (medical costs, lost income, property damage, etc.) so that,income, property damage, etc.) so that,

p(x)Ap(x)A is the is the expected cost of harmexpected cost of harm – dollar value of – dollar value of expectedexpected

harm harm

note that the note that the expected cost of harmexpected cost of harm, p(x)A, decreases as x , p(x)A, decreases as x increases even though A is fixed (since increases even though A is fixed (since ΔΔp(x)Ap(x)A / / ΔΔx x < 0) < 0) because because ΔΔp(x)p(x) / / ΔΔx x < 0 < 0

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A picture of the relationship between the A picture of the relationship between the expected cost of expected cost of harmharm and the amount of and the amount of precautionprecaution

- p(x)A- p(x)A

Expected cost of harmExpected cost of harmp(x)A

xPrecautionPrecaution0

$

p(x)Ap(x)A

Same basic shape as p(x) only shifted by the

amount of A

Note the $ on the vertical axis

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ww is the per unit cost of precaution (time,is the per unit cost of precaution (time, money, convenience, etc.) measured in $’smoney, convenience, etc.) measured in $’s

note we assume that w is fixed, independently of the note we assume that w is fixed, independently of the amount of precaution taken x – we get a straight line as amount of precaution taken x – we get a straight line as a result (not a necessary assumption) a result (not a necessary assumption)

wxwx the total expenditure on precaution the total expenditure on precaution

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A picture of the relationship between the amount of A picture of the relationship between the amount of precautionprecaution taken and the taken and the total cost of precautiontotal cost of precaution

- wx- wx

Total cost of precautionTotal cost of precautionwxwx

xPrecautionPrecaution0

$

wxwxNote the $

on the vertical axis

Straight line because w is

assumed fixed

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In this model there are two types of costs to be considered:In this model there are two types of costs to be considered:

Cost of precautionCost of precaution [wx] [wx]and and

Expected cost of harmExpected cost of harm [p(x)A] [p(x)A]

The Social Costs of accidents is given by the sum of the twoThe Social Costs of accidents is given by the sum of the two

SC = wx + p(x)A SC = wx + p(x)A is the is the expected social costexpected social cost of accidents of accidentsconsidering both the cost of an accident considering both the cost of an accident and the cost of and the cost of trying to avoid the trying to avoid the accident accident

Why SOCIAL COSTS? Because the costs to all members of Why SOCIAL COSTS? Because the costs to all members of society are included –potential victims society are included –potential victims and potential and potential injurersinjurers

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A picture of the relationship between the A picture of the relationship between the social cost of social cost of accidentsaccidents and the amount of and the amount of precautionprecaution

SC = wx + p(x)ASC = wx + p(x)A

xxPrecautionPrecaution0

$$

p(x)Ap(x)A

wxwx

SC = wx +p(x)ASC = wx +p(x)A

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The social cost curve is the vertical sum of the wx and p(x)A The social cost curve is the vertical sum of the wx and p(x)A curvescurves

This curve must be U-shaped because wx is positively sloped This curve must be U-shaped because wx is positively sloped and p(x)A is negatively sloped and p(x)A is negatively sloped andand they cross they cross

Therefore some minimum level of social costs must exist, Therefore some minimum level of social costs must exist, denoted x*.denoted x*.

x* is the level of precaution that minimizes the expected x* is the level of precaution that minimizes the expected social costs. x* is the social costs. x* is the efficientefficient level of precaution level of precaution

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A picture of the relationship between the A picture of the relationship between the social cost of social cost of accidentsaccidents and the amount of and the amount of precautionprecaution

SC = wx + p(x)ASC = wx + p(x)A

xxPrecautionPrecaution0

$$

p(x)Ap(x)A

wxwx

x*

SC = wx +p(x)ASC = wx +p(x)A

xx* * is the ‘efficient’ is the ‘efficient’ level of precautionlevel of precaution

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Some Math – an asideSome Math – an aside

Knowing that SC = wx + p(x)A, how do we determine x*?Knowing that SC = wx + p(x)A, how do we determine x*?

Taking the derivative of SC = wx + p(x)A with respect to x Taking the derivative of SC = wx + p(x)A with respect to x and setting it equal to zero yields:and setting it equal to zero yields:

w + p’A = 0 w + p’A = 0 or or w = -p’(x*)A w = -p’(x*)A

where p’(x*) is the derivative of p(x) at the SC minimizing where p’(x*) is the derivative of p(x) at the SC minimizing level of x, [x*]. level of x, [x*].

OPTIONAL STUFF!OPTIONAL STUFF!

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w is the w is the marginal social costsmarginal social costs of precaution - the cost of one of precaution - the cost of one more, or one less, unit of precaution (x)more, or one less, unit of precaution (x)

-p’(x*)A is the -p’(x*)A is the marginal social benefitmarginal social benefit associated with one associated with one more, or one less, unit of precaution evaluated at x* more, or one less, unit of precaution evaluated at x*

The The marginal social benefitmarginal social benefit being the decrease in the being the decrease in the expected costs of harm resulting from the last unit of expected costs of harm resulting from the last unit of precaution purchased precaution purchased

Go back to the numerical example (Ms. Jones)Go back to the numerical example (Ms. Jones)

In plain EnglishIn plain English

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The social cost of the last unit of precaution (the marginal The social cost of the last unit of precaution (the marginal social cost) should just equal the increase in social benefit social cost) should just equal the increase in social benefit (marginal social benefit) that it produces. Why?(marginal social benefit) that it produces. Why?

If social cost of the last unit of precaution < social benefit it If social cost of the last unit of precaution < social benefit it produces, then it has a net benefit to society. produces, then it has a net benefit to society.

If social cost of the last unit of precaution > social benefit it If social cost of the last unit of precaution > social benefit it produces, then it is a waste.produces, then it is a waste.

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A change in the underlying riskiness (probability function) of an A change in the underlying riskiness (probability function) of an accident accident p(…)p(…)

xxPrecautionPrecaution0

$$

p(x)Ap(x)A

wxwx

x*

SC = wx +p(x)ASC = wx +p(x)A

xx* * the ‘efficient’ level of precaution increases the ‘efficient’ level of precaution increases as the underlying riskiness increasesas the underlying riskiness increases

x*

p(p(xx))AA

SC = wx +SC = wx +p(p(xx))AA

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A change in the cost of an accident AA change in the cost of an accident A

xxPrecautionPrecaution0

$$

p(x)Ap(x)A

wxwx

x*

SC = wx +p(x)ASC = wx +p(x)A

xx* * the ‘efficient’ level of the ‘efficient’ level of precaution increase as A increaseprecaution increase as A increase

x*

p(x)p(x)AA

SC = wx +p(x)SC = wx +p(x)AA

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A change in the cost of an accident cost per unit of precaution wA change in the cost of an accident cost per unit of precaution w

xxPrecautionPrecaution0

$$

p(x)Ap(x)A

wxwx

x*

SC = wx +p(x)ASC = wx +p(x)A

xx* * the ‘efficient’ level of precaution the ‘efficient’ level of precaution decreases as the per unit cost of decreases as the per unit cost of precaution increasesprecaution increases

x*

SC = SC = wwx +p(x)Ax +p(x)A

wwxx

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Potential Victim – Potential InjurerPotential Victim – Potential Injurer

So far we have not indicated who is taking the precaution, the So far we have not indicated who is taking the precaution, the potential injurerpotential injurer or the or the potential victimpotential victim. .

It could be either, or more likely, both (ie. driving It could be either, or more likely, both (ie. driving safely/wearing a seat belt). safely/wearing a seat belt).

We say We say potentialpotential because we are considering the possibility of because we are considering the possibility of a future accident – you can only take precaution in a future accident – you can only take precaution in advance. advance.

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Internalizing externalitiesInternalizing externalities

Imposing risks on others is like blowing smoke in their face – it Imposing risks on others is like blowing smoke in their face – it is an external cost to those around you.is an external cost to those around you.

We will use the above model to assess the incentive effects of We will use the above model to assess the incentive effects of various aspects of tort law.various aspects of tort law.

An incentive will be ‘efficient’ if the decision maker An incentive will be ‘efficient’ if the decision maker internalizesinternalizes the marginal costs and marginal benefits of her the marginal costs and marginal benefits of her actions. actions.

We will want the decision makers to act so as ensure that x* is We will want the decision makers to act so as ensure that x* is the amount of precaution taken.the amount of precaution taken.

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A Generalization of the Efficient A Generalization of the Efficient Precaution ModelPrecaution Model

We can generalize the above story and allow for precaution by We can generalize the above story and allow for precaution by both the potential injurer and the potential victim. both the potential injurer and the potential victim.

Consider, Consider, xxvv the amount of precaution taken by the potential victim the amount of precaution taken by the potential victim wwvv the cost per unit of precaution taken by the potential the cost per unit of precaution taken by the potential

victim victim

xxii the amount of precaution taken by the potential injurer the amount of precaution taken by the potential injurer wwiithe cost per unit of precaution taken by the potential the cost per unit of precaution taken by the potential injurer injurer

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Now p(xNow p(xvv, x, xii) is the probability of an accident ) is the probability of an accident

where p’ < 0 (p is a decreasing function of both xwhere p’ < 0 (p is a decreasing function of both xvv and x and xii))

p(xp(xvv, x, xii)A)A is the expected dollar value of harm faced by is the expected dollar value of harm faced by the potential victim. Note that we are now the potential victim. Note that we are now

considering the case in which both the considering the case in which both the potential injurerpotential injurer and and the the potential victimpotential victim can take precautions that reduce the can take precautions that reduce the probability of accidents – xprobability of accidents – xv v , x, xii

So that the social costs of accidents is given by:So that the social costs of accidents is given by:

SC = wSC = wvvxxvv + w + wiixxii + p(x + p(xvv, x, xii)A)A

Where’s the picture? It is three dimensionsWhere’s the picture? It is three dimensions