On the Role of Sales Taxes for Efficient Compensation of Property Loss Under Strict Liability

2.1 Potential Victim

The well-being of the potential victim can be described by the Cobb–Douglas utility function^[2]

where x is the amount of some composite consumption good and y is the quality level of the good prone to be lost in an accident. The composite good stands for a basket of goods including all commodities other than the good at risk of an accident which the potential victim consumes. We assume that the quantity of the good at risk of an accident is fixed at one so that only the quality of this good is variable. For example, we may imagine that the potential victim surely needs one car and that y represents the car’s quality.

With both products being sold by competitive firms, the goods’ net prices equal their social (resource) costs and amount to p per unit of good x and 1 for each quality unit of good y. A value-added sales tax is imposed on both goods at rate t. The potential victim has exogenous income m and a good of quality y ̄ . The idea of the endowment with a good of quality y ̄ is that it was purchased at some previous point in time and represents the property value that would be lost in an accident (e.g. the potential victim’s car). The quality level y ̄ may no longer constitute the individual’s optimal quality level, for example, because the quality level y ̄ was purchased when the potential victim was still subject to uncertainty regarding income m. However, due to the frictions created by the sales, tax the potential victim might still prefer keeping the good with quality y ̄ to selling the good (receiving its net value) and replacing it with a good of different quality (paying sales taxes). In the following, we focus on this case in which the potential victim prefers keeping y ̄ and spending m on good x in the case without accident; that is, the case in which y ̄ still is reasonable given income m. Given the Cobb–Douglas utility function, this is the case as long as the resulting expenditures for composite good which equal monetary income m are not to different from the gross value y ̄ ( 1 + t ) of the other consumption good. In consequence, this means that the potential victim only chooses the quality level of the good anew after the property was lost in an accident.

More specifically, the potential victim prefers keeping the quality level when the income lies in some range:

The intuition for the existence of an upper and a lower bound for income m is that the potential victim would want to correct the quality level y either if the current quality level turns out too low given high income m or too high given low income m. The upper and the lower bound on income stem from the fact that, when a Cobb–Douglas utility function can describe the decision-maker’s well-being, optimal consumption results from sharing the budget equally on the different goods consumed. When the income level is either high or low, the levels of x and y will be too imbalanced, meaning adjusting y creates a utility gain. Note that the tax level drives the interval width. In the following analysis, we assume that the income level is from this interval.

2.2 Potential Injurer

The risk-neutral injurer can implement care s ≥ 0 in order to influence the accident probability π(s), where 0 < π < 1 and π′ < 0 < π″. The potential injurer’s costs of care amount to C(s) = s(1 + αt); that is, we assume that a share 0 ≤ α ≤ 1 represents care measures that are monetary and thus also subject to the sales tax.^[3] The injurer minimizes the sum of expected liability and care costs and spends the remaining income on some consumption goods.

2.3 Damages

In the event of an accident, the injurer is held strictly liable for the damages amounting to D. Damages always include the net value of property destroyed y ̄ . Concerning the inclusion of sales taxes into the damages awards, we compare four different regimes:

1. Unconditional compensation of only the market value of the lost property excluding sales taxes: D ( a ) = y ̄

2. Unconditional compensation of the market value and the sales tax required to replace the good with the same quality: D ( b ) = y ̄ ( 1 + t )

3. Compensation of the sales tax conditional on the victim repurchasing the same quality level as before:

   D ( c ) = y ̄ ( 1 + t )    if  y = y ̄ y ̄    otherwise .

4. Conditional compensation of sales taxes to the extent to which it is incurred but capped at y ̄ t : D ( d ) = min { y ̄ + y t , y ̄ ( 1 + t ) }

Scenario (d) represents Section 249 of the German Civil Code, whereas the other scenarios represent possible departures.

2.4 Tax Revenue

Both potential victim and injurer spend their entire exogenous income on some goods. Sales tax revenue is therefore independent of the damages award and of whether an accident occurs.^[4]

2.5 Timing

The injurer chooses care s, and Nature determines whether or not an accident occurs. In the event of an accident, the damages award is transferred from the injurer to the victim. Finally, the (potential) victim and the injurer make consumption choices.

3.1 Victim’s Consumption and Utility

In the no-accident state, the potential victim spends m on good x sold at a total price p(1 + t) and obtains utility

In the accident state, the victim uses his income m and the compensatory transfer to decide on the consumption quantity x and the quality level y. For the four different damages regimes considered, we obtain the following:

3.2 Injurer’s Care Incentives

Given the victim’s post-accident decisions, the level of damages in the event of an accident amount to

where we use how victims will behave in Regimes (c) and (d), which the injurer is also assumed to anticipate.

The injurer chooses care to minimize the sum of his expected liability and care costs:

To do so, the injurer chooses care to equate marginal costs and marginal savings in expected compensatory transfers

which, using (5) and the curvature of π, gives rise to the following ranking of injurer care levels across regimes

Compensatory transfers after an accident and the induced care level are lowest in Regime (a) and maximal in Regimes (b) and (c).

3.3 Welfare Comparison of Damages Regimes

Our description of damages regimes identified regime-dependent victim utility and injurer care levels. To assess the welfare implications of the different regimes, we must (i) transform the parties’ utilities to make them comparable and (ii) account for the difference between private and social costs (the former include sales taxes while the latter are relevant for welfare). With sales tax revenue constant across regimes, we can focus on effects on the injurer and the victim. To achieve these objectives, we use a measure of (real) social resource expenditures where one unit of quality of the good y serves as our numeraire (since it has a social cost of one).

In Regime (i), the injurer spends s ⁽ⁱ⁾(1 + αt) on care but s ⁽ⁱ⁾ αt constitutes a tax payment and thus no social cost. The social care cost amount to s ⁽ⁱ⁾. In the event of an accident, the injurer pays damages equal to D ⁽ⁱ⁾. The injurer thereby loses real income in the amount of D ⁽ⁱ⁾/(1 + t) since he would have paid sales taxes equal to tD ⁽ⁱ⁾/(1 + t) on any other taxed expenditure possible for him without damage payment. Taken together, in Regime (i), the injurer incurs a social cost amounting to

Next, we translate the change in victim utility into a real resource cost. For this, we use a concept similar to the expenditure function.^[6] Again, we account for the difference between private expenditures and social costs created by sales taxes, and focus on the latter. Fundamentally, given the utility function U and since good x has a resource cost of p whereas a unit of quality y a resource cost of one, optimal consumption levels ought to be x = R/(2p) and y = R/2 for any level of resources R. Maximum utility amounts to U = R ²/(4p) and, consequently, (minimum) resource expenditures to achieve the utility level U are given by

In an accident, the victim incurs a social cost amounting to the difference between the utility level U y ̄ and U ⁽ⁱ⁾. We express this utility difference by the corresponding difference between R y ̄ and R ⁽ⁱ⁾ to have comparability with the injurer’s social cost. Accordingly, expected social costs incurred by the victim are given by

such that the total expected social costs of accidents and care measures in regime (i) amount to

When applied to the victim’s utility levels in the no-accident state and the different regimes, our approach leads to

and

Using these resource levels, we obtain the following social costs:

and^[7]

First, consider the different levels of social harm in the event of an accident. The social harm is equal in Regimes (a) and (b) and, with

lower than in Regime (c) and, if m L ≤ m < y ̄ ( 1 + t ) , lower than in Regime (d). This is due to the distortion in the victim’s consumption choices in the latter two regimes where, from the victim’s point of view, sales taxes differ for the two goods. This allows us to conclude that Regime (b) dominates Regime (c) in terms of welfare, given that the induced injurer’s care levels are the same in the two regimes, see (7). Similarly, Regime (b) weakly dominates Regime (d) for y ̄ ≤ m ≤ m H .

Next, we turn to the optimality of care decisions in the remaining Regimes (a), (b), and (d), where the latter is considered for m L ≤ m < y ̄ as it is weakly dominated otherwise. With SH denoting social harm in the event of an accident, efficient safety is characterized by

whereas injurer care is determined by (6). The levels of regime-dependent social harm in the event of an accident are ranked as follows:

In other words, the damages award exceeds social harm in all regimes. This implies that care will be inefficient unless the difference in social and private marginal care costs, 1 compared to 1 + αt, perfectly balances this difference between social harm and private damages.

When care is completely non-monetary (i.e. when α = 0) and therefore not subject to the sales tax, the injurer’s care incentives are always excessive. Social and private marginal costs of care are perfectly aligned, but marginal personal benefits of care surpass marginal social benefits. In this case, Regime (a) outperforms Regimes (b) and (d), given the lowest level of social harm and a minor upward distortion in care incentives.

The private marginal costs of care increase in all regimes with the share of monetary precautions that require tax payments. Whereas care incentives remain excessive in Regimes (b) and (d), the injurer may choose lower than efficient care in Regime (a). For high values of α, Regime (a) may be inferior to one of the other regimes.

In summary, we find:

The result is interesting in that it establishes that the policy recommendation for the sales tax case may contrast with the policy recommendation for the income tax case in the event of lost earnings. Whereas the income tax constitutes a social loss due to lower production and should be compensated to instill efficient care incentives (Shavell 1987), a property loss redirects spending, and sales taxes incurred when replacing lost property do not constitute social harm.