Optimal safety standards under moral hazard when accident prevention is a function of both firm - and worker efforts.

Draft paper 13.03.2013

Optimal safety standards under moral hazard when accident prevention is a function of both firm - and worker efforts.

Abstract

A double principle-agent model is developed to analyze optimal safety regulation when accident risks are affected by decisions made by the firm as well as the firm’s employees (workers). The regulator perfectly observes the safety investments of the firm, but can not observe worker safety decisions, while the firm imperfectly observes the worker safety decisions. It is shown, in the case of no regulation, that the first-best is unattainable unless third-party accident costs are absent and the firm uses performance payments. The optimal regulatory standard produces a sub-optimal solution and is shown to depend on safety effort costs, the accident technology, the distribution of accident costs across firms, workers and third-parties, and the firm’s ability to observe worker safety efforts.

Key words: Safety regulation; Standards; Incentive contracts, Ex-ante regulation.

JEL Classifications: D62, D82, K20, L51.

1.  INTRODUCTION  

Standards define specific requirements (minimal or maximal) with respect to inputs, processes or outputs (e.g. pollution emissions, quality standards, safety standards). A precondition for using standards (ex-ante regulation) is that prevention activities are observable, at least to some degree, for the regulator. However, in many situations, this may not be the case. One example would be for organizations where both managers and employees play important roles in preventing accidents.

Managers invest into organizational routines, guidelines, adequate staff training, and safety equipment while their employees utilize the same systems when making their safety decisions. Firm safety investments are often easier to observe ex-ante for a regulator compared with the day-by-day decisions made by the firm’s employees. On the same time the firm management is in a better position, than a regulator, in observe ring the same day-by-decisions. The above features seem relevant for many organizations being concerned with safety reduction activities such as hospitals, airlines, nuclear plants etc.

Situations where several agents within the same organization can influence safety, where some have delegated authority to others, and, where a regulator s’ ability to observe safety efforts vary across agents, raise interesting questions. First, when a regulator observes firm safety investments, but not employee safety decisions, can the use of standards produce first-best outcomes? Second, to what extent does the optimal standard depend on the nature of the contractual relationship between the firm and its’ employees? Third, should optimal standards depend on accident prevention technologies and the distribution of accident risks? These questions will be addressed in the following by employing a (double) principal-agent(s) framework with moral hazard where accident risks depend on the actions of decision-makers being in a contractual relationship. The regulator is assumed to perfectly observe firm safety investments, but can not observe worker safety care, while the firm imperfectly observes worker safety care.

This work extends former works on optimal safety standards and quality standards. Early works were concerned with establishing optimality by equating the marginal benefits from standards with the marginal costs of achieving it (the standard rule). Later works, predominantly concerned with environmental regulation, extended this approach by studying optimal safety standards when enforcement is incomplete.

Viscusi and Zeckhauser (1979) analyze imperfect (exogenous)

compliance. Endogenous enforcement policies (inspection rates and fines) are studied by Neilson and Kim (2001), Arguedas and Hamoudi (2004), Malik (2007) and Arguedas (2008). Other works on standards include imperfect enforcement due to limited enforcement resources (Jones 1989), optimal                                                                                                                      

1

Some works consider optimal enforcement policy for exogenous standards (see e.g., Harford, Gavin and Keeler

and Keeler).

enforcement for different penalty functions (Veljanovski, 1984), and optimal enforcement for various regulator motivations (Garvie and Keeler 1988; Russel 1990; Jones and Scotchmer 1990). Heyes (2002) provide a comprehensive survey on standards and enforcement issues in an environmental context. A somewhat different approach is presented by Leland (1979) studying the licensing of professionals as a minimum quality standard of competence (screening device).

Other branches of the literature are concerned with comparing the efficiency properties of liability and standards [see e.g.

Shavell (1984), Bohm and Russel (1985), Rose-Ackerman (1991), Cropper and Oates (1992) and Schmitz (2000)], and the role influential activities (rent-seeking) may have on obtaining more lenient standards [see e.g. Buchanan and Tullock, 1975).  

The remainder of the paper is organized as follows. In the next section, we present the model together with the first-best solution acting as a reference case to the subsequent analyses. In section 3; we analyze the case of absent regulation (laissez faire), while in section 4 optimal regulatory standards are considered. Section 5 concludes.

2. THE MODEL AND THE FIRST-BEST SOLUTION

Consider a three-agent model consisting of the management of a firm (hereafter denoted the firm), a worker, and, a regulator. All three agents are risk-neutral and both the firm and the worker can influence the probability of an accident. Thus, the accident probability, P(E,e), becomes a function of the efforts of both the firm and the worker, where E denotes firm safety investments and e worker safety care. Assumptions made about the accident probability functions are

;

( A.1)

(A.2)

(A.3)

The cross partial derivative of the accident probability function can be positive, negative or zero.

means that more careful worker behavior is less effective in reducing the                                                                                                                      

2

A main finding is that, if a professional group is allowed to define own standards (self-regulation), standards will be set too high or too low relatively to what is socially desirable.

3

 First and second order derivatives are in the following denoted as follows: ,

etc.

accident probability, the more careful the firm behaves. means that more careful worker behavior is more effective in reducing the accident probability, the less careful the firm behaves. The expected social cost associated with an accident (D) is the sum of the accident costs inflicted upon the firm ( ) accident costs inflicted upon the worker ( and accident costs inflicted upon third-parties ( ) where:

where (A.4)

The worker is assumed to earn his/her expected reservation utility of , where expected utility, U, is an additive function of expected wage-income, , subtracted expected worker accident costs,

, and worker care costs, ke. Thus, expected utility becomes;

where (1)

From (1) we observe that the firm investment level, E, affects the utility of the worker via the accident probability function. Worker care (e) is non-verifiable for the firm, however, following, Holmstrøm (1985), is verifiable , where and . Here linear wage contracts are considered and since is verifiable, the firm decides on a fixed wage component (A) and an incentive component ( ). The expression for worker income becomes;

where ,

which again implies the following expression for expected income;

The firm is assumed to have a fixed income R, and the net revenues is defined as the difference between the budget and the sum of expected firm accident costs, , expected wage expenses,

, and safety investment costs, KE. The objective function of the firm is;

where (2)

The objective function presented in (2) is assumed strictly positive for all values of E and e considered in the forthcoming analyses.

4

This specification of the firm objective function could be revised. An alternative specification would be to let

the firm minimize for R=0, where now is the expected firm costs again being the sum of wage

expenses, accident costs, and safety costs. Such a specification will not change any of the forthcoming findings.

The social planner (regulator), if being perfectly informed, minimizes expected social costs, being the sum of the expected social accident costs, P(E,e)D, and total safety efforts costs, Ke+ke, which yields the following social problem;

(3)

Solving (3) defines the first-best effort levels. From A1-A2, given strictly positive values of K and k , interior solutions follow, and the equation system becomes (superscripts FB will in the following refer to the first-best safety effort levels);

(4a)

= K (4b)

The first-best conditions presented in (4ab) coincide with the standard rules identified in the accident literature. For each safety effort variable, the expected marginal reduction in society’s accident costs is to equal the marginal safety effort cost.

3. THE CASE OF ABSENT REGULATION

In the following we study a model without regulation with informational asymmetry between the firm and the firms ‘workers with respect to e (moral hazard). This relationship is analyzed as a two-stage game where the firm first decides on both investments (E) and the contract (A and b). In stage two of the game, the worker, for given levels of E, A and b, decides on care (e). To solve the problem we apply backward induction. The problem of the worker follows by inserting (2) into (1);

(5)

The first-order condition becomes;

, (6)

5

The minimization problem in (3) corresponds to maximizing the sum of worker utility and the firms’ pay-offs subtracting expected third-party accident costs.

6

The second-order condition for problem (4) and all subsequent problems are presented in the appendix.

7

We will in the following only denote the equilibrium values of the endogenous variables when the same values

are used to compare equilibria across models.

since from A1-A3, (6) can not be strictly negative. We observe that (6) is strictly positive for causing e to approach infinity, which again yields the following condition;

(7) For , (6) is binding implying that;

, (8)

The optimal choice of worker care, defined by (8), is determined by the equality between the expected marginal reduction in the worker-related accidents costs and the difference between marginal costs and the expected marginal income. Eq. (8) defines worker care as a function of E and b (the workers’

response function) and can be expressed as;

(9) where

(10a)

(10b)

Worker care increases with b (see 10a) but increases, decreases or is independent of E, depending on the sign of the cross partial derivative of the accident probability function (see 10b). According to Bulow et al. (1985) we say that e and E are; strategic substitutes if , strategic complements if

, and strategic independent variables if .

Now, consider the first stage of the game where the firm, given the workers’ response function, is to decide on investments and the contract. By inserting (2) into a binding version of (1), and inserting this expression again into (3), we arrive at;

(11)

We observe from (11) that the accident costs inflicted upon the worker ( ) is taken into account by

the firm. This property follows since, for a binding participation constraint, a higher for workers

must be compensated financially by the firm. The conditions that describe the optimal choices of an

unregulated firm now become;

(12a)

(12b)

From A1-A2 it follows that (12a) is strictly positive. For , (7) becomes strictly positive and e approaches infinity, thus (12a) must be strictly negative. However, this result can not be optimal since the marginal benefit is less than the marginal cost when e approaches infinity. Thus in optimum both (12a) and (7) must be binding implying that (12a) can be written as follows;

(13)

The optimal contract is now derived by inserting for (9) into (13) and solving with respect to b. This procedure gives (superscripts AR refer to the case of absent regulation given performance payments);

, (14)

and since .

The optimal size of the incentive component is positive and proportional to k with a factor equal to the ratio between the accident costs inflicted upon the firm and the sum of the accident costs being

inflicted upon both the firm and the worker. The optimal size increases with lower values of and higher values of The external effect that worker care (e) has on the firms’ net revenues is observed to be internalized by using performance payments (b>0). The use of performance payments is optimal if while the use of a fixed wage contract is optimal if . Furthermore it is observed that the level of E has no impact on the design of the contract (see 14). From (13) it follows that the first term of (12b) disappears, and from A1-A2, it follows that (12b) is binding, which again means that (12b) can be written as follows;

(15) By inserting (14) into (8), and the derived expression again into (15), and by rearranging, we arrive at the following system of equations describing the optimal effort levels (AR);

(16a)

(16b)

By comparing (16ab) with the first-best case (5ab), we observe that the right hand sides of (16ab) are inflated when there is third-party accident costs ( ) , confirming that absent regulation yields effort levels that deviate from the first-best levels. If E and e are strategic substitutes or

strategically independent variables , both effort levels will be lower than the first-best levels ( and ). If E and e are strategic complements

, we can not rule out the possibility of one of the effort levels being higher than the corresponding first-best level. However, in situations without third-party accidents costs ( ), the first-best is achieved. This finding follows because the firm now internalizes worker accident costs via the participation constraint and on the same time is able to steer worker care by using performance payments.

Fixed wage contracts (see 14) are only optimal in the absence of firm accident costs ( ), however, this conclusion may change if introducing monitoring costs. Now, assume a fixed cost (C ) associated with measuring in connection with using performance payments (b>0). Now, for , a firm must compare the pay-off given optimal performance payments ( ) with the pay-off for a fixed wage contract ( ) (superscript ar refers to the case of absent regulation given a fixed wage contract).

Define as . Given sufficiently high monitoring costs, , a firm

will choose a fixed wage contract. Optimal firm behavior in this case follows by inserting for into (9) which yields a worker response function equal to (see 10a). The firms’ maximization problem now becomes;

, (17)

The first-order condition for this problem is;

By applying the same procedures as above, we arrive at the following equation system describing optimal effort levels for absent regulation given a fixed wage contract (ar);

(18a)

(18b)

If (18ab) is compared with (16ab), two deviations are observed. First, accident costs inflicted upon the

firm are no longer part of the right hand side (see 18a). This is because worker care, for a fixed wage,

can not be steered by the firm as was the case when using performance payments (see 16a).

Monitoring costs have made the firm to decide not to induce the worker to internalize own external effects. Second, a new term (the adjustment term), being the product of the first derivative of the worker response function and a constant , appears on the right hand side (see 18b). The adjustment term reflects that the firm, when deciding on investments, takes into account the indirect effect E has on e. Such considerations were absent in (16b) because the firms’ desired level of worker care was realized via performance payments. Consequently, an adjustment shows up when the use of performance payments is non-optimal for a firm. We also observe that (18ab) coincide with (16ab) for the case where firm accident costs are absent ( ), thus the optimal effort levels become identical across the two cases.

If comparing the effort levels defined by (18ab) with the first-best effort levels, we observe that their ranking partly differ from the ranking where the model with performance payments was compared with the first-best effort levels. If E and e are strategic substitutes or strategic independent variables

, both effort levels are still lower than the first-best levels ( and ). However, if E and e are strategic complements

we can not rule out the possibility that both effort levels now are higher than the first-best levels.

4. OPTIMAL SAFETY REGULATION

A three-step model of optimal regulation is now considered. The regulator first decides on the safety standard, thereafter the firm decides on the contract ( and A), and, finally the worker decides on care (e). We abstract from enforcement problems by assuming that the regulator observes E perfectly. The problem is solved by applying backward induction. The last step of this game (workers’ care choice) coincides with the problem solved in the preceding chapter that gave the worker response function presented in (9). The second step of the game is concerned with the firm considering the following problem:

, (19)

which yields,

(20)

By inserting for (9) into (20), we arrive at the following expression for the optimal contract;

(21)

We observe that (21) is identical with the optimal contract being derived for the case of absent regulation (see 14).

In the third stage of the game, the regulator decides on the safety standard by minimizing total social costs taking the response functions of the worker (see 9) and (21) into account. This optimization problem becomes;

and (22)

Solving (22), where A1-A2 ensure an interior solution, yields the following first-order condition (superscript R now refers to optimal regulation/optimal safety standard for ;

(23)

The system that describes the optimal safety standard ( ) and the corresponding worker care level ( ) follows by inserting (21) into (8) for arriving at (24a), and by inserting (24a) into (23) for arriving at (24b);

(24a)

(24b)

First, if comparing the case of optimal regulation (24ab) with absent regulation (16ab), we observe that the conditions describing worker care (24a and 16a) are similar while the conditions that describe the optimal safety standard/investment level differ (24b and 16b). The latter finding follows (partly) from the presence of an adjustment term in (24b). In contrast to the case with absent regulation, now the adjustment term shows up also when performance payments are used which again mirrors the inability of the regulator to directly steer worker care. For this reason the regulator must also rely on an indirect mechanism - the effect the setting of E has on e.

Second, by comparing (24ab) with the first-best solution (5ab), we see that the two sets of conditions differ unless there are no third-party accident costs ( ). Consequently, for , optimal regulation yields sub-optimal effort levels. Given strategic independence,

, the adjustment term becomes zero. Now, the rule describing the

optimal safety standard (24b) coincides with the corresponding first-best rule (5b), while the rules for

worker care differ (24a and 5a) implying that and . Consequently, the first-best solution is unattainable. For , both rules (24ab) differ from the first-best rules (5ab). If , the adjustment term (see 24b) becomes strictly positive

implying that and If , the adjustment term

becomes strictly negative implying that and .

Finally, consider the case with high monitoring costs ( ), where the firm chooses a fixed wage contract also in situations where . The three-player game is now reduced to a two-player game of the following type;

s.t. (25)

Solving (25) by following the same procedures as above, yields the following equation system that describes optimal regulation ( superscript r will in the following refer to optimal regulation given a fixed wage regime for );

(26a)

(26b) First, we observe by comparing (26ab) with (24ab), that the two sets of conditions differ across contract types (performance payments or fixed wages). The difference between and

reflects the fact that the choice of worker care level, prior to the regulators’ implementation of a safety standard, will differ across the contracts. For absent regulation and the use of performance payments, the worker takes own accident risk ( ) and the firms’ accident risk into account, while, for absent regulation and a fixed wage contract, only own accident risk is considered. Consequently, the societal marginal benefit from using E to adjust e, will be higher at the margin for a fixed wage regime. For similar reasons, a regulator, ceteris paribus, will prefer performance payments to a fixed wage contract since now a higher share of society’s’ accident costs is internalized.

By comparing (26ab) with the first-best solution (5ab) we arrive at conclusions similar to the ones arrived at when comparing (24ab) with the first-best solution. First-best is unattainable for the

regulator and; (i) and (given strategic independence), (ii) and (given

strategic substitutes), and, (ii) and (given strategic complements).

5. CONCLUSIONS

This paper analyses optimal regulation in the presence of accident externalities and asymmetric information where accident probabilities are affected by two agents being in a contractual relationship.

The regulator observes a firm’s safety investments perfectly, but can not observe worker care, while the firm (imperfectly) observes the care level of the employee (the worker). All three agents are exposed to accident costs and the firm and the worker are confronted with mutual external effects since worker care affects firm net revenues and since firm safety investments affect expected worker utility.

In the absence of regulation we arrive at a sub-optimal solution that most often yields effort levels being lower than the first-best levels. One exception matters for the case where third-party accident costs are absent and the firm uses performance payments (low monitoring costs). Now, first-best is achieved since the remaining external effects are internalized by the two decision-makers. A firm internalizes worker accident risk via the participation constraint, while the worker internalizes the firm accident risk via the use of performance payments.

The optimal regulatory standard will not produce a first-best solution due to the regulators’ inability to observe worker care. The optimal regulatory standard is lower than the first-best standard when the safety effort variables are strategic substitutes. If the safety effort variables are strategic complements, we can not rule out the case that the optimal regulatory standard is higher than the first-best standard.

The size of the optimal regulatory standard depends on factors such as marginal safety costs, the ratio between internalized and non-internalized accident costs, and firm observability (monitoring costs). In situations with performance payments (low fixed monitoring costs), a firm will induce workers to internalize firm accident risks while the third-party accident risk is ignored. For a fixed wage contract (high fixed monitoring costs), the firm is unable to steer worker care directly, thus both firm accident risks and third-party accident risks are now ignored by the workers. In both situations, the setting of the regulatory standard takes into account the indirect effect this choice has on safety care decision taken by the worker.

A final conclusion is that the rule that defines the optimal regulatory standard coincides with the first-

best rule when the safety effort variables are strategically independent. Now, the marginal reduction in

society’s expected accident costs is balanced with the marginal increase in safety costs (the standard

rule). However, the setting of the standard according to this rule will produce a worker care level being

lower than the first-best worker care level for both types of contracts (performance payments and fixed

wages).

APPENDIX: Second-order conditions for problems (3), (5), (11), (17), (19), (22) and (25)

The second-order condition for problem (3) is >0 , where d is the determinant of the Hesse-matrix. is fulfilled from (A1).

The second-order condition for problem (5) is , being fulfilled given A1.

The second-order condition for problem (11) is <0, where is the determinant of the Hesse-matrix implying that . Given that the third derivatives of the accident probability function are equal to zero, and by inserting for optimal worker behavior we arrive at the following expressions;

From (A1) it follows that is strictly negative and from (A3) it follows that is strictly negative.

Thus, the second-order condition is fulfilled.

The second-condition for problem (17) is . Assuming that the third derivatives of the accident probability function is zero and inserting for optimal worker behavior we arrive at the following expression;

, which from (A3) is strictly negative.

The second-order condition for problem (19) is where . From (A1) it follows that this condition is fulfilled.

The second-order condition for problem (22) and problem (25) is . Assuming that the third derivatives of the accident probability function are zero and inserting for optimal worker behavior yields being strictly positive from (A3).

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