Firm’s Protection against Disasters: Are Investment and ... · Technological disasters are catastrophic events which result in contamination of the environment and lead to ﬁnancial,

Firm’s Protection against Disasters: are investment anD insurance substitutes or comPlements?

Documents de travail GREDEG GREDEG Working Papers Series

Giuseppe AttanasiLaura ConcinaCaroline KamateValentina Rotondi

GREDEG WP No. 2018-24https://ideas.repec.org/s/gre/wpaper.html

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Firm’s protection against disasters:

Are investment and insurance

substitutes or complements?

Giuseppe Attanasi⇤, Laura Concina†,

Caroline Kamate‡, Valentina Rotondi§

December 20, 2018

Abstract

We use a controlled laboratory experiment to study firm’s protectionagainst potential technological damages. The probability of a catastrophicevent is known, and the firm’s costly investment in safety reduces it. Thefirm can also buy an insurance with full or partial refund against the conse-quences of the catastrophic event, which ultimately reduces the variance ofthe firm’s investment-in-safety lottery. The firm makes these two choices si-multaneously, after observing the insurance contract proposed by an insurerwho chooses this contract within a set of premium-deductible combinations.

We parametrize the insurer-firm game such that: (i) a risk-neutral insurermaximizes his expected profit by o↵ering an actuarially fair contract withfull insurance; (ii) a risk-neutral firm is indi↵erent between investing in safetyand accepting a fair insurance contract.

We aim at understanding whether investment in safety and insurance aresubstitutes or complements in the firm’s risk management of catastrophicevents. In line with our predictions, the experimental results suggest thatthey are substitutes rather than complements: the firm’s investment in safetymeasures is a↵ected by the insurer’s proposed contract, the latter usually in-volving only partial insurance.

Keywords: Decision under risk; Losses; Small probabilities; Probability re-duction; Technological disasters; Insurance; Deductible.

JEL classification: D81, G22, K32, Q58.

⇤Corresponding author. GREDEG, University of Nice ([email protected]).We thank Roman Meleshko for useful research assistantship. We thank for useful dis-cussions and comments Olivier Armantier, Nikolaos Georgantzis, Yolande Hiriart, ElenaManzoni, Eric Marsden, Nicolas Treich, and the seminar participants at the 2016 Inter-national Conference on Economic and Financial Risks in Niort, and the Foundations ofUtility and Risk (FUR) 2018 Conference in York. The research leading to these resultshas received funding from the French Agence Nationale de la Recherche (ANR), undergrant ANR-17-CE03-0008-01 (project INDUCED).

†FONCSI, Toulouse‡FONCSI, Toulouse§Bocconi University, Milan

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GREDEG Working Paper No. 2018--24Revised version: December 2018

1 Introduction

In 1921 a tower silo storing 4,500 tones of a mixture of ammonium sulfateand ammonium nitrate fertilizer exploded at a BASF plant in Oppau (Ger-many) causing 500 deaths and 2,000 injuries. In 1976 the release of dioxinsin a small chemical manufacturing plant approximately 20 kilometers northof Milan (Italy), resulted in 3,000 dead pets and farm animals, 70,000 ani-mals slaughtered and several long-term health e↵ects (Bertazzi et al., 2001).The disaster led to the Seveso II Directive by the European Communitywhich imposed stronger industrial safety regulations (Zio and Aven, 2013).In 1984 in Bhopal (India) over 500,000 people were exposed to methyl iso-cyanate (MIC) gas and other chemicals, with exposure causing more than20,000 deaths and 570,000 injuries. The Bhopal disaster, also referred to asthe Bhopal gas tragedy, is currently considered the world’s worst industrialdisaster. In 2010 an explosion occurred on the Deepwater Horizon oil rigworking on the Macondo exploration well, causing 11 deaths and releasingmillions of barrels of oil into the Gulf of Mexico.

The above examples are only few of the several technological disastersoccurred in the last years. Technological disasters are catastrophic eventswhich result in contamination of the environment and lead to financial, en-vironmental and human losses. They typically contain a greater threat tohealth than to property (Cutchin et al., 2008) and directly a↵ect firms’ rev-enues by disrupting their production process, their stock (Capelle-Blancardand Laguna, 2010) and their equity prices (Brown et al., 2015). As a result,the economic implications of technological disasters are substantial and of-ten catastrophic in their consequences (Pelling et al., 2002; Capelle-Blancardand Laguna, 2010). Moreover, the damages caused are frequently not paidfor due to the insolvency of the responsible parties (Angelova et al., 2014),which points to the need to elaborate e↵ective risk management and finan-cial security instruments to cope with the failures of such brittle technologies(Marsden, 2014).

Several institutional mechanisms exist that are aimed at controlling therisks generated by hazardous activities. First of all, regulation imposes re-strictions on hazardous activities with the aim to improve safety. Second,liability regimes require firms to compensate the victims of their accidentsand, third, mandatory insurance guarantees that firms are able to compen-sate their victims when the damages exceed their ability to pay (Marsden,2014). Last, self-regulation approaches are aimed at establishing voluntarystandards of behavior in order to improve safety. These institutional mech-anisms are often simultaneously applicable, which raises the question of thepossible complementarity or interference between di↵erent instruments.

In this paper we study firm’s and insurer’s behavior when the firm can in-vest so as to protect itself against potential technological damages in a worldwhere insurance is not legally mandated. Insurance is generally viewed as away to transfer risk from small and risk-averse actors to large and risk-neutral

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ones. Essentially, it constitutes a financial safeguard against the economicconsequences of a bad event. International conventions often require firms tosubscribe to insurance in order to allow victims to obtain a rapid compen-sation. In sectors where liability insurance is not legally mandated, banksoften demand it as a conditio sine qua non for obtaining a loan, therebymaking the insurance almost compulsory. Nevertheless, according to thesigma study by Bevere et al. (2016), in 2015 the amount of financial lossesgenerated by catastrophes and not covered by insurance has been estimatedat around USD 55 billion. As a percentage of GDP, uninsured losses rosefrom 0.08% in 1976–1985 to 0.17% in 2006–2015. Although these figureshide the increasing weight of natural disasters, they still raise a general con-cern regarding under-insurance against man-made catastrophes, especially indeveloping countries.

From a policy perspective, it is therefore crucial to understand firms’ andinsurers’ behavior when confronted with the – usually small – probabilityof incurring in a disaster. More specifically, since insurance has complexe↵ects on firms’ investment in prevention, with the most well-known beingmoral hazard (La↵ont, 1995), it is crucial to understand how and when in-vestment and insurance are substitutes or complements and what mattersmore for firm’s optimal protection against technological damages. At thesame time, given that insurance is proposed by insurers, it is equally crucialto understand when they propose a contract with a high prize and low de-ductible (high insurance) and when they o↵er one with a low price and highdeductible (low insurance).

In order to answer these questions, we implement a controlled laboratoryexperiment with a sequential insurance game with two players: a firm pro-ducing dangerous materials and running the risk of incurring in an accident,and an insurer o↵ering the firm a contract to cover the potential losses.

Firm’s potential accidents are of di↵erent size (small, medium, and largeloss), and the probability of each size of loss is known and decreasing in thedamage size (largest for the small loss, and smallest for the large loss). Thefirm can make a costly investment in safety aimed at reducing the probabilityof each loss. The firm can also buy an insurance with full or partial refundof the loss. Therefore, both ex-ante safety and ex-post insurance reduce theexpected value of a loss, the former through its probability and the latterthrough its amount.

In the insurer-firm game, first the insurer decides which contract to pro-pose to the firm. After having observed the proposed insurance contract,the firm simultaneously decides whether or not to buy it and whether or notto invest in the reduction of the probability of the catastrophic events. Weparametrize the insurer-firm game such that: (i) a risk-neutral insurer max-imizes his expected profit by o↵ering an actuarially fair contract with fullinsurance; (ii) a risk-neutral firm is indi↵erent between investing in safetyand accepting an actuarially fair insurance contract.

We aim at understanding whether investment in safety and insurance

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against negative consequences are substitutes or complements in the firm’srisk management of catastrophic events. Our experimental results suggestthat investment to reduce the probability of damages and insurance againsttheir negative consequences are in general substitutes. In accordance withour theoretical predictions, investment is chosen more frequently than insur-ance. This result is independent of whether the probability of a technologicaldisaster is increasing or decreasing over time.

Investment and insurance show some complementarity only at the begin-ning of the experiment, when firms have not yet perceived that a large lossis very unlikely to occur. Substitutability between insurance and investmentis mostly driven by the fact that insurers usually o↵er contracts with partialinsurance only. This makes the insurance contract lose part of its intrin-sic value of ex-post protection instrument, and so its complementarity withex-ante protection (investment in safety).

These results, which could be generalized to several types of insuranceinvolving a premium and a deductible, are only partially in line with theliterature on safety management, which states that the firm’s investment insafety measures is not a↵ected when an insurance market is available.

The remainder of this paper is organized as follows: Section 2 presentsthe related literature. Section 3 describes the experimental design. Section4 presents our theory-driven behavioral predictions. Section 5 presents anddiscusses the results. Section 6 concludes.

2 Related Literature

This paper relates to two di↵erent strands of the literature on behavioraldecision making in risk management.

The first one concerns the sparse experimental evidence on liability rules.The seminal work in this literature is Kornhauser and Schotter (1990). Theyexperimentally tested the e�cacy of the rules of strict liability and negligencein the context of unilateral care models. Their results suggest that undernegligence rule the subjects’ choice of care levels corresponds to the predictionof the theory (no investment in safety). However this is not true for strictliability, as also shown by Dopuch and King (1992) and King and Schwartz(1999), who studied the special case of liability rules for auditors.1

Recently, Angelova et al. (2014) have compared experimentally the per-formance of liability rules for investment in safety against environmentaldisasters when third parties are harmed and cannot always be compensated.

1See also Dopuch et al. (1997), who used experimental methods to assess how di↵erentlegal regimes a↵ect the frequency and amounts of settlements in a three-person setting (oneplainti↵ and two defendants). In particular, their study explores liability rules applied tothe multi-defendant case, namely proportionate vs. joint and several liability rules. Thelearning of liability rules in a dynamic setting has been investigated by Wittman et al.(1997).

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Their design presents three di↵erent liability rules (i.e., no liability, strict lia-bility, and negligence) and compares these rules to the first best. In line withthe theory, strict liability and negligence are equally e↵ective; both performbetter than no liability; investment in safety is not sensitive to the ability ofthe firm to compensate potential victims. In contrast with theoretical pre-dictions, however, investment in safety, absent liability, is much higher andliability is much less e↵ective.

With regard to this strand of the literature, our paper focuses on thestrict liability regime – the firm is held responsible for losses irrespective ofits investment in safety –, although the injured party is the firm itself (nothird party is involved). As in Angelova et al. (2014), a firm which invests insafety halves the probability of the disaster. However, in our experiment thefirm deals with three di↵erent sizes of losses (and three di↵erent attachedprobabilities) at the same time. Furthermore, di↵erently from all above-mentioned articles, we bring into the picture insurance as an alternative orcomplementary form of protection, and an active role of the insurer, therebymoving from individual decision making to strategic choice.

This introduces the second strand of the literature, namely insurancebargaining under risk.

From a theoretical point of view, Kihlstrom and Roth (1982) studiedinsurance contracts in non-competitive markets where a risk-neutral insurerand a risk-averse individual bargain over the terms of an insurance contract.They showed that in a cooperative Nash bargaining setting the insurer’sexpected profit is higher when bargaining with a more risk-averse individual.Roth (1985) extended the discussion by analyzing the impact of risk aversionusing a non-cooperative model of multi-period bargaining. Again, it is shownthat a risk-neutral insurer’s expected profit is higher when bargaining witha more risk-averse individual (see also Viaene et al., 2002).

From an empirical point of view, several field studies report that peopleunderinsure against low-probability, high-loss events such as floods, fires,earthquakes and windstorms (see, e.g., Kunreuther et al., 1978; Kunreuther,1984). Suggested reasons for underinsurance of low-probability, high-lossevents are (see Laury et al. 2009 for a survey): lack of information (about,e.g., probability of loss, average loss amounts, load factors); transaction costs(costly e↵ort required to obtain optimal insurance); faith in the government(belief that the government will step in and provide coverage in the event of anatural disaster); decision-making heuristics (low probabilities are processedas essentially zero); insurance price being too high. Our experimental studyfocuses on the last two potential sources of underinsurance in a strategicsetting.

Results of field studies on underinsurance against low-probability, high-consequence events can be hard to interpret because of the inherent lack ofcontrol outside the lab. Indeed, Kunreuther et al. (1977) find that manypeople living in hazard areas do not even possess the knowledge of conceptsneeded to make informed decisions regarding hazard insurance purchase de-

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cisions (e.g., premiums, deductibles and subsidies). That is one of the mainreasons why laboratory experiments with objective (and commonly agreedupon) risks are called for.

Experimental evidence on the topic is largely based on three studies:Slovic et al. (1977), McClelland et al. (1993), and Laury et al. (2009).

Slovic et al. (1977) reported a carefully controlled experiment in whichsubjects filled out a questionnaire that elicited their willingness to purchaseactuarially fair insurance in up to eight di↵erent situations. The study wasnot incentivized (hypothetical payments) and losses and insurance prices werereported in terms of points rather than monetary outcomes. They found thatthe percentage of subjects purchasing insurance was relatively low (less than10 percent) when the probability of a loss was very low (and therefore theloss amount was high), and systematically increased as the probability of aloss increased.

McClelland et al. (1993), using real payments, allowed groups of eightsubjects to participate in a fifth-price auction for an insurance. They found abimodal pattern of bidding: when the probability of a loss was low, bids weremost common at zero and twice expected value. Thus, their results supportSlovic et al. (1977), although bidding behavior may not reflect insurancebuying behavior in naturally-occurring markets.2

Laury et al. (2009) reexamined the issue of underinsurance with low-probability risky hazards with an experiment that accounts for possible con-founds in prior insurance experiments. Their results are against the priorexperimental evidence, as they find that subjects buy more insurance forlower-probability events than for higher-probability events, given a constantexpected loss and load factor.

Our study aims at shading light on the issue of underinsurance for catas-trophic risk, by accounting for losses of di↵erent sizes and with di↵erentprobabilities, keeping invariant the expected value of the loss. To the best ofour knowledge, this is the first study that examines this issue in a dynamicsetting (increasing vs. decreasing risk of an accident) and by allowing forstrategic interaction between the insurer and the potentially injured party(the firm). As discussed above, the latter can also invest in safety under astrict liability regime.

2Harbaugh et al. (2002) o↵er evidence that individuals exhibit sharply di↵erent be-havior when they are o↵ered a choice between gambles (a choice between a certain loss –insurance – and a probabilistic loss) and when they are asked to submit a price they arewilling to pay for the same gamble.

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3 Experimental design

3.1 Procedures

The experiment was conducted at the LESSAC (Laboratory for Experimen-tation in Social Sciences and Consumer Behavior) of the Burgundy School ofBusiness, Dijon (France) in April 2016 and April 2017. It was a computer-ized experiment, programmed by means of the z-Tree software (Fischbacher,2007), and the instructions were in English.3

Participants were 150 1st-year undergraduates in Management, attend-ing a course of Theory of Management, taught in English, thus subjects hada good command of English language. We run five experimental sessionsin April 2016 (74 subjects) and six experimental sessions in April 2017 (76subjects), with each student participating in only one session. Each exper-imental session was run before the beginning of the course, and students ofone group did not have the chance to talk to students of another group beforethe experiment.

Average duration of a session was 65 minutes, including instructions andpayment. Payo↵s in the experiment were expressed in terms of “tokens”.Earned tokens were converted into euros at the end of the experiment ac-cording to the exchange rate 100 tokens = 1 euro. Average payment was13.5 Euros, including a 3 Euros show-up fee.

3.2 Design

3.2.1 The Insurer-Firm Game

The experiment consists of three separate phases (A, B, C), each one madeof five periods. In each phase an insurer-firm game is played. Subjects arerandomly assigned the role of insurer or firm at the beginning of the experi-ment, and maintain it until the end of the experiment. At the beginning ofeach phase, each insurer and each firm are randomly assigned to a pair: thepair matching remains unchanged during the whole phase (partner matchingacross periods). At the beginning of a new phase, a new insurer-firm pair israndomly formed (absolute stranger matching across phases).

In each period of the insurer-firm game, the subject in the role of thefirm is given an initial endowment of 1,500 tokens (15 Euros), and told thatthis is the profit from selling his/her product. As the product was producedthrough a hazardous process (e.g., the firm produced dangerous materials),the firm runs the risk of incurring in an accident, which would reduce his/herendowment in the period. Three sizes of damage are possible: small (200tokens) with probability ps, medium (500 tokens) with probability pm, andlarge (1,500 tokens) with probability pl. The damage is not certain (ps +

3The experimental instructions are provided as an electronic supplementary material ofthis paper that can be found at www.giuseppeattanasi.wixsite.com/index/working-papers.

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pm + pl < 1), and the higher the size of the damage the lower its probability,i.e., ps > pm > pl.

In each phase, both the firm and the insurer are given a Damage Estima-tion Table showing the size of the damage (loss) that the accident may causeand the objective probability at which each specific damage is generated.Table 1 shows the Damage Estimation Table in each of the three phases.Notice that the one-period expected value of the loss is essentially invariantto the phase, and corresponds to 20% of the firm’s initial endowment, i.e., 3Euros (310, 305, and 300 tokens in phase A, B, and C, respectively).

Table 1: Damage Estimation Table

Phase A

Loss Small Medium Large

Damage 200 500 1.500Probability 0.30 0.20 0.10

Phase B



Phase C



In each phase, the insurer-firm game is a two-stage game with perfectinformation, where only the insurer is active in the first stage (Insurer Stage),and only the firm is active in the second stage (Firm Stage). In period 1 ofeach phase of the experiment, both stages are implemented, with the InsurerStage played first and then followed by the Firm Stage. In the remainingfour periods, only the Firm Stage is implemented. Therefore, the insurer isonly active in period 1, while the firm is active in each of the five periods ofthe phase.

Insurer Stage In period 1 of each phase, the insurer o↵ers to the firmmatched with him/her in that phase a contract to cover the firm’s potentiallosses during the five periods of the phase. In order to represent a realisticsituation in which the insurance contract cannot be easily modified, the in-surer proposes an insurance contract only once in each phase, i.e., in period

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1. He/she is then constrained to o↵er the same contract in the remainingfour periods of the phase.The insurer chooses the proposed contract in a set of available contracts thatis represented by the Insurance Table. Table 2 shows the Insurance Table foreach of the three phases, with nine available contracts per phase.Each contract is composed of two parts: a premium – the price the firmshould pay to buy the insurance –, and a deductible – the maximum amountof loss that, although the firm buys the insurance, is not taken in charge bythe insurer and, therefore, has to be paid by the firm. Notice that, indepen-dently from the phase:

• contracts of type X include no deductible, contracts of type Y includea deductible equal to the small damage in Table 1, and those of typeZ include a deductible equal to the medium damage in Table 1;

• contracts of type j.1 (with j = X, Y, Z) are actuarially fair, with loadingof 1 on the insurance cost (expected loss of the insured firm), and thoseof type j.2 and j.3 are unfair, with loading of respectively 1.5 and 2.

Firm Stage In period each period of every phase, after having observedthe contract o↵ered by the matched insurer in period 1 of the same phase,the firm simultaneously chooses whether or not to buy that contract andwhether or not to invest in safety.Investment in safety can be made at a given cost, so as to rely on a ModifiedDamage Estimation Table, reported in Table 3 for each of the three phases.By paying the investment cost of around 150 tokens in each treatment –half of the expected loss – the firm halves the probability of each size ofdamage (compare Table 3 to Table 1). The firm’s investment decision isprivate information. Thus, at the end of the period, the insurer will neverknow whether the firm has invested in safety. We see also this assumptionas realistic enough to represent a situation where a firm tries to hide his/hersafety measures to the insurer to avoid give him/her too much information.4

At the end of the period, the computer randomly determines, for each firmindependently, if one out of three possible losses (small, medium, or high)occurs. The fact that at the end of the period a loss has occurred or notis made public information to both the firm and the insurer in each pair,whatever the firm’s investment in safety and purchase of insurance. Notethat a firm is held active in the next period independently of an accidentoccurring or not in the previous one(s).The end of the Firm Stage coincides with the end of the phase.

Insurer’s one-period payo↵ If no accident occurs at the end of a period,the insurer’s payo↵ is the premium, if the firm has bought the insurance

4For example, in real life a firm may want to hide absence of safety measures, in orderto avoid getting an insurance contract with a too high premium or a too high deductible.

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Table 2: Insurance Table

Phase A

Premium Deductible

Contract X.1 310 0Contract X.2 465 0Contract X.3 620 0Contract Y.1 190 200Contract Y.2 285 200Contract Y.3 380 200Contract Z.1 100 500Contract Z.2 150 500Contract Z.3 200 500

Phase B

Premium Deductible


Phase C

Premium Deductible


contract in that period, and zero otherwise. If one of the three accidentsoccurs at the end of a period, the insurer’s payo↵ is the minimum betweenthe size of the damage and the deductible, if the firm has bought the insurance

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Table 3: Modified Damage Estimation Table

Phase A

(Investment Cost = 155)



Phase B




Phase C




contract, and zero otherwise.5 All this holds for each period of a phase.This simplified payment structure (with respect to more complex incentiveschemes for insurers in real life situations) satisfies the condition of a positivecorrelation between the profit of the insurance company and the payo↵ of theinsurer. Indeed, we assume that the insurer works for an insurance company– in our design, the experimenter, who refunds the firm’s damage if insured –incentivizing the insurer to propose the firm a contract with a high premiumif the insurer expects no damage to occur, and with a high deductible ifhe/she expects a damage to occur, with the proposed deductible increasingin the insurer’s expectation that a damage of a larger size will occur. Such aselling strategy minimizes the expected loss of the insurance company.

Firm’s one-period payo↵ If no accident occurs at the end of a period, thefirm’s payo↵ is his/her endowment minus the insurance premium (if he/shehas bought the proposed insurance contract) and the investment cost (ifhe/she has invested in safety). If one of the three accidents occurs at the

5As an example, suppose that the insurer o↵ers the insurance contract Z.2 in period 1of phase A, with premium 150 and deductible 500 (see Table 2). Suppose that the matchedfirm buys this contract in period 1 and that a damage of small size (200) occurs at theend of period 1. Then, the insurer’s payo↵ in period 1 is 200, i.e., the minimum betweendamage of small size (200) and the deductible (500).

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end of a period, the firm’s payo↵ is his/her endowment minus the insurancepremium minus the deductible, if he/she has bought the proposed insurancecontract, and his/her endowment minus the size of the damage otherwise; ineither case, also the investment cost is substracted if the firm has investedin safety. The firm’s payo↵ is private information, in order to prevent theinsurer to know whether the firm has invested or not in safety. All this holdsfor each period of a phase.

3.2.2 Treatments

In each experimental session, subjects play the insurer-firm game describedin Section 3.2.1 for three times – i.e., for phase A, phase B, and phase C – on awithin-subject base. We have run two treatments: Increasing Probability ofDamage (74 subjects), and Decreasing Probability of Damage (76 subjects).6

In both treatments, subjects participate in the three phases sequentially,with instructions of a new phase distributed and read aloud only prior to thatphase. The treatment variable is the order of presentation of the phases:

• Increasing Probability of Damage. The three phases are presentedto subjects according to the order A - B - C. Thus, the total prob-ability of damage in the three phases is equal to, respectively, 0.60,0.77, and 0.94 (see Table 1). Therefore, although the probability ofa large damage decreases over phases, the probability that a firm willhave an accident (of any size) increases over phases. In particular,in the last phase of the experiment (phase C), the firm is almost surethat an accident will occur.

• Decreasing Probability of Damage. The three phases are pre-sented to subjects according to the reverse order C - B - A. There-fore, the probability of a large damage increases over phases, while theprobability that a firm will have an accident decreases over phases(respectively, 0.94, 0.77, and 0.60: see Table 1). In particular, in thefirst phase of the experiment (phase C), the firm is almost sure thatan accident will occur.

3.2.3 Final Questionnaire and Payment

At the end of the experiment one of the three phases and one of the fiveperiods of that phase are randomly drawn for payment from two separateenvelopes.7

6More precisely, we have run: for the Increasing Probability of Damage treatment,2 sessions in April 2016 (resp., 20 and 34 subjects), and 1 session in April 2017 (20subjects); for the Decreasing Probability of Damage, 2 sessions in April 2016 (resp., 18and 22 subjects), and 2 sessions in April 2017 (resp., 16 and 20 subjects).

7See Section 5.3 of Attanasi et al. (2014) for a discussion on the participants’ highertrust in physical rather than computerized instruments when facing lotteries for the final

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Firm’s and insurer’s payo↵s are realized and paid in cash to participants:subjects in the role of firms are paid their earnings in the randomly selectedperiod, while subjects in the role of insurers are paid the sum of their earningsin the five periods of the selected phase, since their proposed contract heldfor all periods of the phase.8

Payment is preceded by a questionnaire. Each subject is asked some stan-dard idiosyncratic features: age, gender, field of study, nationality, and pre-vious participation in an economic or psychology experiment. The question-naire also includes a question about self-assessment of risk attitudes (Dohmenet al., 2011), own and others’ selfishness, own trust on others, and others’fairness. Finally, it poses questions about own insurance decisions in realdomains (home, car, health, life, and travels).

4 Behavioral Predictions

In this section, we derive predictions for insurer’s and firm’s behavior in theinsurer-firm game for every phase where it is played. First, as in Angelovaet al. (2014), we assume that both the firm and the insurer are risk neutral.Then, we generalize our predictions to account for agents’ risk attitudes andfor the fact that in the two treatments the three insurer-firm games wereplayed sequentially with increasing (phases A, B, and C) and decreasing(phases C, B, and A) probability of damage.

Consider a firm (“she”) with revenues w. The firm is exposed to somerisky activity and she can cause damages of di↵erent magnitudes: small (hs),medium (hm) and large (hl), with hk 2 (0, w) for k = s,m, l. The probabilitythat one of these damages occurs is higher the smaller the size of the damage(ps > pm > pl), and the damage is not certain (ps + pm + pl < 1). If anaccident of magnitude hk occurs, the firm pays the amount hk to a thirdparty (strict liability rule).

The firm can invest in safety to reduce the probability that an accidentoccurs (safety measure), by paying the investment cost c. If she invests insafety, the probability of causing the damage is reduced by half. Thus, eachdamage hk occurs with probability pk/2 and the probability of an accidentbecomes (ps+ pm+ pl)/2, as shown in Table 3. As in Angelova et al. (2014),the investment choice in safety is dichotomous (invest or not) and privatelyknown to the firm.

payment of earnings in laboratory experiments. Another motivation behind this designchoice is to make subjects aware that the final (physical) random draw of the phase andperiod to be paid at the end of the experiment is independent from the (computerized)random draws of the size of the damage in each period of the experiment.

8This design choice also minimizes payo↵ di↵erences between subjects in the role offirms (per period expected payo↵ equal to 12 Euros) and subjects in the role of insurers(phase expected payo↵ between 6 and 15 Euros, according to the proposed contract, if thecontract is accepted).

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Before choosing the level of the investment, the firm can insure herselfby agreeing to sign a contract with an insurer (“he”). The insurer proposesto the firm an insurance contract composed by a premium ⇡ < hl and adeductible D 2 [0, hm]. The deductible is the maximum amount of moneythat the firm accepting an insurance contract pays if the accident occurs. Theset of contracts (⇡, D) an insurer can o↵er in the three phases is reportedin Table 2. If D = 0 (full-insurance contracts X.1 � X.3), the insured firmis refunded in full whatever the damage size. If D = hs (partial-insurancesmall-deductible contracts Y.1 � Y.3), the insured firm is refunded nothingif the damage size is hs, and hk � D otherwise, with k = m, l. If D = hm

(partial-insurance medium-deductible contracts Z.1� Z.3), the insured firmis refunded nothing if the damage size is hs or hm, and hl�D otherwise. Forsymmetry with the investment choice, we assume that also the firm’s choiceis dichotomous: she can only decide whether or not to accept the contracto↵ered by the insurer.

The insurer and the firm bargain to settle the contract. First, in theInsurer Stage, the insurer proposes one of the contracts of Table 2. Then, inthe Firm Stage, the firm decides simultaneously whether or not to accept thiscontract and whether or not to invest in safety in each of the five periods,taking into account that a damage can potentially occur at the end of eachperiod. Therefore, from period 2 onward, holding constant the insurer’s pro-posed contract, the firm makes the insurance-investment choice after havingobserved whether a damage has occurred in the previous period(s).

Let us solve the one-period insurer-firm game by backward induction.Given that the insurer is not active in periods 2-5, the one-period analysiseasily extends to the five-period case. First, we focus on the case of risk-neutral agents. Then we add heterogeneous degrees of risk aversion into thepicture.

In the Firm Stage, the one-period expected value of the damage lotteryfor the risk-neutral firm in the case of No investment in safety and Noinsurance is:

EVF (No,No) = w �X

k=s,m,l

pk · hk. (1)

The expected value of the damage lottery for the risk-neutral firm in thecase of Investment in safety and No insurance is:

EVF (Inv,No) = EVF (No,No)� c+1

2

X

k=s,m,l

pk · hk. (2)

Therefore, by comparing (1) to (2), we conclude that the firm invests insafety in the case of no insurance if and only if

c 1

2

X

k=s,m,l

pk · hk. (3)

14

By comparing the Damage Estimation Tables in Table 1, and the ModifiedDamage Estimation Tables in Table 3, it is easy to get that Eq. (3) is satisfiedwith equality for each of the three phases (the insurance cost has been setequal to half of the expected loss). This leads to the following statement:

Proposition 1. Absent insurance, a risk-neutral firm is indi↵erent betweeninvesting or not in safety.

The expected value of the damage lottery for the risk-neutral firm in thecase of No investment in safety but with Insurance is:

EVF (No, Ins) = EVF (No,No)� ⇡ +X

k=s,m,l

pk ·max{hk �D, 0}. (4)

For all actuarially fair contracts X.1, Y.1, and Z.1 in Table 2, it is ⇡ =Pk=s,m,l pk · max{hk � D, 0}. Thus, absent investment, a risk-neutral firm

is indi↵erent between accepting or not an actuarially fair contract o↵eredby the insurer (EVF (No, Ins) = EVF (No,No)). Proposition 1 states thatEVF (No,No) = EVF (Inv,No), then we can conclude that EVF (No, Ins) =EVF (Inv,No). All other contracts j.2 and j.3 with j = X, Y, Z in Table 2are actuarially unfair and therefore EVF (No, Ins) < EVF (Inv,No). Thisleads to the following statement:

Proposition 2. A risk-neutral firm is indi↵erent between investing in safetyand accepting an actuarially fair insurance contract. If o↵ered an actuariallyunfair insurance contract, she prefers to invest in safety.

Finally, the expected value of the damage lottery for the risk-neutral firmin the case of Investment in safety and Insurance is:

EVF (Inv, Ins) = EVF (No, Ins) � c +1

2

X

k=s,m,l

pk · max{hk � D, 0}. (5)

Let us focus on the actuarially fair contracts X.1, Y.1, and Z.1, forwhich ⇡ =

Pk=s,m,l pk · max{hk � D, 0} holds. For them, it also holds

EVF (No, Ins) = EVF (No,No) (see Eq. (4)). Plugging EVF (No, Ins) =EVF (No,No) into Eq. (5) and recalling that Eq. (3) is satisfied with equal-ity for all insurance contracts, we obtain

EVF (Inv, Ins) � EVF (No,No) = �1

2

X

k=s,m,l

pk · [hk � max{hk � D, 0}].

(6)

Since hk > max{hk�D, 0}, then from Eq. (6) it follows EVF (Inv, Ins) <EVF (No,No). This holds a fortiori for all other (unfair) contracts in Table 2,since EVF (Inv, Ins) is lower for these contracts. This leads to the followingstatement:

15

Proposition 3. A risk-neutral firm never invests in safety and accepts aninsurance contract at the same time.

Going backward to the first stage of the one-period insurer-firm game,the insurer anticipates that, due to the absence of complementarity betweeninsurance and investment in safety (Proposition 3), the risk-neutral firm willaccept the proposed contract only if she will not invest in safety. Further-more, the firm would only accept actuarially fair insurance contracts, dueto the substitutability between insurance and investment in safety stated inProposition 2. Recalling that the insurer gets a zero payo↵ if the firm doesnot accept the proposed contract in the second stage, the following holds fora risk-neutral insurer:

Proposition 4. A risk-neutral insurer o↵ers the firm an actuarially faircontract with full insurance against any possible loss.

Therefore, the subgame perfect equilibrium outcome of the one-periodinsurer-firm game with risk-neutral agents consists in the insurer o↵eringcontract X.1 independently of the phase, and the firm either accepting thiscontract or investing in safety. This results extends even when we take intoaccount that the contract proposed by the insurer is o↵ered to the firm forfive periods in each phase.

We now extend Propositions 1-4 to account for the firm and the insurer’srisk aversion.9 We do so by assuming that both the firm’s and insurer’spreferences are represented by Expected Utility (henceforth EU) in choiceunder risk, with u : R ! R being the corresponding utility function overthe outcomes of the damage lottery. This function is assumed to be strictlyincreasing.10

With this, we can extend Eqs. (1), (2), (4) and (5), by writing thefirm’s expected utility for each investment-insurance protection strategy, withk = s,m, l:

EUF (No,No) =

1 �

X

k

pk

!· u(w) +

X

k

pk · u(w � hk), (7)

EUF (Inv,No) =

1 �

X

k

pk2

!· u(w � c) +

X

k

pk2

· u(w � c � hk), (8)

9Subjects usually exhibit risk aversion in economic experiments (see Holt and Laury2002, and a plethora of follow-up papers).

10We assume that the functional form of the von Neumann–Morgenstern utility u(·)is independent from the role in the game, since in our experiment the roles of firm andinsurer are randomly assigned to subjects belonging to the same population.

16

EUF (No, Ins) =

1�X

k

pk

!·u(w�⇡)+

X

k

pk ·u(w�⇡�min{hk, D}),

(9)

EUF (Inv, Ins) =

1�X

k

pk2

!·u(w�⇡)+

X

k

pk2·u(w�c�⇡�min{hk, D}).

(10)

Recall that Eq. (3) holds with equality, i.e., the investment-in-safety costc =

Pk

pk2 . Substituting this into (7) and (8), it is easy to check that for

a risk-averse firm it is EUF (Inv,No) > EUF (No,No), due to the concavityof u(·), for any degree of risk aversion. Therefore, absent insurance, a risk-averse firm invests in safety, for any degree of risk aversion. This extendsProposition 1.

Comparing Eq. (7) to Eq. (9), and again due to the concavity of u(·), awell-known result holds: a risk-averse firm accepts an actuarially fair (⇡ =P

k pk ·max{hk�D, 0}) contract (X.1, Y.1, and Z.1 in Table 2). Furthermore,between a fair contract with full insurance (D = 0) and one with a positivedeductible, a risk-averse firm prefers the former (i.e., X.1 is preferred toY.1, which is preferred to Z.1). Moreover, for a high enough degree of riskaversion, it can still be EUF (No, Ins) > EUF (No,No), so that the firmaccepts the insurance contract also if it is unfair (all other contracts in Table2).

With this, we can compare Eq. (8) to Eq. (9): if o↵ered an actuariallyfair contract with full insurance (X.1), a risk-averse firm accepts the contractand does not invest in safety, for any degree of risk aversion. For all othercontracts, the presence of a deductible and the possibility to halve the overallprobability of damage by investing in safety makes the latter more profitableto the risk-averse firm, apart from very special cases: absent the possibility offair and full insurance, EUF (No, Ins) > EUF (Inv,No) only for high enoughdegree of risk aversion of the firm, and for high enough overall probability of adamage (i.e., more in phase C than in phase A). All this extends Proposition2.

Comparing Eq. (10) to Eqs. (8)-(9), and recalling that c =P

kpk2 , we

can conclude that Proposition 3 still holds for actuarially fair contracts withfull insurance (i.e., for X.1 only). However, absent the possibility of fair andfull insurance, only a highly enough risk-averse firm prefers both forms ofprotection at the same time (complementarity), rather than only one of thetwo. This extends Proposition 3.

Let us now turn to the risk-averse insurer. Recall that he is paid onlyif the contract is accepted by the firm, and in that case he receives the

17

premium if no damage occurs and the minimum between the damage and thedeductible if a damage occurs. Therefore, his expected utility from proposingan insurance contract to the firm is:

EUI(sI , sF ) = sF=(·,Ins)

" 1�

X

k

pk

!· u(⇡) +

X

k

pk · u(min{hk, D})#,

(11)

with sI = (⇡, D) being the insurer’s strategy (proposed contract), and pk 2{pk, pk2 }, depending on whether the firm’s strategy is sF = (No, Ins) orsF = (Inv, Ins), which is unknown to the insurer.

Under the assumption that the insurer knows that the firm is risk-averse,he maximizes the likelihood that sF = (·, Ins) by proposing contract X.1(actuarially fair and with no deductible).

However, because of his own risk aversion, the insurer may prefer to o↵ercontracts with positive deductible (and lower premium), which better spreadhis potential wealth between the two states of the world of no damage andany damage occurring (see Eq. (11)). Let us analyze this case. Then, bybackward induction, in the Insurer Stage the insurer anticipates that therisk-averse firm will never choose both insurance and investment in safetyin the Firm Stage. Therefore, he selects the optimal contract by assumingthat the actual probability of damage if accepted will be the high one (ph),described in Table 1, i.e., he maximizes EUI(sI , (No, Ins)) in Eq. (11).

First of all, note that, for j = X, Y, Z, contract j.3 always gives the in-surer a higher expected utility if accepted, because of same deductible andhigher premium than j.1 and j.2 (see Table 4 in Section 5.1.1). This holdsindependently of the insurer’s risk attitude. Furthermore, if accepted, con-tract Z.3 gives the insurer the highest possible expected value, independentlyfrom the phase.

Therefore, recalling that sF = (No, Ins), and noticing thatX.3 has higherrisk and lower expected value for the insurer than Y.3, we can conclude thatthe set of optimal contracts for a risk-averse insurer shrinks to {Y.3, Z.3} ofTable 2 (Insurance Table) with attached probabilities of Table 1 (DamageEstimation Table). Now consider phase A. Plugging sI = (380, 200) andsI = (200, 500) and into Eq. (11) for contract Y.3 and Z.3 respectively, itis easy to check that there exists a threshold for the insurer’s degree of riskaversion above which he prefers Y.3 to Z.3 in each period of the phase.

This threshold is higher for periods of phase B, and highest for periods inphase C, where for Y.3 it is sI = (224, 200) and for Z.3 it is sI = (200, 500).In fact, in phase A if Y.3 is proposed to and accepted by the firm, the insurerfaces lottery (380, 40%; 200; 60%), while if Z.3 is proposed to and acceptedby the firm, the insurer faces the two-outcome lottery (200; 70%; 500, 30%).In phase C, Y.3 proposed and accepted leads to lottery (224, 6%; 200, 94%),

18

while Z.3 proposed and accepted leads to lottery (80, 6%; 200, 70%; 500,24%). Having assumed EU , the independence axiom holds, and in phase Athe insurer compares (380, 40%) to (200; 10%; 500, 30%), while in phase Che compares (224, 6%; 200, 24%) to (80, 6%; 500, 24%), respectively for Y.3vs. Z.3. Then, with respect to phase A, in phase C: the spread in favor ofY.3 in the state of the world ‘no damage’ decreases, and its probability islower; the spread in favor of Z.3 in the state of the world ‘medium or highdamage’ increases. Therefore, a higher degree of risk aversion is needed forthe insurer to select Y.3 rather than Z.3 in phase C, so as to weigh more thelower variance of Y.3.

Now fix the insurer’s degree of risk aversion, and suppose that it is abovethe threshold to select Y.3 in phase C, hence in all the three phases of theexperiment. Let us introduce strategic uncertainty into the picture: whenproposing Y.3, the (risk-averse) insurer is not certain that the (risk-averse)firm will accept it. Note that, however, Y.3 has the same deductible (200) inthe three phases, but its cost for the firm (premium) is decreasing from phaseA to phase C: it is 380 in phase A, 302 in phase B, and 224 in phase C (seeTable 2). Furthermore, the probability of the high damage – which mostlyinfluences the insurance decision of a risk-averse firm aiming at smoothingwealth across states of the world – is decreasing from phase A to phase C.Both these e↵ects increase the likelihood that Y.3 will be accepted as theinsurer moves from phase A, to phases B and C.

Therefore, a risk-averse insurer solves the trade-o↵ between increasingthe expected utility if the contract is accepted without decreasing too muchthe likelihood that the firm will really accept it, by proposing contract Y.1(actuarially fair) when the objective probability of the high damage is high(phase A), and contracts Y.2 and Y.3 (actuarially unfair) as the objectiveprobability of the high damage decreases (in phases B and C). All this extendsProposition 4.

Despite heterogeneity in behavior between individuals and within individ-uals in choice under risk (see, e.g., Conte et al., 2011), several experimentalstudies of decisions under risk in the laboratory using experimental taskssimilar to ours11 have reported that the majority of subjects exhibit riskaversion within an expected utility framework.12 Therefore, the followingbehavioral predictions that we test in our experiment are elaborated underthe assumption of the majority of subjects in the population of participantsbeing risk-averse, with only few of them being highly risk-averse.

We first elaborate three predictions about the insurers’ behavior in the

11For the use of expected utility to detect and measure risk aversion in laboratoryexperiments with objective lotteries, see the surveys in Hey (1997) and Hey (1999).

12Besides Holt and Laury (2002) and follow-up articles, see the recent articles byCrosetto and Filippin (2016) and Attanasi et al. (2018) using expected utility for a theo-retical and experimental appraisal of di↵erent risk elicitation methods. In all these studies,and independently from the specific risk elicitation task, concavity of the uniparametricutility function is detected for the majority of subjects.

19

first stage of the game (Insurer Stage). The three predictions of this grouprely on the extension of Proposition 4 to risk-averse insurers (who assumethat the matched firm is risk-averse too).

The first two predictions concern proposals of contracts with partial in-surance, and, within this family, whether fair or unfair contracts are mainlyproposed.

Behavioral Prediction 1 Across the three phases, insurers mostly o↵ercontracts with only partial insurance. The proposed deductible coincides withthe small damage (contracts Y.1, Y.2, Y.3). This holds independently of thetreatment.

Behavioral Prediction 2 Actuarially unfair contracts with only partialinsurance (Y.2 and Y.3) are more frequently o↵ered as the overall probabilityof damage increases (i.e., in phase C rather than in phase A). This holdsindependently of the treatment.

The last prediction of this group concerns contracts with full insurance,i.e., X.1–X.3. The first part of this prediction is theory-driven (extension ofProposition 4): A risk-averse insurer rarely proposes these contracts, sincehe is aware that he will be paid nothing if any damage will occur to thefirm. Furthermore, he proposes contracts X.1–X.3 less often the higher theoverall probability of damage. Recall that, under no firm’s investment insafety when she buys the insurance, this probability is only 60% in phase Aand 94% in phase C.

The second part of the prediction is not theory-driven. It rather relies onprevious experimental evidence that shows that firms tend to decrease theirinvestment in protection against disasters as long as they did not experienceextremely negative events in the past (i.e., in our experiment, a large damageoccurring in the first phases, which is quite unlikely).13 Angelova et al. (2014)have shown such firms’ behavior in safety management (i.e., reduction of theprobability of a damage) under a regime of strict liability similar to ours,but with no possibility of insurance. Here we assume that if the insurerpresumes a decrease in the firms’ investment in safety (which he cannotobserve) as the repeated game unfolds, then he increases his expectationsthat the firm will accept full-insurance contracts in the next phases of theexperiment, thereby o↵ering X.1–X.3 more often than before. Note that thise↵ect reinforces the first part of the prediction in the Decreasing treatment,where phase C comes at the beginning and phase A is played at the endof the experiment. Conversely, in the Increasing treatment it weakens the

13The large damage would be perceived as even more unlikely if one allows for under-statement of low probabilities in the loss domain under a Prospect Theory approach (see,e.g., the survey in de Palma et al. (2014) about the black swan e↵ect). More precisely, the(low) probability of the large damage in each phase of our experiment (i.e., 10% in phaseA, 7% in phase B, and 4% in phase C) would be perceived as lower by a Prospect Theorymaximizer. However, this does fit with our approach, since we assume Expected Utility.

20

first part of the prediction, since the phase where X.1–X.3 are predicted tobe proposed less often (phase C, where the overall probability of damage ishighest) comes at the end of the experiment.

With all this, we can elaborate the last prediction for the insurers.

Behavioral Prediction 3 Contracts will full insurance (X.1–X.3) arerarely o↵ered, independently of the treatment. Furthermore, they are lessfrequently o↵ered as the overall probability of damage increases (i.e., in phaseC rather than in phase A). The latter holds more in the Decreasing than inthe Increasing treatment.

We also elaborate three predictions about firms’ behavior in the secondstage of the game (Firm Stage).

The first part of the first two predictions (complementarity between in-vestment and insurance) relies on the extension of, respectively, Propositions1 and 3 to risk-averse firms: absent insurance, both a risk-neutral and a risk-averse firm prefers to invest in safety measures, thereby selecting at least oneprotection instrument in a period (extension of Proposition 1); only highlyrisk-averse firms select both protection instruments in a period (extension ofProposition 3). Assuming that only a few firms are highly risk-averse, thefraction of insured firms with also safety measures should be low.

The second part of both predictions is not theory-driven. In fact, it relieson the same experimental evidence we have reported in order to elaboratethe second part of Behavioral Prediction 3 for the insurers (Angelova et al.,2014): As the repeated insurer-firm game unfolds, given the low probabilityof the high damage, firms rarely experience this extremely negative event inthe first phase(s) of the experiment. Therefore, they tend to decrease theirinvestment in protection in later phases. This should ultimately lead, inthe last phase of the experiment, to an increase in the predicted negligiblefrequency of no protection at all in a period (Proposition 1), and to a decreasein the predicted low frequency of both protection instruments selected in aperiod (Proposition 3). With this, we elaborate the first two behavioralpredictions for the firms.

Behavioral Prediction 4 Across all periods, and independently of thetreatment, the fraction of firms choosing none of the two protection instru-ments (no protection at all) is negligible. This fraction is highest in the lastphase of the experiment (i.e., in phase C of the Increasing, and phase A ofthe Decreasing treatment).

Behavioral Prediction 5 Across all periods, and independently of thetreatment, the fraction of firms choosing both investment in safety and in-surance in the same period (complementarity) is low. This fraction is lowestin the last phase of the experiment (i.e., in phase C of the Increasing, andphase A of the Decreasing treatment).

21

The last prediction (substitutability between investment and insurance)relies on the extension of Propositions 1 and 2 to risk-averse firms. The su-periority of ex-ante investment in safety over ex-post insurance as a firm’sprotection instrument is mainly due to the insurer often proposing the firmcontracts with only partial insurance, i.e., with a positive deductible (exten-sion of Proposition 4). The impossibility to completely eliminate the risk of adamage makes the insurance instrument less profitable in terms of expectedutility for a risk-averse firm. The second part of the prediction relies on theabove-mentioned behavioral argument of a lower perceived need of protec-tion in the last part of the experiment (see the second part of BehavioralPrediction 4): the lower the firm’s overall level of protection, the lower thedi↵erence between frequencies of implementation of the two instruments.

Behavioral Prediction 6 Across all periods, and independently of thetreatment, firms choose to (only) invest in safety more often than to (only)buy the proposed insurance contract. The investment-insurance gap is great-est in the first phase of the experiment (i.e., phase A in the Increasing, andphase C in the Decreasing treatment).

5 Results

The results in this section are presented as follows: we first analyze (Section5.1, “Main results”) (i) the behavior of the insurer, i.e., which contractsdoes he o↵er (Behavioral Predictions 1-3); (ii) the behavior of the firm, i.e.,whether or not the firm undertakes any protection instrument (BehavioralPrediction 4), there exists a complementarity between investment in safetyand insurance (Behavioral Prediction 5), and there exists a trade-o↵ betweeninvestment in safety and insurance (Behavioral Prediction 6). Then we lookat whether and how agents’ risk preferences and social preferences influencethe insurer’s proposed insurance contract, and the firm’s decision to investin safety and to buy the insurance (Section 5.2, “Determinants”).

5.1 Main results

5.1.1 Contracts proposed by the insurers

In this section we focus on insurers’ behavior in the Insurer Stage (first periodof each phase of the experiment). Table 4 reports the distribution of pro-posed contracts for the three phases separately. For each phase, the fourthand the fifth column report the insurer’s expected value of the proposed con-tract (if accepted) in the case of, respectively, no investment and investmentin safety by the firm. The last two columns report the frequencies of pro-posed contracts for the two treatments separately – Increasing Probabilityof Damage (Inc.: from phase A to phase C), and Decreasing Probability ofDamage (Dec.: from phase C to phase A).

22

Table 4: Frequencies of contracts proposed by the insurer, disentangled byphase and by treatment (last two columns)

Phase A

Contract Premium Deductible EVI |ph EVI |(ph/2) Incr. Decr.

X.1 310 0 124 217 0% 21%X.2 465 0 186 326 11% 8%X.3 620 0 248 434 11% 0%Y.1 190 200 196 193 24% 16%Y.2 285 200 234 260 14% 13%Y.3 380 200 272 326 16% 13%Z.1 100 500 250 175 11% 5%Z.2 150 500 270 210 13% 21%Z.3 200 500 290 245 0% 3%No 0.4 0.6Inv 0.7 0.3

Phase B


X.1 305 0 70 188 8% 8%X.2 458 0 105 282 0% 8%X.3 610 0 140 375 8% 3%Y.1 151 200 189 170 8% 37%Y.2 227 200 206 217 14% 18%Y.3 302 200 223 263 38% 5%Z.1 70 500 251 161 5% 5%Z.2 105 500 259 182 14% 11%Z.3 140 500 267 204 5% 5%No 0.23 0.77Inv 0.615 0.385

Phase C


X.1 300 0 18 159 5% 0%X.2 450 0 27 239 8% 5%X.3 600 0 36 318 14% 8%Y.1 112 200 195 153 13% 8%Y.2 168 200 198 183 30% 29%Y.3 224 200 201 213 13% 32%Z.1 40 500 262 151 3% 10%Z.2 60 500 264 162 3% 3%Z.3 80 500 265 172 11% 5%No 0.06 0.94Inv 0.53 0.47

23

In line with Behavioral Prediction 1, the most frequently proposed groupof contracts in Table 4 is Y.1–Y.3. In fact, these three contracts (consideredas a whole) are chosen by at least 54% of the subjects in the Increasingtreatment across the three phases. The corresponding fraction of subjectsin the Decreasing treatment is 54%. The contract maximizing the insurer’sexpected value under no investment (EVI |ph) is instead only rarely proposed.In fact, independently of the treatment, Z.3 is more frequently chosen inphase C, where only 11% and 5% of insurers propose this contract in theIncreasing and the Decreasing treatment, respectively. This leads to thefollowing result:

Result 1: Insurers rarely propose a contract maximizing their expected prof-its conditional on the contract being accepted by the firm. Rather, in linewith Behavioral Prediction 1, contracts Y.1–Y.3, i.e., those maximizing theinsurers’ expected utility, are proposed more than half of the times. All thisholds independently from the phase and the treatment.

Results in Table 4 are also in line with Behavioral Prediction 2. In fact,independently of the treatment, the actuarially fair contract Y.1 is morefrequently o↵ered in phase A than in phase C (24% vs. 13% in the Increasing;16% vs. 8% in the Decreasing treatment), and the two unfair contracts aremore frequently o↵ered in phase C than in phase A (43% vs. 30% in theIncreasing; 61% vs. 36% in the Decreasing treatment). Both these di↵erencesare significant in each treatment (t-test: highest P -value = 0.031).

This leads to the following result:

Result 2: Behavioral Prediction 2 is verified in each treatment: actuariallyfair (resp., unfair) contracts with partial insurance are more frequently o↵eredwhen the overall probability of damage is lower (resp., higher), both when thisprobability increases and when it decreases over time.

Behavioral Prediction 3 concerns the low fraction of proposed contractswith full insurance (X.1–X.3), with this fraction being higher in phase Athan in phase C of each treatment, especially in the Decreasing treatment.

Table 4 shows that, across all phases, around 22% of insurers in theIncreasing and 20% of insurers in the Decreasing treatment o↵er X.1–X.3contracts. This fraction is significantly lower than the one (33%) of a randomguess over the three possible groups of insurance contracts X, Y , and Z (�2

test, P -value = 0.000 for both treatments).The second part of the prediction is verified in the Decreasing treatment,

where X.1–X.3 contracts are proposed by 29% of the insurers in phase Aand by only 13% of them in phase C, this di↵erence being significant (t-test: P -value = 0.063). However, the opposite behavior is detected in theIncreasing treatment, where 22% vs. 27% of the insurers in phase A vs. phaseC propose contracts with full insurance, this di↵erence being significant (t-test: P -value = 0.000). Indeed, this result is in line with our intuition that

24

the phase A-C gap would have more easily emerged in the Decreasing thanin the Increasing treatment. However, we did not expect that the learninge↵ect of a higher frequency of full-insurance contracts in the last than in theinitial phase would have more than o↵set the theoretically predicted higherfrequency of these contracts in phase A than in phase C.

This leads to the following result:

Result 3: In line with Behavioral Prediction 3, a quite low fraction of full-insurance contracts is o↵ered independently of the treatment (highest detectedfrequency of X.1–X.3 proposals: 29% in phase A of the Decreasing treat-ment). Furthermore, in both treatments full-insurance contracts are morefrequently o↵ered at the end than at the beginning of the experiment (resp.,phase A and phase C). This significant order e↵ect is in line with BehavioralPrediction 3 only for the Decreasing treatment.

5.1.2 Complementarity between insurance and investment

In this section and in the following one we focus on the Firm Stage. Weanalyze the firms’ reactions to the insurers’ proposed contracts, by pooling,for each phase, the five insurance-investment choices made in each period ofthe phase.

First, we analyze whether there exists a complementarity between insur-ance and investment in safety measures.

Figure 1 reports the frequency of purchase of none of the two instruments(green histograms) and of both instruments (blu histograms), by phase (A,B, and C) and by treatment (Increasing and Decreasing).

Pooling data of the two treatments, we find that, on average among allfifteen periods of the experiment, around 18% of firms purchase both insur-ance and investment, while 15% of them purchase none of the two with thedi↵erence being almost significant (P -value = 0.104).

We see this as a confirmation of the first part of Behavioral Predictions4 and 5. If we look at di↵erences across phases on average, we find that thefrequency of purchase of both instruments is always higher than the frequencyof purchase of none of the two (respectively, 18.46% vs. 17.33% in phase A,15.46% vs. 13.60% in phase B, 18.66% vs. 13.33% in phase C), with onlythe di↵erence in phase C being significant (t-test: P -value = 0.070).

This figure, however, hides some di↵erences across treatments. The moststriking di↵erence concerns the choices made in phase A (first four bars inFigure 1), where 25% of subjects have purchased both instruments in theIncreasing treatment, while only 12% have purchased both instruments inDecreasing treatment, with the di↵erence between the two treatments beingsignificant at the 5% level (t-test: P -value = 0.030). Recalling that phaseA is the first phase in the Increasing treatment and the last phase in theDecreasing treatment, we see all this as a confirmation of the second part

25

Figure 1: Complementarity of insurance and investment, by phase and treat.

0

.05

.1

.15

.2

.25Fr

eque

ncy

Phase A Phase B Phase CInc. Dec. Inc. Dec. Inc. Dec.

NoneBoth

of Behavioral Predictions 4 and 5. Indeed, with respect to Behavioral Pre-diction 4, the frequency of “None” choices (green histograms in Figure 1)almost significantly increases from phase A to phase C in the Increasingtreatment (15% vs. 19%; P -value = 0.171), while it significantly decreases inthe Decreasing treatment (19% vs. 8%; P -value = 0.000). With respect toBehavioral Prediction 5, the frequency of “Both” choices (blu histograms inFigure 1) significantly decreases from phase A to phase C in the Increasingtreatment (25% vs. 14%; P -value = 0.000), while it significantly increases inthe Decreasing treatment (12% vs. 23%; P -value = 0.000).

All this leads to the following results:

Result 4: In line with Behavioral Prediction 4, in each treatment and phaseof the experiment, the fraction of firms choosing none of the two protectioninstruments in the same period is quite low, although non-negligible (on av-erage across periods, 16% in the Increasing and 14% in the Decreasing treat-ment). As predicted, it is highest in the last phase of the two treatments (19%in both phase C of the Increasing treatment and phase A of the Decreasingtreatment).

Result 5: In line with Behavioral Prediction 5, in each treatment and phaseof the experiment, there is no strong complementarity between insurance and

26

investment in safety (on average across periods, 19% of firms in the Increas-ing and 16% of firms in the Decreasing treatment choose both instruments inthe same period). As predicted, complementarity is highest in the first phaseof the two treatments (25% in phase A of the Increasing treatment, and 23%in phase C of the Decreasing treatment).

Therefore, the overall probability of damage does not seem to play asignificant role in the complementarity of the two instruments. Rather, asthe play unfolds and the firms learn that a large damage (leading to theloss of all their endowment) is very unlikely to occur, they stop using bothinstruments and eventually face the risk of an accident with no protection atall.

5.1.3 Trade-o↵ between insurance and investment

Next, we analyze whether there exists a trade-o↵ between insurance andinvestment in safety measures. Figure 2 reports the frequency of purchaseof either insurance (brown histograms) or investment (yellow histograms) byphase (A, B, and C) and treatment (Increasing and Decreasing). Table 5reports whether the di↵erences shown in Figure 2 are significant when usinga t-test on the equality of means.

Figure 2: Trade-o↵ between insurance and investment, by phase and treat.

0

.1

.2

.3

.4

.5

Freq

uenc

y

Phase A Phase B Phase CInc. Dec. Inc. Dec. Inc. Dec.

InsuranceInvestment

27

Table 5: Di↵erences between frequencies of Investment vs. Insurance

(Only) Investment = (Only) Insurance

Phase A Phase B Phase CIncreasing *** *** **Decreasing * * *Pooled *** *** ***

On average among all fifteen periods of the experiment, 41% of the firmspurchase only investment and 26% of them purchase only insurance. In linewith the first part of Behavioral Prediction 6, the di↵erence is significant forany treatment-phase combination (see Table 5).

The frequency of firms choosing only investment (39% in phase A, 44%in phase B and 41% in phase C) and the frequency of firms choosing onlyinsurance (25% in phase A, 27% in phase B and 27% in phase C) are quitestable across phases, and indeed no di↵erence was predicted.

However, di↵erently from the second part of Behavioral Prediction 6, wedo not find a significantly greater investment-insurance gap in the first phasethan in the last phase of the experiment. In fact, this gap is quite stableacross phases, independently from the treatment.

Finally, and importantly, note that the investment-insurance gap is sig-nificantly greater in the Increasing than in the Decreasing treatment, due toa higher level of investment and a significantly lower level of insurance in theformer treatment. Therefore, although our model does not produce specificpredictions on this issue, we record more investment in safety and less in-surance in an environment where the overall probability of damage increases(and the probability of the large damage decreases) over time.

All this leads to the following results:

Result 6: In line with Behavioral Prediction 6, investment in safety is sig-nificantly more frequent than insurance in each phase of the experiment, andindependently from the treatment. We do not detect the predicted greaterinvestment-insurance gap in the first phase of the experiment, this gap beingquite stable across phases. However, the investment-insurance gap is greaterin the Increasing treatment, especially due to a lower frequency of insurancethan in the Decreasing treatment.

5.2 Determinants

Tables 6–8 in the Appendix report the empirical estimation of the determi-nants of: the proposed deductible (by the insurers), the insurance purchaseand the investment purchase (by the firms). In each estimation, the controlsconcern subjects’ answers to the questions presented in the final question-naire.

28

First, let us consider the determinants of the insurers’ proposed de-

ductible. Table 6 presents the results of a Probit regression model evaluatingthe probability of proposing a contract with partial insurance and deductibleequal to the small damage (contracts Y.1–Y.3). These are the contracts thatrisk-averse insurers (with high enough degree of risk aversion) should pro-pose more often (see the extension of Proposition 4 to risk-averse insurersin Section 4). And in fact Table 6 shows that risk aversion has a positiveimpact on the likelihood of proposing contracts Y.1–Y.3, since the coe�cientof the explanatory variable Risky is negative and significant when poolingall observations.14 Also in line with our predictions, the positive e↵ect ofrisk aversion on Y.1–Y.3 is stronger and more significant in phase C of theexperiment, where the overall probability of damage is close to 1.

A negative e↵ect of the treatment dummy Decreasing is detected in phaseA, which is the last phase in this treatment, where Behavioral Prediction 1suggests that the insurer should rather propose contracts with full insurance(X.1–X.3).

As for idiosyncratic features, the likelihood of proposing EU -maximizingcontracts Y.1–Y.3 is greater for female insurers (gender e↵ect) and for for-eign participants of the experiment. Furthermore, and more importantly, wesee the fact that previous attendance of experiments (variable Expert) posi-tively contributes to the likelihood of proposing Y.1–Y.3 as a support to ourtheoretical analysis: due to previous learning of the experimental rules, weassume that these subjects makes less mistakes in the experiment, therebyshowing a more ‘rational’ behavior.

Finally, proposing partial-insurance contracts seems to be also motivatedby the insurer’s social preferences and beliefs, due to the significant negativee↵ect of the variable Selfishness,15 and the significant positive e↵ect of thevariable Others’ Fairness16 both if pooling all observations and in almost allphases separately considered. Also No Trust in Others17 plays a significantpositive e↵ect, which might be seen as indicating that the insurers did nottrust the firms actually investing in safety measures.

Let us now consider the determinants of the firms’ insurance purchase.As expected, Table 7 shows that the insurer’s proposed insurance premiumis a significant determinant of the firms’ insurance purchase if pooling allobservations. Furthermore, risk aversion plays a positive e↵ect on insurancepurchase in all phases, with this e↵ect being significant only in phase A and

14This variable is elicited in the final questionnaire through the question: How do youjudge yourself: are you generally a risk-loving person, or do you try to avoid risks? (0 notrisk-loving at all; 10 very risk-loving).

15Would you say that most of the time you try to help others or only follow your owninterests? (0 helps others; 6 only follow your own interest).

16Do you think most people would try to take advantage of you if they got a chance, orwould they try to be fair? (0 try to take advantage; 6 try to be fair.)

17Generally speaking, would you say that most people can be trusted or that you can?tbe too careful in dealing with people? (0 can be trusted; 6 can?t be too careful).

29

if pooling all observations (recall that insurance is preferred to investment insafety only for high enough degree of the firm’s risk aversion). No treatmente↵ect is detected.

About idiosyncratic features, we only find a significant e↵ect of beingenrolled in a Degree in Marketing with respect to the control category ofstudents in Management.

As for social preferences and beliefs, we detect again a positive e↵ect ofNo Trust in Others : maybe not trusting other human beings leads a firm alsonot to trust ‘nature’ thereby insuring against potential natural disasters.

We conclude this section by focusing on the determinants of firms’ in-

vestment in safety. Interestingly, Table 8 shows that, by pooling all ob-servations, risk aversion plays the predicted positive e↵ect on the firms’ like-lihood of investing in safety. However, this e↵ect is not significant. Indeed,our model predicts that a risk-neutral firm is indi↵erent between investmentand insurance, prefers investment for moderate risk aversion (the one char-acterizing the majority of subjects), and prefers insurance for high enoughrisk aversion. This non-monotonicity may be responsible for the absence ofsignificant e↵ect of risk aversion over investment in safety. Furthermore, notreatment e↵ect is detected.

As for the idiosyncratic features, previous attendance of experimentsplays a positive e↵ect on investment, again supporting our operational as-sumption that more expert subjects behave more in line with our modelpredictions.

Finally, Others’ Selfishness18 plays a negative e↵ect on the likelihoodto invest in safety. This might be linked to the firms’ perception that thedamage due to absence of investment in safety a↵ects (non-selfish) thirdparties. Therefore, although a third party is not explicitly included in ourexperimental design, believing the others not being selfish pushes the firm tobehave pro-socially, i.e., to invest in safety measures.

6 Concluding remarks

In the last years several industries have undergone an increase in the severityand complexity of their operations with a subsequent increase in the asso-ciated risk of large scale technological hazards (Zio and Aven, 2013). Thisprocess determined a growing concern about safety and protection from dis-asters. The results of this study shed new light on this relevant issue.

In a world with strict liability for the technological damages, there arethree mechanisms that a firm can use so as to cope with the negative con-sequences of technological hazards. First, the firm can simply take all theconsequences when an accidental event related to its activity occurs. This

18Would you say that most of the time people try to be helpful, or that they are mostlyjust looking out for themselves? (0 try to be helpful; 6 mostly looking out for themselves).

30

means that it is completely prone to take risks and it is not willing to reduce,in advance, the probability of incurring in a hazard. Second, the firm canreduce the probability of an accidental event by implementing safety mea-sures. Third, the firm can transfer the consequences of the occurrence of adisaster to third parties, i.e., an insurance, who are better prepared to carrythe risks and to face their consequences. Of course, the firm can also chooseto combine the last two mechanisms together.

The literature on safety management suggests that even when an insur-ance market is available, the firm’s behavior with respect to investing insafety measures is not a↵ected (Abrahamsen and Asche, 2010). To put itotherwise, there is not a trade-o↵ between investment in safety measuresand insurance. Our results, obtained by testing theory-driven behavioralpredictions in a laboratory experiment, are only partially in line with theseprevious findings. In accordance with our theoretical predictions, we observethat investment in safety measures is always more frequently chosen thaninsurance by risk-averse firms. Therefore, our results claim for a superiorityof ex-ante protection (safety measures) over ex-post protection (insurancemeasures) from the objective risk of a technological accident.

Furthermore, they suggest that, even if a technological damage is almostsure without any investment in safety and the investment in safety greatlyreduces the probability of an accident (phase C of our experiment), less than50% of the firms purchase only investment while less than 20%, on average,purchase both, investment and insurance. Therefore, several agents (morethan 30%) are not willing to pay a fair cost to halve (from 94% to 47%) theoverall probability of a technological damage.

Our results also point out that firms’ insurance choices are quite con-stant across di↵erent probabilities of technological damages. In fact, there isnot complementarity between investment and insurance, and instead we findevidence of a substitution e↵ect. This happens because the overwhelmingmajority of insurers o↵er a contract without full insurance to the firms. Thepositive – though low – value of the proposed deductible makes the insurancecontract lose part of its intrinsic nature of ex-post protection instrument.

The main limitation and advantage of this paper lies in the methodologyimplemented. In fact, we relied on an economic experiment based on a finitelyrepeated two-stage game between two subjects in the role of the insurerand the firm. The experiment was realized in a laboratory setting and cantherefore su↵er of some limitations due to the simplification of the reality(e.g., the potentially injured third party was not included in the design).However, this methodology o↵ers the possibility to observe behaviors andvariables that otherwise could not be observed in the field. Furthermore, itallows to set the objective parameters of the insurer-firm experimental game(e.g., probabilities and sizes of damage, insurance premiums and deductibles)in a way that it is possible to elaborate clear-cut predictions about insurers’and firms’ strategic decisions by relying on decision and game-theoretic tools.

31

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Appendix

Table 6: Determinants of Deductibe equal to small damage (200)

(1) (2) (3) (4)Pooled observations Phase A Phase B Phase C

b/se b/se b/se b/seAge 0.040 0.063 0.202*** -0.181**

(0.039) (0.075) (0.074) (0.072)Male -0.165* -0.399** -0.238 0.305*

(0.089) (0.163) (0.165) (0.175)Marketing 0.114 -0.192 0.782*** -0.301

(0.091) (0.164) (0.172) (0.187)Economics -0.125 0.591** 0.341 -1.766***

(0.174) (0.291) (0.319) (0.411)France -0.852*** -1.749*** -0.337 -0.668*

(0.178) (0.341) (0.301) (0.375)Expert -0.426*** -0.380 -1.049*** -0.038

(0.150) (0.279) (0.273) (0.368)Selfishness -0.120*** -0.174*** -0.132*** -0.057

(0.027) (0.049) (0.046) (0.051)Risky -0.034* 0.064** 0.043 -0.301***

(0.018) (0.032) (0.033) (0.041)Others’ Selfishness 0.010 0.039 0.014 -0.071

(0.031) (0.054) (0.057) (0.065)Others’ Fairness 0.112*** 0.195*** 0.136** 0.103

(0.032) (0.059) (0.058) (0.075)No Trust in Others 0.160*** 0.159*** 0.162*** 0.340***

(0.033) (0.058) (0.058) (0.073)Decreasing -0.104 -0.588*** 0.206 0.156

(0.086) (0.158) (0.158) (0.169)Constant -0.127 -0.177 -4.686*** 5.188***

(0.825) (1.608) (1.549) (1.565)N. 1125 375 375 375

Note: Probit (mfx), standard errors in parentheses. * p<0.10, ** p<0.05, *** p<0.01

35

Table 7: Determinants of Insurance purchase


b/se b/se b/se b/sePremium -0.001*** -0.002 -0.002 0.001

(0.000) (0.002) (0.002) (0.001)Age -0.186 -0.196 0.053 0.420*

(0.119) (0.224) (0.328) (0.225)Male 0.451 -0.788 -1.186* -0.285

(0.319) (0.520) (0.617) (0.398)Marketing 0.518* -0.313 -0.225 0.223

(0.289) (0.483) (0.558) (0.567)Economics 0.909 1.040

(0.765) (0.636)France -0.398 -0.267 0.509 -0.911

(0.538) (1.146) (1.060) (0.851)Expert -0.299 0.963 -0.285

(0.564) (0.865) (0.736)Selfishness 0.054 0.016 -0.096 0.039

(0.099) (0.137) (0.141) (0.113)Risky -0.109* -0.166* 0.028 -0.051

(0.058) (0.096) (0.108) (0.082)Others’ Selfishness 0.107 0.283 -0.074 -0.187

(0.120) (0.191) (0.135) (0.141)Others’ Fairness -0.033 0.011 0.258 0.135

(0.091) (0.156) (0.184) (0.171)No Trust in Others 0.220* -0.085 0.111 0.106

(0.114) (0.189) (0.192) (0.144)Decreasing -1.226 0.003 0.381 -0.046

(1.320) (0.728) (0.515) (0.595)Constant 3.936 5.460 -1.277 -7.607*

(2.654) (3.597) (6.254) (4.566)N. 1035 240 230 245

Note: Probit (mfx), standard errors in parentheses. Subject FE. * p<0.10, ** p<0.05,*** p<0.01

36

Table 8: Determinants of Investment in safety


b/se b/se b/se b/sePremium -0.000 0.001 -0.000 0.002

(0.000) (0.002) (0.002) (0.001)Age 0.036 0.108 -0.208 -0.184

(0.110) (0.242) (0.300) (0.227)Male -0.042 0.022 0.163 0.035

(0.290) (0.665) (0.495) (0.468)Marketing 0.083 0.068 -0.222 0.324

(0.271) (0.528) (0.503) (0.545)Economics 0.793 0.946

(0.849) (0.643)France 0.246 -1.707 1.040 -1.449

(0.487) (1.087) (1.088) (1.035)Expert 1.270** -0.533 -0.105 0.433

(0.547) (0.878) (0.989) (0.552)Selfishness -0.026 -0.065 0.147 -0.137

(0.086) (0.118) (0.143) (0.107)Risky -0.071 0.000 -0.083 0.089

(0.053) (0.114) (0.123) (0.104)Others’ Selfishness -0.197* 0.017 -0.044 -0.088

(0.107) (0.124) (0.129) (0.143)Others’ Fairness 0.117 -0.048 0.022 -0.254

(0.087) (0.181) (0.116) (0.157)No Trust in Others -0.109 -0.056 -0.031 -0.009

(0.106) (0.145) (0.154) (0.151)Decreasing 0.517 -0.280 -0.405 -0.613

(1.358) (0.406) (0.488) (0.534)Constant -0.964 -0.192 3.668 5.608

(2.483) (4.663) (6.054) (5.059)N. 1065 265 275 290

Note: Probit (mfx), standard errors in parentheses. Subject FE. * p<0.10, ** p<0.05,*** p<0.01

37

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