Workplace Deviance and Recession

1 Introduction

In this paper we examine whether there is any relationship between workplace deviance (on-the-job crime) and the state of the economy i.e. does such behavior increase or decrease with the anticipation of a recession? This “deviant behavior” could be in the form of shirking at work, stealing from the firm, sabotage, harassing other workers etc. Deviant behavior has consequences for a firm’s profitability. Further, while a deviant employee enjoys current benefits from such behavior, it affects her future employment prospects if caught. Currently, there is no consensus in the literature about whether deviant behavior in the workplace increases or decreases with the state of the economy, nor is there a cogent theoretical framework to analyze such behavior. ^[1] Popular newspaper accounts suggest that the recent recession may have increased employee theft. For instance an article by Needleman (2008) in the Wall Street Journal suggests employee theft has increased in the recession. In the UK, KPMG’s fraud barometer (KPMG 2010) for January 2010 shows employee fraud in book keeping and accounting on the rise in the recent recession. However, one of the few scholarly studies on employee theft (Rickman and Witt 2007) shows strong evidence of employee theft decreasing as unemployment increased in the UK for the period 1999–2000. This suggests that the answer to the question may well depend on the nature or severity of recession and there could well be opposing forces at work in a recession. Further, (as we elaborate below) shirking which is another measure of workplace deviance also does not appear to unambiguously increase or decrease with recession. In order to provide an answer to this question, we build a dynamic model which analyzes how the current market condition as well as expectations about future market conditions affect the intertemporal tradeoffs that people make in deciding how to behave in their current workplace.

We treat workplace deviance as a type of on-the-job crime committed by rational criminal economic agents who derive benefit from their deviant behavior but incur costly sanctions if caught. Thus, we can analyze it using the framework in Becker’s (1968) analysis of crime, though as we discuss, the framework would need to be extended to take into account expectations about future market conditions. To see this, recall that in a standard Beckerian model, if people can choose between work and crime, the impact of unemployment increases crime as it lowers the opportunity cost of crime. ^[2] However, when we modify the model to take account of the fact that people can both work and commit crime which is what we need to consider when analyzing workplace deviance, the state of the economy plays no role in a standard static framework. Instead, workplace deviant behavior is entirely determined by the probability of apprehension and the severity of punishment. However, once we take into account that being caught in deviant behavior has implications for future employment, the state of the economy plays an important role, albeit a complex one.

To see this, we note two effects that act in opposite directions-deviant behavior today causes lowered chances of employment tomorrow but employment prospects additionally also depend on the general state of the economy tomorrow. Intuitively, if the general state of the economy is such that employment prospects are bleak in the future, loss of employment may not act as much of a deterrent and thus people would commit more on-the-job crime (including shirking) when expecting an economic downturn as the opportunity cost of such behavior goes down with a lowered expected wage in the labor market. However, working against that is the fact that when jobs have to be cut, managers may fire the least productive workers, that is, workers with “bad” records may find it harder to get jobs in a recession which makes the marginal value of abstaining from deviant behavior higher in a tight labor market. Further, indulging in workplace crime may incur a setup cost and may be worth paying only if the potential criminal has a chance to benefit from incurring this one time cost. The direction of the net effect is thus far from clear and we identify two types of regimes where there are different impacts of future market conditions on current workplace behavior. Thus we identify how the state of the economy affects firm profitability not only through well recognized channels such as the strength of demand, wage rates etc. but also through its effect on workplace behavior.

While there are no precise estimates of workplace deviance, the empirical evidence on labor effort and the state of the economy is mixed. Some papers find empirical evidence that labor effort increases in a recession (e.g. Franke and Kaul 1978; Stern and Friedman 1980; Wadhwani and Wall 1991; Drago and Heywood 1992; Agell 1994). Others, such as Spitz (1993) find no such relationship. Surveys conducted on managers give contradictory findings. Some believe that shirking goes down during a recession while others believe that it goes up. In an interview of 47 businesses undertaken by Bewley (1999), it was found that 42% believed labor effort goes up during a recession, while 15% believed it goes down and 42% believed it has no impact. The closest data we have on deviant behavior would be data on recorded employee theft. In a study of employee theft in the UK, Rickman and Witt (2007) find that increases in the unemployment rate decrease employee theft. These contradictory findings on deviant behavior suggests the need to build a theoretical framework to provide a precise answer to the question.

We consider a model where the economy can be in a good state (boom) or a bad state (recession). Given the state of the economy, there is a probability distribution over the future state of the economy. Further, we assume that the prospects of employment depend on (a) the state of the economy and (b) one’s past record. During a boom, more people are employed, so an employer may have to employ people even with a “bad” record, while during a recession the employer can afford to be more selective. This bad record can be in the form of having been caught shirking or committing an on-the-job crime including sabotage and harassing other employees. Formally, any kind of deviant behavior which will lead to a “bad” history and sanctions if caught is what we call crime. Thus while crime has benefits, it has costs in terms of sanctions if caught and a lowered probability of future employment. Further, a career in crime has set up costs, so a first time criminal additionally faces a one time fixed cost.

The current state and expectations about the future state are parameters in the model and determine current and future employment. The probability of getting a job from any state to another is however endogenously determined as it depends on current behavior. Further, current behavior in turn determines the crime rate in period 1, which determines the probability of employment in period 2. We assume for simplicity that everyone is employed in a boom, and that only a fraction of the population is employed during a recession; the severity of the recession is measured by labor demand in a recession. A more severe recession is associated with lower labor demand. Our results depend on the characteristics of the equilibrium, and in particular, we find that three possible types of regimes can exist. In one type of equilibrium (Regime 1), only a fraction of people with an unblemished record can find jobs in a recession. Everyone else is unemployed in that case. In another type of equilibrium (Regime 2), all people with an unblemished record find jobs in a recession as do a fraction of currently unemployed people. Finally, in the third type of equilibrium (Regime 3), even a fraction of workers with a bad record get jobs. We analyze what happens when the severity of recession increases, both for small shifts (which does not change the equilibrium regime) as well as for large shifts (which can change the regime). Changes in expectations i.e. changes about the probability of facing a future recession is also analyzed.

We provide an intuitive discussion of our main results. The impact of increased severity of recession (in terms of a lowered probability of finding a job if there is a recession tomorrow) in period 2 on the incentive to commit crime in period 1 depends on the proportion of the workers with a bad record (which is an endogenous variable in our model). Suppose in case of a recession tomorrow, the equilibrium is regime 1 i.e., a situation where in a recession, no one with a criminal record finds a job, but even some employed people without a criminal record cannot find jobs. Given such an equilibrium, any worsening of the severity of the recession (while staying in regime 1) increases the incentive to commit crime in period 1. Since a person with a bad record is anyway not hired in this case, therefore, any change in the severity of the recession will not have any impact on her behavior. However, a change in the severity matters to a person who has a good record, since such a person has a lower likelihood of being employed in a more severe recession. Thus, an increase in the severity of a recession reduces the incentive to maintain a good record. Consequently, the incentive to commit crime should be higher in this case. This is similar to what is predicted by the standard Becker model.

However, it is also possible that the anticipated equilibrium is regime 3. In this case, all people without a criminal record get a job, but some people with a criminal record also get a job. In this case, an increase in the severity of a recession has an opposite effect. Since people with a good record get a job anyway, therefore, any change in the severity of the recession does not affect their behavior. However, a tightening of the labor market reduces the probability of people with bad records of getting a job. This in turn lowers the incentive to commit crime as the marginal value of staying crime free (and thus having a clean record) increases. This goes against the standard Beckerian result as the future value of staying crime free is higher in a tighter labor market where employers can be more selective about who to hire. In regime 2 there is no net impact as the changes in employment affect only the employment prospect of the currently unemployed and not of those employed in period 1.

The setup cost always reduces the incentive to commit crime if the labor market is expected to tighten in the future and will counteract the first effect in regime 1 but reinforce it in regime 3. Thus, in regimes 2 and 3, the net incentive to commit crime goes down with anticipated recession, while in regime 1 it can go either way.

Our results differ both from the theoretical prediction in the standard Becker (1968) model which implies that crime increases with an increase in unemployment as well as the theoretical (and empirical) prediction in Rickman and Witt (2007) who on the contrary find that the rate of unemployment has a negative relationship with the crime rate. Once one accounts for the future consequences of crime and trades it off against current profitability, the relationship between deviant behavior and the unemployment rate (or the severity of the recession) could be positive or negative depending on the type of recession.

In the context of the recent recession, there has been some discussion about policies that could increase confidence in the economy. This paper does not deal with such policies but it can rather be used to determine the impact on workplace discipline if there is an increase in optimism about the future. ^[3] Our model predicts that an increase in optimism i.e. a decrease in the probability of recession in period 2 (holding the severity of recession constant) always increases the incentive to commit crime in period 1 as long as nearly everyone gets a job in a boom. Thus, there is a difference in the predicted impact of a change in the severity of the recession vs. a change in the degree of optimism. The impact of the former can vary depending on the character of the equilibrium, while the impact of the latter stays the same regardless of the nature of the equilibrium provided that people with bad records gain more relative to people without one.

We set up the model in the next section, solve for period 2 equilibrium in Section 3, and analyze the incentive to commit crime in period 1 in Section 4. Section 5 endogenizes the decision making process of firms and Section 6 concludes.

2 The Model

This is a two period model ^[4] (t=1,2) with two possible states s∈H,L during each period with H representing a boom and L representing a recession or bust. Further, the state in period 1 determines the probability of the state in period 2. In particular, let γss′ be the probability that the state in period 2 is s′, given that the state in period 1 is s. For example, if the state in period 1 is H, then the probability of the state being H in period 2 is γHH, and the probability of the state being L in period 2 is γHL. It follows that,

There are a large number of potential workers in our model. We denote the employment status of a person in period t by an indicator variable Ωt that takes a value 1 if the person is employed in period t and takes a value of 0 otherwise. At the end of a period, an employed person receives a wage ws if the state is s. Further, in the spirit of several models of the labor market such as Shapiro and Stiglitz (1984), we assume that wages are lower in a recession, that is wL≤wH. ^[5]

We denote labor demand in state s by es. Further, we assume that

while

Hence, by assumption, no one is unemployed during a boom, while a fraction 1−eL is unemployed during a recession. ^[6],[7] Since the focus of the paper is to explain why the level of deviant behavior in the workplace may be non-monotonically related to the severity of the recession (as opposed to explaining the reason for business cycles), therefore, it suffices for our purpose to assume the level of employment in each state as exogenous. However, Section 5 endogenizes the hiring choice of firms and in the concluding section, we argue that the central idea of the paper would still hold if we had allowed labor demand in period 2 to depend endogenously on the crime rate in period 1.

If employed, an individual can choose to work honestly or indulge in deviant behavior. Throughout the analysis, we use the terms “crime” and “deviant behavior” interchangeably. An unemployed person cannot commit crime in our model because we consider only on-the-job crimes. In contrast, an employed person can choose to either commit a crime or to remain innocent in each period. We denote the set of actions for a generic employed individual by

where Ct denotes that the individual chose to commit a crime in period t and It denotes a decision to remain honest in period t. If an individual commits a crime, she derives a private benefit of B≥0 while her payoff is 0 if she chooses to be innocent. The private benefit from crime (which one may interpret as the individual’s type) is heterogeneously distributed in the population in the interval 0,B‾ following the distribution function GB with density function gB.

An individual incurs a cost of τ>0 in her first crime episode and a cost of 0 in subsequent episodes (which is a normalization). The cost τ can be thought of as a set-up cost of crime. This framework has similarities with Mocan, Billups, and Overland (2005) who allow individuals to allocate their time in the acquisition of either legal human capital or criminal human capital. As they note (see p. 660), “individuals become more skilled criminals through learning-by-doing.” Deutsch, Simon, and Spiegel (1990) provide empirical evidence that there is indeed substantial learning-by-doing in criminal activity. In our model, we allow an individual to decide whether or not to acquire the criminal human capital by paying a one-time fee. The set-up cost can also include any emotional cost associated with committing crime. From the second time onwards, such a set up cost will be lower due to a lowering of the learning cost or any psychic or emotional cost associated with committing a crime. Hence, one could have assumed that the fixed cost associated with crime is τ1 for the first instance of crime and τ2 for the second instance such that τ1>τ2. Since τ2 would unnecessarily complicate the analysis, therefore, we assume it to be 0.

The employer imperfectly monitors actions in the workplace. Only a fraction of deviant behavior is punished with p∈0,1 being the conviction probability. We assume that the conviction probability p is exogenous. Such an assumption allows us to demonstrate the main result (that an increase in the severity of a recession need not lead to an increase in workplace deviance) in the simplest possible manner. Later on, in the concluding section, we discuss the effect of endogenizing this variable. The monetary value of a convicted person’s punishment is denoted by f. It is helpful to think of this monetized value as a fine (perhaps in terms of withheld bonuses) though in the concluding section we discuss what would happen if instead the penalty took the form of imprisonment.

We assume that the enforcement agency publicly releases the record of a person. We denote the record of a person by R. The record of a person can be “bad” if an employed person is convicted, or it can be “good” if she has either not committed a crime or has not been caught in an act of crime and “no record” if a person does not have an employment history. Note once again that an unemployed person cannot be convicted in a period since we consider only on-the-job crimes.

Recall, in any period, employers hire the entire population in a boom and only a fraction eL in a recession. Employers are profit maximizers and as we explain below, first prefer hiring previously hired workers with a good record, then prefer hiring previously unemployed persons and finally prefer previously hired workers with a bad record. Therefore, in the model, the record (and previous employment status) of a person affects her likelihood of being employed if there is a recession in period 2. In the discussion of period 2, we provide a rationale for this preference.

The utility of a person in period 2 is given by

where Λ is an indicator variable that takes the value 1 for the first offense and 0 otherwise. Notice that the term B−Λτ−pf is the net expected benefit from deviant behavior in period 2 while the net benefit from remaining innocent in period 2 is 0. Notice that the utility of an unemployed person in period 2 is 0 since for this person, Ω2=0. Similarly, the utility of an employed person in period 1 is given by

where E1U2A2,s2|Ω1,A1 is the expected utility in period 2 assuming agents take optimal actions in period 2. The optimal actions will depend on the parameters of the economy and on the action in period 1.

3 Analysis of Period 2

We first analyze an individual’s optimal behavior in period 2. In period 2, an employed person commits crime for the first time if and only if expected net benefit from crime is positive

Similarly, an employed person commits crime for the second time in period 2 if

Thus, it follows from the expressions above that B2∗>B2∗∗ i.e., the threshold for committing crime is lower for a person who had already committed a crime previously. This is because such a person does not have to incur the setup cost of crime again.

In order to complete the analysis for period 2, we need to specify the firm’s preference between hiring (i) an experienced worker with a good record, (ii) an experienced worker with a bad record, and (iii) an inexperienced worker. We assume that a firm prefers (i) over (iii) and (iii) over (ii). The firm’s preference of (i) over (iii) is consistent with the observation that employers value experience because previously unemployed workers may have lost some skills due to having been out of the job market. This is what is called the “scar of unemployment” (Arulampalam 2001). As for the comparison between (iii) and (ii), there is a tension between the higher productivity of an experienced worker (albeit with a bad record) and the higher incentive to indulge in deviant behavior of a person who has already incurred the set up cost. We assume that the latter effect outweighs the former. In any case, what matters most for our analysis is that the firm prefers an experienced worker with a good record over an experienced worker with a bad record and the spirit of the argument would not change if we were to assume that the inexperienced workers are the least preferred category.

4 Analysis of Period 1

In order to determine U1, we need to determine the value of E1U2A2,s2,B|Ω1,A1. It follows from eq. [1] that the decision of a person to commit crime or to remain honest in period 2 depends on her private benefit B from crime as well as her expected cost of committing crime in period 1. Note that the expected cost of committing crime in period 1 depends on the expected employment status in period 2. In order to determine the expected utility of a person in period 2, we need to first determine the expected value of Ω2 and this is done below.

Let qA1,s1 be the probability that Ω2 takes a value of 1 in period 2 (that is, it is the probability that a worker is employed in period 2). In this model, q depends on two factors: (i) the action of an individual in period 1 since it stochastically affects the individual’s record R, and (ii) the state of the economy in period 1, given by s1, because it determines the likelihood of the state of the economy being H or L in period 2. Therefore, the utility in period 1 is given by

In the right hand side of eq. [2], the first term is the payoff that a person receives in the first period and the second and third terms capture the continuation payoffs. In period 2, the person receives at least wL+maxB−Λτ−pf,0 if she is employed and an additional wage premium wH−wL if there is a boom in period 2. The chance of being employed in period 2 is qA1,s1 and the chance of a boom is γs1H. Hence, the sum of the second and third terms capture the continuation payoff in period 2, conditional on information available in period 1. ^[8]

It follows from eq. [2] that if a person commits a crime in period 1 i.e., if A1=C, then the benefit from the crime is B plus a lowered cost of committing crime next period (i.e. τ won’t have to be incurred next period) while the cost from the crime depends on τ+pf as well as on the changed probability of employment because of committing crime in period 1 given by qI,s1−qC,s1 which captures the additional benefit of being crime free.

An individual commits a crime in period 1 if her private benefit from crime is greater than or equal to a threshold level and she does not commit a crime if her private benefit is below that threshold level. We refer to this threshold level as the marginal criminal type. Let B˜ denote the marginal criminal type. Then the following conditions must be satisfied: (i) the benefit from committing crime must be equal to the cost of committing crime of the marginal criminal, (ii) the benefit from committing crime must be less than the cost of committing crime if the private benefit is less than the marginal criminal type, and (iii) the benefit from committing crime must be greater than the cost of committing crime if the private benefit is greater than the marginal criminal type. This is illustrated in Figure 1. As shown in the diagram, the marginal criminal type B˜ is indifferent between committing a crime and remaining innocent.

Figure 1:

This diagram compares the cost of crime with its benefit given that the type of the marginal criminal is B˜.

We denote the cost of crime when the benefit is B, the marginal criminal is B˜ and the state is s1 by

Note that Δs1B,B˜ depends on the parameters τ,p and f as well as the change in probability of being employed in period 2 for being deviant in period 1. The right hand side of the above expression will be explained in detail later.

Let λB˜ be the proportion of the workforce (or the workers employed in period 1) with a bad record at the beginning of period 2 given that the marginal criminal type is B˜. Hence, it is given by the product of the probability of being caught and the probability of having a benefit high enough to commit crime i.e.

Similarly, the proportion of workforce (or the workers employed in period 1) with a good record is

if the marginal criminal type is B˜. Let the state in period 1 be s1 where s1=H,L. Hence, if the marginal criminal type in period 1 is B˜, then the fraction of the population with no record is

the fraction of the population with a good record is

while the fraction of the population with a bad record is

We now determine the probability of being employed in period 2.

4.1 Probability of Employment in Period 2

If there is a boom in period 2, then everyone is employed regardless of record. However, if there is a recession in period 2, then only a fraction of the population can gain employment and hence, in this case, a person’s record matters. Suppose there is a recession in period 2. Then, three kinds of regimes can occur: (a) Only a fraction of the population with a good record is employed (Regime 1), (b) all individuals with a good record are employed and only a fraction of those with no record are employed (Regime 2), and (c) all individuals with a good record or no record are employed, while only a fraction of individuals with a bad record are employed (Regime 3). Given the marginal criminal type B˜, Regime 1 occurs if

1−λB˜es1≥eL,

i.e. the number of people not convicted in period 1 is greater than labor demand in a recession. Hence, only a fraction of employed people without a criminal record in period 1 are employed in period 2. Given that these are the most preferred employees, no other category of workers is employed.

Regime 2 occurs if

1−λB˜es1+1−es1≥eL>1−λB˜es1,

i.e., after employing all workers without a criminal record, only some of the workers unemployed in period 1 (given by 1−es1) are employed. No convicted person in period 1 is employed in period 2. Regime 3 occurs if

[3]eL>1−λB˜es1+1−es1.

i.e., in this case, even some period 1 convicts are hired as labor demand exceeds the number of people without a record as well as the number of unemployed. In particular, when there is a boom in period 1, then Regime 1 occurs if the marginal criminal type B˜ is such that 1−λB˜≥eL, Regime 2 does not occur at all (since there are no unemployed people in period 1 if it is a boom) and Regime 3 occurs if eL>1−λB˜. By definition, λB˜ is a decreasing function of B˜, and hence, 1−λB˜ is an increasing function of B˜. Therefore, when there is a boom in period 1, then Regime 1 occurs for relatively high values of B˜ and Regime 3 occurs for relatively low values of B˜. This is depicted in the upper panel of Figure 2. Further, when there is a recession in period 1, then Regime 1 does not occur at all (since in regime 1 there are no people unemployed in period 1), Regime 2 occurs if the marginal criminal type B˜ is such that 1−λB˜≥2−1eL and Regime 3 occurs if 2−1eL>1−λB˜. ^[9] Therefore, when there is a recession in period 1, then Regime 2 occurs for relatively high values of B˜ and Regime 3 occurs for relatively low values of B˜. This is depicted in the bottom panel of Figure 2.

Figure 2:

The upper panel shows the different regimes when there is a boom in period 1 followed by a recession in period 2. The bottom panel shows the different regimes when there is a recession in both periods.

The probability of being employed in period 2 depends upon the action of an individual in period 1, the state in period 1 and the appropriate regime in which the equilibrium occurs in period 1. These probabilities are presented in Table 1. We now explain these probabilities. First, consider the probability qC,H of being employed in period 2 if there is a boom in period 1 and an individual commits a crime in period 1. Such an individual will have a bad record with probability p and a good record with probability 1−p. Suppose Regime 1 occurs, that is, during a recession only a fraction of the individuals with a good record are employed and the others are not employed. An individual who committed a crime in period 1 will be employed in period 2 only when there is either a boom in period 2 or a recession in period 2 and this individual was not convicted in period 1. In the latter case, this individual can be employed only if she is assigned to a job from amongst the pool of workers with a good record. Conditional on a good record the probability of being employed is given by the ratio of employment slots (eL) and the number of people with a good record (1−λB˜). The probability of a boom in period 2 is γHH and the probability of this individual being employed with a good record during a recession is given by the joint probability of a recession (γHL) and not being caught (1−p). Thus, the probability of employment if the state is L in period 2 conditional on crime in period 1 is given by γHL1−peL1−λB˜. Hence, qC,H=γHH+γHL1−peL1−λB˜ under Regime 1. In contrast, under Regime 3, during a recession, only a fraction of individuals with a bad record are unemployed while the others are all employed. If an individual commits a crime, then the joint probability of a recession and the individual having a bad record is γHLp. Notice that the number of “unemployment spots” is 1−eL and these have to be rationed among the number of individuals with a bad record given by λB˜. Hence, the probability that an individual with a bad record will be unemployed in a recession is 1−eLλB˜ and consequently, qC,H=1−γHLp1−eLλB˜. Similarly, we obtain the other expressions. The probabilities are plotted as a function of the marginal criminal type in Figure 3.

Table 1:

The probability of being employed in period 2 as a function of the action and state in period 1.

q(A1, s1)	Regime 1	Regime 2	Regime 3
q(C, H)	γHH+γHL1−peL1−λBˉ	–	1−γHLp1−eLλBˉ
q(I, H)	λHH+γHLeL1−λBˉ	–	1
q(C, L)	–	1−γLLp	1−γLLp1−eLλBˉeL
q(I, L)	–	1	1

Figure 3:

The upper panel shows the probability of being employed in period 2 if there is a boom in period 1. The lower panel shows the probability of being employed in period 2 if there is a recession in period 1.

5 Robustness Checks: Endogenizing the Employment Decision

In the discussion above, the decision making process of the firms were not explicitly modeled. In this section, we therefore endogenize the employment decision of the firms and consider its impact on the main results of the paper.

Consider a perfectly competitive industry with a large number of identical firms. For this industry, the (inverse) demand curve is

and the supply curve is

Hence, in equilibrium, the price and aggregate quantity are

The parameter α varies according to the state of the economy. In particular, it takes a value of αH during a boom and αL during a recession; αH>αL. Each worker produces one unit of output.

In each period, the aggregate employment depends on the state of the economy in that period and the record of the workers. Consider period 1. The gross value of marginal product (excluding deviance costs) of each worker is

where s1 is the state in period 1. In this period, all workers are identical since none of them have any record. However, the firms can anticipate that the probability that a worker will engage in deviant behavior in period 1 is λB˜. Suppose a firm incurs a loss of m for each instance of deviance. Then, the net value of marginal product of a worker in period 1 is

The net value of marginal product curve is depicted by the horizontal lines in Figure 4. The supply curve of labor is determined by a function we where e is the level of employment. The labor supply curve is determined by the labor leisure choice of individual workers; w′e>0.

Figure 4:

The net value of marginal product of a firm is given by the horizontal line whose height depends on the state of the economy. The supply curve of labor is given by w(e).

The employment in the economy is determined by the intersection of the net marginal product curve and the labor supply curve. We assume that

This ensures that everyone (including those with a bad record) is employed if there is a boom in period 1, that is,

If there is a recession in period 1, then it follows from Figure 4 that some workers are unemployed and the rate of employment depends positively on B˜, that is

Let w1s1,B˜ denote the wage rate in period 1 if the state is s1 and the marginal criminal type is B˜. Notice that

that is, in period 1, the wage rate decreases with the incentive to commit crime. This is different from the previous model with exogenous firms because that model assumes that the wage rate depends on the state of the economy but not the crime rate.

Now consider period 2. The net value of marginal product in period 2 is a step function. This function attains its highest value for experienced workers with a good record, followed by inexperienced workers, followed by experienced workers with a bad record. The wage rate is determined by the intersection of the wage schedule with the net value of marginal product schedule. As long as eq. [15] holds, it can be shown that everyone will be employed if there is a boom in period 2. Following similar lines, it can be shown that

and

In period 2, the employment rate in a recession as well as the wage rate depends on the marginal criminal types in both periods. However, the marginal criminal type in period 2 will have no role to play in the analysis, therefore, we suppress it. In this model, the wage rates in the two periods can be different, even for the same state. This model will also require a modification of Figure 2 because e2L,B˜ is now an increasing function of B˜, instead of being a constant function. However, there are still two regimes (for each initial state). A decrease in the severity of a recession in this model is captured by an increase in αL and we show below that in this model, such a change has similar effects as in Propositions 1 and 3.

Suppose there is a boom in period 1. In this model, the dynamic deterrence effect is still given by a similar expression as eq. [6] with the differences being as follows: (i) In the expressions for V2⋅, we replace wL with w2L,B˜, and (ii) in the expressions for qC,H and qI,H, we replace eL with e2L,B˜.

First, consider the case in which the equilibrium is Regime 1 before and after the increase in αL. An increase in αL increases e2L,B˜ and w2L,B˜. These changes in turn increase qI,H−qC,H, qC,H, and V2A1=I but leaves V2A1=C−V2A1=I unchanged. Hence, it follows that both terms of the dynamic deterrence effect eq. [6] increase. Consequently, the net effect of an increase in αL can go either way. This result is same as in Proposition 1.