1. Model
The firm and enforcement agency interact in more than one domain. This may arise because a single agency is responsible for enforcing more than one regulation or because it enforces the same regulation at more than one constituent plant of a multi-plant firm. For simplicity we will assume that the number of domains is two and that they are ex ante identical. In each domain the firm is required to comply with a regulation. If it complies it inflicts no environmental damage otherwise it inflicts damage d, which is commonly observed. The cost to the ith firm of compliance in domain j [ h1, 2j will be denoted cij where ci 1 and ci 2 are independent, privately observed draws from a distribution f(c) with associated cumulative F(c). F is common knowledge.
If the agency observes non-compliance by a firm in either domain it can take that firm to court (‘‘pursue’’ the firm), in which case the firm is subject to a penalty L which is exogenous. Penalties are assumed to be restricted in the sense that F(L) < 1. This implies that a policy of full-pursuit, whereby the agency pursues all 3 violations, will not generate full-compliance. The firm and enforcement agency are both risk neutral and aim to maximise expected profit and minimise expected environmental damage respectively.

can someone explain to me what F(L) < 1 implies?

if you need the context behind this model, please tell me ill explain that as well

• @Glen_b alright i signed up for this website today so i wasnt aware of it, my other question which was legit just get downvoted and i was wondering why, makes sense now, thanks for the heads up – SGd May 1 '13 at 1:47
• You can edit your questions to better follow the guidelines. That may help. – Glen_b -Reinstate Monica May 1 '13 at 1:55

It means that the fine is lower than the compliance cost.

In the case of rational economics entities a firm will maximize its profit. This is not what is observed in reality. In theory, if the fine is lower than compliance cost a rationnal entity will not pay. In reality the fine is lower than compliance cost, but firms pay.

This suggest image concern ( or altruism....)

• Could you explain the stats behind it? (im a first year so im not the best) I understand theres a pdf, f(c) which has a CDF F(c) and then they go onto talk about F(L) what is the function F in F(L) here? is this a cdf? and more importantly, the coursework question is, "what does it imply when F(L) < 0.4" – SGd Apr 30 '13 at 15:46
• I d'ont use these words but yes, i think pdf = distribution and cdf = cumulative function. F is still the cumulative function for C. As $F(infinity) = 1$, $F(L) < 1$ strictly means that $P(c>L)>0$, that there is cost superior to the fine. This does NOT strictly means that the fine is lower to the compliance cost. But as far as I don't have the question I'am not sure wether it's an hypothesis you want to make or something you wnat to show. – lcrmorin Apr 30 '13 at 16:01
• $F(L) < 0.4$ means that c has <40% chance to be below L and >60 to be above. Express what does it mean for the expected profit of the company, the environmental damage. – lcrmorin Apr 30 '13 at 16:27
• hey imorin, ive attached the pdf file here link could you help me figure this out. there are 2 equations i do not understand how they were derived. none of my coursemates have chosen this topic so i cant get help anywhere else. could you explain equation (3) and (4) to me? and could you tell me how i should structure my answer with regards to what F(L) < 0.4 implies? – SGd Apr 30 '13 at 16:39
• I'm sorry but I can reach your pdf. Probably due to my internet proxy – lcrmorin May 2 '13 at 14:24

The cost of compliance for the company is $c_{ij}$ and the penalty for not complying is $\Lambda$. Since $c_{ij}$ is a random variable with cdf $F$, we have that $F(\Lambda)$ is another way of writing $P(c_{ij} < \Lambda)$, or in other words the probability that the compliance cost will be less than the cost of not complying. Or in other words, the probability that the company will comply.

So saying that $F(\Lambda) <1$ is just saying that the probability that the company will comply is less than $1$. Similarly, $F(\Lambda) < 0.4$ means that there is less than a $40\%$ chance that the company will comply.

For your questions in the comments about how to derive (3) and (4), to get (4) you observe that the only way in which a company can do more damage under dealing is if it's in category $\beta$ on page 365 of the paper. This is the same as $\mathrm{max}\{c_{i1}, c_{i2}\} < \Lambda$. Since $c_{i1}$ and $c_{i2}$ are independent, the probability of both of them being less than $\Lambda$ is $$P(c_{i1}, c_{i2} < \Lambda) = P(c_{i1} <\Lambda)P(c_{i2} < \Lambda) = F(\Lambda)^2$$ which gives (4).

To get (3), the company needs to be in category $\alpha$ on page 364, which means that one $c_{ij}$ has to be between $\Lambda$ and $2\Lambda$ and the other $c_{ij}$ has to be greater than $\Lambda$. The desired probability is $$\alpha(\Lambda) = P((\Lambda < c_{i1} < 2\Lambda \text{ and } c_{i2} > \Lambda) \text{ OR } (\Lambda < c_{i2} < 2\Lambda \text{ and } c_{i1} > \Lambda))$$ but when you have the OR" of two events you have to take into account that they might have outcomes in common, so you need to use the formula $P(X \text{ or } Y)=P(X) + P(Y) - P(X \text{ and } Y)$. Here, this gives you $$\alpha(\Lambda) = P(\Lambda < c_{i1} < 2\Lambda \text{ and } c_{i2} > \Lambda) + P(\Lambda < c_{i2} < 2\Lambda \text{ and } c_{i1} > \Lambda) - P(\Lambda < c_{i1} < 2\Lambda \text{ and } c_{i2} > \Lambda \text{ and } \Lambda < c_{i2} < 2\Lambda \text{ and } c_{i1} > \Lambda)$$ which using independence reduces to $$P(\Lambda < c_{i1} < 2\Lambda)P(c_{i2} > \Lambda) + P(\Lambda < c_{i2} < 2\Lambda)P(c_{i1} > \Lambda) - P(\Lambda < c_{i1} < 2\Lambda)P(\Lambda < c_{i2} < 2\Lambda)$$ which gives $$\alpha(\Lambda) = 2(1-F(\Lambda))(F(2\Lambda)-F(\Lambda))-(F(2\Lambda)-F(\Lambda))^2$$ which is (3).