You are a polluter who is faced with the question of whether to comply with the regulations on emissions or to violate them. If you comply with the regulations, your costs are \(k\). If you do not comply with the regulations, your costs are zero. The EPA has a new system whereby it audits your compliance with the regulation and gives you a black mark or gold star. If you have a gold star and are audited and found out of compliance, you pay a \(\$ 50\) fine and get a black mark. If you have a black mark and are audited and found in compliance, the EPA flips a coin and gives you a gold star with probability 0.5. If you have a black mark, are audited, and found out of compliance, you are fined \(\$ 100\). The probability of audit depends on whether you have a gold star (probability 0.1 ) or a black mark (probability 0.5 ). These probabilities and costs are annual.

Based on your costs, four strategies are available to you: never comply, always comply, comply only if you have a gold star, comply only if you have a black star. Call these strategies, \(\mathrm{nn}, \mathrm{cc}\), en, and \(\mathrm{nc}\). One is always best for you, and it depends on your costs of compliance, \(\beta\). Suppose you repeatedly interact with the regulator year after year. Your discount factor is \(k\) (you are indifferent between \(\$ 1\) tomorrow and \(\$ \beta\) today) and is equal to 0.9. Define your expected costs from all future interactions as \(C_{i j}(m)\) where \(i j\) is \(\mathrm{nn}, \mathrm{cc}\), en, or \(\mathrm{nc}\) and \(m\) is gold star or black mark.

Calculate the value of \(C_{i j}(m)\) for both \(m s\) and all four \(i j s\). [write an equation for a particular \(C_{i j}(m)\) in terms of this year's costs and next year's \(C_{i j}(m)\), which must be the same as this year's.]

What is the highest value of \(k\) for which you always comply with the regulations? What is the lowest value of \(k\) for which you never comply with the regulations?