Externality (One Firm)
- V.A.6: Solve for equilibria when there are positive or negative consumption or production externalities and demonstrate that they are inefficient.
- V.A.7: Determine the effects of externalities on efficiency.
- V.A.8: Compare and contrast ways to correct externalities via regulation, creating markets, merging firms, and taxation.
Consider Coase Village, which has 40 people. Everyone works at Collegetown Bagels (CTB), and each person can either make bagels or brew coffee. If $L$ people make bagels, the total number of bagels produced is $f(L) = 30L - L^2$. Assume there are no production costs.
Each bagel sells for 3 dollars each, and each person who brews coffee generates exactly 30 dollars of revenue. Therefore, the profit function is: \[\pi(L) = 3(30L-L^2) + 30(40-L).\]
Use the above graph to answer the following questions, and first assume that CTB collectively decides how to divide its people into bagel production and coffee brewing in order to maximize total profit.
- How many people make bagels? How many bagels are made?
- What is the total revenue from bagels? What is the total revenue from brewing coffee? What is the total revenue for the village?
- Verify your answers to (a) and (b) by solving the village’s profit maximization function.
Now assume that CTB pays each person who makes bagels the average number of bagels made. Each person then earns their profit by selling their bagels for 3 dollars each. Coffee brewers earn the 30 dollars that they generate in revenue.
- What is the average payoff for each bagel maker? Your answer should be in terms of $L$.
- Assuming people are free to decide whether to make bagels or brew coffee, how many people will decide to make bagels? Remember that at equilibrium, no one should have incentive to switch from making bagels to brewing coffee or vice versa. What is the total revenue of the village now?
- Is this a positive or negative externality? What is the change in the total revenue that you calculated in part (b)?
Now suppose Coase Village issues permits to make bagels at price $c$. CTB is still allowing employees to choose whether to make bagels or brew coffee and will pay them the same as above. Anyone who chooses to make bagels must also purchase their permit for $c$ dollars.
- Rewrite the profit function of an individual bagel maker incorporating $c$. What is the new value of $L$? How many bagels are made? What is the new total revenue?
- Is there a value of $c$ that will yield the efficient quantity of bagels and coffee? If so, what is it?
- Suppose the Coase Village wanted to issue permits to brew coffee instead. Rewrite the profit function, but do not solve for $L$. Is there a cost $c$ of a coffee brewing permit that would yield the efficient production quantities? If so, what is it?
- Is there a way Coase Village could subsidize coffee production to yield the efficient production quantities? If so, how much would they have to pay each person that brews coffee?