Exam 33: Externalities

The 130 campers at Bear Creek Campground love their own campfires but hate the smoke from their neighbors' campfires. Each camper's utility function is U = 25f - f² - s, where f is the number of hours her own campfire burns per day and where s is the amount of smoke in the air. It happens that s is 11 times the average amount of hours that campers use their fires. The campground authority could make all campers better off by limiting the number of hours of campfire per day for everyone. How many hours of campfire per day should the authority allow each camper in order to make the typical camper as well off as possible?

The efficient amount of air pollution is in general independent of whether polluters or pollutees pay to reduce pollution.

Two firms in a grimy Ohio town produce the same product in a competitive industry. Each has an old factory using an old technology. It still pays to operate these factories but it would not pay to expand them. The only variable factor used by either firm is labor. Each firm pollutes the other and thus reduces the output of the other firm. The production functions of firms A and B respectively are Q a = L .5 a ( )Q b and Q b = L .5 b ( )Q a, where L .5 a and L .5 b are the square roots respectively of the amount of labor used by firms A and B. The wage rate of labor is $1 and the price of the firms output is $12. a. If the two firms each maximize profits independently, what is their total output and how much quasi-rents do their factories earn? b. If someone buys them both and maximizes joint profits, how much quasi-rent is earned in total?

Suppose that in Horsehead, Massachusetts, the cost of operating a lobster boat is $4,000 per month. Suppose that if x lobster boats operate in the bay, the total monthly revenue from lobster boats in the bay is $1,000(12x - x²). If there are no restrictions on entry and new boats come into the bay until there is no profit to be made by a new entrant, then the number of boats who enter will be X₁. If the number of boats that operate in the bay is regulated to maximize total profits, the number of boats in the bay will be X₂.

The "tragedy of the commons" refers to the tendency for common property to be overused.

Firm 1 produces output x with a cost function c₁(x) = x² + 10. Firm 2 produces output y with a cost function c₂(y, x) = y² + x. Thus, the more that firm 1 produces, the greater are firm 2's costs. Both firms face competitive product markets. The competitive price of x is $20 and the competitive price of y is $40. No new firms can enter the industry and the old ones must remain. The efficient Pigouvian tax on the x good is

Marbella has 101 residents. All wear the same fancy clothes and each has the same utility function, u(m, b, B) = m + 14b - b² - , where m is the amount of macaroni (in kilograms) that he or she eats per day, b is the number of hours that he or she spends on the beach per day, and B is the total number of person-hours spent per day on the beach by other residents of Marbella. Each has an income of $10 per day and macaroni costs $1 per kilogram. The city council is considering a law that would limit the amount of time that any person can spend on the beach. How many hours per day should they allow in order to maximize the utility of a typical Marbellite?

Peter's utility is U(c, d, h) = 2c + 9d - d² - 6h, where d is the number of hours per day that he spends driving around, h is the number of hours per day spent driving around by other people in his home town, and c is the amount of money he has left to spend on other stuff besides gasoline and auto repairs. Gas and auto repairs cost $.50 per hour of driving. All the people in Peter's home town have the same tastes. If each citizen believes that his own driving will not affect the amount of driving done by others, they will all drive D₁ hours per day. If they are all drive the same amount, they would all be best off if each drove D₂ hours per day, where

Two stores are located side by side. They attract customers to each other and to themselves by advertising. The profit functions of the two stores are (45 + x₂)x₁ - 2x²₁ for store 1 and (75 + x₁)x₂ - 2x²₂ for store 2, where x₁ and x₂ are total advertising expenditures by stores 1 and 2 respectively. If each store sets its advertising expenditures independently (as in Nash equilibrium), how much would store 1 spend on advertising?

A mountain village owns a common pasture where villagers graze their goats. The cost to a goat owner of owning and caring for a goat is 4 groschens. The pasture gets overgrazed if too many goats share the pasture. The total revenue from all goats on the common pasture is f(g) = 48g - 2g², where g is the number of goats on the pasture. The town council notices that total profit from the pasture is not maximized if villagers are allowed to pasture goats for free. The council decides to allow a goat to use the common pasture only if its owner buys it a goat license. To maximize total profit (of villagers and council), how many groschens per goat should the council charge?

Peter's utility is U(c, d, h) = 8c + 18d - d² - 6h, where d is the number of hours per day that he spends driving around, h is the number of hours per day spent driving around by other people in his home town, and c is the amount of money he has left to spend on other stuff besides gasoline and auto repairs. Gas and auto repairs cost $.50 per hour of driving. All the people in Peter's home town have the same tastes. If each citizen believes that his own driving will not affect the amount of driving done by others, they will all drive D₁ hours per day. If they are all drive the same amount, they would all be best off if each drove D₂ hours per day, where

Two stores are located side by side and attract customers to each other and to themselves by advertising. Where x₁ and x₂ are the advertising expenditures of stores 1 and 2, the profits of the firms are (48 + x₂)x₁ - 2(x₁)² for store 1 and (54 + x₁)x₂ - 2(x₂)² for store 2. Knowing these functions, one investor buys both stores. In order to maximize his total profits, how much should he spend on advertising for store 1?

Suppose that in Horsehead, Massachusetts, the cost of operating a lobster boat is $4,000 per month. Suppose that if x lobster boats operate in the bay, the total monthly revenue from lobster boats in the bay is $1,000(28x - x²). If there are no restrictions on entry and new boats come into the bay until there is no profit to be made by a new entrant, then the number of boats who enter will be X₁. If the number of boats that operate in the bay is regulated to maximize total profits, the number of boats in the bay will be X₂.

An apiary is located next to an apple orchard. The apiary produces honey and the apple orchard produces apples. The cost function of the apiary is C_H(H, A) = - 3A and the cost function of the apple orchard is C_A(H, A) = , where H and A are the number of units of honey and apples produced respectively. The price of honey is $2 and the price of apples is $7 per unit. Let A₁ be the output of apples if the firms operate independently, and let A₂ be the output of apples if the firms are operated by a single owner so as to maximize total profit.

An airport is located next to a housing development. Where X is the number of planes that land per day and Y is the number of houses in the housing development, profits of the airport are 26X - X² and profits of the developer are 22Y - Y² - XY. Let H₁ be the number of houses built if a single profit-maximizing company owns the airport and the housing development. Let H₂ be the number of houses built if the airport and the housing development are operated independently and the airport has to pay the developer the total "damages" XY done by the planes to the profits of the developer.

Mike's utility function is U(c, d, h) = 8c + 12d - d² - 6h, where d is the number of hours per day that he spends driving around, h is the average number of hours per day spent driving around by other citizens of his town, and c is the amount of money he has to spend on other things than gasoline and auto repairs. There are 1,001 identical citizens in Mike's home town. Mike's expenses for gasoline and auto repairs amount to $.50 per hour for the time he spends driving. If Mike believes that his amount of driving won't affect the amount that others drive, how many hours per day will he choose to drive?

If preferences are quasilinear, then the delineation of property rights has no distributional consequences.

If your consumption of toothpaste produces positive externalities for your neighbors (which you ignore), then you are consuming less toothpaste than is Pareto optimal.

Two stores are located side by side. They attract customers to each other and to themselves by advertising. The profit functions of the two stores are (75 + x₂)x₁ - 2x²₁ for store 1 and (105 + x₁)x₂ - 2x²₂ for store 2, where x₁ and x₂ are total advertising expenditures by stores 1 and 2 respectively. If each store sets its advertising expenditures independently (as in Nash equilibrium), how much would store 1 spend on advertising?

If there are negative externalities in production or consumption, competitive equilibrium is unlikely to be Pareto efficient but positive externalities enhance the efficiency of the market.