Exercises on Externalities
ข้ อ 1
The picturesque village of Horsehead, Massachusetts, lies on a
bay that is inhabited by the delectable crustacean, homarus americanus,
also known as the lobster. The town council of Horsehead issues permits
to trap lobsters and is trying to determine how many permits to issue.
The economics of the situation is this:
1. It costs $2,000 dollars a month to operate a lobster boat.
2. If there are x boats operating in Horsehead Bay, the total revenue
from the lobster catch per month will be f(x) = $1, 000(10x − x2).
(a)
Plot the curves for the average product and the marginal product. In the same
graph, plot the line indicating the cost of operating a boat.
(b)
If the permits are free of charge, how many boats will trap lobsters in
Horsehead, Massachusetts? (Hint: How many boats must enter before there
are zero profits?)
(c)
What number of boats maximizes total profits?
(d)
If Horsehead, Massachusetts, wants to restrict the number of boats to the
number that maximizes total profits, how much should it charge a boat per
month for a lobstering permit?
ข้ อ 2
Suppose that a honey farm is located next to an apple orchard
and each acts as a competitive firm. Let the amount of apples produced
be measured by A and the amount of honey produced be measured by H.
The cost functions of the two firms are
cH(H) = H2/100 and cA(A) =A2/100 − H.
The price of honey is $2 and the price of apples is $3.
(a)
If the firms each operate independently, what would be the equilibrium
amount of honey and apples produced?
(b)
Suppose that the honey and apple firms merged. What would be the profitmaximizing output of honey and apples for the combined firm?
(c)
What is the socially efficient output of honey? If the firms stayed separate,
how much would honey production have to be subsidized to induce an
efficient supply?
ข้ อ 3
In El Carburetor, California, population 1,001, there is not
much to do except to drive your car around town. Everybody in town
is just like everybody else. While everybody likes to drive, everybody
complains about the congestion, noise, and pollution caused by traffic. A
typical resident’s utility function is U(m, d, h) = m+16d−d2−6h/1, 000,
where m is the resident’s daily consumption of Big Macs, d is the number
of hours per day that he, himself, drives, and h is the total amount of
driving (measured in person-hours per day) done by all other residents
of El Carburetor. The price of Big Macs is $1 each. Every person in El
Carburetor has an income of $40 per day. To keep calculations simple,
suppose it costs nothing to drive a car.
(a)
If an individual believes that the amount of driving he does won’t affect the
amount that others drive, how many hours per day will he choose to drive?
(b)
If everybody chooses his best d, then what is the total amount h of driving by
other persons?
(c)
What will be the utility of each resident?
(d)
If everybody drives 6 hours a day, what will be the utility level of a typical
resident of El Carburetor?
(e)
Suppose that the residents decided to pass a law restricting the total number of
hours that anyone is allowed to drive. How much driving should everyone be
allowed if the objective is to maximize the utility of the typical resident?
(Hint: Rewrite the utility function, substituting 1, 000d for h, and maximize
with respect to d.)
(f)
The same objective could be achieved with a tax on driving. How much would
the tax have to be per hour of driving? (Hint: This price would have to equal
an individual’s marginal rate of substitution between driving and Big Macs
when he is driving the “right” amount.)
ข้ อ 4
A clothing store and a jewelry store are located side by side in a small shopping mall. The
number of customers who come to the shopping mall intending to shop at either store
depends on the amount of money that the store spends on advertising per day. Each store
also attracts some customers who came to shop at the neighboring store. If the clothing
store spends $xc per day on advertising, and the jeweler spends $xj on advertising per day,
then the total profits per day of the clothing store are Πc(xc, xj) = (60+xj)xc −2xc2 and
the total profits per day of the jewelry store are Πj(xc, xj) = (105 + xc)xj − 2xj2
(In each case, these are profits net of all costs, including advertising.)
(a)
If each store believes that the other store’s amount of advertising is
independent of its own advertising expenditure, find the equilibrium amount
of advertising for each store. Then find the profit of each store.
(b)
The extra profit that the jeweller would get from an extra dollar’s worth of
advertising by the clothing store is approximately equal to the derivative of
the jeweller’s profits with respect to the clothing store’s advertising
expenditure. When the two stores are doing the equilibrium amount of
advertising that you calculated above, how much extra profit would a dollar’s
worth of advertising by the clothing store give the jeweler store? How much
extra profit would a dollar’s worth of advertising by the jeweller store give the
clothing store?
(c)
Suppose that the clothing store and the jewelry store have the same profit
functions as before but are owned by a single firm that chooses the amounts of
advertising so as to maximize the sum of the two stores’ profits. What would
be the amount of xc and xj, the single firm would choose? Without calculating
actual profits, can you determine whether total profits will be higher, lower, or
the same as total profits would be when they made their decisions
independently? How much would the total profits be?
ข้ อ 5
กำหนดให้ บริ ษัท Charcoal ผลิตถ่ำนจำกฟื นด้ วยฟั งค์ชนั่ กำรผลิต (production
function) y C (x C )  x C โดย y C เป็ นจำนวนถ่ำนที่ผลิตและ x C เป็ นจำนวนฟื นที่ใช้
นอกจำกนันก
้ ำหนดให้ กำรผลิตถ่ำนหนึง่ หน่วยทำให้ เกิดแก๊ ส g เป็ นจำนวน 0.25 หน่วย ส่วนบริษัท
Electric สำมำรถผลิตไฟฟ้ำจำกน ้ำมันหรื อแก๊ ส g โดยมีฟังค์ชนั่ กำรผลิต (production
function) y E (x E )  x E 
g
โดย y E เป็ นปริ มำณไฟฟ้ำที่ผลิตและ
2
x E เป็ นจำนวนน ้ำมันที่ใช้ (สมมุตใิ ห้ มีท่อส่งแก๊ สไปยังบริ ษัท Electric )
นอกจำกนัน้ กำหนดให้ บริษัททังสองผลิ
้
ตสินค้ ำในตลำดแข่งขันสมบูรณ์ซงึ่ มีรำคำถ่ำนเท่ำกับ 20
บำทต่อหน่วย รำคำฟื นเท่ำกับ 1 บำทต่อหน่วย รำคำไฟฟ้ำเท่ำกับ 80 บำทต่อหน่วย
และรำคำน ้ำมันเท่ำกับ 2 บำทต่อหน่วย
i)
ท่ำนคิดว่ำแก๊ ส g เป็ นสินค้ ำประเภทใด จงอธิบำย
ii) หำกบริ ษัททังสองต่
้
ำงผลิตสินค้ ำของตนโดยไม่สำมำรถทำกำรซื ้อขำยใดๆ ระหว่ำงกันได้
แต่ละบริ ษัทจะผลิตสินค้ ำเป็ นจำนวนเท่ำใดและจะมีกำไรเท่ำไร
iii) หำกเจ้ ำของบริษัท Electric ตัดสินใจซื ้อบริษัท Charcoal
จะทำให้ เกิดกำรผลิตถ่ำนและไฟฟ้ำเป็ นจำนวนเท่ำใด และมีกำไรเท่ำไร
iv) จงหำจำนวนกำรผลิตที่มีประสิทธิภำพที่สดุ
v) หำกเจ้ ำของบริษัท Electric ไม่ได้ ซื ้อบริษัท Charcoal
แต่กำหนดบริษัททังสองสำมำรถมี
้
กำรซื ้อขำยแก๊ สระหว่ำงกันได้
จงหำจำนวนกำรผลิตและกำไรของแต่ละบริ ษัท