Download Homework 2 - Instructure

Homework 2::
Hendrik Wolff
Environmental Economics ECON436
Homework 2
Kolstad, Chapter 3
Answer Question 6, part a) and part b). Use Excel
Kolstad, Chapter. 4
1. Consider the market for electricity. Suppose demand (in megawatt hours) is given by
𝑄𝑄 = 50 − 𝑃𝑃 and that the marginal private cost of generating electricity is $10 per megawatt
hour (P is the same units). Suppose further that smoke is generated in the production of
electricity in direct proportion to the amount of electricity generated. The health damage from
the smoke is $15 per megawatt hour generated.
a. Suppose the electricity is produced by competitive producers. What price will be
charged, and how much electricity will be produced?
The profit maximizing firms in the perfectly competitive market each set marginal
revenue equal to marginal cost, where each of their marginal revenue curves are
𝑴𝑴𝑴𝑴 = 𝑷𝑷. As represented by point A on the figure below, 𝒒𝒒∗𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 = 𝟒𝟒𝟒𝟒 and
𝒑𝒑∗𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 = 𝟏𝟏𝟏𝟏.
b. How would your answer in part a change if electricity were produced by an unregulated
monopolist?
The monopolist sets marginal revenue to marginal cost, where their marginal revenue
is 𝑴𝑴𝑴𝑴 = 𝟓𝟓𝟓𝟓 − 𝟐𝟐𝟐𝟐. Thus, as represented by point B in the figure below, 𝒒𝒒∗𝒎𝒎𝒎𝒎𝒎𝒎 = 𝟐𝟐𝟐𝟐
and 𝒑𝒑∗𝒎𝒎𝒎𝒎𝒎𝒎 = 𝟑𝟑𝟑𝟑.
c. In parts (a) and (b), what is the consumer surplus from the electricity generation? What
is the net surplus, taking into account the pollution damage?
i. The net surplus is 200. As presented in the figure below, the net surplus in part
(a) of the question is the sum of consumer surplus (CS) and producer surplus
(PS) minus the pollution damage (Dmg).
𝟏𝟏
𝟐𝟐
CS = Area(AYZ) = × 𝟒𝟒𝟒𝟒 × 𝟒𝟒𝟒𝟒 = 𝟖𝟖𝟖𝟖𝟖𝟖
PS = Area(AY) = 0
Dmg = Area(AWXY) = 𝟏𝟏𝟏𝟏 × 𝟒𝟒𝟒𝟒 = 𝟔𝟔𝟔𝟔𝟔𝟔
ii. The net surplus is 300. As presented in the figure below, the net surplus in part
(b) of the question is the sum of consumer surplus and the monopolist
producer surplus minus the pollution damage.
CS = Area(BTZ) =
𝟏𝟏
×
𝟐𝟐
𝟐𝟐𝟐𝟐 × 𝟐𝟐𝟐𝟐 = 𝟐𝟐𝟐𝟐𝟐𝟐
PS = Area(BTVY) = 𝟐𝟐𝟐𝟐 × 𝟐𝟐𝟐𝟐 = 𝟒𝟒𝟒𝟒𝟒𝟒
Dmg = Area(UVWY) = 𝟏𝟏𝟏𝟏 × 𝟐𝟐𝟐𝟐 = 𝟑𝟑𝟑𝟑𝟑𝟑
Remark: In a market with complete property rights (without pollution externalities), typically
a) In a competitive equilibrium there is no DWL (Dead weight loss), whereas
b) the monopoly generates a DWL because the firm restricts the quantity and charges a higher
price.
In the above example, however, we do not have complete property rights and each unit of production
of q produces marginal damage MD = 15. So what economy system is “better”, competitive or
monopoly? Here in the above example the net surplus of the monopoly is HIGHER (300) compared to
the competitive market (net surplus = 200)., so the monopoly is to be preferred. Is this always the
case? No, it really depends on the amount of MD produced by each unit of q. If MD is small relative to
the marginal cost, then the competitive market can be still be the preferred option. Make sure that
you understand this by moving around supply, MD and demand functions and convince yourself that
the above statement is correct.
2. In a photocopy of Figure 4.1, find and label the following points:
a. A point D such that Z and Y are Pareto preferred to D but S is not.
b. A point E such that the arc BZ is the portion of the Pareto frontier, which are Pareto
improvements on E.
Example:
4. Combine Figure 4.2 and 4.5 into a single figure showing how much garbage disposal and wine
will be produced in an economy consisting of Ivorytower Land Services (ILS—the producer) and
Brewster, our consumer. Show the relative prices of wine and garbage disposal services in the
figure. Assume Brewster’s income is fixed.
The amount of garbage disposal and wine is shown as 𝒒𝒒∗𝑮𝑮 and 𝒒𝒒∗𝑾𝑾 , respectively, on the figure
below. These values are given by the point of tangency between Brewsters highest
indifference curve and the production possibilities frontier. The slope at this point of tangency
represents the relative price ration of the two goods –Pw/PG.
7. Suppose Humphrey and Matilda live together. Humphrey currently smokes 20 packs of
cigarettes per month; Matilda hates the smoke. They currently have no agreement restricting
smoking. Their only joint expense is monthly rent, which they split 50:50. Draw an Edgeworth
box with two goods—smoke and rental payments. Make up some reasonable indifference
curves. Show the initial endowment. What Pareto efficient points might result from bargaining
to restrict smoke? How does the graph show what price per pack Matilda might pay to buy
down Humphrey’s smoking (i.e., show the relative prices on your figure)? How would your
answer change if the status quo is that the two have an agreement for no smoking and
Humphrey would like to smoke as much as 20 packs per month? He must seek Matilda’s
permission to do so. (Hint: For Matilda, redefine Humphrey’s smoking as smoke reduction.)
The figure below depicts an Edgeworth Box with indifference curves for Humphrey and
Matilda. First consider Humphrey: his utility depends on packs of cigarettes (a good) and
rental payments (a bad). He therefore has upward sloping indifference curves of the type
introduced in this chapter of the text. Their concave (downward) shape suggests diminishing
marginal utility of smoking. Matilda's utility depends on her rental payments and Humphrey 's
smoking, both bads. As suggested by the hint in the problem, we can redefine the leftward
direction on the horizontal axis for Matilda as smoke reduction (a good - for Matilda).
Consider the interpretation of the upper-left and bottom-right corners of the box. Point C
represents an arrangement where Humphrey pays 100% of the rent and smokes zero packs of
cigarettes. This is the point of highest utility in the box for Matilda, and the point of lowest
utility in the box for Humphrey. Point D is the opposite: Matilda pays all the rent and
Humphrey smokes 20 packs of cigarettes. This is the feasible point of highest utility for
Humphrey and lowest utility for Matilda.
The two endowment points considered in the problem are labeled A and B in the figure
below. At point A, rent is split 50:50 and Humphrey smokes 20 packs per month. Indifference
curves that pass through A are labeled 𝑼𝑼𝑨𝑨𝑴𝑴 and 𝑼𝑼𝑨𝑨𝑯𝑯 for Matilda and Humphrey, respectively.
These indifference curves are not tangent to one another, suggesting that bargaining over rent
and Humphrey's smoking can lead to a Pareto improvement. More specifically, consider the
lens-shaped area to the south-west of A. All points inside the lens are preferred to A by both
agents; it appears Matilda will be able to "buy down" Humphrey's smoking in an arrangement
that makes both of them better off.
Bargaining would result in an allocation like point E. At E, their indifference curves are
tangent, and no further mutually beneficial trades are possible. At E, the prevailing price of
smoke reduction is reflected in the slope of the dashed line connecting points E and A. This is
the rate at which Matilda must buy down her roommate's smoking.
Point B in the second figure represents the initial allocation where the rent is split
50:50, but a no-smoking arrangement is in place. Here the lens-shaped area of preferred
allocations lies to the north-east, suggesting that bargaining will involve Humphrey paying
Matilda for the right to smoke. An equilibrium arrangement would be supported by a point
like F in the figure. Humphrey could pay his roommate a price per pack given by the slope of
the dashed line that connects points Band F.
Notice that while the first arrangement would be strongly preferred by Humphrey,
and the second strongly preferred by Matilda, both lead through bargaining to allocations that
are Pareto optimal. This is the important result regarding the assignment of property rights to
which the text returns in Chapter 13.
Kolstad, Chapter. 5
3. Consider an air basin with only two consumers, Huck and Matilda. Suppose Huck's demand for
air quality is given by 𝑞𝑞𝐻𝐻 = 1 − 𝑝𝑝 where 𝑝𝑝 is Huck's marginal willingness to pay for air quality.
similarly, Matilda's demand is given by 𝑞𝑞𝑀𝑀 = 2 − 2𝑝𝑝. Air quality can be supplied according to
𝑞𝑞 = 𝑝𝑝 where 𝑝𝑝 is the marginal cost of supply.
a. Graph the aggregate demand for air quality along with individual demands.
Huck and Matilda's demand curves (𝑫𝑫𝑯𝑯 , 𝑫𝑫𝑴𝑴 ) are presented in the figure below along
with the aggregate demand (𝑫𝑫𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 ). We can characterize the inverse aggregate
demand function as
𝑷𝑷𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨
𝟏𝟏
𝟐𝟐 − 𝟏𝟏 𝑸𝑸𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 ,
𝟐𝟐
=�
𝟏𝟏
𝟏𝟏 − 𝑸𝑸𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 ,
𝟐𝟐
𝑸𝑸𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 ∈ [𝟎𝟎, 𝟏𝟏)
𝑸𝑸𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 ∈ [𝟏𝟏, 𝟐𝟐]
b. What is the efficient amount of air quality?
The efficient quantity of air quality is 0.8 and is found where Supply = Aggregate
𝟏𝟏
𝟐𝟐
Demand as seen in the figure below. In this case, 𝑸𝑸 = 𝟐𝟐 − 𝟏𝟏 𝑸𝑸𝑨𝑨𝑨𝑨𝑨𝑨𝑨𝑨 .
4. Consider an airport that produces noise that decays as the distance (d), in kilometers, from the
airport increases: 𝑁𝑁(𝑑𝑑) =
1
.
𝑑𝑑 2
Fritz works at the airport. Fritz's damage from noise is $1 per unit
of noise and is associated with where Fritz lives. His costs of commuting are $1 per kilometer
(each way). The closest he can live to the airport is d = 0.1 km.
a. Write an expression for Fritz's total costs (noise and transportation).
We are given that "noise costs" are 𝑵𝑵 =
𝟏𝟏
𝒅𝒅𝟐𝟐
and transportation costs are 𝑻𝑻 = 𝟐𝟐𝟐𝟐
($1 per kilometer, each way). Daily total costs are therefore given by
𝟏𝟏
Daily 𝑻𝑻𝑻𝑻 = 𝑻𝑻 + 𝑵𝑵 = 𝟐𝟐 + 𝟐𝟐𝟐𝟐
𝒅𝒅
b. What is the distance Fritz will live from the airport in the absence of compensation for
the noise? What are his total costs?
Where would Fritz live? He will choose 𝒅𝒅 to minimize total costs from noise and
transportation. Differentiating 𝑻𝑻𝑻𝑻 with respect to 𝒅𝒅 and setting this expression equal
to zero, the condition for minimization of the function in part (a) is
𝝏𝝏𝝏𝝏𝝏𝝏
𝝏𝝏𝝏𝝏
=−
𝟐𝟐
𝒅𝒅𝟑𝟑
+ 𝟐𝟐 = 𝟎𝟎
⇒
𝒅𝒅𝟑𝟑 = 𝟏𝟏
⇒
𝒅𝒅∗ = 𝟏𝟏km
Daily total costs are minimized where 𝒅𝒅 = 𝟏𝟏 km and Fritz's total cost is 3.
c. Suppose Fritz is compensated for his damage, wherever he may live. How close to the
airport will he choose to live? How much will he be compensated?
(Hint: Solve graphically or using calculus.)
Fritz is now told he will be compensated for any noise damage he suffers. Total costs
are now given by
𝟏𝟏
𝟏𝟏
− 𝟐𝟐 = 𝟐𝟐𝟐𝟐
𝟐𝟐
𝒅𝒅
𝒅𝒅
The problem is therefore reduced to one of minimizing travel costs. Fritz would move
as close as possible to the airport, which in this case is 𝒅𝒅 = 𝟎𝟎. 𝟏𝟏 km. Compensation, in
turn, is maximized at
𝟏𝟏
𝟏𝟏
Compensation = 𝑵𝑵 = 𝟐𝟐 =
= $𝟏𝟏𝟏𝟏𝟏𝟏
(𝟎𝟎. 𝟏𝟏)𝟐𝟐
𝒅𝒅
This outcome demonstrates what is known as "moving to the nuisance," and
represents a strong argument against compensation based on damages for victims of
externalities.
𝑻𝑻𝑻𝑻 = 𝟐𝟐𝟐𝟐 +
5. Two types of consumers (workers and retirees) share a community with a polluting cheese
factory. The pollution is nonrival and nonexcludable. The total damage to workers is 𝑝𝑝2 where 𝑝𝑝
is the amount of pollution and the total damage to retirees is 3𝑝𝑝2 . Thus marginal damage to
workers is 2𝑝𝑝 and marginal damage to retirees is 6𝑝𝑝. According to an analysis by consulting
engineers, the cheese factory saves 20𝑝𝑝 − 𝑝𝑝2 by polluting 𝑝𝑝, for a marginal savings of 20 − 2𝑝𝑝.
a. Find the aggregate (including both types of consumers) marginal damage for the public
bad.
We have the individual marginal damage functions for the two types of residents:
𝑴𝑴𝑫𝑫𝑾𝑾 = 𝟐𝟐𝟐𝟐
𝑴𝑴𝑫𝑫𝑹𝑹 = 𝟔𝟔𝟔𝟔
Because pollution is a nonrival bad, the aggregate marginal damage function is the
vertical sum of these individual damage functions:
𝑴𝑴𝑫𝑫𝑻𝑻 = 𝟐𝟐𝟐𝟐 + 𝟔𝟔𝟔𝟔 = 𝟖𝟖𝟖𝟖
b. Graph the marginal savings and aggregate marginal damage curves with pollution on the
horizontal axis.
The graph of marginal savings and aggregate marginal damage is below.
c. How much will the cheese factory pollute in the absence of any regulation or
bargaining? What is this society's optimal level of pollution?
In the absence of any regulation, the firm would pollute as long as the marginal
savings from pollution are positive. That is, the uncontrolled level of pollution, 𝒑𝒑𝟎𝟎 , is
found where marginal savings equals zero:
𝟐𝟐𝟐𝟐 − 𝟐𝟐𝟐𝟐 = 𝟎𝟎 ⇒
𝒑𝒑𝟎𝟎 = 𝟏𝟏𝟏𝟏
The socially optimal level of pollution, 𝒑𝒑∗ , is defined by the level at which marginal
savings from pollution are equal to the marginal damages from pollution:
𝟐𝟐𝟐𝟐 − 𝟐𝟐𝟐𝟐 = 𝟖𝟖𝟖𝟖 ⇒
𝒑𝒑∗ = 𝟐𝟐
d. Starting from the uncontrolled level of pollution calculated in part (c), find the marginal
willingness to pay for pollution abatement, A, for each consumer class. (Abatement is
reduction is pollution; zero abatement would be associated with the uncontrolled level
of pollution.) Find the aggregate marginal willingness to pay for abatement.
The uncontrolled level of pollution has been identified as 𝒑𝒑𝟎𝟎 = 𝟏𝟏𝟏𝟏. The question asks
to turn the problem around and find marginal willingness to pay (demand) for
pollution abatement. Let abatement equal A, so that 𝒑𝒑 + 𝑨𝑨 = 𝟏𝟏𝟏𝟏. At the uncontrolled
level, marginal damages to workers are 20. Marginal willingness-to-pay for pollution
reduction at 𝒑𝒑𝟎𝟎 is therefore 20. Marginal damages fall by 2 per unit as pollution is
decreased (abatement is increased), and are zero where = 𝟎𝟎 (𝑨𝑨 = 𝟏𝟏𝟏𝟏). To transform
the marginal damage (from pollution) function to a marginal willingness-to-pay (for
abatement) function, substitute for pollution using 𝒑𝒑 = 𝟏𝟏𝟏𝟏 − 𝑨𝑨
⇒ 𝑴𝑴𝑴𝑴𝑴𝑴𝑷𝑷𝑾𝑾 (𝑨𝑨) = 𝟐𝟐(𝟏𝟏𝟏𝟏 − 𝑨𝑨) = 𝟐𝟐𝟐𝟐 − 𝟐𝟐𝟐𝟐
⇒ 𝑴𝑴𝑴𝑴𝑴𝑴𝑷𝑷𝑹𝑹 (𝑨𝑨) = 𝟔𝟔(𝟏𝟏𝟏𝟏 − 𝑨𝑨) = 𝟔𝟔𝟔𝟔 − 𝟔𝟔𝟔𝟔
⇒ 𝑴𝑴𝑴𝑴𝑴𝑴𝑷𝑷𝑻𝑻 (𝑨𝑨) = 𝟖𝟖(𝟏𝟏𝟏𝟏 − 𝑨𝑨) = 𝟖𝟖𝟖𝟖 − 𝟖𝟖𝟖𝟖
The marginal willingness to pay (aggregate) can also be found by the vertical
summation of the Workers' and Retirees' marginal willingness to pay:
𝑴𝑴𝑴𝑴𝑴𝑴𝑷𝑷𝑻𝑻 (𝑨𝑨) = (𝟐𝟐𝟐𝟐 − 𝟐𝟐𝟐𝟐) + (𝟔𝟔𝟔𝟔 − 𝟔𝟔𝟔𝟔) = 𝟖𝟖𝟖𝟖 − 𝟖𝟖𝟖𝟖
e. Again starting from the uncontrolled level of pollution, what is the firm's marginal cost
of pollution abatement? What is the optimal level of A?
f.
Since we know 𝒑𝒑 = 𝟏𝟏𝟏𝟏 − 𝑨𝑨 we can also substitute this identity into the Marginal
Savings function to get the Marginal Cost of Abatement for the firm:
𝑴𝑴𝑴𝑴(𝑨𝑨) = 𝟐𝟐𝟐𝟐 − 𝟐𝟐(𝟏𝟏𝟏𝟏 − 𝑨𝑨) = 𝟐𝟐𝟐𝟐
The socially optimal level of abatement is 8 and is defined as the point at which the
marginal cost of abatement is equal to the aggregate marginal willingness to pay:
𝑴𝑴𝑴𝑴(𝑨𝑨) = 𝟐𝟐𝟐𝟐 =
𝟖𝟖𝟖𝟖 − 𝟖𝟖𝟖𝟖 = 𝑴𝑴𝑴𝑴𝑴𝑴𝑷𝑷𝑻𝑻 (𝑨𝑨)
⇒ 𝑨𝑨∗ = 𝟖𝟖
Are the problems of optimal provision of the public bad (pollution) and the public good
(abatement) equivalent? Explain why or why not.
The answers to parts (c) and (e) are the equivalent. The problems of optimal provision
of public bads (pollution in part (c)) and public goods (pollution abatement in part (e))
are logically the same. Problems of this type are cast in one way or the other for
convenience, but the underlying objective in choosing A or p is the same: to maximize
social welfare.