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Suppose the inverse demand function expressed in dollars or a product is P = 80 – q, and the marginal cost in dollar of producing it is MC = lq, where P is the price of the product and q

Chapter 2 (page 37)

1 One convenient way to express the willingness-to-pay relationship between price and quantity is to use the inverse demand function In an inverse demand function, the price consumers are willing to pay is expressed as a function of the quantity available for sale Suppose the inverse demand function

(expressed in dollars) or a product is P = 80 – q, and the marginal cost (in dollar) of producing it is MC = lq, where P is the price of the product and q is

the quantity demanded and/or supplied (a) How much would be supplied in a static efficient allocation? (b) What would be the magnitude of the net benefits (in dollars)?

2 In the numerical example given in the text the inverse demand function for the

depletable resource is P = 8 – 0,4q and the marginal cost of supplying it is

$2,00 (a) If 20 units are to be allocated between two periods, in a dynamic efficient allocation how much would be allocation to the first period and how much to the be in the two period when the discount rate is zero? (b) What would the efficient price be in the two periods? (c) What would be marginal user cost be in each period?

3 Assume the same demand condition as stated in question 2, but for this question let the discount rate be 0,10 and the marginal cost of extraction be

$4,00 How much would be produced in each period in an efficient allocation? What would the marginal user cost be in each period? Would the static and dynamic efficiency criteria yield the same answer for this problem? Why?

4 Compare two versions of the two-period depletable resource model which differ only in the treatment of marginal extraction cost Assume that in the second version the constant marginal extraction cost is lower in the second period than the first (perhaps due to the anticipated arrival of a new, superior extraction technology) The constant marginal extraction cost is the same in both periods in the first version and is equal to the marginal extraction cost in the first period of the second version In a dynamic efficient allocation how would the extraction profile in the second version differ from the first? Would relatively more or less be allocation to the second period in the second version than in the first version? Would the marginal user cost be higher or lower in the second version?

APPENDIX………

The Simple Mathematics od Dynamic Efficiency*

Assume that the demand curve for a depletable resource is linear and stable over time

Thus the inverse demand curve in year t can be written as

P t = a - bq t

The total benefits from extracting and amount qt in year t is then the integral of this

function (the area under the inverse demand curve):

t

2

t

q

t

b

a bq dq aq  



Further assume that the marginal cost of extracting that resource is a constant c and therefore the total cost of extracting nay amount qt in year t can be given by

(Total cost)t = cqt.

If the total available amount of this resource is Q, then the dynamic allocation of a

resource over n year is the one which satisfies the maximization problem:

Max qt

2

i - 1

2

1

i

b

Assuming that Q is less than would normally be demanded, the dynamic efficient allocation must satisfy

1 i - 1 0

i

 , i = 1, …, n,

1

0

n

i

i



We can illustrate the use of these equations with the two-period example dealt with in

the text The following parameter values are assumed in that problem: a = 8, c = $2, b

= 0,4, Q = 20; and r = 0,10.

Using these, we obtain

8 – 0,4q1 – 2 – λ = 0,

8 - 0, 4 – 21

0 1,10

q



q 1 – q 2 = 20.

It is now readily verified that the solution (accurate to the third decimal place) is

q 1 = 10,238, q2 = 9,762, λ= $1,905.

We can now demonstrate the propositions discussed in the text

1 Verbally, equation (7) states that in dynamic efficient allocation the present

value of the marginal net benefit in period 1 (8 – 0,4q1 – 2) has to equal λ.

Equation (8) states that the present value of the marginal bet benefit in period 2sould also equal λ Therefore, they must equal each other This demonstrates the proposition shown graphically in Figure 2.7

2 The present value of marginal user the cost is represented by λ Thus equation

(7) states that price in the first period (8 – 0,4q1) should be equal to the sum of

marginal extraction cost ($2) and marginal use cost (1,905) Multiplying (8) by

1 + r, it becomes clear that price in the second period (8 – 0,4q2) is equal to the marginal extraction cost ($2) plus the higher marginal user cost [λ (1 + λ (1 + r) =

(1,905).(1,10) = $2,095] in period These results show why the graphs in Figure 2,8 have the properties they do They also illustrate the point that, in this case, marginal user cost rises over time

Chapter 3 (page 65)

1 Suppose the state is trying to decide how many miles of a very scenic river it should preserve There are 100 people in the community, each of whom has an

identical inverse demand function given by P = 10 – 0,1q, where q is the number of miles preserved and P is the per mile price he or she is willing to pay for q miles of preserved river (a) If the marginal cost of preservation is

$500 per mile, how many miles would be preserved in an efficient allocation? (b) How large are the net benefits?

2 (a)- Compute the consumer surplus and producer surplus if the product described by the first problem in Chapter 2 were supplied by a competitive industry Show that their sum is equal to the efficient net benefits

(b)- Compute the consumer surplus and the producer surplus assuming this

same product was supplied by a monopoly (Hint: The marginal revenue curve

has twice the slope of the demand curve)

(c)- Show that when this market is controlled by a monopoly, producer surplus

is larger, consumer surplus is smaller, and net benefits are smaller than when it

is controlled by competitive industry

3 Suppose you were asked to comment on a proposed policy to control oil spills Since the average cost of an oil spill has been computed as $X, the proposed policy would require any firm responsible for a spill to immediately pay the government $X Is this likely to result in the efficient amount of precaution against oil spills? Why or why not?

4 “In environmental liability cases, courts have some discretion regarding the magnitude of compensation polluters should be forced to pay for the environmental incidents they cause In general, however, the large the required payments the better” Discuss

(page 90)

1 In Mark A Cohen “The Costs and Benefits of Oil Spill Prevention and

Enforcement”, Journal of Environmental Economics and Management 13

(June 1986), an attempt was made to quantify the marginal benefits and marginal costs of U.S Coast Guard enforcement activity in the area of oil spill prevention His analysis suggests (p 185) that the marginal per gallon benefit from the current level of enforcement activity is $ 7,50 while the marginal per gallon cost is $ 5,50 Assuming these numbers are correct, would you recommend that the Coast Guard increase, decrease, or hold at the current level their enforcement activity? Why?

2 In his book Reducing Risks to Life: Measuring the Benefits, Martin Bailey

estimates that the cost per life saved by current government risk-reducing programs ranges from $72.000 for kidney transplants to $624.976.000 for a proposed standard to reduce occupational exposure to acrylonitrile

(a) Assuming these values to be correct, how might efficiency be enhanced in these two programs?

(b) Should the government strive to equalize the marginal costs life saved across all life-saving programs?

Chapter 8 (page 197)

1 Suppose a product can be produced using virgin ore at a marginal cost given by

MC 1 = 0,5q1 and with recycled materials at a marginal cost given by MC2 = 5 + 0,1q2 (a) If the inverse demand curve were given by P = 10 – 0,5(q1 + q2), how

many units of the product would be produced with virgin ore and how many

units with recycled materials? (b) If the inverse demand curve were P = 20 – 0,5(q2 + q 1), what would your answer be?

2 When the government allows private firms to extract minerals offshore or on public lands, two common means of sharing in the profits are bonus bidding and production royalties The former awards the right to extract to the highest bidder, while the second charges a per ton royalty on each ton extracted Bonus bibs involve a single, up-front payment, while royalties are paid as long as minerals are being extracted

(a) If the two approaches are designed to yield the same amount of revenue, will they have the same effect on the allocation of the mine over time? Why or why not?

(b) Would either or both be consistent with an efficient allocation? Why or why not?

(c) Suppose the size of the mineral deposit and the future path of prices are unknown How do these two approaches allocate the risk between the mining company and the government?

3 “As society’s cost of disposing of trash increases over time, recycling rates should automatically increase as well” Discuss

Chapter 12 (page 297)

1 Assume that the relationship between the growth of a fish population and the

population size can be expressed as g = 4P – 0,1P 2 , where g is the growth in tons and P is the size of the population (in thousands of tons) Given a price of

$100 a ton, the marginal benefit of smaller population size (and hence, large

catches) can be computed as 20P – 400 (a) Compute the population size that is

compatible with the maximum sustainable yield What would be the size of the annual catch if the population were to be sustained at this level? (b) If the marginal cost of additional catches (expressed in term of the population size) is

MC = 2(160 – P), what is the population size which is compatible with the

efficient sustainable yield?

2 Assume that a local fisheries council imposes an enforceable quota of 100 tons

of fish on a particular fishing ground for one year Assume further that 100 tons per year is the efficient sustained yield Once the 100th ton has been caught, the fishery would be closed for the remainder of the year (a) Is this an efficient solution to the common-property problem? Why or why not? (b) Would your answer be different if the 100-ton quota were divided up into 100 transferable

quotas, each entitling the holder to catch one ton of fish, and distributed among the fishermen in proportion to their historical catch? Why or why not?

3 In the economic model of the fishery developed above, compare the effect on fishing effort of an increase in cost of a fishing license with an increase in a per-unit tax on fishing effort that raises the same amount of revenue Assume the fishery is private property Repeat the analysis assuming that the fishery is a free-access common-property resource

Chapter 14 (page 350)

1 Two firms can control emissions at the following marginal costs: MC1 =

$200q1, MC2 = $100q2, where q1 and q2 are, respectively, the amount of

emissions reduced by the first and second firms Assume that with no control at all, each firm would be emitting 20 units of emissions or a total of 40 units for both firms

(a) Compute the cost-effective allocation of control responsibility if a total reduction of 21 units of emissions is necessary

(b) Compute the cost-effective allocation of control responsibility if the ambient standard is 27 ppm, and the transfer coefficient which translate a unit

of emissions into a ppm concentration at the receptor are, respectively, a1 = 2,0 and a2 = 1,0.

2 Assume that the control authority wanted to reach its objective in 1 (a) by using

an emission charge system

(a) What per unit charge should be imposed?

(b) How much revenue would the control authority collect?

APPENDIX………

The Simple Mathematics of Cost-Effective Pollution Control

Suppose that each of N polluters would emit un of emission in the absence of any control Furthermore suppose that the pollutant concentration KR at some receptor

R in the absence of control is:

K R =

1

N

n n n







Where B is the background concentration and an is the transfer coefficient This

K R is assumed to be greater than ɸ, the legal concentration level The

 

1

2 t 0

2

i - 1

i - 1

1 1

0

2

2

1

0

1

0

0, 4 – 2

0 1,10

t

n n

n n

N

n n n q

t

i

i

n i i

C q

P a q

b

a bq dq aq Q

b

r

r

q























Ω

Regulatory problem therefore is to choose the cost-effective level of control qn for each of the n sources Symbolically this can be expressed as minimizing the following Lagrangian with respect to the Nqn control variables:

–

Where Cn (q n ) is the cost of achieving the q n level of control at the nth source

and λ is the Lagrangian multiplier

The solution is found by partially differentiating (2) with respect to λ and the

Nq n ’s This yields

 

0

n n

n

C q

a q





    , n = 1, …, N,

1

N

n



Solving these equations produces the N-dimensional vector qo and the scaiar λo

Notice that this same formulation can be used to reflect both the uniformly mixed and non-uniformly mixed, single-receptor case In the uniformly mixed case

the an ’s all = 1 This immediately implies that the marginal cost of control should be equal for all emitters who are required to engage in some control (The first N

equations would hold as except for any source where the marginal cost of controlling the first unit exceeded the marginal cost necessary to meet the target) For the non-uniformly mixed, single-receptor case, in the cost-effective allocation the control responsibility would be allocated so as to ensure that the ratio of the marginal control

costs for two emitters would be equal to the radio of their transfer coefficients For J

receptors both λo and ɸ would become J-dimensional vectors.

Policy Instruments

A special meaning can be attached to λ If transferable permits were being used, it would be the market clearing price of a permit In the uniformly mixed case λ would be the price of a permit to emit one unit of emission In the non-uniformly mixed case λ would be the price being allowed to raise the concentration at the receptor location one unit In the case of taxes, λ represents the value of the cost-effective tax

Notice how firms choose emissions control when permit price or tax is equal to

λ Each firm want to minimize its costs Assume that each firm is given permits of Ωn

where the regulatory authority ensures that

1

N

n n

n



 Ω ɸ

for the set of all emitters Each firm would want to

min Cn (q n ) + Po[λ (1 + Ωn – an(un – q n)].

The minimum cost is achieved by choosing the value of qn (q n ) that satisfies

 

0

n n

n n

C q

P a q



This condition (marginal cost equals the price of a unit of conventration

reduction) would hold for each of the N firms Because Po would equal λo and the number of permits would be chosen to ensure the ambient standard would be met, this

allocation would be cost-effective Exactly the same result achieved by substituting To,

the cost-effective tax rate, for Po

Chapter 15 (page 378)

1 The marginal control cost curves for two air pollutant source affecting a

single receptor are MC1 = $0,3q1 and MC2 = $0,5q2, where q1 and q2 are controlled emissions Their respective transfer coefficients are a1 = 1,5 and a2 =

1,0 With no control they would emit 20 units of emission apiece The ambient standard is 12ppm

(a) If an ambient permit system were established, how many permits would be issued and what price would prevail?

(b) How much would each source spend on permits if they were auctioned off? How much would each source ultimately spend on permits if each source were initially given, free of charge, half of the permits?

Chapter 16 (page 403)

1 Explain why an acid-rain policy using emissions charge revenue to provide capital and operating subsidies for scrubbers is less cost-effective than an emission charge policy alone

2 The transfer costs associated with an emissions charge approach to controlling chlorofluorocarbon pollution are unusually large in comparison to other pollutants What circumstances would be favorable to high transfer costs?

Chapter 18 (page 456)

1 Consider the situation posed in Problem 1(a) in Chapter 14

(a) Compute the allocation which would result if 10 emission permits were given to the second source and 9 were given to the first source What would be the market permit price? How many permits would each source end up with after trading? What would the net permit expenditure be for each source after trading?

(b) Suppose a new source entered the area with a constant marginal cost of control equal to $1600 per unit of emission reduced Assume further that it would add 10 units in the absence of any control What would be the resulting

allocation of control responsibility? How much would each firm clean up? What would happen to the permit price? What trades would take place?

Problem Set Answers

CHAPTER 2 (page 576)

1 a Net benefits are maximized where the demand curve intersects the marginal

curve-cost curve Therefore, the efficient q would occur when 80 – lq = lq.

Thus the efficient q = 40 units

b Draw the diagram Draw a horizontal line from the place where the demand curve intersects the marginal cost curve to the vertical axis This intersection will take place at a price of 840 The net benefits can now be computed as the sum of the upper right triangle (the area under the demand curve and over this line) and the lower right triangle (the area under the demand curve and over this line) The area of a right triangle is x base x height Therefore, the net

benefits are x 840 x 40 + x 840 x 40 = $1600

2 a Ten units would be allocated to each period

b P = 88 – 0,4q = $8 - $4 = $4

c User cost = P – MC = $4 – 2 = $2

3 Because in this example the static allocations to the two periods (those which ignore the effects on the other period) are feasible within the 20 units available, the marginal user cost would be zero With a marginal cost of $4,00, the net benefits in each period would independently be maximized by allocating 10 units to each period In this example no intertemporal scarcity is present, so price would equal $4,00 marginal cost

4 Refer to Figure 2.7 In the second version of the model the lower marginal extraction cost in the second period would raise the marginal net benefit curve

in that period (since marginal net benefit is the difference between the

unchanged demand curve and the lower MC curve) This would be reflected in

Figure 2.7 as a parallel leftward shift out of the curve labeled “present value of marginal net benefits in period 2.” This shift would immediately have two