Energy & Environmental Economics - Unibg 2015 Public... · 2015-12-14 · Examples Pure public...

Energy & Environmental EconomicsPublic Goods, Externalities and welfare

Università degli Studi di Bergamo

a.y. 2015-16

(Institute) Energy & Environmental Economics a.y. 2015-16 1 / 29

Public Goods

What is a public good ?

Pure public goods: goods which are:

- Non-rival: use of a unit of the good by consumer j does not reduce thepossibility for another consumer i to derive utility from the same unit ofthat good.- Non-excludable: it is not possible or too expensive, to exclude anyonefrom the access to the goodWhile non-rivalry is an intrinsic feature of the good under examination,excludability may depend on the institutional and technological setting.Non rivalry may exist up to a certain degree of utilization, beyond whichcongestion prevails.

Club Goods non-rival, but excludable goodsCommons: non excludable, but rival goods.


Examples

Pure public goods: a lighthouse; a planet�s temperature rangingbetween limits consistent with human life; clean air.

Club Goods: ideas and blueprints; a swimmming pool, a landscape;Commons: timber, game, and other products from an uncontrolledforest; �shing.


The optimal rule for private good provision

Suppose there are two goods, both private. Assume a very simpletechnology: 1 unit of good 1 can be converted into 1 unit of good 2, andviceversa. So, the marginal cost of good 1 in terms of good 2 is equal to 1. We already know what is optimal in this case: each consumer equates itsmarginal rate of substitution to the relative price

∂u∂x1∂u∂x2

=p1p2

and under perfectly competitive conditions,

p1p2= mc = 1

so that∂u∂x1∂u∂x2

= mc


The intuition behind this rule is:the allocation is e¢ cient when any individual is willing to give up, inexchange for one more unit of good 1, no more and no less of good2 than it is strictly necessary to produce that additional unit ofgood 1


.


The optimal rule for public good provision

With public goods the situation changes. Suppose now that good 1 is purepublic and good 2 is private, with the same production technology. Then ifyou increase production of good 1 by one unit, you still have to give upone unit of good 2, so that the marginal cost of good 1 in terms of good 2is still one. The di¤erence lies in the e¤ects on utility. Since good 1 isnon-rival, that single additional unit positively in�uences the utility of allindividuals, with the sacri�ce of just one unit of good 2 overall.A theorem by Samuelson (1953) states that in this case, the optimal rule is

n

∑i=1

∂u i∂x1∂u i∂x2

= mc = 1

The intuition behind this rule is now:the allocation is e¢ cient when the individuals, taken together arewilling to give up, in exchange for one more unit of good 1, no moreand no less of good 2 than it is strictly necessary to produce thatadditional unit of good 1.


Contributing to a public good: the free riding problem

Suppose that the public good is a lighthouse and the population consistsof sailors who all bene�t from the lighthouse. The taller is the lighthouse,the better is lighting and the higher is individual utility. Obviously, a tallerlighthouse requires larger quantitites of bricks, and all sailors are requiredcontribute by bringing as many bricks as they want. Besides bricks, theother good consumed by sailors is bread, and the relative price of bricksand bread is 1


Then each sailor maximizes his utility function

max uj xj ,Gj +∑

i 6=jGi

!where xj is bread consumed by sailor j , Gj is the number of brickscontributed by sailor j and ∑i 6=j Gi is the sum of bricks contributed by theother individuals (the "quantity" of publc good is measured by the numerof bricks, i.e. by the height of the lighthouse) The budget constraint forthe individual is:

y j � xj � GjSince the individual is maximizing his own utility, he equates his marginalrate of substitution to the price ratio

∂uj

∂G∂uj

∂xj

= 1


and since this is true of all individuals, we have

∑∂uj

∂G∂uj

∂xj

> 1

The provision of the public good is less than optimal (it would be possibleto increase all the utilities by increasing further the height of thelighthouse).The reason for this is that each individual bene�ts not only from the"bricks" he himself contributed, but also from those contributed by others,and he cares only about how his own contribution in�uences his ownutility, not the utility of the others. The outcome of this "sel�sh"behaviour is that everyone is worse o¤ than with a more "sociallycommitted" pattern of choice.


Declaring your preferences

Suppose now that lighthouses are of a �xed height, so that, the only thingto do is to choose between having one lighthouse (1) or none (0). De�newj as the j-th individual willingness to pay to have the lighthouse. It isdetermined as follows

uj�y j � wj , 1

�= u

�y j , 0

�i.e. the maximum the individual is willing to pay is the amount whichwould make him indi¤erent between having the lighthouse or not.Suppose also that everyone is asked to state how much he would be willingto pay to have the lighthouse built.The lighthouse will be built if bwj +∑

i 6=jbwi � c

where c is the production cost of the �xed-size lighthouse and bwj is thewillingness to pay declared. by the individual. As we shall see, bwj does notnecessarily coincide with wj .


Suppose thatn

∑i=1wi � c

i.e., should individuals tell the truth about their preferences, and shouldeach pay not more than his true willingness to pay, the lighthouse wouldbe built and everyone would be indi¤erent (or better o¤) . with thelighthouse than not.


Shall individuals reveal their true preferences ?

Suppose �rst that, should the lighthouse be built, each individual will becharged a uniform contribution t = c

n . If for some individual j

wj < t

he then does not want the lighthouse to be built, and he is going todeclare bwj = 0, hoping that the lighthouse project does not pass. If forsome other individual

wi � the will probably tell the truth (in this case .bwi = wi ) Then it mighthappen that

n

∑i=1bwi = ∑

i2Swi < c �

n

∑i=1wi

where S is the set of individuals who reveal their true preferences.In thiscase the lighthouse project will be stopped, even if it would be optimalwhen everyone pays no more than his true willingness to pay.


Suppose now that should the lighthouse be built, the j � th individual willbe charged a contribution t j = bwj . Then there is an incentive to declareless than one�s own true willingness to pay, in the hope that the overalloutcome is building the lighthouse anyway.Obviously the individual behaviour in this case depends to a large extenton what each one expects the others to declare. Suppose for instance thatthe j � th individual thinks that

∑i 6=jbwi � c

Then under the rule t j = bwj , he will declare bwj = 0.If instead he thinks:∑i 6=jbwi < c

he will declare bwj = c �∑i 6=j if

uj y j �

c �∑

i 6=jbwi!, 1

!� u

�y j , 0

�(Institute) Energy & Environmental Economics a.y. 2015-16 13 / 29

Social Choice

In general, public goods are decided upon by means of social choicemechanisms. Social choice theory is a very sophisticated district of theland of economic theory, which we cannot cover in this course. Just togive an idea, think of one of the best known social choice mechanisms,i.e. voting over alternative projects under a majority rule. Here is anexample of the so-called Condorcet paradox .


I choice II choice III choiceIndividual 1 Police station School BridgeIndividual 2 School Bridge Police stationIndividual 3 Bridge Police station School

Police st. vs School: Police st. winsPolice st. vs Bridge: Bridge winsSchool st. vs Bridge: School winsSocial preferences are not transitive !.


Externalities

In general, externalities appear when

1 The behaviour of individuals and �rms have a direct impact on other�rms�production functions and costs or on other individuals�utility

2 Agents who su¤er (enjoy) the externality are not compensated(charged) for it, nor agents who cause the externality are charged(compensated), as it usually happens

For this reason sometimes externalities are de�ned as "absence of markets"


Classi�cations

Positive externalities: the behaviour of one subject has a favourableimpact on the pro�t/utility of another subject. Examples: theneighbour�s �owers, on-the-job training

Negative externalities: the behaviour of one subject has a noxiousimpact on the pro�t/utility of another subject. Examples: theneighbour�s �owers (if I�m allergic), an ill-frequented pub, thegreenhouse e¤ect

Firm-to-Firm: chemical industry polluting the lake where a �sheroperates, on-the-job-training

Firm-to-individual: Fumes, wastes, noises, again training

Individual-to-individual: the neighbour�s �owers, the saxophone player

Individual-to-�rm ?


An example of Firm-to-Firm externality

Consider a chemical plant producing an output yK and, as a by-product, apolluting waste which is dispersed in a lake. Call Z the amount ofpollution and assume

Z = Z (yK )

with Z 0 (yK ) > 0. Instead

CK (yK )

is the cost function of the chemical factoryOn the same lake operates a �shery �rm, whose cost is

CF (yF ,Z )

where yF is �sh catch (the output of the �shery), andC py (yp ,Z ) ,C

pZ (yp ,Z ) > 0 (the larger the pollution, the larger is time,

e¤ort, fuel, etc. needed to catcha given quantity of �sh)(Institute) Energy & Environmental Economics a.y. 2015-16 18 / 29

The socially optimal solution

Assume that the only magnitude of interest are pro�ts, and that there areno other interested parties than the two �rms. Which is the maximumpro�t that the two �rms together may generate ?Assuming for simplicity prices equal to 1 both for �sh and chemicals, jointpro�t is

yK + yF � CK (yK )� CF (yF ,Z )or

yK + yp � CK (yK )� CF (yF ,Z )The �rst-order conditions for maximizing the above pro�t are:

1 =dCK (yK )dyK

+∂CF (yF ,Z (yK ))

∂ZZ 0 (yK )

1 =∂CF (yF ,Z )

∂yF


The market solution

If instead each �rm maximizes its own pro�t, we have

max yK � CK (yK )

with the �rst-order condition

dCK (yK )dyK

= 1

andmax yF � CF (yF ,Z )

with �rst-order condition

1 =∂CF (yF ,Z )

∂yF


The immediate conclusion we get from the comparison is that the marketsolution does not coincide with the socially optimal solution: this meansthat, although each �rm maximizes its own pro�t in the market setting, asa whole they might earn more if they behaved according to the sociallyotpimal solution.The di¤erence lies in the �rst-order condition concerning chemicalproduction:

1 =dCK (yK )dyK

+∂CF (yF ,Z (yK ))

∂ZZ 0 (yK )

(socially optimal)

1 =dCK (yK )dyK

(market)


It is easy to see that in the socially optimal solution, the marginal cost ofproducing chemicals has two components: one which stems out of thecosts of the chemical plant (Private Marginal Cost dC K (yK )

dyK), the other

stemming out of �sh production (Social Marginal Cost∂C F (yF ,Z (yK ))

∂Z Z 0 (yK )). The latter component disappears as we turn to themarket solution, since the chemical �m disregards the damaging e¤ects ofchemical production on the �shery industry.


PMC

PMC+SMC

P=1

YK(social) YK(private)


A2

a1

A6

A3

A7

A5

A9 A10

A4

A8

In the market solutionConsumer surplus: a1+ a2+ a3+ a4Pro�t: a6+ a7+ a8+ a9+ a10Social cost: a3+ a4+ a10+ a7+ a8+ a5Net Welfare:(a1+ a2+ a3+ a4) + (a6+ a7+ a8+ a9+ a10)�(a3+ a4+ a10+ a7+ a8+ a5) =(a1+ a2) + (a6+ a9)� a5


In the optimal solutionConsumer surplus: a1Pro�t: a2+ a3+ a6+ a7+ a9+ a10Social cost: a3+ a7+ a10Net Welfare: a1+ (a2+ a3+ a6+ a7+ a9+ a10)� (a3+ a7+ a10) =a1+ a2+ a6+ a9


The bargaining solution to the externality problem

Suppose that the �shery �rm is now entitled to sell to the chemical �rmrights to pollute,at some price w per unit of pollution. The pro�t functionfor the �shery �rm now changes as follows

max yF � CF (yF ,Z ) + wZwhere Z here is both the amount of pollution and the numer of pollutingrights sold to the chemical �rm (the latter can pollute the lake only to theextent that it has bought a corresponding number of rights). The�rst-order conditions are

1 =∂CF (yF ,Z )

∂yFand

w =∂CF (yF ,Z )

∂Zthe �shery sells polluting rights until the marginal revenue from sellingrights equals the increase in �shing cost due to additional pollution.


The pro�t function for the chemical �rm now changes as follows

max yK � CK (yK )� wZ (yK )

1 =dCK (yK )dyK

+ wZ 0 (yK )

1 =dCK (yK )dyK

+∂CF (yF ,Z )

∂ZZ 0 (yK )

At an equilibrium

1 =dCK (yK )dyK

+∂CF (yF ,Z )

∂ZZ 0 (yK )

i.e. the socially optimal solution


The interesting thing is that the same outcome appears if we assign Rrights to pollute to the chemical �rm, and let the �shing �rm buy suchrights order to prevent pollution. The pro�t function for the chemical �rmbecomes

max yK � CK (yK ) + w�R � R

�where R are the pollution rights that the chemical �rm keeps for itself.The following constraint must hold

R = Z (yK )

The �rst-order condition is

1 =dCK (yK )dyK

+ wZ 0 (yK )


The �shing �rm maximizes

max yF � CF (yF ,Z )� w�R � R

�with the �rst-order condition

1 =∂CF (yF ,Z )

∂yF

and

w =∂CF (yF ,Z )

∂Z


Energy & Environmental Economics - Unibg 2015 Public... · 2015-12-14 · Examples Pure public...

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