8

SectionSectionSectionSection 1: 1: 1: 1: Neoclassical welfare economics Neoclassical welfare economics Neoclassical welfare economics Neoclassical welfare economics

(see Chapter 5 of the textbook: Perman, Ma, McGilvray, Common) - the starting point of most economic analysis is the ‘neoclassical general equilibrium theory’ - this is the benchmark against which economists usually compare any types of inefficiency or market failure, e.g. related to imperfect competition or monopoly, incomplete information, or public good problems and externalities (e.g. related to environmental issues)

- the ‘nice’ thing about perfect competition in the neoclassical framework is, that it leads to Pareto efficiency, hence, a situation where nobody can be made better off unless somebody else is made worse off (“First Fundamental Theorem of Welfare Economics”)

- this is also a situation that society may want to achieve, because if somebody can be made better off without making someone else worse off, the social “well-being” can unambiguously be improved, hence, the original allocation was clearly not the best possible outcome

- unfortunately, a Pareto efficient allocation can be very ‘unfair’, e.g. if some individuals are particularly poor and others rich

- the problem is that Pareto efficiency alone says nothing about equity - to judge which of the many (in fact infinitely many) Pareto efficient allocations is ‘optimal’ (the most desirable one from a social point of view), society (or philosophers) must come up with some definition of ‘social welfare’

- whatever this definition might be, another nice thing about the neoclassical theory is, that it predicts that the resulting socially optimal allocation (one specific allocation among the continuum of Pareto efficient allocations) can always be achieved under market conditions, when the government uses lump-sum taxes and transfers to redistribute wealth

- this is a result of the “Second Fundamental Theorem of Welfare Economics”, which states that to every Pareto efficient allocation, there corresponds a competitive market equilibrium, obtained under a particular distribution of initial endowments and, thus, wealth (“wealth” is the value of a consumer’s initial endowment, evaluated at market prices)

- what do we learn from this? : if the assumptions of the neoclassical theory are (approximately) fulfilled, then the government must do only two things to maximize social welfare: 1. make sure that perfect competition prevails in all markets (hence, establish a functioning market economy): this assures Pareto efficiency, and 2. redistribute wealth for equity reasons to achieve not only economic efficiency, but also social optimality

- to put this in other words: theory suggests that, in policy analysis, problems of efficiency and equity can be dealt with independently

- furthermore, whenever economists think that there is a “problem” (e.g. global warming), it should be possible to identify one or several market failures; the assumptions of the

9

neoclassical theory are, then, violated; otherwise, there would be no “problem”, and perfect competition would automatically lead to an efficient allocation

- hence, in this course on environmental economics, we will often try to identify or characterize the reasons why an unregulated market economy does not lead to efficiency (environmental externalities are one possible source of market failure, but there are many others)

- furthermore, we will try to find out how a regulator can correct for these market failures, in order to achieve an efficient or optimal outcome, e.g. by using taxes

- remember: if there is no market failure, then there is no reason why the government should do anything (except redistributing wealth for equity reasons); therefore, as economists, we need to identify sources of market failure in order to justify a policy intervention

1.1 Welfare economics and efficiency in the static framework* - let us review the conditions of efficiency in a static framework (many of these things may be familiar to you)

- suppose, there are only two individuals in our model economy (A and B), two goods or services (consumption quantities X and Y), and two inputs or resources K and L (think e.g. of capital and labor) that exist in fixed quantities (this is the endowment of the economy)

- we assume away any externalities (hence, consumption or production of a good has no positive or negative side-effects on other consumers or firms)

- and we assume that all goods are private (not public), hence, if somebody owns a commodity (has property rights over it), nobody else can consume it or use it for production

- suppose, preferences over bundles of goods can be represented by utility functions:

( , )A A A AU U X Y= , ( , )B B B BU U X Y=

- if the output quantity X depends only on the input quantities used in the production of this

output ( XK and XL ), and similarly for output Y, the technological possibilities can be expressed by the following production functions (assuming efficient production):

( , )X XX X K L= , ( , )Y YY Y K L=

- when there are several firms, we may assume that all firms have the same technology; however, here, we look at efficiency at a more abstract level, and do not discuss any specific institutional arrangements (such as the existence of firms or markets)

- let /A A A

XMU U X= ∂ ∂ be A’s marginal utility from consumption of good X (similarly for

good Y, and for consumer B...)

- let /X X

KMP X K= ∂ ∂ be the marginal product of the input K in production of good X

10

- let AMRUS be consumer A’s marginal rate of utility substitution (the rate at which X can be substituted for Y at the margin), given that A’s utility remains constant

- to compute it, totally differentiate the utility function:

( , ) (...)A A A AA A A A A A A

X YA A

U X Y UdU dX dY MU dX MU dY

X Y

∂ ∂= + = ⋅ + ⋅

∂ ∂

- set this equal to 0 since utility is held constant (we move along an indifference curve), to get:

AAA X

A A

Y

MUdYMRUS

dX MU= = −

- this is the slope of A’s indifference curve at the location ( , )A AX Y (hence, AMRUS is not a

constant but depends on the location in the A AX Y− - space)

- let XMRTS be the marginal rate of technical substitution in the production of X (the rate at

which K can be substituted for L at the margin), given that the output quantity X remains constant

- totally differentiate the production function ( , )X XX X K L= to get:

( , ) (...)X XX X X X X X

K LX X

X K L XdX dK dL MP dK MP dL

K L

∂ ∂= + = ⋅ + ⋅

∂ ∂

- set this equal to 0 since the output X is held constant (we move along an isoquant), to get:

XX

KX X X

L

MPdLMRTS

dK MP= = −

- this is the slope of the isoquant at the location ( , )X XK L (an isoquant connects all input

combinations that yield a fixed output quantity X)

- let LMRT be the marginal rate of transformation of labor for the commodities X and Y: this

is the rate at which the output of one commodity can be “transformed” into the other by marginally shifting labor from one line of production to the other

- hence, LMRT is the increase in the output Y by shifting a marginal amount of labor from use

in the production of X to use in the production of Y - overall, the marginal rate of transformation (MRT) states the rate at which the output of one commodity (X) can be transformed into the other (Y) by shifting both inputs in an efficient

way; using the production functions ( , )X XX X K L= and ( , )Y YY Y K L= , it can be written

as (use the total differentials dX and dY ) :

11

Y YY Y Y YY YK L

X X X XX X K L

X X

Y YdK dL

MP dK MP dLdY K LMRTX XdX MP dK MP dL

dK dLK L

∂ ∂+

+∂ ∂= = =∂ ∂ +

+∂ ∂

- note, that since K and L are fixed, we have Y XdK dK= − and Y XdL dL= −

- when XK and YK are constant (we now shift only labor from use in production of X to Y),

the above formula simplifies to (similarly for KMRT ):

Y

LL Y X X

L

MPdY Y XMRT

L LdX MP

∂ ∂= = − = −

∂ ∂

- the marginal rate of transformation is the slope of the production possibility frontier (combinations of X and Y that the economy can generate with the given endowments)

- with these definitions, we can formally state the requirements for Pareto optimal allocations - there are three requirements: 1. efficiency in consumption, 2. efficiency in production, and 3. product mix efficiency

1.) Efficiency in consumption: A BMRUS MRUS=

Endowment of the

economy: K L

Input mix:

Output quantities:

Consumption bundles:

Utility:

YL YK XL XK

X Y

AX AY BX BY

AU BU

MRUSA

MRTSX

MRT

12

- when the marginal rates of utility substitution are equal for the two individuals A and B, it is not possible to rearrange the allocation of the (given) output quantities X and Y between them, so as to make one consumer better off without making the other one worse off

- hence, there is no more scope for voluntary trade

- A BMRUS MRUS= implies that the consumers’ indifference curves are tangent:

- position a in the above Edgeworth-box indicates some initial allocation of X and Y among the consumers, and position b is a Pareto efficient allocation after (voluntary) exchange took place (under market conditions, the connecting line between the dots represents the consumers’ budget lines, and the slope is the (negative) of the price ratio; however, we do not assume any market mechanism here, so the line is only for illustrative purposes)

2.) Efficiency in production: X YMRTS MRTS=

- when the marginal rates of technical substitution are equal for both commodities X and Y, it is not possible to reallocate inputs to production so as to produce more output of one commodity, without producing less of the other (otherwise production would be inefficient)

- X YMRTS MRTS= implies that the isoquants for X and Y are tangent:

AX

BY

AY

BX

AU

BU

a

b

XK

XL

YK

X

bY a

b

aY

YL

13

- compare position a and b in the above box; each of them shows a different use of the (fixed amount of) input factors capital and labor (K and L) in the production of the two outputs X and Y; in position a, more capital is used in the production of Y instead of X, and more labor in the production of X than Y

- however, the allocation of capital and labor in a is inefficient; moving along the isoquant X , the output quantity X remains constant; however, as we see in the figure, we move from the

isoquant aY to the isoquant bY , that corresponds to a higher output quantity Y; hence, the

allocation of the input factors K and L in point a is inefficient, that in point b is efficient - note, however, that this condition is not sufficient to reach full efficiency in the economy; it only assures that any (given) output quantities X and Y are produced efficiently with the inputs

- but there is a whole continuum of efficient points (such as point b in the figure), each of which implying a different combination of the output quantities X and Y; these combinations of X and Y are called production possibility frontier (output combinations that the economy can produce with its available resources)

- to reach full economic efficiency, one must determine which point on the production possibility frontier yields the highest utility; therefore, we need one more efficiency condition

3.) Product-mix efficiency: A B

L KMRT MRT MRUS MRUS= = =

- when the marginal rates of utility substitution are equal for the two individuals:

A BMRUS MRUS= , we can imagine for the moment that both individuals have the same

utility function (this is of course not correct, but given that A BMRUS MRUS= holds, the corresponding indifference curves have the same slope, and this is what really matters)

- hence, think for the moment of a representative consumer, so we will only plot one indifference curve

- to the same plot, we add the production possibility frontier:

X

Y

U

a

b

Production possibility frontier

14

- consider allocation a; it is located on the production possibility frontier, hence, the inputs K and L are used efficiently in the production of the output quantities X and Y

- however, by producing more X and less Y (moving towards allocation b), the representative consumer reaches a higher utility level

- at the point of tangency between the production possibility frontier and the indifference

curve U (point b), the representative consumer’s utility is maximized, given the available resources in the economy

- the slope of the production possibility frontier is the marginal rate of transformation (note,

that at an efficient allocation, it holds that L KMRT MRT MRT= = )

- an economy attains a fully efficient static allocation of resources if all three of the above conditions are satisfied simultaneously; the output levels X and Y, then, maximize the representative consumer’s utility, given the available resources of the economy

- these results generalize to economies with many inputs, many goods, and many individuals; for any pairwise combination of individuals, goods and inputs, the above conditions must, then, hold

15

1.2 Efficiency and optimality in the static framework* - we characterized conditions that must hold for economic efficiency of an allocation - if a given allocation is not efficient, this means that we can (by reallocating resources) make at least one consumer better off, without making any other consumer worse off

- however, these efficiency conditions do not determine a unique allocation - there is a continuum of efficient allocations, and it is a matter of fairness or social preferences which of these efficient allocations is socially desirable; the most desirable allocation is referred to as the “social optimum”

- before we discuss how the optimum can be defined, let us illustrate why there is a continuum of efficient allocations

- suppose first that the output quantities X and Y are fixed - they can be allocated in a variety of different ways between the individuals A and B

- the efficient allocations satisfy A BMRUS MRUS= and are located on the “contract curve”:

- similarly, in the production of the two outputs, there is not just a single allocation of capital and labor in the two lines of production that is efficient and, hence, satisfies

X YMRTS MRTS= , but a whole continuum of efficient allocations

- hence, there are many combinations of X and Y that are efficient in production, and for any particular combination, there are many allocations between A and B that are consistent with allocative efficiency

- intuitively, the entire stock of resources could be used only for individual A, while individual B gets nothing; efficiency, then, still requires that some efficient combination of X and Y is produced, and this also requires an efficient allocation of capital and labor in the

AX BY

AY

BX

contract

curve

16

production of X and Y; let max

AU be A’s resulting utility, and suppose, B’s utility in this case

is zero - alternatively, the entire stock of resources could be used only for individual B, while A gets

nothing; let max

BU be B’s resulting utility under efficiency, and suppose, A’s utility is zero

- in between these two extreme cases, there is a continuum of other efficient allocations that yield positive utility to both consumers

- the resulting frontier in the utility-space is called the “utility possibility frontier”:

- the utility possibility frontier represents all utility levels AU and BU that are consistent with Pareto efficiency

- however, some of the allocations on this frontier assign a high utility level to one individual, and a low one to the other

- until now, we do not know which one is the “social optimum” (the “best” allocation among all Pareto efficient allocations from a social point of view), because to determine this, we must be able to make interpersonal comparisons

- to this end, a “social welfare function” may be defined, that permits a ranking of alternative allocations:

( , )A BW W U U=

- this function formalizes how society values utility of different individuals

- two extreme cases are: { }min ,A BW U U= , which requires equal utility levels for all

individuals, or A BW U U= + - similarly as a utility function, also the social welfare function can be depicted using indifference curves (see the dotted line in the above figure)

- optimality requires that, in addition to the above efficiency conditions, the following optimality condition holds (a formal derivation or explanation is omitted):

AU

BU

Social optimum

Utility possibility frontier

Social welfare indifference curve

max

BU

max

AU

17

/

/

B BA

X Y

B A A

X Y

MU MUW U

W U MU MU

∂ ∂= =

∂ ∂

- unfortunately, there exists no widely agreed upon specification of a social welfare function - this is ultimately an ethical or cultural issue - therefore, economists usually avoid this, and try to focus on efficiency issues; if there is a policy of reallocation, the economist usually focuses on the question whether the gains exceed the losses; if they do, the policy can be recommended on efficiency grounds, since (in theory) the beneficiaries of the policy can compensate the losers

- it is a separate matter for government to decide whether compensation should actually occur

1.3 Allocation in a market economy (given ideal conditions)* - the “ideal conditions” under which a market economy leads to a Pareto optimal allocation (First Fundamental Theorem of Welfare Economics) are: markets exist for all goods and services, there is perfect competition (price-taking behavior), perfect information, private property rights are assigned over all resources and commodities, there are no externalities, no public goods, and utility and production functions are “well-behaved” (preferences and technologies are convex)

- here, we do not go through all the details of the neoclassical general equilibrium model; we only briefly illustrate why the equilibrium conditions in a market economy coincide with the above efficiency conditions of welfare economics

- as you should know from other classes, utility maximization implies that the slope of the indifference curve in the consumer’s optimum equals the slope of the budget line:

- the slope of the budget line is the negative of the price ratio /X YP P (the rate at which good

X can be substituted for good Y at a constant expenditure – hence, the opportunity cost of consuming one more unit of one good, measured in units of the forgone consumption of the other good)

AX

AY

AU (indifference curve)

optimal bundle

budget line

18

- this holds for all consumers, since all face the same prices in a market economy; hence:

A B X

Y

PMRUS MRUS

P= = −

- comparison with our earlier results shows that the consumption efficiency condition is, thus, fulfilled under market conditions

- now consider a firm that minimizes its production costs (given a target output quantity of X) - cost minimization requires that the slope of the isocost-line equals the slope of the isoquant at the optimum:

- the slope of the isoquant is the marginal rate of technical substitution, and the slope of the

isocost line is /K LP P− (the negative of the input price ratio)

- this holds for all firms and commodities produced, hence:

KX Y

L

PMRTS MRTS

P= = −

- therefore, the production efficiency condition we saw earlier is fulfilled - now consider the choice of the target output quantity X - in choosing the optimal amount of the input labor, a firm compares the benefits of a

marginal increase in XL with the resulting additional costs - the benefit is the marginal product of labor (hence, the additional output of X) times the

output price XP : X

X LP MP⋅

- the marginal cost is the price of labor, hence LP

- profit maximization requires that labor is used until the point where the marginal benefit of an increase in labor equals the marginal cost, hence:

XK

XL

X (isoquant)

optimal input mix to

produce X

isocost line

19

X

X L LP MP P⋅ =

- this holds also for capital, and also in the production of good Y, hence:

X

X L LP MP P⋅ = , X

X K KP MP P⋅ = , Y

Y L LP MP P⋅ = , Y

Y K KP MP P⋅ =

- eliminating the prices of capital and labor, we obtain:

Y

X L

X

Y L

P MP

P MP= and

Y

X K

X

Y K

P MP

P MP= , hence: X

K L

Y

PMRT MRT

P= = −

- combined with the condition A B X

Y

PMRUS MRUS

P= = − , this implies that the profit-

maximizing output levels under market conditions satisfy the product mix efficiency condition

- this completes the demonstration that in an ideal market system, the necessary conditions for allocative efficiency are satisfied

- combined with the Second Fundamental Theorem of Welfare Economics, we, thus, learn that a social optimum can always be implemented in such an ideal market economy by simply redistributing the initial endowments or wealth

1.4 Partial equilibrium approach - the above results reflect a general equilibrium approach (for illustrative purposes shown for two consumers, two commodities, and two inputs/resources)

- in general, there are many consumers, commodities, and resources, but we may only be interested in one particular market (e.g. the market for electricity)

- to reduce the complexity of the analysis, economists often aggregate all other goods (except the good of interest) into a “combined commodity”

- the good of interest may, thus, be good X, and the combined commodity (all other goods) be good Y

- if a consumer’s expenditure on good X is small relative to the expenditure on all other goods (good Y), saturation effects (declining marginal utility of good Y) will typically be small when the consumer increases or reduces its consumption of good X

- hence, we may assume that a consumer’s utility is (approximately) linear in Y; we, thus, obtain quasi-linear utility functions of the type:

( , ) ( )PU X Y U X Y= +

, where PU denotes the utility function used in the partial equilibrium model

20

- the price of the combined good Y is normalized to 1 (numeraire good, or simply “money”)

- the condition of tangency for utility maximization reads: X X

Y Y

MU P

MU P=

- since 1YP = and 1YMU = , this simplifies to (in the following, we simply use “P” for XP ):

( ) ( ) ( )PP X MU X MU X= =

� the price of good X equals its marginal utility (this holds for all consumers) - note, that demand for good X depends only upon P , but not on the consumer’s income � the market for good X can effectively be analyzed separately (there are no income effects)

- assuming a representative consumer, ( )P X is the inverse market demand function

- the above condition then implies that the inverse demand curve of a good is simply the MU-curve

- now define consumer surplus (CS) as the willingness-to-pay for X units minus the expenditure

- the willingness to pay for X units is given by the area under the inverse demand curve, and the expenditure for good X is PX , a rectangle in a price-quantity diagram:

inverse demand curve ( )P X

- a similar interpretation can be given to a supply curve in a partial equilibrium model

- profit maximization for a given cost function requires: max ( ) ( )X

X PX C Xπ = − (the cost

function C(X) is derived by solving the firm’s cost minimization problem, given the input

prices KP and LP ; it contains essentially the same information as that which is embedded in

the production function)

- FOC: ( ) ( )P X MC X= : each firm chooses its output quantity such that price equals

marginal cost

- here, ( )P X is the inverse supply function

P

CS

X X

P

21

- the inverse supply curve, thus, coincides with the MC-curve - solving for X, we obtain the firm’s supply function; aggregating over all firms in this market, we get the market supply curve

- the market clearing price is the price that equates demand and supply:

- remember, that the inverse demand curve (D) reflects the marginal utility of the representative consumer, hence, the marginal benefit of an increase in X

- the inverse supply curve (S) reflects the marginal cost of an increase in X - hence, the point where demand and supply are equal, is also the point where the marginal benefit of an increase in X is equal to the marginal cost:

MU MC P= =

- the net benefit is defined as the total benefit minus the total cost; it corresponds to the shaded area in the above figure (interpretation of the net benefit: it is the utility given the efficient provision of the good X, net of the utility when good X is not produced at all: X=0)

- if we assume that all other markets are perfectly competitive, but the quantity X (that may be a consumption good, or a bad such as pollution) is chosen by the regulator (a benevolent social planner or the government), then the curves S and D in the above figure can be reinterpreted as marginal benefit function and marginal cost function (resp. marginal damage function – depending on the problem)

P

X

X

D S

P

X

22

Section 2: Market failure, public goods, externalitiesSection 2: Market failure, public goods, externalitiesSection 2: Market failure, public goods, externalitiesSection 2: Market failure, public goods, externalities

(see Chapter 5 of the textbook)

2.1 Market failure, public goods - we already mentioned conditions that must hold in an “ideal” market economy, under which the market outcome coincides with the allocation that a social planner would choose, hence, the “social optimum”

- if any of these conditions is not fulfilled, the market outcome will be inefficient, and we say there is a “market failure”; let us summarize these ideal conditions:

1. Markets exist for all goods and services produced and consumed 2. All markets are perfectly competitive 3. All transactors have perfect information 4. Private property rights are fully assigned in all resources and commodities 5. No externalities exist 6. All goods and services are private goods (there are no “public goods”) 7. All utility and production functions are “well-behaved” 8. All agents are maximizers (utility/profit maximizers).

- let us discuss a few examples where one or several of the above conditions are violated - e.g., private property rights often do not exist for renewable resources - an example is ocean fishery; if anyone can go out an fish, the exploitation of this resource is uncontrolled, and there will generally be over-exploitation

- another example are “stock-pollution problems”, where the earth or the atmosphere is used as a waste sink, e.g. for carbon dioxide

- generally, no private property rights are assigned, so the atmosphere is an open-access resource; note, that air pollution causes a negative externality

- an important distinction that is often made in the literature is between private and public goods

- private goods are characterized by rivalry and excludability - rivalry refers to whether one agent’s consumption is at the expense of another agent’s consumption (think, e.g., of ice cream)

- excludability refers to whether agents can be prevented from consuming - a “pure” public good is e.g. national defense; no citizen can be excluded from enjoying the benefits of it, and the consumption is clearly non-rival

- some public goods are non-rival, but excludable; they are usually referred to as “congestible resources”; an example are wilderness areas; enjoying wilderness by one individual is

23

generally not at the expense of another agent’s joy (unless it is used to such an extent that congestion occurs); however, using fences, individuals can be excluded from consumption

- other public goods are rival in consumption, but non-excludable; they are usually referred to as “open-access resources”; an example is the ocean-fishery; consumption is rival, because each fish can only be caught once, so the more boats are fishing, the harder it gets to fish; however, it is difficult to exclude anyone from fishing

- let us now analyze the efficient allocation of a public good formally - to this end, we go back to our earlier model-economy with two consumers (A and B), and two goods (X and Y)

- the top-level product-mix condition for allocative efficiency was given by:

A BMRUS MRUS MRT= = - intuitively, MRT describes how (given an efficient use of the inputs to production) good Y can be transformed into good X, and vice versa; MRT, thus, describes how costly the production of good X is in terms of forgone output of good Y

- MRUS describes at what rate a consumer would exchange good Y for good X, given that the consumer’s utility remains unchanged; MRUS, thus, describes how valuable the consumption of good X is in terms of forgone consumption of good Y

- efficiency requires that these relative benefits / costs must be equal at the margin; otherwise, production or consumption of one of the goods may be raised, and the consumer who becomes better off can compensate the consumer who gets worse off such that both consumers benefit from the voluntary exchange (hence, a Pareto improvement is possible)

- now suppose, X is a public good, and Y a private good; this means that consumers do no longer care about their individual consumption of good X, but only about the aggregate

consumption: A BX X X= + - hence, the utility functions are given by:

( , )A A AU U X Y= , ( , )B B BU U X Y= , where A BX X X= +

- it can be shown that in this case, the above top-level efficiency condition becomes:

A BMRUS MRUS MRT+ = - intuitively, this means that for efficiency, not the individual marginal benefit of consumption of good X matters, but the aggregated marginal benefit of all consumers; in general, this implies that more of good X will be provided in the optimum than under market conditions (the market provides to little of the public good – the reverse holds true if X is a public bad, such as pollution)

The derivation and interpretation of the above efficiency condition is easier in the context of a partial equilibrium model; let us go through the details:

24

- in a partial equilibrium model, the utility functions are quasi-linear, and can be written as (the superscript P stands for Partial equilibrium model):

, ( )A P A AU U X Y= + , , ( )B P B BU U X Y= +

- let ( )Y f X= be the production possibility frontier (combinations of X and Y that the

economy can generate, given an efficient use of the available resources) - to determine the efficiency condition, set up the Lagrangian

- the target function is ,P AU , A’s utility, while B’s utility is held constant (this reflects the idea of Pareto efficiency: maximize A’s utility, without making B worse off)

- the technological constraint is ( )A BY Y Y f X= + = ; for simplicity, we substitute for BY ,

using this constraint, to obtain:

( ), ,( ) ( ) ( ( ) )P A A P B A BL U X Y U X f X Y Uλ= + − + − −

- the first-order conditions (FOC) are:

0 '( )A BMU MU f Xλ λ= − − and 0 1 λ= +

- using '( )dY

f X MRTdX

= = , and 1λ = − (second condition), the first condition becomes:

A BMU MU MRT+ = −

- MRT is the slope of the production possibility frontier (note, that it is negative):

- it is convenient to rewrite MRT as a cost function - note, that Y is the numeraire good, hence, its price is 1, and can be interpreted as “money”

- the cost of good X in terms of forgone production of good Y is, thus, simply (0)f (all

resources are used to produce only good Y) minus ( )f X (the quantity of good Y when X

units of the public good are produced); hence, the cost function is given by:

X

($)Y

(0)f

Production possiblily frontier

( )f X

X

25

( ) (0) ( )C X f f X= −

- this yields for the marginal cost of X:

( ) '( )MC X f X MRT= − = −

- the marginal cost of good X in terms of good Y is, thus, the absolute value of the marginal rate of transformation

- the above first-order condition can, thus, be rewritten as follows (this is the partial equilibrium version of the top-level product-mix efficiency condition):

A BMU MU MC+ =

- this means that the public good X should be provided to the point where the aggregated marginal utility equals the marginal cost; since the marginal utility corresponds to the marginal willingness-to-pay, we can also say this as follows: supply the public good at the level where aggregate marginal willingness-to-pay is equal to marginal cost

- under market conditions, the inverse demand functions of consumers A and B are given by:

( )A AP X MU= and ( )B BP X MU=

- if consumer A has a higher willingness-to-pay, this looks as follows:

- under market conditions, only consumer A (the one with the higher willingness-to-pay) will pay for the public good, and it is supplied up to the point where A’s individual marginal utility equals the marginal cost of the public good; consumer B does not pay anything for the public good; because it is non-rival and not excludable, this consumer can nevertheless enjoy the same quantity of the public good as consumer A

- under efficiency, the public good is provided until the point where the aggregated marginal

willingness-to-pay A BMU MU+ equals the marginal cost, hence, a higher quantity is provided than under market conditions (this illustrates the existence of a market failure)

A BMU MU+ MC

X

AMU BMU

P

26

2.2 Externalities - an externality (external effect) occurs when the production or consumption decisions of one agent have an impact on the utility or profit of another agent in an unintended way, and when no compensation is made by the generator of the impact

- externalities can be classified according to whether they arise during production processes or during consumption (e.g. smoking), and whether they affect firms or other consumers’ utility / consumption

- externalities can be harmful (negative) or beneficial (positive) to the affected agents - an example of a positive externality is knowledge generated via R&D by one firm, that partially “leaks” to other firms and, thus, positively affects their productivity (knowledge-spillovers)

- externalities are a source of market failure; in the presence of a negative externality (e.g. pollution), the good or by-product is oversupplied; in the presence of a positive externality (e.g. knowledge spillovers), private agents face insufficient incentives to generate the good because they can not fully appropriate the rents, so it tends to be undersupplied under market conditions

- consider for example a situation where some agent’s activity generates an external effect (e.g. a firm that pollutes the environment, or a smoker)

- let the agent’s marginal benefit (e.g. marginal profit in case of a firm) be MB(Z) if Z is the level of the activity that causes the externality, and let the marginal external cost to society or to some other individual be MEC(Z); the situation is illustrated in the following figure:

- if the external costs are not taken into consideration (market outcome without policy intervention), the polluter chooses Z until the point where the private marginal benefit goes to zero; depending on the shape of the MB-curve, this can be a very high quantity

- if the external costs are taken into consideration (e.g. because of government intervention, or when private property rights can be assigned to the environment), hence, when the

MEC(Z)

Z

( )MB Z

P ($)

27

externality is internalized in the polluter’s decision-making, the efficient activity level is reached, located at the point where the marginal benefit equals the marginal external costs

The Coase Theorem: - the problem that arises in the presence of external costs or benefits is that the involved individuals and firms do not own property rights over the polluted environment (e.g. the clean air in case of a smoker), or can not fully appropriate the rents generated by their R&D activities (knowledge spillovers)

- suppose, it is possible to assign such property rights (to pollute the air, or property rights over clean air, over knowledge and ideas…)

- the Coase theorem (Coase, 1960) states that, when property rights are assigned, individuals can bargain over the amount of pollution or other activity causing the externality, and the outcome of this bargaining process will be Pareto efficient, irrespective of the initial allocation of the property rights

- hence, in a shared flat with one smoker and one non-smoker, the outcome of the bargaining process will be Pareto efficient, no matter whether rights of clean air are assigned to the non-smoker, or whether rights to pollute are assigned to the smoker

- note, however, that since a continuum of Pareto efficient allocations exists in a general equilibrium framework, nothing is said about which Pareto efficient allocation will be reached, and whether it is the socially optimal one; hence, although both cases lead to economic efficiency, the initial allocation of property rights can affect the resulting allocation via wealth effects

- in a Partial equilibrium framework (as in the above figure), the amount of pollution will, however, not depend on the initial allocation of property rights, because there are no income effects; the agent who gets the property rights benefits from a higher wealth, but this only leads to a higher consumption of the numeraire good (the location of the MB and the MEC – curves are independent of the initial endowments or wealth of the individuals)

- this argument can also be applied to the issue of the initial allocation of emission certificates in a cap-and-trade scheme (grandfathering versus auctioning); we will come back to this issue later in this course

- finally consider an example where the pollution is also costly to the polluter; e.g., if we discuss CO2-emissions that result from productive activities, then the costs to the firm are clearly correlated with the emissions (more output – more pollution)

- hence, in contrast to the above figure, we may distinguish between private marginal costs of some activity Z (that causes an externality), and social marginal costs (obtained when the external costs are added to the private costs of the activity Z):

28

- if the polluter is a competitive firm, Z is the output, and the output price is P, the firm will produce until the point where PMC equals the output price (PMC = P)

- however, efficiency requires that the firm produces until the social marginal cost equals P - to achieve this, the regulator can introduce a tax on the activity Z equal to the marginal external cost (t = MEC)

- such a tax is often referred to as a “Pigouvian tax”, because it leads to an internalization of the external effect; the Pigouvian tax leads to Pareto efficiency

- as a final note, consider situations that are characterized by multiple market failures (hence, more than just one of the above conditions for an ideal market economy is violated)

- it may sometimes be possible to correct some of these inefficiencies, but not all of them - in that case, it is not necessarily in the interest of society to correct the inefficiencies that can be corrected, given that others can not be corrected

- sometimes, different market failures can distort the outcome in reverse directions (e.g. an environmental externality can imply too much economic activity, while knowledge spillovers can imply too little)

- hence, when only one market failure is corrected, the outcome can actually be worse than without any government intervention

- the problem to find an optimal policy, given that some market failures can not be corrected, is often referred to as the “second-best problem” (because the first-best outcome – Pareto efficiency – can not be obtained when some market failures can not be corrected)

PMC(Z)

Z

P

SMC(Z) = PMC+MEC P ($)

29

2.3 Abatement of emissions

- “abatement” means that less greenhouse gases are emitted under an environmental policy than without the policy

- we usually refer to an outcome or a projection obtained in the absence of any environmental policy intervention as a BAU – “business as usual” outcome or scenario

- suppose, all other markets in the world economy are perfectly competitive, but there exists no market for CO2-emissions (these emissions are a public “bad”)

- hence, the assumptions of the neoclassical general equilibrium model are not fulfilled, and an inefficiency arises

- the absence of a market for emissions implies that there will generally be an inefficiently high level of emissions, hence, there is a market failure (due to the negative externality they constitute)

- to correct for this market failure, the regulator can simulate a market by introducing a price for CO2-emissions (e.g. using a carbon tax or a cap-and-trade scheme)

- alternatively, suppose the regulator is a benevolent social planner who can directly control the amount of carbon used by all individuals and firms (this is less plausible, but corresponds to our earlier discussion of welfare economics); hence, the planner directly controls the abatement A (and how much each individual or firm must abate)

- when determining A, the regulator must compare the benefits with the costs of CO2-abatement

- suppose, if the world economy abates an amount A of emissions (relative to baseline emissions – BAU), then costs of C(A) are incurred (welfare losses)

- at the same time, the problem of climate change becomes less severe, as less greenhouse gases are emitted than under BAU; suppose, the reduction in economic and social damages, measured in monetary units, when the amount A of emissions is abated, equals B(A) (B for “benefit”)

- suppose further that C(A) is convex, reflecting the idea that abatement costs rise more than proportionally; marginal abatement costs C’(A) are, thus, increasing

- B(A) is assumed to be concave; the marginal benefit function B’(A) is, thus, downward sloping (this reflects the idea that when, say, 50 percent of total emissions are abated, a certain benefit is achieved; if the remaining 50 percent are also abated, the additional benefit (due to less severe climate change) is lower than before, because climate change is already much less severe than under BAU)

30

- the vertical distance between B(A) and C(A) is the net benefit of the abatement A Now what is the optimal amount of abatement (which maximizes the net benefit)? If, starting from any given abatement level, another unit of emissions is abated (A is marginally increased), then the costs of this marginal abatement are C’(A). The marginal benefit equals B’(A). If the marginal benefit exceeds the marginal abatement cost, then it is overall beneficial to society to abate more (as the benefit of a marginal increase of A exceeds the cost). If B’(A) is smaller than C’(A), then the benefit of a marginal increase in A is lower than its cost. Therefore, it is instead beneficial to reduce the amount of abatement (so there can be “too much” abatement). The efficient amount of abatement is reached when:

'( ) '( )B A C A=

, hence, when the marginal benefit equals the marginal abatement cost:

- the green shaded area is the net benefit B(A) – C(A); it corresponds to the welfare gain due to the abatement A (relative to a BAU allocation)

- if the optimal abatement is to be achieved using a carbon tax, the regulator should set the tax at the level t as shown in the above figure; firms will, then, “internalize” the environmental damages caused by their CO2-emissions, and an allocation will be achieved as if CO2 were a private good (hence, Pareto efficiency is achieved under market conditions)

- to see this more clearly, suppose there are J firms

B‘(A)

C‘(A)

Optimal Abatement

t

Abatement A

B(A)

C(A)

Abatement A

net

benefit

31

- firms operate in different sectors, countries, and with different technologies

- therefore, each firm has an individual abatement cost function ( )j jC A , where jA is the

amount of abatement by firm j given the carbon tax t (relative to BAU) - from the point of view of a firm, the marginal revenue from abatement is equal to the tax,

because when the amount jA is abated, tax expenditures of jt A⋅ are avoided

- firm j’s marginal abatement cost is '( )j jC A ; if this is lower than t, the firm benefits from

abating more emissions, because the tax savings exceed the abatement costs (at the margin)

- therefore, a profit maximizing firm abates until the following condition holds: '( )j jC A t=

- since the condition '( )j jC A t= holds for each firm j, it follows that under a uniform carbon

tax, the marginal abatement costs of all firms are equalized:

'( ) '( )i i j jC A C A= , for all ,i j

This is also required for efficiency (and, hence, from a welfare perspective). Suppose to the contrary that the marginal abatement costs were not equalized in the optimum. Then, there is a firm (say, firm 1) with higher marginal abatement costs than another firm (say, firm 2). This means that if firm 1 abates a bit less, and firm 2 abates an equal amount more, the aggregated (total) abatement costs are reduced, because at the margin, it is cheaper to abate for firm 2. Therefore, a situation where (any two) firms have different marginal abatement costs can not be Pareto efficient. - note, that this discussion yields an interesting and powerful conclusion: if a carbon tax is introduced to correct for the market failure caused by environmental externalities of the emissions, the tax should be identical for all firms, in all sectors of the economy, and in all countries around the world

- otherwise, Pareto efficiency will not be achieved

'( )j jC A

Abatement Aj

Optimal Abatement

t

32

- a similar conclusion holds for a cap-and-trade (emission trading) scheme, but this will be discussed in more detail in a later section

- in the real world, however, the assumptions of the neoclassical model may not be fulfilled for many markets and for many reasons; the above conclusion must, thus, be treated with caution; e.g., a CO2-price of 50 Dollars per ton may be problematic in certain low-income countries, while it may not be a problem in high-income countries

Review of the Lagrange multiplier method of constrained optimization: - the Lagrange multiplier method is widely used in resource and environmental economics - the multipliers have a very useful interpretation as “shadow prices” on the constraints: the value of a multiplier tells us what the effect on the maximized value of the objective function would be for a marginal relaxation of the corresponding constraint

- let us briefly review this method and the interpretation of the constraints, using a simple example of an emission abatement problem

- suppose, there are two firms, with the abatement cost functions:

2

1 1 1( )C A A= , 2

2 2 2 2( ) 2 2C A A A= +

- the total abatement A is constrained to be 10 units of emissions, and the regulator tries to find the cost-minimizing allocation of emissions / of abatement, given this constraint

- hence, the problem is: 1 2minC C+ , s.t. 1 2 10A A+ =

- Lagrangian: 1 2 1 2( 10)L C C A Aλ= + − + −

- optimality conditions: 1

1

0 2L

AA

λ∂

= = −∂

, 2

2

0 2 4L

AA

λ∂

= = + −∂

- eliminate λ to get: 1 22 2 4A A= + ; using 1 2 10A A+ = , this yields:

*

1 7A = , *

2 3A = , and 12 14Aλ = =

- interpretation: firm 1 has lower abatement costs, hence, in the optimum, this firm will be assigned a higher abatement level (if both firms had the same emissions before the regulation, firm 1 has lower emissions than firm 2 after the regulation)

- the value of the Lagrange multiplier: 14λ = tells us, that the total abatement cost 1 2C C+

would rise by 14 units of money (since 1C and 2C are measured in units of money), if the

total abatement level A would be raised by one unit (this result holds at the margin)

33

- alternatively, the problem can be restated in terms of the emission levels of firm 1 and firm 2; if the total amount of emissions is allowed to rise by one (marginal) unit, then the total

abatement costs would fall by 14λ = units of money - knowledge of the value of the Lagrange multiplier is useful, because the regulator can achieve the cost-minimizing outcome (computed above) by simply imposing a tax on

emissions equal to the value of the multiplier, hence: 14t = - each firm then abates to the point where the marginal abatement cost equals the tax

- to see this, suppose 1M and 2M are the BAU emissions of firm 1 and firm 2 (given no tax

or other regulation)

- under the tax, firm 1 minimizes: 2

1 1 1( )A t M A+ − , which yields the FOC: 10 2A t= − , hence,

the same as above when λ is replaced by t; the same holds for firm 2, so for * 14t λ= = ,

firms choose the cost-minimizing abatement levels *

1A and *

2A

- if the regulatory authority instead issues tradable emission permits, such that total emissions

fall from 1 2M M+ to 1 2 10M M+ − ; the equilibrium permit price would also be 14