Welfare economics An introduction
Economics is both a descriptive science that explains the functioning of the economy and a normative science, which tries to make prescriptions a. the prescriptions are based on value judgements which are made explicit b. the most important value judgements are i. consumer sovereignty, i.e. individuals’ preferences are to be count ii. Pareto criterion, i.e. if at least one is better off but no one worse off, the economy is better off iii. direct judgements on distribution of well-being this normative science is called welfare economics Society’s objective is to maximize the well-being of its citizens. For that we must be able to represent the well-being in a practical way and we need a way of comparing different individuals. We will start with the first issue, namely on the impact on human well-being from changes in quantity and quality of resources. We will therefore in the next section focus on individual wellbeing. In order to do that we need a way to represent mathematically the well-being of individuals.
Individual preferences
It is assumed that the individuals have preferences over all bundles of goods and services that they come across. The preferences will be represented by the symbol xšy which interpretation is that the bundle x is at least as good as the bundle y in the views of the individual. If x š y but not y š x, then x is strictly preferred to y, and we write x ™ y. and if he is indifferent between the two bundles (x š y and y š x) we write y - x. The set {x; x - y} is called an indifference set. In two dimensions, an indifference set is an indifference curve.
We need some assumptions on the properties of the preferences.
Transitivity If
then
Reflexivity
xšy and zšx, zšy x-x
Completeness or connectedness For all x and y, either
xšy or yšx
Continuity
The preference order is continuous if the set A ={y; yšx} is closed. This means that if a sequence of y 0 A, i = 1, 2, ... converges to z, then z 0 A, lim y = z | z 0 A. i
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Utility functions
For practical reasons we would like a simpler representation of the preferences. Utility functions offer such opportunity.
The existence of a utility function
If there exists a function u mapping commodity bundles to real numbers such that x š y Y u(x) $ u(y) and y - x Y u(x) = u(y), then the function u is the utility function of the individual. The utility function is a representation of preferences! Note that it is wrong to say that an individual prefers x to y because x gives higher utility. The correct way to express the statement is that if the individual prefers x to y, then the utility of x is higher than the utility of y. If u(x) > u(y), we will also interpret that as meaning that the individual is better off with x than with y, otherwise he would not prefer x to y! This is the assumption of consumer sovereignty: Only the individual can judge what is good for him. Thus we can use the utility function to measure the well-being of an individual as he himself judges it. Usually we impose on the utility function various structures. Doing environmental valuation requires for example special structures. If the preference order is continuous, then there exists a utility function. The following example illustrates a preference order that cannot be represented by a utility
function!
The lexicographic order The lexicographic order is not continuous and it is impossible to represent this order with a utility function Assume two commodities, x1 and x2. The preference order ĹĄ is lexicographic if (x1, x2) ĹĄ (x1', x2') when x1 > x1' or when x1 = x1' and x2 $ x2' For this preference order no indifference curves exist. The set of bundles better than or indifferent to (x1', x2') are the points to the right of the blue whole line and points on the black broken line. The only point indifferent to (x1', x2') is itself!
Monotonic transformations of utility
There is no unique representation of a preference order in terms of a utility function! If x ˜ y implies that u(y) $ u(x), then any monotonic increasing function of the utility function is also a utility function. For example, if u = xy, then ln u = ln x + ln y or u2 = x2y2 are also utility functions representing the same preferences. They are equally valid representations. Any monotonic increasing transformation of a utility function is a valid utility function representing the same preferences. If f(.) is a monotonic increasing function, f(u(x)) is a utility function. The trick is to find an f such that the transformed utility is as simple as possible! We will later introduce the expenditure function or cost of utility function as such a transformation.
Ordinal utility
Because any monotonic increasing transformation of a utility function also is a utility function, utility is ordinal.
Utility functions and well-being
! ! ! !
Utility is a representation of preferences. The individual knows best his own well-being - consumer sovereignty. We therefore also interpret utility as the well-being of the individual. Utility is a non-unique non-linear index of well-being.
Properties of utility functions
! ! !
Utility is increasing in goods and decreasing in bads Utility is continuous (except in some cases such as lexicographic preference orders) Utility is quasi-concave, that is the set {x;u(x)$"} is convex for all ". (More on convexity later.)
Preferences among risky choices
Let L be a lottery with outcomes c and with associated probabilities B , i = 1, 2, ...n. Let L’ be another lottery with the same probabilities B but with outcomes c’ . Then there exist a utility function u such that if L is preferred to L’, then E B u(c ) > E B u(c ’) E B u(c ) is the von Neuman and Morgenstern expected utility. Note that expected utility is only invariant to linear transformation and is therefore cardinal. i
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Existence of the expected utility function
Assume that preferences satisfy ! conditions on preference order for certain outcomes, i.e. " completeness " transitivity " reflexivity " continuity ! and two new conditions " probabilistic equivalence " probabilistic independence Then there exists an expected utility index. Probabilistic equivalence
Individuals are indifferent between two lotteries offering the same outcomes with the same probabilities. This means that the individual does not care about the process by which outcomes and probabilities are generated but only about the final outcomes and the final probabilities. Probabilistic independence
The individual’s preferences between two lotteries that offer the same outcomes in some states of the world must be independent of these outcomes. Many individuals
How do we measure social well-being when there are many individuals?
The Pareto criterion
A change that makes at least one individual better off but no one worse off satisfies the Pareto criterion. By the Pareto criterion is meant A partial order on the set of allocations of resources such that one allocation A is socially preferred to another allocation B if at least one individual is better off with A and no one is worse off with A. Let the utility of individual h in allocation A be uhA and the utility in B uhB. Then A is better than B according to the Pareto criterion if uhA $ uhB with strict inequality for at least one individual. This is a partial order, because not all allocations can be compared.
Pareto optimality
A change that makes at least one individual better off but no one worse off satisfies the Pareto criterion. By Pareto optimality is meant An allocation such that it is not possible to improve the situation of one or more individuals without harming other individuals. A Pareto optimal allocation is thus a “maximal� allocation with regard to the Pareto criterion. Note that a given Pareto optimal allocation may not be socially desirable at all because the distribution of well-being, that is the utilities may be very unevenly distributed.
Utility possibility frontiers
The red curve gives the maximum utility to individual 2, given 1's utility. The curve is known as the utility possibility frontier.
Competitive equilibrium
Competitive markets consist of 1. producers, that cannot affect prices, that maximize profits and of
2. 3.
consumers that cannot affect prices, and maximize utility within their budgets, and of equilibrium prices such that consumers’ total demand for each good and service is not greater than the total supply of the same goods and services from the producers.
The first theorem of welfare economics if
all goods and services can be bought and sold on markets, and if the economy is in a competitive equilibrium, then the competitive equilibrium is a Pareto optimum. This theorem does not say much because the resulting Pareto Optimal allocation may not be desirable. The second theorem addresses this problem.
Second theorem of welfare economics If
production possibilities are convex and continuous, each individual has a continuous utility function which is quasi-concave, all goods and services can be bought and sold on a market, Then, each possible Pareto optimum can be achieved as a competitive equilibrium after initial endowments have been correctly redistributed. Note the three conditions: 1 Convexity 2 Continuity 3 Private property rights
Convexity
A set is convex if two arbitrary points can be connected with a straight line that belongs completely to the set.
A set i strictly convex if the boundary does no contain any straight line segments.
Convex production sets Convexity in production means loosely that we have diminishing returns. More strictly we can represent convexity as follows: In this figure a production function is drawn. The production set is the set of all combinations of inputs and outputs that are on or below the production function. Take two arbitrary points in that set, for example A and B and connect them with a line. If that line is completely in the set, the set is convex. With two inputs, the production set is the set of combinations of labour, capital and output that is on or below the surface in the figure above. Once again, take two arbitrary points in this set and draw a line
Preferences are convex if the indifference curves look as they usually do in textbooks, that is as is depicted in the diagram below.
Quasi concave functions
The shaded area, above and to the right of the indifference curve is convex. The function is strictly quasi concave if the set is strictly convex. U(x) is quasi concave if {x; u(x)$ "} is convex for all ". Preferences are convex if the area above an indifference curve is convex (that is if a line drawn between two arbitrary points in that area is wholly inside the area).
We can get an intuitive understanding of the economic meaning of convexity and a feeling for the two theorems if we look at Robinson Crusoe who can produce two goods, x and y. Robinson is alone but we will try to think of him as two persons: a producer and a consumer. As a producer, Robinson must choose a bundle of the two goods that is feasible, i.e., that belong to the production set in the figure.
This set is defined by the transformation curve T-T and the area below it. Note that it has been drawn as convex area. Robinson, the consumer has preferences that are represented by the indifference curves I-I. Obviously, the best for Robinson is to choose to
produce and consume at the point where the indifference curve is tangent to the transformation curve. At this point, the two curves have a common tangent, P-P. This is the role played by convexity. It guarantees that the indifference curve and the production set can be separated by a straight line. Without convexity, we cannot guarantee that. The slope of this tangent can be interpreted as the relative price of y in terms of x.
Separation theorems
two convex sets, of which at least one has interior points, and with no interior points in common can be separated by a hyperplane.
Decentralization If Robinson the producer maximizes profit, he should choose the production bundle that corresponds to the tangency of the price line P- P and the transformation curve T-T. If Robinson the consumer is maximizing utility, he will choose the consumption bundle that corresponds to the tangency of the price line P-P and the indifference curve I-I. These points are the same. Therefore, we can decentralize the decisions: we let the producer choose the production bundle, only knowing the relative prices and let the consumer choose the consumption bundle, only knowing the relative prices. Thus the assumptions of convexity of preferences and convexity of production allow decentralized decision making that still achieve the social optimum. This is the intuitive background for the second welfare economics theorem.
Non-convexity
Let us also see what happens if the production set is not convex. In the diagram below, the production set defined by the curve T-T is not convex. Robinson’s optimum is at point A, where the indifference curve I-I and the transformation curve T-T are tangent to each other.
However, at the prices defined by the slope of the tangent at A, Robinson, the producer could make higher profits at other output combinations, for example he could make a profit corresponding to the dotted line if he produces at B. Thus, with this non-convexity, the production and the consumption decisions cannot be separated by using prices, and the second theorem of welfare economics does not hold. Thus, if there are non-convexities, it may not be possible to decentralise decisions between consumers and producers and still achieve a social optimum in equilibrium! Non-convexity may arise from many different reasons: 1. increasing returns to individual factors of production 2. increasing returns to scale 3. synergistic effects in natural systems 4. satiation
Continuity
Continuity is almost completely a mathematical device and very often (but not always) without significant economic meaning and we will not dwell on this concept.
Property rights
Without well defined property rights, markets will not be established for all goods and services and incentives will be distorted as a result. The reason is that well-defined property rights define responsibilities in such a way that the owner of the rights has incentives to manage the resource in a socially efficient way. A person who owns an asset, such as a piece of land, will have strong incentives to manage that asset efficiently because he will himself bear the cost of mismanagement. When the property rights are not well defined, someone else will bear this cost.
We do not have private ownership of the atmosphere. I will therefore not bear the full cost of the environmental damage when I pollute the air. Due to the lack of property rights, an externality has been created. The result will be too much pollution. If there are no well-defined property rights to a particular grazing land, there will be overgrazing, because no single herder has to take the full social cost of bringing cattle to the land into account. If the grazing land would be a common property with access only for the members of a particular community, there would still be overgrazing because each member of the community would have incentives to bring too many cattle to graze. However, if the common land is managed by social norms that are well anchored in the community, then the grazing land may very well be managed efficiently. Similarly, if the land is divided into pieces and there would be individual ownership of the land, each land owner would limit the number of animals in such a way as to be able to use the land in the future. Once again, we will achieve efficient management. Thus, environmental problems are to a large extent due to property rights failures. There are two basic reasons for the absence of well-defined property rights - policy decisions and costs of establishing property rights. In many countries, some resources are regarded as publically owned. For example, land that has not been claimed by anyone else can be regarded as open for anyone interested in using the land. Because of that there are incentives to overuse this land. This is an example of failures in the property right structure, introduced by bad policies. However, for many resources, the costs of introducing property rights are very high and sometimes it is impossible even to define individual rights. For example, it is difficult to think of how the global climate could be assigned individual property rights. The reason is that a change in the climate that affects me will also affect you. The climate is a public good (or bad, depending how you look at it). Of course, we could think of individual rights to emit green house gases, but the climate itself will continue to be a public good.
Environmental economics
Environmental economics is to a very large extent the analyses of two of the three conditions under which the second theorem holds: convexity and property rights. Convexity cannot be taken for granted when we study environmental problems. There are two reasons for this: first, nature may not be convex, that is the ecosystems may not produce bundles of services to man that can be described by convex production sets, and second, the absence of property rights can generate non-convex production sets. This is of utmost importance for the design of institutions that can manage environmental resources well.
Decision making
Valuation is a tool for organising information in an efficient way. We can look at a stylised picture of decision making in the following way.
Alternatives
The first thing to remember is that we have a set of actions, and that the whole decision making problem is to choose one action which is “best� in one way or the other. So it is absolutely necessary to define the set of actions, i.e. the choice alternatives. ! The alternatives - or the definition of commodities - must be very precise! ! The alternatives must be very well understood! ! The implications of the alternatives, including the uncertainties, must be fully understood! This set can sometimes be very simple and consist of only two alternatives - build a project or don’t. This presumes that the project is well defined and cannot be altered. In other situations, the set of actions may be much more complex. If the problem is about choosing a structural adjustment plan to increase economic efficiency, there are many parameters such as tax rates, subsidy rates, exchange rates, trading rules, etc that must be defined and most of these parameters can take almost an infinite number of values. Irrespectively of the complexity of the action set, it must be defined, in order to make the valuation exercise interesting. Too often, one can find studies that are technically brilliant but completely devoid of any meaning because the decision making situation has not bee defined. Mappings
The next step is to make an impact statement. What are the consequences of a particular action?
consequences: x = N(a) N corresponds to environmental impact assessment. Note that x may be a probability distribution Estimation of N requires ecological, geophysical, geological, meteorological, economic and technological knowledge. Sometimes, this is rather simple. If the problem is to decide whether the beach should be cleaned once a year or never, the impact statement is basically in the case of cleaning the beach a description of the clean beach and the cost of cleaning it. Sometimes, one can go further and add the possible ecological side effects a clean beach may have if there are any “outside� effects. In the other alternative, with no cleaning, the impact is a description of the polluted beach, perhaps with some information on the ecological side effects. The cost of cleaning is obviously zero. In other cases it may be much more difficult to make an impact statement. In the case with structural adjustment, we must estimate the effects from the plan on production in different sectors, the consequences on the environment from these changes in production structures, the impact from price changes on different socioeconomic groups etc.In general this would require the use of a Computable General Equilibrium (CGE model in order to trace all the primary, secondary , tertiary etc. effects of one particular structural adjustement plan. Obviously, the CGE model must be constructed in such a way that it incorporates the property rights failures that cause environmental problems. In many cases, it will be impossible to come up with one particular impact from a chosen action. There are many reasons for this. For example, if one action leads to higher air pollution, this has effects on morbidity and mortality. However, it will in general be impossible to say who will be affected. In the best case, we may be able to say something about exposure and the increased risk for deseases and even for death because of the increased air pollution. Thus, we have to represent the impact by a probability distribution of different impacts. In other cases, the scientific knowledge is not enough to predict precisely the consequences from an action. Once again we may have to rely on information in terms of probabilities of different impacts. Valuing the alternatives
Once the impacts of different actions have been mapped, one should rank the different actions by valuing the impacts. In other words, we want to give values W = W(x) to different impacts. If the change affects only one individual one can use the utility function: W = U(x), An impact y is better than an impact x if U(y) > U(x) If several individuals are affected the problem is less simple. Valuing the alternatives: several individuals
Some may be better off, others worse off, from choosing one alternative instead of another. The gains for the winners must be compared with the losses for the losers. Two approaches to do that: ! Compensation tests ! Social welfare functions
Compensation tests
Kaldor criterion
Consider a change from social state A to social state B. Individual 1 will gain and individual 2 will loose from the change. Kaldor criterion says that potential welfare will increase if 1 can compensate 2 for the loss he has incurred and still be better off. Note that the compensation is purely hypothetical.
Hicks criterion
Consider the same change from A to B. If in B, 2 cannot bribe 1 to accept to move back to A, and not being worse off compared to B, B has higher potential welfare.
Utility possibility curves
Assume we are at point A. Then by redistributing income, all utility levels on the curve can be attained. This curve is known as the utility possibility curve.
Compensation tests
Consider the change from A to B. Individual 2 can compensate individual 1 by moving from B to D along the utility possibility curve through B. Kaldor’s test is satisfied. But 1 can also bribe 2 to abstain from the change by moving to C where both are better than at B. Hicks’ test shows that change from A to B reduces the potential welfare!
Problems with compensation tests
Such tests result in most cases to intransitivities. They consider hypothetical compensations - so there may be ethical problems
Social welfare functions
Assume the existence of a social welfare function: W(u1, ..., uH) Does such a social welfare function exist?
Arrow’s (im)possibility theorem
When can we aggregate individual preferences into a social welfare function in a reasonable way? Define reasonability as follows: ! Unrestricted domain, i.e. all possible profiles of individual preference order can be aggregated. ! Independence of irrelevant alternatives, i.e. if we only consider choices between two alternatives, A and B, then the aggregated order should only depend on the individual preferences on these two alternatives. ! Weak Pareto principle, i.e. if for any pair A and B, all individuals prefer A to B, then the social ordering should also prefer A to B ! No dictatorship, i.e. there is no individiual h such that for all possible profiles of preference orders, the social order coincides with the preference order of individual h.
Theorem
Assume that there are at least three individuals and there are at least three alternatives over which society must make decisions. Then there is no mechanism by which individual preferences can be aggregated and satisfying simultaneously the four conditions.
Condorcet’s paradox
It is an example of Arrow’s impossibility theorem ! Let there be three individuals, 1, 2, 3, who have to choose between three alteratives, A, B, and C. Assume the individuals rank the alternatives as follows ! Individual 1 A, B, C 2 B, C, A 3 C, A, B ! In majority voting, A would win against B, B would win against C, and C would win against A. Majority voting does not create a social order.
Bergson-Samuelson social welfare function
The existence of a social welfare function
W(u1, ..., uH) will violate one of the Arrow conditions. It implies comparability of different individuals’ utilities and therefore it expresses value judgements of some special group of individuals - the policy makers. Thus its main role is as an objective function for public policy making. It comprises the ethical values on the distribution of well-being. We will assume that W satisfies the Pareto criterion, i.e. MW/Muh > 0 for all h. Social welfare functions
The welfare function
W(u1(x), ..., uH(x)) is defined for a particular choice of utility representation of the underlying preferences. If we change this representation by making a monotonic increasing transformation of the utility functions, we have to make corresponding transformation of the welfare function in order to make the social preferences invariant. If W(u1, ..., uH) represents a particular social order and we make monotonic transformations of the utilities, f(.), then the welfare function must be transformed accordingly: W =W(f-1(f(u1)), ..., f-1(f(uH))) in order make the social ordering invariant for these mononic transformations. Thus, we transform uh(x) to vh(x)=f(uh(x)). The new welfare function is W=W(f-1(v1(x)), ...,f-1(vH(x))))
Linear approximation
Consider a change in society from A to B. This change implies that the individual utility levels will change from uhA to uhB, and that the social welfare will change with )W =W(u1B, ..., uHB) - W(u1A, ..., uHA) We make a linear approximation )W = Eh wh )uh, where wh = MW/Mu w , h=1, 2, ..., H are the income distribution weights h
h
Social welfare measurements
How can the W function be identified and estimated? How do individual utilities )u change? h
Measurements of individual utility changes
The diagram shows the idea behind measuring individual utility. We search for the income, which at given, constant prices can support a certain utility level. In order to carry out this argument more rigorously we need to discuss demand analysis. Choose a simple representation of the preference order!
Utility maximization
The individual chooses the bundle of commodities that he finds best. This is represented by utility maximization under a budget constraint: max u(x , ..., x , Q) subject to px px #I where p , i = 1, ..., n are the prices, Q the supply of public goods which the individual has no control over and I the (lump sum) income. If the utility function is strictly quasi-concave, there will be a unique solution and that solution will obviously depend on prices, public goods, and income: x = x (p , ..., p , Q, I) i = 1, ...., n These are the Marshallean demand functions. 1
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Demand functions The demand functions have four properties. The first two are: ! satisfying the budget constraint ! homogenous of degree zero in prices and income (no money illusion) We will come back to the remaining properties later. The indirect utility function
If we substitute for the x’s (the Marshallean demand functions) in the utility function we obtain u(x1(p,Q, I), ..., xn(p, ..., Q, I), Q) / v(p, Q, I) v is known as the indirect utility function. One can easily prove:
Roy’s theorem:
x (p1, ..., pn, Q, I) / - [Mv/Mpi]'[Mv/MI] i
Individual utility changes
The expenditure function is calculated in the following way: m = min 31n pixi s.t. u(x, Q) $ u’ To solve this problem we form the Lagrangean L = 31n pixi - µ(u(x, u Q) -u’). Necessary conditions:
µ Mu/Mxi = pi m is the expenditure function or the cost of utility function
Hicksean compensated demand functions
The solution to the cost minimization yields the Hicksean compensated demand functions x = xc(p, Q, u’) This gives the demand for different prices and different supplies of public good if the individual is so compensated that he remains on the same indifference curve (or at the same utility level).
Marshallean and Hicksean compensated demand functions
Properties connecting compensated and non-compensate demand functions xc(p, Q, u) / x(p, Q, m(p, Q, u)), xc(p, Q, v(p, Q, I)) / x(p, Q, I)
Properties of compensated demand functions
The compensated demand functions have several properties: ! symmetry: Mxci/Mpj = Mxj/Mpi ! negativity: Mxci/Mpi #0 ! homogeneity of degree 0 in p ! adding-up property: 3ipixic = m(p,u) u(xc(p, Q, u) / u Differentiate with respect to pj 3i(Mu/Mxi )(Mxic/Mpj) = 0 From necessary conditions for cost minimization follows Mu/Mxi = (1/µ)pi put 1/µ= λ (µ Lagrangean multiplier) and therefore 3i pi(Mxic/Mpj) = 0
The expenditure or the cost of utility function
The minimum expenditure to achieve utility level u’ is given by m(p, Q, u’) = 31n pixic(p, Q, u’). m is known as the expenditure function or the cost of utility function. ! m(p, Q, u) / 3pixic(p, Q, u) ! m is homogenous of degree one in prices ! m is increasing in u ! m is concave in prices ! Mm/Mpi = xci(p, Q, u) / xi(p, Q, m) (Because 3i pi(Mxic/Mpj) = 0)
Slutsky equation From
xci(p, Q, u) / xi(p, Q, m(p, Q, u)) follows after differentiating with respect to pj Mxci/Mpj = Mxi/Mpj + Mxi/MI Mm/Mpj or Mxci/Mpj = Mxi/Mpj + xj Mxi/MI This is the Slutsky equation In the same way Mxcj/Mpi = Mxj/Mpi + xi Mxj/MI and therefore Mxi/Mpj + xj Mxi/MI = Mxj/Mpi + xi Mxj/MI
The indirect utility function and the expenditure function
The expenditure function can also be defined from the indirect utility function: v(p, Q, m) = u’ yields m = m(p, Q, u’)
Properties of the demand functions
We have earlier seen that ! demand functions satisfy the budget constraint ! they are homogenous of degree zero in income and prices We can now add the following two properties ! the substitution effect is symmetric, Mxi/Mpj + xj Mxi/MI = Mxj/Mpi + xi Mxj/MI ! the matrix of substitution effects, Mxi/Mpj + xj Mxi/MI , is negative semi-definite (concavity of the expenditure function in prices
Consumer surplus
Consumer surplus is meant to be an approximate way of recovering preferences from revealed behaviour, i.e. demand functions. Price equals marginal utility. At prices high (p”) the marginal utility is then given by the demand curve. When quantity increases, the marginal utility falls. Therefore, the total “excess” utility when price = p’ is equal to the shaded area. This is the Marshallean consumer surplus.
Recovery of the expenditure function
Assume we have only two goods, x and y, and that we have estimated econometrically the demand functions for them: x = x(px, py, I) y = y(px, py, I) Can we recover the expenditure function from this information? Yes, if the demand functions are such that the substitution effects are symmetrical , i.e. Mxi/Mpj + xj Mxi/MI = Mxj/Mpi + xi Mxj/MI and the matrix of substitution effects is negative semi-definite. We do this by solving the system of differential equations: Mm/Mpx = x(px, py, m) Mm/Mpy = y(px, py, m) The symmetric substitution effects guarantee a solution. The negative semi-definitness of the matrix of substitution effects guarantees that the solution has the properties of the expenditure function. Recovery of the preferences
By observing individual behavior we can, at least in theory, reveal their preferences and therefore also define utility functions for them. We will next see how this is done in practice. Utility representations
The previous analysis is not rigorous because it does not consider income effects. The following will correct for that. Keep p and Q fixed at levels p’ and Q’. m(p’, Q’, u) is then an monotonic increasing transformation of u. Therefore,
m(p’, Q’, u) is a valid utility function, representing the same underlying preference order as u.
M with fixed prices and supply of public goods measures the income corresponding to the parallel price lines Equivalent variation
Consider a change from A(p’, Q’, I’) to B(p”, Q”, I”). The utilities in A and B are u’ and u” respectively. Taking A for the fixed prices and fixed supply of public goods, the utility change can be written m(p’, Q’, u”) - m(p’, Q’, u’) = m(p’, Q’, u”) - I’ = m(p’, Q’, u”) - I” + (I” - I’) = m(p’, Q’, u”) - m(p”, Q”, u”) + ) I = EV (equivalent variation)
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EV is the change in income that would give the individual the utility change in A as would the change from A to B. ! If u” > u’, then EV is the minimum willingness to accept in compensation the status quo, that is situation A. ! If u” < u’, then EV is the maximum willingness to pay for avoiding the change. Let p’ = 1. Then y’=m(p’, Q’,u’), y”=m(p’, Q’, u”) and EV=y’-y” y
The original equilibrium is at A. After an increase in price, we are at C. The same indifference curve could have been reached by reducing income with y”-y’ = -EV Remember that Mm/Mp = x . Assume only p changes from p ' to p ". Then EV = m(p ", Q’, u”) - m(p ', Q’, u”) = = IMm/Mp dp = Ix dp c
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where the integration is from p ' to p ". The shaded area in the figure below is therefore the equivalent variation. 1
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Compensating variation
Use the prices and supply of public goods in B (the final position) for measuring utility. Then the utility change is m(p”, Q”, u”) - m(p”, Q”, u’) = I” - m(p”, Q”, u’) = I’ + )I - m(p”, Q”, u’) = m(p’, Q’, u’) - m(p”, Q”, u’) + )I = -CV CV is the Compensating variation
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Compensating variation is the amount of change in income necessary to keep the individual at the same indifference curve as in A after the change to B. If u” < u’, then CV is the minimum willingness to accept the change. If u” > u’, then CV is the maximum willingness to pay for the change.
Only price change is on x. y’=m(p’, Q’, u’) y”=m(p”, Q’, u”) CV=y”-y’. The original equilibrium is A. After an increase in price the new equilibrium is at B. In order to be at the same indifference curve with new price, income must be y”. The individual needs y”-y’=CV in compensation for the price fall.
Remember that Mm/Mp = x . Assume only p changes from p ' to p ". Then CV = m(p ", Q’, u’) - m(p ', Q’, u’) = = IMm/Mp dp = = Ix (p , u')d p1, where the integration is from p1' to p1". The compensating variation CV is equal to the shaded area in the figure above. c
j
j
1
1
1
1
1
1
1
c
1
Consumer’s surpluses
1
We can now compare EV, CV and Marshallean consumer’s surplus for a price increase. The EV is equal to the squared area. The CS (consumer’s surplus) is equal to the squared plus the waved area. The CV is the sum of all filled areas.
Further comparisons between CV, EV, and CS
The reason why there are two compensated demand curves is the inco me effect. In the original point A real income and therefore also utility are greater than in the final point B. So the compensated demand curve through A is further to the left than the one through point B. The waved and dotted areas are therefore determined by the income effect or the income elasticities. If they are small, the three measures will be approximately equal.
Contents Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 An introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Individual preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Completeness or connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The existence of a utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The lexicographic order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Monotonic transformations of utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Ordinal utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Utility functions and well-being . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Properties of utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Preferences among risky choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Existence of the expected utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Probabilistic equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Probabilistic independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Many individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Pareto criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Pareto optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Utility possibility frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Competitive equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The first theorem of welfare economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Second theorem of welfare economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Convex production sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Quasi concave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Separation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Decentralization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Non-convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Property rights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Environmental economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Decision making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Valuing the alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Valuing the alternatives: several individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Compensation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Kaldor criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Hicks criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Utility possibility curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Compensation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Problems with compensation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Social welfare functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Arrowâ&#x20AC;&#x2122;s (im)possibility theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Condorcetâ&#x20AC;&#x2122;s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Bergson-Samuelson social welfare function . . . . . . . . . . . . . . . . . . . . . . 15 Social welfare functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Linear approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Social welfare measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Measurements of individual utility changes . . . . . . . . . . . . . . . . . . . . . . 16 Utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Demand functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 The indirect utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Individual utility changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Hicksean compensated demand functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Marshallean and Hicksean compensated demand functions . . . . . . . . . . . . . . . . 18 Properties of compensated demand functions . . . . . . . . . . . . . . . . . . . . . . . . . . 18 The expenditure or the cost of utility function . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Slutsky equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 The indirect utility function and the expenditure function . . . . . . . . . . . . . . . . . 19 Properties of the demand functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Consumer surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Recovery of the expenditure function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Recovery of the preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Utility representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Equivalent variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Compensating variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Consumerâ&#x20AC;&#x2122;s surpluses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Further comparisons between CV, EV, and CS . . . . . . . . . . . . . . . . . . . . . . . . . 25 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26