CHAPTER 5
NORMATIVE OR WELFARE ECONOMICS, DECISIONS AND GAMES, AND BEHAVIORAL ECONOMICS
Our efforts thus far have been primarily devoted to perfect competition. When perfect competition conditions exist, it turns out that no individual consumer or firm can increase their objective function or payoff without decreasing the payoff for other firms and/or consumers. When imperfect competition (externalities, public goods, taxes and subsidies to selected firms or consumers, and other conditions) exist, the rules of perfect competition do not generally apply. Improvements in the accuracy of the results obtained by assuming perfect competition are then often possible, as well as improvements (or at least changes) in the results that accrue to firms and consumers. In this chapter we will examine concepts of optimality and economic efficiency of production and distribution under such conditions. We will focus on the allocation of land, capital, and resources in ways that generate the maximum benefit to society (or the community or nation). There are two fundamental issues underlying this: the allocation of the factors of production and the distribution of these products among different individuals. We will examine efficient, effective, and equitable ways in which we can accomplish this allocation. Then we will discuss the concept of social welfare and the conditions that must exist to determine optimum behavior under these social welfare conditions. This will lead us in the concluding part of this chapter to a discussion of behavioral economics. It is potentially important to be able to apply this thinking to markets where providers are not all identical and, less constraining, consumers are not all identical. Differences in production functions, perhaps in terms of differing fixed and variable cost structures, can be central to gaining competitive advantage. We will examine some of these issues in Chapters 6, 8, 9, and 10.
5.2 PARETO OPTIMALITY UNDER PERFECT COMPETITION CONDITIONS
Concepts of Pareto optimality serve as the basis for much of modern welfare economics. An allocation of resources is said to be Pareto optimal if there is no reallocation of production or distribution of the products of production that will increase the economic utility of one or more households without decreasing the economic utility of other households. We may also define a non-Paretian allocation of resources as one in which it is possible to increase the utility of one household without reducing the utility of any other. In most cases Pareto optimality cannot be achieved. But it is an obviously desirable criterion, in the sense that no one is harmed by a move in the direction of Pareto optimality. Often improvement in the utility of one household can occur only through reduction in the utility of others. Decisions that have to be made under such circumstances are difficult indeed. Thus when the conditions for Pareto optimality exist, we are fortunate; when they do not, issues such as income distribution and redistribution must be resolved.
We will now obtain the necessary conditions for the existence of Pareto optimal economic solutions. First we will consider Pareto optimal production and then Pareto optimal consumption. Finally we will combine production and consumption, and discuss Pareto optimal equilibrium.
We will first consider the case of two firms using two factor inputs to produce two commodities, and then generalize our arguments to higher order cases.
We assume the production functions
where the total factor input quantities are constrained, perhaps by availability concerns, such that
A problem of the firm in the short run is to maximize output. But Pareto optimal production efficiency requires a firm to maximize its output subject to the constraint that there is no reduction in the production output of any other firm. Thus the problem of firm 1 is to maximize its output such that the output of firm 2 remains at some fixed level Q2. The Lagrangian for the problem of maximizing production output for firm 1 subject to the constraints of Equations 5.3 and 5.4 and the constraint that q2 = Q2 is given by
where λ, γ, and ν are Lagrange multipliers. The necessary conditions for optimality are given by
We now combine Equations 5.6 and 5.8 and obtain the expression
We combine Equations 5.7 and 5.9 and obtain the expression
Here, 
is the marginal productivity of firm i with respect to the jth factor. Dividing these two relations results in the interesting and useful fact that the ratio of the marginal productivities of any one firm for two inputs xi and xj is the same for all firms. In doing this, we have the relation
This relation is important for efficient production. The ratio of marginal productivities is generally called the marginal rate of technical substitution or substitution, as we know from Equation 2.17. We may write Equation 5.15 as
Therefore, the marginal rate of technical substitution for the two firms must be equal if Pareto optimality exists. If this is not the case, then the produced output of one commodity can be increased without reducing the produced output of the other commodity. We have shown in Chapter 2 that the ratio of marginal productivities is the same as the ratio of wages for the factor involved if profit is to be maximized. For Pareto optimality, we require in addition that Equation 5.16 holds true. We can show that Pareto optimal production efficiency requires that the marginal rates of technical substitution for two input factors i and j be the same for all productive units that use these two factor inputs to production. Precisely these same results are obtained by maximizing the output of firm 2 in a Paretian fashion. Thus we see that
since the wages are the same for all firms for a given factor. These are very classic and very restrictive results.
Example 5.1:
Suppose that the production equations for two firms are
The marginal rates of substitution are
Figure 5.1 illustrates the isoquants for the two firms.
Figure 5.1. Isoquants for the two production functions of Example 5.1.
Much of the preceding discussions concerning Pareto optimality of the firm apply directly to Pareto optimality of the household. Before considering a more general case, let us consider the case of two consumers who must share, in a Paretian fashion, two commodities that are in restricted supply. The consumers have the utility function 
and 
. The total supply of commodity 1 is Q1, so that
The total supply of commodity 2 is Q2, so that
To maximize the utility of consumer or household 1, with the constraints that the utility of consumer 2 is fixed at U2 and the commodity supply is restricted, we define the Lagrangian as
where λ, γ, and ν are Lagrange multipliers. The necessary conditions for optimality are given by
We combine Equations 5.23 and 5.25 to obtain the expression
We combine Equations 5.24 and 5.26 to obtain
Dividing these two equations to eliminate λ yields the expressions
These three relations are important. Each is a statement of the requirement for efficient consumption under these fairly restrictive classic assumptions.
We have just derived the important result that under Paretian optimality of the household, the marginal rate of commodity substitution for the two households is the same. The marginal rate of commodity substitution is, as we have defined in Equation 3.7, the ratio of the marginal utilities for the commodities in question. If the relations of Equation 5.33 or 5.34 are not satisfied, it means we have not satisfied conditions for Pareto optimality. We recall from Chapter 3 that under conditions of optimality of the individual households, the ratio of marginal utilities is just the ratio of the prices for the commodities involved. Also, we can show that the results of Equation 5.33 are valid for any two consumers and any two commodities in the utility function of the two consumers. Thus we have the expression
since the price of a commodity is the same for one consumer as for any other consumer under the conditions assumed here.
Example 5.2:
Suppose that the utility functions of the consumers are
The marginal rates of commodity substitution for the two consumers are given by
Figures 5.2 and 5.3 illustrate the isoquants of constant utility for the two consumers.
Figure 5.2. Isoquants of constant utility for consumer 1.
Figure 5.3. Isoquants of constant utility for consumer 2.
From the equality of the MRCS under Pareto optimality, we see from Equations 5.38 and (5.39) that
Thus from Equation 5.37, we have for the utility of consumer 2,
From Equations 5.36 and 5.40, we have for the utility of consumer 1
By comparing Equations 5.41 and 5.42, we obtain the Pareto frontier or utility possibility curve equation
This represents the maximum possible combination of consumer levels of satisfaction or utility that yields a Pareto optimal solution.
It is relevant to ask at which values of U1 and U2 satisfying Equation 5.43 should equilibrium occur. The answer depends, of course, on the interaction of the firms with consumers in the marketplace, as we will see soon. Hopefully, we can write a social welfare function W(U1, U2) to describe the joint utility of consumers 1 and 2 along the utility possibility curve. We will examine some notions of social welfare after we have examined Pareto optimal equilibrium under several competition forms.
The relationships between equilibrium in perfect competition and Pareto optimality are of considerable interest. We will first consider the case of two consumers each possessing a single factor input to production and two firms each producing a single product. Thus we assume the production functions of Equations 5.1 and 5.2, the factor inputs to production of Equations 5.3 and 5.4, the utility functions 
and 
, and the produced commodities of Equations 5.28 and 5.29.
The long-term problem of firm 1 is to maximize profits, as given by
where the production equation of firm 1 is the relation
The long-term problem of firm 2 also is to maximize profits, as given by
where the production equation of firm 2 is the relation
The variables in these relations are those that we have been using in our efforts thus far. The terms w1 and w2 are the wages paid by the consumers who furnish factor inputs 
and 
to firm 1, and 
and 
to firm 2. Here, p1 and p2 are the price paid by the firms for the two commodities they produce, Q1 and Q2.
The problem of consumer 1 is to maximize the utility
This is subject to a constraint on the income of consumer 1, as given by
The problem of consumer 2 also is to maximize utility, as given by
This is subject to a constraint on the income of consumer 2, as given by
Again we use the familiar symbols: 
and 
for the commodities purchased by consumer 1; 
and 
are the commodities purchased by consumer 2. We assume that the nonsatiation axiom applies and consequently that the consumer spends the entire income from labor as well as from fractional ownership of the firms.
We recognize that the problem we have formulated is a simple case of the general equilibrium problem of Table 4.1. As with the general problem, we have factor market clearing equations that here become
and the commodity market clearing equations
The sum of the fractional ownership by consumers of the firms must be equal to unity. So we have
We will now determine equilibrium requirements under perfect competition conditions. Then we will compare these results to the conditions for Pareto optimality. The necessary conditions for optimality of firm 1 are straightforwardly given by
Similarly, the necessary conditions for optimality of firm 2 are given by
By combining these four relations, we obtain Equation 5.15, which is the necessary requirement for Pareto optimality of the two firms. Although we will not show it here, this conclusion applies to the general case of F firms using M factor inputs and N commodity inputs. Thus perfect competition conditions ensure Pareto optimality of the production of F firms. Solving Equations 5.58 through 5.61 results in a specific production, whereas solving Equation 5.22 results in a Pareto optimal relationship between Q1 and Q2.
To maximize the utility of consumer 1 subject to the budget constraint, we form the Lagrangian
We can obtain the necessary conditions for optimality, under the assumption of zero profits or constant profits, such that we avoid the need to take partial derivatives of Π1 and Π2, as
The necessary conditions for optimality of consumer 2 are obtained from the Lagrangian for consumer 2, which is given by
as
Combining Equations 5.63, 5.64, 5.68, and 5.69 easily results in Equation 5.32. Thus we are led to suspect that the requirements for consumer maximization of utility under perfect competition satisfy the conditions for Pareto optimality.
To see this fully, we need to determine the conditions for Pareto optimality of the consumers, since we now have factor inputs to the utility functions of Equations 5.48 and 5.50. To determine Pareto optimality of consumer 1, we maximize the utility of consumer 1 subject to equality constraints, thereby ensuring that
a. the utility of consumer 2 is U2,
b. the equation relating the production of firms 1 and 2 is satisfied, and
c. the commodity and factor market clearing equations are satisfied.
Thus the Lagrangian, which we use for determining the conditions for Pareto optimality of consumer 1, is given by
where λ, γ1, γ2, ν1, ν2, η1, and η2 are the Lagrange multipliers. We set
and
We obtain from Equation 5.73, after some obvious combinations of terms,
Careful inspection of these relations shows that they are just those that follow from the necessary conditions for equilibrium under perfect competition conditions: Equations 5.58 through 5.61, Equations 5.63 through 5.66, and Equations 5.68 through (5.71). Solving these equations results in a set of relations between commodities and factors that is Pareto optimal but does not determine specific optimal values of these commodities and factors. Such determination is accomplished using the necessary conditions for perfect competition equilibrium. We will return to this point when we consider social welfare functions. This result is true not only for the case of two commodities and two firms but also for all other cases in general. Thus we conclude that satisfaction of the conditions of perfect competition and associated equilibrium attainment ensure Pareto optimality. Doubtlessly this result has done much to confirm the belief that perfect competition is a desirable economic result. Adam Smith formulated, in the eighteenth century, his invisible guiding hand principle, which asserted that free individual private decisions in a competitive economy will be socially optimum in that these decisions lead to that which is most beneficial in the general interest and serve a desirable societal purpose not part of the original intention. To be sure, perfectly competitive conditions do ensure Pareto optimality.
However, the converse is not true. Perfect competition is not necessary (although it is sufficient) for Pareto optimality, and Pareto optimality can result without any form of competition at all, such as in a socialist or dictatorial government.
To achieve Pareto optimality, we need to satisfy three fundamental relations. These are as follows: (1) the efficient production relation of Equation 5.15, which ensures that the marginal rate of substitution among the factors of production is the same for all producers; (2) the efficient consumption relation of Equation 5.32, which ensures that each consumer places the same relative economic value (at the margin) on all of the products of production; and (3) the efficient product mix relations, which are contained as part of Equations 5.75–5.77. These equations necessarily specify the efficient consumption and efficient product mix relations. In addition, the product mix relations require that the ratio of the economic value of two products is given by the ratio of their marginal costs.
We have from Equations 5.75 through 5.77
which is the production mix relationship for the case where consumer utility does not depend on input factors to production, and
which are the additional relations that must hold when consumer utility does depend on the factor input to production. By solving these relations for efficient production, consumption, and product mix, we obtain necessary conditions for Pareto optimality. To ensure equity, we should also maximize a (scalar) social welfare function W(U1, U2). We will turn our attention to techniques to accomplish this in Section 5.5.
In this section, we have considered Pareto optimality under perfect competition conditions. We demonstrated the very useful result that equilibrium under perfect competition conditions ensures Pareto optimality. We did not examine the sufficiency conditions for Pareto optimality, which require that the second-order conditions for each consumer and firm delineated in the previous chapters are satisfied. If some of these conditions are not satisfied, Paretian optimality is not ensured. For example, suppose that one consumer is satiated. Then commodities may be taken from this consumer and given to others. The satiated consumer will suffer no loss of utility by this redistribution, whereas other consumers will increase their utility. Also, we require that there be no externalities, either in production or in consumption. These externalities may be beneficial, such as the effect of a firm’s labor-training program on other firms and consumers. They may be detrimental, such as pollution associated with an otherwise good product of a firm. In many cases these externalities will exist. Also there will be many cases in which imperfect competition conditions exist.
Four basic assumptions are necessary for a freely competitive market to exist. When these conditions or assumptions do not exist, a freely competitive economy will not be optimum in the sense of being Pareto optimal or maximizing social welfare. In these cases, some form of regulations or other government intervention is potentially desirable. The four conditions that ensure perfect competition are the following
1. No increasing returns to scale for any firms. If increasing returns to scale exist along the production frontier, they must be exhausted before the supply–demand equilibrium is reached. We have seen in Chapters 2 and 4 that increasing returns to scale lead to monopolistic production and, therefore, to a noncompetitive economy in which firms will be price setters and not price takers.
2. No technological external effects. This is needed to ensure relevant prices and market efficiency. There are at least three forms of external effects: one person’s consumption affects another person’s utility, one firm’s production affects another firm’s utility, and one firm’s production affects another firm’s production.
3. No ill-effects due to lack of certainty and perfect information. All firms and consumers must have perfect knowledge of their own utility and production functions, and perfect awareness of market conditions.
4. Correct ownership of all production factors. The distribution of ownership of the factors of production enables each consumer to purchase the commodity bundle that corresponds to social welfare maximization.
The first three conditions are required for Pareto optimality to exist. The fourth condition is needed, as we have seen, to maximize social welfare. We will now turn our attention to a detailed consideration of these, and other, factors that alter or prevent perfect competition.
5.3 EXTERNAL EFFECTS AND IMPERFECT COMPETITION: PUBLIC GOODS
Our assumption of perfect competition requires that there exist no external effects in production or distribution. For example, we assume that the cost of production for a given firm does not depend on the production level of any other firm. Also, we assume that the utility of one consumer does not depend on the utility of other consumers. When there are externalities present, the conditions for Pareto optimality will generally not be satisfied. Also, imperfect competition will generally make Pareto optimal allocation of resources impossible. In this section, we will examine the effect of external effects and imperfect competition on Pareto optimality. We will also consider regulations on the economy that will result in Pareto optimal allocation.
A very important type of commodity or good is one that can be shared by all members of the public rather than consumed by individual members. The transportation system is one example of a public good, as shared commodities are generally called. Here the notion of a public good is very broadly defined such that there is no need to consider a public bad. In our initial analysis of public goods, we assume two consumers: a single bundle of ordinary goods and a single bundle of public goods. We assume the utility functions
where the total consumption of the ordinary good is given by
and where g is the public good. To maximize the utility of consumer 1 subject to the constraint that the utility of the second consumer be fixed at some amount U2, we form the Lagrangian
The necessary conditions for optimality are obtained from the Lagrangian as
We combine Equations 5.82 and 5.83 to eliminate the Lagrange multiplier γ and then obtain
Taking any two components of the commodity bundle of ordinary goods in Equation 5.87, we can obtain the same results as in Equations 5.32 through 5.34:
Thus we see that the marginal rate of commodity substitution is the same for the ordinary goods components, just as it was in our earlier analysis in Section 5.2. The efficient consumption relation, as it applies to ordinary goods, is not affected by the presence of public goods.
A similar operation using any two components of the public good bundle in Equation 5.84 leads us to the relation
or the equivalent statement that
Thus we see that the marginal rate of public good substitution is the same for the two consumers under conditions of Pareto optimality.
However, when we calculate the ratio of one component from the ordinary good relation of Equation 5.87 and one from the public good relation of Equation 5.84, we obtain the relation
We note that these results are different from those obtained for Pareto optimality of two ordinary goods in Equations 5.32 through 5.34. Since the ratio of marginal utilities of a consumer for two commodities is, under conditions of utility maximization, the ratio of their prices, we see that the conditions for Pareto optimality when both ordinary and public goods are considered will not correspond to utility maximization with these goods. This means that we must distinguish between private and public goods in any realistic economic systems analysis and assessment.
Example 5.3:
Suppose that consumer utilities are represented by
Here 
and 
are ordinary goods and q3 is a public good. The optimum commodity bundle to maximize utility subject to a budget constraint is easily determined, assuming for the moment that each consumer can demand a different quantity of the public good and that each demand can be satisfied, as the expressions
It can be straightforwardly shown that the sufficiency condition for a maximum of Ui, that ∂2Ui/∂q2 is negative definite along Ii = pTq, requires 
. For this particular utility function, we see that unless the income of all consumers is greater than p3, the demand by each consumer for the public good is different and the entire income of each consumer is spent on it. This is a contradiction since the public good can exist only in a given quantity for all consumers. If all consumers have incomes greater than the relation 2p1p2/p3, then the demand for the public good is the same for all consumers, and Pareto optimality becomes possible. This occurs in this example only because the demand for the public good vanishes for a sufficiently large consumer income. In the range 2p1p2/p3 ≥ Ii ≥ p3, the consumers also demand differing quantities of the public good. Again, this is an impossible demand to fulfill, as only one quantity of a given public good can be made available.
The marginal utilities for this example are given by
The required ratios for Pareto optimality, for ordinary goods 1 and 2, are given by
Between ordinary good 1 and public good 3, we obtain
Thus we will certainly satisfy Equation 5.94 with the quantities of the ordinary goods demanded in Equation 5.93 for all Ii > p3. The ratio of marginal utilities is then just p1/p2, the ratio of prices for consumer optimality. Equation 5.95 will generally not be satisfied unless Ii = I2 or 
, when it is routinely satisfied. In this particular example, q3 = 1 for all Ii = p3. So if all consumers have an income equal to precisely this amount, we satisfy Equation 5.95. In this situation, where p3 = 1, each consumer demands all public goods (Ii < 2p1p2) or all private goods (Ii > 2p1p2). If the incomes are equal, but not necessarily equal to p3, then the consumer demands for like goods, public or private, is the same, and it is possible to achieve Pareto optimality with p3 ≤ Ii ≤ 2p1p2/p3 and with consumers demanding both private and public goods. We can also argue that conditions for Pareto optimality are satisfied for Ii > 2p1p2/p3. This satisfaction is somewhat moot as there is no demand at all for the public good in this case.
A somewhat unrealistic feature of the results obtained thus far in this section is that there is no mechanism to alter the production of commodities in accordance with consumer demand. There is no difficulty in obtaining Paretian optimality of the firms when there are public goods present, so we do not need to consider this case. Profit maximization, under conditions of perfect competition, will result in Pareto optimality of the firm. We will introduce a production function into the relations determining the behavior of the consumer to allow for a more realistic determination of the commodities demanded and the primary factors delivered from consumers in terms of the utility functions of consumers in a society. We will restrict our discussion to the case of two consumers, since the extension to the case of an arbitrary number of consumers is straightforward. For n consumers, the consumer will maximize personal utility subject to the constraint that the utility of other consumers is unchanged. The resulting equations are solved simultaneously to determine Pareto optimal behavior.
We assume that there exists a disaggregation of a produced vector of commodities, Q, into two components q1 and q2. There is a single shared public goods bundle g. The two consumers possess primary factors r1 and r2 and these combine to form the factor input to production x. Here we use r1 and r2 as the factor inputs from consumers in keeping with the notion used in Section 4.3. We assume a single firm, which may well represent an aggregation of a number of firms, with production function φ(Q, g, x) = 0. The problem of consumer 1 is to maximize personal utility subject to the constraint that the utility of consumer 2 remains unchanged and constraints on market clearing equations and the production function. We have the utility functions
Also, we have the production equation
The market clearing (distribution) equations or, more precisely, the Pareto optimal production frontier distribution constraints are given by
The Lagrangian for consumer 1 is
Here λ, γ, v, and g are Lagrange multipliers. The necessary conditions for Pareto optimality are obtained from the Lagrangian in the usual way, where we drop arguments of Ui and φ for notational simplicity:
We can obtain a very similar set of 11 necessary condition equations for Pareto optimality of consumer 2, which we will not formally write down here, primarily to save space.
We can combine Equations 5.102 and 5.103 to eliminate the Lagrange multiplier γ and then take the ratios of any two scalar components of the resulting vector equation to obtain the same result as previously obtained. This shows that efficient consumption of commodities given by
is satisfied, in that the marginal rates of commodity substitution (ordinary goods) are the same for any two consumers, and in general the same for all consumers. In this restrictive sense, we might conclude that the two consumers have the same behavior. By combining Equation 5.102 or (5.103) with Equation 5.105 and taking the ratios of any two scalar components, we obtain
This indicates that the marginal rate of (ordinary) commodity substitution for any consumer is equal to the marginal rate of product transformation (MRPT) for the firms under conditions of Pareto optimality. Thus the efficient product mix relation is satisfied for (ordinary) commodities, even where there are public goods.
We can combine Equations 5.102, 5.103, and 5.105 to eliminate γ such that we obtain
A specific component of this vector equation is
This can be used to obtain Equation 5.114. A specific component of Equation 5.104 is
We may rearrange the two preceding equations to eliminate the Lagrange multiplier such that we obtain
This relation shows that the sum of the marginal rates of substitution of the public good j for the ordinary commodity i for the two consumers is equal to the marginal rate of product transformation from producing ordinary commodity Qi to producing public good gj. Unlike the requirement for marginal rates of commodity substitution, the rates of substitution of public goods for ordinary commodities need not be equal. Consumers do not control producers and there is generally no way in which they can force the producers to adjust their production so that Equation 5.116 is satisfied and such that consumer Pareto optimality will result.
Two other useful results, which can be obtained from Equations 5.102 through 5.112, are
and
These results indicate that the marginal rate of substitution of a factor input from consumers for an ordinary good must, for each consumer, equal the marginal rate of productivity of the factor input to production of the ordinary good. Also, these results show that the sum of the marginal rates of substitution of a primary factor input for a public good is, for all consumers, equal to the marginal productivity of the factor input to production of the public good.
Example 5.4:
Suppose there is one ordinary good, one public good, and one factor input and that consumer utility, for i = 1,2, is
Here we assume that qi, g, and ri are each greater than zero and that ri is also less than the endowed value 
. The firm’s production will be assumed to be
Here ai, b, and c are assumed to be known.
The set of necessary conditions for Pareto optimality for consumer 1, from Equations 5.102 through 5.112, are given by a set of 11 equations of the form
By solving these 11 equations in 11 unknowns, we can obtain the desired operating conditions for Pareto optimality of consumer 1 in terms of the assumed fixed utility of consumer 2, U2. We can formulate a similar set of 11 equations in 11 unknowns and can obtain the desired operating conditions for Pareto optimality of consumer 2 in terms of an assumed fixed utility of consumer 1, U1. We obtain the following additional equations:
Here we use the overbars to denote the Lagrange multipliers for the problem of consumer 2. These equations easily show that the corresponding Lagrange multipliers for the two problems are the same. We then have the following 11 equations:
Solving these 11 relations for the Pareto optimal equilibrium, we obtain
These represent most of the Pareto optimum solution components for this example and the ones of most interest. We note that we cannot obtain explicit solutions for q1 and q2 here. From utility optimality of each consumer, we can assume a wage–income–expenditure relation that may be somewhat unrealistic only in the sense that consumers do not usually pay directly for public goods. This relation is
where p0 is the price of the ordinary good and pp the price of the public good. We arbitrarily assume that each consumer pays half the cost of the public good. These can be used to determine qi. If p0 and pp are not given, they can be obtained from economic optimization of the firm and simultaneous solution of the resulting equations together with the equations for Pareto consumer optimality.
The marginal substitution rates are of considerable interest in this example. From Equation 5.116 we obtain
Thus we see, again, that we must impose a requirement in the firm’s production equation to ensure Pareto optimality. From Equation 5.117 we obtain
Also, from Equation 5.118 we have
These three equations are not independent and can, of course, be obtained directly from the optimum solutions obtained previously.
We have seen that the presence of public goods makes attainment of Pareto optimality difficult owing to requirements that are imposed on the production characteristics of the firm. It may require, for example, that firms sublimate production of automobiles by production of mass transit systems when the economic incentives to the firm for such a transition are not present. Public goods are generally financed by public agencies from taxation and not paid for directly by consumers. Ultimately the consumer must pay for a public good, but this payment will be through a, perhaps complicated, tax structure and not by such a simple scheme as dividing the cost of the public good by the number of consumers. We now focus on other externalities, such as taxes, and imperfections associated with production.
5.4 EXTERNAL EFFECTS AND IMPERFECT COMPETITION: NONINDEPENDENT PRODUCTION AND CONSUMPTION
Imperfections in the production process such as monopoly, oligopoly, monopsony, oligopsony, and the like generally prevent the attainment of Pareto optimality. This also occurs when consumer utility functions are nonindependent. We will first derive conditions of Pareto optimality for the consumer for a case where there are no external or imperfect competition effects. We assume two consumers, two commodity bundles and factor inputs, and a single production equation. This is precisely the problem posed in Equation 5.101 if we simply force the public good bundle equal to zero. Equations 5.113 and 5.114 result for the perfect competition case. We know that the ratio of marginal utilities is just the ratio of prices for the commodities. Thus we have from Equations 5.113 and 5.114 the efficient commodity consumption and efficient product mix relations
Also, we have from Equations 5.106 through 5.108 the efficient factor supply and efficient factor mix relations
From Equations 5.102, 5.103, and 5.105 through 5.108, we obtain the efficient product and factor mix relations
One condition for profit maximization of the firm is obtained by setting the derivative of the profit with respect to factor inputs equal to zero. From this we obtain the relation that the marginal factor productivity of production is just the ratio of wages for the factor to the price of the good produced. Also, we can take the derivative of profit with respect to the quantity produced and obtain the result that the price of the product is just the marginal cost of producing the product. Generally those firms operating under imperfect competition will not obey these rules derived for the perfect competition case. We have shown that when firms and consumers operate under the rules of perfect competition, and there are ordinary goods only, Pareto optimality will result. We should expect that, unless a firm operates under perfectly competitive conditions, we cannot obtain Pareto optimality. However, as we have previously indicated, Pareto optimality is certainly possible even though firms and consumers do not operate under conditions of perfect competition. In this sense, perfect competition is a sufficient, but not necessary, condition for Pareto optimality.
Often the utility of one consumer will depend on the utility, and hence the level of consumption, of other consumers. One consumer’s utility may increase when neighbor Smith buys a new Mercedes, whereas that of another consumer may decrease dramatically because of this same phenomenon. Under these circumstances, to show the dependence of their utilities on the consumption of the other consumer, the utility functions of two consumers may be written as
where the total bundle is constrained by
To maximize the utility of consumer 1 subject to an equality constraint on the utility of consumer 2 and the size of the commodity bundle, we define the Lagrangian
The necessary conditions for maximization of the utility of consumer l are given by
We eliminate the Lagrange multiplier γ by combining Equations 5.127 and 5.128 to obtain
We find two scalar components of this vector equation and form the ratio of these equations to eliminate λ. We obtain
as one of the necessary conditions for Pareto optimality. Equation 5.132 is not the same as Equation 5.120, which is the equation we obtain for perfect competition. Thus we see that perfect competition, which results in Equation 5.120, will not result in Pareto optimality when the consumers have nonindependent utility functions.
Example 5.5:
Suppose that utility functions of two consumers are given by
such that consumer l’s utility is reduced, for fixed 
and 
, as the consumption of consumer 2 increases.
To maximize the utility of consumer 2, we instruct consumer 2 to use income I2 to purchase the commodity bundle
The optimum commodity bundle for consumer 1, who suffers from a keeping up with the Jones’s phenomenon (for fixed purchases by consumer 2), is given by
When we substitute the values of the commodity bundle actually purchased by consumer 2 into the two preceding relations, we obtain
These expressions are independent of the results obtained for consumer 2 only if a = b, or I2 = 0, which is degenerate. This is a very special case that occurs because consumer l’s utility is symmetrically reduced with that of consumer 2 for a = b.
The requirement for Pareto optimality (Eq. 5.132) for this example becomes
If we substitute the results of utility optimization with an income constraint, Equations 5.135, 5.136, 5.139, and 5.140, into Equation 5.141, we obtain a = b as the requirement under which individual maximization of utility under a budget constraint leads to Pareto optimality. For example, with 
and 
, we have U2 = 1250. This is the optimum utility of consumer 2 under the perfect competition utility maximization. If we assume Q1 = 112.5 and Q2 = 43.75, the same total consumption as under individual utility optimization, then the necessary conditions for Pareto optimality of consumer 1 (Eqs. 5.127–5.130) become
Solving these equations leads to the following conditions for Pareto optimality: 
, and 
. Under both Pareto optimality and individual utility maximization, we have U2 = 1250. Under Pareto optimality we have U1 = 721.31, whereas under individual utility maximization we have U1 = 703.13. As expected, we can get a better result for one consumer (consumer 1) under Pareto optimality by readjusting the distribution of commodities to the other consumer (consumer 2) while maintaining constant utility for consumer 2. This occurs because the utility functions are specified such that Pareto optimality requirements do not correspond to the requirements for maximization of each individual’s utility.
An imposed or negotiated Pareto optimal solution results from aggregation of the utility functions of Equations 5.133 and 5.134 into a single utility function given by
We maximize this subject to the two equality constraints on income. The Lagrangian for this problem is given by
Following the usual optimization approach, we obtain the optimum purchased commodities as 
and 
. The consumer utilities corresponding to these purchased commodities are U1 = 800 and U2 = 1200. The sum of the utility functions is greater than that examined thus far for any other case. Unfortunately, however, the utility of consumer 2 has been reduced. If there were some way to redistribute the gain in utility of consumer 1 to enhance the satisfaction of consumer 2, then, since the total utility is higher, there would have been some net societal gain. We will not pursue this point further in this example. However, some of our later discussions in this chapter concerning taxes and social welfare functions will address distribution problems such as this.
We will conclude our efforts in this section by illustrating how taxes and subsidies can be used to obtain Pareto optimality under external disturbance conditions when each economic unit maximizes its own objective function. We could consider a consumer tax and/or subsidy, but will instead focus on the behavior of two firms, each producing a single product and subject to external effects possibly including taxation and subsidy. We assume production cost functions for the two firms, PC1 (q1, q2) and PC2 (q1, q2), such that the profit of the firms is
If each firm maximizes profits by setting its own production level, the necessary conditions for perfectly competitive optimal behavior are
and
A form of negotiated or imposed Pareto optimality may be obtained by forming an aggregate profit function1 given by
and then adjusting q1 and q2 for optimality. In this case, we obtain the results
and
The necessary conditions obtained for perfect competition optimality (Eqs. 5.145 and 5.146) will generally be different from those obtained for the imposed Pareto optimality (Eqs. 5.148 and 5.149). We can compensate for the differences by adding a tax or subsidy to the production cost function for each firm. We will assume a unit tax or subsidy, although an ad valorem tax or subsidy is certainly possible. Thus we have the modified cost functions including the tax or the subsidy, PC1 (q1, q2) + q1 TS1 and PC2 (q1, q2) + q2 TS2, where TS is positive for a tax and negative for a subsidy. The tax or subsidy is generally determined such that when the firms individually optimize their profits, they obtain the production levels corresponding to Pareto optimal conditions. The profit functions with the tax and/or subsidy are given by
The new necessary conditions for perfectly competitive optimality are
We wish to obtain the same production levels from the results of Equations 5.152 and 5.153, and Equations 5.148 and 5.149. Thus we see that the taxes and/or subsidies are obtained by solving
evaluated at the Pareto optimal q1 and q2.
Total profits under conditions of Pareto optimality will generally be greater than profits under perfect competition conditions when market imperfections exist. These profits can be distributed to the firms such that their profits return to the original level as obtained from the production levels in Equations 5.145 and 5.146. There will be a surplus or social dividend, since Pareto optimal profits are greater than perfect competition profits under market imperfections. This social dividend should be used in some beneficial way. This process and results can be best illustrated with an example.
Example 5.6:
We consider the production cost functions
This indicates that each firm experiences diseconomies due to the presence of the other firm. We assume that p1 = 20 and p2 = 25. From Equations 5.145 and 5.146 we obtain
Now, under individual profit optimization we obtain q1 = 5 and q2 = 10. The profits of the firms are Π1 = 15 and Π2 = 80, and the total profit is 95.
Under imposed Pareto optimality, we have the necessary conditions of Equations 5.148 and 5.149, which result in
This leads to the optimal results q1 = 3 and q2 = 9.5. The profits of the firms are now Π1 = 11.5 and Π2 = 87.75, and the total profit is 99.25, some 4.25 units greater than the earlier profit.
From Equations 5.154 and 5.155, we compute the unit taxes to be added to the production costs of the firms as TS1 = 4 and TS2 = 1. We now determine the profits of the firms with the tax imposed from Equations 5.150 and 5.151 as Π1 = −0.50 and Π2 = 78.25. Thus we must make a lump sum subsidy of 15.5 units to firm 1 to bring it back to the original operating-level profit of Π1 = 15, and a subsidy of 1.75 units to firm 2 to bring it back to the original operating-level profit of Π2 = 80. But where does the total subsidy of 17.25 units come from? The government has collected taxes totaling TS1q1 + TS2q2 = 21.5 units. A total of 17.25 units of this tax must be returned to the firms to bring them back to their original profit levels, and 4.25 units represent social dividends and can be used for other purposes.
We should not delude ourselves with the thought that we have just found the Pareto optimal solution. All we can really find is a Pareto frontier or a production possibility set. There are two optimization problems to determine Pareto optimality. The first problem is to maximize the profit of firm 1 subject to the constraint that the profit of firm 2 is constant. The second problem is to maximize the profit of firm 2 subject to the constraint that the profit of firm 1 is constant. We will now illustrate the complete Paretian solution to this problem.
For the first problem we define the Lagrangian
and then obtain the necessary conditions for optimality as
For the second problem we define the Lagrangian
Then, we obtain the necessary conditions for optimality of firm 2 as
By comparing Equations 5.159, 5.160, 5.163, and 5.164, we easily see that the relationship between the Lagrange multipliers is λγ = 1. Thus Equation 5.159 is equivalent to Equation 5.163, and Equation 5.160 is equivalent to Equation 5.164. Equations (5.159)–(5.161) and (5.164) represent four equations in five unknowns. We obtain from these equations
These five equations represent solutions for all negative values of λ that yield all positive solutions. For λ = −1 we obtain the negotiated or imposed Pareto optimal solution just discussed. For λ = −1 the total profit is increased, and if the taxes and subsidies are designed properly, then total profit is increased and no firm should complain if the gain of one firm is used to compensate the loss of another by adopting this imposed solution.
In this example we have seen how taxes and subsidies can be used to entice firms to production levels that enhance social welfare. The results of this example would have been far more dramatic had we allowed firm l’s production to add to the cost of firm 2’s production and firm 2’s production to be deducted from the cost of firm l’s production.
In this section and in Section 5.3 we examined departures from perfect competition. We saw that this will generally create a situation such that non-Paretian allocation of resources will result if firms and consumers optimize according to the rules of perfect competition. We saw how taxes and subsidies can be used to obtain Paretian results from firms (and consumers) that follow the rules of perfect competition.
An interesting question is, what if some constraint prohibits fulfillment of one or more conditions of Pareto optimality? Is there any benefit in adjusting the remaining conditions for Pareto optimality? The answer is, it depends. A sensitivity analysis of pertinent conditions would appear to be the best approach to providing an answer to this question. The theory of the second best can be used to show that a solution with three equations different from the Pareto optimal equations may be better than a solution with two equations that are different from the ones that would result in true Pareto optimality.
Thus we have seen here and in our previous discussions that the notion of Pareto optimality is a very fundamental one and enters in many ways in economic systems analysis and assessment efforts. It is strongly related to other approaches as we have seen in our discussions. We need to strive for Pareto optimality such that there will be no possible allocation of effort or reallocation of organizational resources that will increase product differentiation without increasing costs, and no reallocation that will decrease costs and which will not also decrease differentiation. Generally, an organization should select a single perspective for competitive advantage, that is to be a low cost producer or a high differentiation producer, and then adopt a competitive strategy that is supportive of this. This will insure that we deal with issues involving cost, value, and competition in information and knowledge intensive systems, organizations, and enterprises in an efficient and effective manner.
5.5 WELFARE MAXIMIZATION AND SOCIAL CHOICE
As we have seen, the attainment of Pareto optimality results in situations in which it is not possible to make anyone better-off without making someone else worse-off. Thus Pareto optimal solutions are efficient. There is no guidance concerning equity in that any shift along a Pareto optimal frontier is bound to make someone better-off at the expense of making others worse-off. Thus it should be easy to get agreement on making all distributions Pareto efficient, but there is every reason to expect difficulty in getting agreement concerning which of the many Pareto efficient solutions to implement.
To resolve this difficulty, we pose a scalar social welfare function that expresses a social or normative utility of a, perhaps large, set of individual utility functions. Since increasing the utility of any individual while holding that of all other individuals constant must result in an increase in social utility, we see that the social welfare function must be convex. A social welfare function maximum must, of course, be Pareto efficient. The converse is also true.
We can verify the statement that if X maximizes social welfare, it is Pareto efficient by showing the negative statement is false. If X is not Pareto efficient, then there must exist some other resource allocation 
such that 
for all j with strict inequality holding for at least one j. But if this is the case, then it is not possible for xj and the associated X to maximize social welfare. Demonstrating that every Pareto efficient solution maximizes a (not necessarily the) social welfare function is relatively simple as well. All we need to do is assume that X is Pareto efficient and that individual utilities Uj(xj) are continuous, concave, and monotonic in xj. Then for the linear social welfare function
the particular choice of weights wj such that resource allocation X maximizes W subject to resource constraints 
is given by wj = 1/λj, where λj is the jth consumer’s marginal utility of income and R is the resource constraint. This is equivalent to
This result (see Problems 26 and 27), that the weights wj for the linear social welfare function are the reciprocals of the marginal utility of income, has a very interesting economic interpretation. If a person has a large income, then that person’s marginal utility for additional income will typically be smaller than that for a person with a lower income. The weight corresponding to those with large incomes will be larger than that for those with smaller incomes. We must here remember that this is an implicit social welfare function associated with a perfect competition result involving Pareto efficiency. This shows that a perfectly competitive market will give efficient allocations. Nothing at all is said here about equitable allocations except perhaps the statement that under perfect competition conditions, those with larger incomes will have greater weight in determining the social welfare function than those with lower incomes.
The type of social welfare function needed for normative economics is a cardinal welfare or utility function. It is often difficult to establish group cardinal welfare functions, as they involve interpersonal comparisons of values. It would be preferable to use only ordinal utility functions, which provide numerical representations that allow preference comparisons to be made for each individual. Generally, this cannot be done. Let us now elaborate a little on this point.
We can use ordinal utilities to determine the commodity bundle that would be purchased by a consumer with a fixed income. For example, we can find the commodity bundle that maximizes
subject to the income constraint
for a known price vector. The utility function here need only be an ordinal utility function. Such an ordinal utility function would not be sufficient to allow us to determine by how much the change from commodity bundle A to commodity bundle B would be more or less preferred over the change from commodity bundle C to commodity bundle D. This is generally needed if we have to cope with uncertainty or the chance associated with the outcomes that will result from particular allocations. Also, one needs to compare the relative pleasure that two consumers receive from changes in resource allocations.
It is this latter case that is of concern to us here. If we could work with ordinal preferences only, then it should be possible for us to combine the ordinal preferences of three consumers to obtain a social choice among alternatives. Suppose that there are three possible social distributions of resources XA, XB, and XC. Also suppose that the preferences of the three individuals are as follows:
| Person | Preference structure |
| 1 | XA > XB > XC |
| 2 | XB > XC > XA |
| 3 | XC > XA > XB |
What is the preferred equitable ordering of society’s resources in this case? There is simply no answer to this question. We surely cannot use voting, for two people have preferences XA < XB, two have preferences XB > XC, and two have preferences XC > XA. If society were transitive for these ordinal preferences, we would require that if XA > XB and if XB > XC, then XA > XC. This result does not imply in any sense that society is intransitive; it only says that we may well obtain intransitive group results for transitive group members if we insist on using ordinal preference or ordinal utility functions.
In 1951, Kenneth Arrow demonstrated this result in a very elegant way. He assumed that individual and social preference orderings will satisfy two very reasonable and weak axioms:
Axiom 1. Completeness or Total Ordering.
For all alternative pairs X and Y, either XℜY or YℜX, where ℜ may stand for strong preference >, weak preference ≥, or indifference ∼.
Axiom 2. Transitivity.
For all X, Y, and Z, if XℜY and Yℜ Z, then XℜZ.
Next, Arrow postulated five very reasonable conditions:
1. Complexity and free choice. There are at least three alternatives under consideration and all possible orderings are allowed.
2. Positive association of individual and social values. If an individual increases his or her preference for an alternative, then society increases its preference for that alternative.
3. Independence of irrelevant alternatives. Introduction of a dummy alternative will not influence individual or societal choices, that is, ordinal preference orderings, for the alternatives originally considered.
4. Sovereignty of individual citizens. The social preference ordering cannot be imposed by society such that XA > XB, even if XB > XA by all members of society. This is simply a weak version of Pareto efficiency that says it must be possible to reach the Pareto frontier.
5. Nondictatorship. There must not be an individual such that if he or she has preference XA > XB, then society prefers XA > XB no matter what the individual preference orderings of society are.
Then Arrow showed, by a rather complex argument, that there exists no social welfare function that is guaranteed to satisfy the two axioms and five conditions. This is known as the impossibility theorem of Arrow. It shows us that all voting methods based on ordinal rankings may be irrational. This is not guaranteed, however. Indeed, it can be shown that if the alternatives can be arranged such that all individual preferences among alternatives are single peaked, then majority rule is transitive. But to require that individual preferences be single peaked, such that there exists a common point set that represents alternatives and where preferences decrease with increasing distance from the most preferred alternative, results in a violation of the free-choice condition.
Condition 3 is the most vulnerable of Arrow’s conditions. It in fact involves two parts. The first part requires the irrelevance of dummy alternatives, and that, by itself, is entirely reasonable. The second part requires that this irrelevance be true when we have knowledge only of ordinal preferences or orderings. Many believe that this requirement is unreasonable for cases in which we have only ordinal preference relations. If we allow cardinal utilities or preferences, then we can obtain a possibility theorem using Arrow’s axioms and conditions. To allow this requires that we should obtain revealed preference intensities or cardinal utilities for individuals and combine them into a scalar choice function. To fully pursue this topic would, sadly, take us far away on a lengthy journey that would be somewhat removed from our present goals. Suffice it to say that we advocate taking a cardinalist perspective with respect to individual utility and social choice functions.
Our goal in this chapter has been to extend the concepts concerning consumers, firms, and market equilibriums to cases where perfectly competitive economic assumptions do not apply. We introduced the important concept of Pareto optimality and showed some very simple results from multiple objective optimization. We indicated that Pareto optimal surfaces are efficient but not necessarily equitable and introduced the concept of a social welfare function that could, in principle, ensure equity. Finally we indicated that ordinal preferences are generally insufficient to result in meaningful social choice and that cardinal preference intensities must be used. We will extend these concepts of normative economics to cost–benefit and cost–effectiveness analyses in Chapter 6.
PROBLEMS
1. Derive the Pareto production curve or Pareto frontier in Example 5.1.
2. In a two-consumer, two-producer economy, the utility and production functions are given by
where
are the quantities produced. Capital and labor are constrained by
Show that the efficient production, efficient consumption, and efficient product mix equations are given by
where
3. What are the relations for Problem 2 that ensure that the utility of each consumer is maximized subject to fixed income and known prices? Does the efficient consumption relation follow?
4. What are the relations for Problem 2 that ensure maximum profit on each firm, assuming fixed wages for capital and labor? Does the efficient production relation follow?
5. Two consumers in a closed system have the following utility functions:
The initial endowment for consumer 1 is one unit of commodity 1; the initial endowment for consumer 2 is one unit of commodity 2. What is the Paretian efficient redistribution of commodities? What are the market clearing prices that should exist for equivalence to the economic theory of the consumer?
6. Repeat Problem 5 with 
.
7. In an economy of 50 consumers and 3 commodities, consumer 6 has a Cobb–Douglas utility function
Consumer 6’s commodity bundle under a certain Paretian efficient allocation is 
. What are the perfectly competitive equilibrium prices of these commodities? By extrapolating on the implications of this example, can you establish a related method to determine parameters for a utility function of fixed form from the knowledge of the prices and the size of the commodity bundle? What is the role of Pareto optimality in this?
8. Suppose that there are two firms and two consumers in an economy whose behaviors are described by
where
a. What are the conditions for Pareto optimality for simultaneous consumption and production?
b. How do these compare with Paretian optimality conditions for production only, and for consumption only?
c. What are the Paretian production and distribution frontiers for the case where
d. How do the results in part (c) compare with the general economic equilibrium results that you obtain for the specifications of part (c)?
9. Suppose that the production functions of two firms are given by
Initially one-half of the total capital and labor is associated with each firm. Can capital and labor be reallocated to enhance production?
10. Suppose that in Problem 9 there is a single consumer with utility function U(q1, q2) = q1q2. How should capital and labor be allocated to maximize utility?
11. Suppose that a person may divide his or her labor L between two activities q1 and q2. The person’s utility function is U(q1, q2) = q1q2 and the production equations are 
and 
. What is the optimum division of labor among the two production processes? How does the efficient product mix relation apply here?
12. What are the relevant efficient production, consumption, and production mix equations for a model such as that in Problem 2 where there are three consumers and three firms?
13. The efficient production frontier in a certain economic system is given by
There are three consumers with identical utility functions given by
The actual consumption of the three consumers at present is given in the table below.
What is the Pareto optimal reallocation of the commodities?
14. Investigate the Pareto optimal allocation of n commodities among m consumers with utility functions
for an economy in which the production frontier is given by
15. Derive Equations 5.120 through 5.122.
16. Determine the conditions for Pareto optimality of the two firms whose profit equations are 5.143 and 5.144. Use this result to determine the efficient production frontier in general terms and for the specific production functions used in Example 5.6.
17. Suppose that there are two firms that produce products q1 and q2. The production cost equations are
The price of each product is $10.
a. What are the requirements for the profit maximization of each firm?
b. What are the requirements for Pareto optimality?
c. How can an imposed Pareto optimality to maximize the sum of the profits of the two firms be obtained?
18. In this example, we consider the case where Paretian efficient consumption does not exist owing to presence of a single monopolistic firm. Assume that the consumer demand function is p = D(q) and that the production cost function is PC = c(q). Generally the price charged by the monopolist is too high and the produced quantity too low for Pareto optimality of consumers to exist. Show that a unit subsidy added to the profit of the monopolist may encourage a production level and subsidy that are those of perfect competition and that therefore satisfy (consumer) Pareto optimality. What is the amount of this subsidy?
19. With respect to the issue posed in Problem 18, suppose that the consumer demand and production cost function are given by p = 3 − q and PC = 0.5q2. What are the imperfect competition price, the quantity produced, and the profit? What are the Pareto optimal price, the quantity produced, and the profit? What are the amounts of the subsidy per unit product and the lump sum tax on consumers?
20. The production cost functions for two firms are 
and 
. The market prices are p1 = 3 and p2 = 5. Determine the optimum profit for each firm. What is the optimum profit for the firms if the sum of the profits of the individual firms is maximized? What are the taxes and subsidies that will result in this profit?
21. Suppose that the utility of consumer i can be described by
where there are m consumers and n commodities. What are the requirements for Pareto optimality? How can they be imposed?
22. Investigate the hypothesis that efficiency and equity can be disaggregated in the sense that we can maximize efficiency by a correct allocation of factors (capital and labor) between production inputs. Then, as a separate problem, we can determine a unique allocation of these produced goods among people such as to ensure equity. You may assume, to simplify your efforts, that an efficient production frontier F(q) = 0 has been determined and is assumed to be known. As a particular vehicle for your discussion, you may wish to use the utility functions for two consumers
the social welfare function
and the resource constraint
23. Did your response in Problem 22 depend on the knowledge of the production frontier? In other words, is the efficient production frontier a function of the social welfare function?
24. Suppose that the utility function of two individuals in a perfectly competitive closed economy is
and that societal utility is
where
The production function for the two goods is
Suppose that person 1 is initially endowed with one-fourth of the total labor. What portion of the capital should be owned by consumer 1 if we wish to maximize social welfare? How is this influenced by Ξ? Provide an interpretation to this.
25. Consider the two-household social welfare function
where Ui is the utility of the ith household. The production characteristic for the two firms is given by
where
a. Show that the optimal solution to maximize social welfare is Pareto optimal.
b. Suppose that
Find the optimum distribution of commodities as a function of a. How could a be specified?
c. Suppose that the factors are owned by households and that each of the two households maximizes its utility Ui subject to a budget constraint 
, i = 1,2. Find the economic equilibrium conditions and contrast and compare them with your results in part (b).
26. Verify Equation 5.167 for a perfectly competitive economy.
27. Suppose that two consumers have the utility functions
and that the initial allocation of commodities is 
a. What are the market equilibrium prices?
(a) Reformulate this problem in terms of conventional utility functions and incomes.
b. What are the resulting weights for the social welfare function of Equation 5.166?
28. Suppose that there are 50 consumers and 3 goods in a simple economy. Consumer 1 has a Cobb–Douglas utility function 
and one Pareto efficient allocation is x1 = [5, 2, 2]T. What are the perfectly competitive prices? How do these change with changing the Pareto efficient allocation and what does this say with respect to the income of consumer 1? Suppose that the utility of consumer 1 is given by 
and that the social welfare function is 
. What is the equitable distribution of commodities? How would this differ from the distribution under perfect competition?
29. Suppose that the production function in an economy is linear and given by
and that the social welfare function is
What is the optimum set of commodities X to maximize social welfare?
30. Suppose that there are three alternatives to choose from, and a group of three people elects to vote to determine the preference ordering among alternatives. Each individual is transitive and the preferences of the individuals in the group are such that one person prefers a to b to c, one prefers b to c to a, and one prefers c to a to b. Majority rule decides. Does the majority prefer a to b or b to c or a to c? How could an interpersonal comparison of preferences using a cardinal scale over the interval 0 to 1 resolve fundamental difficulties encountered here?
31. In attempting to rank alternatives, we might assign points in accordance with preferences. For example, our first choice might get five votes, our second choice four votes, etc. Suppose that in ranking five alternatives, two people assign votes as follows:
Now suppose that alternatives C and E are, for some reason, eliminated. Show that the new voting is such that B is assigned a higher priority than A and preferences for the first two alternatives are reversed. This results from the nonindependence of irrelevant alternatives, one of Arrow’s conditions for the existence of an ordinal social welfare function. Show that if the preferences had been scaled in a cardinal fashion over the interval 0 to 1, we would not have this problem. What potential difficulties exist in obtaining this interpersonal comparison of values?
BIBLIOGRAPHY AND REFERENCES
Three references that provide excellent and detailed coverage of welfare economics are as follows:
Layard PRG, and Walters AA. Microeconomic theory. New York: McGraw-Hill; 1978.
Mishan EJ. Introduction to normative economics. New York: Oxford University Press; 1981.
Ng YK. Welfare economics: towards a more complete analysis. New York: Macmillan; 2004.
The above references provide much historical perspective and many references to original journal sources. The following reference is of interest with respect to a survey of modern theories of social choice and systems engineering and management science applications
Sage AP. Behavioral and organizational considerations in the design of information systems and processes for planning and decision support. IEEE Trans Syst Man Cybern 1981;SMC-11(9):640–678.
A brief excellent philosophical text concerning social welfare is
MacKay AF. Arrow’s theorem: the paradox of social choice. New Haven Connecticut: Yale University Press; 1984.
The celebrated works of Arrow, which we have mentioned in Section 5.5, are discussed in the aforementioned texts and in
Sage AP. Methodology for large scale systems. New York: McGraw-Hill; 1977.
1This is not the general result we obtain by maximizing Π1 with an equality constraint on Π2 and maximizing Π2 with an equality constraint on Π1.