IV.- RESULTADOS Y DISCUSIÓN
4.2. Subcuencas Tributarias
Figures 5.1 to 5.4 illustrates these four types of pricing schedules. In the horizontal axis are represented the quantities consumed X and in the vertical axis we have the households' total
11
bill (TB)^ for i= P(poor) ,R(rich). The curves of total bill are different according to the type of pricing schedule being used:
i) in the case of type A, the total bill for the poor and for the rich are the straight lines O-POOR and 0-RICH respectively, with constant marginal prices P ^ ^ and P ;
IP IR
ii) in the case of type B, the total bill is the curve OABC, with a constant marginal price P^ up to (in the segment OA) and a constant, higher, marginal price P^’ for quantities superior to X^ (in the segment BC). The curve OABC shows a discontinuity in the segment AB explained by the upward movement ot the total bill curve when all units are charged at Pj^ ' instead of at P^ ;
iii) in the case of type C, the total bill OAB shows in thesegments OA and AB the same marginal prices P^ and P^ ’ shown by type B; however, instead of having a discontinuity in A, the total bill curve shows a kink at this point since the higher price P^* applies only to the marginal quantities;
iv) in the case of type D, the total bill curve (TB)^ is a continuous curve with tangents that increase with the quantity
consumed since higher prices are used for charging each marginal quantity.
Let us examine now some possible consumer equilibria when these types of pricing schedules are used to charge households for
4
Type D can be thought as an approximation of type C, for an infinite number of blocks, each block comprising one unit consumed.
their consumption of commodity 1. In figure 5.1 to 5.4 we will be using indifference curves relating quantities consumed of commodity 1 to the total bill these consumptions entail. In the appendix to this chapter we discuss the transformation of the original households' indifference curves defined in terms of the quantities consumed of commodities 1 and 2 to ones defined in terms of the quantity consumed of commodity 1 and the respective household’s expenditure made to buy this quantity. This change is convenient since it is important to show how the equilibrium will be reached for different expenditure levels.^ Figure 5.1 shows two indifference curves, C** and C^. These curves are related to two utility functions U (TB) ,X for i=P(poor household), R(rich household), where is the quantity consumed of commodity 1 by household i , and (TB)^ is its respective total bill for consuming that quantity, that is, (TB)^ = P^^X^^ ; we are assuming that SU^/9(TB)^ < 0 and ôU^/ôX^^ > 0. We are also assuming they are concave, that is, the disutility of an increase in the total bill requires a larger increase in the quantity consumed to generate the same level of utility to household i. The most preferred indifference curves are the lowest ones: for a given X , the ^ ^ 6 largest level of utility is given by the lowest level of (TB)^. Curve is steeper than C^, what means that U ’ > U ’ for all X ,
R p 1
that is, the marginal utility of consuming an additional quantity of X^ is greater for the rich than for the poor. In other words, the rich’s marginal willingness-to-pay is always greater than the
7 poor’s.
Note that Sharkey and Sibley (1993,pp.202-205) also found it necessary to use expenditure-quantity indifference curves to discuss the self-selection problem.
^ For figures 5.1 to 5.4, the depicted curves and C^ will be assumed to be lowest ones that can be reached when a given pricing schedule is used.
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This means that the rich is a high-demand and the poor is a low-demand consumer; they have non-crossing demands for commodity 1 and the rich’s demand is above the poor’s.
Figure 5.1 exhibits a full information equilibrium, that is, a quantity equilibrium that is produced when a type A pricing schedule is used, that is, when the households' social conditions are known to the public utility and the discriminatory prices and are accordingly applied.^ At these prices (constant for any quantity consumed) the chosen quantities demanded by the poor and the rich are and respectively. Note that when this type of pricing schedule is used, there is not possibility of arbitrage, that is, for instance, the rich consuming the same quantity consumed by the poor in order to pay the lower price.
Total Bill (TB)^ RICH POOR IR IP 0 Quantity X X X IP I R 11
Figure 5.1: Discriminatory prices and full information equilibrium.
Figures 5.2 and 5.3 show a possible quantity pooling equilibrium when a block-price schedule type B or C is used, respectively. In both figures we illustrated a situation in which the
g
We are employing Spence’s (1974) terminology used in his analysis of the job market signaling.
rich's self-selection constraint induce them to select the same quantity consumed by the poor, paying the same lower price. Of course, their choice frustrates the distributional objective the price schedule may have whem it was assumed that the rich would select a higher quantity, paying consequently a higher price. It should be noted that the pooling quantity equilibrium is allowed by the discontinuity or the kink at point A of the total bill curve: the indifference curve is steeper than C^; then they could not touch the total bill curve in the same point if this point had not that special condition. Total Bill (TB)^ 0 Quantity X X X IR IP 11
Figure 5.2: Type B block-quantity pricing schedule and a quantity pooling equilibrium.
Total Bill (IB)
X X Quantity X
IP IR 11
Figure 5.3: Type C block-quantity pricing schedule and a quantity pooling equilibrium.
Figure 5.4 illustrates a quantity separating equilibrium which occurs when the the pricing schedules is of type D, that is, when the total bill curve is a continuous smooth curve as depicted in
the figure. The equilibrium quantities chosen by the poor and by the rich will be necessarily different since their indifference curves show different degrees of steepness.
(TB)
Quantity X
X X
I P IR 11
Figure 5.4:Type D price schedule and a quantity separating equilibrium.
In the event of a generalized personal arbitrage as that one illustrated in figures 5.2 and 5.3, then instead of observing a set of strictly progressive discriminatory prices being practiced (assuming a continuous income distribution), some households with different incomes could bunch together in terms of the quantities consumed, bringing about price constancy for them, although their incomes differ. As a consequence, the conceived social policy built in the public utility's price schedule (that is, price growing with income, as in the Brazilian case) would be frustrated.
It seems clear that a possible failure of a block quantity price schedule in generating an effective progressive pricing in terms of households incomes is not due to the fact that households are cheating their socio-economic conditions (actually, when the price schedule is defined in terms of the quantity consumed, consumers are not supposed to reveal their incomes), but because there may be no compatibility between the higher quantities they are assumed to consume and the corresponding higher prices they should pay, given their consumption preferences. In other words, the signal
(in the Spence’s (1974) sense) they are required to give, the quantity consumed, is attached to such a high price that this makes
them prefer transferring their demands to a lower level of consumption and, consequently, paying a lower price. The next section will deal with this problem, deriving a price for which the problem of adverse selection is eliminated.
5.4 - Derivation of an Adverse Selection-Free Optimal Price.
The nature of the problem is that, although consuming a larger quantity of would make the rich household enjoy a higher level of welfare, this expanded quantity would mean a higher total bill and in their comparison of the additional welfare with the additional costs of obtaining it, these households come to the conclusion that it is not worth.
The higher total bill the households should pay results from the larger quantity of consumption by the fact that the price schedule sets a progressive tariff in terms of the quantity consumed.
It is important to clarify the nature of the problem we are going to examine in this section. These are the main, simplified, features of the problem:
i) We assume two groups of households, the poor and the rich or non-poor; there are n^ households in the poor group and n^ in the rich group. The poor are low-demand consumers and the rich are high-demand consumers.
ii) The public utility uses a price schedule that offers two bundles: , respectively the quantity the poor is supposed to consume and the unit price they should pay for this quantity; (X ,P ), directed to the rich, where X and P are the quantity
IR IR IR IR ^ ^
the are supposed to consume and the price for each unit they consume, respectively.
iii) The choice of the prices P^^ and P^^ by the public utility has to take into account the individual rationality constraints and the incentive compatibility constraints of each of those two types of households, in addition to the objective of maximization of social
welfare and the financial balance constraint.
Let us derive the prices P and P that satisfy constraints
^ IP IP
(5.7) to (5.10) in addition to the financial balance constraint; let W be the social welfare function and TC - R 3 D be the public utility's financial balance constraint, where C is its total cost function, R is its total revenue and D is the level of deficit financed by the government.
To derive P^^ for i= P (poor), R (rich) we need to maximize W subject to those conditions. The maximand function is
L = W+ kJ D - C + R ] + - UptO.Y^)] +
where for i=l,...,5 is, respectively, the Lagrange parameter for the financial balance constraint, for the poor’s individual rationality and the incentive-compatibility constraints, and for the rich’s individual rationality and incentive compatibility constraints.
The Kuhn-Tucker necessary conditions for a maximum of L are:
dL/dX :s 0 , X 2:0 and X .dL/dX = 0
IR IR IR IR
ôL/9p^^ 0 , and p^.5L/9p^= 0 , for i=l 5.
We have that
a w / a x = - n (T X a p ’ / a x for i = p , R (5.1 2)
li i l l ! 11 11
where is the marginal social utility for household i;
aC/aX^^= n^ m (5.13)
ÔR/aX = n P (1 - l/e ) for i=P,R
11 i 11 11 (5.14)
where e^^ is the household i’s demand price elasticity for commodity
1; (5.15)
[“
a U (X ,Y -P X ) /a x = n U ’(X ,Y -P X ) R IP R IP IP IP P R IP R IP IP (5.16) (5.17) i"p(\R'VlR^lR)]''«\R = "r " ; « 1 R - V ^ I R ^r’ (5. 18)For short, let us call the derivatives that appear in the right-hand side of expressions (5.15) to (5.18), respectively, (X^^), ' U ’ (X ) and U ’(X ).