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PUBLIC S ECTOR R ATIONING AND PRIVATE S ECTOR S ELECTION
SIMONA GRASSI University of Lausanne
CHING-TO ALBERT MA Boston University
Abstract
We study the interaction between nonprice public rationing and prices in the private market. Under a limited budget, the public supplier uses a rationing policy. A private firm may supply the good to those consumers who are rationed by the public system. Consumers have different amounts of wealth, and costs of providing the good to them vary. We consider two regimes. First, the public supplier observes consumers’ wealth information; second, the public supplier observes both wealth and cost information. The public sup- plier chooses a rationing policy, and, simultaneously, the pri- vate firm, observing only cost but not wealth information, chooses a pricing policy. In the first regime, there is a con- tinuum of equilibria. The Pareto dominant equilibrium is a means-test equilibrium: poor consumers are supplied while rich consumers are rationed. Prices in the private market increase with the budget. In the second regime, there is a unique equilibrium. This exhibits a cost-effectiveness ra- tioning rule; consumers are supplied if and only if their cost–benefit ratios are low. Prices in the private market do not change with the budget. Equilibrium consumer utility is
Simona Grassi, Faculty of Business and Economics, Department of Economics and Econo- metrics, and Institut d’Economie et de Management de la Sant´e, University of Lausanne, Building Internef, CH-1015 Lausanne, Switzerland (simona.grassi@unil.ch). Ching- to Albert Ma, Department of Economics, Boston University, Boston, MA, United States (ma@bu.edu). We thank many seminar and conference participants for their comments and sugges- tions. We also thank the editor John Conley and two referees for their advice. Various parts of the research here were done while the authors were at the Universidad Carlos III de Madrid; we are grateful to their hospitality. The first author received partial financial support from the Italian Fulbright Foundation.
Received January 12, 2009; Accepted September 10, 2010. ©C 2012 Wiley Periodicals, Inc. Journal of Public Economic Theory , 14 (1), 2012, pp. 1–34.
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higher in the cost-effectiveness equilibrium than the means- test equilibrium.
1. Introduction
Many governments and public organizations provide or subsidize goods and services such as health care and education. Free or subsidized public provi- sion often coexists with a private market. In this paper, we study the inter- action between rationing policies in the public sector and profit-maximizing prices in the private sector. We derive equilibria of games between the pub- lic and private sectors, and compare equilibrium prices and aggregate con- sumer utilities. All public programs operate under limited budgets. Unable to cover all intended consumers, a public supplier must use a rationing rule. A va- riety of rationing and subsidy practices exist. Current Medicaid policies in the United States provide health insurance to indigent individuals. In the pending U.S. health care reform, families up to 400% of the Federal Poverty Level receive either free health insurance or substantial subsidies. These ra- tioning policies are a means-test mechanism, which allocates public supply to poor individuals. In Canada and many European countries, public health systems ration care according to illnesses, patients’ medical conditions, and treatment costs. This form of rationing is based on a cost-benefit or cost- effectiveness mechanism, which allocates public supply to individuals for whom it is worthwhile, in some qualified sense. Rationed individuals nevertheless may consider purchasing from the pri- vate market. Here, an important issue is selection of profitable consumers by a private firm. A focus of our paper is on how a private firm’s profit- maximizing strategy will react to the public rationing mechanism. For ex- ample, when an individual does not qualify for Medicaid, his income should not be very low; neither should be his willingness to pay for health insurance. On the other hand, if a patient does not qualify for a certain treatment ac- cording to the public sector cost-effectiveness measure, a private firm may not infer about his willingness to pay. Our model consists of a set of consumers, a public supplier, and a private firm. Each consumer would like at most one unit of an indivisible good (a medical treatment, a course of education, etc.). Consumers have different wealth or income levels, and the costs of providing the good to them also differ. We use wealth heterogeneity to model differences in consumers’ val- uations of the good. Rich consumers are more willing to pay for the good than poor consumers. Cost heterogeneity arises because a consumer’s char- acteristics may determine how much it costs to supply the good to him. For example, the cost of a medical treatment depends on illness severity, and the cost of helping a student to achieve an academic standard depends on the student’s ability and aptitude. Variations in these characteristics affect provision costs.
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and ignores consumers’ wealth information. The cost-effectiveness criterion emerges in equilibrium. The private and public sectors will appear to be sep- arated, with the public sector serving consumers with low cost-benefit ratios, while the private sector serving those with high cost–benefit ratios. Equilib- rium private market prices for available consumers remain the same as if the public sector were inactive, and equilibrium prices are independent of the budget. At any given budget level, equilibrium aggregate consumer utility is higher when rationing is based on the cost-effectiveness criterion than the means-test criterion. What explains these results? Richer consumers have higher willingness to pay than poor consumers. In the private market, consumers face prices that are dependent on their costs, not wealth. Hence, richer consumers obtain higher incremental surplus from trades in the private sector. This trade-surplus effect is common across both information regimes. When ra- tioning is based on wealth, releasing richer consumers to the private market allows them to realize more incremental trade surplus there than poorer con- sumers. This motivates the public supplier to ration the rich. Simultaneously, when public supply is for poor consumers, the firm knows that only richer consumers are in the market. These consumers have higher willingness to pay, so the firm raises prices. In the second information regime, cost information is available. Here, absent the private sector, a cost-effectiveness principle applies: public supply is for consumers with a low cost–benefit ratio. The cost-effectiveness princi- ple continues to apply when there is a private market. For a given cost level, a consumer obtains more surplus from free public supply than the private market. Rationing low-cost and rich consumers cannot occur in equilibrium. If such consumers were rationed, the private firm would understand that low-cost consumers must be rich, and would raise prices accordingly. The trade-surplus effect cannot be implemented. Our goal is to study the effect of a price-reactive private sector, so we have rejected a perfectly competitive market where prices would always be marginal costs (but for completeness, we have included related results). We use the monopoly setup, but all results extend to Cournot competition. We have let the monopolist observe costs, but not wealth. If the firm would also observe wealth, it would be able to extract all surplus, and the public ra- tioning policy would not affect consumers’ trade surplus in the private mar- ket. In any case, firms seldom possess information on wealth. Given that the good is indivisible and a consumer buys at most one unit, nonlinear prices cannot be implemented. It would be a less interesting model if the firm did not observe consumers’ costs; selection and cream-skimming issues would be assumed away. 2 Most other papers assume that the government is a first mover. Gener- ally, an ability to commit to a rationing rule is valuable. However, as we show
(^2) Also, when the price is based on the expected cost, the firm may renege once it learns that the cost turns out to be higher than the price it has charged.
Public Sector Rationing 5
in a companion paper, Grassi and Ma (2011), the public supplier’s Stackel- berg rationing rule is time-inconsistent. By committing to ration some poor consumers, the public supplier can implement lower prices because the firm will want to sell to poor consumers when costs are low. However, given lower prices, the public supplier would exploit the trade-surplus effect by reneging, supplying poor consumers and rationing rich consumers. We have adopted a simultaneous-move game, and the public supplier and the private firm have symmetric commitment power. This is a long-term perspective between the players because both players choose mutual best responses. Finally, our fo- cus is on nonprice rationing. Free public provision is the main mechanism in public health and education systems, so we have not included taxes and subsidies in the public supplier’s strategies. We study interactions between public rationing and a noncompetitive private market, while models in the literature usually assume a competitive private market, or exogenous pricing rules. Barros and Olivella (2005) focus on public physicians referring patients to their own private practices. The public sector uses waiting-time rationing, and physicians refer patients when the patients’ costs are low. Iversen (1997) considers the effect of a private sector on waiting time in the public sector. Hoel and Sæther (2003) consider supplementary private health care when public health care is subject to wait- ing time rationing. In the above papers, the price in the private market is fixed. Hoel (2007) derives the optimal cost-effectiveness rule when patients have access to a competitive private market, where prices do not respond to the public sector’s allocation rule. By contrast, in our model, prices in the private sector respond to public policies. Rationing is similar to transfers in kind. Blackorby and Donaldson (1988) show how transfers in kind may solve asymmetric information prob- lems. The literature has also studied redistribution effects. In Besley and Coate (1991), the government uses a poll tax to provide for free a good at a low quality. Rich consumers optimally choose the good at a high quality from a competitive private market, while poor consumers do not. The gov- ernment in effect taxes rich consumers to subsidize poor ones. Segregation between rich and poor consumers is an equilibrium of our game when ra- tioning is based on wealth, but this stems from a trade-surplus effect in the private market. In Besley and Coate (1991) the budget is endogenous while we abstract from financing issues, and assume an exogenous budget. Our formal model is like a common agency model. The public supplier and the private firm are two principals whose actions will affect the con- sumer, who is the agent; see Bernheim and Whinston (1986). In line with the common agency model, we use a symmetric setup, so that both sectors react against each other’s strategy. We are unaware of a paper that models how public sector rationing and private sector pricing strategies mutually react. We also depart from the mechanism design literature on the provi- sion of public goods (see Norman 2004, and Hellwig 2003 on excludable public goods), where incentive-compatible, individually rational and budget- balanced schemes are derived. In Norman (2004) the public supplier may set
Public Sector Rationing 7
generates the same utility increment, independent of the consumer’s wealth. If the consumer with wealth w pays a price p to consume the good, his utility is U ( w − p ) + 1; his utility is U ( w ) if he does not consume the good. We discuss the nonseparable utility function in Appendix B. If a consumer with wealth w is indifferent between paying τ for the good and the status quo, we have
U ( w − τ ) + 1 = U ( w ). (1)
This equation implicitly defines a willingness-to-pay function τ : [ w , w ] → R+^ for consumers with various wealth levels. Because U is concave, hence almost everywhere differentiable, the willingness-to-pay function is differen- tiable. From total differentiation of (1), we have
dτ d w
U ′( w ) U ′( w − τ )
A consumer’s willingness to pay for the good is strictly increasing in wealth due to the strict concavity of U. We will assume that the lowest willingness to pay τ ( w ) is larger than the lowest cost c. This assumption ensures that there is some scope for any consumer to benefit from a trade in the private market. We illustrate our description of consumer preferences and costs with ex- amples in the health market. The good may refer to a surgical procedure (for example, a hip replacement). Patients differ in their illness severity levels (some hip replacements are more difficult than others). For a fixed amount of improvement in health, interpreted as a unit increment of utility (for ex- ample, the ability to walk about without pain), sicker patients require more resources, and richer patients are more willing to pay. In our setup, consumer preferences do not directly depend on the pro- vision cost c. In the health care example, this means that patients with differ- ent severity levels obtain the same incremental utility from the good. One interpretation is that the good provides a standardized unit of improve- ment in well-being. In other situations, consumers obtain different incre- mental utilities depending on their severity levels. Consumer preferences then may depend on cost, and we will discuss this alternative assumption in Appendix B.
2.2. The Public Sector and Rationing
The public sector has a budget B which is insufficient to supply the good to all consumers for free, so 0 < B < γ. We consider two information regimes. First, only consumers’ wealth information is available to the public supplier, and second, consumers’ wealth and cost information is available. In each case, nonprice rationing will be used to allocate the budget for providing the good to consumers. In the first regime, the public supplier’s rationing rule is a function θ : [ w , w ] → [0, 1]. For w ∈ [ w , w ], the public supplier provides consumers
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with wealth below w a total of
∫ (^) w w (1^ −^ θ( x ))^ f^ ( x ) d x^ units of the good. The rationing rule θ splits the density f so that at w , [1 − θ( w )] f ( w ) of con- sumers are supplied at zero price, and θ( w ) f ( w ) of consumers are rationed. Because wealth and cost are independently distributed, the cost c among rationed consumers remains distributed according to G. In the second regime the rationing rule is a function φ : [ w , w ] × [ c , c ] → [0, 1]. It has the same interpretation as in the first regime. For consumer ( w , c ), the density φ( w , c ) f ( w ) g ( c ) is available to the private firm. 3 In each regime, the public supplier’s objective is the sum of consumer utilities. The rationing schemes θ and φ correspond to random rationing, but can be implemented by waiting times. We can add to the consumer prefer- ence specification a new parameter, say δ, a random variable whose distri- bution depends on wealth, cost, or both. The utility of a consumer is now U ( w ) + 1 − δ t if he gets the good after a delay of t units of time. The pa- rameter δ describes the consumer’s marginal waiting cost. An impatient con- sumer (one with a high value of δ) may decide against the public system if he expects a long delay. By setting the delay t , the public supplier determines the fraction of consumers within a wealth group or a wealth-cost group who choose to wait for the good in the public sector.
2.3. Benchmark Optimal Rationing Policies with an Inactive Private Market
For now suppose that the public sector is the sole provider. Consider the first information regime where rationing is based on wealth. For a rationing rule θ, total consumer benefit from the public supply is
∫ (^) w w (1^ −^ θ( w^ ))^ f^ ( w^ ) d w as each unit of consumption increases a consumer’s utility by one unit. The consumer welfare index, which the public supplier maximizes, is
V (θ) ≡
∫ (^) w
w
U ( w ) d F +
∫ (^) w
w
[1 − θ( w )] f ( w ) d w. (3)
The rationing rule must satisfy the budget constraint
γ
∫ (^) w
w
[1 − θ( w )] f ( w ) d w ≤ B , (4)
which says that the expected cost must not exceed the available budget.
(^3) We restrict rationing rules to those that leave the functions θ( w ) f ( w ) and φ( w , c ) f ( w ) integrable, so that
∫ (^) w w θ( x )^ f^ ( x ) d x^ and^
∫ (^) w w φ( x ,^ c^ )^ f^ ( x ) d x^ at each^ c^ are well defined for w ∈ [ w , w ]. We can restrict the public provider to supply to either all or none of the con- sumers within a wealth class or a wealth-cost class. Rationing schemes are then functions that map [ w , w ] to { 0 , 1 }^ and [ w , w ] × [ c , c ] to { 0 , 1 }. The general rationing functions can now be interpreted as mixed strategies. For ease of exposition, we do not use the mixed strategy interpretation.
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or the marginal consumer w by setting the price τ ( w ). A quantity function is denoted by w ̂ : [ c , c ] → [ w , w ]. We present the profit functions under the two rationing rules. First, sup- pose that rationing is based on wealth. At cost c the density of consumers available to the firm is θ( w ) f ( w ). At a price τ ( w ), consumers with wealth higher than w will buy, and the profit is
π( w ; c , θ) =
∫ (^) w
w
θ( x ) f ( x ) d x [τ ( w ) − c ]. (7)
Here, the integral is the total quantity purchased, and τ ( w ) − c is the price- cost margin. Second, suppose that rationing is based on wealth and cost. At cost c the density of consumers available to the firm is φ( w , c ) f ( w ). At a price τ ( w ), consumers with wealth higher than w will buy, and the profit is
π( w ; c , φ) =
∫ (^) w
w
φ( x , c ) f ( x ) d x [τ ( w ) − c ]. (8)
2.5. Interaction between the Public and Private Sectors
We study the following games and look for their subgame-perfect equilibria:
Stage 1: Nature draws ( w , c ) according to the distributions F and G , respec- tively, for each consumer. The private firm observes c. The public supplier observes either w , or both w and c.
Stage 2: In each information regime, the public supplier chooses a rationing rule, θ or φ, and the private firm chooses a quantity function ̂ w.
Stage 3: Consumers supplied by the public sector get the good for free, and consumers not supplied by the public sector may purchase from the private firm at prices set in Stage 2.
3. Equilibrium Rationing and Prices When Rationing
is Based on Wealth
We begin with the private firm’s profit-maximizing prices and consumers’ utilities. Then we present an equilibrium in which the public supplier uses the entire budget on consumers with lower wealth levels. Next, we present a continuum of equilibria, and show that the one we have presented is Pareto dominant. Finally, we discuss some comparative statics on the budget, as well as Cournot and perfect competition in the private market.
3.1. Profit-Maximizing Prices and Consumer Utilities
To characterize an equilibrium, we need to refer to a profit-maximizing quantity function when the firm has access to all consumers. Let this function
Public Sector Rationing 11
Figure 1: Quantity functions ̂w m^ ( c ) and ̂w ( c ).
be ̂ w m^ ( c ). Suppose that the public supplier rations all consumers, so in (7) we set θ( w ) = 1 for all w. The function ̂ w m^ : [ c , c ] → [ w , w ] is given by
̂ w m^ ( c ) ≡ argmax w
∫ (^) w
w
f ( x ) d x [τ ( w ) − c ]. (9)
We assume that the profit function in (9) is concave, and that ̂ w m^ ( c ) is single- valued. By the Maximum Theorem ̂ w m^ ( c ) is continuous. We further assume that as c varies over [ c , c ], the marginal consumers vary over a proper subset of [ w , w ], so w < ̂ w m^ ( c ) < ̂ w m^ ( c ) < w. This requires that variation in wealth is sufficiently large relative to variation in costs. The optimal quantity ̂ w m^ ( c ) is given by the usual marginal-revenue-equal-marginal-cost condition. The quantity function ̂ w m^ ( c ) is strictly increasing. As marginal cost increases, the optimal quantity is adjusted to achieve a higher level of marginal revenue. This means setting a higher price and selling to less consumers. 5 By the con- cavity of the profit function for any w > ̂ w m^ ( c ), the derivative of profit with respect to w is negative:
d d w
∫ (^) w
w
f ( x ) dx [τ ( w ) − c ] < 0 for w > ̂ w m^ ( c ), any c. (10)
The solid line in Figure 1 illustrates such a quantity function. We have assumed that the budget is insufficient to cover all consumers at zero cost ( B < γ ). For a very small budget, the interaction between the public and
(^5) The sign of the derivative of ̂ w m (^) ( c ) is the same as the sign of the cross partial of ∫^ w w f^ ( x ) d x (^) [τ ( w ) − c (^) ], which is positive.
Public Sector Rationing 13
Figure 2: Consumer density under rationing scheme θ.
optimal quantity stays at w ˜. 6 Under the rationing scheme, in Figure 1 the optimal quantity function becomes the horizontal, dotted line when cost falls below ˜ c. When will an equilibrium quantity function fail to be continuous? Sup- pose that θ( w ) = 0 for w ∈ [ w 1 , w 2 ] where w < w 1 < w 2 < w , and θ( w ) = 1 otherwise. The public sector supplies only to consumers with medium wealth. Figure 2 illustrates the density of consumers available to the pri- vate firm. The profit-maximizing quantity function ̂ w ( c ) is in Figure 3. For c < c 1 or c > c 2 , the profit-maximizing quantity is unique. For c ∈ ( c 1 , c 2 ), the price remains constant because all consumers with wealth in [ w 1 , w 2 ] are supplied by the public sector. At cost c 1 , the firm makes equal amounts of profit whether it charges a price τ ( w 2 ) selling to consumers with w > w 2 , or τ ( w 0 ) selling to consumers with w between w 0 and w 1 and above w 2. Finally, some quantities may never be chosen; in Figure 3, the firm never sets ̂ w ( c ) to any w ∈ [ w 0 , w 2 ). We need to write down aggregate consumer utility given rationing functions and quantity functions that are weakly increasing and possibly
(^6) Any w < ˜ w yields a profit
[ ∫ (^) ˜ w
w
θ( x ) f ( x ) d x +
∫ (^) w
˜ w
θ( x ) f ( x ) d x
] [τ ( w ) − c ]
=
[ ∫ (^) w
˜ w
f ( x ) d x
] [τ ( w ) − c ] <
[ ∫ (^) w
˜ w
f ( x ) d x
] [τ (˜ w ) − c ]
so setting w = ˜ w is optimal.
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Figure 3: The quantity function ̂w ( c ) and its “inverse” ̂ c (w).
exhibiting upward jumps. Given a quantity function, ̂ w ( c ), rationed con- sumer ( w , c ) buys from the private firm if and only if w ≥ ̂ w ( c ). In Figure 3, this is the set above the graph of ̂ w ( c ). It is more convenient to view the set of purchasing consumers as one indexed by a function ̂ c : [ w , w ] → [ c , c ] that is like an “inverse” of ̂ w. Define ̂ c ( w ) = sup { c : w ≥ ̂ w ( c )}; if there is no c ∈ [ c , c ] such that w ≥ ̂ w ( c ), set ̂ c ( w ) = c. Such a function ̂ c is illustrated in Figure 3. While the function ̂ w gives the wealth of the marginal consumer in terms of his cost, the function ̂ c gives the threshold cost level below which a con- sumer with wealth w will buy at price τ (̂ w ( c )). Whenever ̂ w is strictly increas- ing and continuous, the function ̂ c is its inverse. When ̂ w is constant on an interval, then ̂ c exhibits discontinuities at the two ends of the interval. Finally, ̂ c ( w ) becomes c when the firm does not sell to consumer ( w , c ). Clearly ̂ c ( w ) is increasing whenever its value is not c. The set of consumers who purchase are those with ( w , c ) below the graph of ̂ c ( w ), and this differs from those above the graph of ̂ w ( c ) at most for a set of measure zero. Functions ̂ w and ̂ c are two equivalent ways of keeping track of consumer types who purchase from the private firm. Given a quantity function ̂ w (and its equivalent ̂ c ), and a rationing scheme θ, the welfare index V (θ) is ∫ (^) w
w
[1 − θ( w )] f ( w )[ U ( w ) + 1] d w
∫ (^) w
w
θ( w ) f ( w )
∫̂ (^) c ( w )
c
{ U ( w − τ (̂ w ( c ))) + 1 } g ( c ) d c
∫ (^) c
̂ c ( w )
U ( w ) g ( c ) d c
d w. (^) (13)
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Under rationing policy θ, aggregate consumer utility is
∫ (^) w
w
[ U ( w ) + (1 − θ( w ))] f ( w ) d w
∫ (^) w
w E
θ( w )
∫ (^) c E
c
[
U ( w − τ ( w E^ )) + 1 − U ( w )
]
g ( c ) d c
∫̂ (^) c m (^) ( w )
c E
[
U ( w − τ (̂ w m^ ( c ))) + 1 − U ( w )
]
g ( c ) d c
f ( w ) d w.
(15)
The welfare index is described as follows. The integral on the first line of (15) is the sum of the base utility U ( w ) plus the utility increase from the public supply. The second line is the private market incremental surplus of rationed consumers. Only rationed consumers with wealth above w E^ will buy from the private market. For these consumers, if their costs are below c E^ , they purchase at price τ ( w E^ ), and obtain the incremental surplus in the integral with limits between c and c E^ ; if their costs are above c E^ , they purchase at price τ (̂ w m^ ( c )) if their costs are below ̂ c m^ ( w ), and obtain the incremental surplus in the integral with limits between c E^ and ̂ c m^ ( w ). A rationing policy θ is a best response if it maximizes (15) subject to the budget constraint (4). We consider the trade-off in rationing a consumer with wealth w. The benefit of rationing a consumer is the saving of expected cost γ , a constant. The cost of rationing a consumer depends on the con- sumer’s wealth level. If w is below w E^ , this consumer does not buy from the private market, so the cost is one unit of utility due to nonconsumption. If w is above w E^ , this consumer may gain some incremental surplus from the private market (the second line in (15)), so the cost of rationing him is less than one unit of utility. Rationing a rich consumer is less costly precisely be- cause the rich consumer has the opportunity to buy from the private market. Therefore, it is optimal to ration richer consumers, those with wealth above w E^. This equilibrium is similar to many practical schemes in which poor con- sumers receive free supplies while the rich do not, but this means-test equi- librium is not due to an equity concern. The public supplier selects among consumers with different wealth levels to participate in the private market. Wealthy consumers realize larger gains in trade in the private market, so they are rationed. The private market fully anticipates that poor consumers are unavailable, so even when cost decreases, the equilibrium price stops falling.
3.3. Characterization of a Continuum of Equilibria In this subsection, we characterize all equilibria. We will show that in equi- librium the firm’s quantity function is the monopoly quantity function for consumers with cost higher than a threshold, and a constant otherwise, but
Public Sector Rationing 17
this threshold must be higher than the one in Proposition 1. In equilibrium the public supplier must ration rich consumers, but may also ration some poor consumers. To characterize the equilibrium rationing policies, we let the public sup- plier choose the net density of rationed consumers θ f , and impose the re- quirement that 0 ≤ θ f ≤ f. The consumer welfare index (14) is linear in θ f , and for each w its first-order derivative with respect to θ f is
∂ V ∂θ f
∫̂ (^) c ( w )
c
{ U ( w − τ (̂ w ( c ))) + 1 − U ( w )} g ( c ) d c − 1. (16)
This expression measures the change in aggregate consumer utility at wealth level w. It is the expected incremental surplus from consumer with wealth w buying from the firm (the integral) less the unit incremental utility of con- sumption at zero cost. We establish a monotonicity in the supplier’s prefer- ences.
LEMMA 2: The first-order derivative (^) ∂θ∂ V f in ( 16 ) is increasing in w. It is strictly increasing in w ∈ [ w 1 , w 2 ] unless ̂ c ( w ) = c for each such w.
Lemma 2 says that the public supplier favors rationing the consumer over supplying as the wealth level increases. This is a basic principle in our model. Prices in the private market depend on cost, so consumer ( w , c ) gets more surplus from a trade at price τ (̂ w ( c )) as w increases: U ( w − τ ( w ̂ ( c ))) + 1 − U ( w ) is increasing in w. The public supplier’s marginal util- ity from rationing, (16), is strictly increasing in w for consumers who buy from the firm. When consumers do not buy from the firm, there is no in- cremental surplus, so the integral in (16) is 0, and the derivative in (16) becomes −1, independent of w. Lemma 2 does not take into account the budget, the consideration of which is our next step. Against a quantity function ̂ w ( c ) (and the correspondinĝ c ( w )), the pub- lic supplier chooses θ f to maximize (13) subject to the budget constraint (4). Using pointwise optimization, we consider the Lagrangean
θ( w ) f ( w )
∫̂ (^) c ( w )
c
{ U ( w − τ (̂ w ( c ))) + 1 } g ( c ) d c
∫ (^) c
̂ c ( w )
U ( w ) g ( c ) d c
- [1 − θ( w )] f ( w ) [ U ( w ) + 1] − λ[γ (1 − θ( x )) f ( x ) − B ],
where λ is the multiplier. The first-order derivative of the Lagrangean with respect to θ f is
∂ V ∂θ f
∫̂ (^) c ( w )
c
{ U ( w − τ (̂ w ( c ))) + 1 − U ( w )} g ( c ) d c − 1 + λγ. (17)
Public Sector Rationing 19
Figure 4: Equilibria in which some poor consumers are rationed.
θ( w ) = 1 for w E^ + δ < w ≤ w ,
where > 0 and δ > 0 are both small numbers. In this rationing rule the supplier shifts some resources from those with wealth just above the lowest value w to those consumers with wealth just above w E^. Figure 4 shows the density of consumers available to the private firm in such an equilibrium. Values of and δ can be so chosen that the new scheme satisfies the budget: B = γ [ F ( w E^ + δ) − F ( w + )]. Against this rationing scheme, the private firm sets a quantity function equal to ̂ w m^ ( c ) for c > c E^ + η and ̂ w m^ ( c E^ + η) for c < c E^ + η, where c E^ is the cost threshold in Proposition 1, and η > 0 satisfies ̂ w m^ ( c E^ + η) = w E^ + δ. In this equilibrium, the public supplier gives the good to some con- sumers with wealth slightly higher than w E^ , but rations consumers with wealth close to the lowest level. These rationed consumers have such low willingness to pay that the firm will not reduce price to sell to them even when cost is lowest. Furthermore, because consumers with wealth slightly higher than w E^ are now supplied by the public, the private firm’s price will not fall all the way to τ ( w E^ ). In Appendix A, we provide a formal proof for this equilibrium. Infinitely many equilibria can be constructed in a similar fashion. As long as the private firm does not find it profit-maximizing to reduce price to sell to consumers with low willingness to pay, a quantity function like the one in Figure 1 remains a best response. In all these equilibria the public supplier rations some consumers with low wealth, but must ration all consumers with wealth above a threshold.
20 Journal of Public Economic Theory
The equilibrium in Proposition 1 is focal. This is the one that achieves the highest welfare index for the public supplier. This is because it has the widest range of price reduction as cost decreases. The equilibrium also allows the private firm to make the highest equilibrium profit. Any equilibrium dif- ferent from the one in Proposition 1 would have fewer transactions in the private market.
PROPOSITION 2: The equilibrium in Proposition 1 achieves the highest equi- librium consumer utility, and the highest equilibrium profit for the private firm. In any other equilibrium, the public supplier sets θ( w ) = 1 , for w > ˜ w e^ , where ˜ w e^ > w E^ (defined by F ( w E^ )γ = B in Proposition 1) and the firm sets a price equal to τ (̂ w m^ ( c )) for c > ˜ c e^ , and a price equal to τ (˜ w e^ ) for c < ˜ c e^ , where ̂ w m^ (˜ c e^ ) = ˜ w e and ˜ c e^ > c E^ (defined by w E^ = ̂ w m^ ( c E^ ) in Proposition 1).
How are consumers’ utilities affected by the public supply and the price reaction in the private market? In Proposition 1, consumers with wealth above w E^ are rationed, where γ F ( w E^ ) = B , and the firm’s equilibrium prices range between τ ( w ) and τ ( w E^ ). Suppose that the budget increases by B , then w E^ will increase by w E^ , where γ F ( w E^ + w E^ ) = B + B , so the minimum price in the private market becomes higher. In Figure 1, the new equilibrium is obtained by shifting the dotted horizontal line upward. An increase in the budget will be used in equilibrium to supply consumers with wealth just above w E^ so available consumers in the private market are wealthier, and the firm reduces its price less when cost falls. Define c E^ by ̂ w m^ ( c E^ + c E^ ) = w E^ + w E^. Consider consumers with wealth above w E^ + w E^ , those that remain rationed after the budget increase. They still are offered the monopoly prices when their costs are above c E^ + c E^ , so their utilities remain unchanged, but those with costs below c E^ + c E^ face a higher price τ ( w E^ + w E^ ) although all of them still prefer to purchase. Wealthy and low-cost consumers are hurt by the budget increase, while more poor consumers benefit.
COROLLARY 1: Under rationing based on wealth, private market equilibrium prices are higher when the public supplier’s budget increases. Consumers who remain rationed after the budget increase face a strictly higher price when their costs are low.
We have assumed a monopolistic private sector. The extension to an im- perfectly competitive sector poses no conceptual problem. For our model of a homogeneous good, we consider a Cournot model. Let there be N firms in the private sector. Given a rationing scheme θ, let each firm choose a quan- tity function ̂ q (^) i ( c ), where i = 1 ,... , N. The total supply is q ( c ) =
∑ N
i = 1 q^ i^ ( c^ ). For the market to clear the marginal consumer is ̂ w ( c ) where
∫ (^) w ̂ w ( c ) θ( w^ )^ f^ ( w^ ) d w = q ( c ), and the price in the private sector is τ ( w ̂ ( c )). All results derived above continue to hold for any given number of firms in the private sector.
