Why do nonprofits exist in some sectors and not in others?
Sera Linardi
Division of Humanities and Social Sciences, California Institute of Technology
Abstract
This paper investigates whether concentration of nonprofits can be explained by its competition with for-profits. Under the assumption that nonprofits maximize consumer welfare, it solves for the equilibrium producer of the Cournot and Stackelberg competition between a nonprofit and a for-profit (mixed duopoly) under various cost structure. The result predicts that nonprofits will be more prevalent in constant cost industries and will coexist with for-profits in industries with binding capacity constraints. The equilibrium under mixed duopoly also predicts more sectoral specialization than under regular duopoly (competition between two profit maximizers). The results, when compared with domestic data on concentration of nonprofits, are consistent with broad industry categories (e.g. manufacturing, social services), but does not explain detailed breakdown (e.g types of educational services) as well as the existing theory (Glaeser, 2001).
Introduction
In economics, nonprofit organizations are seldom studiedand not well understood. This is a significant omission; not only does the nonprofit sector makes up nearly one-tenth of the United States economy, but it also addresses many socially important problems. International nonprofits, or
non-governmental organizations (NGOs), include some of the world’s most recognizable global service providers, such as Oxfam and Doctors Without Borders. The focus of this paper lies mostly within domestic nonprofits, although it will discuss international NGOs whenever appropriate.
In the United States and most industrialized nations, nonprofits exist almost exclusively in the service industry. Several explanation have been offered, the most theoretically developed of which focuses on information asymmetry. The idea that nonprofits exist in sectors where ex-post contract enforcement is costly for consumers was first advanced by Arrow (1963) and most recently developed by Glaeser and Shleifer (2004). Another major strand of thought, introduced by Weisbrod (1977) and further developed by Ben-ner (2004), argues that nonprofits exist to fulfill the residual demand left from
government’s provision of public good.
These explanations overlook a simple but important piece of the puzzle: concentration of firms in an industry must be examined within the context of strategic intra-industry competition. Regulatory restrictions aside, production in an industry can take place in three kinds of firms: a for-profit private firm, a non-profit private firm, or the government (as a public firm). The identity of the first mover and the action it takes to further its objectives can change the attractiveness of that industry to late comers and potentially affect the industry concentration.
In this paper I focus on strategic interaction between nonprofits and for-profits in oligopolistic markets. To learn about the effect of cost structure on the identity of the producer(s) in equilibrium, I will model their interaction as a simultaneous quantity competition (Cournot) and as a quantity competition with a leader and follower (Stackelberg). This analysis suggests that the historical domination of for-profits in manufacturing and the existence of fixed cost may partially explain why domestic nonfor-profits are mostly in the service industry and not at all present in the production of consumer goods.
Even though it comes as no surprise that cost structure affect the composition of firms in an industry, this approach has not been used in the nonprofit sector. Klemm (2004) alludes to the nonprofit sector in a welfare analysis of constant and fixed cost quantity competition between a profit maximizing firm and a quantity maximizing firm. Klemm departs from our assumption of non-operation when budget
constraint is not met; his model assumes that the quantity maximizing firm will produce for maximum profit when the budget constraint is violated. There are also studies of mixed oligopoly between a government firm (as a total welfare maximizer) and multiple for-profit firms, again focusing on total welfare (Beato and Mas-Colell (1984), De Fraja and Delbono (1989)). This paper, on the other hand, is focused on industry concentration predicted by a mixed oligopoly with a consumer welfare maximizing firm and a profit maximizing firm.
There are several reasons the industrial organization approach has not been widely used with nonprofits. First, academics have yet to reach an agreement on a nonprofit objective function. Section II discuss this central debate in more detail. Second, it is difficult to be specific about the costs faced by nonprofits. Beyond the usual consideration of technology and management, there are multiple forces that affect the nonprofit firm’s costs. Volunteerism, donations, and tax exemption are assumed to lower costs, while the lack of profit motive is assumed to lower innovation and hence lead to higher costs. However, these relationships are yet to be established.
In this paper I will examine the equilibrium producer of the quantity competition between a nonprofit and a for-profit firm. I will show that in most increasing returns to scale industries, only one kind of firm will be in production. The entrance of the nonprofit firm reduces producer surplus to such an extent that the for-profit will almost always attempt to prevent the nonprofit from entering. However, when the technology exhibits decreasing returns to scale, neither firm will be able to individually satisfy demand and hence they will have to coexist. This is consistent with the fact that the manufacturing
industry, which is well represented by increasing returns to scale technology, is not a mixed sector. On the other hand, in the hospital industry, where capacity is capped by number of beds, the share of nonprofit and for-profits are nearly even.
Even though this analysis has not been extended to two period games, I do not forsee many changes in the results by expanding into a limit-entry model. In the fixed cost cases, the Stackelberg leader in our model always produces alone in equilibrium. The benefit of becoming a monopolist in the
second period will only lessen the effect of capacity constraints in limiting what the Stackelberg leader can do to prevent the other firm from entering.
The paper will proceed as follows: Section I gives a literature review and presents the data on the types of producer in various industries. Section II describes the model and explains the reason behind our choice of framework. Section III presents the detailed results and compares it to the data. Section IV lays out open questions and preempts further discussion.
I. Background
The Role of Nonprofits
Nonprofits and its international variant, the non-governmental organization (NGO), play a large and increasing role in domestic and international development. Wilkipedia.org recently stated that there are 2 million nonprofits in United States, most of them formed in the past 30 years and employing 10% of the labor force. India has 2 million NGOs and creates dozens more daily; in Kenya, a country with a labor force of only 9 million, 240 nonprofits are created every year. Today nonprofits and NGOs deliver more official development assistance than the entire UN system combined, excluding the World Bank and the International Monetary Fund (Matthews, 1997). In many countries with faltering government, nonprofits are the active players in urban and rural community development, education, and healthcare.
Problems from globalization such as global poverty and migration have brought NGOs to the forefront when policymakers realize that those problems could not be solved within a nation. International organizations, such as the World Trade Organization, can only serve as a partial solution to these global problems since they were perceived as being too centered on industry interests. NGOs are developed to counterbalance this trend by emphasizing humanitarian issues, developmental aid, and sustainable development.
The academic study of nonprofits can roughly be categorized into several areas of focus: taxation (the legal system), volunteerism and donation (altruism), public good provision, and information
asymmetry. The bulk of research of domestic nonprofits is done through in-depth empirical investigation of specific sectors, such as Sharon Kagan’ study of day care center (1991), Henry Hansmann’s analysis of the arts industry (1980), and Mark Pauly’s work in health economics (1987). On the other hand, the study of NGOs in development is dominated by international affairs and public policy case studies and data gathering through development agencies such as the United Nations or the World Bank. Beyond work by economists such as Maithreesh Gathak (2003, 2006), there are nearly no theoretical framework developed specifically for NGOs. We currently know too little about international NGOs to confidently evaluate whether the models developed for domestic nonprofits is appropriate for them.
Industry Concentration
The persistent puzzle in the not for profit sector has been why there are nonprofits in some
industries and not in others. This question has mostly been asked in the context of domestic nonprofits in industrialized nations and has been plagued by data problems: many nonprofits are small and unregistered and even among those that are
registered it is sometimes unclear whether they are advocacy organization or producers. I will use 1987 data from the John Hopkin Comparative Nonprofit Sector Project, which is the largest academic project to collect and classify international nonprofit data. Comparing this data with a 2005 study of Indiana’s nonprofit sector revealed that
even though the exact percentages vary across states and time, the identity of the dominant producer in each industry remains the same.
According to employment data from New York state in 1987 (Table 1), nonprofits dominate social services (87%), museum, zoos & gardens (92%), and membership organizations (61%). For-profits are the dominant in non-service industry – in agricultural, mining, construction, transport, commerce, printing, electricity, gas, and sanitation, sectors where the nonprofit share is nonexistent. Interestingly, for-profits are also dominant in certain service industry, such as financial services (51%), amusement (79%), and legal services (97%). The two sectors coexist evenly in education and health care, however, in education (Table 2) there is a very distinct product specialization where four year colleges and research universities are more often nonprofit, while non collegiate educational services are usually for-profits.
Table 2. Detailed Industry Concentration
Source: Rose-Ackerman, S., “Altruism, Nonprofits, and Economic Theory”, Journal of Economic Literature, June 1996, Vol.34-2, pp. 701-728.
Literature Review
The early models to explain the presence of nonprofits suggested that in a society with heterogeneous taste, government's provision of public good at the median voter's level leaves many unsatisfied demanders, who then donate to nonprofits to get the level of good that they want (Weisbrod (1979), Bergstrom, Blume and Varian (1986)). However, it is not entirely convincing to attribute the absence of nonprofits to homogeneous demand and most importantly, this theory does not answer why residual demand in certain industry are served solely by the private sector. For example, the private industry, not the nonprofit sector, produces cars to serve those not satisfied by public transport. In postal service, we also have for-profit providers such as UPS and FedEx. On the other hand there are nonprofit and for-profit substitutes to government hospitals and public education.
The currently prevailing theory is based on information asymmetry between producer and consumer. After Kenneth Arrow introduced it in 1963, most economists have used this approach to discuss the nonprofit sector (Easley and O'Hara, 1982, James and Rose-Ackerman, 1989, Steinberg and Gray, 1993). It suggests that nonprofits would dominate in industries where the opportunity for ex-post expropriation is high; in other words, in industries where it is difficult for the purchaser to contract with the providerof the good or service. Glaeser and Shleifer (2001) observed that the industries where nonprofits exist involve one time purchases, or purchases that are difficult to reverse. For example, hospitals deal with issues of life and death, and schools do not often lose students due to transfers to better schools. The nonprofit’s inability to distribute profits forces it to weigh the loss of reputation more heavily than for-profits, which results in higher quality production and ability to charge higher prices.
Glaeser proposed this as a model of research universities – the non-distributional constraint in nonprofit universities signals that nonprofits are more like to use funds to raise quality and to improve job benefits (such as provide opportunity for research) than for-profits. Students for whom quality is
important pay a higher price to attend these schools and research-oriented professors apply to work there in hope to enjoy perquisites such as time off for research.
Glaeser’s discussion of the professors illustrates the supply side of this information asymmetry model. Preston (1983) observed that employees in the nonprofit sector often have a civic-minded interest in service and consider the level or quality of service important. Her labor donation hypothesis says that these workers are attracted to nonprofits because it increases the likelihood that their labor will affect the quality of service. Rose-Ackerman (1996) further generalizes this argument for donor-nonprofit
relationship.
However, the information asymmetry approach does not explain why there are no nonprofits separating out the “lemons” in markets where consumer are famously uninformed, such as the market for used cars, or auto-service. It also does not explain the low share of nonprofits in finance and legal services when the quality of financial and legal advice are as hard to verify as the quality of service from a hospital or university and the cost of ex-post expropriation to the customers is just as high (e.g, jail time).
These contracting problem based theories have also been criticized for not being able to explain why reputations do not similarly work for for-profit firms in sectors where nonprofit firms are widespread (Francois, 2004). Arguing that sinceempirical evidence (Engle 1980, Dubin and Navarro 1988,
Hansmann 1987a) shows a high concentration of nonprofits in services with a public good component, Avner Ben-Ner (2002) proposed that "the nonprofit sector exists because it can solve better than for-profit firms problems associated with the provision of products with publicness attributes". He then argued that there is a correlation between level of non-rivalry and non-excludability of the goods provided to the presence of nonprofits. Casual observation of domestic nonprofits seems to confirm this relationship, however, nonprofits in developing countries deals with the distribution of excludable goods. For example, many people in Indonesia and Uruquay buy their food from local cooperatives, and in China mortuary services are done exclusively by nonprofits. This seems to suggest that present day composition of a sector can partially be explained by the type of firm that had a historical head start.
The missing piece from the explanations above is the fact that when the nonprofits act as producers, their market shares will depend on competition from other firms, which in most cases are for-profit firms. My results show that while a nonfor-profit can potentially produce in any field, it will do
especially well in constant average cost industries, where even a for-profit Stackelberg leader is
indifferent about its entry. On the other hand, a for-profit Stackelberg leader will defend its turf fiercely in a fixed cost industry and hence exclude nonprofits from production.
II. The Basic Model
Nonprofit Firm’s Objective Function
What is a nonprofit firm’s objective function?This question has been a central puzzle in modeling nonprofits -- the exclusion of profit maximization has left a space that no alternative objective function has been able to fill. This lack of objective function has hampered the study of nonprofits, for example, when we ask in this paper why there are nonprofits in some industries and not in others we first need to know where we expect to see them, which require us to know what the nonprofits are trying to accomplish.
Although nonprofits are mission driven, these goals are too varied to be modeled. The problem of using stated mission to infer a nonprofit’s objective is further complicated by the fact that the firm may not be actively pursuing its mission. A large portion of nonprofit funding comes from government grants or private firm donations, which may cause the nonprofit to act as the donor’s agent instead.
In line with Newhouse (1970) and Pauly (1987), this paper posits consumer welfare maximization as the nonprofit objective function. Newhouse’s model of nonprofit hospital maximizes quality and quantity since “consumers have a right to medical care. It is the basis for granting hospitals certain tax privileges.. it seems to be the raison d’etre for philanthropy.” Our reasoning is similar: since nonprofit firms are usually started by concerned citizens, it is reasonable to assume that they were created to maximize consumer welfare. Whether they actually do so or not is a whole topic upon itself; for example, James, 1990, and Winston, 1999 proposed that nonprofit educational institutions pursue “prestige
Another attractive feature of consumer welfare maximization is that it provides a way to conceptually distinguish nonprofits firms from private for-profits firms and public firms / government. The lines between these three types of firms are hazy in real life and we may be able to separate them in the normative sense: the for-profit firm maximizes its own profit (which in the case of one firm is producer welfare), the public firm maximizes total welfare, while the nonprofit firm maximizes the consumer surplus. This conceptualization lays the groundwork for future work on nonprofits, such as analyzing the impact of the replacing governments with nonprofits in public service delivery.
The Model
This inquiry starts from the assumption that the only difference between a nonprofit firm and a for-profit firm is that the former maximizes consumer welfare instead of profits. I study the problem within the following simplified framework: consider an industry in which two firms, a Nonprofit firm (N) and a For-profit firm (P), face the same technology (represented by the cost structure) and produce a homogeneous commodity to maximize their objectives (un and up) . Let qn and qp be the amount produced
by N and P, respectively. Firms can not operate when profits are negative. I adopt a static, partial equilibrium analysis, assume complete knowledge on the part of all agents, and rule out any principal-agent problem or presence of any potential entrants.
For simplicity, the firms face linear demand with a choke point at A>0 : p (qn, qp) = A - (qn+qp)
Assume A>c, F>c, and c>0. I want to discover the identity of the producer(s) in the equilibrium of oligopolistic quantity competition under the following settings:
• Constant Returns to Scale technology, represented by Constant Average Cost (CAC) cost function: C(q)=c.q, which I will treat as the baseline case.
• Increasing Returns to Scale technology, represented by Decreasing Average Cost (DAC) cost function: C(q)=F+cq, which captures industry with large entry cost, such as those which requires extensive infrastructures.
• Decreasing Returns to Scale technology, represented by Increasing Average Cost (IAC) cost function: C(q)=c.q2, which describes capacity constraints.
• U-shaped Average Cost industries, represented by the cost function: C(q)=F+c.q2. This is considered
the “most realistic” representation of technology since it captures both startup cost and capacity constraints.
I will use the standard industrial organization model for quantity competition, the Cournot and Stackelberg model, which implies that nonprofits charge market prices. This is not completely
representative of the real world: nonprofits in an industry are varied in their decision and freedom to choose whether to charge or not. Advocates of the information asymmetry approach will argue that price discrimination and/or product differentiation is a more appropriate model. Price discrimination through the standard Bertrand model will not yield interesting results since the nonprofit’s production will simply shifts the demand function downward and leaves the for-profit to be the monopolist of the residual demand. I readily agree that a model of nonprofit-for-profit competition through product differentiation is necessary, however, for now the Cournot and Stackelberg model provides the simplest way to start analyzing this interaction.
III. Results
The equilibrium producer in the Cournot and Stackelberg game between a Nonprofit (N) and a For-profit (P) for the four different cost structures I outlined above is summarized in Table 3 below. I will refer to this game as the Mixed Duopoly, to distinguish this from Regular Duopoly, the standard for-profit competition. Derivation of the results is given in Appendix A and B.
The basic intuition of the result is as follows. Let qN(0) be the Nonprofit’s ideal production point,
the intersection of average cost and demand, and let qP (0) be the For-Profit’s ideal point, the intersection
(such as a fixed cost) or satisfying the entire market (such as capacity constraints), the total quantity produced will be qN(0) and the mixed duopoly market can only return zero profits, which is not so
attractive for For-profits firms . This is illustrated by the Constant Average Cost case as the baseline case (see Lemma 1 in Appendix B).
However when fixed cost is present (see Lemma 2, Appendix B), the For-profit firm can earn strictly positive profits by preventing Nonprofit’s entry. How it does so depends on the market size. In large markets, qP (0) will leave enough demand for the Nonprofit to recoup the fixed cost and is therefore
not enough to prevent its entry. For-profit then have to produce more than its profit maximizing amount, which may lower its profits but will still leave it with strictly positive profits.
When capacity constraints binds, both types of firms will coexist regardless of Stackelberg leadership. This is because in order to prevent the other firm from entering, one firm has to serve the entire market, which it can not do while satisfying its nonnegative profit constraint. We see this in the Increasing Average Cost case (Lemma 3, Appendix B) or U-shape average cost case where demand intersects average cost beyond the minimum efficient scale point.
This model suggests that nonprofits will be more prevalent in the service industry than in the production of goods for two reasons. First, the cost structure in the service industry most resembles the CAC case, while the production of goods most resembles the DAC case. For example, in social services the addition of each child represents a nearly constant marginal cost due to the fixed requirement of face-to-face time with social worker, while the production of an extra unit of good is nearly costless to the factory. Second, the social services were historically provided by nonprofits (e.g, churches) while manufacturing in western countries has historically been fueled by technological innovation from the private sector. However, costs do not explain why there are no nonprofits in the legal industry, which is a person to person service industry like social services. This may be explained by Mark Pauly (1987) suggestion that for-profit firms are more likely to locate where market conditions permit high profit and prices for any firm.
This model also suggests that the reason nonprofits and for-profits coexist in the health case and education industry may be because both are large U-shape average cost markets. Hospitals are capacity constrained by the number of beds, while schools are limited by classroom size. However, cost structure does not explain the intra-sectoral breakdown, such as the specialization of nonprofits in research institutions and for-profits in community colleges and distance learning programs. It may be that the cost structure is better at explaining broad industry concentration while information asymmetry models such as the Glaeser and Shleifer’s is better at explaining intra-industry breakdown.
There is another interesting insight from this result. Academic research has implicitly taken the position that existing models can adequately account for the nonprofit case with only minor adaptations. However, the comparison of the equilibrium under mixed duopoly to the equilibrium under regular duopoly (Table 4) suggests more sectoral specialization in a nonprofit for-profit competition than
competition between for-profits. In Mixed Cournot, only IAC and Large U-AC predicts coexist, while in Regular Cournot all but Small DAC and Small U-AC predict coexistence. Mixed Stackelberg also only predicts coexistence in IAC and Large U-AC, while Regular Stackelberg predicts coexistence in CAC, Large DAC, IAC, Medium to Large U-AC, This is consistent with our observation that nonprofits and
for-profits coexist in few industries, while for-profit firms coexist with each other in any industries. This may suggest that there are fundamental differences between businesses and nonprofits that make the prediction of existing models problematic, and that we need to address these shortcomings with a new approach to understanding nonprofit sector.
In all cases, consumer welfare under mixed duopoly is always higher than that in regular duopoly while producer welfare is always lower. The pull of the nonprofit towards average cost pricing in mixed duopoly minimizes the area of the deadweight loss triangle between the regular duopoly quantity, price, and average cost. Note that total welfare in a regular duopoly is calculated by adding the surplus that goes to the consumer and the two producers, while total welfare in a mixed duopoly is comprised only of the consumer surplus and the for-profit firm’s profit. It is unclear how to account for the nonprofit firm’s welfare – it seems that we can not simply double the consumer surplus even though it also represent the nonprofit’s utility.
Since I am only interested in the identity of producer(s) in equilibrium, I assume that the nonprofit is receiving no grants or donations to finance its production. I can easily extend this model to account for donations just by adding a variable to the budget constraint. Assuming the donation is always positive, this extension will strengthen the nonprofit’s position, further increasing the cases where only the nonprofit will produce.
Note that these results are only evaluated against domestic data. The figure in the next page describes the differences in the share of nonprofits in seven industrialized nations. It will be interesting to get the full breakdown of nonprofit- for-profit -government shares in the industries discussed in this paper and see whether the correlation between cost function and nonprofit concentration holds in those
IV. Comments and Future Directions
There are many potential extensions for this simple model. First of all, it seems that this cost structure approach complements the information asymmetry model rather well. As stated earlier, the cost structure approach may be better at explaining broad industry concentration while information asymmetry models may explain intra-industry breakdown better. It will be interesting to integrate the two approaches together.
Second, this model needs to build up a supply side and explain why the nonprofit is there in the first place. While it is clear why the for-profit would be in the market (to make a profit), it is not clear where this benevolent nonprofit firm comes from. The nonprofit firm and its employees might be gaining some altruistic satisfaction from increasing other’s welfare, or they may be increasing their reputation by engaging in social activities. In any case, the choice of firm needs to be endogenized and the role of these
intrinsic motivations needs to be formalized. One way to do this is by borrowing from other models (such as Easley and O’Hara) where an entrepreneur choose between the nonprofit form and the for-profit form and make it the first stage of the game before the Cournot or Stackelberg competition begins. The hope is that this model will present a more comprehensive answer to the question of industry concentration by linking the supply of nonprofits to the effect of competition with for-profits.
Third, the role of the government needs to be accounted for. A true model of production in a mixed economy should have all three sectors: private for-profit, private non-profit, and the government. One way to model this is to have the government produce a public good at the level demanded by the median voter and have the for-profit firm produce the private substitute to the public good at a profit maximizing level. The nonprofit will then have to choose whether to produce a supplementary level of public good or to compete with the for-profit sector by producing the private substitute. It then chooses price, quantity and quality as it did previously.
In conclusion, this model provides a new insight into an old problem: it shows that we can not afford to ignore the for-profit sector when answering question about the nonprofit sector, and that cost structure and historical market leadership affects the feasibility of nonprofit firms in an industry. This model can be easily extended to accommodate current approaches in the field such as information
asymmetry, and can also serve a basic framework for new approaches such as the creation of a model that incorporate all three producers in our economy: the government, the private industry, and the nonprofit sector.
Appendix A: Reaction Functions
Firms face linear demand: p(q) =A−q
Assumec>0 andF>0. We want to compute Cournot reaction functions for the Nonprofit firmqn(qp)and the For-profit firmqp(qn)under the following cost
structures:
1. Constant Average Cost (CAC):c(q) =c.q
2. Decreasing Average Cost (DAC): c(q) = F+c.q i f q>0
0 otherwise.
3. Increasing Average Cost (IAC):c(q) =c.q2
4. U-shaped Average Cost (UAC): c(q) = F+c.q2 i f q>0
0 otherwise.
TheNonprofit firm, N, maximizes consumer surplus subject to nonnegative quantity and nonnegative profit:
un= (p(0)−p(q)). q
2 s.t.qn≥0and p(q).qn−c(qn)≥0 Substituting p(q) and simplifying:
max qn un=max( (qn+qp)2 2 ) s.t.qn≥0and(A−qn−qp).qn−c(qn)≥0 Because ∂u2n
∂2qn >0,qn(qp)is defined only by its constraints.
For CAC, the budget constraints is:
(A−c−qn−qp).qn≥0
hence the Nonprofit reaction function is: qn(qp) = A−c−qp i f qp≤A−c
0 otherwise. (1)
For DAC, the budget constraints is: −q2n+ (A−c−qp).qn−F≥0
hence the Nonprofit reaction function is: qn(qp) = A−c−qp+ √ (A−c−qp)2−4F 2 i f A−c−qp≥2 √ F 0 otherwise. (2) 18
For IAC, the budget constraints is: −c.q2n+ (A−qn−qp).qn≥0
hence the Nonprofit reaction function is: qn(qp) =
A−qp
c+1 i f qp≤A
0 otherwise. (3)
For UAC, the budget constraints is: −c.q2n+ (A−qn−qp).qn−F≥0
hence the Nonprofit reaction function is: qn(qp) = A−qp+ √ (A−qp)2−4(c+1)F 2(c+1) i f A−qp≥2 p (c+1)F 0 otherwise. (4)
TheFor-profit firm, P, maximizes its profit subject to nonnegative quantity and profit: up=p(q).qp−c(qp) s.t.qp≥0and p(q).qp−c(qp)≥0
L(qp,λ1,λ2) = (1+λ2).((A−qp−qn)qp−c(qp)) +λ1(qp)
First order conditions are:
[qp]: ∂L ∂qp = (1+λ2)(A−2qp−qn−c0(qp)) +λ1=0 [λ1]: ∂L ∂ λ1 =λ1.qp=0 [λ2]: ∂L ∂ λ2 =λ2.((A−qp−qn)−c(qp)).qp=0
Resolve the Kuhn-Tucker conditions: Assumeλ1=λ2=0, then
By[qp],A−2qp−qn−c0(qp) =0 ⇒ qp(qn) =
A−qn−c0(qp)
2 Assumeλ1>0 andλ2=0, then
By[λ1]and[qp],qp=0 ⇒qn−A−c0(0) =λ1>0 ⇒qn>A−c0(0)
Assumeλ1=0 andλ2>0, then
By[qp],qp(qn) =
A−qn−c0(qp)
2
and by[λ2],qp(qn)also has to solve(A−qp−qn).qp−c(qp) =0
Assumeλ1>0 andλ2>0, then
By[λ1]and[qp],qp=0 ⇒qn−A−c0(0) = λ1
1+λ2>0 ⇒qn>A−c
0(0)
and by[λ2],(A−0−qn)0−c(0) =0 ⇒ c(0) =0
Summarizing the Kuhn Tucker boundaries, we arrive at the general For-profit reaction function: qp(qn) = A −qn−c0(qp) 2 i f(A−qp−qn).qp−c(qp)≥0 and qn≤A−c0(0) 0 otherwise.
Substituting CAC cost function into the budget constraint inqp(qn)yields the
For-profit reaction function:
qp(qn) = A−2qn i f qn≤A−c
0 otherwise. (5)
Likewise, the DAC cost function gives the For-profit reaction function: qp(qn) = A−2qn i f A−c−qn≥2
√
F
0 otherwise. (6)
The IAC cost function gives the For-profit reaction function: qp(qn) = 2A(−c+q1n) i f qn≤A
0 otherwise. (7)
And from the UAC cost function, we get the For-profit reaction function: qp(qn) = 2A(−c+q1n) i f A−qn≥2
p
(c+1)F
0 otherwise. (8)
Appendix B: Proofs
To simplify notation, letA0=A+candc0=c+1.
Proposition 1.a
If Nonprofit and For-profit have identical constant average cost and choose quantity simultaneously, there is a unique equilibrium where only the Nonprofit produces. Proof.
Substitute P’s reaction function Eq.(5) into N’s reaction function Eq.(1) we get: q∗n(qp(qn)) =A0− A0−qn 2 =A 0=A−c q∗p(q∗n) =A 0−A0 2 =0
The Cournot equilibrium quantity isq∗n=A−candq∗p=0. u
Proposition 1.b
If Nonprofit and For-profit have identical constant average cost and one chooses quantity before the other, the number of equilibria and identity of producer(s) depend on the first mover.
Proof.
When the Nonprofit is a Stackelberg leader, the For-Profit treatsqnas a constant in its
objective function, resulting in the same reaction functionqp(qn)as in the Cournot
case (5). On the other hand, the Nonprofit substitutesqp(qn)into its objective function unand solves: max qn un= (qn+ A0−qn 2 ) 2/2s.t.q n≥0and(A0−qn−qp).qn≥0 Since ∂u2n
∂2qn >0,qn(qp)is defined by the same constraints as the Cournot case,qp(qn)=
Eq.(1). Hence the Stackelberg equilibrium where the Nonprofit leads is the same as the Cournot equilibrium:q∗n=A−candq∗p=0.
Now suppose For-profit is the market leader. Solving backwards, Nonprofit takesqp
as a constant andqn(qp)is the Cournot reaction function (1). The For-profit optimizes up(qn(qp),qp): max qp up=max qp (A0−qp−(A0−qp)).qps.t qp≥0and up≥0
Since maxup=0,qp(qn)depends only on the constraints.
qp(qn) =A0−qn≥0
The For-Profit is indifferent between any output 0≤qp≤A−csince it will always
make zero profit. In equilibrium, 0≤q∗p≤A−c, andq∗n=A−c−q∗p. u
Proposition 2.a
If Nonprofit and For-profit have identical decreasing average cost and choose quantity simultaneously, the number of equilibria and identity of producer(s) depend on the size of demand relative to cost.
1. When demand issmall, or A<2√F+c, no firms will produce. 2. When demand ismedium, or2√F+c≤A<4√F+2c, there are two
equilibria, one where only the Nonprofit produces, and another where only the For-profit produces.
3. When demand is exactly4√F+2, there are three equilibria: only Nonprofit, only For-profit, and coexistence.
4. When demand islarge, or A>4√F+2c, only the Nonprofit produces.
Proof.
Case 1: First assumeA0<2√F. We want to show thatq∗n=q∗p=0 is the only equilibrium. Suppose it is not andq∗n>0. Then from N’s reaction function (Eq.(2)), A0−qp≥ 2
√
Fmust hold. ButA0<2√F ⇒ A0−qp<2
√
F∀qp≥0, which is a
contradiction. The same argument holds forq∗p>0. Henceq∗n=q∗p=0 is the only equilibrium.
Now we want to show Case 2-4. AssumeA≥2√F.
Let us rewrite Eq.(2) and Eq.(6) to qn(qp) = A0−qp+ √ (A0−q p)2−4F 2 i f qp≤A 0−2√F 0 otherwise. (9) qp(qn) = A−2qn i f qn≤A0−2 √ F 0 otherwise. (10)
Because of the discontinuities in the reaction function, there are four possible cases that have to be analyzed:
A: Supposeq∗p>A0−2√Fandq∗n>A0−2√F. Thenq∗p=qp(q∗n) =0 ⇒ q∗p<A0−2 √ F. Contradiction. B: Supposeq∗p>A0−2√Fandq∗n=0. Thenq∗p=qp(0) =A 0 2 >A 0−2√F ⇒ A0<4√Fandq∗ n(q∗p) =0. C: Supposeq∗p=0 andqn∗>A0−2√F. Thenq∗n=qn(0) =A 0+√A02−4F 2 ;>A 0−2√F ⇒ √A0−4F>A0−4√F.
This is true for allA0>2√Fwhich implies thatq∗p=qpqn(0) =0.
Hence for allA0>2√F,q∗n>A0−2√F andq∗p=0 is an equilibrium. D: Supposeq∗p≤0 andqn∗≤A0−2√F.
Substituting Eq.(10) into Eq.(9), we get: 2.q∗n(qp(qn)) = A0+q∗n 2 + r (A 0+q∗ n 2 ) 2−4F q∗n=A 0+√A02−8F 2 A0+√A02−8F 2 ≤A 0−2√F ⇒ A02−8F≤A02−8√FA+16F ⇒ 0≤24F−8p(FA ⇒ A≤3√F Substituting Eq.(9) into Eq.(10), we get:
q∗p(qn(qp)) = A0+A 0−q p+ √ (A0−q p)2−4F 2 2 q∗p=A 0+√A02−8F 4 A0+√A02−8F 4 ≤A 0−2√ F ⇒ A02−8F≤9A02−8.3.2√FA+64F ⇒ 0≤A02−6 √ FA+9F ⇒ A≥3 √ F
This impliesq∗p≤A0−2√Fandqn∗≤A0−2√Fonly whenA=3√F. As we stated in the proposition, our equilibrium depends on the size of demand relative to costs:
• Case 2:When2√F+c<A<4√F+2c, both (q∗p>A0−2√F, q∗n=0) and (q∗p=0, q∗n>A0−2√F) are Nash equilibria.
• Case 3:When A0=3√F, there is a third possible equilibrium: q∗p=A0+ √ A02−8F 4 = √ F , q∗n=A0+ √ A02−8F 2 =2 √ F.
• Case 4:When A0>4√F, only (q∗p=0, q∗n>A0−2√F) is a Nash equilibria. u
Proposition 2.b
If Nonprofit and For-profit have identical decreasing average cost and one chooses quantity before the other, in equilibrium only the first mover will produce.
Proof.
WhenA0<2√F,q∗n=q∗p=0 (Proof in Proposition 2.a). Therefore we assume A0≥2√Fand use the rewritten reaction function Eq.(9) and Eq.(10). When the Nonprofit is a Stackelberg leader, For-profit’s reaction functionqp(qn)is just Eq.(10).
The Nonprofit’s problem becomes: max qn un= (qn+ A0−qn 2 ) 2/2s.t.q n≥0and(A0− A0−qn 2 −qn).qn−F≥0 q∗n=A 0+√A02−8F 2 >A 0−2√F so q p(q∗n) =0
When the Nonprofit leads, it becomes the only producer in the Stackelberg equilibrium.
Now suppose For-profit is the market leader. Nonprofit takesqpas a constant and qn(qp)is just the Cournot reaction function (9). Substitutingqn(qp)intoup, For-profit
finds that: up(qn(qp),qp) =0 when qp≤A0−2 √ F, = (2√F−ε).(A0−2√F+ε)>0 when qp>A0−2 √ F Henceq∗p=max(A0−2.√F+ε,A 0 2)andq ∗
n=0. When the market is large enough for
more than one firm, the For-Profit will produce more than it’s profit maximizing amount to prevent Nonprofit from producing. u
Proposition 3
If Nonprofit and For-profit have identical increasing average cost and compete by choosing quantities, both will produce in equilibrium.
Proof.
From IAC reaction functions Eq.(3) and Eq.(7) we can see that the only situation in which either firm will produce zero quantity is when the other firm produce A units. However, no firms can produce that amount (even as a Stackelberg leader) since the budget constraint limits them to k.A wherek≤ 1
c+1<1. u
U-shaped average cost industry.
I analyzed the U-shaped average cost industry informally using numerical evaluations. For the Cournot model, when all firms have identical U-shaped average cost and choose quantity simultaneously, the number of equilibria and identity of producer(s) depend on the size of demand relative to cost. Recallqn(qp) =Eq.(4). Let
A=k√Fc0. When the size of demand,k√Fc0, is such thatqn(0)<2
p F(c+1): qn(0)<2 √ Fc0 ⇒ k √ Fc0+p (k2−4)Fc0 2c0 <(k−2) √ Fc0
then the demand curve intersects the average cost curve beyond the minimal efficient scale point and the market becomes a decreasing returns to scale industry. By Proposition 3, both firms will coexist when the demand size A is larger thank√Fc0 that solve the quadratic above. On the other hand, when the size of demand, A, is such thatqn(0)≤2
p
F(c+1), the demand curve intersects the average cost curve before the minimal efficient scale point and cuts off the increasing part of the U-average cost curve. The equilibrium for markets whereA≤k√Fc0is equal to the equilibrium under Decreasing Average Cost shown in Proposition 2.a.
The numerical evalution for the Stackelberg model also suggests the existence of a cutoff demand sizekpF(c+1). WhenA≤kpF(c+1), the firms are facing a decreasing average cost industry, and whenA>kpF(c+1)the firms are in an increasing average cost industry. In other words:
If Nonprofit and For-profit have identical U-shaped average cost and one chooses quantity before the other, the first mover produces alone in medium-sized markets and shares large markets with the other firm.
When the Nonprofit is a Stackelberg leader, For-profit’s reaction functionqp(qn)is
Eq.(8). The Nonprofit’s problem becomes: max qn un= (qn+ A−qn 2.c0 ) 2/2.c0 s.t.qn≥0 and(A−A−qn 2.c0 −c 0.q n).qn−F≥0 q∗nsolves the quadratic (2c
02−1 2c0 )q 2 n−( 2c0−1 2c0 )qn+F≤0
We want to find A such thatq∗n>A−2√c0F. LetA=kn
√
c0F. Numerically solving forknthat satisfy the following equation:
(2c 0−1 2c0 )± r (2c 0−1 2c0 ) 2−4(2c02−1 2c0 )F/(2.( 2c02−1 2c0 )) = (kn−2) √ c0F
we find thatkn(c0)is an exponential decay function that approach 3.5 asc0→0 and
levels to 2 asc0→∞. The equilibrium is therefore:
• When demand is medium, orA<kn
p
(c+1)F, we are in the DAC case (Proposition 2.b). Only Nonprofit will produce since
q∗n>A−2p(c+1)F,q∗p=0.
• When demand is large, orA≥kn
p
(c+1)F, we are in the IAC case (Proposition 3) where both firms will produce.
Now suppose For-profit is the market leader. Nonprofit’s reaction functionqn(qp)is
Eq.(4). Forprofit solves: max qp up=max(A−( A−qp+ p (A−qp)2−4c0F 2c0 )−qp)qp−c.q 2 p−F st qp≥0and up≥0
Again, the size of the demand will determine whether we fall into the DAC
(Proposition 2.b) or IAC(Proposition 3) case. Whenq∗p>A−2√c0F, Nonprofit’s best
response is not to produce. This impliesq∗p>2cA0 >A−2
√
c0F, which happens when
A<2c4c0−01.4
√
c0F. Through numerical evalution ofkp(c0) = 4c 0
2c0−1we find thatkp(c0)
approaches 4 asc0→0 and levels to 2 asc0→∞. The equilibrium is therefore:
• When demand isA<kp
p
(c+1)F, we are in the DAC case (Lemma 2.b) where only the leader produces.q∗p>A−2p(c+1)F,q∗n=0.
• When demand isA≥kp
p
(c+1)F, we in the IAC case (Lemma 3) where both firm produce positive quantities.
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