3.1. Free-Riders’ Influence on the Public Goods
In case of market failure, the balance of the perfect market cannot maximize social welfare any longer. As for public goods, a kind of market failure, there will be losses of welfare (dead weight loss) because its non-rivalry and non-excludability makes it difficult for a market to form, and even if there is a market, there will be under-production due to the free-rider problem causing welfare losses, as production would not reach the optimal level.
The perfectly competitive market equilibrium achieved by the price mechanism maximizes social welfare. This can be demonstrated by showing that social welfare decreases when actual production is less than or exceeds the production Q* in perfect competition equilibrium.
Figure 1 shows the situation in which social welfare loss occurs when the production in the actual market falls below the production Q*, maximizing social welfare due to market failure. Production Q* in perfect competition equilibrium maximizes consumer and producer surplus, and consequently, social welfare. If actual production becomes Q
1, less than Q* due to market failure, consumer and producer surplus will decrease and there will be social welfare loss of △ABC. This loss of welfare is referred to as “dead weight loss” [
24].
In the theory of finance, non-rivalry and non-excludability are cited as major characteristics of public goods as opposed to private goods [
2,
8,
25]. Due to non-rivalry of the public goods, meaning use by one individual does not reduce its availability to others, these goods can be effectively consumed simultaneously by more than one person within the range of the benefits. Also, public goods have non-excludability, meaning individuals cannot be excluded from use for not paying for them, so the so-called free-rider problem occurs when individuals free ride on the contribution of others to enjoy benefits. Due to these characteristics, if the provision of public goods is at the hand of the free market, efficient allocation of resources cannot be achieved, and the more free-riders, the more distorted social decision making is, which causes inefficiency of resource allocation (dead weight loss) and, consequently, market failure. There have been a number of scholars who proved the existence of this free-ride phenomenon with various methodologies. Recently, the focus has been on confirming the possibility of free-riding through mathematical induction or social experiments [
16,
26,
27,
28,
29].
Out of many theories and models which analyze market failure due to the free-rider problem, the four representative models are the prisoners' dilemma model by the same utility functions; prisoners' dilemma model by the different utility functions; public goods game social experiment based on prisoner’s dilemma model; and analysis of consumer surplus of free-riding [
30]. Consumer surplus analysis explains market failure due to free-riding well by introducing realistic assumptions that individuals have different utility functions, supply and demand of public goods are not fixed and divided, and marginal benefits vary depending on changes in supply and demand [
30]. In this chapter, we analyze the case of free-riding without dead weight losses through consumer surplus analysis in contrast with prior research on free-riding with public goods.
3.2. Welfare Economics Perspective of Free-Riding
Private goods do not have non-rivalry or non-excludability as factors, and their consumption by one consumer prevents simultaneous consumption by other consumers, which means an individual cannot free ride on the consumption of these goods to enjoy their benefits. Therefore, market demand at a given price is the sum of consumption by individual consumers. Market demand would be the horizontal summation of all the market participants' individual demand while the market demand curve is represented by the horizontal summation of all the individual demand curves.
On the other hand, as presented in
Figure 2, the relationship between individual demand curves and the market demand curve of public goods is different from that of private goods. As previously mentioned, the consumption of public goods are non-rivalrous, and it is possible for different consumers to consume the same amount of public goods simultaneously. Therefore, the benefit of society as a whole from a certain quantity of public goods is the sum of the marginal benefits of individual consumers, so the social demand curve (social marginal benefit curve) is equal to the sum of individual demand curves. In
Figure 2, the demand curve means the marginal benefit curve and the height of the demand curve measures the consumer's marginal willingness to pay (WTP) and the size of the marginal benefit.
As suggested by Bowen [
1], we intend to analyze the free-rider problem and dead weight loss through a partial equilibrium model [
1,
23]. In the situation where the market demand curve is obtained by the vertical summation of the individual consumers’ marginal benefit curves, there is a gap between the social optimal production amount and the market equilibrium production realized in the actual market, and the welfare loss caused by this shows the market failure caused by free-riders. Let us consider a typical case where a dead weight loss is incurred by the free ride problem through a consumer surplus analysis model.
Assume there are two consumers; consumer A with relatively higher benefit to gain from a public good, meaning the height of the demand curve is higher than B, and consumer B with lower benefit from a public good, meaning the height of the demand curve is lower than A. For convenience’s sake, A is referred to as a high-benefit consumer, while B is a low-benefit consumer. The marginal cost curve of the public goods production becomes the supply curve of public goods. In this case, the amount of dead weight loss can be obtained by comparing social optimum (F) and market equilibrium when there are free-riders (D, E). Let us examine the dead weight loss incurred by an individual free rider. The demand curve of high-benefit consumer A has a higher demand curve than that of consumer B (see
Figure 3). In this case, actual market equilibrium, when B free rides on the contribution of A, and A pays for the provision of public goods alone, is point D where A’s demand curve and supply curve meet, and social optimal equilibrium is F, where social demand curve, which reflects the true preferences of the two consumers, intersects the social supply curve. Therefore, when comparing point F, where social welfare is maximized, and the actual market equilibrium D, it is evident that social welfare decreases by △ADF at point D.
On the other hand, when A free-rides on B’s contribution and B pays for all the provision cost of the public good, actual market equilibrium is point E, where B’s demand curve and supply curve meet. Thus, comparison of point F, where social welfare is maximized, and the actual market equilibrium E, shows that social welfare decreases △HEF at point E due to B’s free ride. In this case, the dead weight loss incurred when a high-benefit consumer A free rides is greater than that of a low-benefit consumer B (ΔADF > ΔHEF).
In short, classical welfare economics conclude that when a consumer free-rides, concealing his or her benefit and distorting social benefits, it leads to under-provision of public goods at the level lower than is required to maximize social welfare, incurring dead weight losses. This result justifies the argument of the traditional public economics that the government should intervene with policy measures to reach the optimal social welfare level (point F in the figure) to address dead weight losses associated with the free-rider problem [
2,
7,
31]. However, further research is required to understand if free-riding always incurs dead weight losses, or, in other words, causes undesirable results in terms of economic efficiency. The next section examines the cases of free-riding, where there is no loss in economic efficiency or dead weight losses, from the perspective of welfare economics.
3.4. Possibility of Free Riding without Dead Weight Losses
Shin [
12] suggests that there are cases in which free-riding does not lead to dead weight loss, unlike the conventional argument. Let us assume that low-benefit consumer C free-rides on the contribution of high-benefit consumer A, who pays for all the provisional costs of a public good (see
Figure 4). In this case, the bending point of the social benefit curve (point Y) is located to the right of market equilibrium and is reached when there is free-riding (point E) and the location and shape of demand-supply curves for the public good becomes different than usual. When point E, where public goods supply curve intersects with high-benefit consumer A’s demand curve, is located to the right of the bending point (Y), point E becomes the actual market equilibrium
1 and social optimum (welfare maximization), simultaneously, although low-benefit consumer C free-rides. Unlike the case explored in
Figure 4, social welfare can be maximized at the market equilibrium where consumer C free-rides and only consumer A’s demand is expressed. That is, as social welfare is maximized at the actual market equilibrium, there are no dead weight losses.
Figure 4 shows that the actual market equilibrium E, where demand curve for A—whose demand is expressed—and public goods supply curve meet when C free-rides, is located to the right of the bending point Y of the social marginal benefit curve. In this case, unlike in the case illustrated in
Figure 3, the social benefit curve bends in the left of market equilibrium E and there is no gap between the actual market equilibrium and social optimum welfare, the two points are identical because high-benefit consumer A’s demand curve corresponds to the social demand curve to the right of the bending point Y.
On the other hand, in the case where low-benefit consumer C’s demand is expressed and high-benefit consumer A free-rides, social welfare loss becomes larger than that of
Figure 4. This is because when high-benefit consumers free-ride, there is a gap between the actual market equilibrium G and social optimum E, and production of public goods at point G does not reach that of the social optimum point E (See notes in
Figure 5 for the size of dead weight loss).
Figure 5 shows
Figure 3, where free-riding incurs dead weight loss, and
Figure 4, the case of free-riding without dead weight loss, simultaneously. If a high-benefit consumer is A and low-benefit consumer who free-rides is B, the social marginal benefit curve bends at point X. Therefore, actual market equilibrium is inconsistent with social optimum (E ≠ F), and the production of public goods at the actual market equilibrium Q
E is smaller than social optimum production Q
F, creating dead weight loss of △EFH. On the other hand, when low-benefit consumer C free-rides, the social marginal benefit curve bends at point Y. Therefore, social welfare is at its maximum at the actual market equilibrium (E) where only A’s demand curve is considered, because social marginal benefit curve is the same as high-benefit consumer A’s demand curve on the right of point Y. In other words, if the low-benefit consumer who free-rides is C, rather than B, the Q
E that is supplied from the actual market equilibrium becomes the production maximizing the social welfare immediately. Thus, dead weight loss is not incurred. On the other hand, if A free-rides on C, a larger dead weight loss is incurred than when A free-rides on B, which has already been explained in
Figure 4.
In conclusion, whether dead weight loss will be created due to free-riding or not, depends on the relative position of the bending point on the social marginal benefit curve (demand curve for the public good), whether it is located to the left or right of the actual market equilibrium E. Given this, what are the conditions for low-benefit consumers’ free-riding not to incur social welfare loss? X-coordinate for bending points (Y, X) on the social marginal benefit curve means demand where low-benefit consumers’ marginal benefit (or WTP) becomes 0 (QC, QB), F means social optimum where social marginal benefit curve intersects marginal cost curve and the social optimum production is QF. Under the circumstances, the conditions for low-benefit consumer C’s free-riding not to incur dead weight loss, can be formulated as discussed below.
Let’s assume the marginal benefit curve of consumer A enjoying a relatively high benefit from a certain public good, consumer C enjoying a relatively low benefit, and producer, as MBA = MBA(Q), MBC = MBC(Q), MC = MC(Q), respectively. Market equilibrium E satisfies MBA(Q) = MC(Q). And where low-benefit consumer C free-rides without dead weight loss and high-benefit consumer A bears all the cost, the production at market equilibrium E is QE. Meanwhile, production where low-benefit consumer C’s marginal benefit becomes 0 (MBC(Q) = 0), is QC, which is also production at point Y where the social marginal benefit curve bends.
As mentioned above, when low-benefit group C free-rides on the contribution of high-benefit group A and there is no dead weight loss, point E corresponds to the actual market equilibrium and social optimum for maximization of social welfare and should be located to the right of bending point Y on the social benefit curve. This means the production that makes low-benefit consumer C’s marginal benefit 0, needs to be the same as social optimum production or smaller, as represented as QC ≤ QE. From the discussion above, it is evident that the incurring of dead weight loss is determined by the relative position of demand (X-intercept on the demand curve) making low-benefit consumer’s WTP 0, which is the shape of the demand curve (marginal benefit curve) of low-benefit consumer, rather than supply side factors such as the slope or position of the supply curve (marginal cost curve) for the public good. And the Condition for free-riding without dead weight losses is as follows.