Abstract
In this paper we study a particular case of “multiple” externalities associated to the production of a good/activity, whose external effects can change from positive to negative depending on the level of output (intersecting externalities). To analyze their impact on the public policy we propose a very simple two-agent partial equilibrium model in the technological context of externalities. In a static framework, the centralized solution always implies an optimal policy, which may consist of taxation or subsidization depending on the individual optimum and on the technology parameters. In a dynamic model with local knowledge of the efficiency function and instantaneous output adjustments, such an optimal policy can be structurally stable or unstable. In the latter case, under small changes of the parameters the policy may switch from low taxation/subsidization to high taxation/subsidization or vice versa, or even jump discontinuously from taxation to subsidization or vice versa. Furthermore, the decentralized solution based upon “tradable rights” can be economically equivalent to the centralized solution in the form of taxation policy but the two solutions may be not politically equivalent.
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Notes
See Pigou (1932).
“In the late 1960s, on-line systems became popular as a vehicle for providing timely update and access to large databases; this greatly increased the end user’s network externality. As online systems proliferated through the 1970s, congestion externalities often slowed response, or made response so unreliable, that beneficial network effects were cancelled by deleterious congestion effects”, Westland (1992, p. 1007). For more details see, for example, Economides (1996), Shapiro and Varian (1999), Shy (2000).
Economides (1993) defines network externality as a function that measures the increase in the aggregate willingness to pay because of the existence of the network externality.
“In particular, a work of art may at first be considered to be a negative externality but over time people often start to enjoy it, so that it produces a positive externality. Whether a piece of art or an artistic production generates positive or negative externalities is the outcome of a social process”. See (Frey 2000, p. 131).
“Jean-Jacques Rousseau defined music as ’the art to combine the sounds in pleasant way to the ear’, putting so in evidence a correlation between music and pleasure (…). But, as all we know, any pleasure, if administrated in massive doses or against the wish of the receiver, can become motive for trouble, uneasiness and even desperation” (Our translation). See Magrini (1997).
In particular Westland (1992) analyzes the short run pricing problem for a firm that has to internalize every type of multiple externalities in a static context of partial economic equilibrium, whereas we consider an economic policy problem for a policy maker and two firms that do not internalize external effects of their production, both in a static and dynamic context of partial economic equilibrium. For more details see Westland (1992).
This will be made more precise in Sect. 2. Here we note that in our model the function h X (x) represents the external effect per unit output, so that H X (x): = xh X (x) denotes the total effect. Therefore, from the perspective of Westland’s multiple externalities, and using linear specifications for f X (x) and g X (x), namely f X (x) = −f 0−f 1 x, g X (x) = g 0 + g 1 x (where f 0, f 1, g 0, g 1 are non negative parameters), the condition which produces intersecting externalities is g 0 > f 0, g 1 < f 1. The negative and positive effects per unit output exactly offset one another for x = (g 0−f 0)/(f 1−g 1), and here a change of sign of h X (x) takes place. Alternatively, if one considers the marginal effects, the balancing of the positive and negative components occurs for x = (g 0−f 0)/[2(f 1−g 1)], which implies a change of sign of H ′ X (x).
In general, the output (x) and the outcome (h X (x)) are the combined results of a single process of production; therefore the theoretical framework of the analysis would be the theory of the joint production. In particular, in this paper we suppose that the technology of production (x, h) is decomposable in two independent technologies, q X (k) and h X (x), which are correlated through the output x: h X (q X (k)).
The problem of the existence and of the unique equilibrium has been studied in the context of general economic model and its solution requires standard convexity assumptions in the aggregate production set. See, for example, Laffont (1988). However, in this paper we want to analyze the problem in a framework of partial equilibrium approach where the aggregate production set is not necessarily convex. Particularly, we are interested in the conditions for the occurrence of multiple local maxima. For more details on welfare analysis and competitive general equilibrium in a non-convex environment see Mas-Colell et al. (1995) too.
By Laffont (1988), in the framework of the Arrow–Debreu model in which agents and markets are well defined but externalities exist (general equilibrium model) the types of public intervention to restore Pareto efficiency (assuming that the cost of intervention is zero) are the following: (1) creation of markets by specifying property rights; (2) optimal taxation; (3) integration of firms or elimination (reduction of the level) of the externality. In terms of government’s information both the first and the second type of intervention are considered forms of decentralization.
Without our hypothesis on switching costs, the actual cost to modify Pigou’s policy with an administrative act depends only on the difference between the values of indirect controls and it is never determined by the change of policy involved in switching from tax to subsidy or vice versa. On the contrary, we guess that the latter is more problematic than the former from a political point of view.
We model, for example, a case of business to business network externality.
Without loss of generality we consider a reduced form instead of a more structural model because our focus is the technology of the externalities. Therefore the externality has no impact on the other firm’s output but only on the other firm’s profit through an additive shift of its cost function. For an analogous formulation see Mas-Colell et al. (1995).
The function F(x) can be interpreted as the profit function of firm 1 adjusted in a way to “internalize” the externality.
The economic policy function (σ) is obviously determined by the condition: arg maxΠ1(x;σ) = arg max F(x).
By unimodal here we mean that H(x) attains a unique maximum, with no further (strong) local maxima, over the closed and bounded domain [0, M]. It follows that we label as unimodal also a monotonic function which attains its maximum on the boundary.
An example is provided by the rational function H(x) = μx/(λ + x 2), μ, λ > 0, which is strictly positive for x > 0 and unimodal, with H′(x) > 0 (resp. H′(x) < 0) for \(x < \sqrt{\lambda}\) (resp. \(x > \sqrt{\lambda}).\) In this case the average external effect is given by h(x) = μ/(λ + x 2), which is strictly positive (and decreasing) for x ≥ 0.
In this case a change of sign of the marginal effect, H′, will be followed by a change of sign of the average effect per unit output, h. Of course there will be some intermediate output range where H is positive, but decreasing, and therefore h and H′ have opposite sign.
More generally, according to Corollary 1, any case where h(x) is positive and monotonically increasing [for instance a positive externality specified through the increasing quadratic function h(x) = k + αx + γx 2, k > 0, α, γ ≥ 0] is similar to case (1) discussed above, while cases where h(x) is negative and monotonically decreasing are similar to case (2).
For the case of intersecting externalities, x o will in fact be interior to [0, M] provided that M is large enough.
This phenomenon will be further analyzed in a dynamic framework in Sect. 4.
Π2 is now regarded as a (additively separable) function of both y and x.
The symmetric case of positive externalities, namely h(x) = k + γx with k > 0, γ ≥ 0, results in a negative equilibrium price (firm 1 owns the rights) and in an equilibrium output larger than the private optimum of firm 1.
See, e.g., Hale-Koçak (1991) for a more formal definition.
Other specifications are possible for the third dynamic equation of (19). The one we use neglects the actual distance (x o−x) between the current and the desired output level. More suitable specifications should include a correction term that takes into account such a deviation (and that vanishes for x = x o). Notice however that in the remainder of this section we will assume in particular η→ +∞ (instantaneous output adjustment of firm 1), which results in x(t)≡x o(t), for any t. In such a case both the third equation in (19) and other more sophisticated variants would all reduce to x o′(t) = θ ∂W(x o(t), y(t))/∂x o(t).
Different from the static case, in this dynamic setting x o no longer represents the efficiency optimal output, but the one which is “believed” optimal by the policy maker at time t. As explained later, the optimal output in this framework is among the points, denoted by \(\overline{x}^{o},\) which represent the steady state levels of x o.
Namely, if \((\overline{x}^{o},y^{\ast})\) is a local maximum or a local minimum of W, then \(x=x^{o}=\overline{x}^{o},\) y = y * is a steady state of (19).
See again Hale-Koçak (1991) for a deeper discussion.
From a dynamic perspective, the switching occurs because the qualitative shape of W has now changed from that depicted in Fig, 3b to that in Fig. 3c, so that the gradient process (20) starts to push the policy towards x o u . The speed of such a dynamic adjustment depends on the parameter θ in (20).
With non intersecting externalities the economic system is never efficient while the system with intersecting externalities can still be efficient under very particular conditions. Then the economic efficiency without externalities is a general property of the market equilibrium, in the sense that it holds over the whole space of the parameters. In contrast, the economic efficiency with non-Pigou externalities is a local property which may hold along a particular subspace. For instance, it may be satisfied for a particular price, for arbitrary values of the cost and externality parameters.
This includes the case of non intervention (i.e. σ = 0), when the parameters of the function h(x) are such that k = 2x *γ, or k = x *(3γx *−2α), in the first- and second-order specification, respectively.
If the production function of the output and the technology of the externalities are non separable it is necessary to choose the combination between externalities and output that maximizes the general efficiency. On the contrary, if the production function of the output and the technology of the externalities are separable, as in the present paper, it is sufficient to correct the process that generates the externalities to maximizes the general efficiency.
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Acknowledgments
We would like to thank Renato Balducci, Corrado Benassi, Alberto Bucci, Roberto Cellini, Paolo Figini, Antonello E. Scorcu and Laura Vici for helpful comments on a previous version of the paper. We also wish to thank an anonymous reviewer for his helpful comments. The usual disclaimer applies.
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Appendix
Appendix
This appendix provides further details on the numerical examples chosen to illustrate different cases of intersecting externalities, the graphical representation of which is included in Sect. 4. For each example we report the corresponding parameter set and provide further description.
The numerical example chosen to illustrate case 2 of Sect. 2.2 is obtained with parameters k = 10, α = 3, γ = 1, so that h(x) = 10 + 3x−x 2. The average external effect h(x) and the externality function H(x) are represented in Fig. 1a. The parameter selection used to illustrate case 3 of the same section is k = 10, α = −5.5, β = 1, γ = 0.05, which implies h(x) = 10−5.5x + x 2−0.05x 3. Figure 1b represents h(x) and H(x) in this case.
Figure 2 illustrates the parameter dependence of the policy maker’s optimal choice in the cases in question. In particular, Fig. 2a represents the social optimal output x o in case 2 as a function of the parameter α [according to (7)] assuming that the average external effect is specified as h(x) = 10 + αx−x 2 and setting p x = 10, n = 5, m = 1, ρ = 1.5. The private optimal output in this case is x * = 5, while the social optimal output turns out to be \(x^{o}=\alpha/3-0.25+(\sqrt{(2\alpha-1.5)^{2}+210})/6.\) The optimal policy is “switchable”: for example, x o≃4.73 < x * for α = 6, and x o≃5.27 > x * for α = 7. Figure 2a can also be regarded as a plot of the steady state of the dynamical system (20) versus the parameter α, when the externalities are specified as in case 2. Figure 2b is a qualitative picture of the (locally) optimal policies in case 3, where the average external effect is specified as h(x) = k + αx + βx 2−γx 3, as a function of the parameter α [according to the implicit equation (8)]. This reveals that the policy function is both switchable and discontinuous. The figure can also be interpreted as a qualitative bifurcation diagram of the dynamical system (20) when the externalities are specified as in case 3, with the solid (dashed) line representing stable (unstable) steady states. The private optimal output x * may in general be located in any position on the vertical axis of the graph of Fig. 2b. Footnote 37 As remarked in Sect. 4.2, in the qualitative case x * = x * A (let us denote it as case A) the threshold α = α2 corresponds to a jump in the size of the subsidy, while in the qualitative case x * = x * B (case B) the same threshold is associated to a jump in the size and type of the intervention of the policy maker. The situation reported in Fig. 2b, in case B, can be obtained with p x = 10, n = 5, m = 1, ρ = 1.5, k = 10, β = 1, γ = 0.05 so that h(x) = 10 + αx + x 2−0.05x 3, with α ranging in the interval −6.5 ≤ α ≤ −5.5 (and consequently x o ranging from 0 to 12). A picture qualitatively similar to Fig. 2b in case A can be numerically obtained, for instance, under the alternative parameter set: p x = 10, n = 5, m = 10, ρ = 1.5, k = 5, β = 0.75, γ = 0.025, so that h(x) = 5 + αx + 0.75x 2−0.025x 3, with α ranging in the interval 0 ≤ α ≤ 4 (and 0 ≤ x o ≤ 18). Very similar pictures can be obtained in cases with ρ = 1, i.e. a = b (for instance by taking ρ = 1 in both the above examples), for which the centralized solution coincides with the decentralized one: therefore these bifurcation phenomena concern the decentralized solution as well.
With the same parameter setting as in the previous example (in case A) and assuming p y = 50, r = 25, q = 1, Fig. 3 represents the social efficiency W as a function of x, the output level of firm 1, under the assumption that firm 2 is at its optimal output level y * = (p y −r)/q, i.e.
with α chosen in three different regimes 0 < α = 0.25 < α1 (Fig. 3a), α1 < α = 1.25 < α2 (Fig. 3b), and α = 2.75 > α2 (Fig. 3c). By comparing Fig. 3 with the qualitative Fig. 2b (and assuming that the output x of firm 1 is at the desired level x o) we notice that the minimum point of W(x, y *) corresponds to the unstable steady state x o c , while the maximum points of W(x, y *) correspond to the stable steady states x o l and x o u .
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Candela, G., Castellani, M. & Dieci, R. Economics of externalities and public policy. Int Rev Econ 55, 285–311 (2008). https://doi.org/10.1007/s12232-008-0045-6
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DOI: https://doi.org/10.1007/s12232-008-0045-6