Abstract
In this chapter, we deal with the efficiency and welfare properties of an SRPE. Since each SSPE is also an SRPE, the results also hold for the former.
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Notes
- 1.
An analogous comment applies to the definition of a natural monopoly (Definition 4.2) below.
- 2.
See [Baumol (1977)] for the characterization of the functions that exhibit both the economies of scale and the economies of scope.
- 3.
The information about a natural oligopsony (natural monopsony) I ∗ (i ∗ ) in the analyzed market has the greatest value when the latter remains a natural oligopsony (natural monopsony) even after adding any finite number of replicas of all firms in I ∗ (i ∗ ) to I. In this case, the sum of the costs of selling the quantities of goods given by z cannot be decreased by any change in the number of buyers.
- 4.
Following the microeconomics textbooks (e.g., [Varian (1992), p. 215]) that define a competitive firm as a price taking firm, we can use the term “perfect competition on the supply side of the analyzed market.” Nevertheless, according to Assumption 4.5 below, all goods in J are different. Hence, each type of good is sold in the analyzed market by only one firm (although it can be sold in other markets by other firms that are potential competitors of the firm in J selling it in the analyzed market). Since perfect competition is associated with a large number of firms producing a homogeneous good, we prefer the term “price taking behavior on the supply side of the analyzed market.”
In the analysis of non-collusive behavior (similarly as in the game Γ(δ)), we allow the firms to leave the analyzed market. Nevertheless, we do not consider the entry of firms (that could drive the economic profits of the price taking firms down to zero). This is another reason why we do not use the term “perfect competition on the supply side of the analyzed market.”
- 5.
Our results in this subchapter continue to hold when function U#(J)+1 is not differentiable on some subset of its domain but has right-hand side and left-hand side partial derivatives satisfying Assumption 4.5 on it. A stage game, in which the monopsonist is a retailer and each inverse demand function is linear on the subset of its domain on which it has a positive functional value, is an example of this. With a slight abuse of language, when subsets, at which Assumption 4.5 does not hold, have zero measure, we say that Assumption 4.5 is satisfied.
- 6.
The profit of each j ∈ J from its competitive supply is strictly increasing in price on the set of prices that are higher than cj ′(0) and no higher than cj ′(χj). For
$${c}_{j}^{{\prime}}(0) < {p}_{ j}^{{\prime}} < {p}_{ j} \leq {c}_{j}^{{\prime}}({\chi }_{ j})$$we have
$$\begin{array}{rcl}{ p}_{j}{\psi }_{j}({p}_{j}) - {c}_{j}[{\psi }_{j}({p}_{j})]& >& {p}_{j}{\psi }_{j}({p}_{j}^{{\prime}}) - {c}_{ j}[{\psi }_{j}({p}_{j}^{{\prime}})] \\ & >& {p}_{j}^{{\prime}}{\psi }_{ j}({p}_{j}^{{\prime}}) - {c}_{ j}[{\psi }_{j}({p}_{j}^{{\prime}})]\end{array}$$ - 7.
The changes in the monopsonist’s expenditure on goods in J− are the same in both replacements. Assumption 4.5 implies that
$$\begin{array}{rcl} & & \frac{\partial [{U}_{\#(J)+1}(\widehat{x} + {z}^{+}) - {c}_{\#(J)+1}(\widehat{x} + {z}^{+})]} {\partial {z}_{j\#(J)+1}} \\ & & \quad \geq \frac{\partial [{U}_{\#(J)+1}(\widehat{x}) - {c}_{\#(J)+1}(\widehat{x})]} {\partial {z}_{j\#(J)+1}},\forall j \in {J}^{-}\end{array}$$We can view the replacement of \(\widehat{x} + {z}^{+}\) by \(\widehat{x} + {z}^{+} + {z}^{-}\) (as well as the replacement of \(\widehat{x}\) by \(\widehat{x} + {z}^{-}\)) as a sequence of “small” changes. Then, for each of them, we can approximate a change in the functional value of the function U#(J)+1 − c#(J)+1 by the scalar product of its gradient and a vector of the changes in its arguments.
- 8.
This can be seen as follows. Let v be the vector of the firms’ profits generated by \((\widehat{x} +\widehat{ z},\widehat{p} +\widehat{ r})\). Suppose that there exists \((\widetilde{z},\widetilde{r})\neq (\widehat{z},\widehat{r})\) such that \((\widehat{x} +\widetilde{ z},\widehat{p} +\widetilde{ r})\) gives a vector of the firms’ payoffs \(\widetilde{v}\) that weakly Pareto dominates v. Clearly, there exists j ∈ J with \(\widetilde{{v}}_{j} > {v}_{j}\). (Otherwise, we would have a contradiction with the fact that \((\widehat{z},\widehat{r})\) solves (4.18)–(4.23).) Thus, by decreasing \(\widetilde{{r}}_{j}\) for each j ∈ J with \(\widetilde{{v}}_{j} > {v}_{j}\), we obtain a vector of firms’ profits \(\overline{v}\) with \({\overline{v}}_{j} = {v}_{j}\) for each j ∈ J and \({\overline{v}}_{\#(J)+1} > {v}_{\#(J)+1}\). This contradicts the fact that \((\widehat{z},\widehat{r})\) solves (4.18)–(4.23) and completes the argument.
- 9.
Let \(\overline{r}\) result from the changes in the prices that leave each j ∈ J with profit \(\widehat{{v}}_{j}\). Suppose that \((\overline{r},z)\) does not solve (4.18)–(4.23). Take solution \((\widehat{z},\widehat{r})\) of (4.18)–(4.23). Then, the sum of the components of the vector of the profits generated by \((\widehat{x} +\widehat{ z},\widehat{p} +\widehat{ r})\) exceeds the sum of the components of the vector of the profits v generated by \((\widehat{x} + z,\widehat{p} + r)\) (which equals the sum of the components of the vector of the profits generated by \((\widehat{x} + z,\widehat{p} + \overline{r})\)). Therefore, we can change the prices given by \(\widehat{p} +\widehat{ r}\) in such a way that we obtain a vector of the profits that weakly Pareto dominates v. This contradiction with the assumption that v is strictly Pareto efficient completes the argument.
- 10.
A reader may find this choice of prices strange. It is motivated by the desire to have prices that are not periodic numbers and have at most two non-zero decimal points (and hence, they can be paid by the banknotes and coins in circulation).
- 11.
In this example, ζ{1,3} = ζ{2,3} = 9. Thus, the firms’ profits in the collusive outcome satisfy (3.22). Therefore, the described collusive outcome can be sustained in an SSPE of Γ(δ) for any δ ∈ (0, 1). (See Proposition 3.2.)
- 12.
With the original fixed cost a long-run non-collusive equilibrium would not exist. The producers’ profits in the solution of the maximization program (4.13) would be negative. Nevertheless, the stage game in example 3.1 with original fixed cost can be used as an example that a collusion can make viable production that is not viable under a non-collusive arrangement.
- 13.
I am grateful to Martin Gregor for his comments on this section.
- 14.
We can extend the analysis (at the expense of making it more cumbersome) to the situation where for each firm i ∈ I and every k ∈{ 1,…,m}, a positive output requires a positive input of at least one good in J(k) but the goods in the latter group are close substitutes instead of being identical and their quantities are aggregated using the fixed coefficients (in general, different from one) that express the fixed (i.e., independent of their quantities) marginal rates of substitution between them.
- 15.
For i ∈ I and λi ∈ (0,λi max], we have
$${C}_{i}^{{\prime}}({\lambda }_{ i}) ={ \sum }_{k=1}^{m}\frac{\partial {c}_{i\Sigma }({\lambda }_{i}{\underline{Q}}^{(i)})} {\partial ({\lambda }_{i}{\underline{Q}}_{k}^{(i)})}{ \underline{Q}}_{k}^{(i)}$$and, using part (iii) of Assumption 4.8,
$${C}_{i}^{{\prime\prime}}({\lambda }_{ i}) ={ \sum }_{k=1}^{m}({\underline{Q}}_{ k}^{(i)}{ \sum }_{n=1}^{m} \frac{{\partial }^{2}{c}_{i\Sigma }({\lambda }_{i}{\underline{Q}}^{(i)})} {\partial ({\lambda }_{i}{\underline{Q}}_{k}^{(i)})\partial ({\lambda }_{i}{Q}_{n}^{(i)})}{\underline{Q}}_{n}^{(i)}) \geq 0.$$ - 16.
The stage game in Example 3.3 satisfies all assumptions made in this section. Nevertheless, (due to the same form of buyers’ production functions and symmetry of inverse demand functions for their products) inverse demand functions for the analyzed market (functions Φ 1 −1 and Φ 2 −1 in the notation used below in the text) do not exist in it.
- 17.
This ensures that in the Cournot oligopoly in the analyzed market, each producer’s payoff function is continuous in the whole vector of arguments and concave (and hence, quasi-concave) in his/her own output.
- 18.
Given the assumptions already made in this section, a sufficient condition for the existence of a Cournot equilibrium in the market for the buyers’ products is that U i is concave in x i and c i is convex in the whole vector of arguments for each i ∈ I. Nevertheless, in order to obtain continuous inverse demand functions for the types of goods in the analyzed market, we need for each vector of the prices of the types types of goods either Cournot equilibria in the market for the buyers’ products that give the same vector of aggregate demands for the types of goods or continuous selection from the equilibrium correspondence. Moreover, in order to apply the standard sufficient condition for the existence of a pure strategy Nash equilibrium of a strategic form noncooperative game (see, e.g., [Osborne & Rubinstein (1994), Proposition 20.3, p. 20]) to a Cournot oligopoly on the supply side of the analyzed market, the inverse demand functions for the analyzed market would have to be such that each producer’s profit is quasi-concave in his/her output.
- 19.
In a non-collusive benchmark equilibrium, the upper bounds on the prices in the analyzed market are determined by the production capacities of the firms in J and the inverse demand functions for the types of goods. Nevertheless, these upper bonds need not be high enough for the validity of the following analysis (see Steps 1–4 below).
- 20.
We have
$$\begin{array}{rcl} & & \frac{d{\sum }_{\gamma =1}^{m}{\sum }_{j\in {J}^{(\gamma )}:\widehat{{x}}_{ ji}>0}{c}_{j}({\theta }_{j}(\widehat{x}) -\widehat{ {x}}_{ji} + \frac{\hat{{x}}_{ji}} {{Q}_{\gamma }^{(i)}} \widehat{{\lambda }}_{i}{\underline{Q}}_{\gamma }^{(i)})} {d{\lambda }_{i}} \\ & & \quad ={ \sum }_{\gamma =1}^{m}(\frac{{\underline{Q}}_{\gamma }^{(i)}} {{Q}_{\gamma }^{(i)}}{ \sum }_{j\in {J}^{(\gamma )}:\widehat{{x}}_{ ji}>0}{c}_{j}^{{\prime}}({\theta }_{ j}(\widehat{x}))\widehat{{x}}_{ji}) \\ & & \quad \leq {\sum }_{\gamma =1}^{m}({\underline{Q}}_{\gamma }^{(i)}\max \{{c}_{ j}^{{\prime}}({\theta }_{ j}(\widehat{x}))\mid j \in {J}^{(\gamma )},\widehat{{x}}_{ ji} > 0\})\end{array}$$Recall that in (4.41), for each γ ∈{ 1,…,m}, j(γ) can be any producer belonging to the set J (γ).
- 21.
This argument is based on an infinitesimal change in λi. Nevertheless, taking into account (4.30), part (ii) of Assumption 4.9, parts (ii) and (iii) of Assumption 4.7, (4.28), and Assumption 4.10, the difference between the right- and the left-hand side of (4.41) is nonincreasing in λi. Thus, an infinitesimal decrease in λi does not decrease this difference. Therefore, we can view any decrease in λi as a sequence of infinitesimal decreases, apply the argument in the text to each element of this sequence, and reach the conclusion stated in the text. An analogous comment applies to the other arguments in this section that are based on the signs of the derivatives.
- 22.
Suppose that for γ ∈{ 1,…,m}, producer j ∈ J(γ) has share ωj ∈ (0, 1] in i’s increased purchase of type γ of good. Let
$$Z = {\theta }_{j}(\widehat{x}) -\widehat{ {x}}_{ji} -{\sum }_{k\in {J}^{(\gamma )}\setminus \{j\}}\widehat{{x}}_{ki}$$Then, we have
$$\begin{array}{rcl} \frac{d{c}_{j}(Z + {\omega }_{j}\widehat{{\lambda }}_{i}{\underline{Q}}_{\gamma }^{(i)} + (1 - {\omega }_{j}){Q}_{\gamma }^{(i)})} {d{\lambda }_{i}} = {c}_{j}^{{\prime}}({\theta }_{ j}(\widehat{x})){\omega }_{j}{\underline{Q}}_{\gamma }^{(i)}& & \\ \end{array}$$and
$$\begin{array}{rcl} {\sum }_{j\in {J}^{(\gamma )}:{\omega }_{ j}>0}{c}_{j}^{{\prime}}({\theta }_{ j}(\widehat{x})){\omega }_{j}{\underline{Q}}_{\gamma }^{(i)} \leq {\underline{Q}}_{\gamma }^{(i)}\max \{{c}_{ j}^{{\prime}}({\theta }_{ j}(\widehat{x}))\mid j \in {J}^{(\gamma )},{\omega }_{ j} > 0\}.& & \\ \end{array}$$Thus, we can still use (4.41).
- 23.
We have considered the specific changes in the prices in the analyzed market for which the change in i’s expenditure on good j ∈ J equals the change in firm j’s cost. Nevertheless, the prices in the analyzed market do not affect the sum of the profits of all firms in J ∪ I. An analogous comment applies also to the following arguments in this section concerning the sum of the profits of all firms in J ∪ I.
- 24.
In Step 4, we have shown that when only the increases in traded quantities take place, we can increase the profit of buyer i (without violating any constraint in (4.43)–(4.44)) by more than when also the decreases in traded quantities take place. Here, we show that when only the increases in traded quantities take place, we can give each firm a profit no lower than its profit when also the decreases in traded quantities take place and we can give at least one firm a higher profit.
- 25.
- 26.
We compute the Cournot equilibrium using the buyers’ reaction functions obtained from the solutions to the optimization problem (4.37) for i = 3, 4. The formulae given in the text are valid for prices for which they give nonnegative outputs. This condition is satisfied in the non-collusive benchmark equilibrium. Therefore, here, we do not deal with the prices for which the above formulae are not valid. An analogous comment applies to the demand functions and the inverse demand functions for the analyzed market given below.
References
Baumol, W. J.: “On the Proper Cost Tests for Natural Monopoly in a Multiproduct Industry,” American Economic Review 67 (1977), 809-822.
Baumol, W.J.: “Contestable Markets: An Uprising in the Theory of Industry Structure,” American Economic Review 72 (1982), 1-15.
Cournot, A.: Research into the Mathematical Principles of the Theory Wealth. Edited by N. Bacon. New York: MacMillan, 1897.
Osborne, M.J. and A. Rubinstein: A Course in Game Theory. Cambridge, MA: MIT Press, 1994.
Spengler, J.J.: “Vertical Integration and Antitrust Policy,” Journal of Political Economy 58 (1950), 347-352.
Varian, H.R.:Microeconomic Analysis. Third Edition. New York, W.W. Norton, 1992.
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Horniaček, M. (2011). Efficiency of an SRPE and an SSPE. In: Cooperation and Efficiency in Markets. Lecture Notes in Economics and Mathematical Systems, vol 649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19763-5_4
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