Abstract
The purpose of this chapter is to show, by examples, how our canonical model can be extended to address some well-known issues and how the possibility of considering competitive toughness as a continuous parameter may open new perspectives and new insights. As a first extension, important for applications in macroeconomics or international trade, we explicitly introduce a labour market and examine in particular the possibility of “involuntary” unemployment (in Keynes’ sense). This possibility is shown to result, with a fully adjustable wage, from the oligopolistic character of output markets. In the second extension we introduce a model with overlapping generations and study its distinct dynamic properties under strategic and non-strategic investment in capital. In the third extension, in a model of localised competition, we give another interpretation of our approach in terms of a delegation game.
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Notes
- 1.
See, for instance, Christiano et al. (2005), where the representative household “makes a consumption decision and a capital accumulation decision, and it decides how many units of capital services to supply” (p. 11). Admittedly, their “assumption that households make the capital accumulation and utilization decisions is a matter of convenience. At the cost of more complicated notation, [they] could work with an alternative decentralization scheme in which firms make these decisions” (p. 13, n. 8). The results of such an alternative scheme may however be significantly different if strategic interaction is allowed for.
- 2.
See d’Aspremont et al. (1979).
- 3.
“The first characteristic […] is the fact that money has […] a zero, or at any rate a very small, elasticity of production. […] Money, that is to say, cannot be readily produced—labour cannot be turned on at will by entrepreneurs to produce money in increasing quantities as its price rises in terms of the wage-unit. […] Obviously […] the above condition is satisfied, not only by money, but by all pure rent-factors, the production of which is completely inelastic” (Keynes, 1936, pp. 230–231).
- 4.
Dos Santos Ferreira (2014) suggests a formalisation of Keynes’ approach to involuntary unemployment, as developed in the General Theory.
- 5.
More generally, if \(\widehat {D}\left ( P,1\right ) >0\) for \(P<\overline {P} <\infty \) and \(\widehat {D}\left ( \overline {P},1\right ) =0\), we assume instead: \(\lim _{P\rightarrow \overline {P}}\widehat {\sigma }\left ( P\right ) >1 \).
- 6.
We use the following specification of the inverse demand: \(1/\left ( X^{8}+10X\right ) \), and take n = 4 and L∕c = 0.5.
- 7.
This OLG extension of our analysis was further exploited by d’Aspremont et al. (1995b) to discuss policy implications of involuntary unemployment in the above sense, and by d’Aspremont et al. (1995c) to show how producers’ market power can favour the emergence of coordination failures and endogenous fluctuations.
- 8.
A complementary development can be found for the Cournotian case in d’Aspremont et al. (2015).
- 9.
With these parameter values, the matrix equation (2.22) can indeed be rewritten as
$$\displaystyle \begin{aligned} \left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right] \left[ \begin{array}{c} d\ln k_{t} \\ d\ln n_{t} \end{array} \right] =\left[ \begin{array}{cc} 1 & -2 \\ 1 & -1 \end{array} \right] \left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right] \left[ \begin{array}{c} d\ln k_{t-1} \\ d\ln n_{t-1} \end{array} \right] \text{.} \end{aligned}$$By a direct computation, we then obtain the system (2.27).
- 10.
Hence, the firm with a larger captive segment (less exposed to competition) sets, ceteris paribus, a higher price and is, in this sense, less “aggressive” (see Narasimhan, 1988, p. 435).
- 11.
The figure is computed with the parameter values v = 1 and t 1 = t 2 = t = 0.4 and the competitor j strategy p j = x j = 0.8.
- 12.
For an approach based on “pricing schemes” see d’Aspremont et al. (1995a).
- 13.
References to such observation in empirical and experimental studies of behavioural economics can be found in the survey of Armstrong and Huck (2010).
- 14.
- 15.
Of course the same objective can be attained through an appropriate monetary compensation, independently of the personality of the manager. The model is valid for the two interpretations.
- 16.
We follow d’Aspremont et al. (2016).
- 17.
Contrary to the conventional presentation of the Stackelberg duopoly as a two-stage game, where a pre-determined leader plays at the first stage and a pre-determined follower at the second (see, e.g., Vives, 1999), we do not find sequentiality in Stackelberg (1934) either. Instead, “A views the behaviour of B either as dependent on or independent of his own behaviour. […] Since each supplier can have each of the two types of position, the price formation structure […] is thus imperfect” (Stackelberg, 1934 [2011], p. 17). The supplier that views his rival’s behaviour as dependent “will examine all profit maximisation options and implement the best one. We can say that [he] dominates the market” (ibid., p. 16). The Stackelberg leader is not the first mover but the supplier exerting market dominance (“Marktherrschaft”).
- 18.
It can be noted that the Cournot solution coincides with the collusive solution in the symmetric case. This can be checked by examining the first-order conditions (3.5), where the partial derivatives with respect to p i of the two demand functions are both multiplied by \( p_{i}^{\ast }\). In the first-order conditions for joint profit maximisation, they are each multiplied by the own price (hence, ∂ i D j by \( p_{j}^{\ast }\)). However, in the symmetric case where the two prices are equal, this difference is immaterial.
- 19.
Using the words of Deneckere et al. (1992), the firm with the largest captive segment becomes a price leader “with a large loyal consumer base” and provides a “price umbrella” to the follower.
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d’Aspremont, C., Dos Santos Ferreira, R. (2021). Extensions. In: The Economics of Competition, Collusion and In-between. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-63602-9_4
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