Abstract
In this chapter economic space comes into play. This is done in a framework analogous to the flexible one used by Greenhut et al. (1987) to investigate spatial oligopoly in commodity markets. The main implication of economic space is that agents face significant travel/transportation cost. First, we will discuss the basic assumptions employed by this framework.
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Notes
- 1.
Of course, there are places where workers receive the same income from two firms and are therefore indifferent between working for the one firm or the other. For instance, this holds at the border of the adjoint market areas of two firms. However, since we are considering the real line as economic space, the set of these places is a null set with mass zero, and so all workers indeed work for only one employer in a well-defined sense.
- 2.
Note that all the following results do not hinge on l | (r, ϕ)’s smoothness. They also hold if l | (r, ϕ) is just twice continuously differentiable, i.e., \(l{\vert }_{(r,\phi )} \in {\mathcal{C}}^{2}\), where \({\mathcal{C}}^{k}\) with k ≥ 1 denotes the set of functions with k continuous derivatives. For analytical convenience, though, we will regard l | (r, ϕ) as smooth below. In the following, we shall also not bother with explicitly distinguishing the individual labour supply function from its restriction on (r, ϕ) whenever there is no danger of confusion or no need for such rigour but just sloppily refer to both as individual labour supply.
- 3.
If otherwise r > ϕ, there would be no gains from trade and both firms and workers would not be willing to participate in the labour market, preferring inactivity instead.
- 4.
In equilibrium, firms indeed choose the same wage and thus have the same market radius due to the symmetry of the framework. This will be discussed in some detail in Section 3.6.
- 5.
It should be noted though that this equilibrium concept is static by nature. We will discuss this point later in some detail when dealing with the long-run equilibrium’s stability in Section 4.4.
- 6.
Note that the Riemann integral given on the right-hand side of (3.6) exists because l is smooth and therefore continuous, which implies l’s Riemann integrability. Note further that for circular markets, the firm’s aggregate spatial labour supply would be given by
$$L(w,X) = D{\int \nolimits \nolimits }_{0}^{2\pi }\!\!\!\!{\int \nolimits \nolimits }_{0}^{X}l(w - tx)x\,dx\,d{\vartheta} = 2\pi D{\int \nolimits \nolimits }_{0}^{X}l(w - tx)x\,dx.$$(For a derivation of this result, see Ohta, 1988, pp. 16–19.) As already stated in Section 3.1, considering two-dimensional economic space is therefore both straightforward and messy, so that, for simplicity, we shall stick to one-dimensional economic space instead.
- 7.
Obviously, individual labour supply and market areas must be positive, while wages must be no more than workers’ marginal revenue product in economically relevant situations, so that we will restrict attention to cases with r < z(w, X) < ϕ or (w, X) ∈ Z = { (w, X) ≫ 0 | r < z(w, X) < ϕ}, respectively, where \((w,X) \gg 0\, :\Leftrightarrow \, (w,X) \in \{ (w,X)\vert w > 0,X > 0\} ={ \mathbb{R} }_{ +}^{2}\).
- 8.
First of all, note that Π as given by (3.7) is smooth and therefore differentiable because l is smooth, so that the standard approach to profit maximisation is viable. Next, similar to our discussion of simple spaceless monopsony in Chapter 2 (see footnote 6 on page 11), assume that there exists a unique positive solution of (3.8). This is, for instance, satisfied if ∂Π(w, X) ∕ ∂w > 0 for all w smaller than the wage solving (3.8) and ∂Π(w, X) ∕ ∂w < 0 for all w larger than this wage. In particular, this is implied by strict concavity of Π with respect to w. Besides, by drawing attention to (w, X) ∈ Z we also avoid dealing with corner solutions.
- 9.
The Leibniz rule for differentiation under the integral sign states that
$$\frac{\partial } {\partial y}{\int \nolimits \nolimits }_{a(y)}^{b(y)}z(x,y)\,dx ={ \int \nolimits \nolimits }_{a(y)}^{b(y)}\frac{\partial z(x,y)} {\partial y} \,dx + b'(y)z[b(y),y] - a'(y)z[a(y),y]$$under certain conditions; in particular, this holds if z, a, and b are continuously differentiable real functions.
- 10.
Note that Definitions 3.1–3.3 only make sense if l is at least twice differentiable, so that ℓ′(z) actually exists for all r < z < ϕ. Since we assumed in Section 3.1 that l is smooth, it is convenient to work with these definitions. One can, however, state more general definitions of log-linearity, -concavity, and -convexity that do not require the function under consideration to be twice differentiable. For example, using Prékopa’s (1973) definitions, an individual labour supply function is strictly log-concave (log-convex) if \(l[{\vartheta}{z}_{1} + (1 - {\vartheta}){z}_{2}] < l{({z}_{1})}^{{\vartheta}}l{({z}_{2})}^{1-{\vartheta}}\) (\(l[{\vartheta}{z}_{1} + (1 - {\vartheta}){z}_{2}] > l{({z}_{1})}^{{\vartheta}}l{({z}_{2})}^{1-{\vartheta}}\)) for all r < z 1, z 2 < ϕ with z 1≠z 2 and all \(0 < {\vartheta} < 1\); and log-linear if \(l[{\vartheta}{z}_{1} + (1 - {\vartheta}){z}_{2}] = l{({z}_{1})}^{{\vartheta}}l{({z}_{2})}^{1-{\vartheta}}\). If l is twice differentiable, these definitions are equivalent to those given by us in Definitions 3.1–3.3, which should not be surprising given the analogy to the well-known concept of a function’s concavity, convexity, and linearity.
- 11.
This will perhaps become even clearer if one considers Prékopa’s (1973) less restrictive definition of a function’s log-linearity (that does not require l to be twice differentiable) and compares it to the definition of a function’s linearity. While a function l 1 is log-linear if \({l}_{1}[{\vartheta}{z}_{1} + (1 - {\vartheta}){z}_{2}] = {l}_{1}{({z}_{1})}^{{\vartheta}}{l}_{1}{({z}_{2})}^{1-{\vartheta}}\) for all r < z 1, z 2 < ϕ with z 1≠z 2 and all \(0 < {\vartheta} < 1\), a function l 2 is linear if \({l}_{2}[{\vartheta}{z}_{1} + (1 - {\vartheta}){z}_{2}] = {\vartheta}{l}_{2}({z}_{1}) + (1 - {\vartheta}){l}_{2}({z}_{2})\). Consider now l 1. Taking the logarithm of l 1, we see at once that \({\mathcal{l}}_{1}[{\vartheta}{z}_{1} + (1 - {\vartheta}){z}_{2}] = {\vartheta}{l}_{1}({z}_{1}) + (1 - {\vartheta}){l}_{1}({z}_{2})\) holds. Hence, ℓ 1 is linear, while l 1 = expℓ 1 is log-linear and therefore as convex as an exponential.
- 12.
By the way, \(\exp [\tan (z - \pi /2)]\) is an example of a convex function with strictly log-convex parts (for \(\pi /2 < z < \phi = \pi \)) and strictly log-concave parts (for \(r = 0 < z < \pi /2\)).
- 13.
Contrary to Remark 3.3, Greenhut et al. (1987, p. 25) claim that Definitions 3.1–3.3 are not mutually exclusive. Hence, there should be functions that fit to more than just one of these definitions. Actually, it is clear that there exist functions that do not fit either of Definitions 3.1–3.3. However, it is impossible for a function to fit two or even all of them, for (restrictions of) functions and their logarithms can be either strictly convex, linear, strictly concave, or neither of these three possibilities. The reason for Greenhut et al.’s claim might be Stevens and Rydell’s (1966, p. 197) remark that Definitions 1–7 in their paper are not mutually exclusive. Their Definitions 1–3, however, are the standard definitions of strict convexity, strict concavity, and linearity, whilst Definitions 4–6 correspond to convexity relative to an exponential. As shown above, upward-sloping affine functions are always less convex than an exponential, so that Definitions 2 and 4 in Stevens and Rydell (1966, p. 196) are indeed not mutually exclusive, whereas our Definitions 1–3 on their own certainly are.
- 14.
Note that we shall speak of r as the reservation income in the following. However, as noted above in Section 3.1, r can also be thought of as the lower bound of the support of workers’ reservation incomes depending on the interpretation of the individual labour supply function as either workers’ individual labour supply curve or as the c.d.f. of their reservation incomes.
- 15.
To be precise, (3.19) does not converge pointwise to exponential individual labour supply if γ → 0 because this limit is not defined. It is defined, however, if we add a one to the term in parentheses. Then (3.19) indeed converges pointwise to exponential individual labour supply, which is shown in Appendix A.2. For expositional convenience, however, we shall not bother with this detail here.
- 16.
This claim follows immediately from ℓ(z) = 0 for all r < z < ϕ, so that ℓ′(z) = 0 for all r < z < ϕ.
- 17.
Furthermore, one should stress that the linear case can also serve as an approximation for the more general case with arbitrary individual labour supply functions as given by (3.4). For instance, Beckmann (1976, p. 619) states that ‘[l]inearity is not much of a restriction as long as the range of prices considered is small. This will be true, in turn, when transportation costs per unit distance are small relative to mill prices.’
- 18.
For an analysis of spatial monopsony with stochastic individual (labour) supply, see Löfgren (1992).
- 19.
More formally, this can be established by using equation (3.10) to arrive at
$$\begin{array}{rcl} {\lim}_{{X}_{M}\rightarrow 0}{e}_{M}(w,{X}_{M})& {=\lim}_{{X}_{M}\rightarrow 0}\frac{w{\int \nolimits \nolimits}_{0}^{{X}_{M}}l'(w-tx)\,dx} {{\int \nolimits \nolimits}_{0}^{{X}_{M}}l(w-tx)\,dx} & \\ & {=\lim }_{{X}_{M}\rightarrow0}\frac{wl'(w-t{X}_{M})} {l(w-t{X}_{M})} & \\ & = \epsilon (w, 0),& \\ \end{array}$$where the second equality follows after applying both de l’Hôpital’s and the Leibniz rule.
- 20.
- 21.
As Ivaldi et al. (2007, p. 4) put the state of affairs: ‘Tacit collusion refers to a group of oligopolists’ ability to coordinate, even in the absence of explicit agreement, to raise price or more generally increase profit at the detriment of consumers. … “Tacit collusion” need not to involve any “collusion” in the legal sense, and in particular need involve no communication between the parties.’ By using the term ‘tacit collusion,’ it is just emphasised that the outcome (e.g., parallel wage-setting behaviour of spatial competitors) resembles the pattern that would have been observed had there been explicit collusion.
- 22.
It is interesting to note that in a spaceless dyopoly setting GO conjectures give rise to perfectly competitive outcomes (e.g., Giocoli, 2005). This also holds in a spaceless oligopsony setting (e.g., Naylor, 1996). Therefore, it seems particularly appropriate to refer to GO conjectures as representing a more competitive environment than HS conjectures.
- 23.
To deal with the second criticism several authors have worked out the concept of consistent conjectural variations, which was first introduced by Harrod (1934 b), influential contributions including Bresnahan (1981), Perry (1982), Boyer and Moreaux (1983 a; 1983b), and Kamien and Schwartz (1983). For instance, Capozza and Van Order (1989) investigate consistent conjectural variations within a linear oligopoly model completely analogous to the linear oligopsony model to be presented in Section 4.5. Another related problem consistent conjectural variations aim to alleviate is that imposing arbitrary conjectures may yield a theory of economic behaviour that is not refutable in the Popperian (2005) sense. This may be the case as it allows for (almost) any observed behaviour to be ‘explained’ by a suitable choice of the conjectural variations (cf. Figuières et al., 2004, p. ix). Furthermore, one should add that the first criticism – vigorously stated for the first time by Harrod (1934 a), Kahn (1937), Stigler (1940), and Fellner (1949) – has not to be lethal at all. For example, Dockner (1992), Cabral (1995), and Figuières et al. (2004, pp. 33–64) discuss under which conditions the conjectural variations approach gives a reduced form of the equilibrium of an (unmodelled) dynamic game. In particular, it is well-known that collusive behaviour or tacit collusion, i.e., Löschian conjectures, can be in some cases supported even in noncooperative dynamic settings by trigger strategies (cf., e.g., Friedman, 1971; 1983, pp. 123–134). An interesting historical account of the conjectural variations approach and the problem of imposing consistency conditions on firms’ conjectures is given by Giocoli (2005).
- 24.
Of course, for this interpretation of a decreasing market radius as competitive entry to hold, we have to impose a costless relocation assumption like the one imposed in the long run (see basic assumption (A7) in Section 3.1). Then competitive entry actually yields a symmetric reduction in firms’ market radius (see the discussion of the short-run dynamics in Section 3.6 and the discussion of free entry and costless relocation in Section 4.4).
- 25.
This is shown in Appendix A.4.
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Hirsch, B. (2010). Short-Run Spatial Monopsony. In: Monopsonistic Labour Markets and the Gender Pay Gap. Lecture Notes in Economics and Mathematical Systems, vol 639. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10409-1_3
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