Oligopolistic vs. monopolistic competition: Do intersectoral effects matter?

Abstract

Recent extensions of the standard Dixit–Stiglitz (Am Econ Rev 67:297–308, 1977) model, that go beyond the CES sub-utility assumption while maintaining monopolistic competition, have mainly emphasized the role of intrasectoral substitutability. We argue that introducing oligopolistic competition can be an alternative extension, still tractable, allowing to restore the role of intersectoral substitutability and reinforcing the general equilibrium dimension of the model. For this purpose, we define a comprehensive concept of oligopolistic equilibrium to give account of a large set of competition regimes with varying competitive toughness. For two particular regimes, price competition and quantity competition, we show how, with strategic interactions, pro-competitive or anti-competitive effects now depend on the intersectoral elasticity of substitution as compared to the intrasectoral elasticity of substitution.

Appendix

1.1 The elasticities of substitution

The concept of elasticity of substitution was introduced by Hicks (1932) to measure the response of the input quantity ratio to a variation in the corresponding price ratio, under price taking and cost minimization by the producers. The original concept concerned the case of the sole two inputs, capital and labor, of some production function, but it was later generalized in several ways to the case of an arbitrary number of inputs (see e.g., Blackorby and Russell 1989).

1.1.1 The intrasectoral elasticity of substitution

In this paper, while referring to the sub-utility function X, we consider the elasticity of substitution $s_{i}$ of good i for the composite good (rather than the elasticity of substitution $s_{ij}$ between two goods i and j). It is defined, for a bundle of differentiated goods $\mathbf {x} =\left( x_{1},\ldots ,x_{i},\ldots ,x_{n}\right)$, as the absolute value of the elasticity of the ratio $x_{i}/X(\mathbf {x})$ with respect to the marginal rate of substitution $\partial _{i}X(\mathbf {x})$, which coincides here with the marginal (sub-)utility of $x_{i}$:

$$\begin{aligned} s_{i}\equiv -\frac{d\left( x_{i}/X(\mathbf {x}) \right) }{d\left( \partial _{i}X(\mathbf {x}) \right) }\frac{\partial _{i}X(\mathbf {x})}{x_{i}/X(\mathbf {x})}. \end{aligned}$$

(47)

By differentiating $x_{i}/X(\mathbf {x})$ and $\partial _{i}X(\mathbf {x})$ with respect to $x_{i}$, we obtain the following expression for the intrasectoral elasticity of substitution of good i:

$$\begin{aligned} s_{i}=\frac{1-\partial _{i}X(\mathbf {x}) x_{i}/X(\mathbf {x})}{-\partial _{ii}^{2}X(\mathbf {x}) x_{i}/\partial _{i}X(\mathbf {x})}. \end{aligned}$$

(48)

Notice that $s_{i}$ can be seen as the reciprocal of the degree of curvature of the graph $\left( x_{i}/X(\mathbf {x}),X(\mathbf {x}) \right)$.

The elasticity of substitution between goods i and j is:

$$\begin{aligned} s_{ij}\equiv & {} -\frac{d(x_{i}/x_{j})}{d\left( dx_{j}/dx_{i}\right) }\frac{dx_{j}/dx_{i}}{x_{i}/x_{j}} \\= & {} {\tiny -}\frac{\partial _{i}X(\mathbf {x}) \partial _{j}X(\mathbf {x}) \left( x_{i}\partial _{i}X(\mathbf {x}) +x_{j}\partial _{j}X(\mathbf {x}) \right) }{x_{i}x_{j}\left( \partial _{ii}^{2}X(\mathbf {x}) \left( \partial _{j}X(\mathbf {x}) \right) ^{2}-2\partial _{ij}^{2}X(\mathbf {x}) \partial _{i}X(\mathbf {x}) \partial _{j}X(\mathbf {x}) +\partial _{jj}^{2}X(\mathbf {x}) \left( \partial _{i}X(\mathbf {x}) \right) ^{2}\right) } \nonumber \end{aligned}$$

(49)

(see Nadiri 1982). If sub-utility X is symmetric and if we consider any pair (i, j) of goods such that $x_{i}=x_{j}=x$,

$$\begin{aligned} s_{ij}=\frac{1}{-\partial _{ii}^{2}X(\mathbf {x}) x/\partial _{i}X(\mathbf {x}) +\partial _{ij}^{2}X(\mathbf {x}) x/\partial _{i}X(\mathbf {x})}, \end{aligned}$$

(50)

which converges to the same value as our elasticity $s_{i}$ as n tends to infinity, implying that $\partial _{i}X(\mathbf {x}) x_{i}/X(\mathbf {x}) \rightarrow 0$ and $\partial _{ij}^{2}X(\mathbf {x}) x/\partial _{i}X(\mathbf {x}) \rightarrow 0$ (cf. Parenti et al. 2014, where the set of the differentiated goods is a continuum). Also, symmetry and constancy of the elasticity $s_{ij}$ ($s_{ij}=s$ for any $\mathbf {x}$ and any pair $\left( i,j\right)$ of goods), corresponding to the CES sub-utility function $X(\mathbf {x}) =\left( \sum _{i=1}^{n}a_{i}x_{i}^{(s-1) /s}\right) ^{s/(s-1)}$, naturally lead to the coincidence of both elasticities: $s_{i}=s_{ij}=s$. By applying (48), we indeed obtain at $\mathbf {x}$:

$$\begin{aligned} s_{i}=\frac{1-a_{i}\left( x_{i}/X(\mathbf {x}) \right) ^{(s-1) /s}}{\left( 1-a_{i}\left( x_{i}/X(\mathbf {x}) \right) ^{(s-1) /s}\right) /s}=s. \end{aligned}$$

(51)

Thus, the elasticity $s_{i}$ of intrasectoral substitution to which we refer in this paper coincides with the standard elasticity $s_{ij}$ in two of the most used cases in the literature, namely the case of the CES sub-utility function and the case of a “large group” of differentiated goods (justifying the monopolistic competition approach), when the sub-utility function X and the observed basket $\mathbf {x}$ are both symmetric.

The intrasectoral elasticity of substitution $s_{i}$ can alternatively be computed in terms of prices (under consumer price taking and expenditure minimizing). To do that, denoting $\underline{X}=X(\mathbf {x})$, we refer to the shadow price of the composite good $\underline{P}\equiv \partial _{X}e\left( \mathbf {p,}\underline{X}\right)$, and use the expression of $x_{i}$ given by Shephard’s lemma (3):

$$\begin{aligned} s_{i}=-\frac{d\left( x_{i}/\underline{X}\right) }{d\left( p_{i}/\underline{P} \right) }\frac{p_{i}/\underline{P}}{x_{i}/\underline{X}}=-\frac{d\left( \partial _{p_{i}}e\left( \mathbf {p,}\underline{X}\right) /\underline{X} \right) }{d\left( p_{i}/\underline{P}\right) }\frac{p_{i}/\underline{P}}{\partial _{p_{i}}e\left( \mathbf {p,}\underline{X}\right) /\underline{X}} . \end{aligned}$$

(52)

By differentiating $\partial _{p_{i}}e\left( \mathbf {p,}\underline{X}\right) /\underline{X}$ and $p_{i}/\underline{P}$ with respect to $p_{i}$, we obtain

$$\begin{aligned} s_{i}=\frac{-\partial _{p_{i}p_{i}}^{2}e\left( \mathbf {p,}\underline{X} \right) p_{i}/\partial _{p_{i}}e\left( \mathbf {p,}\underline{X}\right) }{1-\partial _{p_{i}X}^{2}e\left( \mathbf {p,}\underline{X}\right) p_{i}/\partial _{X}e\left( \mathbf {p,}\underline{X}\right) } \end{aligned}$$

(53)

and, by using the definition of the Hicksian demand $H_{i}$ in (3), together with the first-order condition (2), we finally get

$$\begin{aligned} s_{i}=\frac{-\epsilon _{p_{i}}H_{i}\left( \mathbf {p,}\underline{X}\right) }{1-\left[ \epsilon _{i}X(\mathbf {x}) \right] \left[ \epsilon _{X}H_{i}\left( \mathbf {p,}\underline{X}\right) \right] }. \end{aligned}$$

(54)

1.1.2 The intersectoral elasticity of substitution

To define the intersectoral elasticity of substitution of good i, we refer to the quantity ratio $x_{i}/Y$ (instead of $x_{i}/\underline{X}$),¹¹ implicitly taking into account the variation of the Marshallian demand D (rather than sticking to the mere share adjustment as measured by $-\epsilon _{p_{i}}H_{i}\left( \mathbf {p,}\underline{X}\right)$). The corresponding price ratio is of course $p_{i}/1=p_{i}$. So, the intersectoral elasticity of substitution $\sigma _{i}$ of good i can be expressed as follows:

$$\begin{aligned} \sigma _{i}\equiv -\left. \frac{d(x_{i}/Y) }{dp_{i}}\right| _{X(\mathbf {x}) =D(\mathbf {p},Y)}\frac{p_{i}}{x_{i}/Y}=-\frac{\left( 1/Y\right) \partial _{p_{i}}D\left( \mathbf {p},Y\right) }{\partial _{i}X(\mathbf {x})}\frac{p_{i}}{x_{i}/Y}= \frac{-\epsilon _{p_{i}}D(\mathbf {p},Y)}{\epsilon _{i}X(\mathbf {x})}. \end{aligned}$$

(55)

To illustrate, take the case of the utility function introduced in Sect. 2.4:

$$\begin{aligned} U(X(\mathbf {x}),z)= & {} \frac{b^{1/\sigma }}{1-1/\sigma }X(\mathbf {x})^{1-1/\sigma }+z, \\ \text {with}\;\;X(\mathbf {x})= & {} \left( \sum _{i=1}^{n}x_{i}^{(s-1) /s}\right) ^{s/(s-1)},\quad \sigma >1\;\;\text {and}\;\;b>0. \nonumber \end{aligned}$$

(56)

The corresponding Marshallian demand for the composite good is the isoelastic function

$$\begin{aligned} D(\mathbf {p},Y) =bP(\mathbf {p})^{-\sigma },\quad \text {with }\quad P(\mathbf {p}) =\left( \sum _{i=1}^{n}p_{i}^{1-s}\right) ^{1/(1-s)}, \end{aligned}$$

(57)

so that we obtain the following expression for the intersectoral elasticity of substitution, as defined by (55):

$$\begin{aligned} \sigma _{i}=\sigma \frac{\left( p_{i}/P(\mathbf {p}) \right) ^{1-s}}{\left( x_{i}/X(\mathbf {x}) \right) ^{(s-1) /s}}=\sigma , \end{aligned}$$

(58)

since $x_{i}=\left( p_{i}/P(\mathbf {p}) \right) ^{-s}X(\mathbf {x})$ when the $x_{i}$’s are chosen optimally. Thus, the intersectoral elasticity of substitution is symmetric and constant in this case ($\sigma _{i}=\sigma$ for any i), equal to the elasticity of the demand for the composite good.