## Abstract

We develop revealed preference tests for models of multi-product oligopoly, building on the work in Carvajal et al. (Econometrica 81(6):2351–2379, 2013). We analyze a Cournot model with multiple goods and show that it has testable restrictions when at least one good is produced by two or more firms. We also develop a revealed preference test for Bertrand oligopoly in a setting where each firm produces a single differentiated good, and these goods are potentially substitutes for each other. Our tests require qualitative assumptions on the shape of the demand curves and (in the Bertrand case) their evolution across observations, but they do not rely on the estimation of market demand.

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## Notes

For a more recent treatment of Afriat’s Theorem see Fostel et al. (2004).

Note that we are

*not*saying that firms that fail Cournot or Bertrand rationalizability must necessarily be colluding. A discussion of the distinction between testing for Cournot rationalizability and testing for collusion can be found in Carvajal et al. (2013).This mirrors the setup in CDFQ in the sense that data are generated by demand shocks rather than cost shocks and, of course, the demand shocks are never firm specific since all firms produce the same good.

This notion is well defined in a supermodular game (see Milgrom and Roberts 1990).

Some restriction on the shape of the firms’ cost functions is necessary for there be nontrivial testable restrictions on the data; this is formally shown in Theorem 1 of CDFQ. For a property on cost functions that is weaker than convexity but still leads to observable restrictions on data, see the working paper version of CDFQ, Carvajal et al. (2010).

These conditions are stated slightly differently in CDFQ. Condition (ii) is stated in CDFQ as the

*common ratio property*and condition (iii), which guarantees the convexity of the firms’ cost functions, is captured by the*co-monotone property*. These conditions have been restated here in a way that is more clearly related to the multi-product generalization in Theorem 2.For the use of this condition in the context of multi-product oligopolies, see Vives (1999). The micro-foundations of this property have been extensively studied; see Quah (2003) and the survey of Jerison and Quah (2008). The literature usually considers demand as a function of price, rather than the inverse demand function considered here. However, the two cases are equivalent: if \(\partial \bar{P}_t(q)\) is negative definite, then \(\bar{P}_t\) is locally invertible, and its inverse, the demand function \(\bar{Q}_t\), has a negative definite matrix at \(\bar{P}_t(q)\).

Instead of requiring \(\bar{P}_t\) to obey the law of demand, we could require that, for all \(i\in \mathcal I\) and \(t\in \mathcal T\), the derivative matrix \(\partial \bar{P}_t\) is negative definite when restricted to the subset of goods \(\mathcal{K}_i\), which is weaker than simply requiring \(\partial \bar{P}_t\) to be negative definite. To characterize Cournot rationalizable data sets with this weaker requirement on demand, condition (i) in statement [B] must be modified. It should now require \(\varLambda _t\) to have the following property: for all \(i\) and \(t\), the restriction of \(\varLambda _t\) to \(\mathcal{K}_i\) is negative definite. This modification of Theorem 2 is possible because the weaker property on \(\varLambda _t\) is nonetheless sufficient to enable the construction of a rationalizing inverse demand function \(\bar{P}_t\) such that each firm’s profit function is concave (see Lemma 1).

The original procedure of Tarski and Seidenberg, and the alternative of Cylindrical Algebraic Decomposition are known to be doubly exponential. More recent developments in Quantifier Elimination may yield a procedure that is singly exponential: see Basu et al. (2006, Ch. 13).

This property guarantees the positive definiteness of the symmetric matrix \(\varLambda +\varLambda ^T\), which is equivalent to the positive definiteness of \(\varLambda \); see Mas-Colell et al. (1995, Appendix M.D.) for more on diagonal dominance.

Even though cross-price effects are zero, each firm is producing more than one good and the market for each good can be served by more than one firm, so this is

*not*formally identical to the case of an industry where each firm is producing a good different from that produced by other firms. In the latter scenario, as we have already pointed out, observable restrictions do not exist if there are no cross-price effects.If we impose the condition that all the goods are substitutes, then Cournot rationalizability requires that all observed prices be nonnegative: if \(P_t^{\bar{k}}< 0\) then any firm that is producing good \(\bar{k}\) is strictly better off if it reduces its output of \(\bar{k}\). In the case when the goods are not necessarily substitutes, the model allows for the possibility that some observed prices are negative: firms can optimally pay for a good to be consumed in order that it may raise the demand for some other good.

In the previous section, superindices were used to denote commodities, while firms were indicated by the first of two subindices. In this section, each firm produces only one product, so we shall dispense with the superindex.

An issue that all revealed preference tests have to contend with is that they are binary: a data set either passes or fails the test. Thus, implementing such a test on a large data set will often be problematic, unless a way of accounting for errors is formally included. This is one reason why data sets used in revealed preference tests generally tend not to be very big. In studies of consumer demand, it is typical to implement these tests (for example, Afriat’s Theorem or its variations) on a large number of subjects, with the number of observations for each subject being fairly modest (fewer than 15 or even 10 observations are not uncommon). In other words, there is a large number of tests, with each test having a small data set, and the frequency of passing the test in this collection of data sets is used as a measure of the model’s success. The empirical strategy proposed here, and used in CDFQ, has a broadly similar pattern.

Notice that (ii) is equivalent to \((P_t-\delta _{i,t})/Q_{i,t}=(P_t-\delta _{i,t})/Q_{i,t}>0\) for any two firms \(i\) and \(j\) and for all \(t\). In CDFQ, this is called the

*common ratio property.*The profit function \(\bar{\Pi }_i\) of firm \(i\) is log-concave if, and only if,

$$\begin{aligned} -\bar{\Pi }'_i(p_i) = \bar{C}_i'(\bar{Q}_i(p_i)) \bar{Q}_i'(p_i)-\bar{Q}_i(p_i) - p_i \bar{Q}_i'(p_i) \end{aligned}$$is a single crossing function of \(p_i\). (We are suppressing the dependence of \(\bar{Q}_i\) on \(p_{-i}\) in the notation.) Since \(\bar{C}_i'(\bar{Q}_i(p_i)) \bar{Q}_i'(p_i) - \bar{Q}_i(p_i)\) and \(-p_i\bar{Q}_i'(p_i)\) are both single crossing functions of \(p_i\) (indeed, the first is a negative-valued function and the second a positive-valued function), it suffices to show that ratio

$$\begin{aligned} \frac{C_i'(\bar{Q}_i(p_i))\bar{Q}_i'(p_i)-\bar{Q}_i(p_i)}{p_i \bar{Q}_i'(p_i)}=\frac{1+\bar{\varepsilon }_i(p_i)C_i'(\bar{Q}_i(p_i))}{p_i\bar{\varepsilon }_i(p_i)} \end{aligned}$$is decreasing in \(p_i\) (see Quah and Strulovici 2012). This is true because \(\bar{C}_i'\) is nondecreasing in \(q_i\) while \(\bar{\varepsilon }(p_i)\) and \(\bar{Q}_i\) are both nonincreasing in \(p_i\).

To construct the demand system, we choose, for each \(i\), a function \(\epsilon _{i}:\mathbb {R}^I_{++}\times A\times B_i\rightarrow \mathbb {R}\) such that \(\epsilon (p,\alpha ,\beta _i)\) is strictly positive, continuous in \(p\), nondecreasing in \(p_i\), nonincreasing in \(p_{-i}\) and nondecreasing in \((\alpha ,\beta _i)\) such that \(\epsilon _{i}(P_t,\alpha _t,\beta _{i,t})=\lambda _{i,t}\). This is possible because of the new condition (iv). The log-demand function for good \(i\) is chosen to be \(\bar{L}_{i}(p,\alpha ,\beta _i)=-\int _{0}^{p_i}\epsilon _{i}(z,p_{-i},\alpha ,\beta _i)\, \text{ d }z +k_{i}(\alpha ,\beta _i)\) where the function \(k_i\) is chosen so that \(\bar{Q}_{i}(P_t,\alpha _t,\beta _{i,t})=\exp (\bar{L}_i(P_t,\alpha _t,\beta _{i,t})=Q_{i,t}\) (for all \(t\in \mathcal{T}\)). \(\bar{Q}_i(p,\alpha ,\beta _i)\) has all the required properties of a demand function for product \(i\) and also satisfies

$$\begin{aligned} -\frac{1}{Q_{i,t}}\frac{\partial \bar{Q}_i}{\partial p_{i}}(P_t,\alpha _{t},\beta _{i,t})=\lambda _{i,t}. \end{aligned}$$We could think of \(w_i\) as the price of a factor, with a higher price leading to a higher marginal cost. Note that, instead of a one-dimensional parameter \(w_i\), we could allow for a multi-dimensional parameter which is guaranteed to raise the marginal cost function of a firm only if it is higher in all dimensions; such a formulation is more complicated but can be handled in a similar way. The reader can consult the treatment of this issue in Carvajal et al. (2010).

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The financial support of the ESRC to this research project, through Grants RES-000-22-3771 (Andrés Carvajal) and RES-000-22-3187 (John Quah) is gratefully acknowledged. Rahul Deb and James Fenske would like to acknowledge the financial support they received from their Leylan fellowships. Part of this paper was written while John Quah was visiting the Economics Department at the National University of Singapore and while Andrés Carvajal was visiting CORE–Université catholique de Louvain; they thank these institutions for their hospitality. We are grateful to Dirk Bergemann, Don Brown, Greg Crawford, Luis Corchon, Françoise Forges, Enrico Minelli, Margaret Slade, Mike Waterson, Mike Whinston and Glen Weyl for helpful comments.

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Carvajal, A., Deb, R., Fenske, J. *et al.* A nonparametric analysis of multi-product oligopolies.
*Econ Theory* **57**, 253–277 (2014). https://doi.org/10.1007/s00199-014-0843-x

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DOI: https://doi.org/10.1007/s00199-014-0843-x

### Keywords

- Cournot equilibrium
- Bertrand equilibrium
- Revealed preference
- Observable restrictions
- Product differentiation
- Supermodular games