Revealed preference tests for price competition in multi-product differentiated markets

Assumptions of competitive structure are often crucial for marginal cost estimation and counterfactual predictions. This paper introduces tests for price competition among multi-product firms. The tests are based on the firm’s revealed preference (revealed profit function). In contrast to other approaches based on estimated demand functions such as conduct parameter estimation, the proposed tests do not require any instrumental variables, even though the models can accommodate structural error terms. In this paper, I employ a demand structure introduced by Nocke and Schutz (Econometrica 86(2):523–557, 2018. https://doi.org/10.3982/ECTA14720), the discrete/continuous choice model, which nests the multinomial logit demand and CES demand functions. Any price and quantity data can be rationalized by price competition under a discrete/continuous choice model and increasing marginal costs. Adding more assumptions to the demand functions, such as logit, CES, or the co-evolving and log-concave property produces some falsifiable restrictions.


Introduction
In the industrial organization literature, we often assume specific competitive structures such as price competition or quantity competition.In many cases, competitive structure assumptions are crucial for empirical research.For instance, we often back out marginal costs from first-order conditions based on estimated demand functions and competitive structures.Results of counterfactual analysis, which often provide the main policy implication in research with structural models, also depend on the imposed competitive structures.Even though we can obtain parameter estimates in a structural model that fits the data best, the structural model itself could possibly not fit the data.That is, the data might not be rationalized by the model for any possible parameters.Furthermore, the data might not be rationalized by any realizations of structural error terms.This is because some data points might be outside the support of the structural model.In this paper, I provide a systematic method to detect such inconsistency between the data and price competition among single-or multi-product firms under a certain class of demand functions.
Regarding consumer behavior, Afriat (1967) shows that finite data satisfy GARP if and only if they are rationalized by utility maximization given a finite set of price vectors.That is, if the data violate GARP, then they cannot be explained by any (locally non-satiated) utility functions.Brown and Matzkin (1996) extend this idea to the general equilibrium framework.Carvajal et al. (2013) apply the idea to Cournot competition, and show that Cournot rationalizability can be checked by the existence of parameters that satisfy some inequalities.Carvajal et al. (2014) introduce a few variants of Carvajal et al. (2013): a test for multi-market-contact Cournot competition and a test for price competition in a differentiated market.However, they only focus on price competition where each firm produces a single product.Price competition with multi-product firms is often examined in the empirical industrial organization literature (e.g., Berry et al. 1995;Goldberg 1995).One of the main difficulties in extending Carvajal et al. (2014)'s test to competition among multi-product firms arises from the substitution effects among the same firm's products (or cannibalization effects).We can circumvent such a difficulty by employing an important class of demand structure, namely the discrete/continuous choice model introduced by Nocke and Schutz (2018), which nests the multinomial logit demand function and constant elasticity of substitution (CES) demand function as special cases.
In order to test the competitive structure, we can also estimate the conduct parameter.Bresnahan (1982) shows that we can identify the conduct parameter in an industry from a rotation of the demand functions over time [see Bresnahan (1989) for its estimation and applications].Researchers have recently estimated the extent to which firms internalize other firms' profits, a measure that is closely related to the conduct parameter (e.g., Miller and Weinberg 2017;Sullivan 2016).Alternatively, if we have data on the cost structure, we could compare marginal costs backed out from the model with the actual cost data since different competitive structures yield different first order conditions and, in turn, different estimates of marginal costs (e.g., Wolfram 1999).The revealed preference test examined in this paper provides an alternative approach with advantages and disadvantages.The first advantage is that the revealed preference test in this paper does not require any IVs while both conduct parameter estimation and the Wolfram (1999) approach require appropriate IVs.The reason we do not need IVs is that we do not estimate parameters to test the model, but rather check restrictions that are valid for any parameter values.This is analogous to Afriat (1967)'s theorem, which characterizes a set of data restrictions satisfied if consumers maximize their own utility, regardless of the underlying utility function specification.Second, we only need the market-level price and quantity data, and no other product characteristics, to implement the test.Due to this parsimonious data requirement, this test could be used as a pretest/sanity check before detailed estimation.
A disadvantage of the test is that it is a joint test of competitive structure and demand/cost functions.Therefore, rejection of the model might imply other types of competition under a discrete/continuous demand structure, price competition under other demand functions, or other types of competition under other demand functions.Second, even though the discrete/continuous choice model is general enough to include the logit and CES demand functions as special cases, it still has the independence-toirrelevant-alternative (IIA) property.Therefore, the main theorem in this paper does not hold for the random coefficient logit model (e.g., Berry et al. 1995).Another issue is that cost functions are assumed to be invariant over time in the tested model, even though the constant marginal costs are not implied since cost functions are assumed to be convex.Therefore, the test should be implemented for short-panel data, for which the cost structure is not supposed to change during the time range.In practice, if a researcher has a long panel, then the data can be divided into many short panels, the test can be implemented for each short data segment, and the rejection ratio can be reported (e.g., Carvajal et al. 2014).Incorporating the observed cost shifter alleviates this issue, as shown in Sect.3.
In terms of the power of tests, any data would satisfy the rationalizability condition of the price competition under the general discrete/continuous demand functions.This result is inconsistent with the findings of Carvajal et al. (2013) and Carvajal et al. (2014).The key difference is that they consider demand changes due to a common shock for different firms. 1 Naturally, we can obtain falsifiable restrictions by imposing a similar additional restriction that is compatible with the discrete/continuous demand functions.We can also obtain falsifiable restrictions by restricting the underlying demand functions to a sub-class of discrete/continuous demand structures, which still nests both logit and CES as its special cases.This also implies that price competition under logit or CES demand functions is falsifiable.
In general, we can check the set of restrictions by evaluating a loss function similar to those for moment inequality estimations.In principle, therefore, the revealed preference tests in this article share some computational issues with moment inequality estimations.However, we can characterize the set of restrictions as a set of linear constraints over parameters by focusing on the logit demand functions and considering a slightly modified data requirement, which must always be satisfied when researchers estimate logit demand functions.Then, we can implement the test through standard algorithms for linear constraints.
The remainder of the paper is organized as follows.I introduce the main model and its special cases in Sect. 2. I first exemplify a revealed preference test under logit demand functions and then formalize and generalize the result.In Sect.3, I discuss some extensions of the test with (i) additional demand restrictions discussed in earlier research, (ii) observed cost shifters, (iii) the possibility of collusive conduct among firms, and (iv) non-separable cost functions.I discuss the algorithms for the tests in Sect.4, and provide a summary in Sect. 5.

The model
In this paper, I consider a standard competition framework for a differentiated market, where each firm produces different products.The products can be similar but not completely the same.The demand functions are assumed to change over time, potentially because of a change in consumer tastes or product characteristics.The characteristics might or might not be observed by the econometrician.I denote J = {1, 2, . . ., J } as a set of products and Q j,t : R J + → R + as a demand function of product j ∈ J at time t ∈ {1, . . ., T } ≡ T .The demand functions are assumed to be in a class of discrete/continuous models, which is explained later.Firm f ∈ {1, . . ., F} produces a set of products The cost function of product j ∈ J , C j : R + → R, is assumed to be convex, strictly increasing, and continuously differentiable.2In this section, I focus on time-invariant cost functions, which serve a similar purpose as time-invariant preference in Afriat (1967). 3Then, the profit function for firm f at time t is written as Let the observed price and quantity be as follows: { p, q} where x = x 1 , . . ., x T and xt = x1,t , . . ., xJ,t for x = p, q.4In the following, I introduce tests of whether a set of data { p, q} can be rationalized by price competition.
Definition 1 A set of data { p, q} is Bertrand-rationalizable if, for any t ∈ T , { pt , qt } is generated as a result of Nash equilibrium of price competition, in which each firm f ∈ F solves max for some demand and cost functions, Q j,t (•) and C j (•).
I primarily utilize first-order conditions of profit functions and cost convexity to derive testable data restrictions that should be satisfied regardless of the parameter values and structural error terms.Using the profit functions defined above, the first-order condition w.r.t.p j is written as (1)

Logit demand function
Before proceeding to the main result with a general specification, I demonstrate that some data cannot be explained by price competition with the logit demand functions, which is a special case of the discrete/continuous model.By using logit demand functions, for some M t , α ∈ R + and v j,t j∈J ∈ R J , the first-order condition is rewritten as By rearranging it, we obtain the following equation: Example 1 Between-Firm Restriction.
The first example shows that the logit specification generates between-firm restrictions on data to be rationalized.To emphasize this, I focus on single-product firms 1 and 2, each of which produces product 1 and 2, respectively.Then, the first-order condition becomes for j = 1, 2. By cancelling 1/α, we obtain the following necessary condition for equilibrium: Suppose that we observe the following data: p j,τ , q j,τ j=1,2, τ=s,t s.t.p j, t = p j, s = p j for j = 1, 2 and q2, t > q1, t = q1, s > q2, s .The RHS for time t and s gives (5) However, the LHS for time t and s gives Eqs. ( 5) and ( 6) show that the data do not satisfy Eq. ( 4).Thus the data cannot be explained by price competition under logit demand functions.The explanation is as follows.In the logit specification, a change in price and quantity over time can be explaiend by change in values of the products, v j,t , or market size, M t .If v j,t changes, the price p j, t changes in the same direction as v j, t .If prices do not change over time, as in the example, a change in quantity can be explained by the market size.However, change in the market size is common to both products.Therefore, the quantities of both products should move in the same direction.
The second example shows that the logit demand functions generate within-firm restrictions.In Eq. (2), the RHS applies in common to the goods produced by the same firm.Therefore, must hold at equilibrium.In this paper, I call this property the "common mark-up property." 6The model is rejected by the common mark-up property together with the increasing marginal cost assumption, given the following data: p j,τ , q j,τ j=1,2, τ=s,t s.t.{1, 2} ⊂ J f , p1,s > p1,t , p2,s < p2,t , q1,s < q1,t , and q2,s > q2,t .That is, the price and quantity of good 1 and those of good 2 move in the opposite direction (see Fig. 1).Suppose that the data satisfy Eq. (7) at time s.If the marginal costs are (weakly) increasing with own quantity, then p1,t − C 1 q1,t < p1,s − C 1 q1,s = p2,s − C 2 q2,s < p2,t − C 2 q2,t .Therefore, Eq. ( 7) cannot be satisfied at time t.Thus, these data, p j,τ , q j,τ j=1,2, τ=s,t , cannot be explained by (a repetition of static) price competition under logit demand functions.This means these data cannot be explained by any set of parameters, α, m t , and v j,t and non-parametric cost functions, C j (•).

Properties of revealed preference tests
The above examples highlight some important features of revealed preference tests for competitive structures.
IVs not needed With the logit specification, one can better understand an underlying mechanism of the revealed preference test by comparing it with an alternative procedure to check the competitive structure.A parameter of the demand functions, α, is often estimated from aggregated data (with the use of IVs to address unobserved heterogeneity potentially correlated with prices), and marginal costs are backed out from the first-order condition.Subsequently, we could check whether the obtained marginal costs are reasonable.Alternatively, in the revealed preference test, a similar procedure is used for any possible α > 0, instead of an estimated α determined by IVs.Therefore, we do not need IVs for unobserved heterogeneity that is potentially correlated with price or some other variables.
Interpretation of rejection Note that rejection or acceptance is not probabilistic even if the model has (only) the structural error term in the logit demand functions.When we estimate logit demand functions from aggregate data, v j,t is decomposed as v j,t = x j,t β+ξ j,t where x j,t is a vector of product j's observed characteristics, ξ j,t represents the unobserved characteristics, and β is a vector of parameters.In the logit demand estimation, ξ j,t is treated as a structural error term.However, Eq. ( 7) should be satisfied regardless of what values of ξ j,t are realized, as long as firms compete on price under logit demand functions (recall that I did not impose any assumptions on v j,t ).This is because each firm (but not the econometrician) is assumed to know what ξ j,t j∈J is realized, as is often assumed in the empirical IO literature.Therefore, any rejection of the model cannot be attributed to a peculiar realization of structural error terms.
(No) data restrictions in each assumption In this article, revealed preference tests are joint tests of demand and cost specifications and the competitive structure.However, it is worth noting that each of them by itself cannot be rejected by any data, { p, q}, but can only be rejected together.Assuming only logit demand functions, for any data p j,t , q j,t j∈J at each t, we can back out the corresponding v j,t j∈J by an inversion of the market share functions as in Berry (1994).Thus, logit demand can fit the data since any changes in data over time can be captured by changes in v j,t j∈J over time.Regarding the assumption of price competition and cost functions, any data can be rationalized by price competition under more general demand functions and convex time-invariant cost functions, as explained in Sect.2.2.This emphasizes that each assumption in this article is not trivially restrictive, especially when we have only price and quantity data.
In the following part, I provide a set of inequalities as a systematic method to detect data inconsistent with price competition, and show that such conditions are sufficient for rationalization by price competition.Instead of the logit demand functions, I employ a class of demand functions by Nocke and Schutz (2018) that nests the logit demand functions and CES demand functions.

Discrete/continuous demand function
In the following, I employ the discrete/continuous demand functions introduced by Nocke and Schutz (2018), where the demand function for product j is written as , where h j (•) is decreasing and log-convex for every j, and m is a positive constant.An important example of this demand function is the logit model h j p j = exp v j − α p j and m = M/α, where v j ∈ R is the value of good j, α > 0 is the coefficient for prices, M > 0 is the size of the market, and h 0 is the exponentiated value of the outside option. 7Another important example is the CES model h j p j = a j p 1−σ j and m = I / (σ − 1), where I is the income level of the consumer and σ is the elasticity of substitution (σ > 1).
In this paper, I utilize the fact that we can express the partial derivatives of the discrete/continuous demand functions in a simple form: It is worth noting that −m t h j,t p j /h j,t p j − Q j,t (p) is positive because of the log-convexity of h j (•).8With the above expression, the FOC w.r.t.p j is written as follows: Therefore, if the data { p, q} are generated by price competition with (unknown) discrete/continuous demand functions, there exists α jt , δ jt , which corresponds to −h j,t p j /h j,t p j and C j q j,t , respectively, such that On the other hand, since δ j,t corresponds to C j q j,t and C j (•) is assumed to be increasing, δ j,t must be greater than δ j,s (s = t) if q j,t is greater than q j,s .This is summarized as an inequality: 0 ≤ δ j,s − δ j,t q j,s − q j,t . (9) Combining Eqs. ( 8) and ( 9), we obtain a set of necessary conditions for the data to be rationalized by the model.Furthermore, the conditions are also sufficient for rationalization.They are summarized in the following theorem.
Theorem 1 (Discrete/Continuous) The set of observations { p, q} is Bertrandrationalizable under convex and strictly increasing cost functions and a discrete/continuous demand functions if and only if, for any t ∈ T , f ∈ F, and j ∈ J f , there exist real numbers α j,t , δ j,t , and m t such that the following hold: The first set of conditions is derived from the underlying specifications of the demand and cost functions: α j,t > 0 from the assumption that h j is decreasing and logconvex, δ j,t > 0 from the increasing cost functions, and m t > 0 from the assumption that the quantity of each good is non-negative.The proof of sufficiency consists of two steps.First, given α j,t and δ j.t , which satisfy the conditions, I construct demand functions Q j, t (•) and cost functions C j (•) to satisfy − h j,t p j / h j,t p j = α j,t and C j q j,t = δ j.t .9Then, the data { p, q} satisfy the first-order conditions under the reconstructed demand and cost functions.In the second step, I show that the first-order conditions are a sufficient condition for profit maximization given the other firms' prices and the reconstructed demand cost functions.This result is not trivial since the profit functions do not satisfy quasi-concavity.In this paper, sufficiency is proved by the unique solution of the first-order conditions.The uniqueness is derived from the unique "common ι-markup" and a mapping from ι-markup to price vectors as in Nocke and Schutz (2018).See the "Appendix" for the full proof.10

Special cases: logit and CES
For more restrictive specifications, such as logit or CES demand functions, we can easily derive the necessary condition for data to be rationalized by the models, by simply adding restrictions to the second condition in the above tests.Sufficiency of the restriction, however, is less trivial.In the proof of sufficiency in Theorem 1, I reconstruct the demand functions as Q j, t (•) , which still nest the logit demand functions, but not CES.Such a reconstruction is sufficient for Theorem 1 since the reconstructed demand functions Q j, t (•) are in the class of demand functions we are interested in.To test a model with the logit demand, we can apply a similar reconstruction of Q j, t (•) and the remaining is proved analogously.In contrast, such a reconstruction is no longer valid for a model with CES demand.Therefore, demand functions are reconstructed in a different way, and the sufficiency of the first-order condition is proved by a slightly different method due to the different construction of the demand functions.The modified tests mentioned above are articulated in the following propositions (the specifications and results are summarized in Table 1).
Proposition 1 (Logit) The set of observations { p, q} is Bertrand-rationalizable under convex and strictly increasing cost functions and logit demand functions if and only if, for any t ∈ T , f ∈ F, and j ∈ J f , there exist real numbers α t , δ j,t , and m t such that the following hold: qk,t ; and 3. 0 ≤ δ j,t − δ j,t q j,t − q j,t .
In the above statement, α t is allowed to vary over time for the sake of generality.We can easily replace α t with time-invariant α.Such a simplified version is proved analogously to Proposition 1.
Proposition 2 (CES) The set of observations { p, q} is Bertrand-rationalizable under convex and strictly increasing cost functions and CES demand functions if and only if, for any t ∈ T , f ∈ F, and j ∈ J f , there exist real numbers σ t , δ j,t , and m t such that the following hold: p j,t m t σ t + k∈J f pk,t − δ k,t qk,t ; and 3. 0 ≤ δ j,t − δ j,t q j,t − q j,t .
As explained earlier, this necessary condition is derived as a special case of (the necessity part of) Theorem 1.However, the sufficiency is not derived from Theorem 1 since the reconstructed demand functions should be also CES instead of arbitrary discrete/continuous demand.See the "Appendix" for the proof.

Falsifiability
Theorem 1 characterizes the necessary and sufficient condition for the data to be rationalized by the model of price competition under discrete/continuous demand func-tions and time-invariant convex cost functions.Meanwhile, readers might wonder how restrictive the conditions are.It turns out that the restriction in Theorem 1 is so loose that any data can be rationalized by the model with the general discrete/continuous demand functions.This may be surprising, considering that even the general discrete/continuous choice model satisfies the IIA property.Any changes in price and quantity along a fixed discrete/continuous demand function must satisfy the IIA property, while demand functions themselves are allowed to change over time with the model in Theorem 1.To clearly understand how any data satisfy the restrictions, consider the following.For any given δ j,t and m t , the remaining parameter α j,t , which characterizes the demand functions, can be determined only through the first-order condition w.r.t.p j, t , independently of the first-order condition w.r.t.p k, s , where k = j or s = t.11Thus, for any data, we can find corresponding α j,t , δ j,t , m t , and then, the sufficiency implies that any data are rationalized by the model.
Corollary 1 Any data, { p, q}, are Bertrand-rationalizable under convex and strictly increasing cost functions and discrete/continuous demand functions.
Even though price competition under the general discrete/continuous choice model is not falsifiable, a model can be falsifiable under a more restrictive demand model such as the logit demand functions as shown in the example.This naturally raises a question: How general is this falsifiability?In the following, I show that a subclass of discrete/continuous demand functions that nests both the logit and CES demand is falsifiable.Consider a discrete/continuous demand function generated by h j,t (•) such that h j,t p j −h j,t p j = 1 a t p j + b t for some 1 > a t ≥ 0 and b t ≥ 0. Now, we can express the logit and CES demand function by setting a t = 0 and setting b t = 0, respectively.I call such a demand function a discrete/continuous demand function with HARA h since h is characterized as analogous to a hyperbolic absolute risk averse vNM utility function, which nests CARA and CRRA as special cases.First, I introduce a modified version of the necessary and sufficient condition for data to be rationalized by price competition under the modified specification.
Proposition 3 The set of observations { p, q} is Bertrand-rationalizable under convex and strictly increasing cost functions and discrete/continuous demand functions with HARA h if and only if, for any t ∈ T , f ∈ F, and j ∈ J f , there exist real numbers a t , b t , δ j,t , and m t such that the following hold: qk,t ; and 3. 0 ≤ δ j,t − δ j,t q j,t − q j,t .
Note that the modified model is falsifiable, i.e., the model can be rejected given certain data.By the second condition in the set of restrictions, the data must satisfy for all j, k ∈ J f and all t ∈ T .The following example shows how this applies.
Example 3 Discrete/Continuous Demand with HARA h Consider a case in which one firm produces products 1, 2, and 3 and generates the data { p, q} such that p j,t = p j,s ≡ p j and q j,t = q j,s ≡ q j for j = 1, 2 and for some t, s ∈ T , and p3,t < p3,s and q3,t > q3,s .
Then, the above equality is rewritten as follows: and Note that the equalities for goods 1 and 2 in Eqs. ( 10) and ( 11) imply that b t /a t = b s /a s ≡ b/a.Therefore, all the terms in Eqs. ( 10) and ( 11) must be the same.Thus, for good 3, must hold.This contradicts p3,t < p3,s and q3,t > q3,s .

Extensions
This section introduces the following extensions of the revealed preference tests: (i) additional assumptions regarding the demand functions introduced by Carvajal et al. ( 2014), (ii) observable cost shifters as discussed in Carvajal et al. ( 2014), (iii) collusive price competition, which can also work as an alternative hypothesis for the above tests, (iv) cost functions that are not additively separable for different products.

Additional restrictions on demand
Even though the above results provide some testable restrictions, the general model is not falsifiable.We can obtain a stricter constraint by combining the demand assumption introduced by Carvajal et al. (2014).
In order to define the additional restrictions, I first introduce some notations.I denote jt (p) : R J + → R as the relative decrease in the demand of good j at time t in response to an infinitesimal increase in its price.That is, given the demand function Q jt for good j at time t, Therefore, the own price elasticity is expressed as p j jt (p).
I then define the following properties of the demand functions.Definition: A system of demand functions satisfies the co-evolving property if, for any s and t ∈ T , either js (p) ≥ jt (p) for all p ∈ R J + and all j ∈ J , or js (p) ≤ jt (p) for all p ∈ R J + and all j ∈ J (13) The co-evolving demand property captures the idea of common demand shock in Carvajal et al. (2013), which is a key component to obtain non-trivial data restrictions in their work.As seen in the above equations, if the relative slope of demand is higher for firm j in market t, then so are the relative slopes for other firms k = j.That is, we can construct a well-defined order of demands over T , which is common to all firms according to the relative slopes.

Example 4 Co-evolving Property
The power of the co-evolving property is emphasized by two products produced by different firms.Consider the same prices and quantities as Example 2 (Within-firm restrictions), but the two goods are produced by different firms: p j,τ , q j,τ j=1,2, τ=s,t s.t.J 1 = {1}, J 2 = {2}, p1,s > p1,t , p2,s < p2,t , q1,s < q1,t , and q2,s > q2,t .12Since the two goods are produced by different firms, Eq. ( 7) is no longer satisfied.However, the co-evolving property gives us an alternative restriction even under the general discrete/continuous model (instead of the logit demand functions).If they are singleproduct firms, the first-order condition is re-written as follows: p j − C j Q j,t (p) = 1/ j,t (p).Since the marginal costs are increasing, we can obtain an inequality of the profit margins: pt ) for firm 1.Similarly, we also have 1/ 2,s ( ps ) < 1/ 2,t ( pt ).Therefore, the data imply 1,s ( ps ) < 1,t ( pt ) and 2,s ( ps ) > 2,t ( pt ).Assuming that j,s (•) is nondecreasing in the own price and decreasing in the other's price, we have 1,s (p) < 1,t (p) but 2,s (p) > 2,t (p), which contradicts the co-evolving property.In the following, I describe j,t (•) that is non-decreasing in the own price as log-concave, following Carvajal et al. (2014). 13efore stating the proposition, I demonstrate that the discrete/continuous model can incorporate the co-evolving property without any conflicts.For example, given multinomial logit demand, the co-evolving property is satisfied when v jt and v kt move almost in parallel over time.Since the logit demand functions require that jt (p) = α − α Q j,t (p) /M t , jt (p) ≤ js (p) holds if and only if Q j,t (p) /M t ≥ Q j,s (p) /M s holds.The co-evolving property under the logit demand functions require that Q j,t (p) /M t ≥ Q j,s (p) /M s if and only if Q k,t (p) /M t ≥ Q k,s (p) /M s .This can be satisfied when v jt and v kt move almost in parallel over time or a change in M t is dominant.Log-concavity (of Q j,t (p)) is also satisfied if −h j p j /h j p j is non-decreasing in p j .In the following proposition, I combine the discrete/continuous choice model and the co-evolving property to derive a set of necessary conditions for the data to be rationalized by price competition.

Proposition 4
The set of observations { p, q} is Bertrand rationalizable under convex and strictly increasing cost functions and discrete/continuous demand functions with log-concavity and co-evolving property only if there is a permutation of T , denoted by the function σ : T → T , and real numbers α j,t , δ j,t , and m t for any s, t ∈ T , f ∈ F, and j ∈ J f , such that the following hold: 1. α j,t > 0, δ j,t > 0, m t > 0; 2. 0 = m t − p j − δ j,t m t α j,t + k∈J f pk − δ k,t qk,t ; 3. 0 ≤ δ j,t − δ j,t q j,t − q j,t ; and 4. if p j,t ≤ p j,s , p− j,t ≥ p− j,s and σ (t) < σ (s), then α j,t − m −1 t q j,t ≤ α j,s − m −1 s q j,s See the "Appendix" for the proof.The last condition arises from the co-evolving property and log-concavity, which characterize the common order of jt (p) over time.Under the discrete/continuous demand model, The permutation, σ , is constructed to provide a common order for jt (p) (if one exists).Notably, the co-evolving property is defined by comparing jt (p) and js (p) for all p, but we only observe values corresponding to jt ( pt ) and js ( ps ), where pt and ps can take different values.To address this subtlety, the inequalities " p j,t ≤ p j,s , p− j,t ≥ p− j,s " are added to the last condition.14

Observed cost shock
One of the important assumptions in the above tests is the time-invariant cost functions.However, in reality, the cost functions shift over time, such as because of a change in input prices.We can accommodate such a shift in the revealed preference tests if cost shifters are observed.Now, assume the following cost functions: C j q j , w j where ∂C j q j , w j /∂q j is increasing both in q j and w j .Assume also that we observe the cost shifter w in addition to price and quantity.Denote the observed price, quantity, and cost shifter as follows: { p, q, w} where x = x 1 , . . ., x T and xt = x1,t , . . ., xJ,t for x = p, q, w.Then, the restriction is modified as follows.
Remark 1 The set of observations { p, q, w} is Bertrand-rationalizable under marginal cost functions increasing in own quantity and a cost shifter and discrete/continuous demand functions only if, for any t ∈ T , f ∈ F, and j ∈ J f , there exist real numbers α j,t , δ j,t , and m t such that the following hold: 1. α j,t > 0, δ j,t > 0, m t > 0; 2. 0 = m t − p j,t − δ j,t m t α j,t + k∈J f pk,t − δ k,t qk,t ; and 3. δ j,t ≥ δ j,t whenever q j,t , w j,t > q j,t , w j,t .
In the above claim, I use a partial order for the third condition since, if q j,t > q j,t and w j,t < w j,t , then we cannot tell which marginal cost is higher.Tests for price competition under logit, CES, and HARA h, can be also derived analogously.

Tests of collusion
We can also test a perfect collusion model by interpreting that all products are produced by a single large cartel.Therefore, we obtain the same result as in Sect. 2 with a different definition of a firm. 15To be more specific, I write an immediate corollary of Theorem 1 (characterization of the test) and Corollary 1 (unfalsifiability) for collusion under a general discrete/continuous demand model.In the following, I call { p, q} collusionrationalizable if, for any t ∈ T , { pt , qt } is generated as a result of profit maximization of a collusive group of firms that solves max for some demand and cost functions, Q j,t (•) and C j (•).

Corollary 2
The set of observations { p, q} is collusion-rationalizable under convex and strictly increasing cost functions and discrete/continuous demand functions if and only if there exist real numbers α j,t , δ j,t , and m t for any t ∈ T and j ∈ J such that the following hold: 1. α j,t > 0, δ j,t > 0, m t > 0; 2. 0 = m t − p j,t − δ j,t m t α j,t + f ∈F k∈J f pk,t − δ k,t qk,t ; and 3. 0 ≤ δ j,t − δ j,t q j,t − q j,t .Furthermore, any data, { p, q}, are collusion-rationalizable under convex and strictly increasing cost functions and discrete/continuous demand functions.
By restricting a class of demand functions to the logit, we can also obtain a falsifiable model, which is summarized as an immediate corollary of Proposition 1.
Corollary 3 (Logit) The set of observations { p, q} is collusion-rationalizable under convex and strictly increasing cost functions and logit demand functions if and only if there exist real numbers α t , δ j,t , and m t for all t ∈ T and j ∈ J such that the following hold: 1. α t > 0, δ j,t > 0, m t > 0; 2. 0 = m t − p j,t − δ j,t m t α t + f ∈F k∈J f pk,t − δ k,t qk,t ; and 3. 0 ≤ δ j,t − δ j,t q j,t − q j,t .
It is worth noting that a perfect collusion is strictly more restrictive than a price competition (and more restrictive than a collusive price competition discussed in the "Appendix") because the common mark-up property applies for all products in the market under the perfect collusion case but not for price competition.

Example 5 Price Competition v.s. Collusion
For example, suppose that we observe the following data on products 1 and 2, which are produced by firm 1 and 2, respectively: p j,τ , q j,τ j=1,2, τ=s,t s.t.p1, t = p1, s = p2, t = 1, p2, s = 0.95, q1, t = q1, s = 1, q2, t = 2, q2, s = 3.Then, such data can be rationalized by a Bertrand competition with convex and strictly increasing cost functions such that C 1 (1) = 0.6, C 2 (2) = 0.2, C 2 (3) = 0.3, and market sizes for each time M t = 4 and M s = 11, which correspond to m t = 0.12, m s = 4, α t = 100/3, and α s = 11/4. 16Note that the example here is not a special case of Example 1 in Sect.2.1 because quantities of product 2 are now always greater than the quantity of product 1, while an inequality q2, t > q1, t = q1, s > q2, s holds in Example 1.In contrast, the data cannot be rationalized by perfect collusion because the data contradict the common mark-up property between product 1 and 2, which should hold under a perfect collusion.

Non-separable cost functions
In this part, I further generalize the model to price competition with convex and strictly increasing cost functions that are not additively separable.Each firm f ∈ {1, . . ., F} produces a set of products At the same time, firm f ∈ {1, . . ., F} faces a cost function, C f : R J f + → R, which is assumed to be convex and continuously differentiable.Denote a vector of demand functions for firm . Then, the profit function for and the 16 Like Eq. ( 22), the first-order conditions can be reduced to which are easier to verify than the original first-order conditions.
The only difference in the first-order condition from that with additively separable cost functions is that 1)).Therefore, we have to find δ j t , which now corresponds to . The transformation of the first-order conditions using the discrete/continuous demand structure is applied as in Sect. 2. The convexity of the cost functions imply where q f ,t = q j,t j∈J f .Combining these conditions, we obtain a test for price competition with non-separable and convex cost functions.Theorem 2 (Non-separable cost functions) The set of observations { p, q} is Bertrandrationalizable under non-separable, convex, and strictly increasing cost functions and discrete/continuous demand functions if and only if, for any t ∈ T , f ∈ F, and j ∈ J f , there exist real numbers α j,t , δ j,t , and m t such that the following hold: 1. α j,t > 0, δ j,t > 0, m t > 0; 2. 0 = m t − p j,t − δ j,t m t α j,t + k∈J f pk,t − δ k,t qk,t ; and 3. 0 ≤ δ f ,t − δ f ,t q f ,t − q f ,t , where x f ,t = x j,t j∈J f for x = δ, q.
A proof for sufficiency of conditions 1, 2, and 3 is in the "Appendix".It is worth noting that we can replace the assumption of additively separable cost functions in Propositions 1-4 with non-separable cost functions by replacing the third condition as in Theorem 2.

Implementation
The existence of parameters satisfying the inequality constraints can be checked by minimizing a loss function over a set of parameters, given the data observed, and checking whether the minimized value is close to zero.For instance, for an inequality g (θ ; p, q) ≥ 0, we can construct a loss function (min {0, g (θ ; p, q)}) 2 .Similarly, for a vector of equality constraints h (θ ; p, q) = 0 and a vector of inequality constraints g (θ ; p, q) ≥ 0, we can construct a loss function h (θ ; p, q) T h (θ ; p, q) + g (θ ; p, q) T g (θ ; p, q) , where g (θ ; p, q) = min {0, g i (θ ; p, q)} i .In general, this minimization faces computational issues similar to those of estimation with moment inequalities. 17With logit demand and a slightly modified data requirement, the inequalities are written as linear constraints on parameters so that we can check the existence of parameters using off-the-shelf tools for linear constraints.In the following, I assume that the market size Mt t , prices p j, t , and quantities q j, t are observable, as is always the case when the logit demand functions can be estimated.The market shares of products at each time s j, t are also observable since s j, t = q j, t

Mt
. Then, considering that m = M α under logit demand, and with the replacement of 1 α = α, the data restrictions for price competition under the logit demand functions are characterized by a set of linear constraints on parameters α j, t , and δ j, t .
Corollary 4 (Logit) The set of observations p, q, M is Bertrand-rationalizable under convex and strictly increasing cost functions and logit demand functions if and only if, for any t ∈ T , f ∈ F, and j ∈ J f , there exist real numbers αt , δ j,t , and m t such that the following hold: 1. αt > 0, δ j,t > 0; 2. 0 = Mt αt − p j,t − δ j,t + k∈J f pk,t − δ k,t qk,t ; and 3. 0 ≤ δ j,t − δ j,t q j,t − q j,t .
Proof The proof is an immediate corollary of Proposition 1.Thus, we can use standard algorithms for linear constraints to check the constraint.

Summary
In this paper, I modify a Bertrand assumption test introduced by Carvajal et al. (2014) to allow it to be implemented for multi-product firms.To address difficulties caused by cannibalization effects, I employ the discrete/continuous demand functions introduced by Nocke and Schutz (2018), which includes the multinomial logit demand functions and the CES demand functions as special cases.In the main theorem, I provide the necessary and sufficient condition for data to be rationalized by Bertrand competition among multi-product firms under the discrete/continuous model.The test is implementable without any IVs, and rejection by it deterministically implies misspecification of the model rather than a peculiar realization of structural error terms.Under the general discrete/continuous model, any data would satisfy the necessary and sufficient condition to be rationalized by price competition, while some data are not rationalized by price competition under more restrictive demand specifications such as the logit demand functions, CES demand functions, or discrete/continuous demand functions with HARA h.I also discuss additional restrictions on the demand functions discussed in previous research, a test with observed cost shifters, a test for collusive price competition, and a simple implementation of the logit demand functions.The tests can also be applied for price competition, collusive price competition, and collusion with cost functions that are not additively separable.
Matsumura, Moritz Meyer-ter-Vehn, Susumu Sato, and seminar participants at UCLA and University of Tokyo.I also thank the editor, the accosiate editor, and the anonymous referee for valuable comments.All remaining errors are my own.

Declarations
Conflict of interest The author has no competing interests to declare that are relevant to the content of this article.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Appendix
Proof of Theorem 1 For sufficiency, we only need to construct cost and demand functions for each firm whose profit function is maximized at p j,t , q j,t .
First, consider a reconstruction of demand functions.If the data satisfy conditions 1 and 2 in Theorem 1, we should be able to find α j,t > 0 which corresponds to h j,t ( p j,t ) −h j,t ( p j,t ) for any j and t, where h j,t : R + → R represents the true data-generating process.For reconstruction of demand functions, I consider h j,t : R + → R s.t.
for any p j ∈ R + .Note that this condition holds for any p j ∈ R + , but not only for p j,t .Since the constant h j,t (pj ) h j,t (pj ) imply that h j,t p j can be represented as a CARA function with risk averseness α j,t , we can represent h j,t p j = exp v jt − α jt p j for some v jt .Then, we can construct a demand function, for some h0, t .Here, I denote the reconstructed demand functions as { Q j,t (p)} in order to distinguish it from the demand functions in the true data-generating process, {Q j,t (p)}.Now, v jt can be chosen to satisfy a system of |J | equations, for all j.Since δ jt , q jt satisfies condition 3 in Theorem 1, the co-monotone property, we can use monotone cubic interpolation to reconstruct the increasing and continu-ously differentiable C j (•).Then, we can reconstruct C j (q) = q 0 C j (x) dx, which is convex and twice continuously differentiable.Note that cost functions in the true data generating process {C j (q)} do not have to be twice continuously differentiable.Still, we construct twice continuously differentiable cost functions { C j (q)} because is useful for the following proof of the unique solution of the first order conditions.
The final step is to prove that ( p, q) is an equilibrium under the reconstructed demand and cost functions.By construction of Q j,t (p) and C j (q), the first-order conditions are satisfied at the observed prices, p.Therefore, we only need to show that a solution to the first-order conditions for each firm is unique given the other firms' strategies. 18 In the following, I omit the subscript for time t considering that the static NE is repeated and the following logic is applied for each t.Then, I denote the reconstructed demand function for good j as and h j p j = −α j exp v j − α j p j , h j p j = α 2 j exp v j − α j p j , and We now consider a maximization problem of a specific firm, f , with respect to Therefore, I omit p − f with slight abuse of notation for the demand function for good j ∈ J f : where H 0 = h0 + k / ∈J f hk ( p k ) and J f = {1, . . ., n} without loss of generality.By the FOC, we have the following for any j 18 To do so, I use the common ι-markup property examined in Nocke and Schutz (2018).The following part is closely related to the proofs in Nocke and Schutz (2018) (especially Lemma F), despite a few differences.First, we do not need to prove the existence of an equilibrium since we already have data as an equilibrium candidate.Therefore, we only need to show that those data are an equilibrium.Second, we consider a more general cost specification than Nocke and Schutz (2018).It complicates the inversion from a common ι-markup for firm f , μ f , to price vectors since marginal costs are not constants, but functions of product quantities.Third, the reconstructed demand functions are a special case of the demand functions in Nocke and Schutz (2018).Therefore, we can circumvent the difficulty arising from general cost functions by specifying the shape of the demand functions.
Since the RHS is same for any j ∈ J f , the solution of a system of equations defined by Eq. ( 14) for any j ∈ J f satisfies for some positive constant μ f , which corresponds to the RHS of Eq. ( 14).Let ν p f = ν 1 p f , . . ., ν n p f .Then, p f = ν −1 1μ f ≡ r μ f ≡ r 1 μ f , . . ., r n μ f at the solution of Eq. ( 14).Then, we can rewrite the condition Eq. ( 14) as Then, the uniqueness of the solution of the first-order condition is proved by the strict monotonicity of ψ μ f .Again, the existence of a solution can be omitted since the data satisfy the first-order condition by the construction of Q j (•) , C j (•) j∈J f .By taking a derivative w.r.t.μ f , Since μ f > 0 and m > 0, we only need to show that B is negative semi-definite.
To examine the derivatives of ν −1 (m), I first consider the derivative of ν.Recall that ν j p f ≡ p j − C j Q j ( p) α j .Then, the partial derivatives are Note that those are well-defined and continuous because C j (•) is constructed to be twice continuously differentiable.Now, the partial derivatives are summarized as follows: is negative definite because a discrete/continuous demand system is derived from the utility maximization of a representative consumer with a quasi-linear preference.(See Nocke and Schutz 2018 for details.) Therefore, B is negative definite, so that ψ μ f < 0.

Proof of Proposition 2 and 3
In this part, I show a reconstruction of demand functions and prove the uniqueness of the solution of the first-order condition.The notations in the following are for Proposition 3 (1 > a t > 0 and b t ≥ 0), but the same proof can be applied for Proposition 2 by setting b t = 0.When a t = 0, the proof is analogous to Theorem 1. Suppose that we have 1 > a t > 0 and b t ≥ 0.Then, we can reconstruct h j, t p j = v j, t p j + b t a t 1− 1 a t for some v j, t .Subsequently, the demand function for good j at time t is reconstructed as Note that there exist v j, t 's satisfying for any j ∈ J .It remains to prove the uniqueness of the solution to the first-order conditions.In the following, I omit time-index, t, for simplicity.Now, similar to Theorem 1, by the first-order condition, we have the following for any j ∈ J f : The RHS is rewritten as Then, Eq. ( 16) is decomposed into the following system of equations: where H 0 = k / ∈J f hk ( p k ).By Eq. ( 18), Then, by Eq. ( 17), Note that p j is uniquely determined by Eq. ( 17) given H f The solution is denoted as p j H f .Then, the aggregator in Eq. ( 19) is written as Now, we prove that Eq. ( 20) has a unique solution.It is sufficient to show that the derivative of the RHS of Eq. ( 20) is always less than 1, i.e., Observe that Then, the RHS of Eq. ( 20) becomes Thus, the observed prices of firm f 's products are the unique solution of the set of the firm f 's first-order conditions, given the other's prices.

Proof of Proposition 4
We need to derive the final condition in this proposition.By the co-evolving property, we can find a permutation such that σ (t) < σ (s) implies j,t ( p) ≤ j,s ( p) for all j ∈ J and for all p.If pi,t ≤ pi,s , p−i,t ≥ p−i,s and σ (t) < σ (s), then α j,t − m −1 t q j,t = j,t p jt , p− jt ≤ j,t p js , p− jt ≤ j,t p js , p− js ≤ j,s p js , p− js = α j,s − m −1 s q j,s .Thus, α j,t − m −1 t q j,t ≤ α j,s − m −1 s q j,s

Collusive price competition
This section discusses revealed preference tests of collusive price competition.Each firm is assumed to choose their own price while (partially) internalizing the effect on the other firms as in Miller and Weinberg (2017) and Sullivan (2016).More specifically, firm f at time t maximizes an objective function given the others' prices at time t, where φ f , f ∈ [0, 1] is firm f 's weight on firm f 's profit.The first-order condition w.r.t.p j is written as follows: This first-order condition is simplified using the derivatives of discrete/continuous demand functions and dividing both sides by the market share of product j.Then, the first-order condition gives the following data restriction: Then, I state a modified version of the revealed preference test as follows: Remark 2 The set of observations { p, q} is rationalized by collusive price competition under convex and strictly increasing cost functions and logit demand functions only if there exist real numbers α t , δ j,t , and m t , for all t ∈ T and j ∈ J and φ f , f for any f , f ∈ F such that the following hold: 1. α t > 0, δ j,t > 0, m t > 0, φ f , f ∈ [0, 1]; 2. 0 = m t − p j,t − δ j,t m t α t + f ∈F φ f , f k∈J f pk,t − δ k,t qk,t ; and 3. 0 ≤ δ j,t − δ j,t q j,t − q j,t .
In the above test, the demand functions are assumed to be logit demand functions since any data are rationalized by the general discrete/continuous model, and the additional parameter φ f , f would weakly loosen the restrictions.In contrast, under logit demand functions, the common markup property still holds, so the data shown in Example 2 are not rationalized by the collusive price competition.Now, a natural question is to what extent the additional parameter φ f , f loosens the data restrictions.In fact, it loosens the data restrictions less than might be expected.For example, consider the following variant of the above test.Suppose that there are two firms a and b, producing products a and b, respectively, and one attempts to test whether a set of observed data { p, q} is rationalized by collusive price competition under logit demand functions.If the firms are competing in prices, the data { p, q} must satisfy the following conditions for some α t , δ j,t , m t , and φ f , f : 0 = m t − pa,t − δ a,t m t α t + pa,t − δ a,t qa,t + φ a,b pb,t − δ b,t qb,t 0 = m t − pb,t − δ b,t m t α t + pb,t − δ b,t qb,t + φ b,a pa,t − δ a,t qa,t This implies the following condition: Note that for any m t α t , δ j,t , and φ f , f satisfying the above equation, we can determine the corresponding m t and α t from the original first-order equation, Eq. ( 21).Therefore, constraints characterized by Eq. ( 22) are equivalent to constraints characterized by Eq. ( 21).Now, focusing on symmetric φ f , f 's, i.e., φ a,b = φ b,a ≡ φ, as is often the case, data restrictions for price competition, φ = 0, and restrictions for collusive price competition, φ ∈ [0, 1), are equivalent.That is, if there exists a set of {α t } t , {m t } t , δ j,t j, t , φ for φ ∈ [0, 1) satisfying Eq. ( 22), we can also find { α t } t , { m t } t , δ j,t j, t combined with φ = 0 satisfying Eq. ( 22). and α t ≡ mα t / m t .Thus, the data can also be explained by the price competition (φ = 0).It is also worth noting that asymmetric φ f , f ∈ [0, 1] generates strictly loose conditions versus symmetric φ ∈ [0, 1].

Proof of Theorem 2
The only remaining part of the proof is sufficiency of the conditions, 1,2, and 3. First, I construct a preliminary cost function for firm f , C f q f = max t V f ,t + δ f ,t • q f for some V f , t such that C f q f , t = V f ,t + δ f ,t • q f ,t (i.e., ∇ C f q f , t = δ f ,t ).This cost function is convex, but not differentiable everywhere.We can construct a convex and differentiable cost function by smoothing this.For example, we can use a convolution-based smoothing technique: where μ (z) is a density function for a uniform distribution around 0 with radius > 0. For a sufficiently small > 0, we construct a convex and differentiable cost function while keeping ∇ C f q f , t = δ f ,t .
For the uniqueness of the equilibrium, most arguments in the proof hold by replacing C j Q j, t p f with ∂ C f Q f , t p f /∂q j .The remaining task is to prove that the derivative of ν j p f j is negative definite where The partial derivatives are where Note that p f is positive semi-definite because C f is convex.Now, the same algebra as Theorem 1 applies by re-interpreting p f in the proof of Theorem 1.

Table 1
Summary of results The construction of { α t } t and { m t } t is as follows.Suppose that {α t } t , {m t } t , δ j,t j, t , φ satisfies Eq. (22): Now, observe that { mα t } t , δ j,t j, t such that mα t = m t α Furthermore, we can construct the corresponding m t and α t as m t ≡ pa,t − δ a,t mα t − pa,t − δ a,t qa,t t / (1 − φ) satisfies pa,t − δ a,t pb,t − δ b,t = mα t − qb,t mα t − qa,t for any t.