Revealed Preference Tests of the Cournot Model

Transcription

1 Andres Carvajal, Rahul Deb, James Fenske, and John K.-H. Quah Department of Economics University of Toronto

2 Introduction Cournot oligopoly is a canonical noncooperative model of firm competition. In this model, the firms set quantities and the market determines price. Examples: crude oil, copper, electricity etc. In this paper, we design revealed preference tests for strategic behavior in a Cournot oligopoly. In other words, can we deduce that firms are best responding from minimal finite data?

3 Background: Afriat s Theorem Suppose we have a set T = {1, 2,..., T } of observations drawn from a single consumer. Each observation consists of a price vector p t = (p 1 t, p 2 t,..., p l t) and a consumption bundle x t = (x 1 t, x 2 t,..., x l t) chosen by the consumer at p t. When are the observations {(p t, x t )} t T consistent with a utility-maximizing consumer? Formally, we wish to test the hypothesis H: there exists an increasing function U : R l + R such that x t = argmax Bt U(x), where B t = { x R l + : p t x p t x t }.

4 Background: Afriat s Theorem Definition (GARP) Given arbitrary data set D = {(p t, x t )} T t=1. For any two consumption bundles x t and x t we say x t R 0 x t if p t x t p t x t. We say x t P x t if p t x t > p t x t. Finally we say x t R x t if for some sequence of observations (x t1, x t2,..., x tm ) we have x t R 0 x t1, x t1 R 0 x t2,..., x tm R 0 x t. In other words relation R is the transitive closure of R0. The data D satisfies GARP if x t R x t = x t P x t x t, x t

5 Background: Afriat s Theorem Definition (GARP) Given arbitrary data set D = {(p t, x t )} T t=1. For any two consumption bundles x t and x t we say x t R 0 x t if p t x t p t x t. We say x t P x t if p t x t > p t x t. Finally we say x t R x t if for some sequence of observations (x t1, x t2,..., x tm ) we have x t R 0 x t1, x t1 R 0 x t2,..., x tm R 0 x t. In other words relation R is the transitive closure of R0. The data D satisfies GARP if x t R x t = x t P x t x t, x t Afriat s Theorem: The set of observations {(p t, x t )} t T is consistent with H if and only if it obeys the general axiom of revealed preference (GARP).

6 Background: Afriat s Theorem Afriat s Theorem: The set of observations {(p t, x t )} t T is consistent with H if and only if it obeys the general axiom of revealed preference (GARP). Various generalizations of Afriat s result (by Varian and many other authors) and also applications to data.

7 Background: Afriat s Theorem Afriat s Theorem: The set of observations {(p t, x t )} t T is consistent with H if and only if it obeys the general axiom of revealed preference (GARP). Various generalizations of Afriat s result (by Varian and many other authors) and also applications to data. Example of GARP violation:

8 Revealed Preference in the Cournot Model In practice, it is hard to observe the firm s costs and it is hard to estimate market demand. Suppose we have a set T = {1, 2,..., T } of observations drawn from an industry producing a homogeneous good. Each observation consists of the market price p t and the output vector (q i,t ) i I, where I is the set of firms and q i,t is the output of firm i in observation t.

9 Revealed Preference in the Cournot Model In practice, it is hard to observe the firm s costs and it is hard to estimate market demand. Suppose we have a set T = {1, 2,..., T } of observations drawn from an industry producing a homogeneous good. Each observation consists of the market price p t and the output vector (q i,t ) i I, where I is the set of firms and q i,t is the output of firm i in observation t. Suppose that firms cost functions are unchanged across observations and the observations are generated by changes to the market demand function.

10 Revealed Preference in the Cournot Model In practice, it is hard to observe the firm s costs and it is hard to estimate market demand. Suppose we have a set T = {1, 2,..., T } of observations drawn from an industry producing a homogeneous good. Each observation consists of the market price p t and the output vector (q i,t ) i I, where I is the set of firms and q i,t is the output of firm i in observation t. Suppose that firms cost functions are unchanged across observations and the observations are generated by changes to the market demand function. In this case, what restrictions on the data would we expect, if any?

11 Revealed Preference in the Cournot Model: Hypothesis In this paper, we test the following hypothesis: H: Given observed data {[p t, (q i,t ) i I ]} t T, there exist cost functions C i for each firm i, and downward sloping demand functions P t for each observation t such that (i) P t (Q t ) = p t ; and { (ii) q i,t argmax qi 0 q i P t (q i + } j i q j,t) C i (q i ) ; where q i,t is the output of firm i and Q t = i I q i,t is the total output.

12 Revealed Preference in the Cournot Model: Hypothesis In this paper, we test the following hypothesis: H: Given observed data {[p t, (q i,t ) i I ]} t T, there exist cost functions C i for each firm i, and downward sloping demand functions P t for each observation t such that (i) P t (Q t ) = p t ; and { (ii) q i,t argmax qi 0 q i P t (q i + } j i q j,t) C i (q i ) ; where q i,t is the output of firm i and Q t = i I q i,t is the total output. Condition (i) says that the inverse demand functions agree with the observed price and industry output at each observation. Condition (ii) says that, at each observation t, firm i s observed output level q i,t maximizes its profit given the output of the other firms.

13 Revealed Preference in the Cournot Model: Hypothesis In this paper, we test the following hypothesis: H: Given observed data {[p t, (q i,t ) i I ]} t T, there exist cost functions C i for each firm i, and downward sloping demand functions P t for each observation t such that (i) P t (Q t ) = p t ; and { (ii) q i,t argmax qi 0 q i P t (q i + } j i q j,t) C i (q i ) ; where q i,t is the output of firm i and Q t = i I q i,t is the total output. Condition (i) says that the inverse demand functions agree with the observed price and industry output at each observation. Condition (ii) says that, at each observation t, firm i s observed output level q i,t maximizes its profit given the output of the other firms. If the hypothesis is true then we say the data is rationalizable.

14 First Order Condition from Profit Max Suppose that at observation t, the market inverse demand function is P t. Then the first order condition for profit maximization for firm i is C i (q i,t ) = P t (Q t ) + q i,t P t(q t ).

15 First Order Condition from Profit Max Suppose that at observation t, the market inverse demand function is P t. Then the first order condition for profit maximization for firm i is Re-arranging, we obtain C i (q i,t ) = P t (Q t ) + q i,t P t(q t ). P t(q t ) = P t (Q t ) C 1 (q 1,t) = = P t (Q t ) C I (q I,t). q 1,t q I,t This implies that if q i,t > q j,t then C i (q i,t) < C j (q j,t). In other words, a firm with the larger share has the lower marginal cost.

16 First Order Condition from Profit Max Suppose that at observation t, the market inverse demand function is P t. Then the first order condition for profit maximization for firm i is Re-arranging, we obtain C i (q i,t ) = P t (Q t ) + q i,t P t(q t ). P t(q t ) = P t (Q t ) C 1 (q 1,t) = = P t (Q t ) C I (q I,t). q 1,t q I,t This implies that if q i,t > q j,t then C i (q i,t) < C j (q j,t). In other words, a firm with the larger share has the lower marginal cost. Conclusion: if every firm has constant marginal costs (i.e., constant with respect to its output), then their rank cannot change across observations.

17 Testable Restrictions: Increasing Marginal Costs A similar observable restriction holds when firms have increasing marginal costs. Suppose at observation t, firm i produces 20 and firm j produces 15. At another observation t, firm i produces 15 and firm j produces 16. This is not rationalizable with a Cournot model with increasing marginal costs.

18 Testable Restrictions: Increasing Marginal Costs A similar observable restriction holds when firms have increasing marginal costs. Suppose at observation t, firm i produces 20 and firm j produces 15. At another observation t, firm i produces 15 and firm j produces 16. This is not rationalizable with a Cournot model with increasing marginal costs. Proof: Observation t tells us that C i (20) < C j (15). If firm i and j both have increasing marginal costs then C i (15) C i (20) < C j (15) C j (16). But observation t tells us that C i (15) > C j (16). QED

19 Testable Restrictions: Increasing Marginal Costs A similar observable restriction holds when firms have increasing marginal costs. Suppose at observation t, firm i produces 20 and firm j produces 15. At another observation t, firm i produces 15 and firm j produces 16. This is not rationalizable with a Cournot model with increasing marginal costs. Proof: Observation t tells us that C i (20) < C j (15). If firm i and j both have increasing marginal costs then C i (15) C i (20) < C j (15) C j (16). But observation t tells us that C i (15) > C j (16). Note: the restriction does not even rely on price information! QED

20 Testable Restrictions: Increasing Marginal Costs Definition: A set of observations is {[p t, (q i,t ) i I ]} t T is rationalizable with a Cournot model with constant (increasing) marginal costs if there are linear (convex) cost functions C i : R + R for each firm i I and downward sloping inverse demand functions P t : R + R for each t T, such that (i) P t (Q t ) = P t ; and (ii) argmax qi 0[ q i P t ( q i + j i q j,t) C i ( q i )] = q i,t. Identical to our original hypothesis except in the above definition, the cost functions must satisfy additional conditions.

21 Linear Costs Recall that P t(q t ) = P t (Q t ) C 1 (q 1,t) = = P t (Q t ) C I (q I,t). q 1,t q I,t Clearly, if {[p t, (q i,t ) i I ]} t T is compatible with a Cournot model with constant marginal costs then there must be {λ i } i I 0 such that, 0 < p t λ 1 q 1,t = p t λ 2 q 2,t =... = p t λ I q I,t for all t T. (1)

22 Linear Costs Recall that P t(q t ) = P t (Q t ) C 1 (q 1,t) = = P t (Q t ) C I (q I,t). q 1,t q I,t Clearly, if {[p t, (q i,t ) i I ]} t T is compatible with a Cournot model with constant marginal costs then there must be {λ i } i I 0 such that, 0 < p t λ 1 q 1,t = p t λ 2 q 2,t =... = p t λ I q I,t for all t T. (1) Theorem: A set of observations is {[p t, (q i,t ) i I ]} t T is rationalizable with a Cournot model with constant marginal costs if and only if there exists {λ i } i I 0 such that (1) is satisfied.

23 Convex Costs P t(q t ) = P t (Q t ) C 1 (q 1,t) = = P t (Q t ) C I (q I,t). q 1,t q I,t If {[p t, (q i,t ) i I ]} t T is compatible with a Cournot model with increasing marginal costs then the following must be satisfied: A there exists {λ i,t } (i,t) I T such that, 0 < p t λ 1,t q 1,t = p t λ 2,t q 2,t =... = p t λ I,t q I,t for all t T (we refer to this as the common ratio condition) and B for each firm i, the coefficients {λ i,t } t T are co-monotonic with its output, i.e., λ i,t λ i,t if q i,t > q i,t.

24 Convex Costs A there exists {λ i,t } (i,t) I T such that, 0 < p t λ 1,t q 1,t = p t λ 2,t q 2,t =... = p t λ I,t q I,t for all t T B for each firm i, the coefficients {λ i,t } t T are co-monotonic with its output, i.e., λ i,t λ i,t if q i,t > q i,t. Theorem: A set of observations is { [p t, (q i,t )] i I is rationalizable }t T with a Cournot model with increasing marginal costs if and only if conditions [A] and [B] are satisfied.

25 Convex Costs: Proof Proof of sufficiency: Condition [A] says there exists {λ i,t } (i,t) I T such that, 0 < pt λ1,t q 1,t = pt λ2,t q 2,t =... = pt λ I,t q I,t for all t T. (2) Construct a cost function for firm i such that C i (q i,t ) = λ i,t. Because of condition [B], C i can be chosen to have increasing marginal cost.

26 Convex Costs: Proof Proof of sufficiency: Condition [A] says there exists {λ i,t } (i,t) I T such that, 0 < pt λ1,t q 1,t = pt λ2,t q 2,t =... = pt λ I,t q I,t for all t T. (2) Construct a cost function for firm i such that C i (q i,t ) = λ i,t. Because of condition [B], C i can be chosen to have increasing marginal cost. For each t, let the demand function be P t(q) = a t b tq, where b t = (pt λ i,t ) q i,t and choose a t to solve a t b tq t = P t, so P t is compatible with the observation at t.

27 Convex Costs: Proof Proof of sufficiency: Condition [A] says there exists {λ i,t } (i,t) I T such that, 0 < pt λ1,t q 1,t = pt λ2,t q 2,t =... = pt λ I,t q I,t for all t T. (2) Construct a cost function for firm i such that C i (q i,t ) = λ i,t. Because of condition [B], C i can be chosen to have increasing marginal cost. For each t, let the demand function be P t(q) = a t b tq, where b t = (pt λ i,t ) q i,t and choose a t to solve a t b tq t = P t, so P t is compatible with the observation at t. Assume that for every j i, firm j s output is q j,t. The firm i s chooses q i to maximize q i P t( q i + j i q j,t) C i ( q i ). The first order condition for this problem is [ q i b t + a t b t( q i + ] j i q j,t) C i ( q i ) = 0. This is satisfied at q i = q i,t because q i,t b t + p t λ i,t = 0.

28 Convex Costs: Proof Proof of sufficiency: Condition [A] says there exists {λ i,t } (i,t) I T such that, 0 < pt λ1,t q 1,t = pt λ2,t q 2,t =... = pt λ I,t q I,t for all t T. (2) Construct a cost function for firm i such that C i (q i,t ) = λ i,t. Because of condition [B], C i can be chosen to have increasing marginal cost. For each t, let the demand function be P t(q) = a t b tq, where b t = (pt λ i,t ) q i,t and choose a t to solve a t b tq t = P t, so P t is compatible with the observation at t. Assume that for every j i, firm j s output is q j,t. The firm i s chooses q i to maximize q i P t( q i + j i q j,t) C i ( q i ). The first order condition for this problem is [ q i b t + a t b t( q i + ] j i q j,t) C i ( q i ) = 0. This is satisfied at q i = q i,t because q i,t b t + p t λ i,t = 0. Since the firm s objective is a concave function (of q i ), the first order condition is also sufficient to guarantee that q i = q i,t is optimal for firm i. QED

29 Shifting Costs The assumption of cost functions being constant over time is quite restrictive. Assume that, we also observes some parameter α i that has an impact on firm i s cost function. α i is partially ordered, for example, it can belong to a subset of the Euclidean space. For example, α i can represent a vector of input prices for example. Higher values of α i are associated with higher marginal costs. Formally, if ᾱ i ˆα i, then C i (q i; ᾱ i ) C i (q i; ˆα i ) for all q i > 0.

30 Shifting Costs In this context, a set of observations takes the form {[p t, (q i,t ) i I, (a i,t ) i I ]} t T, where a i,t is the observed value of α i at observation t. We say that this data set is Cournot rationalizable with shifting convex cost functions respecting {a i,t } (i,t) I T if there exists convex cost functions C i ( ; a i,t ) (for each firm i at observation t), and downward sloping demand functions P t for each observation t such that (i) P t (Q t ) = p t ; { (ii) q i,t argmax qi 0 q i Pt (q i + j i q j,t) C } i (q i ; a i,t ) ; and (iii) C i ( ; a i,t) C i ( ; a i, t) if a i,t > a i, t and C i ( ; a i,t) = C i ( ; a i, t) if a i,t = a i, t.

31 Shifting Costs Corollary: A set of observations {[p t, (q i,t ) i I, (a i,t ) i I ]} t T is Cournot rationalizable with shifting convex cost functions respecting {a i,t } (i,t) I T if there exists a set of positive scalars {λ i,t } (i,t) I T satisfying the common ratio property, with λ i,t (=) λ i,t whenever q i,t (=) q i,t and a i,t (=) a i,t.

32 The Multi-product Cournot Oligopoly Suppose there are N products (forming the set N ) in this industry. The price of good k depends on the total output vector Q R N +, so we write it as P k (Q). The inverse demand system P = (P k ) k N obeys the law of demand if the derivative matrix is negative definite for all Q. This is equivalent to [ ] P k Q l (Q) (k,l) N N (Q Q ) (P(Q) P(Q )) < 0 for distinct Q and Q.

33 The Multi-product Cournot Oligopoly: Demand The law of demand is the multi-product extension of downward sloping demand curves. Micro-foundations of this property are well established. The different goods in this setting can be thought of as distinct goods or distinct markets. The latter would be an example of third-degree price discrimination.

34 The Multi-product Cournot Oligopoly: Costs Firm i s cost depends on the output vector it chooses to produce; formally, we denote the cost of producing q i R N + by C i (q i ). Cost functions are not restricted to be additive, hence, this allows for synergies in production. Observation t consists of the price vector p t = (p k t ) k N and output vectors q t i R N +, for each firm i. What conditions are necessary for { [p t, (q i,t )] i I }t T rationalizable? to be Cournot

35 The Multi-product Cournot Oligopoly Theorem: A set of observations is { [p t, (q i,t )] i I is rationalizable }t T with a Cournot model with convex cost functions for every firm and inverse demand functions P t k obeying the law of demand if and only if there exists a solution to a system of polynomial inequalities with parameters derived from the data.

36 The Multi-product Cournot Oligopoly Theorem: A set of observations is { [p t, (q i,t )] i I is rationalizable }t T with a Cournot model with convex cost functions for every firm and inverse demand functions P t k obeying the law of demand if and only if there exists a solution to a system of polynomial inequalities with parameters derived from the data. In fact, like the single good case, the inverse demand function for each good can be chosen to be linear in the output vector Q whenever a rationalization exists. Simple examples of data can be constructed which cannot be rationalized.

37 Collusion Let { [p t, (q i,t )] i I }t T be a set of observations. We would like to see if these observations are consistent with the set of firms acting as a single profit maximizing entity.

38 Collusion Let { [p t, (q i,t )] i I }t T be a set of observations. We would like to see if these observations are consistent with the set of firms acting as a single profit maximizing entity. Choose a number m such that 0 < m < p t for all t. Then it is clear that there are inverse demand functions P t, with P t (Q t ) = p t such that Q t = argmax { q P t ( q) m q }. q>0 The above profit maximization problem corresponds to collusion when firms have identical linear cost functions with marginal cost m.

39 Collusion Let { [p t, (q i,t )] i I }t T be a set of observations. We would like to see if these observations are consistent with the set of firms acting as a single profit maximizing entity. Choose a number m such that 0 < m < p t for all t. Then it is clear that there are inverse demand functions P t, with P t (Q t ) = p t such that Q t = argmax { q P t ( q) m q }. q>0 The above profit maximization problem corresponds to collusion when firms have identical linear cost functions with marginal cost m. In other words, any set of observations is consistent with collusion. So while Cournot interaction is refutable, collusion is not.

40 Conjectural Variations Bowley (1924), Iwata (1974, Ecta), Bresnahan (1981, AER). Firm i conjectures that dq i dq i = θ i 1. In other words, Firm i conjectures that, if it increases quantity by a single unit the other firms will decrease production by 1 θ i units. Therefore, if Firm i decides to increase production by one unit the total market quantity supplied will increase by θ i units.

41 Conjectural Variations Definition: Suppose the inverse demand function is P t. Then (q i,t ) i I constitutes a θ-cv equilibrium (with θ = {θ i } i I 0) if argmax [ qi P t (θ i ( q i q i,t ) + Q t ) C i ( q i ) ] = q i,t. q i 0

42 Conjectural Variations Definition: Suppose the inverse demand function is P t. Then (q i,t ) i I constitutes a θ-cv equilibrium (with θ = {θ i } i I 0) if argmax [ qi P t (θ i ( q i q i,t ) + Q t ) C i ( q i ) ] = q i,t. q i 0 If firms in the industry are price-taking, then θ = (0, 0,..., 0). A Cournot equilibrium is a θ-cv equilibrium with θ = (1, 1,..., 1). Collusion corresponds to the case where θ > (1, 1,..., 1).

43 Conjectural Variations Definition: A set of observations {[P t, (q i,t ) i I ]} t T is θ-cv rationalizable (with θ = {θ i } i I 0) if we can find a downward sloping demand function, P t, for each observation t, and cost functions, C i, for each firm i, such that (i) P t ( i I q i,t) = P t and (ii) (q i,t ) i I obeys argmax [ qi Pt (θ i ( q i q i,t ) + Q t ) C i ( q i ) ] = q i,t. q i 0

44 Conjectural Variations Definition: A set of observations {[P t, (q i,t ) i I ]} t T is θ-cv rationalizable (with θ = {θ i } i I 0) if we can find a downward sloping demand function, P t, for each observation t, and cost functions, C i, for each firm i, such that (i) P t ( i I q i,t) = P t and (ii) (q i,t ) i I obeys argmax [ qi Pt (θ i ( q i q i,t ) + Q t ) C i ( q i ) ] = q i,t. q i 0 The first order condition for firm i, at the equilibrium is So we obtain P t (Q t ) + θ i q i,t P t(q t ) C i (q i,t ) = 0. P t(q t ) = P t C 1 (q 1,t) = = P t C I (q I,t). θ 1 q 1,t θ I q I,t

45 Conjectural Variations Theorem: A set of observations is { [p t, (q i,t )] i I }t T is {θ i} i I -CV rationalizable with increasing marginal costs if and only if the following hold: A there exists {λ i,t } (i,t) I T such that 0 < p t λ 1,t θ 1 q 1,t = p t λ 2,t θ 2 q 2,t =... = p t λ I,t θ I q I,t for all t T B for each firm i, the coefficients {λ i,t } t T obey: λ i,t λ i,t if q i,t > q i,t.

46 Conjectural Variations Theorem: A set of observations is { [p t, (q i,t )] i I }t T is {θ i} i I -CV rationalizable with increasing marginal costs if and only if the following hold: A there exists {λ i,t } (i,t) I T such that 0 < p t λ 1,t θ 1 q 1,t = p t λ 2,t θ 2 q 2,t =... = p t λ I,t θ I q I,t for all t T B for each firm i, the coefficients {λ i,t } t T obey: λ i,t λ i,t if q i,t > q i,t. Observation: If the observed data is {θ i } i I -CV rationalizable then it is also {µθ i } i I -CV rationalizable for any positive constant µ > 0.

47 Conjectural Variations Corollary: A set of observations is Cournot rationalizable if and only if it is ( θ, θ,.., θ)-cv rationalizable for any positive scalar θ. In this sense, a rejection of the Cournot model in this context is very strong because it is equivalent to a rejection of every symmetric-cv model.

48 Conjectural Variations Corollary: A set of observations is Cournot rationalizable if and only if it is ( θ, θ,.., θ)-cv rationalizable for any positive scalar θ. In this sense, a rejection of the Cournot model in this context is very strong because it is equivalent to a rejection of every symmetric-cv model. However, relative market power as measured by θ i is testable. Potentially, a set of observations could be consistent with, say, θ = (1; 1) but not with θ = (1; 2).

49 Non-convex Costs Consider once again, the standard, single product Cournot setting. Proposition: Any set of observations { [p t, (q i,t )] i I is rationalizable }t T with a Cournot model (with not necessarily increasing marginal costs). Without any restriction on the cost curves, any data set can be rationalized. In the paper, we suggest settings which are more general than convex costs but do not allow ill-behaved cost functions.

50 Conclusion Revealed preference tests are attractive due to their minimal assumptions. We provide test for a variety of models within the Cournot setting. In addition, we show that our test has real bite on actual data. Revealed preference tests in game theoretic settings are often hard to derive. Other than this paper, some notable exceptions are Sprumont (2000, JET), Ray and Zhou (2001, GEB) and Deb (2009, JET).