Monopolistic Competition when Income Matters

Transcription

1 Monopolistic Competition when Income Matters Paolo Bertoletti and Federico tro University of Pavia and Ca Foscari University, Venice Hitotsubashi University, March 6, 2014

2 Purpose We propose an alternative microfoundation to models of imperfect competition and product differentiation with and without endogenous entry

3 Purpose We propose an alternative microfoundation to models of imperfect competition and product differentiation with and without endogenous entry Alternative to the Dixit-Stiglitz (1977) microfoundation.

4 Purpose We propose an alternative microfoundation to models of imperfect competition and product differentiation with and without endogenous entry Alternative to the Dixit-Stiglitz (1977) microfoundation. Separable utility à la Dixit-Stiglitz is widely applied in trade (Krugman, 1980; Melitz, 2003) and macroeconomics (New-Keynesian models, ndogenous entry models) under CS preferences.

5 Purpose We propose an alternative microfoundation to models of imperfect competition and product differentiation with and without endogenous entry Alternative to the Dixit-Stiglitz (1977) microfoundation. Separable utility à la Dixit-Stiglitz is widely applied in trade (Krugman, 1980; Melitz, 2003) and macroeconomics (New-Keynesian models, ndogenous entry models) under CS preferences. It is also useful to study Cournot competition with product differentiation.

6 Background D-S assume additively separable preferences ( direct additivity ): U = n j=1 u (x j )

7 Background D-S assume additively separable preferences ( direct additivity ): U = n j=1 u (x j ) Separability is restrictive, because there are many other symmetric but non-separable preferences U = U (x)

8 Background D-S assume additively separable preferences ( direct additivity ): U = n j=1 u (x j ) Separability is restrictive, because there are many other symmetric but non-separable preferences U = U (x) Direct additivity implies that the Marginal Rate of Substitution u (x i ) /u (x j ) between any two varieties does not depend on the consumption of other varieties x k.

9 Background D-S assume additively separable preferences ( direct additivity ): U = n j=1 u (x j ) Separability is restrictive, because there are many other symmetric but non-separable preferences U = U (x) Direct additivity implies that the Marginal Rate of Substitution u (x i ) /u (x j ) between any two varieties does not depend on the consumption of other varieties x k. Most applications focus on CS preferences with θ (1, ): U = n j=1 x θ 1 θ j

10 Monopolistic competition à la Dixit-Stiglitz quilibrium in the CS case: p = θc θ 1, n = L θf, (θ 1) F q = c where p = price, n = number of firms and q = xl = firm production, = income, L = population, c = marginal cost and F = fixed cost. A double marke size (number of consumers) generates double number of goods, with same price (and quantity per firm), but only with CS. Income does not affect price and quantity, not just with CS

11 Monopolistic competition à la Dixit-Stiglitz quilibrium in the CS case: p = θc θ 1, n = L θf, (θ 1) F q = c where p = price, n = number of firms and q = xl = firm production, = income, L = population, c = marginal cost and F = fixed cost. A double marke size (number of consumers) generates double number of goods, with same price (and quantity per firm), but only with CS. Income does not affect price and quantity, not just with CS General case (Zhelobodko et al., 2012,.; Bertoletti-pifani, 2012; Bertoletti-tro, 2014, con. Bulletin): p = θ(x)c θ(x) 1, n = L θ(x)f, where θ(x) = u (x)/xu (x). (θ(x) 1) F q = c

12 Monopolistic competition à la Dixit-Stiglitz quilibrium in the CS case: p = θc θ 1, n = L θf, (θ 1) F q = c where p = price, n = number of firms and q = xl = firm production, = income, L = population, c = marginal cost and F = fixed cost. A double marke size (number of consumers) generates double number of goods, with same price (and quantity per firm), but only with CS. Income does not affect price and quantity, not just with CS General case (Zhelobodko et al., 2012,.; Bertoletti-pifani, 2012; Bertoletti-tro, 2014, con. Bulletin): p = θ(x)c θ(x) 1, n = L θ(x)f, (θ(x) 1) F q = c where θ(x) = u (x)/xu (x). With Cournot or Bertrand competition and CS, an additional competition effect for trade (tro, 2013, Scand.J..) and RBC (tro-colciago, 2010, con.journ.)

13 The Model We consider a different microfoundation, based on different preferences

14 The Model We consider a different microfoundation, based on different preferences We look at the indirect utility (dual approach)

15 The Model We consider a different microfoundation, based on different preferences We look at the indirect utility (dual approach) and assume additively separable indirect utility ( indirect additivity ):

16 The Model We consider a different microfoundation, based on different preferences We look at the indirect utility (dual approach) and assume additively separable indirect utility ( indirect additivity ): V = n ( pj ) v j=1 with v > 0, v < 0 and v > 0 and some regularity conditions. is income of the consumer.

17 The Model We consider a different microfoundation, based on different preferences We look at the indirect utility (dual approach) and assume additively separable indirect utility ( indirect additivity ): V = n ( pj ) v j=1 with v > 0, v < 0 and v > 0 and some regularity conditions. is income of the consumer. By Hicks (1969) and Samuelson (1969) we know that direct additivity and indirect additivity represent two distinct classes of well-behaved preferences with only one case in common: CS.

18 Direct demand function Indirect additivity ( n j=1 = n 0 V = v if you like): ( pj )

19 Direct demand function Indirect additivity ( n j=1 = n 0 V = v if you like): ( pj ) The Roy identity generates the direct demand function of each consumer: x i = V / p i V / = v ( pi ) v ( p j ) pj

20 Direct demand function Indirect additivity ( n j=1 = n 0 V = v if you like): ( pj ) The Roy identity generates the direct demand function of each consumer: x i = V / p i V / = v ( pi ) v ( p j ) pj Indirect additivity implies that the relative demand of two varieties x i /x j does not depend on the price of other varieties p k.

21 Direct demand function Indirect additivity ( n j=1 = n 0 V = v if you like): ( pj ) The Roy identity generates the direct demand function of each consumer: x i = V / p i V / = v ( pi ) v ( p j ) pj Indirect additivity implies that the relative demand of two varieties x i /x j does not depend on the price of other varieties p k. The denominator µ = v ( p j monopolistic competition. ) pj < 0 is taken as given in

22 Direct demand function Indirect additivity ( n j=1 = n 0 V = v if you like): ( pj ) The Roy identity generates the direct demand function of each consumer: x i = V / p i V / = v ( pi ) v ( p j ) pj Indirect additivity implies that the relative demand of two varieties x i /x j does not depend on the price of other varieties p k. The denominator µ = v ( p j monopolistic competition. ) pj < 0 is taken as given in Market demand is given by q i = x i L where L is number of consumers (marke size)

23 Some examples of direct demands CS: v(p) = p 1 θ delivers: q i = p θ i L p 1 θ j

24 Some examples of direct demands CS: v(p) = p 1 θ delivers: q i = p θ i L pj 1 θ XPONNTIAL: v(p) = e τp delivers log-linear demand: q i = τpi e L e τp j p j (notice the difference from the Logit, which has no income effects)

25 Some examples of direct demands ADDILOG: v(p) = (a p) 1+γ delivers the linear perceived demand when γ = 1: q i = ( a p i ) L ( a p ) j pj

26 Some examples of direct demands ADDILOG: v(p) = (a p) 1+γ delivers the linear perceived demand when γ = 1: q i = ( a p i ) L ( a p ) j pj DISPLACD CS: v(p) = (p + b) 1 ϑ delivers: q i = ( pi + b) ϑ L ( p i + b) ϑ p j

27 Monopolistic Competition (Dual) Monopolistic competition implies that there are so many firms that the impact of each price on the marginal utility of income is negligible: µ is taken as given

28 Monopolistic Competition (Dual) Monopolistic competition implies that there are so many firms that the impact of each price on the marginal utility of income is negligible: µ is taken as given Profit can be written as: π i = (p ( i c)v pi ) L F µ where c > 0 and F > 0 are respectively marginal and fixed costs.

29 Monopolistic Competition (Dual) Monopolistic competition implies that there are so many firms that the impact of each price on the marginal utility of income is negligible: µ is taken as given Profit can be written as: π i = (p ( i c)v pi ) L F µ where c > 0 and F > 0 are respectively marginal and fixed costs. The demand elasticity is θ(p i / ) v p i v p i /, not on µ and L. > 0: it depends on

30 Pricing The FOC is: p e c p e = 1 θ ( p ) e

31 Pricing The FOC is: p e c p e = 1 θ ( p e The optimal price is always independent from the number of varieties. )

32 Pricing The FOC is: p e c p e = 1 θ ( p e The optimal price is always independent from the number of varieties. However, if θ > (<) 0, the optimal price grows (decreases) with income because firms face a more (less) rigid demand. )

33 Pricing The FOC is: p e c p e = 1 θ ( p e The optimal price is always independent from the number of varieties. However, if θ > (<) 0, the optimal price grows (decreases) with income because firms face a more (less) rigid demand. Rationale for procyclical markups in macro, for pricing to market in trade )

34 ndogenous ntry quilibrium Dual: p e c p e = 1 θ ( p e ), n e = L F θ ( p e θ ), q e = F ( p e c ) 1

35 ndogenous ntry quilibrium Dual: p e c p e = 1 θ ( p e Notice that ), n e = L F θ ( p e θ ), q e = F ɛ pl = ɛ ql = 0 and ɛ nl = 1 ( p e c ) 1 which generalizes the classical result by Krugman (1980) concerning market size with CS preferences: pure gains from variety without any competitive effect on prices and firm size.

36 ndogenous ntry quilibrium Dual: p e c p e = 1 θ ( p e Notice that ), n e = L F θ ( p e θ ), q e = F ɛ pl = ɛ ql = 0 and ɛ nl = 1 ( p e c ) 1 which generalizes the classical result by Krugman (1980) concerning market size with CS preferences: pure gains from variety without any competitive effect on prices and firm size. Other results (pricing to market and undershifting): ɛ p 0 and ɛ n 1 iff θ (p e / ) 0 ɛ pc 1 and ɛ nc 0 iff θ (p e / ) 0

37 Two new examples with closed form solutions The (negative) exponential demand q i = e τp i L/µ delivers: p e = c + τ, ne = 2 L F (cτ + ), qe = F τ

38 Two new examples with closed form solutions The (negative) exponential demand q i = e τp i L/µ delivers: p e = c + τ, ne = 2 L F (cτ + ), qe = F τ The linear demand case q i = ( a p ) i L/µ delivers: p e = c + a 2, n e = (a c) L F (a + c), qe = 2F a c

39 Two new examples with closed form solutions The (negative) exponential demand q i = e τp i L/µ delivers: p e = c + τ, ne = 2 L F (cτ + ), qe = F τ The linear demand case q i = ( a p ) i L/µ delivers: p e = c + a 2, n e = (a c) L F (a + c), qe = 2F a c The displaced CS case q i = ( p i + b) ϑ L/µ delivers: p e = ϑ (c + b ), n e (c + ϑb ) L = ϑ 1 ϑf (c + b ), F (ϑ 1) qe = c + b

40 Direct Utility Functions How did the direct utility look like? Not separable, but how?

41 Direct Utility Functions How did the direct utility look like? Not separable, but how? We can recover it by duality..

42 Direct Utility Functions How did the direct utility look like? Not separable, but how? We can recover it by duality.. The exponential demand derives from the direct utility: ( U = x i exp τ + ) n j=1 x j ln x j n j=1 x j

43 Direct Utility Functions How did the direct utility look like? Not separable, but how? We can recover it by duality.. The exponential demand derives from the direct utility: ( U = x i exp τ + ) n j=1 x j ln x j n j=1 x j The linear demand case derives from the direct utility: U = (a x j 1) 2 x 2 j

44 Direct Utility Functions How did the direct utility look like? Not separable, but how? We can recover it by duality.. The exponential demand derives from the direct utility: ( U = x i exp τ + ) n j=1 x j ln x j n j=1 x j The linear demand case derives from the direct utility: U = (a x j 1) 2 x 2 j The displaced CS case derives from the direct utility: U = ( x ϑ 1 ϑ j ) ϑ ϑ b x j

45 Social Optimum and ineffi cient entry The best allocation solves the following problem: ( p ) max n v n,p under the resource constraint L n (cq + F ).

46 Social Optimum and ineffi cient entry The best allocation solves the following problem: ( p ) max n v n,p under the resource constraint L n (cq + F ). FOCs deliver: p c p = 1 + η 1 ( p ), n = F L [ 1 + η ( p )] where η(p/ ) v p v > 0 is the elasticity of v( ).

47 Social Optimum and ineffi cient entry The best allocation solves the following problem: ( p ) max n v n,p under the resource constraint L n (cq + F ). FOCs deliver: p c p = 1 + η 1 ( p ), n = F L [ 1 + η ( p )] where η(p/ ) v p v > 0 is the elasticity of v( ). xcess entry arises if and only if η > 0, as in the exponential and linear examples (CS delivers the optimal equilibrium)

48 Bertrand competition and endogenous entry Suppose that the number of firms is limited and strategic interactions play a role

49 Bertrand competition and endogenous entry Suppose that the number of firms is limited and strategic interactions play a role In a Bertrand setting, considering the actual demand, each firm i chooses its price p i to maximize: π i = (p ( i c)v pi ) L v ( F p j ) pj n j=1 where the denominator is not taken as given.

50 Bertrand competition and endogenous entry Suppose that the number of firms is limited and strategic interactions play a role In a Bertrand setting, considering the actual demand, each firm i chooses its price p i to maximize: π i = (p ( i c)v pi ) L v ( F p j ) pj n j=1 where the denominator is not taken as given. In a symmetric Bertrand equilibrium: p B c p B = 1 + [θ(pb /) 1]F L), n B = L F ) + 1 θ F θ ( p B ( p B

51 Bertrand competition and endogenous entry Suppose that the number of firms is limited and strategic interactions play a role In a Bertrand setting, considering the actual demand, each firm i chooses its price p i to maximize: π i = (p ( i c)v pi ) L v ( F p j ) pj n j=1 where the denominator is not taken as given. In a symmetric Bertrand equilibrium: p B c p B = 1 + [θ(pb /) 1]F L), n B = L F ) + 1 θ F θ ( p B ( p B n B > n e and thus excess entry is more likely in Bertrand than in monopolistic competition. The competitive effect of L is restored.

52 Aggressive Leaders and implications for Competition Policy The model belongs to the class of "aggregative" games with endogenous entry (tro, 2006, Rand; 2008, J): neutrality of the price/commitments of Stackelberg leaders on µ and the strategy of followers.

53 Aggressive Leaders and implications for Competition Policy The model belongs to the class of "aggregative" games with endogenous entry (tro, 2006, Rand; 2008, J): neutrality of the price/commitments of Stackelberg leaders on µ and the strategy of followers. Leaders always choose p e < p B, thereby reducing the equilibrium number of firms with respect to n B.

54 Aggressive Leaders and implications for Competition Policy The model belongs to the class of "aggregative" games with endogenous entry (tro, 2006, Rand; 2008, J): neutrality of the price/commitments of Stackelberg leaders on µ and the strategy of followers. Leaders always choose p e < p B, thereby reducing the equilibrium number of firms with respect to n B. This is neutral on consumer welfare with CS preferences (tro, 2008; Anderson et al., 2012), but raises (decreases) consumers welfare if η > (<) 0.

55 Aggressive Leaders and implications for Competition Policy The model belongs to the class of "aggregative" games with endogenous entry (tro, 2006, Rand; 2008, J): neutrality of the price/commitments of Stackelberg leaders on µ and the strategy of followers. Leaders always choose p e < p B, thereby reducing the equilibrium number of firms with respect to n B. This is neutral on consumer welfare with CS preferences (tro, 2008; Anderson et al., 2012), but raises (decreases) consumers welfare if η > (<) 0. Applications to competition policy: vertical contracts with low wholesale price below the marginal cost, bundling to strengthen price competition in the secondary market, other incentive contracts increase CS with η > 0, mergers to increase prices reduce CS with η > 0

56 xtensions and applications 1. Outside good à la D-S: ( ) γ ( ( pj )) 1 γ V = p 0 v

57 xtensions and applications 1. Outside good à la D-S: ( ) γ ( ( pj )) 1 γ V = p 0 v All goes through.

58 xtensions and applications 1. Outside good à la D-S: ( ) γ ( ( pj )) 1 γ V = p 0 v All goes through. The first best requires marginal cost pricing.

59 xtensions and applications 1. Heterogenous consumers: ( ) pj V h = v h h

60 xtensions and applications 1. Heterogenous consumers: V h = v h ( pj h ) All goes through with p e c p e = ( ) 1 p θ (p e, C ) with θ (p, C ) θ h ω h dc (h) h h

61 xtensions and applications 1. Heterogenous consumers: V h = v h ( pj h ) All goes through with p e c p e = ( ) 1 p θ (p e, C ) with θ (p, C ) θ h ω h dc (h) h h a) market size is neutral b) if θ > 0, a change of the distribution according to the likelihood-ratio dominance raises prices and number of firms more than with respect to the increase of the average income c) a mean preserving spread of the income distribution decreases (raises) prices and the mass of active firms if the demand elasticity is convex (concave) in the price.

62 xtensions and applications 1. Heterogenous costs à la Melitz: market size is neutral, but changes in income induce selection effects

63 xtensions and applications 1. Heterogenous costs à la Melitz: market size is neutral, but changes in income induce selection effects c is distributed according to G (c). ntry cost F e

64 xtensions and applications 1. Heterogenous costs à la Melitz: market size is neutral, but changes in income induce selection effects c is distributed according to G (c). ntry cost F e Condition for marginal active firm: [p(ĉ) ĉ] v (p(ĉ)/ )L µ = F

65 xtensions and applications 1. Heterogenous costs à la Melitz: market size is neutral, but changes in income induce selection effects c is distributed according to G (c). ntry cost F e Condition for marginal active firm: [p(ĉ) ĉ] v (p(ĉ)/ )L µ = F Condition for endogenous entry: ĉ c [π v (c) F ] dg (c) = F e

66 xtensions and applications 1. Heterogenous costs à la Melitz: market size is neutral, but changes in income induce selection effects c is distributed according to G (c). ntry cost F e Condition for marginal active firm: [p(ĉ) ĉ] v (p(ĉ)/ )L µ = F Condition for endogenous entry: ĉ c [π v (c) F ] dg (c) = F e market size is neutral but higher income increases all prices and makes less productive firms able to survive (an anti-selection effect) if θ > 0

67 xtensions and applications 1. Two-country model à la Krugman (assume θ > 0) First case: no transport costs, different countries

68 xtensions and applications 1. Two-country model à la Krugman (assume θ > 0) First case: no transport costs, different countries firms adopt a higher price in the richer country, and international trade does reduces the mass and increases the size of firms in the richer country Second case: identical countries with transport costs

69 xtensions and applications 1. Two-country model à la Krugman (assume θ > 0) First case: no transport costs, different countries firms adopt a higher price in the richer country, and international trade does reduces the mass and increases the size of firms in the richer country Second case: identical countries with transport costs trade opening reduces the markup on the exported goods and the mass of firms in each country relative to autarky

70 xtensions and applications 1. Two-country model à la Krugman (assume θ > 0) First case: no transport costs, different countries firms adopt a higher price in the richer country, and international trade does reduces the mass and increases the size of firms in the richer country Second case: identical countries with transport costs trade opening reduces the markup on the exported goods and the mass of firms in each country relative to autarky a reduction in transport costs reduces the price of exports but increases their markups, and therefore induces the creation of new traded goods

71 xtensions and applications 1. Two-country model à la Krugman (assume θ > 0) First case: no transport costs, different countries firms adopt a higher price in the richer country, and international trade does reduces the mass and increases the size of firms in the richer country Second case: identical countries with transport costs trade opening reduces the markup on the exported goods and the mass of firms in each country relative to autarky a reduction in transport costs reduces the price of exports but increases their markups, and therefore induces the creation of new traded goods richer countries trade between themselves more than poorer countries

72 xtensions and applications 1. New-Keynesian macroeconomics à la Blanchard-Kyotaki: nominal rigidities are amplified with Bertrand competition

73 xtensions and applications 1. New-Keynesian macroeconomics à la Blanchard-Kyotaki: nominal rigidities are amplified with Bertrand competition 2. ndogenous quality à la Sutton: price and quality still independent of market size

74 xtensions and applications 1. New-Keynesian macroeconomics à la Blanchard-Kyotaki: nominal rigidities are amplified with Bertrand competition 2. ndogenous quality à la Sutton: price and quality still independent of market size 3. Generalized MS under any symmetric non-separable preferences

75 Conclusions The dual assumption of Indirect Additivity introduces a new setting into monopolistic competition. Its results generalize properties of the CS case concerning the impact of the market size. The neutrality of income disappears and a rationale for pricing to market emerges. New simple models of price competition with and without free entry can be derived from indirectly additive preferences (exponential and linear demand).