Monopolistic Competition Model with Endogeneous Investments: the case of Closed Economy

Transcription

1 Monopolistic Competition Model with Endogeneous Investments: the case of Closed Economy I.Bykadorov (together with E.Zhelobodko, S.Kokovin) November, 29th, 2011

2 Abstract Model - (Theoretical?): in monop. competition; - (Model novelties): variable elasticity of substitution (VES), each firm chooses investment in decreasing marginal cost or increasing quality; - (Results in closed economy): [Growing market pushes each firm s R&D investment down, cost and markup up ] utility with increasing relative love for variety (decreasing elasticity of demand). Number of firms and total R&D investment in economy go up always and...

3 Outline Model 1 Model 2

4 Motivation, topic Empirics on international trade show: firms operating in bigger markets have lower markups (Syverson-2007); firms are larger in larger markets (Campbell-Hopenhayn-2005); larger economies export higher volumes of each good, a wider set of goods, and higher-quality goods (Hummels-Klenow-2005); within an industry there can be considerable firm s heterogeneity: firms differ in efficiency, in exporting or not (associated with high/low efficiency), in wages (Reddings-2011). Various explanations in literature. Monopolistic competition idea suggests that a bigger market invites more R&D investment from a firm because of economies of scale.

5 Literature, topic Basic idea of Monopolistic Competition: many firms - price-makers producing varieties, free entry, fixed and variable costs => increasing returns. Chamberlin (1929), Dixit and Stiglitz (1977), for trade Krugman (1979). MC model with CES or quadratic utility was recently generalized to any (VES) utility: Zhelodobko, Kokovin & Thisse (2010), Zhelodobko, Kokovin, Parenti & Thisse (2011) Oligopolistic choice of technology in quasilinear setting: Vives (2008): firm s R&D investment in economy go up with market size always, and number of varieties can increase or decrease. Our goal is to combine choice of technology like in Vives - with monop. competition under VES (under CES combining was non-interestiong: zero effects).

6 MC model: assumptions Increasing returns to scale in a firm, due to investment cost f and marginal costs c(f ). Firms are identical. Each firm i produces one variety as a price-maker, but its demand x i (p i,p j...) is influenced by other varieties. Each demand function results from additive utility function U = i N u(x i)di. Degree of u concavity (i.e., elasticity of demand or substitution among varieties) - determines intensity of competition. Number of firms is big enough to ignore firm s influence on the whole industry/economy. Free entry drives all profits to zero. Labor supply/demand is balanced.

7 Model of 1x1x1 economy. Consumers A diversified sector has an interval [0, N] of firms=varieties i-th brand is i-th firm, i [0,N], L identical consumers, each has 1 of labor and chooses an (infinite-dimensional) consumption vector x( ) : [0,N] R + i.e., a non-negative integrable function x: N 0 N u(x i )di max, p i x i di 1. x(.) 0 Here: utility function u(.), price vector p(.) : [0,N] R + ; p(i) p i is price for i-th variety, x(i) x i is demand for i-th variety. Lagrange multiplier λ, labor endowment is 1. From FOC, inverse demand for i-th variety is: p i (x i,λ) = u (x i ) λ.

8 Model: Producers, FOC i-th firm knows its inverse-demand function p i (x i,λ), sells Q = L i x i and maximizes profit π = Lx i [p i (x i,λ) c(f i )] f i max. x i,f i R + L is consumers mass, c is marginal cost and f is fixed cost measured in labor (total cost is cx i L + f ), λ is marginal utilities of money. Symmetric equilibrium is (x,f,p,n,λ) satisfying all FOC and budget, free entry and labor balance:

9 Model: Equilibrium (x,f,p,n,λ) Consumers FOC and budget constraint: p = p ( x) = u ( x)/λ Producers FOC: Np x = 1 π( x, f ) f = 0, Zero-profit condition (free entry): π( x, f ) x = 0 π = (p ( x) c( f )) xl f = 0. Labor balance: ( f + c( f ) x)n = L This system can be reduced to 2 equations (x,f ):

10 Equilibrium equations for (x, f ) We use the Arrow-Pratt measure of concavity defined for any function g : r g (z) = zg (z) g (z). Proposition. Equilibrium consumption/investment (x,f ) in one country with endogenous technology is the solution to under SOC conditions r u (x)x 1 r u (x) = f Lc(f ), (1 r lnc (f ) + r c (f ))(1 r u (x)) = 1, r u (x) < 1, (2 r u (x))r c (f ) > 1. Differentiating this system w.r.t. L we got Theorem of comparative statics.

11 About E g,r g,r g,r lng,r g, etc. Definition of elasticity: E g (z) = zg (z) g(z) Elasticity of the product is the sum of elasticities: E gh (z) = E g (z) + E h (z) The interconnection between elasticity and Arrow-Pratt measure: r g (z) = zg (z) g (z) = E g (z) One has: r g (z) = zg (z) Moreover g (z), r lng (z) = z (lng(z)) (lng(z)) = E g (z) + r g (z) r g (z) z = ( 1 + r g (z) r g (z) ) r g (z) E g (z) z = ( 1 E g (z) + E g (z) ) E g (z) = (1 r lng (z))e g (z) It is important to note: If g(z) is CES then r g (z) = E g (z) = 1 r lng (z) = 0

12 Theorem: signs of elasticities w.r.t. market size L: case IES CES case DES (like x + a) r u < 0 r u = 0 r u > 0 r lnc > 1 r lnc 1 r lnc > 1 r lnc = 1 r lnc < 1 E f = E Lx r c E p = E Lx r u + 0 E p c p = E x xr u r u E x 0 + E Nf E N > 1 = 1 (0,1) = r u < 1 < 1 (Only) under DES-utility a growing market increases firm size Lx and R&D investment f, but decreases prices p. Cost elasticity r lnc(f ) > 1 influences per-variety consumption x and markup p c p. Shortly, DES+larger market bigger firms higher productivity.

13 The results of ZKT(2010), ZKPT (2011), Krugman (1979) Zhelodobko, Kokovin & Thisse (2010), Zhelodobko, Kokovin, Parenti & Thisse (2011): r u < 0 r u = 0 r u > 0 E p + 0 E Lx + 0 E x E N > 1 = 1 (0,1) Krugman (1979) (Nobel Prize, 2008): r u < 0 (without strict proofs) E p + E Lx + E x E N +

14 Comparison with of ZK(P)T(2010,2011), Krugman (1979) r u < 0 r u = 0 r u > 0 r lnc > 1 r lnc 1 r lnc > 1 r lnc = 1 r lnc < 1 E f E p = E Lx r u E p c p E x 0 + E Nf E N > 1 = 1 (0,1) = r u < 1 < 1 ZKT (2010), ZKPT(2011) Krugman(1979) r u < 0 r u = 0 r u > 0 r u < 0 E p + 0 E p + E Lx + 0 E Lx + E x E x E N > 1 = 1 (0,1) E N +

15 Another interpretation: Quality q i = quality of i th variety, x i = consumption, z i = q i x i = satisfaction with variety. N U(z) = 0 u(q i x i )di, p(x i,q i,λ) = q iu (q i x i ). λ i th cost function (marginal and fixed costs), auxiliary c,f : c(q i )Lx i + f (q i ), f i = f (q i ), c (f i ) = c(q i(f i )) q i (f i ). Naturally, c (q i ) > 0, f (q i ) > 0 (higher quality requires spendings). Then the setting and the effects similar as those for productivity: ( u ) (z i ) c (f i ) Lz i f i max λ. z i 0,f i 0 Theorem: Under DES, larger market bigger firms higher quality. CES zero effects.

16 Outline Model 1 Model 2

17 Let c = c(f,α) where α > 0 is technological progress parameter: c f < 0, 2 c f 2 > 0(??), c α < 0, 2 c f α < 0. Theorem. Markets classification for technological progress: IES CES DES r u < 0 r u = 0 r u > 0 E x/α E f /α E Nf /α = E p c p /α 0 + E N/α? E p/α Thus, technological progress always stimulates R&D investment and pushes prices down, but impact on markup differs for DES and IES cases.

18 Outline Model 1 Model 2

19 In symmetric case optimality means that x opt, f opt and N opt are the solution of optimization problem Nu(x) max N,x,f N(c(f )xl + f ) = L One has by substituting N from labour balance equation Lu(x) c(f )xl + f max x,f

20 Equilibrium and optimality Theorem. The interconnection between optimal consumption (x opt ) and equilibrium consumption (x ), optimal fixed costs (f opt ) and equilibrium fixed costs (f ), optimal mass of firms (N opt ) and equilibrium mass of firms (N ), are as follows r lnu < 1 r lnu = 1 r lnu > 1 x opt < x x opt = x x opt > x f opt < f f opt = f f opt > f N opt > N N opt = N N opt < N The interconnection between optimal total investments (Nf ) opt = N opt f opt and equilibrium total investments (Nf ) = N f, one has (1 r lnu )(1 r lnc ) < 0 (1 r lnu )(1 r lnc ) = 0 (1 r lnu )(1 r lnc ) > 0 (Nf ) opt > (Nf ) (Nf ) opt = (Nf ) (Nf ) opt < (Nf )

21 Comparative statics for optimality Theorem. For optimality the signs of elasticities of x opt, f opt and N opt w.r.t. market size L are IES CES DES r lnu < 1 r lnu = 1 r lnu > 1 r lnc > 1 r lnc 1 r lnc > 1 r lnc = 1 r lnc < 1 E x opt = 1 = 0 + E Lx opt = 0 (0;1) = 1 > 1 E f opt = 0 + (0;1) + opt > 1 = 1 (0;1) = 1 > 1 E N opt > 1 = 1 (0;1) (0;1) < 1 E N opt f

22 Comparative statics for equilibrium and optimality r u < 0 r u = 0 r u > 0 Elasticity r lnc > 1 r lnc 1 r lnc > 1 r lnc = 1 r lnc < 1 E x 0 + E Lx E f E N f E N > 1 = 1 (0,1) = r u < 1 < 1 r lnu < 1 r lnu = 1 r lnu > 1 Elasticity r lnc > 1 r lnc 1 r lnc > 1 r lnc = 1 r lnc < 1 E x opt = E Lx opt 0 (0;1) = 1 > 1 E f opt 0 + (0;1) + opt > 1 = 1 (0;1) = 1 > 1 E N opt > 1 = 1 (0,1) (0;1) < 1 E N opt f

23 Outline Model 1 Model 2

24 Heterogenity: the study of expected profit i-th firm s individual marginal cost function: c(f, i) Firms are ordered: c(f,i) > c(f,j) i > j Firms appear in the market randomly, with distribution function Γ(i) on [0, ) with probability density γ(i). Prior to entering market, each firm does not know its actual total costs c(f,i)y + f + f e where y is output, f e is expenditure to study expected profit (the cost to order business plan). The problem of representative consumer: î U = u(x i )NdΓ(i) max 0 x i s.t. î 0 p i x i NdΓ(i) = w 1 [0,î] set of different varieties, x i consumption of the variety of type i

25 Heterogenity: the study of expected profit Short run: the cutoff number î is exogenous, firms having a marginal cost equal to c(f,î) earn zero profits. All firms, operating in the market, have a type î. Firms of type > î do not produce. Long run: î is endogenously determined by free entry.

26 : the equilibrium The inverse demand function: p i (x i ) = u (x i )/λ The operating profit of a type i-firm: [ u π(i,x ] (x i ) i,f i,λ,l) = c(f i,i) Lx i f i λ π(i,x i,f i,λ,l) max x i,f i, x i (λ,l), f i (λ,l) π(i,λ,l) = π(i,x i (λ,l),f (λ,l),λ,l) The condition for boundary producer: π(î,λ,l) = 0 The condition for firm to enter market: the expected profit (i.e. average profit of the firms already entered market) must be non less than the cost to order business plan: î 0 π(i,λ,l)dγ(i) = f e. Thus we have the system of two equations with two variables, î and λ.

27 Conclusions, main directions of study In a larger economy firm size and productivity (quality) is higher DES-utility; Techn. progress increases R&D investment always, markup DES-utility]; but [decreasing Hypothesis 1: In welfare analysis, socially-optimal solutions show similar comparative statics as equilibria, and only under CES equilibria are optimal (?); Hypothesis 2: Under heterogenous firms a-la Melitz, market size yields similar effects (?); Thank you