New Trade Theory I. Part A: Simple monopolistic competition model. Robert Stehrer. The Vienna Institute for International Economic Studies - wiiw

Transcription

1 Part A: Simple monopolistic competition model The Vienna Institute for International Economic Studies - wiiw May 15, 217

2 Introduction 1 Classical models 1 Explanations based on technology and/or factor endowment differences 2 Predict complete trade specialisation, i.e. one good is imported the other is exported 3 Perfectly competitive markets 2 New trade theory 1 Bulk of trade takes place across countries with similar technology levels and factor endowments 2 Two-way trade (intra-industry trade) is important 3 Imperfect competition

3 Related issues 1 Importance of increasing returns 2 Market size matters 1 In small countries firms cannot operate at most efficient scale 2 Access to larger markets allows firms to exploit economies of scale 3 Consumers gain from larger variety of (same) product

4 Assumptions Preliminaries 1 Important modelling strategies 1 CES preferences (allow for love for variety ) 2 Monopolistic competition model 2 Assumptions 1 Number of workers and technology (productivity) fixed 2 Wage rate is numeraire 3 One-industry model 3 Extensions possible 1 More types of production factors 2 More industries 3 Relation to endogenous growth theory

5 Monopol with profits Preliminaries p p m Average costs Marginal costs = average variable costs q m Demand q Marginal revenue

6 Demand (iso-elastic) Inverse demand Revenue q = A p ε Preliminaries p = A 1 ε q 1 ε r = p q = A 1 ε q 1 ε q = A 1 ε q 1 1 ε Marginal revenue = marginal costs (constant) ( 1 1 ) A 1 ε q 1 ε = c ε ( 1 1 ) A 1 ε c 1 = q 1 ε ε Monopoly pricing p = A 1 ε q = ( 1 1 ) ε A c ε ε ( 1 1 ) 1 ( ) A 1 ε ε c = c ε ε 1

7 Monopolistic competition Preliminaries p Rents = fixed costs p m Average costs Marginal costs = average variable costs q m Residual demand q Marginal revenue

8 Demand structure Demand structure Constant-elasticity of substitution demand function 1 Monopolistic competition model with a continuum of products 1 Other variants use (large) discrete number of products 2 Specific functional form: CES ( Dixit-Stiglitz model ) 1 Implies love-for-variety 2 Also used to model input functions (Ethier)

9 Demand structure Constant-elasticity of substitution demand function Constant-elasticity of substitution (CES) utility function [ n U = ] 1 q(i) ρ ρ di = [ n ] σ q(i) σ 1 σ 1 σ di with < ρ < 1 or σ > 1 Marginal utility is U q(j) = 1 [ n ] 1 q(i) ρ ρ 1 [ n ] 1 di ρq(j) ρ 1 = U q(i) ρ di q(j) ρ 1 ρ and the (absolute value) of the marginal rate of substitution (MRS) between any pair of products is MRS = U q(i) U q(j) = q(i)ρ 1 q(j)1 ρ q(j)1/σ = = q(j) ρ 1 q(i) 1 ρ q(i) 1/σ

10 Demand structure Constant-elasticity of substitution demand function Budget constraint: y = Consumption patterns n p(i)q(i)di 1 Minimal costs (expenditures) to achieve given level of utility 2 Maximise utility given budget constraint Cost-minimisation problem min n Utility maximisation problem p(i)q(i)di s.t. [q(i) ρ di] 1 ρ = Ū [ 1 ] 1 max q(i) ρ ρ di s.t. n p(i)q(i)di = y

11 Utility maximisation Demand structure Constant-elasticity of substitution demand function [ 1 ] 1 max q(i) ρ ρ di s.t. n First order conditions (derived from Lagrangian) Leads to q(i) ρ 1 p(i) = q(j) ρ 1 p(j) q(i) ρ 1 q(j) ρ 1 = p(i) p(j) p(i)q(i) 1 ρ = p(j)q(j) 1 ρ p(i)q(i)di = y p(i)q(i) = p(j)q(j) 1 ρ q(i) ρ

12 Inserting U ρ = y ρ P ρ y = p(j)q(j) 1 ρ y ρ P ρ Integrating Introduction n p(i)q(i) = q(i) ρ p(j)q(j) 1 ρ Demand structure Constant-elasticity of substitution demand function n p(i)q(i)di = p(j)q(j) 1 ρ q(i) ρ di y = p(j)q(j) 1 ρ U ρ The proper price index satisfying U = y/p is given by (see Appendix) [ n ] 1 P = p(i) 1 σ 1 σ di q(j) 1 ρ = p(j)y ρ 1 P ρ q(j) = p(j) 1 ρ 1 yp ρ ρ 1 = p(j) 1 1 ρ yp ρ 1 ρ

13 Demand structure Constant-elasticity of substitution demand function Uncompensated demand function (Marshallian demand function) Rewriting q(j) = p(j) σ P 1 σ y q(j) = p(j) σ p(j) ( [ n p(i)1 σ di ] ) 1 1 σ y = σ n 1 σ p(i)1 σ di y p(j) 1 σ 1 = n p(i)1 σ di p(j) y q(j) = α(j) y p(j) with α(j) denoting the nominal expenditure share dependent on prices Note: Cobb-Douglas case with σ = 1: α(j) becomes a constant

14 Demand structure Constant-elasticity of substitution demand function Price index (of compensated demand function): [ n P = ] 1 p(i) 1 σ 1 σ di If p(i) = p for all i [, n] the price index becomes [ n P = ] 1 p 1 σ 1 σ n ] 1 di = [p 1 σ 1 σ 1 di = p n 1 σ 1 Price index is decreasing in n for σ > 1 2 Costs (expenditures) of achieving a certain utility level Ū is decreasing with n e = P Ū 3 Utility which can be achieved at given income is increasing in n U = y P 1

15 Labour demand: l(i) = f + 1 ϕ q(i) 1 f... fix costs 2 ϕ... productivity Cost function Marginal costs c(i) = w l(i) = w f + w 1 ϕ q(i) MC(i) = c(i) q(i) = w 1 ϕ

16 Demand for product i q(i) = p(i) σ n p(j)1 σ dj E Derive indirect demand function (price as function of output) Revenues Marginal revenues MR(i) = r(i) q(i) = ( n 1 p(i) σ = q(i) 1 p(j) dj) 1 σ E p(i) = q(i) 1 σ ( n r(i) = q(i) p(i) = q(i) 1 1 σ ( 1 1 σ ) 1 p(j) 1 σ σ 1 dj E σ ( n ) ( n q(i) 1 1 σ ) 1 p(j) 1 σ σ 1 dj E σ ) 1 ( p(j) 1 σ σ 1 dj E σ = 1 1 ) p(i) σ

17 Profit maximisation w 1 ( ϕ = 1 1 σ Pricing rule (Monopolistic) p(i) = p = MC(i) = MR(i) ) ( σ 1 p(i) = σ ( ) σ w 1 σ 1 ϕ ) p(i) 1 Increasing in w 2 Decreasing in ϕ 3 Decreasing in σ 1 Higher σ implies more competition, drives down mark-up

18 Zero-profit condition π(i) = p q(i) w f w 1 ϕ q(i) = ( ) σ = w 1 σ 1 ϕ q(i) w f w 1 ϕ q(i) = ( ) σ = σ 1 1 w 1 ϕ q(i) w f = w f = 1 σ 1 w 1 ϕ q(i) f = 1 σ 1 1 ϕ q(i) q(i) = f (σ 1) ϕ

19 Equilibrium In equilibrium: q(i) = q(j) Labour demand l(i) = f + 1 ϕ q(i) = f + 1 f (σ 1) ϕ = f σ ϕ Full employment condition h = n l(i) = n f σ n = h f σ

20 Summary Introduction 1 Exogenous parameters: σ, ϕ, w, f, h 2 Endogenous variables: p and P, q, n 1 Real wage w p = ( σ 1 σ ) ϕ 2 Price and price index ( ) σ p = 1 σ 1 ϕ w and P = p n 1 σ 1 3 Firm size 4 Number of operating firms q = f (σ 1) ϕ n = h f σ

21 Numerical example Let the paramater σ = 5., productivity is equal to ϕ = 1., the fix costs are f = 1., labour endowment is h = 1 and the nominal wage rate equals w = 1.. Applying above formulas gives p = 1.25, real wage is w/p =.8, firm output (size) is q = 4., and the number of varieties is n = 2. The price index is P =.591; real income is w/p =

22 Introduction 1 with identical country 1 Note: In classical models there is no reason for trade 2 Firms can export to other countries 3 Consumers can buy varieties from other countries 4 Special case: Constant elasticity 1 Gains from trade only because consumers have more varieties ( love for variety ) n + n 2 All other variables constant

23 Numerical example (contd.) Let the paramater σ = 5., productivity is equal to ϕ = 1., the fix costs are f = 1., labour endowment is h = 1 and the nominal wage rate equals w = Autarky: Applying above formulas gives p = 1.25, real wage is w/p =.8, firm output (size) is q = 4., and the number of varieties is n = 2. The price index is P =.591; real income is w/p = Free trade: Assume that the other country is characterised by exactly the same parameters. Applying above formulas gives p = 1.25, real wage is w/p =.8, firm output (size) is q = 4., and the number of varieties is n = 4. The price index is P =.497; real income is w/p = 2.12.

24 What happens if elasticity of substitution is allowed to change (general case)? 1 Countries face mutually competition from foreign firms 2 Competition is increasing, therefore elasticity σ is increasing 1 Prices decrease 2 Real wage increases 3 Firm size increases ( Scale effect ) 1 Firms exploit economies of scale (produce at lower average costs) 2 As prices are lowered, more has to be produced to cover fix costs 3 Some firms (in each countries) are forced to exit ( Selection effect ) 1 Number of varieties produced in each country falls 2 Total number of varieties for consumers still increasing 3 Gains from varieties 4 Real income effects can be ambiguous.

25 Numerical example (contd.) Let the paramater σ = 5., productivity is equal to ϕ = 1., the fix costs are f = 1., labour endowment is h = 1 and the nominal wage rate equals w = Autarky: Applying above formulas gives p = 1.25, real wage is w/p =.8, firm output (size) is q = 4., and the number of varieties is n = 2. The price index is P =.591; real income is w/p = Free trade I: Assume that the other country is characterised by exactly the same parameters. Applying above formulas gives p = 1.25, real wage is w/p =.8, firm output (size) is q = 4., and the number of varieties is n = 4. The price index is P =.497; real income is w/p = Free trade IIa: Assume additionally, that the elasticity of substitution increases to σ = 5.5. Applying above formulas gives p = 1.22, real wage is w/p =.818, firm output (size) is q = 4.5, and the number of varieties is n = 36. The price index is P =.55; real income is w/p = Free trade IIb: Assume additionally, that the elasticity of substitution increases to σ = 6. Applying above formulas gives p = 1.2, real wage is w/p =.833, firm output (size) is q = 5., and the number of varieties is n = 33. The price index is P =.595; real income is w/p = 1.68.

26 Various extensions 1 Differences in country characteristics (size, productivity,...) 2 Linear demand curves; other demand patterns 3 Discrete number of products 4 More industries 5 More factors of production 6 Transport costs 7 etc.

27 Appendix: Derivation of Hicksian price index

28 Cost-minimisation problem Cost-minimisation problem min n p(i)q(i)di... Lagrangian... First order conditions Leads to [ n s.t. q(i) ρ 1 p(i) = q(j) ρ 1 p(j) q(i) ρ 1 ρ 1 p(i) = q(j) p(j) q(i) = ( p(i) q(j) p(j) = ( p(j) q(j) p(i) ] 1 q(i) ρ ρ di = Ū ) 1 ρ 1 ) 1 1 ρ

29 Substitute in constraint Ū = = = [ n [ n [ ] 1 q(i) ρ ρ di ( ) ρ ] 1 ρ p(j) q(j) ρ 1 ρ di p(i) q(j) ρ p(j) ρ 1 ρ n [ n = q(j)p(j) 1 1 ρ ( ) ρ ] 1 ρ 1 1 ρ di p(i) ] 1 p(i) ρ ρ ρ 1 di

30 Compensated demand function (Hicksian demand function) [ n Ū = q(j)p(j) 1 1 ρ [ n q(j) = p(j) 1 ρ 1 = p(j) σ [ n ] 1 p(i) ρ ρ ρ 1 di ] 1 p(i) ρ ρ ρ 1 di Ū ] σ p(i) 1 σ 1 σ di Ū

31 Minimum costs of achieving given utility level Ū n q(j) = p(j) σ [ n p(j)q(j) = p(j) 1 σ [ n p(j)q(j)dj = e = = ] σ p(i) 1 σ 1 σ di Ū ] σ p(i) 1 σ 1 σ di Ū [ n ] [ n p(j) 1 σ dj [ n p(i) 1 σ di [ n ] 1+ σ 1 σ ] 1 p(i) 1 σ 1 σ di Ū ] σ p(i) 1 σ 1 σ di Ū Ū

32 Defining the price index as this can be rewritten as [ n P = ] 1 p(i) 1 σ 1 σ di e = P Ū i.e. total expenditure to achieve utility level Ū (quantity composite) equals the price index times Ū Compensated demand for variety j can then be written as ( ) σ p(j) q(j) = Ū P