Monopolistic competition models

models Robert Stehrer Version: May 22, 213

Introduction Classical models Explanations for trade based on differences in Technology Factor endowments Predicts complete trade specialization i.e. no intra-industry trade Competitive firms However: Most trade takes place across countries with Similar factor endowments Similar technology Intra-industry trade (two-way trade) is more important Imperfect competition

Related to this Increasing returns to scale are important Already recognized in Ohlin (1933) and Graham (1923) Market size matters Policy aspect: In a small market firms cannot operate at most efficient scale, i.e. produce at lower costs Access to larger markets (via free trade) allows firms to exploit these economies of scale (efficiency gains) However, this might also imply exit of firms Consumers additionally gain from larger variety of products

Intra-industry trade Imports and exports of goods that have similar characteristics Types of intra-industry trade (see Rivera-Batiz and Oliva, 23) Horizontal IIT: Exchange of differentiated products produced with identical factor intensities, featuring same product quality and carrying the same price Vertical IIT: Quality-differentiated products utilizing different factor intensities and sold at different prices Vertical IIT is a more important component of intra-industry trade Calculate ratio of unit value of exports to unit value of imports in bilateral trade for given variety; if ratio differs more than a threshold (e.g. 2 %) then trade is classified as vertical IIT

Stylized facts Higher income countries trade a greater number of varieties Higher income countries secure higher price for exports Increase of trade to income ratio over time seems greater than what can be accounted for by reduced trade barriers Trade models thus have to incorporate Differentiated products Endogenous quality Trading costs

An overview of models Armington assumption: Products are differentiated by production location (e.g. products are differentiated by nationality) Main approaches to horizontal IIT: Love for variety (Krugman, 1979; Helpman and Krugman, 1985) Explain trade in differentiated products under increasing returns and love for variety HO-Ricardo approach (Davis, 1995) Reciprocal dumping models (Brander, 1981; Brander and Krugman, 1983) Main approaches to vertical IIT: Product characteristics approach (Lancaster, 1979) Products of varying quality (Falvey and Kierzkowski, 1987; Flam and Helpman, 1987) models Market with large number of firms Each firm produces unique variety of a differentiated product With free entry and exit

Imperfect competition Monopolistic pricing and production decisions Figure Profit maximization yields marginal revenues equal marginal costs: MR = MC ( ) Monopoly pricing: p = MC ε with ε denoting price elasticity of demand ε 1 If price is lower average costs, p < AC, firm is earning monopoly profits

Profits will attract other firms to enter Oligopolistic market structures (Cournot, Bertrand, etc.) Complex and controversial issue as firms are interdependent, i.e. depend on other firms strategies Special case: Captures key elements of economies of scale and product differentiation Relatively easy to analyze also in international settings Key assumptions: Product differentiation Firms ignore consequence of their own price decisions on rival firms (no strategic interaction)

Standard monopolistic competition model (Krugman, 1979; etc.) Demand side Product differentiation Horizontal: Many varieties are available which do not vary in quality (e.g. soaps, etc.) Vertical: Significant differences in quality (e.g. cars, etc.) Dixit-Stiglitz preferences model horizontal product differentation All varieties enter utility in a symmetric way Utility increases with number of varieties ( Love for varieties ) Production side Firm sells more the larger is total demand for its industry s products Firm sells more the higher are prices of rivals Firm sells less the larger the number of rivals Firm sells less the higher is price as compared to rivals

Demand side: Product varieties i = 1,..., N (with N to be determined endogenously) L consumers (and workers) Utility function (for each individual) N U = v(c i ) with v >, v < i=1 Note: Symmetric over product varieties Labor income w Budget constraint N w = p i c i i=1

First order condition where λ is the Lagrange multiplier v (c i ) = λp i Assume that number of varieties is large Implies low budget share of each variety Effect of price change of p i on λ can be ignored v dc i = dp i λ dc i dp i Implies elasticity of demand for variety i as η i = dc i dp i p i c i = λ v < = λ v v /λ c i = v c i v > Assume that elasticity rises with falling consumption, i.e. dη i /dc i < e.g. as in linear demand function

Production side: Labor is only input to produce output y i L i = α + βy i with α... fixed labor input β... marginal labor input Marginal costs are constant MC i = wβ Average costs are decreasing in y i AC i = wl i y i = wα y i + wβ Two key equilibrium conditions Each firm maximizes profits: MR i = MC i Free entry: P = AC (i.e. zero profits in the long run) Additionally assume symmetric equilibrium Prices and quantities are the same across varieties

Equilibrium conditions Symmetry implies y i = Lc i and y i = y, etc. MR = MC ( p 1 1 ) = wβ p ( η ) η w = β η 1 P = AC If η is decreasing in firm output, then the lower is firm output, the higher is elasticity, the lower is mark-up Upward sloping in consumption per worker c More firms, more competition, higher elasticity, lower mark-up, lower price Downward sloping in number of products N p = wα y wα + wβ = Lc + wβ p w = α Lc + β The larger is firm output, the lower are average costs (and price) downward sloping in c More firms, lower output per firm, higher average costs Upward sloping in number of products N

Equilibrium number of products Intersection determines inverse of real wage p/w and consumption per worker of each variety c Intersection determines average costs and prices and number of varieties (firms) in market Full employment assumption N N L = L i = (α + βy i ) = N(α + βy) = N(α + βlc) i=1 i=1 Equilibrium number of products is thus N = 1 α/l + βc = L α + βy (Note: y... supply of each good; Lc... demand for each good)

Comparative static analysis of increase in market size, L Number of varieties N is increasing Existing firms increase output y = Lc Consumption of variety per worker decreases (spread over more varieties) Scale effects imply lower average costs and therefore lower prices Implies higher real wages (p/w falls)

Autarky equilibrium I: Real wage and consumption p/w P=AC MR=MC p/w P=AC (p/w) (p/w) MR=MC c c c c Elasticity rising with falling consumption Constant elasticity

Autarky equilibrium II: Price and number of varieties

Effect of free trade between two countries Identical countries Move from autarky to free trade In classical models there would be no reason for trade In monopolistic competition models: Firms produce differentiated product Start exporting to other country But also face competition from firms abroad Increase in competitors will lower equilibrium price; real wage rises as elasticity is increasing Total number of product varieties increases However, number of varieties produced in each country falls But surviving firms have larger output i.e. free trade implies that firms exit in each country, while remaining firms expand output (taking advantage of economies of scale)

Model (with varying elasticity) suggests two effects of free trade on firm productivity Scale effect: Surviving firms expand output Selection effect: Firms are forced to exit Little evidence for scale effect Evidence for selection effect If least productive firms are forced to exit, overall productivity is increasing However outside of scope of this model (see New New Trade Theory) Special case: Constant η Firm scale will not change if elasticity of substitution is constant, i.e. when modeled e.g. via a CES function (see below) No change in average price and real wages Number of varieties produced will not change (i.e. no selection effect) Number of varieties consumed will increase; workers/consumers are better off because love for variety

Introduction with a continuum of products () Constant-elasticity-of-substitution (CES) utility function: Marginal utility is [ 1 n U = q(i) di] ρ ρ with < ρ < 1. U q(j) = 1 [ n ] 1 q(i) ρ ρ 1 [ n 1 di ρq(i) ρ 1 = U q(i) di] ρ q(j) ρ 1. ρ The absolute value of the marginal rate of substitution is thus U q(i) / U q(j) = q(i)ρ 1 = MRS 1 q(j) ρ 1 which yields q(i) q(j) = MRS 1 1 ρ.

Taking logs gives ln q(i) q(j) = 1 ln MRS. 1 ρ The elasticity of substitution σ is defined as σ = q(i) d ln q(j) d ln MRS = 1 1 ρ.

Budget constraint: Introduction n M = p(i)q(i)di. For minimizing costs of attaining U we have to solve the minimization problem n min p(i)q(i)di [ n s.t. q(i) ρ 1 ρ di] = U. The first order conditions to this expenditure minimization problem for any pair of goods are q(i) ρ 1 p(i) = q(j) ρ 1 p(j). This leads to 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 ρ 1 ) 1 1 ρ.

We substitute q(i) in the constraint to get U = = = [ n ] q(i) ρ 1 ρ di [ n q(j) ρ( p(j) p(i) n [q(j) ρ p(j) ρ 1 ρ = q(j)p(j) 1 1 ρ [ n ) ρ 1 ρ di] 1 ρ ( 1 ) ρ 1 ρ 1 ρ di] p(i) p(i) ρ ρ 1 di] 1 ρ.

From this we can easily derive the compensated demand function (i.e. the Hicksian demand function) for variety j as U = q(j)p(j) 1 1 ρ q(j) = p(j) 1 ρ 1 [ n = p(j) σ[ n [ n p(i) ρ ρ 1 di] 1 ρ p(i) ρ ρ 1 di] 1 ρ U ] p(i) 1 σ σ 1 σ di U where we used the definition σ := 1/(1 ρ) and its variants. Note that σ > 1.

The minimum costs of attaining U are calculated as Defining q(j) = p(j) σ[ n p(j)q(j) = p(j) 1 σ[ n n p(j)q(j)dj = we can write this expression as = = ] p(i) 1 σ σ 1 σ di U ] p(i) 1 σ σ 1 σ di U [ n ][ p(j) 1 σ n ] dj p(i) 1 σ σ 1 σ di U [ n ] 1+ p(i) 1 σ σ 1 σ di U [ n ] p(i) 1 σ di 1 σ 1 U [ n P := ] p(i) 1 σ 1 1 σ di M = P U, i.e. total expenditures equals the price index times the quantity composite (utility level).

Compensated demand for variety j can thus be written as ( p(j) ) σu. q(j) = P Now we can solve that consumers maximize utility U subject to the budget constraint, max U s.t. P U = E. Using this gives q(j) = = ( p(j) ) σ E P P ( p(j) σ ) 1 σe P

The uncompensated demand function (Marshallian demand function) can also either be derived directly (e.g. using Lagrange multiplier techniques) or by using the expression above. To derive the uncompensated demand function we start with 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) = q(i) ρ p(j)q(j) 1 ρ.

Integrating both sides yields n p(i)q(i)di = n p(j)q(j) 1 ρ q(i) ρ di E = p(j)q(j) 1 ρ U ρ E = p(j)q(j) 1 ρ E ρ P ρ q(j) ρ 1 = p(j)e ρ 1 P ρ q(j) = p(j) ρ 1 1 EP ρ ρ 1 = p(j) 1 1 ρ EP ρ 1 ρ = p(j) σ P 1 σ E

The uncompensated demand function can be derived by inserting for P, q(j) = p(j) σ ( [ n p(i)1 σ di ] 1 1 σ p(j) σ = n p(i)1 σ di E p(j) 1 σ = n p(i)1 σ di E/p(j) = α(j)e/p(j) ) 1 σ E where α(j) = p(j) 1 σ / n p(i)1 σ di denotes the nominal expenditure share. If p(j) = p for all j the price index becomes [ n P = ] p 1 σ 1 di 1 σ = [p 1 σ n ] 1 di 1 σ = pn 1 σ 1.

Labor used is a linear function of output, l(i) = f + 1 ϕ(i) q(i), where ϕ denotes productivity. The cost function is then and the marginal cost function is c(i) = wl(i) = wf + w 1 ϕ(i) q(i), c(i) = w/ϕ(i). To calculate revenues we first derive the indirect demand function, q(j) = p(j) σ n p(i)1 σ di E p(j) σ = q(j) 1( n 1E p(i) di) 1 σ ( p(j) = q(j) σ 1 n p(i) di) 1 σ 1 σ E σ 1.

Total revenues are then given by ( r(j) = p(j)q(j) = q(j) 1 σ 1 n p(i) di) 1 σ 1 σ E σ 1. from which marginal revenues can be calculated: r(j) = = ( 1 1 ) ( q(j) σ 1 n σ ( 1 1 ) p(j). σ p(i) 1 σ di) 1 σ E 1 σ Setting marginal revenues equal to marginal costs yields the pricing rule ( 1 1 ) p(j) = w/ϕ(j) σ σ p(j) = σ 1 w/ϕ(j) = 1 w ρ ϕ(j).

Let us now assume that ϕ(j) = ϕ for all j. The zero profit condition yields π(i) = pq(i) wf w 1 ϕ q(i) = 1 w ρ ϕ q(i) wf w ϕ q(i) = wf = f = ( 1 ρ 1 ) w ϕ q(i) ( 1 ) 1 σ 1 ϕ q(i) and thus q(i) = f (σ 1)ϕ. In equilibrium all firms have the same size. Labor demand in a particular firm is l(i) = f + 1 ϕ q(i) = f + 1 f (σ 1)ϕ = f σ. ϕ

The full employment condition then determines the number of firms in the market, h = nl(i) = nf σ, or n = h f σ.

Other models Lancaster (198), Intra-industry trade under perfect monopolistic competition, JIE, 1, 151-176. Falvey, R. (1981), Commercial policy and intra-industry trade, JIE 1, 495-511. Brander-Krugman model: intra-industry trade in homogenous products under oligopolistic market structure.

The elasticity of substitution The marginal rate of substitution (MRS) measures the slope of an isoquant. dx 2 dx 1 = f / x 1 f / x 2 The elasticity of substitution measures the percentage change in the quantity ratio divided by the percentage change in the MRS (with utility held fixed), i.e. σ = (x 2 /x 2 ) x 2 /x 1 MRS MRS or the curvature of an isoquant. I.e. how does the ratio of quantities change as the slope of the isoquant changes? If a small change in slope gives a large change in the quantity ratio, the isoquant is relatively flat which means that the elasticity of substitution is high.

Taking the limit as the expression becomes σ = MRS d(x 2 /x 1 ) (x 2 /x 1 ) dmrs Using logarithmic derivation (provided that x i > ) this can be expressed as σ = d ln(x 2/x 1 ) d ln MRS

General For x, y > this can be written as ɛ = dy/y dx/x = dy x dx y ɛ = d ln y d ln x as d ln y = 1 y dy and d ln x = 1 x dx More rigourously shown using chain rule.

Example 1: Cobb-Douglas function MRS = α x 2 1 α x 1 or Taking logs This implies x 2 = 1 α x 1 α MRS = 1 α α MRS ln x 2 = ln 1 α + ln MRS x 1 α σ = d ln(x 2/x 1 ) d ln MRS = 1

Example 2: CES function ( ) y = a 1 x ρ 1 + a 2x ρ 1/ρ 2 or Taking logs and thus ( x1 ) ρ 1 MRS = x 2 x 2 x 1 = MRS 1/(1 ρ) for a 1 = a 2 = 1 ln x 2 = 1 ln MRS x 1 1 ρ σ = d ln(x 2/x 1 ) d ln MRS = 1 1 ρ ρ = 1 σ : Perfect substitutes ρ σ 1: Cobb-Douglas ρ σ : Perfect complements (Leontief function)