Monopolistic Competition in Trade The DixitStiglitz Model Monopolistic

Monopolistic Competition in Trade • The Dixit-Stiglitz Model • Monopolistic Competition and Trade in a One-Sector Model • Monopolistic Competition and Trade in a Two-Sector HO Model • Transport Costs and the Home-Market Effect • Economic Geography © J. P. Neary Saturday, October 31, 2020 1

Monopolistic Competition Due to Chamberlin; key features: 1. Differentiated products: reflecting a “taste for variety” – Hotelling approach (used by Helpman, JIE 1981): each consumer has an “ideal type” - difficult! – Dixit-Stiglitz “taste for variety” approach is now standard – Both approaches have identical implications for positive questions (but not for normative ones) 2. Increasing returns (due to fixed costs perhaps) – Otherwise, every conceivable variety could always be produced, in tiny amounts 3. Free Entry => No long-run profits – Just like perfect competition 4. No strategic behaviour: Firms ignore their interdependence when taking their decisions. 2

1. Tastes: Dixit-Stiglitz Utility Function • • • A symmetric CES function xi is the consumption of variety i n, the number of varieties, is given to consumers, but endogenous in equilibrium The index q is a measure of substitutability, and must lie in [0, 1] As we will show, it is related to the elasticity of substitution s : {0 <q < 1} <=> {1 < s < ¥} A. Preference for Variety/Diversity Proof: Assume all varieties have the same price p and are consumed in equal amounts, so total expenditure is I = npx (This is the indirect utility function in symmetric equilibria) Logarithmically differentiating, with I and p fixed: i. e. , utility rises with diversity, and by more so the lower is s QED 3

The indirect utility function can be inverted to get the expenditure function in symmetric equilibria: The unit expenditure function P is a true price index for the industry. It is decreasing in n (again, because consumers like variety) and to a greater extent the lower is s 4

B. Derive the Elasticity of Substitution Rewrite utility function as uq and differentiate w. r. t. pi: (from consumer’s FOC) Now, take ratio for goods i and j: {0<q<1} <=> {1<s<¥} C. Marshallian Demands Solve for xj, multiply by pj, sum over j and substitute into the budget constraint, S pjxj = I, to obtain: D. Industry Price Index Substitute into u(x) to get the indirect utility function V(p, I): So: 5

Finally, as in the symmetric case, this can be inverted to get the expenditure function: The demand functions can be expressed more simply in terms of the unit expenditure function P : (The Marshallian demand function is log-linear in relative price and real income, both defined with respect to P) 6

Production: Profit Maximisation by Firms maximise profits given the (perceived) demand curve (i. e. , they take income and the industry price index as given): Hence their total and marginal revenue curves are (suppressing i): So the demand MR curves are iso-elastic, with the latter a fraction q of the former. Costs Homotheticity: Production uses a composite input, at unit cost W Overheads require F units; and production c units per unit output: TC = (F+cx)W (Set W=1 for now) Hence: MC = c AC = c+F/x (a rectangular hyperbola) 7

Dixit-Stiglitz (symmetric CES) preferences: p -1/s ==> Demand: p = Ax Marginal Revenue: q p D MR x 8

Total Costs: C = F+cx p ==> Marginal Costs: c Average Costs: c+F/x AC c MC x 9

Profit Maximisation MC=MR => c = q p i. e. , p = c /q (p independent of A, F) Alternatively: p /c = 1/q = s /(s -1) (the price-cost margin is decreasing in s) Free Entry AC=AR => c + F/x = p + {MC=MR} => c + F/x = c / q => x = (s -1) F/c In equilibrium: i. e. , Equilibrium A is also independent of P, I, and therefore n 1

p Equilibrium: Chamberlinian Tangency • Profit maximisation: MR=MC • Free entry: p=0 ==> • Firm output depends only on F, c, s • Industry output adjusts to demand shocks via changes in n only c/q D c AC MC MR (s-1)F/c x 1

Technical Digression As drawn, the AC curve is more convex than the demand curve. Proof that this must be so: So, s > 1 is necessary and sufficient for AC to be more convex. 1

Equilibrium Firm Size In equilibrium, x depends only on c, F and s: all adjustment to changes in other exogenous variables is via changes in n How to avoid this implausible property? 1. Relax CES assumption. [Krugman, JIE 1979] 2. Assume more than one factor with non-homothetic costs: [Lawrence/Spiller, QJE 1983; Flam/Helpman, JIE 1987; Forslid/Ottaviano, JEG 2003] TC = rf + wax x = (s -1)rf/wa 3. Assume heterogeneous firms: [Melitz, Em 2003] xi = (s -1)Fi/ci 1

Role of s High s: • different varieties are close substitutes for each other (preference for diversity is not so strong) • p close to c : so p and MR curves are flat and close together • x large: economies of scale are highly exploited • fewer varieties, higher output of each Low s: • different varieties are less close substitutes (greater preference for diversity) • p >> c : so p and MR curves are steep and far apart • x small: economies of scale are not highly exploited • more varieties, lower output of each 1

p B A MR (low s) AC MR MC (high s) x Effects of Changes in the Elasticity of Substitution 1

Monopolistic Competition and Trade in a One-Sector Model [Krugman, JIE 1979] 2 countries; 1 sector; labour the only factor of production (W=w=1); identical technology and tastes in both countries. Full employment: L = n. Li = n(F+cx) => n = L /(F+cx) But: i. e. , number of varieties is linear in the size of the economy Autarky: L, L* => n. A , n. A*, x Trade: • All trade is intra-industry • Trade is unrelated to comparative advantage (Both countries have the same autarky prices, since they are identical except for size, which has no effect given identical homothetic tastes) • Trade is welfare improving (since it increases the number of varieties available) • Volume of trade is maximised when countries are of equal size 1

Monopolistic Competition and Trade in the Two-Sector Heckscher-Ohlin Model We extend the two-sector HO model by assuming that one sector has a Dixit-Stiglitz monopolistically competitive structure. 2 sectors: • X 1 “Food”: perfectly competitive, output homogeneous, p=1 • X 2 “Manufactures”: monopolistically competitive Tastes: An example of “two-stage budgeting”: • the utility function is Cobb-Douglas in food and manufacturing; • the manufacturing sub-utility function in turn has the Dixit-Stiglitz form. Expenditure function: 1

Economic Geography Standard features of Dixit-Stiglitz type models: 1. Demand intercept A depends: • positively on industry price index P • positively on expenditure I 2. Industry price index depends: • negatively on number of firms at home and abroad • positively on trade costs Additional feature of Venables model: Each firm uses the output of every other as an input. So: • Expenditure I depends positively on n • Input costs depend positively on P [Now we need to make W explicit: replace F and c by FW and c. W] Implications for stability of diversified equilibria: 1

p Effects of entry by one new firm: 1. P, P* fall => p fall 2. Cost linkage: P falls => W falls => p rise 3. Demand linkage: Demand rises => p rise 3 c. W/q 1 c. W D MR (s-1)F/c AC MC 2 x 1

l 1 Core Dispersal 0. 5 Periphery TB TS T Figure 2: Agglomerated and Dispersed Equilibria as a Function of Trade Costs 2

Heterogeneous Firm heterogeneity in monopolistic competition: [Melitz (Em 2003), Helpman-Melitz-Yeaple (AER 2004)] • Firms pay a sunk cost to reveal their productivity Draw c from g(c) with positive support over (0, ) • Given their productivity, they calculate their expected profits and choose to produce or exit Exit if c < ce where p(ce) = 0 or r(ce) = f. • If exporting and/or FDI require an additional fixed cost, only high-productivity firms will engage in them • Predictions are consistent with micro-empirical evidence 2