The Ricardian Continuum Model

The Ricardian Continuum Model Notes for Graduate Trade Lectures J. Peter Neary University of Oxford October 12, 213 J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 1 / 17

Plan of Lectures 1 The Model 2 Comparative Statics 3 Calculate Welfare Explicitly 4 Extensions to Many Countries J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 2 / 17

Assumptions The Model Assumptions Ricardo (1817), Dornbusch, Fischer and Samuelson (AER 1977) Two countries: Home (H), Foreign (F)* Continuum of sectors: z [, 1] Technology: Labour only, fixed coefficients: International differences in technology: a(z) at home, a (z) abroad Factor endowments: L at home, L abroad Labour fully mobile domestically: w independent of z Labour fully immobile internationally: w w Perfect competition Prices: p (z) = c (z) = wa (z) J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 3 / 17

The Model Comparative Advantage Comparative Advantage International differences in technology: a(z) at home, a (z) abroad Define: A (z) a (z) a(z) - home comparative advantage in sector z. Assume A < : i.e., sectors ranked by declining home comparative advantage. N.B. Foreign coeff. on top; also ratio of home to foreign lab. productivity. Production in free trade: All at home IFF: c (z) c (z) wa (z) w a (z) Define: ω w w ; hence the CA locus, which gives the threshold sector z as a function of ω is defined by: z : ω = A ( z) (1) J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 4 / 17

The Model The Comparative Advantage Locus The Comparative Advantage Locus (1) A(z) Home C.A. Foreign C.A. 1 z Figure 1: The Home Comparative Advantage Locus J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 5 / 17

The Model Preferences and Demand Preferences and Demand Utility: ln U = 1 β (z) ln x (z) dz, 1 β (z) dz = 1 1 Max ln U s.t. p (z) x (z) dz I 1 x (z) = β (z) λp(z) [λ: the marginal utility of income] Solving for this, λ = 1 I x (z) = β (z) I p(z) Indirect Utility Function: ln U = ln I ln P where: ln P 1 β (z) ln p (z) dz B, B 1 β (z) ln β (z) dz In equilibrium: ln U = ln (wl) ln P or ln (U/L) = ln (w/p ) i.e., utility p.c. is just real wage. J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 6 / 17

Full Employment The Model Full Employment Labour-Market Equilibrium: L = L D = z ld (z) dz l D (z) = a (z) y (z) y (z) is domestic production = a (z) [[x (z) + x (z)] ] equal to world consumption = a (z) β (z) I+I p(z) = a (z) β (z) wl+w L wa(z) = β (z) ωl+l ω ωl = z β (z) (ωl + L ) dz = (ωl + L ) θ ( z) where: θ ( z) z β (z) dz (share of world spending on home goods) ω = θ ( z) L 1 θ ( z) L [ dω ] d z LMEL θ ( z) = β( z) >. i.e., LMEL is upward-sloping. J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 7 / 17 (2)

Equilibrium Combine the two loci: The Model Equilibrium (2) * ( ~ z ) L 1 ( ~ z ) L ~ (1) A(z) z~ 1 z Figure 2: Comparative Advantage and Full Employment Determine the Equilibrium Wage and Extensive Margin J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 8 / 17

Comparative Statics Comparative Statics Note: CA locus (1) depends only on technology; LMEL locus (2) only on tastes and endowments. Latter would not hold e.g. with CES preferences. Effects of Population Growth Suppose foreign population grows CA is unaffected, LMEL shifts up So: foreign produces more (since the extensive margin z falls) and home relative wage rises. Comparative versus Absolute Advantage Suppose foreign industries become twice as efficient LMEL is unaffected, CA shifts down So: foreign produces more (since the extensive margin z falls) and home relative wage falls. Conclusion: Absolute advantage matters a lot, including for resource allocation;... but not for the ranking of sectors (which depends only on CA). J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 9 / 17

Effects of Tariffs Comparative Statics Effects of Tariffs For home imports: c (z) (1 + τ)c (z) ω = (1 + τ)a ( z) For home exports: (1 + τ)c (z) c (z) ω = A ( z ) /(1 + τ) i.e., non-traded goods in range: z [ z, z] Amended LMEL: L = z l D (z) dz + z z l D (z) dz = z a (z) [x (z) + x (z)] dz + z a (z) x (z) dz z = z β (z) ωl+l ω dz + z β (z) Ldz z ωl 1 z β (z) dz = L z β (z) dz ω = 1 θ ( z ) L 1 θ ( z) L where θ ( z) is the share of home expenditure spent on home-produced goods and θ ( z ) is the share of foreign expenditure spent on foreign-produced goods. (3) J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 1 / 17

Comparative Statics Effects of a Transfer with Different Tastes Effects of a Transfer with Different Tastes Suppose foreign transfers T (measured in foreign labour) to home. Hence budget constraints are: I = ωl + T (H) and I = L T (F) Full Employment: L = L D = z ld (z) dz l D (z) = a (z) y (z) = a (z) [x (z) + x (z)] as before = a(z) p(z) [β (z) I + β (z) I ] N.B. β (z) β (z). = 1 ω [β (z) (ωl + T ) + β (z) (L T )] ωl = z β (z) (ωl + T ) dz + z β (z) (L T ) dz = (ωl + T ) θ ( z) + (L T ) θ ( z) where: θ ( z) z β (z) dz, θ ( z) z β (z) dz (shares of home and foreign spending on home goods) ω = θ ( z) 1 θ ( z) L + θ ( z) θ ( z) T (4) 1 θ ( z) L i.e., transfer per capita raises home wages for given z, IFF home residents spend more on home goods than foreigners. This is the Samuelson (EJ 1952) condition on relative marginal propensities to import for a transfer to generate a secondary burden. J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 11 / 17 L

Calculate Welfare Explicitly Calculate Welfare Explicitly Autarky wages: Substituting in LMEL does not work: L = 1 ld (z) dz = 1 a (z) x (z) dz = 1 a (z) β (z) w AL p A (z) dz = L 1 β (z) dz = L!!! Instead: Calculate the Autarky Price Index ln P A = 1 β (z) ln [w Aa (z)] dz B = ln w A + 1 β (z) ln a (z) dz B So: ln U A ln L = ln w A ln P A = 1 β (z) ln a (z) dz + B Ratio of home to foreign autarky utility per head and real wages: ln ( ( U L )A ln U L = ln )A ( ( w P )A ln w P )A = 1 β (z) ln a (z) dz + 1 β (z) ln a (z) dz = 1 β (z) ln A (z) dz i.e., Home autarky utility p.c. is higher than foreign if a β-weighted geometric mean of CA coefficients across sectors exceeds one. More important to have a CA in sectors whose outputs have larger budget shares. J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 12 / 17

Calculate Welfare Explicitly Free Trade Welfare and Real Wage Free Trade Welfare and Real Wage Note that relative welfare depends only on comparative advantage: ln ( ( U L )F ln U L = ln )F ( ( ) w P )F ln w P = ln ( w w = ln A ( z) F )F Next, to evaluate gains from trade: ln P F = z β (z) ln [w F a (z)] dz + 1 z β (z) ln [w F a (z)] dz B = θ ( z) ln w F + [1 θ ( z)] ln wf + z β (z) ln a (z) dz + 1 z β (z) ln a (z) dz B ln ( wf P F ) + B = [1 θ ( z)] ln ω F 1 β (z) ln a (z) dz 1 ( ) ( ) β (z) ln A (z) dz z ln wf P F ln wa P A = [1 θ ( z)] ln ω F 1 β (z) ln A (z) dz; z ( ) ( ) U U 1 ln ln = β (z) [ln A ( z) ln A (z)] dz (5) L F L A z i.e., gains from trade equal a β-weighted integral of excess comparative advantage! This depends on absolute advantage too, since the absolute level of a (z)/a(z) for z > z matters. J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 13 / 17

Gains from Trade Calculate Welfare Explicitly Gains from Trade ~ z~ 1 z Figure 3: The Shaded Area denotes Home s Gains from Trade in the special case where is the same for all Goods [i.e., when consumers view all goods as symmetric] J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 14 / 17

Calculate Welfare Explicitly Gains from Trade (cont.) Gains from Trade (cont.) Gains from trade expression shows clearly that home gains only on its import goods (z > z): export goods are produced with the same real product wage at home as in autarky: z [, z] : [ ] [ ] w w = = 1 p (z) F p (z) A a (z) whereas the real wage in terms of import goods is (weakly) higher: [ ] w w z [ z, 1] : = p (z) w a (z) 1 [ ] w a (z) = p (z) F This implies that countries gain more from trade the more they import; i.e., the smaller they are. A (6) (7) J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 15 / 17

Calculate Welfare Explicitly Small Countries Gain More from Trade Small Countries Gain More from Trade A i Foreign is small and gains a lot ~ LME 1 Home is large and gains little z~ 1 z Figure 4: Small Countries Gain More from Trade J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 16 / 17

Extensions to Many Countries Extensions to Many Countries How to allow for more than two countries? Wilson (Em 198) View each country as having a net supply of labour World equilibrium can then be solved in terms of national excess labour supplies But: cumbersome Eaton and Kortum (Em 22) Assume each country purchases each good from the lowest-cost supplier Country productivities are stochastic Allows for a tractable derivation of a gravity-type equation But: Special functional forms: CES preferences; Fréchet distribution Costinot (Em 29) Expresses CA in terms of log-supermodularity of sectoral revenue functions J.P. Neary (University of Oxford) Ricardian Continuum Model October 12, 213 17 / 17