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Chapter 4: Increasing returns to scale

Chapter 4: Increasing returns to scale. 4.1 Empirical observations 4.2 Homogeneous products 4.2.1 External economies of scale 4.2.2 Contestable markets 4.2.3 Oligopolies 4.3 Differentiated products 4.3.1 Love of variety 4.3.2 Ideal variety. 4.1 Empirical observations.

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Chapter 4: Increasing returns to scale

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  1. Chapter 4: Increasing returns to scale 4.1 Empirical observations 4.2 Homogeneous products 4.2.1 External economies of scale 4.2.2 Contestable markets 4.2.3 Oligopolies 4.3 Differentiated products 4.3.1 Love of variety 4.3.2 Ideal variety

  2. 4.1 Empirical observations • Volume of trade largest between „similar“ countries within OECD (more than 50% of world trade). • Substantial intra-industrial trade (more than 25%) • Trade grows faster than world income • Growing importance of transactions within multi-national corporations • Trade liberalization and free trade associations should have led to a restructuring of production and changes in factor income – not observed, all factor incomes grow at the same pace.

  3. 4.2 Homogeneous products 4.2.1 External economies of scale Production function of representative firm: xi = f(vi,) (1) where  denotes the positive external effects which are beyond the control of firm i (e.g. domestic output of the commodity, world output, total domestic income etc.)

  4. Justifications: • Cheaper inputs for larger industries (however, begs the question of where cheaper inputs come from) • Approximation of model with price equal to average cost in equilibrium (however, compare to contestable markets) • Technical progress cannot be appropriated (however, begs the question where it comes from). • Learning by doing – larger well trained labor force.

  5. Autarky equilibrium Unit cost function: bj = bj(w,), bj < 0, bjw = aj. aj = (a1j,…,aij,…aMj) = vector of inputs per unit of output of good j. Equilibrium conditions: pa = b(wa,a) (2) v = Axa (3) Trade equilibrium p  b(w,) (4) v = Ax (5)

  6. Welfare gains from trade: Graham (1923): Free trade may reduce welfare through reallocation of resources from industries with scale economies to those without. Kemp and Negishi (1970): Reallocation in opposite direction ensures gains from trade. Helpman and Krugman (1985): Sufficient condition for gains from trade: pjfj(vja,)  pjfj(vja,a), (6) i.e. at world market prices „on average“ factor productivity is greater than in autarky, given the factor allocation of autarky and  instead of a.

  7. Proof: Idea: Show that (6) implies feasibility of ca = xa in the free trade equilibrium, then apply WARP: pxa pjfj(vja,)  G(p,v,) = GDP-function. Alternative sufficient condition for gains from trade: bj(w,a)xja  bj(w,)xja (7) i.e. at world market prices costs of production of autarky outputs are smaller in the free trade equilibrium than in the autarky equilibrium. The proof is similar to the one above:

  8. Proof: (4) and (7) imply pxa bj(w,)xja  bj(w,a)xja , and by definition bj(w,a) = waj(w,a)  waj(wa,a), hence pxa waj(wa,)xja = waj(wa,)xja = vw = G(p,v,), Thus xa = ca is feasible in the free trade equilibrium which cannot be inferior to the autarky equilibrium.

  9. Trade structure In addition to different technologies (Ricardo) and/or different factor endowments (Heckscher-Ohlin) different utilization of economies of scale as cause of comparative advantage: Predictions about trade structure become even more difficult. Sufficient condition for traditional results to hold: external effects are the same for all countries in the free trade equilibrium.

  10. Examples: • Ricardo: Production functions: fj(vj,) = j()vj/aj, Fj(Vj,) = j()Vj/Aj (8) The domestic country will export good one if a1/a2 < A1/A2. This does not rule out the possibility that in autarky we get [2(a)a1]/[1(a)a2] > [2(A)A1]/[1(A)A2], hence predictions about trade patterns may not be possible by using only information about autarky.

  11. b)Heckscher-Ohlin All countries have identical production functions: xj = fj(vj,) (9) i.e. external effects are the same and have the same effects in all countries in the free trade equilibrium – predictions about trade patterns using information after trade takes place are possible. c) Counterexample: sector- and country-specific externalities: xj = xjavj/aj (10) Firm perceives infinite costs without domestic production.

  12. Factor price equalization: Approach: determination of FPE-set (allocations of factor endowments between countries which allow replication of integrated equilibrium through free trade equilibria). Focus on country- and sector specific external scale economies – assumed to exist for subset IE of all goods. Their production functions are xi = fi(vi,xi), i IE (11) Subset of goods produced with constant returns to scale: IC.

  13. Integrated equilibrium conditions: pi = bi(w,xi) (12) vi = aji(w,xi)xi (13) where vi denotes the vector of factor inputs in industry i and aji is the input of factor j per unit of output of industry i. The set of all factor allocations compatible with FPE (denoted as ) is defined analogously to the standard case with the additional requirement that the integrated equilibrium output of goods with increasing returns to scale can be produced in one country:

  14. = {v1,...,vJ|ij  0, jij =1iI, ij{0,1} iIE} vj = iijvi iJ (14) where ij denotes the share of country j of the vector of factor inputs in industry i in the integrated equilibrium, and vj denotes the vector of factor endowments of country j. For two countries and three industries, one of which has external economies of scale, the set  is shown in figure 4.1.

  15. Figure 4.1 FPE-set for industry 1 being capital intensive and enjoying increasing returns to scale.

  16. Remarks: • The diagonal need not belong to . • The equilibrium is not unique with respect to the structure of production. • Increasing returns to scale will generate division of labor and trade even between identical countries. • It need not be the relatively capital rich country that produces the capital intensive good, absolute size also matters. • Net factor import is uniquely determined.

  17. Figure 4.2: Indeterminacy with respect to the location of industry 1

  18. Non-uniqueness Integrated economy, two goods, one factor, production function x1 = x1v1/a1. Let v = 1,v1 < 1/2. The following figure (4.3) shows  for this case: 0___________Q__________Q‘___________0* vh measured from left to right. Let 0Q = Q‘0* = v1. vh between Q and Q‘: integrated equilibrium is reproduced, location of industry 1 indeterminate. vh < v1: integrated equilibrium reproduced if industry 1 is in foreign country. Second equ.: Industry 1 in home country, foreign country firms perceive infinite production costs.

  19. Non-uniqueness cont’dLet v1 > 1/2.  is shown in figure (4.3):0___________Q’_ _ _ _ _ _Q___________0*  consists of the solid lines 0Q‘ and Q0*. • vh between Q‘Q: integrated equilibrium cannot be reproduced. • vh in 0Q‘: integrated equilibrium reproduced, industry 1 in foreign country. • Industry 1 in home country: no FPE. • Both countries produce good 1: FPE, but no reproduction of integrated world equilibrium.

  20. 4.2.2 Contestable markets Extension of Bertrand model to industries with sub-additive cost-functions: c(xi) < c(xi) Reasons: Economies of scale, fixed costs. Crucial assumptions: no sunk costs, no entry or exit costs. Dasgupta-Stiglitz: Theory well funded, but not well founded. Main difference to previous model: scale economies are internal for firms and known to them.

  21. Integrated world equilibrium: additional condition:ci(w,xi(pi))  pi  pi < pi*rules out „inefficient“ equilibria (see Figure 4.5 below)

  22. Sufficient condition for gains from trade:ci(w,xi)xia   ci(w,xia)xia Non-existence of free trade equilibrium:Integrated economy, two goods, one factor, production function x1 = x1v1/a1. Let v = 1,v1 < 1/2. The following figure (4.6) shows  for this case: 0___________Q__________Q‘___________0* vh measured from left to right. Let 0Q = Q‘0* = v1. vh between Q and Q‘: w greater in country that produces good 1  provokes entry from other country, but two firms cannot co-exist in a contestable market.

  23. 4.2.3 OligopoliesOne-shot Cournot-model • Market concentration and trade: Even if two countries are identical, free trade increases competitive pressure and lowers prices as compared to autarky • Oligopoly and transport costs: may lead to asymmetric oligopoly in each country, but above effect still possible.

  24. 4.2.3.1 Concentration in partial equilibrium k identical consumers, n firms, market demand: X(p) = kx(p). Firm i maximizes i(xi) = xip([jixj + xi]/k) – c(w,xi), (15) FOCs: p + p‘xi/k = cx(w,xi) (16) Identical firms  xi = kx(p)/n Elasticity of demand of representative consumer:  = - x‘(p)p/x

  25. Substituting this into (16) yields p[1 – 1/n(p)] = cx(w,kx(p)/n) = pR-1 (17) Suppose there is free trade between two perfectly identical countries (same number of identical consumers and firms). Instead of (17) we get p[1 – 1/2n(p)] = cx(w,kx(p)/n) (18) i.e. the price goes down even though no commodity flows will be observed. Sufficient condition for gains from trade: output in oligopolistic industries grows (pro-competitive effect of trade).

  26. The direction of trade Depends not only on costs and on pre-trade prices, but also on the number of firms and consumers: • Identical countries except for the number of firms: country with larger number of firms exports (but also had lower pre-trade price). • Identical costs, but different number of firms and consumers: net exporter is country with larger firms/consumers ratio. • Identical countries except for costs: country with smaller costs is net exporter.

  27. In general no clear prediction on the basis of only one variable possible: cost advantages may be over-compensated by market size, etc. Reason: Less efficient firms may still be active in a Cournot-equilibrium.

  28. 4.2.3.2 General equilibrium trade patterns Equilibrium condition for a Cournot-oligopoly: p[1 – 1/n(p)] = cx(w,kx(p)/n) = pR-1 (17) implying R = [1 – 1/n(p)]-1 (19) “Wedge” between price and marginal cost: 1 − R-1 Let Ri denote the „mark-up” perceived by firm i. Constant returns to scale imply that resources are allocated s.t. Ri-1pixi is maximized (Helpman 1984).

  29. Factor abundancy: If all firms in a country perceive the same Ri then differences in output will depend solely on differences in supply functions, hence with identical production functions on differences in factor endowments. Ri‘s most likely to be identical across countries with factor price equality.

  30. Factor price equalization Integrated world economy: Two types of industries: Io oligopolies, Ic competitive industries. Equilibrium conditions: := cxx/c….elasticity of costs, measure for increasing returns to scale. := 1/ = c/cx, c = average cost. Consequently, pR-1 = cx can be written as pR-1 = c/.

  31. Collecting terms, the equilibrium conditions are • A(w)xc + A(w)xO = v • Factor price equalization (and reproduction of integrated world equilibrium) requires that factor allocation is compatible with number of firms of the oligopoly in each country.

  32. Figure 4.7: FPE-set, industry 1: oligopoly, industries 2 and 3: competitive.

  33. Net import of all factors‘s services: possible for a country with a large share in oligopolistic industries. Suppose sj (share of country j in world GDP) satisfies sj > v1j/v1 = ...= vNj/vN i.e. share in world GDP is greater than share in factor endowments: possible due to profits.

  34. Figure 4.8: capital rich country exports embodied capital services and imports embodied labor services. C’: share in factor income, C”: share in world GDP.

  35. Figure 4.9: capital rich country imports embodied capital services and embodied labor services. C’: share in factor income, C”: share in world GDP.

  36. 4.2.3.3 Market segmentation Introduction of transport costs t: Example: two identical countries, each with one firm. Home market: 2 firms, asymmetric Cournot model: domestic firm has marginal costs c, foreign firm has marginal costs c + t. Equilibrium conditions: p(xd + xf) + xdp‘(xd + xf) = c, p(xd + xf) + xfp‘(xd + xf) = c + t. Equilibrium with xd > xf > 0 exists for t < pm – c. pm = monopoly price.

  37. Welfare effects: • w = u(xd + xf) – c(xd + xf) – txf • dw = u’(xd + xf)(dxd + dxf) – c(dxd + dxf) – tdxf– dtxf • Normalize u(x) s.t. u‘(x) = p: • dw = (p – c – t)(dxd + dxf) + tdxd– dtxf • Note that dt < 0  dxd < 0 < dxf • xfdt...cost reduction  w, but xf = 0 if t = pm – c. • (p – c – t)(dxd +dxf)  w (export increase), but 0 if t = pm – c. • tdxd...  w (replacement of domestic shipments through exports). • dw/dt > 0 at t  p – c.

  38. Figure 4.10: Gains and losses due to changes of t.

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