The General Neoclassical Trade Model

Transcription

1 The General Neoclassical Trade Model J. Peter Neary University of Oxford October 15, 2013 J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

2 Plan of Lectures 1 Review of Consumer Theory 2 Producer Behaviour 3 Some Special Models 4 The Trade Expenditure Function 5 The Small Open Economy: All Goods Traded 6 SOE: Some Goods Non-Traded 7 The Small Open Economy with Tariffs J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

3 Review of Consumer Theory Plan of Lectures 1 Review of Consumer Theory 2 Producer Behaviour 3 Some Special Models 4 The Trade Expenditure Function 5 The Small Open Economy: All Goods Traded 6 SOE: Some Goods Non-Traded 7 The Small Open Economy with Tariffs J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

4 Review of Consumer Theory Review of Consumer Theory Maximisation of utility function u (x) subject to a budget constraint p.x I Marshallian demand functions: x (p, I) Substitute into u Indirect utility function: V (p, I) u [x (p, I)] Invert V (p, I) = u 0 to get the Expenditure Function e(p, u 0 ) = I: e ( p, u 0) [ Min ] x p.x : u (x) u 0 e is increasing in p Shephard s Lemma: Hicksian demand functions: x c (p, u) = e p (p, u) e is concave in p x c p = e pp is negative semi-definite; i.e., Hicksian demand functions cannot slope upwards N.B. Homothetic tastes: u (x) = f [g (x)] where f > 0 and g is homogeneous of 1 V (p, I) = IṼ (p) e (p, u) = up (p) [P (p) is the true ( Konüs ) price index.] ln V ln p = ln Ṽ ln p i.e., budget shares independent of I. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

5 Producer Behaviour Plan of Lectures 1 Review of Consumer Theory 2 Producer Behaviour The Production Possibility Frontier The GDP Function Properties of g as a Function of p Properties of g as a Function of p (cont.) Properties of g as a Function of v Properties of g as a Function of v (cont.) Properties of g as a Function of p and v 3 Some Special Models 4 The Trade Expenditure Function 5 The Small Open Economy: All Goods Traded J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

6 Producer Behaviour The Production Possibility Frontier Producer Behaviour The Production Possibility Frontier y 2 y' p' y 1 The PPF is defined by the concave function F (y; v) = 0 It encloses a convex set of feasible y The competitive process ( invisible hand ) maximises the value of GDP, p.y, given the PPF In competitive equilibrium: dy 2 dy 1 = p 1 p 2 J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

7 Producer Behaviour The GDP Function The GDP Function Assume a competitive economy, producing n goods: Net outputs: y Elements of y can be negative: e.g., intermediate imports (oil) Prices: p Endowments of m factors are given: v Production must be feasible: y given v and technology. Non-increasing returns to scale (necessary for perfect competition) a convex set of feasible {y, v} Competitive behaviour leads to the maximisation of the value of GDP: p.y is maximised given feasibility The solution to this problem is the vector of competitive supply functions y(p, v) Define the function: g(p, v) Max y [p.y : (y, v) feasible] This is the GDP function [or profit function or (Dixit/Norman) revenue function ] Note: g(p, v) = p.y(p, v) J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

8 Producer Behaviour Properties of g as a Function of p Properties of g as a Function of p Analogous to the expenditure function: g is increasing in p Hotelling s Lemma: Competitive supply functions: y (p, v) = g p (p, v) g is convex in p y p = g pp is positive semi-definite; i.e., Competitive supply functions cannot slope downwards Skip Proofs Proofs: Consider some price vector p and let y be the corresponding optimal output vector: g(p, v) = p.y Since y is optimal for some p, then it is feasible for all p i.e., g SB (p, v) p.y is always feasible [ SB for Sleeping Beauty!] So: g is at least linear in p If there is any possibility of changing outputs optimally in response to price changes, then GDP must be higher still: g(p, v) p.y So: H(p) g(p, v) p.y is non-negative everywhere; zero at p = p i.e., this function (actual less SB GDP) is minimised at p = p. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

9 Producer Behaviour Properties of g as a Function of p GDP is Convex in Prices GDP g(p,v) g SB (p,v) p i 0 p i J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

10 Producer Behaviour Properties of g as a Function of p (cont.) Properties of g as a Function of p (cont.) Proofs (cont.): If g is twice differentiable, necessary conditions for a min. of H: (i) First derivative must be zero at p = p : g p (p, v) y = 0 Since this holds at any point: y(p, v) = g p (p, v). This is Hotelling s Lemma: the price derivatives of the GDP function equal the competitive GE supply functions. (ii) Second derivative matrix g pp must be positive semi-definite. i.e., g is convex in p: matrix of price derivatives y p is pos. semi-def. Implications: (a) Supply curves cannot slope downwards: y i / p i 0 i (b) Supply curves for composite commodities cannot slope downwards: Let p = p 0 P be the price vector for any subset of goods; P a scalar; Define Y p 0.y : fix p 0 so Y is a scalar composite commodity ; Then: Y/ P = p 0. y/ P = p 0.{ y/ p}p 0 0 (c) Price changes and supply responses are positively correlated: For any p, p : p.y = g(p, v) p.y, p.y = g(p, v) p.y p.y = g(p, v) p.y Hence: (p p ).(y y ) 0 J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

11 Producer Behaviour Properties of g as a Function of p (cont.) The Law of Supply y 2 p" y' y" p' p".(y"-y') ( " p'.(y'-y") ( ' y 1 J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

12 Producer Behaviour Properties of g as a Function of v Properties of g as a Function of v g must be concave in v, because technology is convex; Proof: Consider any 2 endowments v and v, with net outputs y and y : Since the set of feasible {y, v} is convex, the average 1 2 (y + y ) is feasible given average endowments 1 2 (v + v ). Therefore GDP of 1 2 [g(p, v ) + g(p, v )] is attainable; and maximum GDP may be even greater. i.e., the function cannot lie below its chord and so is concave. Interpretation of slope g v at v : At the original v, each factor was allocated optimally, earning the same marginal product [ MP ] in all uses. Adding an extra unit of v i and rearranging production optimally yields (to first order) extra GDP equal to this common MP. So, for all factors: w(p, v) = g v (p, v) Given v, these are GE inverse factor demand functions. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

13 Producer Behaviour Properties of g as a Function of v (cont.) Properties of g as a Function of v (cont.) Concavity second derivative matrix g vv is negative semi-definite. i.e., matrix of price derivatives w v is neg semi-def. Implications: a. Any j: w j / v j 0 i.e., factor demand curves cannot slope up. b. Let v = v 0 V be the endowment vector for any subset of factors; V a scalar; define W v 0.w : a scalar composite factor ; fix v 0. Then: W/ V = v 0. w/ V = v 0.{ w/ v}v 0 0 i.e., demand curves for composite factors cannot slope upwards. c. For any two v: g(p, v ) g(p, v ) + (v v ).w and g(p, v ) g(p, v ) + (v v ).w Hence: (v v ).(w w ) 0 i.e., endowment changes and factor-price responses are negatively correlated. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

14 Producer Behaviour Properties of g as a Function of p and v Properties of g as a Function of p and v First derivatives: g p (p, v) = y(p, v), g v (p, v) = w(p, v) Hessian of g: [ gpp g vp g pv g vv ] = [ y p w p y v w v ] [ (1) (3a) (3b) (2) 1 g pp is PSD: Normal Price-Output Response. Back 2 g vv is NSD: Normal Endowment-Factor-Price Response. Back Not ND : g vv = 0 is the case of local factor-price equalisation. 3 Reciprocity : These matrices are transposes of each other: (3a) g pv : Rybczynski derivatives - endowment-output responses. (3b) g vp : Stolper-Samuelson derivatives: price-factor-price responses. What else do we know about g pv and g vp? Not much in general. Homogeneity of g in p: p g pv = w, so they are positive on average. Elements of g pv can be used to define factor intensities in GE. i.e., good i is intensive in factor j IFF g ij is positive. {g ij > 0 g ji > 0}: Rise in p i raises w j IFF good i uses j intensively. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28 ]

15 Some Special Models Plan of Lectures 1 Review of Consumer Theory 2 Producer Behaviour 3 Some Special Models Specific-Factors Model Heckscher-Ohlin [-Samuelson] Model 4 The Trade Expenditure Function 5 The Small Open Economy: All Goods Traded 6 SOE: Some Goods Non-Traded 7 The Small Open Economy with Tariffs J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

16 Some Special Models Specific-Factors Model Some Special Models The Specific-Factors Model: n goods, each using one specific and one mobile factor Call the mobile factor labour (L, w), the others capital (K i, r i ) 1 g pp : All goods are substitutes: y i / p i < 0, i i 2 g vv : Labour is friends with all specific factors: w/ K i > 0, i Each specific factor is friends with labour: r i / L > 0, i... but enemies with other specific factors: r i / K i < 0, i i 3 g pv : Goods are friends with labour and their own specific factor; but enemies with other specific factors: y i / L = w/ p i > 0, i y i / v i = r i / p i > 0, i y i / v j = r j / p i < 0, i j Magnification Effect for specific factors only: ˆp 1 > ˆp 2 ˆr 1 > ˆp 1 > ŵ > ˆp 2 > ˆr 2 J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

17 Some Special Models Heckscher-Ohlin [-Samuelson] Model Some Special Models The Heckscher-Ohlin Model: 2 goods (M and F ), each using 2 mobile factors (K and L) M uses (say) K intensively: k M > k F 1 g pp : Goods are substitutes: x i / p i < 0, i i 2 g vv = 0! Local factor-price equalization: w j / v j = 0, j and j 3 g pv : Goods are friends with factor they use intensively only y F / L = w/ p F > 0; and magnified; y F / K = r/ p F < 0, etc. Magnification Effect for both factors: Stolper-Samuelson: ˆp M > ˆp F ˆr > ˆp M > ˆp F > ŵ Rybczynski: ˆL > ˆK ŷf > ˆL > ˆK > ŷ M J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

18 The Trade Expenditure Function Plan of Lectures 1 Review of Consumer Theory 2 Producer Behaviour 3 Some Special Models 4 The Trade Expenditure Function 5 The Small Open Economy: All Goods Traded 6 SOE: Some Goods Non-Traded 7 The Small Open Economy with Tariffs J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

19 The Trade Expenditure Function The Trade Expenditure Function Define: E(p, u, v) e(p, u) g(p, v) Properties: 1 E p (p, u, v) = e p (p, u) g p (p, v) = x c (p, u) y(p, v) These are GE net import demand functions: = m(p, u, v). 2 e(p, u) p.x and g(p, v) p.y E(p, u, v) = p.m(p, u, v) p.m with equality where p = p i.e., E is a minimum value function, with similar properties to e. 3 e is concave, g is convex in p: hence E is concave in p. So: E pp = m p is NSD: GE import demand functions slope down. 4 E u = e u : marginal cost of utility; inverse of marginal utility of income. So: E pu = e pu = x Ie u; proportional to income derivatives of demand. 5 E v = g v = w; so E vp = g vp : Stolper-Samuelson derivatives (etc.) Note: Derivatives with respect to u and v are independent of each other, reflecting the assumption of inelastic factor supplies. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

20 The Small Open Economy: All Goods Traded Plan of Lectures 1 Review of Consumer Theory 2 Producer Behaviour 3 Some Special Models 4 The Trade Expenditure Function 5 The Small Open Economy: All Goods Traded 6 SOE: Some Goods Non-Traded 7 The Small Open Economy with Tariffs J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

21 The Small Open Economy: All Goods Traded The Small Open Economy: All Goods Traded Equilibrium in a SOE: E(p, u, v) = b (1 equation in 1 unknown: u) Effects of exogenous shocks: e u du = db E pdp E vdv = db m dp + w dv e u du: change in real income (utility in expenditure units) 1 b: Shadow price of foreign exchange [e u du/db] is unity 2 p: Change in terms of trade is minus a trade-weighted sum of world price changes. (So u rises when export prices rise.) 3 v: Endowment increases raise welfare by the amount of the factor s marginal reward. i.e., shadow prices of factors of production [e udu/dv] equal their market prices [w] All these results are unsurprising! But useful benchmarks for later. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

22 SOE: Some Goods Non-Traded Plan of Lectures 1 Review of Consumer Theory 2 Producer Behaviour 3 Some Special Models 4 The Trade Expenditure Function 5 The Small Open Economy: All Goods Traded 6 SOE: Some Goods Non-Traded Dutch Disease and Balassa-Samuelson 7 The Small Open Economy with Tariffs J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

23 SOE: Some Goods Non-Traded SOE: Some Goods Non-Traded Now TEF depends on 2 sets of prices: E(p T, p N, u, v) = b Equilibrium in markets for non-traded goods [ NTGs ]: E N (p T, p N, u, v) = 0 (Now: 2 equations in 2 unknowns: u and p N ) As before: e u du = db m dp T + w dv [m = E T ] Now: effects of shocks on p N : dp N = ( E NN ) 1 [E NT dp T + x NI e u du g Nv dv] = ( E NN ) 1 [x NI db + (x NI w g Nv )dv + (E NT x NI m )dp T ] General: Expression in square brackets gives the effects of exogenous shocks on excess demand for NTGs. If this rises, then p N must also rise to restore equilibrium. Strictly, this is only true for scalar p N ; vector case requires more care. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

24 SOE: Some Goods Non-Traded Dutch Disease and Balassa-Samuelson SOE with Non-Traded Goods (cont.) Effects of a Transfer (db > 0) u rises and p N rises (dp N x NI db) A real exchange rate appreciation: The Dutch Disease Effects of Economic Growth (dv > 0) u rises; change in p N the sum of spending and resource-movement effects: dp N (x NI w g Nv )dv. Time series: Dutch Disease outcome if g Nv not too positive; i.e., if growth at initial prices either lowers output of non-traded goods or does not raise it too much. Cross-section: Balassa-Samuelson Effect i.e., higher-gdp countries have higher relative prices of non-traded goods. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

25 The Small Open Economy with Tariffs 7 The Small Open Economy with Tariffs The Tariff Multiplier Effects of Transfers and Growth Effects of a Change in Tariffs J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28 Plan of Lectures 1 Review of Consumer Theory 2 Producer Behaviour 3 Some Special Models 4 The Trade Expenditure Function 5 The Small Open Economy: All Goods Traded 6 SOE: Some Goods Non-Traded

26 The Small Open Economy with Tariffs The Tariff Multiplier The Small Open Economy with Tariffs Direct Effects of Tariffs: Prices: p = p + t (composite export good is numéraire) Redistribution: Tariff revenue t m; assumed redistributed lump-sum. Equilibrium: E(p, u, v) = b + t m and m = E p (p, u, v) Effects of a transfer of the numéraire good (p, t and v fixed): Totally differentiate: E pdp + E u du = db + m dt + t dm E pdp and m dt are equal and cancel: costless redistribution from public to private sectors, and dp = dt since p is fixed. Also: dm = E pu du = x I e u du Combining and simplifying: e u du = µdb where: µ 1 1 t x I tariff multiplier (p x I: marginal propensity to consume importables) or shadow price of foreign exchange (greater than market price, 1) Must be positive for stability (Hatta REStud 1977). So, a transfer of 1 raises utility by more than 1! Why? It allows more imports, which are too low because of the tariff. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

27 The Small Open Economy with Tariffs Effects of Transfers and Growth Effects of Transfers and Growth Effects of a transfer of non-numéraire goods: Replace b by b + p x. Transfers reduce the need for imports, so: m = E p (p, u, v) x. Totally differentiate: E u du = p d x + t dm and dm = E pu du d x Simplifying: µ 1 e u du = p d x. So: the shadow prices of traded goods equal their world prices. Not their domestic prices, since these are distorted by tariffs. This is the Little-Mirrlees Rule. Effects of Growth: E u du + E vdv = t dm dm = E pu du + E pv dv (1 t x I ) e u du = (w t g pv )dv So: Shadow prices of factors are less than their market prices w if importables use them intensively. Why? Because in that case growth expands production of importables. This reduces imports, exacerbating the trade distortion. Shadow prices of factors can be negative: growth is then immiserizing J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28

28 The Small Open Economy with Tariffs Effects of a Change in Tariffs Effects of a Change in Tariffs Effects of a change in tariffs: Totally differentiate: E u du = t dm and dm = E pp dt + E pu du Simplifying: µ 1 e u du = t E pp dt. With only one tariffed good, RHS must be negative (E pp < 0), so welfare is monotonically decreasing in the tariff. With more than one, t E pp dt need not be negative. Hence the problem of the second best : Cutting one tariff may lower welfare if another tariff remains in place. [Pictures: Neary (IER 1995)] Only a few results are available: 1 Uniform Radial Reduction Rule: dt = tdα (dα a scalar). Now: t E ppdt/dα = t E ppt; quadratic form in a NSD matrix, so 0. 2 Concertina Rule : Cutting the highest tariff raises welfare if the good is substitutable with all others. Anderson and Neary (JIE 2007) generalise these results. J.P. Neary (University of Oxford) Neoclassical Trade Model October 15, / 28