CHAPTER 1. BASIC ISSUES 3 ma+pam2<mi+p.? Combining the equality and the inequality, using that m=0 and that p'mt=0, we have that p2-p)m2<0. Thus the home country will import good 2 if and only if its relative autarchy price is higher than in the trade equilibrium. With two countries, the same is true for the foreign country. Since trade must be balanced, as a corollary we get that the free-trade price must be between the two autarchy price ratios Of course, to determine the equilibrium price, we also need to know demand patterns 1.2 Explaining comparative advantage In the pure exchange model above, there might be two reasons why autarchy prices differ across countries. One is demand and the other endowments Example: same endowment but different taste, or same taste but different endowment. In a more general production model, we can look at tastes technology and factor abundance. Tastes work the same way, so let us deal irst, technology. This is the Ricardian explanation for trade and can be illustrated with one factor, labor. Suppose that consumers want to con- sume both goods in positive quantities. Then in autarchy, a country has to produce both goods 1 and 2. Let a, indicate the unit labor requirement to produce good j, and let w stand for the wage rate. Competition and the re- quirement that both goods are produced ensures that price equals marginal (and average)cost in both sectors a1a=1 p Dividing the second equation by the first, we get that:
CHAPTER 1. BASIC ISSUES 3 it: ma 1 + p ama 2 < mt 1 + p amt 2 . Combining the equality and the inequality, using that ma = 0 and that p tmt = 0, we have that: (p t − p a )mt 2 < 0. Thus the home country will import good 2 if and only if its relative autarchy price is higher than in the trade equilibrium. With two countries, the same is true for the foreign country. Since trade must be balanced, as a corollary we get that the free-trade price must be between the two autarchy price ratios. Of course, to determine the equilibrium price, we also need to know demand patterns. 1.2 Explaining comparative advantage In the pure exchange model above, there might be two reasons why autarchy prices differ across countries. One is demand and the other endowments. Example: same endowment but different taste, or same taste but different endowment. In a more general production model, we can look at tastes, technology and factor abundance. Tastes work the same way, so let us deal with the other two. First, technology. This is the Ricardian explanation for trade and can be illustrated with one factor, labor. Suppose that consumers want to consume both goods in positive quantities. Then in autarchy, a country has to produce both goods 1 and 2. Let aj indicate the unit labor requirement to produce good j, and let w stand for the wage rate. Competition and the requirement that both goods are produced ensures that price equals marginal (and average) cost in both sectors: a1w a = 1 and a2w a = p a . Dividing the second equation by the first, we get that: p a = a2 a1
CHAPTER 1. BASIC ISSUES thuscompa arative advantage is determined by the relative efficiency of a coun- try to produce goods. The equilibrium free-trade price vector will be between a2/an and A2/A1. This is Ricardo's famous insight: trade patterns are de- termined by relative, and not absolute, advantage Second. factor abundance. Assume identical technologies. two factors (we need at least two) and fixed technologies. Let bij indicate the amount of factor i to produce one unit of good j, and let vi be the amount of factor i available and a the autarchy production vector. Assuming that both factors are fully employed, we have that r1b1+x2b12=01 Divide the second equation by the first and solve for the ratio r2/n to get 22 It is easy to see that d(r2/=) b11b22-b12b21 d(2/v1) This expression is positive if and only if the numerator is positive, which can be rewritten as b22/b12>b21/b11. In words, the production of good 2 relative to good 1 will be a positive function of the relative endowment of factor 2 if and only if its production is relatively intensive in factor 2. Without loss of generality we can assume this to be the case The final step in the chain of argument that relates factor endowment and autarchy prices comes from demand. We rule out demand differences in order to focus on factor abundance. This is, however, not enough. The problem is that the consumption ratio(which in autarchy must equal the production ratio) is a function of not just the relative price, but also income Thus to avoid complications with income effects, we have to assume identical homothetic preferences. Then c2/ci will be a decreasing function of p alone, and thus Pa will depend on v2/v negatively. Thus we can conclude that with identical homothetic preferences, a country will have a comparative advantage in producing a good that uses its abundant factor intensively
CHAPTER 1. BASIC ISSUES 4 thus comparative advantage is determined by the relative efficiency of a country to produce goods. The equilibrium free-trade price vector will be between a2/a1 and A2/A1. This is Ricardo’s famous insight: trade patterns are determined by relative, and not absolute, advantage. Second, factor abundance. Assume identical technologies, two factors (we need at least two) and fixed technologies. Let bij indicate the amount of factor i to produce one unit of good j, and let vi be the amount of factor i available and x a the autarchy production vector. Assuming that both factors are fully employed, we have that: x a 1 b11 + x a 2 b12 = v1 and x a 1 b21 + x a 2 b22 = v2. Divide the second equation by the first and solve for the ratio x a 2 /xa 1 to get: x a 2 x a 1 = b11v2/v1 − b21 b22 − b12v2/v1 . It is easy to see that: d(x a 2 /xa 1 ) d(v2/v1) = b11b22 − b12b21 (b22 − b12v2/v1) 2 . This expression is positive if and only if the numerator is positive, which can be rewritten as b22/b12 > b21/b11. In words, the production of good 2 relative to good 1 will be a positive function of the relative endowment of factor 2 if and only if its production is relatively intensive in factor 2. Without loss of generality we can assume this to be the case. The final step in the chain of argument that relates factor endowments and autarchy prices comes from demand. We rule out demand differences in order to focus on factor abundance. This is, however, not enough. The problem is that the consumption ratio (which in autarchy must equal the production ratio) is a function of not just the relative price, but also income. Thus to avoid complications with income effects, we have to assume identical homothetic preferences. Then c a 2 /ca 1 will be a decreasing function of p a alone, and thus pa will depend on v2/v1 negatively. Thus we can conclude that with identical homothetic preferences, a country will have a comparative advantage in producing a good that uses its abundant factor intensively
CHAPTER 1. BASIC ISSUES Notice how much weaker this statement is than its Ricardian counterpart We need homothetic preferences, an unambiguous definition of factor intensi- ties(not trivial when input coefficients are not fixed), and two factors. Even with these, we will see that the factor abundance theory does not readily generalize to higher dimensions. Some other features of the theory, such as factor price equalization, however, do
CHAPTER 1. BASIC ISSUES 5 Notice how much weaker this statement is than its Ricardian counterpart. We need homothetic preferences, an unambiguous definition of factor intensities (not trivial when input coefficients are not fixed), and two factors. Even with these, we will see that the factor abundance theory does not readily generalize to higher dimensions. Some other features of the theory, such as factor price equalization, however, do
Chapter 2 Analytical tools 2.1 The revenue function An extremely useful tool in trade theory is the revenue (or gDP) function is an envelope function defined as follows r(p, u)=maxp l (a, v)EYI, where y is the production possibility set for the economy, a is the production vector, v is the factor endowment vector and p indicates prices. In words the revenue function indicates the maximum amount of gdP a country can chieve given its factor supply and prices The following properties are easy to prove nction p r(p, u)is convex in p. Take any P1, P2 and let Pa= ap1+(1-a)p2. Also let 11, I2 and Ta be the corresponding optimal output vectors. Then =ap1xa+(1-a)pa≤ (1-a)p (P2,) If r is differentiable in p, then a =Tp(p, a)-the Envelope Theorem r(p, v) is homogenous of degree one in p, so that prp(p, u)=r(p, v) Follows from the definition of r If r is twice differentiable in p, rpp is positive semi-definite(convexity)
Chapter 2 Analytical tools 2.1 The revenue function An extremely useful tool in trade theory is the revenue (or GDP) function. It is an envelope function defined as follows: r(p, v) = max x {p x|(x, v) ∈ Y }, where Y is the production possibility set for the economy, x is the production vector, v is the factor endowment vector and p indicates prices. In words, the revenue function indicates the maximum amount of GDP a country can achieve given its factor supply and prices. The following properties are easy to prove. First, for r(p, v) as a function of p: • r(p, v) is convex in p. Take any p1, p2 and let pα = αp1+(1−α)p2. Also, let x1, x2 and xα be the corresponding optimal output vectors. Then r(pα, v) = αp1xα + (1 − α)p2xα ≤ αp1x1 + (1 − α)p2x2 = αr(p1, v) + (1 − α)r(p2, v). • If r is differentiable in p, then x = rp(p, v) – the Envelope Theorem. • r(p, v) is homogenous of degree one in p, so that prp(p, v) = r(p, v). Follows from the definition of r. • If r is twice differentiable in p, rpp is positive semi-definite (convexity) and rppp = 0 (homogeneity). 6
CHAPTER 2. ANALYTICAL TOOLS .(P1-P2)(1-12)20-supply functions are positively sloped. Follows from P121 > Pi2 and 2 C2> P21 Now fix p and look at u If Y is convex, r(p, v) is concave in v. Proof similar to above, just note that if (a1, U1) Y and ( 2, U2)E Y then(aT1+[1-a 2, au+[1 alv2)∈Y If r(p, u) is differentiable in u, than T,(p, u)=w. Thus the shad prices of factors (which in competitive markets equal actual fa prices)are given by the gradient ru. Envelope Theorem If Y has constant returns to scale(Y is a cone), r(p, u) is linearly homogenous in v and vT =r(p, v). For any A>0, suppose that Ar(p, u)>r(p, Au), or Ap r(p, u)>r(p, Au). But c.r.s means that Arp, ul, Av)E Y, so that r(p, Au) cannot be optimal-a contradiction Other direction follows similarly If r is twice differentiable in u, Toy is negative semidefinite. When Y is crs, rv=0 .(U1-u2(w1-w2)<0, that is factor demand curves have negative If r(p, u)is twice differentiable, we have ui=xi. Follows from r w(p, u) is linearly homogenous in p, and thus pp prop w(p, u) Proof: w(Ap, u)=T,(Ap, u)=Ar,(p, u)=Aw(p, u) If Y is c r S, then a(p, v) is linearly homogenous in v, and thus ur,= 2.2 The cost function Notice that the revenue function is defined for a very general production structure. We worked with the production possibility set, which allows for
CHAPTER 2. ANALYTICAL TOOLS 7 • (p1 −p2)(x1 −x2) ≥ 0 – supply functions are positively sloped. Follows from p1x1 ≥ p1x2 and p2x2 ≥ p2x1. Now fix p and look at v: • If Y is convex, r(p, v) is concave in v. Proof similar to above, just note that if (x1, v1) ∈ Y and (x2, v2) ∈ Y then (αx1 + [1 − α]x2, αv1 + [1 − α]v2) ∈ Y . • If r(p, v) is differentiable in v, than rv(p, v) = w. Thus the shadow prices of factors (which in competitive markets equal actual factor prices) are given by the gradient rv. Envelope Theorem. • If Y has constant returns to scale (Y is a cone), r(p, v) is linearly homogenous in v and vrv = r(p, v). For any λ > 0, suppose that λr(p, v) > r(p, λv), or λp x(p, v) > r(p, λv). But c.r.s means that (λx[p, v], λv) ∈ Y , so that r(p, λv) cannot be optimal – a contradiction. Other direction follows similarly. • If r is twice differentiable in v, rvv is negative semidefinite. When Y is c.r.s., rvvv = 0. • (v1 − v2)(w1 − w2) ≤ 0, that is factor demand curves have negative slope. Finally, for cross effects: • If r(p, v) is twice differentiable, we have ∂wi ∂pj = ∂xj vi . Follows from rpv = rvp. • w(p, v) is linearly homogenous in p, and thus pwp = prvp = w(p, v). Proof: w(λp, v) = rv(λp, v) = λrv(p, v) = λw(p, v). • If Y is c.r.s, then x(p, v) is linearly homogenous in v, and thus vxv = vrp,v = x(p, v). 2.2 The cost function Notice that the revenue function is defined for a very general production structure. We worked with the production possibility set, which allows for