Rybczynski Redux. E. Fisher. 17 June Department of Economics California Polytechnic State University Visiting CES

Transcription

1 Rybczynski Redux E. Fisher Department of Economics California Polytechnic State University Visiting CES 17 June 2010

2 Outline

3 Understanding Technological Differences In the first lecture, we saw that the theory of factor content falls short because countries have different technologies This idea was proposed by Leontief and it has not been worked out satisfactorily yet. In essence, we must answer the question, What is a German worker worth, in terms of the United States technology? Marshall and I have developed two answers to this question 1 The factor content in the USA of the Rybczynski effects in Germany of an extra worker 2 The wage in the USA actually maps best onto a linear combination of all factor prices in Germany

4 Simplest Rybczynski Effect y_2 y_1

5 Heckscher-Ohlin-Vanek theory fails because countries have different technologies We know the output vector produced by Germany What would Germany s endowment have to be if she produced using the American technology? This is called Germany s virtual endowment, when the USA is reference

6 The national revenue function Let v be a vector of primary factors in fixed supply The set of feasible outputs F(v) R n is parameterized by v Let p R n + be output prices The national revenue function r(p, v) = max y F (v) p T y The classic reference is Dixit and Norman, Theory of International Trade, Cambridge University Press, 1980

7 Properties of the national revenue function r(p, v) is homogeneous of degree one in p Assume that r(p, v) is differentiable All gradients are row vectors The supply vector is r p (p, v) = y This function is homogeneous of degree 0 in prices The vector of factor prices is r v (p, y) = w Its Hessian r pv is the n f matrix of Rybczynski effects The transpose of the Rybczynski matrix is the Stolper-Samuelson matrix

8 Rybczynski and Stolper-Samuelson effects 2 r(p, v) = y i p i v j v j This term shows how the output of good i changes when the supply of factor j changes, holding goods prices and thus factor rewards as fixed 2 r(p, v) = w j p i v j p i This term shows how factor price j changes when the price of good i changes, holding factors in fixed supply. This relationship shows the duality between Rybczynski effects and Stolper-Samuelson effects, and it is one of the deepest ideas in trade theory

9 Simplest example where 2 = n > f = 1 Ricardian model with two goods and labor a i is the unit input requirement for sector i { p1 L/a r(p, v) = 1 if p 1 /p 2 a 1 /a 2 p 2 L/a 2 otherwise This function is not differentiable

10 National revenue function is convex r(p,v) (p_1/a_1)l (p_2/a_2) L a_1/a_2 p_1/p_2

11 When it is differentiable There is no problem deriving the Rybczynski matrix in this case { (0, L/a2, ) if p r p (p, v) = 1 /p 2 < a 1 /a 2 (L/a 1, 0) if p 1 /p 2 > a 1 /a 2 { (0, 1/a2 ) r pv (p, v) = T if p 1 /p 2 < a 1 /a 2 (1/a 1, 0) T if p 1 /p 2 > a 1 /a 2

12 The technology matrix and its uses The technology matrix A = [ a1 Prices satisfy p = Aw They lie in the column space of A Full employment conditions v = A T y Endowments lie in the row space of A a 2 ]

13 Solutions to Ax = b A is n f and has rank r Three cases 1 There is a unique solution 2 There are many solutions 3 There is no solution since the equations are inconsistent. This case is called econometrics, The case where n > f = r is of practical interest to us. Only special prices will allow several goods to be sold in positive quantities, and the output supply is a correspondence. It is not single-valued

14 Moore-Penrose pseudo inverse x = A + b + (I A + A)z, where z R f The term A + b is the particular solution The term I A + A is the homogeneous solution If n f = r, then it is often the case that I A + A = 0

15 Four properties define this pseudo inverse 1 AA + A = A 2 A + AA + = A + 3 (A + A) T = A + A 4 (AA + ) T = AA +

16 First example 1 3x 1 = = 1/3 3 x 1 = (1/3)4 + (1 1)z

17 Second example 1 3x 1 + x 2 = 4 [ ] [ ] = 0.1 [ ] [ ] [ x = x 2 ] [ z1 z 2 ]

18 The solution of minimum norm x_2 (A+)^T 4 = (1.2,0.4) Slope = -3, intercept = 4 x_1

19 Third example x 1 = = [ ] 3 x 1 = (1 1)z

20 What is ?

21 Calculating the Moore-Penrose pseudo inverse If A T A has full rank, then A + = (A T A) 1 A T So you can calculate this in Excel This generalized inverse always has dimension f n The Moore-Penrose inverse is the regular inverse for a square matrix The Moore-Penrose inverse has an important symmetry property (A + ) T = (A T ) +

22 The Moore-Penrose Inverse of the technology matrix [ ] a1 A = a 2 [ A + = a1 2 + a2 2 a 1 a 2 a a2 2 ]

23 The transpose of A + is the Rybczynski matrix Full employment is v = A T y y = (A T ) + v + (I (A T ) + A T )z where now z R n y = (A + ) T v + (I (A + ) T A T )z is the complete supply correspondence

24 Jones JPE, 1965 The technology matrix really is a function of local factor prices A(w) But cost minimization implies that, for small changes in factor prices, Adw + (da)w = Adw because for each good i, f da if w f = 0 by the envelope theorem Hence every technology is locally a fixed coefficients Leontief technology I think of A + as a Stolper Samuelson matrix for price changes dp that lie in the column space of A(w)

25 Supply correspondence y = (A T ) + v + (I (A T ) + A T )z r(p, v) = p T y = p T (A T ) + v + w T A T (I (A T ) + A T )z r(p, v) = p T y = p T (A T ) + v since A T (I (A T ) + A T ) = 0 The bottom line is that r(p, v) = p T (A + ) T v, a simple quadratic form.

26 Factor prices are overdetermined v = A + p + (I A + A)z r(p, v) = v T w = v T A + p + y T A(I A + A)z r(p, v) = v T w = v T A + p since A(I A + A) = 0 The bottom line is that r(p, v) = v T A + p, the same quadratic form.

27 (A + ) T is the Hessian of r(p, v) This quadratic form is everywhere differentiable We can recover the entire supply correspondence using the homogeneous term A T was designed by Leontief to show how many extra resources were need to produce y (A + ) T gives the change in the output vector that arises from v

28 The Rybczynksi Effect and Movements along the flat y_2 (A+)^T v y Any movement along the PPF has no effect on national revenue y_1

29 The real world We live in a world where there are more goods than factors The output vector of the German economy is one particular value from a correspondence In real world applications, almost every good is produced and traded in every country Hence price changes lie in a restricted column space spanned by each local technology matrix!

30 Numerical example First column is capital and second column is labor 1 1 A = /2 4/3 (A + ) T = 0 1/3 1/2 2/3

31 Numerical example, row stochastic matrix (r, w) = (1, 1) Θ = (Θ + ) T = The Rybczynski matrix is column stochastic. This fact echoes the national income identity.

32 Fisher and Marshall, forthcoming Review of International Economics

33 Estimating factor rewards In this work, we did not use consistent data on factor uses. We wanted to study types of labor By assumption Aw = 1 w = A (I A + A)z, but the homogeneous term disappears Hence ŵ = (A T A) 1 A T 1 We know that we measure the true factor prices with error.

34 Fisher and Marshall, forthcoming Review of International Economics

35 The local factor content of a foreign Rybczynski matrix I hope you find these Rybczynski matrices useful Think about the American (Country 2) factor content of another piece of capital in Germany (Country1) y 1 = (A + 1 )T v 1 + (I (A + 1 )T A T 1 )z where z Rn I will write the homogeneous term as u 1. Remember it has no factor content in Country 1 A T 2 y 1 = A T 2 (A+ 1 )T v 1 + A T 2 u 1 The factor conversion matrix is A T 2 (A+ 1 )T

36 Two interpretations of a factor conversion matrix A T 2 (A+ 1 )T is the f f matrix that translates country 1 factors into those in country 2 Only in rare cases will it be diagonal A + 1 A 2 is the matrix that translates factor prices in country 2 into those in country 1 Normally the wage in country 2 corresponds to a linear combination rent and wage in the country 1

37 Leontief s idea of factor-specific differences The first column is capital and the second is labor. 1 1 A 1 = A 2 = It is obvious that the first country has very good capital and also good workers

38 Example of a factor conversion matrix (A 2 ) T (A + 1 )T = [ ] [ ] 1/2 4/3 0 1/3 1/2 2/3 = One piece of capital in country 1 equals 10 country 2 pieces of capital and no country 2 workers. One country 1 worker equals 2 country 2 workers and no pieces of capital.

39 The factor conversion matrix from Germany to USA The first column is capital, the second is labor, and the third is social capital This matrix is column stochastic $1 of capital in Germany corresponds to $0.67 of US capital. $0.23 of US labor, and $ 0.10 of US social capital

40 Defining a virtual endowment We know the output vector of country i. We also know the technology of the reference country A 0 The virtual endowment of country i as ṽ i = A T 0 y i It depends on the reference country 0 Now the measured factor content of trade is A 0 (x i m i ) and the predictions are based upon i ṽ i We have imposed the pure HOV world

41 Traditional HOV, USA Reference Traditional HOV, USA Reference Country (millions of 2000 dollars) 50,000 40,000 30,000 Measured Factor Content 20,000 10, ,000-30,000-10,000 10,000 30,000 50,000-10,000-20,000-30,000 K L G -40,000-50,000 Predicted Factor Content

42 , USA Reference Country Fig. 2: Virtual Endowments, USA Reference Country (millions of 2000 dollars) 50,000 40,000 30,000 Measured Factor Content 20,000 10, ,000-30,000-10,000 10,000 30,000 50,000-10,000-20,000-30,000 K L G -40,000-50,000 Predicted Factor Content

43 , Korea Reference Country Fig. 3: Virtual Endowments, Korea Reference Country (millions of 2000 dollars) 50,000 40,000 30,000 Measured Factor Content 20,000 10, ,000-30,000-10,000 10,000 30,000 50,000-10,000-20,000-30,000 K L G -40,000-50,000 Predicted Factor Content

44 What have we learned? 1 Measurement error does not matter 2 Homothetic preferences are in the data 3 No home bias in consumption 4 Every good is traded 5 Trade costs do not matter 6 No need to adjust for trade in intermediate inputs 7 Constant returns to scale are in the data

45 Objections to this approach We have assumed away differences in technology We are only testing the demand side of the model The tests using virtual endowments are tautologies

46 World endowments World ldendowments in Barycentric Coordinates G CHN K IDN RUS TUR USA CHE L

47 , USA is the reference Virtual Endowments with ihusa as Rf Reference G USA K L

48 Virtual endowment and the factor conversion matrix The vector y i = (A + i ) T v i + (I (A T i ) + A T i )z i for some z i. So country i s virtual endowment is ṽ i = A T 0 (AT i ) + v i + A T 0 (I (A+ i ) T A T i )z i So its virtual endowment is its actual endowment converted into factors in the reference country, plus an error term that has no factor content in country i

49 Tests using factor conversion matrices Fig. 5: HOV without FPE (millions of 2000 dollars) 50,000 40,000 30,000 Measured Factor Content 20,000 10, ,000-30,000-10,000 10,000 30,000 50,000-10,000-20,000-30,000 K L G -40,000-50,000 Predicted Factor Content

50 National revenue function Rybczynski theory Rybczynski matrices