rybczynski redux - ifo institute for economic research · visiting ces 17 june 2010 e. fisher...

IntroductionThe national revenue function and its uses

The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology

Factor conversion matricesVirtual endowments

Summary

Rybczynski Redux

E. Fisher

Department of EconomicsCalifornia Polytechnic State University

Visiting CES

17 June 2010

E. Fisher Rybczynski Redux




Summary

Outline

1 The national revenue function and its uses

2 The Moore-Penrose Pseudo inverse

3 The revenue function for a Leontief technology

4 Factor conversion matrices

5 Virtual endowments





Summary

Understanding Technological Differences

In the first lecture, we saw that the theory of factor contentfalls short because countries have different technologiesThis idea was proposed by Leontief and it has not beenworked out satisfactorily yet.In essence, we must answer the question, “What is aGerman worker worth, in terms of the United Statestechnology?Marshall and I have developed two answers to thisquestion

1 The factor content in the USA of the Rybczynski effects inGermany of an extra worker

2 The wage in the USA actually maps best onto a linearcombination of all factor prices in Germany





Summary

Simplest Rybczynski Effect

y_2

y_1





Summary

Virtual endowments

Heckscher-Ohlin-Vanek theory fails because countrieshave different technologiesWe know the output vector produced by GermanyWhat would Germany’s endowment have to be if sheproduced using the American technology?This is called Germany’s virtual endowment, when theUSA is reference





Summary

The national revenue function

Let v be a vector of primary factors in fixed supplyThe set of feasible outputs F (v) ⊂ Rn is parameterized byvLet p ∈ Rn

+ be output pricesThe national revenue function r(p, v) = maxy∈F (v)pT yThe classic reference is Dixit and Norman, Theory ofInternational Trade, Cambridge University Press, 1980





Summary

Properties of the national revenue function

r(p, v) is homogeneous of degree one in pAssume that r(p, v) is differentiableAll gradients are row vectorsThe supply vector is rp(p, v) = y This function ishomogeneous of degree 0 in pricesThe vector of factor prices is rv (p, y) = wIts Hessian rpv is the n × f matrix of Rybczynski effectsThe transpose of the Rybczynski matrix is theStolper-Samuelson matrix





Summary

Rybczynski and Stolper-Samuelson effects

∂2r(p, v)

∂pi∂vj=

∂yi

∂vj

This term shows how the output of good i changes whenthe supply of factor j changes, holding goods prices andthus factor rewards as fixed∂2r(p, v)

∂vj∂pi=

∂wj

∂pi

This term shows how factor price j changes when the priceof good i changes, holding factors in fixed supply .This relationship shows the duality between Rybczynskieffects and Stolper-Samuelson effects, and it is one of thedeepest ideas in trade theory





Summary

Simplest example where 2 = n > f = 1

Ricardian model with two goods and laborai is the unit input requirement for sector i

r(p, v) =

{p1L/a1 if p1/p2 ≥ a1/a2p2L/a2 otherwise

This function is not differentiable





Summary

National revenue function is convex

(p_2/a_2) L

r(p,v) (p_1/a_1)L

p_1/p_2

a_1/a_2





Summary

When it is differentiable

There is no problem deriving the Rybczynski matrix in thiscase

rp(p, v) =

{(0,L/a2, ) if p1/p2 < a1/a2(L/a1,0) if p1/p2 > a1/a2

rpv (p, v) =

{(0,1/a2)T if p1/p2 < a1/a2(1/a1,0)T if p1/p2 > a1/a2





Summary

The technology matrix and its uses

The technology matrix A =

[a1a2

]Prices satisfy p = Aw They lie in the column space of AFull employment conditions v = AT y Endowments lie inthe row space of A





Summary

Solutions to Ax = b

A is n × f and has rank rThree cases

1 There is a unique solution2 There are many solutions3 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 inpositive quantities, and the output supply is acorrespondence. It is not single-valued





Summary

Moore-Penrose pseudo inverse

x = A+b + (I − A+A)z, where z ∈ Rf

The term A+b is the particular solutionThe term I − A+A is the homogeneous solutionIf n ≥ f = r , then it is often the case that I − A+A = 0





Summary

Four properties define this pseudo inverse

1 AA+A = A2 A+AA+ = A+

3 (A+A)T = A+A4 (AA+)T = AA+





Summary

First example

1 3x1 = 42 3+ = 1/33 x1 = (1/3)4 + (1− 1)z





Summary

Second example

1 3x1 + x2 = 4

2[

3 1]+

=

[0.30.1

]3

[x1x2

]=

[0.30.1

]4 +

[0.1 −0.3−0.3 0.9

] [z1z2

]





Summary

The solution of minimum norm

x_2

(A+)^T 4 = (1.2,0.4)’

x_1

Slope = -3,

intercept = 4





Summary

Third example

1

123

x1 =

345

2

123

+

=[

0.0714 0.1429 0.2143]

3 x1 = 1.8571 + (1− 1)z





Summary

What is 1.8571?





Summary

Calculating the Moore-Penrose pseudo inverse

If AT A has full rank, then A+ = (AT A)−1AT So you cancalculate this in ExcelThis generalized inverse always has dimension f × nThe Moore-Penrose inverse is the regular inverse for asquare matrixThe Moore-Penrose inverse has an important symmetryproperty (A+)T = (AT )+





Summary

The Moore-Penrose Inverse of the technology matrix

A =

[a1a2

]A+ =

[ a1

a21 + a2

2

a2

a21 + a2

2

]





Summary

The transpose of A+ is the Rybczynski matrix

Full employment is v = AT yy = (AT )+v + (I − (AT )+AT )z where now z ∈ Rn

y = (A+)T v + (I − (A+)T AT )z is the complete supplycorrespondence





Summary

Jones JPE, 1965

The technology matrix really is a function of local factorprices A(w)

But cost minimization implies that, for small changes infactor prices, Adw + (dA)w = Adw because for each goodi ,∑

f daif wf = 0 by the envelope theoremHence every technology is locally a fixed coefficientsLeontief technologyI think of A+ as a Stolper Samuelson matrix for pricechanges dp that lie in the column space of A(w)





Summary

Supply correspondence

y = (AT )+v + (I − (AT )+AT )zr(p, v) = pT y = pT (AT )+v + wT AT (I − (AT )+AT )zr(p, v) = pT y = pT (AT )+v since AT (I − (AT )+AT ) = 0The bottom line is that r(p, v) = pT (A+)T v , a simplequadratic form.





Summary

Factor prices are overdetermined

v = A+p + (I − A+A)zr(p, v) = vT w = vT A+p + yT A(I − A+A)zr(p, v) = vT w = vT A+p since A(I − A+A) = 0The bottom line is that r(p, v) = vT A+p, the samequadratic form.





Summary

(A+)T is the Hessian of r(p, v)

This quadratic form is everywhere differentiableWe can recover the entire supply correspondence usingthe homogeneous termAT was designed by Leontief to show how many extraresources were need to produce ∆y(A+)T gives the change in the output vector that arisesfrom ∆v





Summary

The Rybczynksi Effect and Movements along the flat

y_2

(A+)^T ∆v

Any

movement

along the PPF

has no effect

on national

y_1

on national

revenue∆y





Summary

The real world

We live in a world where there are more goods than factorsThe output vector of the German economy is one particularvalue from a correspondenceIn real world applications, almost every good is producedand traded in every countryHence price changes lie in a restricted column spacespanned by each local technology matrix!





Summary

Numerical example

First column is capital and second column is labor

A =

1 12 13 1

(A+)T =

−1/2 4/30 1/3

1/2 −2/3





Summary

Numerical example, row stochastic matrix

(r ,w) = (1,1)

Θ =

0.5 0.50.67 0.330.75 0.25

(Θ+)T =

−1.21 3.080.66 −0.261.55 −1.82

The Rybczynski matrix is column stochastic. This factechoes the national income identity.





Summary

Fisher and Marshall, forthcoming Review ofInternational Economics





Summary

Estimating factor rewards

In this work, we did not use consistent data on factor uses.We wanted to study types of laborBy assumption Aw = 1w = A+1 + (I − A+A)z, but the homogeneous termdisappearsHence w = (AT A)−1AT 1We know that we measure the true factor prices with error.





Summary

Fisher and Marshall, forthcoming Review ofInternational Economics





Summary

The local factor content of a foreign Rybczynski matrix

I hope you find these Rybczynski matrices usefulThink about the American (Country 2) factor content ofanother piece of capital in Germany (Country1)∆y1 = (A+

1 )T ∆v1 + (I − (A+1 )T AT

1 )z where z ∈ Rn

I will write the homogeneous term as u1. Remember it hasno factor content in Country 1AT

2 ∆y1 = AT2 (A+

1 )T ∆v1 + AT2 u1

The factor conversion matrix is AT2 (A+

1 )T





Summary

Two interpretations of a factor conversion matrix

AT2 (A+

1 )T is the f × f matrix that translates country 1 factorsinto those in country 2Only in rare cases will it be diagonalA+

1 A2 is the matrix that translates factor prices in country 2into those in country 1Normally the wage in country 2 corresponds to a linearcombination rent and wage in the country 1





Summary

Leontief’s idea of factor-specific differences

The first column is capital and the second is labor.

A1 =

1 12 13 1

A2 =

10 220 230 2

It is obvious that the first country has very good capital andalso good workers





Summary

Example of a factor conversion matrix

(A2)T (A+1 )T =

[10 20 302 2 2

] −1/2 4/30 1/3

1/2 −2/3

=[10 00 2

]One piece of capital in country 1 equals 10 country 2pieces of capital and no country 2 workers. One country 1worker equals 2 country 2 workers and no pieces of capital.





Summary

The factor conversion matrix from Germany to USA

The first column is capital, the second is labor, and thethird is social capital 0.6758 0.1124 −0.0363

0.2259 0.8769 1.05890.0983 0.0107 −0.0225

This matrix is column stochastic$1 of capital in Germany corresponds to $0.67 of UScapital. $0.23 of US labor, and $ 0.10 of US social capital





Summary

Defining a virtual endowment

We know the output vector of country i . We also know thetechnology of the reference country A0

The virtual endowment of country i as vi = AT0 yi

It depends on the reference country 0Now the measured factor content of trade is A0(xi −mi)and the predictions are based upon

∑i vi

We have imposed the pure HOV world





Summary

Traditional HOV, USA Reference

10,000

20,000

30,000

40,000

50,000

Me

asu

red

Fa

cto

r C

on

ten

t

Traditional HOV, USA Reference Country(millions of 2000 dollars)

K

-50,000

-40,000

-30,000

-20,000

-10,000

0

-50,000 -30,000 -10,000 10,000 30,000 50,000

Me

asu

red

Fa

cto

r C

on

ten

t

Predicted Factor Content

K

L

G





Summary

Virtual endowments, USA Reference Country

10,000

20,000

30,000

40,000

50,000

Me

asu

red

Fa

cto

r C

on

ten

t

Fig. 2: Virtual Endowments, USA Reference Country(millions of 2000 dollars)

K

-50,000

-40,000

-30,000

-20,000

-10,000

0

-50,000 -30,000 -10,000 10,000 30,000 50,000

Me

asu

red

Fa

cto

r C

on

ten

t


K

L

G





Summary

Virtual endowments, Korea Reference Country

10,000

20,000

30,000

40,000

50,000

Me

asu

red

Fa

cto

r C

on

ten

t

Fig. 3: Virtual Endowments, Korea Reference Country(millions of 2000 dollars)

K

-50,000

-40,000

-30,000

-20,000

-10,000

0

-50,000 -30,000 -10,000 10,000 30,000 50,000

Me

asu

red

Fa

cto

r C

on

ten

t


K

L

G





Summary

What have we learned?

1 Measurement error does not matter2 Homothetic preferences are in the data3 No home bias in consumption4 Every good is traded5 Trade costs do not matter6 No need to adjust for trade in intermediate inputs7 Constant returns to scale are in the data





Summary

Objections to this approach

We have assumed away differences in technologyWe are only testing the demand side of the modelThe tests using virtual endowments are tautologies





Summary

World endowments

W ld E d i B i C diWorld Endowments in Barycentric Coordinates

G

CHN

CHEIDNRUS

TUR

USA

K L





Summary

Virtual endowments, USA is the reference

Vi l E d i h USA R fVirtual Endowments with USA as Reference

G

USA

K L





Summary

Virtual endowment and the factor conversion matrix

The vector yi = (A+i )T vi + (I − (AT

i )+ATi )zi for some zi .

So country i ′s virtual endowment isvi = AT

0 (ATi )+vi + AT

0 (I − (A+i )T AT

i )zi

So its virtual endowment is its actual endowmentconverted into factors in the reference country, plus anerror term that has no factor content in country i





Summary

Tests using factor conversion matrices

10,000

20,000

30,000

40,000

50,000

Me

asu

red

Fa

cto

r C

on

ten

t

Fig. 5: HOV without FPE(millions of 2000 dollars)

K

-50,000

-40,000

-30,000

-20,000

-10,000

0

-50,000 -30,000 -10,000 10,000 30,000 50,000

Me

asu

red

Fa

cto

r C

on

ten

t


K

L

G





Summary

Summary

National revenue function

Rybczynski theory

Rybczynski matrices

Factor conversion matrices

Virtual endowments


rybczynski redux - ifo institute for economic research · visiting ces 17 june 2010 e. fisher...

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