rybczynski redux - ifo institute for economic research · visiting ces 17 june 2010 e. fisher...
TRANSCRIPT
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
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual 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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
Simplest Rybczynski Effect
y_2
y_1
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
Four properties define this pseudo inverse
1 AA+A = A2 A+AA+ = A+
3 (A+A)T = A+A4 (AA+)T = AA+
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
First example
1 3x1 = 42 3+ = 1/33 x1 = (1/3)4 + (1− 1)z
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
]
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
The solution of minimum norm
x_2
(A+)^T 4 = (1.2,0.4)’
x_1
Slope = -3,
intercept = 4
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
Third example
1
123
x1 =
345
2
123
+
=[
0.0714 0.1429 0.2143]
3 x1 = 1.8571 + (1− 1)z
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
What is 1.8571?
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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 )+
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
The Moore-Penrose Inverse of the technology matrix
A =
[a1a2
]A+ =
[ a1
a21 + a2
2
a2
a21 + a2
2
]
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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)
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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.
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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.
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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!
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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.
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
Fisher and Marshall, forthcoming Review ofInternational Economics
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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.
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
Fisher and Marshall, forthcoming Review ofInternational Economics
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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.
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
Predicted Factor Content
K
L
G
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
Predicted Factor Content
K
L
G
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
World endowments
W ld E d i B i C diWorld Endowments in Barycentric Coordinates
G
CHN
CHEIDNRUS
TUR
USA
K L
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
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
Predicted Factor Content
K
L
G
E. Fisher Rybczynski Redux
IntroductionThe national revenue function and its uses
The Moore-Penrose Pseudo inverseThe revenue function for a Leontief technology
Factor conversion matricesVirtual endowments
Summary
Summary
National revenue function
Rybczynski theory
Rybczynski matrices
Factor conversion matrices
Virtual endowments
E. Fisher Rybczynski Redux