Transcript Leontief Matrix

Leontief Matrix
Robert M. Hayes
2002
Nobel Prize in Economics
 The following slides list the persons who have received
the Nobel Prize for Economics since its inception in
1969.
 In making the awards, the Prize Committee appears to
have attempted to balance several aspects of economic
theory:
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Market-oriented vs. Public-sector oriented
Quantitative vs. Qualitative
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2001 George A. Akerlof, A. Michael Spence, Joseph E. Stiglitz
2000 James J. Heckman, Daniel L. McFadden
1999 Robert A. Mundell
1998 Amartya Sen
1997 Robert C. Merton, Myron S. Scholes
1996 James A. Mirrlees, William Vickrey
1995 Robert E. Lucas Jr.
1994 John C. Harsanyi, John F. Nash Jr., Reinhard Selten
1993 Robert W. Fogel, Douglass C. North
1992 Gary S. Becker
1991 Ronald H. Coase
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1990 Harry M. Markowitz, Merton H. Miller, William F. Sharpe
1989 Trygve Haavelmo
1988 Maurice Allais
1987 Robert M. Solow
1986 James M. Buchanan Jr.
1985 Franco Modigliani
1984 Richard Stone
1983 Gerard Debreu
1982 George J. Stigler
1981 James Tobin
1980 Lawrence R. Klein
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1979 Theodore W. Schultz, Sir Arthur Lewis
1978 Herbert A. Simon
1977 Bertil Ohlin, James E. Meade
1976 Milton Friedman
1975 Leonid Vitaliyevich Kantorovich, Tjalling C. Koopmans
1974 Gunnar Myrdal, Friedrich August von Hayek
1973 Wassily Leontief
1972 John R. Hicks, Kenneth J. Arrow
1971 Simon Kuznets
1970 Paul A. Samuelson
1969 Ragnar Frisch, Jan Tinbergen
Wasily Leontief
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His birth in Germany and move to Russia
His education
His early career
His move to the United States
His appointment at Harvard
His visit to Russia in ?
He is awarded the Nobel Prize in 1973
He generalizes the Input-Output Model
He moves to NYU in 1975
His views concerning American economists
His death in 1999
The Impact of Wasily Leontief
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The Leontief Matrix
Use in National Defense
Use in Economic Policy
The Motivation
Emphasis on Data rather than Theory
The Potential value of I-O Accounts
Improved Methodology
Supplemental Accounts
His connection with BEA
Bibliography
The Structure of the Leontief Matrix
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Sectors
Variables
Matrices
The heart of the idea
Schematic of Inter-Sector Transactions
The Fundamental Equation
 The fundamental equation is:
X = A*X +D
where the matrix A represents the requirement for input
(from each sector into each sector) that will generate the
output to serve the needs in production of output X. The
resulting “internal consumption” is represented by A*X.
 In the example given above, output vector is X = (1, 1, 1),
consumer demand vector is D = (0.5,0.2,0.4) and internal
consumption vector is A*X = (0.5,0.8,0.6)
Use of the Fundamental Equation
 Let’s suppose that the input-output matrix is constant, at
least for a range of consumer demands reasonably close to
the given one, which was (0.5,0.2,0.4), from output of (1,1,1).
 What would be needed to meet a different consumer
demand?
 From the basic equation X - A*X = D, the answer requires
solving the linear equation (I - A)*X = D, where I is the
identity matrix.
 In the example, if the consumer demand for sector 3 output
were to increase from 0.4 to 0.5, the resulting sector output
vector would need to be: (1.0303, 1.0417, 1.1591). The
internal consumption (i.e., that output consumed in
production) would be (0.5303,0.8417,0.6591), and the
difference between the two is (0.5000,0.2000,0.5000).
Dynamic Equation
 This becomes really interesting if the production process is
viewed as a progression in time.
 In static input-output models, the final demand vector
comprises not only consumption goods, but also investment
goods, that is, additions to the stocks of fixed capital items
such as buildings, machinery, tools etc.
 In dynamic input-output models investment demand cannot
be taken as given from outside, but must be explained
within the model.
 The approach chosen is the following: the additions to the
stocks of durable capital goods are technologically required,
given the technique in use, in order to allow for an
expansion of productive capacity that matches the
expansion in the level of output effectively demanded.
Dynamic Leontief Models
 A simple dynamic model has the following form
XTt (I - A) - (XTt+1 - XTt )B = DTt,
where I is the nxn identity matrix, A is the usual Leontief
input matrix, B is the matrix of fixed capital coefficients,
X is the vector of total outputs and D is the vector of final
deliveries, excluding fixed capital investment; t refers to the
time period. It deserves to be stressed that in this approach
time is treated as a discrete variable. The coefficient bij in
the matrix B defines the stock of products of industry j
required per unit of capacity output of industry i and is thus
a stock-flow ratio.
THE END