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Transcript of Leontief Matrix.ppt
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Leontief Matrix
Robert M. Hayes2002
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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:
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
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Wasily Leontief
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
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The Impact of Wasily Leontief 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
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The Structure of the Leontief Matrix
Sectors
Variables
Matrices
The heart of the idea
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Schematic of Inter-Sector Transactions
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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. Theresulting “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)
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Use of the Fundamental Equation
Let’s suppose that the input-output matrix is constant, atleast for a range of consumer demands reasonably close tothe given one, which was (0.5,0.2,0.4), from output of (1,1,1).
What would be needed to meet a different consumerdemand?
From the basic equation X - A*X = D, the answer requiressolving the linear equation (I - A)*X = D, where I is theidentity matrix.
In the example, if the consumer demand for sector 3 output
were to increase from 0.4 to 0.5, the resulting sector outputvector would need to be: (1.0303, 1.0417, 1.1591). Theinternal consumption (i.e., that output consumed inproduction) would be (0.5303,0.8417,0.6591), and thedifference between the two is (0.5000,0.2000,0.5000).
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Dynamic Equation
This becomes really interesting if the production process isviewed as a progression in time.
In static input-output models, the final demand vectorcomprises not only consumption goods, but also investmentgoods, that is, additions to the stocks of fixed capital itemssuch as buildings, machinery, tools etc.
In dynamic input-output models investment demand cannotbe taken as given from outside, but must be explainedwithin the model.
The approach chosen is the following: the additions to thestocks of durable capital goods are technologically required,given the technique in use, in order to allow for anexpansion of productive capacity that matches theexpansion in the level of output effectively demanded.
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Dynamic Leontief Models
A simple dynamic model has the following form
XTt (I - A) - (XT
t+1 - XT
t )B = DT
t,
where I is the nxn identity matrix, A is the usual Leontiefinput matrix, B is the matrix of fixed capital coefficients,X is the vector of total outputs and D is the vector of finaldeliveries, excluding fixed capital investment; t refers to thetime period. It deserves to be stressed that in this approach
time is treated as a discrete variable. The coefficient bij inthe matrix B defines the stock of products of industry jrequired per unit of capacity output of industry i and is thusa stock-flow ratio.