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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 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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Use of the Fundamental Equation

Lets 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 consumer

demand? From the basic equation X - A*X = D, the answer requires

solving 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 investment

goods, 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 approachtime 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.

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THE END