Leontief Matrix.ppt

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Leontief Matrix

Robert M. Hayes2002



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



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



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



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

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 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

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. 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)



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).



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.



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.



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