72 Application to economics: Leontief Model
7 2 Application to economics: Leontief Model Wassily Leontief won the Nobel prize in economics in 1973 The Leontief model is a model for the economics of a whole country or region In the model there are n industries producing n di erent products such that the input equals the output or, in other words, consumption equals production One
Matrix Operations & the Leontief Model
Use an Excel spreadsheet to work the Leontief Input-Output Model Economist Wassily W Leontief (1906 - 1999) was awarded the 1973 Nobel Prize in Economics for the development of the input-output method and for its application to important economic problems He was born in Saint Petersburg, Russia, and educated at the universities of Leningrad
27: Leontief Input-Output Model - Texas A&M University
The Leontief Input-Output Model can be described by the equation X = AX +D where X is the production matrix (total output), A is the input-output matrix and D is the consumer demand matrix The calculation AX = X ¡ D is the internal use of the model The production matrix can be solved by the formula: X = (I ¡A)¡1D: °
Leontief Matrix - hlbhattaraifileswordpresscom
Leontief Matrix In an economy of three industries P, Q, R, the data given below (All figures in million of Nu ) are available: P Q R Consumer Demand Total Out put P 80 100 100 40 320 Q 80 200 60 60 400 R 80 100 100 20 300 Determine the output if the final demand changes to 60 for P, 40 for Q, and 60 for R Answer:
The Leontief Input-Output Model
The Leontief Input-Output Model Text Reference: Section 2 6, p 155 The purpose of this set of exercises is to provide three more examples of the Leontief Input-Output Model in action The basic assumptions of the model and the calculations involved are reviewed first Refer to Section 2 6 of your text for more complete information
Aggregation in Leontief Matrices and the Labour Theory of Value
as a by-product of the analysis of Leontief matrices It has been shown,2 for instance, that in a Leontief model the price of a commodity in terms of labour ("wage-price") will equal its Marxian "value" under certain conditions which include (i) competitive long-run equilibrium, i e , zero profits in each
The Leontief Open Model Leontief Open Model
The Leontief Open Model The Leontief Open Model is a simpli ed economic model for an economy in which consump-tion equals production, or input equals output Internal Consumption (or internal demand) is de ned to be the amount of production consumed within the industries themselves, whereas
Leontief Input-Output Model
Leontief Input-Output Model We suppose the economy to be divided into nsectors (about 500 for Leontief’s model) The demand vector d~2Rn is the vector whose ith component is the value (in dollars, say) of production of sector idemanded
A Note on Exchange Market Equilibria with Leontief’s Utility
Arrow-Debreu problem with Leontief’s utilities, a more di–cult exchange market problem, is the Perron-Frobenius eigen-vector to a scaled Leontief utility matrix, and the equilibrium vector is a solution to a system of linear equations and inequalities of the original data
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7.2Applicationtoeconomics:LeontiefModel
distinguishestwomodels: byexternalbodies.Problem:Findrelativepriceofeachproduct.
TheopenLeontiefModel
describedbyan input-outputgraph ofindustrySi. 1Thefollowingequationsaresatised:
horseindustry:y=0:01x+2000 (in1000kmhorserides). respectively.Then 8 :x1=a11x1+a12x1++a1nx1+b1
x2=a21x1+a22x1++a2nx2+b2
x n=an1x1+an2x1++annxn+bn; industrySi. Let A=0 B @a11:::a1n.
a n1:::ann1 C A;B=0 B @b 1. b n1 C A;X=0 B @x 1. x n1 C A matrixequationX=AX+B:
IntheopenLeontiefmodel,AandB6=0
B @0 01 CAaregivenandtheproblemisto
determineXfromthismatrixequation.Wecantransformthisequationasfollows:
I nXAX=B (InA)X=BX=(InA)1B
Forourexamplewehave:
A= 0:050:5
0:10! ;B= 8;0002;000!
;X= x y! 2Weobtainthereforethesolution
X=(I2A)1B
= 10 01!0:050:5
0:10!!
1 8;000
2;000!
0:950:5
0:11!1 8;000
2;000!
1 9 10519:5! 8;000
2;000!
10;000
3;000!
Iftheexternaldemandchanges,ex.B0= 7;300
2;500!
,weget x y! 0 =(I2A)1B0=1 9 10519:5! 7;300
2;500!
= 9;5003;450!
i.e.,onedoesn'tneedtorecompute(I2A)1. omy?Typically,X=0 B @x 1. x n1 CAisknown,B=0
B @b 1. b n1 CAisknownand(aijxj)i;j=1;:::n
byxjforj=1,:::,ntogetA. isgivenbythetable consumptionRSexternal
IndustryRproduction505020
IndustrySproduction
6040100
thenewproductionlevels.X= 120
200!,B= 20 100!
,A= 50
1205020060
12040200!
,andB0= 100 100!.The solutionis X
0=(I2A)1B0=1
419630
6070! 100
100!= 307:3
317:0!
3TheclosedLeontiefModel
X=AX; (Otherwise,theonlysolutionwouldbeX=0 B @0 01 C A.)Theinput-outputgraphlooksnowasfollows:
Thereisonlyinternalconsumption.
labor.Thecorrespondingmatrixequationis:
0 B @x y z1 C A=0 B @0:050:50:50:100:1
0:40:11331
18001C A0 B @x y z1 C A 4 x=1;000;y=2900
11263:63;z=18000111636:36:
denominator.ThisisamatrixeA,suchthat0
B @1 11 C A=eA0 B @1 11 CA.Therecipeis:Dividethei-th
rowofAbythei-thcomponentofA0 B @1 11 CA(thatisthesumofthei-throw).
Forourexample,wehave
A0 B @1 1 11 C A=0 B B @21 20 1 5 223118001
C C A; leadingtothematrix e A=0 B B @1
2110211021
1 2012720