7.2 Application to economics: Leontief Model


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1 7 Application to economics: Leontief Model Wassil Leontief won the Nobel prize in economics in 97 The Leontief model is a model for the economics of a whole countr or region In the model there are n industries producing n different products such that the input equals the output or, in other words, consumption equals production One distinguishes two models: open model: some production consumed internall b industries, rest consumed b eternal bodies Problem: Find production level if eternal demand is given closed model: entire production consumed b industries Problem: Find relative price of each product The open Leontief Model Let the n industries denoted b S, S,, S n The echange of products can be described b an inputoutput graph Here, a ij denotes the number of units produced b industr S i necessar to produce one unit b industr S j and b i is the number of eternall demanded units of industr S i Eample: Primitive model of the econom of Kansas in the 9 th centur
2 The following equations are satisfied: Production of Total output Internal consumption + Eternal Demand farming industr (in tons: horse industr: + (in km horse rides In general, let,,, n, be the total output of industr S, S,, S n, respectivel Then a + a + + a n + b a + a + + a n + b n a n + a n + + a nn n + b n, since a ij j is the number of units produced b industr S i and consumed b industr S j The total consumption equals the total production for the product of each industr S i Let a a n, B b, X A a n a nn b n n A is called the inputoutput matri, B the eternal demand vector and X the production level vector The above sstem of linear equations is equivalent to the matri equation X AX + B In the open Leontief model, A and B determine X from this matri equation We can transform this equation as follows: I n X AX B (I n AX B X (I n A B are given and the problem is to if the inverse of the matri I n A eists ((I n A is then called the Leontief inverse For a given realistic econom, a solution obviousl must eist For our eample we have: ( 5 5 A, B ( 8,,, X (
3 We obtain therefore the solution X (I A B (( ( ( 5 5 8,, ( ( ,, ( ( 5 8, 9 95, (,,, ie,, tons wheat and Million( km horse ride 7, If the eternal demand changes, e B, we get, 5 ( (I A B ( ( ( 5 7, 9, , 5, 5 ie, one doesn t need to recompute (I A One difficult with the model: How to determine the matri A from a given econom? Tpicall, X is known, B is known and (a ij j i,j, b n n is known One takes therefore the matri (a ij j i,j, n and divides the jth column b j for j,, n to get A Eample: An econom has the two industries R and S The current consumption is given b the table consumption R S eternal Industr R production 5 5 Industr S production 6 Assume the new eternal demand is units of R and units of S Determine the new production levels Solution: ( The total production ( is ( units for R and units for S We obtain 5 ( 5 X, B, A 6, and B The solution is X (I A B ( ( ( The new production levels are 7 and 7 for R and S, respectivel b n,
4 The closed Leontief Model The closed Leontief model can be described b the matri equation X AX, ie, there is no eternal demand The matri I n A is usuall not invertible (Otherwise, the onl solution would be X The inputoutput graph looks now as follows: There is onl internal consumption Eample: Etended model of the econom of Kansas in the 9 th centur including labor The corresponding matri equation is: z z
5 If X is a solution, also t X for ever t > is a solution (Usuall, one gets a one parameter famil of solutions If, we can assume, b choosing the appropriate parameter t One obtains then the solution,, 9 66, z For this computation, it is important to use rational numbers (ie, fractions as matri entries since otherwise the approimation to the matri I n A usuall will be invertible and onl the trivial uninteresting solution,, and z will eist This is also the reason, wh the entr a has large numerator and denominator In a closed econom, the absolute units of output are less interesting More important is the relative consumption of a product We can normalize therefore the matri A such that the sum of ever row is This is a matri Ã, such that Ã The recipe is: Divide the ith row of A b the ith component of A For our eample, we have A (that is the sum of the ith row 5 8, leading to the matri Ã 7 8, Ã The entries of the matri Ã (ã i,j i, j,, n have the following meaning: ã ij is the relative consumption of the product of industr S i b industr S j Market prices The consumption of products is regulated b prices All income of an industr is used for buing other (or the own products, ie, income equals ependiture Let P (p,, p n the price vector; p i is the relative price of the product of industr S i We can draw the flow of mone into the inputoutput graph, the mone flows in echange for the products: 5
6 One has p a p + a p + + ã n p n p ã p + ã p + + a n p p n ã n p + ã n p n + + ã nn p n, since ã ij p i is the amount paid b industr S j for products produced b industr S i The total income of industr S j equals the total price S j has to pa to all other industries Again, one can write this as a matri equation: P A P This equation can be transformed in the following wa P I n P Ã P (I n Ã (,, The matri I n Ã is (similar as I n A not invertible, since (I n Ã One can show that this implies that there is also a solution P Since with P also t P for t > is a solution, onl the relative price between the different products has a welldefined meaning Eample (continued: Assume p $, One gets p $ 6 $ 69 and p $ $ 9677 We can compare these relative prices with the production levels measured b the original units and obtain the following relative prices per unit: p / for one ton of wheat, p / 69 6 for km horse ride, and p /z for one manear Since the above matri equation for P is not of the usual form which we have studied so far, we make a final modification We define Ã (ãi,j i, j,, n, where ã i,j ã j,i 6
7 This gives us (just b switching the rôle of rows and columns the price equation P Ã P, where ã i,j is now the relative consumption of industr S j b industr S i, so that the sum of each column is, and P p p n is the price column vector In the tetbook, our matri Ã is again denoted b A and our P is denoted b X The price equation is therefore X A X However, one has to keep in mind that this matri A is different from the inputoutput matri A we used in the open Leontief model! Eample: Let A Compute all wages, given that the wages for the rd product is $, Solution: Let X be the different wages with z, We have to z solve X AX (I AX, This sstem of linear equations for and has the solution, and, 5 The wages for the first and second product are therefore $,, and $, 5, respectivel 7
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