Input-Output-Models and Markov-Chains

Transcription

1 W. Peschek, University of Klagenfurt Input-Output-Models and Markov-Chains I want to present two economic models, of which at least the first one is found in some schoolbooks for commercial high schools ( HAK ) and general high schools ( AHS ). I will start with an eample for the: Interconnection of Materials Eample: A factory uses three components B, B, B for producing two intermediate products Z, Z, from which in turn they produce two end products E and E. The required amounts for production are shown in the graph below: Interconnection matri: (B Z) ; (Z E) If you need i.e. quantitiy units of Z and quantitiy units of Z (demand N Z ), you will need the following production demand X B of the components B, B and B : B : + B : + B : + In matri notation: X B (B Z) N Z with NZ corresponding: X Z (Z E) N E - -

2 If you want to directly calculate a certain demand X B of components, for a certain need N E of end products, you need the interconnection matri (B E). As you can easily see from the graph, it is true that: (B E) (B Z) (Z E) and thus X B (B E) N E (B Z) (Z E) N E In our eample we get a need of E and E X B 9 This step by step treatment of the matter is getting a bit confusing if, apart from the demand for end products, an additional demand for intermediate products and/or components i.e. for a spare parts depot has to be satisfied and if this fails, because there is a backflow in the interconnection of materials. In such cases it s better to present the whole interconnection of materials in one interconnection matri: For the production of a quantity unit of B B B Z Z E E B B B Z Z E you need of (in quantitiy units) E If you denote this interconnection matri with A and the production demand of B, B,..., E with X, so A X indicates the internal ( secondary ) demand, resulting from the interconnection of materials. - -

3 If you denote the eternal demand ( primary demand ) for B, B,..., E with N, so it s true that: X N + A X Total demand (in quantitiy units) primary demand (in quantitiy units) + secondary demand (in quantitiy units) In our eample, for an eternal demand of quantitiy units B, quantitiy units B, quantitiy units Z, quantitiy units Z, quantitiy units E and quantitiy units E we get: + This system of equations is easily and directly solved because A is a triangular matri. You can also find a general solving strategy by transforming the matri equation above: X N + A X X A X N (E A) X N X (E A) - N In our eample: A) (E A E 9 9 N A) (E X - -

4 Open Leontief Models An economic system, i.e. a national economy, consists of several interdependent parts. They are dependent on each other because for production each part needs services from their own area as well as services from other parts ( secondary demand ) as input. Moreover, there is usually also an eternal demand ( primary demand ) of services from the different parts. As within the interconnection of materials, we are here dealing with the question, which services (here measured in money units) each part has to perform, so that the primary and the secondary demand are satisfied. Similar as for the interconnection of materials, it is true that: total service (in money units) primary demand (in money units) + secondary demand (in money units) Eample: A national economy may consist of three parts A, B, C. From the following table you can see, which services (in money units) each part needs, to be able to perform services itself, for a value of money units: from the parts A B C services with a value in (money units) for services with A,,, the value of money B,,, units you need C,, Within a certain period of time units we epect a demand of products from A, B, C with a value of,, money units from all other economic sectors (which do not belong to A, B or C). Which services (in money units) have to be perforquantitiy unitsd in A, B and C? As with the interconnection of materials, it s true that: X N + A X that is X (E A) - N,, whereby A,,, and N,,, thus A,9 B, C,,,,,,

5 Linear models of the interconnection of materials and open Leontief-models have the same structure: There is an eternal demand (primary demand) for each product (services from each part), i.e. altogether a demand vector N. The interconnection of materials (the production dependencies) can be presented in one matri, A (a ij ), whereby a ij is the delivery (the output) of the product (of the part) i for a product (the part) j, that is, conversely the input from j out of i. The amount (service) needed by each product (part) is represented by the vector X. These models are called input-output-models and it s true that: X N + A X that is X (E A) - N The interconnection matri A is thus also called input-output-matri, the matri (E A) - is called Leontief-matri (Technology-matri, Leontief-inverse). Markov-Chains Markov-Chains are results from random tests, whereby - the test result can only be one state (of many possible) - the test result is dependent on the initial state, as well as (known) probabilities for the transition of one state to another Eample: There is a competition of three providers (brands) A, B, C on the market. From the following table you can see that from the buyers, who own a product of the brand A today, % will buy A again within a certain period of tiquantitiy units, % will change to brand B and % to brand C. The saquantitiy units is true for B and C. to A B C A,,, from B,,, C,,, We want to find out the share in the market that the brands A, B and C will level off in the long run (if they will do so at all!), when the recent market shares for A, B and C are %, % and %. - -

6 Initial situation: A %; B %; C % After one period of time the market shares are: A (new ),, +,, +,, % B (new ),, +,, +,, % C (new ),, +,, +,, % New initial situation: A %; B %; C % After another period of time the market shares are: A (new ),, +,, +,,,% B (new )...,% C (new )...,% In matri notation: X t+ P T X t whereby P T is the transpose of the matri P (p ij ) and X t resp. X t+ are the vectors of the market shares at the points of time t resp. t+. From repeated iteration we assume that the market shares will tend towards a balanced state: A,% B,9% C,% That means, that for all products and each (further) period of time it s true that: market share old market share new that means X P T X I: A, A +, B +, C II: B, A +, B +, C III: C, A +, B +, C Obviously we have a homogeneous system of equations; we cancel the parts, but we add, that the sum of the market shares has to make %: IV: A + B + C so that:, A, B, C -, A +,9 B, C A + B + C A, % ; B,% ; C,% In our eample, the achieved balanced state is independent from the choice of the initial values we are talking about ergodic Markov-chains. - -

7 This doesn t always have to be the case. In the following, graphically depicted Markov-chain, we also get a balanced state for every initial state, this however, is dependent on the choice of the initial state we are talking about non-ergodic Markov-chains. Eamples. Given is the following graph ( Gozinto-Graph ) of an interconnection of materials: Develop a matri A (a ij ) which indicates, which amounts R i are needed for the production of a quantity unit of Z j (first production stage) and a matri B (b ij ), which indicates, which amounts Z i are needed for the production of a quantity unit of P j (second production stage). Calculate from A and B a matri C (c ij ) which indicates, which amounts R i are needed for the production of a quantity unit of P j. Calculate which amounts of raw materials R and which amounts of intermediate products Z are necessary to produce quantitiy units of P and quantitiy units of P! In addition to the end products P we need spare parts of,, quantitiy units of Z, Z resp. Z. Calculate all required amounts with the help of the Leontief-matri!. Given is the structure plan of quantities of a chemical process. For A we have an eternal demand of quantitiy units, for B and C an eternal demand of quantitiy units each. Calculate the production amounts required in A-F!. A mineral oil company is able to depose products with a value of. money units daily. For the production of products with a value of one money unit they need coal with a value of. money units, electricity with a value of. money units and diesel oil (from their own production) with a value of. money units. - -

8 An electric power station has an eternal demand of. money units per day. To perform services with a value of one money unit they require fuel with a value of. money units, coal with a value of. money units and power (from their own production) with a value of. money units. Finally, the coalmine, for producing coal with the value of one money unit, needs fuel with the value of. money units and power with a value of. money units. The daily, eternal demand is. money units. With the help of the Leontief-inverse, calculate the daily amounts of production, so that the internal and the eternal demand of these three companies can be satisfied!. (Eample by Bürger: Mathematik Oberstufe ) We divide the national economy of a country in the following sectors, agriculture, trade and commerce, industry and transportation. The production and the use of goods (within a certain period of time) are depicted in the figure (in money units) aside. We have the following eternal demand (in money units) of goods from agriculture: trade and commerce: industry: transportation: Transportation With the help of the Leontief-inverse, calculate the amount, that has to be produced in each sector! Interpret the results!. Calculate the market shares to which the companies R, S and T will level off in the long run, when the consumer behaviour corresponds to a Markov-chain with the given probabilities of changeover. Follow the development of the market shares for the Markov-chain given in the graph, through three periods of time, where we start from a) %, %, %, % b) %, %, %, % market shares for A, B, C resp. D! Try to forecast a balanced state, independent from the initial state! What do you realize from it? - -