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Module 1: Leontief's Input/Output Analysis

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The Leontief Input/Output Model

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We are now starting with a new topic, we are going to talk about input-output analysis and its application to energy systems. We have looked at, at the level of the different projects, how to do the economic calculation to look at the environmental impacts, we now want to see what is an overall impact at a larger scale, at the societal level and for a city for a region for a country and this, there are different ways in which we do energy-economic models. (Refer Slide Time: 0:57) So, we have different types of energy-economic models, the questions that we would like to see is that what if we replaced all our thermal power plants with renewables? What would be the impact not just on the energy sector, but overall on the economy? What would it mean in terms of the investments, what would it mean in terms of the prices, what would it mean in terms of the jobs, what would be the kind of macroeconomic methods? So, then there are different kinds of models which are available in the literature. And there could be models by which we are looking at the energy economic interactions and these types could be classified the simplest kind of model is the input-output model, which is what we will study. There are also optimization models and simulation models. There are models like Markal and there are models which are computable general equilibrium models and then there are models for estimating the demand based on end-use accounting and the econometric models. So, there are a whole host of different models and in this course, we will have time to only look at one type of model. We will talk about the input-output model, which will give us a way in which we can analyze the impacts of the energy sector on the rest of the economy. (Refer Slide Time: 2:27) The models can be classified depending on the purpose, are we using it for one particular sector, are we using it for the overall economy do we want to see what happens if there are different growth rates. We can look at it in terms of short medium long term and, in the short term, what would happen is all the coefficient to remain more or less constant, in the medium term, we can make changes in a variety of things and the long term, many more things can be changed. The models can be also classified as top-down versus bottom-up. Top-down means that we look at an aggregate for the entire country as a whole or the entire world as a whole or the state and then make an estimation, then based on that, we then work out what will be the impacts at different sections. A bottom-up is where we start looking at different end-users, the different sectors, look at the residential, the commercial, industrial, and for each one we have assumptions of different technologies and then build up by taking an aggregation what is the overall picture. Then models can be also classified as simulation versus optimization. In the case of simulation, everything is specified and we would just like to know what if, what if we did it this way, look at all the technology and the systems and then work out what would be the cost etc. and optimization is where we have some degree of freedom and there are decision variables that we can choose and then we can see what is optimal. We can minimize the total sum of costs or minimize the emissions have maximized the revenue and things like that. So, this is another way in which we can classify. We can also classify based on geographical coverage. At the highest level is the world model, we can have regional models, we can have national models, state models and local models. (Refer Slide Time: 4:20) I talked to you about this model, which is there, this is called the market allocation model, which is a bottom-up kind of model, starts from with a reference energy system with the primary energy, then the conversion technologies they induce and then the demand and then you can have either with some assumptions, it results in a linear programming kind of framework or it could have based on if there are nonlinearities then we can have a mixed-integer kind of if there are discrete variable. So, there are various ways in which we can optimize and then detailed modelling can be done and you can see, there are many papers where this has been applied to India, for the world for many different countries of the world. (Refer Slide Time: 5:03) The model that we are going to talk about is the input-output analysis and this was proposed by Wassily Leontief way back in the 1930s, where he initially proposed it, and then he used this methodology to extend the date to develop an input-output model for the US and this was done, there is a paper in Scientific American, it is available in the public domain and you can take a look at it. This will give you an idea of exactly how the original work was done, where he talked about the entire industry flows. And Leontief got the Nobel Prize in Economics for his work and this was given in the 1970s, 1973 and you can see the Nobel lecture that he proposed, where he created a simple aggregate model of the world economy and divided it into developed countries and less developed countries, and then saw what would happen in terms of investments and pollution, and looking at the possibilities of trying to reduce pollution and the investments in the industry as well as in pollution. So, they had created a set of interesting scenarios. This is also available in the public domain and I would urge you to look at both these papers that will give you an idea of the historical development of this method. So, I am going to quickly go through some of these data and tables which were shown in these papers, which will give you that initial idea and then we will, from first principles, develop the theory of the input-output analysis and show how it can be used for the energy sector. (Refer Slide Time: 7:05) So, this is the sequence in which we… So, this is the paper, the Scientific American paper input-output economics and he said that we are concerning a new method which can portray both an entire economy and its fine structure by plotting the production of each industry against its consumption from every other sector. (Refer Slide Time: 7:39) So, typically the input-output method, input-output analysis as proposed by Leontief, finally, results in a set of n linear equations in n unknowns. And that is, the beauty of the method is its simplicity, we can say start with what Leontief said is that there was data of the economic activity of and we can look at a region which could be a country, it could be a state, it could be a city or it could be a region. Typically, of course, this would have to be data, an entity for which the data is usually available. So, usually at the country level is where the data is available. So, in any economy there will be the flow of products or goods and services, that means goods and services. So, this is, these flows are also called these will be from the producer or the seller to the consumer or the buyer and even at that time when he did this paper in the 50s and the 1940s, the economy was being tracked. So, what we have to do is we have to take this, this will be the inter-sectoral or the inter-industry flows or transactions which are observed and this is observed for a period. This is observed for a period and typically that period is a year, is an annual. So, this could be either the calendar year or in many cases, for instance in the Indian case we will talk about the financial year. The financial years starts from 1st April to March 31st. So, you will say 2018-19 2019-20 and so on. And so, based on this, we will have different producers and sellers, different consumers and buyers and every good, if we talk about a particular good, for instance, if you look at steel, steel is being manufactured by the steel industry, that steel is being used by different sectors, let us say in the automobile industry. (Refer Slide Time: 10:35) And so, we can replay we can talk about this in either physical units or monetary units and, if you think about it, if we are talking of so many tons or so many for an economy over the year, so many million tons of steel which are being produced. And then we will talk about so many million tons of cement which are being produced and so many million-kilowatt hours of electricity which is being produced and so on. But when we compare the different things and we add them all up, it is difficult to have multiple physical units. So, one of the best ways to do that is taking the physical unit, multiply it by the price or the value which is there, so you get it all in terms of monetary terms. And that is typically how these transactions are put. So, essentially what we have is we can put each transaction as Z ij, which is the monetary value of the annual transaction from sector I, which is the producer, to sector j. And so, if you look at one sector, if you are looking at steel, steel is being used for the power sector, steel is being used for the cement sector. So, there are inter-industry, internally the output of one sector is being used in the other sectors. In addition to this, there are sales to purchasers, who are exogenous, purchasers who are external to the industrial sector. That means purchasers who are not having any production, who are exogenous to the, and that will be the final demand, exogenous to the industrial sectors. This will be the external demand and this will, who are not, they are not producers. (Refer Slide Time: 13:25) So, these would typically, these sectors would be households, government or maybe you are exporting it, foreign trade. So, these, this is the external demand. So, if we look at xi as the total output or production of the sector I or production of the sector I and fi is the total final demand for the sector I's product, we can write a balanced equation which is xi is Z i1, from I to the first sector and then there are n such sectors Zi2 + n so on Zin + fi. (Refer Slide Time: 15:04) So, we can write this as xi, Z ij plus fi, where Zij is the inter-industry flows, transactions in money terms. Inter-industry flows or transaction. (Refer Slide Time: 15:46) So, let us see how this was represented in the paper by Leontief in Scientific American, you can this is not very clear here, there are small items and we will explain this. You can see this in the paper, a large number of sectors and in each of these from one sector to the other sectors, these are the kind of industry flows. (Refer Slide Time: 16:06) So, if you look at the types of sectors, we are talking about agriculture and fisheries, food and Kindred products, textile mills, apparel and so on and each of these sectors, these are the I's which we are talking. From each sector the agricultural products are used in the other sectors and so that those transactions are represented in this matrix. (Refer Slide Time: 16:32) And then, we also talked about the final demand and the final demand if you see foreign countries, government, households and the private capital formation. (Refer Slide Time: 16:46) And this is a sort of in more detail, you can see each of these sectors and from the sector to the other sector, this is the transaction matrix. From agriculture to agriculture and fisheries, some of the products are used internally. For instance, if we look at the electricity sector and we look at the electricity which is used within the electricity sector that would be like the auxiliary consumption of the power plants. (Refer Slide Time: 17:15) And so, this is how these transactions and then we talked about the final demand. And when we sum this all up, this will be equal to the total gross output or the xi that we had. And this is the final demand, these are the internal demands. (Refer Slide Time: 17:31) And similarly, we had this kind of curve. With this, what the paper showed is that for some of the sectors, it illustrates, what it can do, and this was done, this is a 1950 paper using the data for 1939. It is the tons of steel for a certain amount of output, which is there and you can see tons of steel ingot per thousand dollars of production of each of these sectors. So, if you look at the construction sector, in the metal fabrication, the motor vehicles and sector, these are the three main sectors and relatively less for the others. (Refer Slide Time: 18:09) We can also look at, for the automobile industry, what are the per thousand dollars of the output of the auto industry, how much is the input and you can see the ferrous metals is the main input and then you have all of these. So, these are some of the illustrations of the kind of things. (Refer Slide Time: 18:30) And then Leontief used this for static economy-wide, US-wide mapping of all the inter-industry transactions and then he wanted to illustrate that what happens if we have a 10 per cent increase in the salaries or the wages and how would that affect the overall economy and then showed the impact on different sectors. (Refer Slide Time: 18:59) Leontief's other paper, which was part of the Nobel Economics Prize Talk, he talked about in this case, this was a talk given in 1973, he estimated and built up an input-output framework for the world as a whole. For the world, he divided it into developed and developing countries. And in this, he aggregated it in terms of extraction industry, other production, and then pollution and then the employment and value add and then looked at the transactions in billions of dollars from each of these sectors. (Refer Slide Time: 19:35) And similar kind of thing was done for the less developed countries and then based on this, he created different scenarios. And there was one scenario for the less developed countries where you had not that much production. (Refer Slide Time: 19:49) The other one was where you had a large amount of pollution control in the less developed countries and with these scenarios used, showed the power of the method. And I would suggest you look at the details of this paper and that would give you an idea of how this methodology can be used. (Refer Slide Time: 20:09) n general finally, when we look at the input-output table that is there, this is from the book by Blair and Miller, you can look at the book on input-output analysis, the second edition, we will see different kinds of producers and then the final demand. And in addition to this, so we look at this it typically agriculture, mining, construction, manufacturing, all of this will have, you have a matrix where it goes agriculture to agriculture, agriculture to mining and so on. In addition to this is the salaries that we pay, the taxes that we pay to the government and anything in terms of the profits, etc. So, all of this together, if you look at the entire transactions we can get, if you look at overall, this will give us any indication of an estimation of the gross domestic product. So, let us look at, let us derive this, move forward with this. We are looking at x1 as Z11, Z12 plus Z1j Z1n plus f1. Then we have for the ith row, this will be Zi1, Zi2, Zij, this is Z1n Zin plus fi, and xn would be Zn1 Zn2 Znj, Znn plus fn. So, this can be written in matrix form, whatever we have written so far this can be written in matrix form. (Refer Slide Time: 1:29) So, that we have these following matrices. X is x1, x2 and so on to xn. Z matrix is Z11 to Z1n and so on, Zi1 to Zin and then f is equal to f1, f2 and so on to fn. This can be written as X is equal to Zi plus f, where i is a column matrix with 1,1,1,1,1, all the identity 1 value. (Refer Slide Time: 2:42) So, this is how we can write this, we can also see, this is the equation which is there and this is the values. (Refer Slide Time: 2:52) Now, the basis of this method, the fi, if you see the basis of this method is that we are going to write this in the form of the amount that we need from each of these. Finally, which we are looking at is going to be dependent. (Refer Slide Time: 3:15) The Zij will be a function of xj. That means the amount that we need from the ith sector to the jth sector will depend on the total output that we have from the jth sector and one way the input-output method, the fundamental assumption is that the inter-industry flow from the ith sector to the jth sector depends entirely on the total output, depends entirely on the total output of the jth sector, entirely on the total output of j for that period. So, which will mean that we say that aij, this is the coefficient that we will define, is Zi j by x j. In the input-output method, this coefficient is assumed to be constant, this is a technical coefficient, this is also known as a direct coefficient or the direct coefficient. (Refer Slide Time: 5:14) So, for instance, if we are looking at aluminium being used for aircraft production, so this will be aluminium input by aircraft output. Now what will be the units, this will be in millions of rupees, crores of rupees, so, it is going to be in the monetary units rupees per rupees. So, it is a ratio and so this will be, aij will be defined here as the value of aluminium bought by the aircraft producers in the last year, in the year we are looking at, divided by the value of aircraft production. Now, can we say anything about air? So, aij has to be between 0 and 1, it cannot be negative, which is a physical amount of quantity that is required. It cannot be greater than 1 because finally, the total value that is there in that sector has to be a combination of all the value adds of different components. And since all the, since none of them can be negative when we add it up, this is going to be there. So, this aij xj is equal to Zij, this is the basis. These coefficients are constant, which means that economies of scale are ignored and this operates under constant returns to scale. In the Leontief system, the entire basis is that the product operates under constant returns to scale. (Refer Slide Time: 8:48) So, we can now write this as, if we look at this matrix a11, a12, a1n, Zij, if you remember the Zij's which we had, we can write the expression, let us start from the… Let us write down let us look at the expression, that we had got earlier, which was in terms of x, xi being equal to that Zi plus fi. We can write the Zij, as we said is going to be a combination of an into x. (Refer Slide Time: 9:05) So, we are going to have X is equal to the Zij, Zi plus f is what we had and this is going to be nothing but A into your X. So, this is an X plus f. So, we can take X into the identity matrix, we have identity matrix I minus A, once we take it on this side into X is equal to f and now we can get X, what we want to do is if we know the final demand, what will be the values of X that we will get. So, X will be, we can take the inverse of this I minus A inverse into f. (Refer Slide Time: 10:16) So, essentially what we had is we started with Ax plus f is equal to x, x minus Ax is equal to f. So, this x is the identity matrix, identity matrix will be for 2 by 2, it is 1,0,0,1, it will be 1,0,0,0,1,0,0,0,1 for 3 by 3, this is I minus A into x is equal to f and so x is equal to I minus A inverse into f. This value I minus A inverse is called the Leontief inverse and this can be written as Li j. (Refer Slide Time: 11:26) So, we can write this as x is equal to x1 is equal to L11 f1 plus L12 f2, L1n fn, and so on xi is equal to Li1. Similarly xn, so we have to calculate in each case the Leontief inverse and so, by essentially, for instance, you may remember, if you look at 2 by 2 matrix, if you have A is equal to a, b, c, d, we can calculate A inverse will be nothing but 1 over module determinant of A and this is d minus b minus c, a and we have a determinant of A as ad minus bc and of course, determinant of A should not be equal to 0. With the 2 by 2 is something we can do by hand, but if we want to calculate this f or 3 by 3, 4 by 4, 5 by 5, n by n we can use any, you can use MATLAB or you can use Excel and you can do the matrix inversion and get the Leontief. (Refer Slide Time: 13:23) So, essentially what happens is we can take this and we would have the different sectors, the buying and selling sectors and then we will get, we also said that the final demand is a combination of the different sectors, the consumption, the government consumption, the exports. (Refer Slide Time: 13:37) And then there are these payment sectors which we talk of, which is, in the columns, these are the additional sectors that we are talking of where we are paying the amount which is going to the other like wages and to government and any other services, labour, government services and inputs. And this will add up to the 2 outlays, the rows will add up and so will the columns. And so, typically, if we are talking of 2 processing sectors and some payment sectors, this is what we will get and this is finally the kind of input-output table. (Refer Slide Time: 14:17) These are the balance equations that we already talked of for the 2 sectors that are X is equal to x1 plus x2 plus L, this is a balanced equation for the row and the balanced equation for the column and with this, this you will get this kind of calculation. (Refer Slide Time: 14:37) So, if we look at these sectors, L is the labour services employment, N all other value-added and M are imported, all of these will come under each sector in the column. And in the row, if you look at it, there are additional final demands, which will come in terms of consumption, investment goods, government and export. So, F will be equal to C plus I plus G plus E. And this is the payment sector, which is the additional thing which will come in the column. (Refer Slide Time: 15:10) With this, we will take an example, which is from the book by Blair and Miller. It is a 2 sector example and we will take that example and then process it and then see what will happen when we make, what are the coefficients, how do we take Leontief inverse, what is the implication of the Leontief inverse and how can we use it to see what if in case there is a growth or there is some change in the sectoral demand. (Refer Slide Time: 15:40) So, if we look at this sector, let me just write down this. We have essentially, let us say there is an agricultural sector and a manufacturing sector. And so, if this is agricultural, you have an agricultural sector, and then you have a manufacturing sector, and in this case, this is 1 and 2 and then here also we have agriculture and manufacturing and this is some 150 some units, billion rupees, million dollars, the financial units, manufacturing is 500 and the total final demand fi for this is 300. So, the total which is there, the xi which is there to total final demand is going to be 150 plus 500 plus, this is 350. So, this is total output, total output. This is the transaction which we have noticed that means, from agriculture of the output of agriculture 150 million rupees is being used in agriculture, 500 million rupees is used in manufacturing and 350 is the final demand. So, the total output of agriculture is 1000 million rupees and from the case of manufacturing, 200 is going here and 100 going to the manufacturing sector. The final demand for all the manufacturing products is 1700. So, if you add this up this comes out to be 2000. And then there is this payment sector as we said wages, taxes, profits whatever else we are looking at. So, remember this has to balance out, so the total outlay which is xi's, total outlays must balance out, so this must be equal to 1000 which will mean that this is 650. And this is again this will be equal to 2000 and this will be 1400. The final demand, in this case, is for the payments 1100. So, if you add this up this is 28 and this is 3150, this is 3150 total, which is the total value add in the economy is 6150 appropriate financial units. Now, let us look at how do we calculate the aij's. So, if we look at a11, a11 is what is the amount per unit of agriculture what is the amount of agriculture use. So, this will be equal to 150 by 1000 which is 0.15. That means per unit of agricultural output 0.15 times of that is the ratio of what is used within this sector itself. a12 is the percentage of agriculture which is used in the agriculture… If you look at the transaction from agriculture to manufacturing is 500 units and this will depend on the output per unit of manufacturing. So, here this is going to be 500 divided by 2000. This will be divided by xj in this case x2, so this is going to be 0.25. Similarly, this is going to be 200 divided by 1000 which is 0.2 and this is 100 divided by 2000 which is 0.05. (Refer Slide Time: 20:30) These are the technical coefficients, we have the A matrix which is point 0.15, 0.25, 0.2 and 0.05, this is the A matrix and we can then put down if you see this is the A matrix, the f matrix is 350 and 1700 and the value of x is 1000 and 2000. Now, the question is that what if instead of this kind of output we had a chance, where the agricultural output decreased and both of these… If the agricultural output supposes instead of the final demand for agriculture instead of 350 if that increases to f new, if we say that instead of this we are converting, we increase it to 600 and the industrial demand decreases. So, suppose we move from here to here, we want to know how what will be the changes in the economy and how much of each of these products would be calculated. So, this is what we want to do in terms of x new. (Refer Slide Time: 22:10) So, in doing this, the first thing which we can do is calculate I minus A, if you remember this is A, so 1 minus point 0.85 and here is 0 minus 0.25, so it is minus 0.25. This is 0 minus 0.2. So, this point 0.2, 1 minus 0.05, so it is 0.95. This is I minus A, and we want to calculate the inverse of this. And you can see the determinant of this I minus A. You can check this out, it comes out to be 0.7575. And so, the Leontief inverse is, I minus A inverse is 1 by 0.7575 and this is now 0.95, 0.25, 0.2, 0.85. You will see that this comes out to be 1.2541, 0.330 and 0.2640 and this is… So, this is what was told, we calculated this. (Refer Slide Time: 24:01) And so the interesting thing to see is if you look at this values that we have of the Leontief matrix 1.2541, 0.3300, 0.2640, 1.1221, you will notice that all the coefficients which are there in the diagonal these are greater than 1 and that is essential which means that in the per unit of, we had said there is a direct coefficient which is what is the amount of agricultural output increase per unit of agriculture. But if you look at for a particular value of output, if we look at x, how much is the total direct plus indirect requirement these values in the diagonal will always be greater than 1 and that is by the nature of this. (Refer Slide Time: 25:14) So, we can now take, if we want to calculate the value of x new, we can just take L into f new and this is 1.2541, 0.3300, 0.2640, you multiply the two matrices 600, 1500 we get 1247.52 1841.58 and what are the Z new? Z new can be the new inter-industry transaction will just be A into X new. (Refer Slide Time: 26:21) And if you do this you will find that Z new is 187.13, 460.40, 249.5 and then if we look at this, we can get the new final output input-output table. (Refer Slide Time: 26:52) And that will now be agriculture, manufacturing 187.13, 460.4 fi is 600, 1247.52, we could round it off also, this is 249.5, 92… Payment sector. Now to calculate the payment sector, you will see that this total is 1247.52 subtract from that these two and you will get this as 810.89, this is 1289.11, this will remain unchanged, the total will be 3200, total outlays, I can add this will be 1247.52, 1841.58, 3200, 6289.10. Let us look at what it was earlier and then we can compare these two. (Refer Slide Time: 28:28) So, you see to what has happened here is that the total output in the agricultural sector even though the final demand of the agricultural, the final demand of the agricultural sector has increased from 350 to 600 with the result that the total output of the agriculture sector to meet this demand has increased from 1000 units to 1247 units. And in the case of industry the demand has reduced final demand has reduced from 1700 to 1500, it has reduced by 200 units, but the total output has also reduced but not in the same amount. It has reduced, it has reduced to 1841. And when we look at the addition of this, the earlier the total output of the economy, the 2 sector economy was 6150 overall now, the economy has increased to 6289. And so, we can see that increase in agriculture decrease in industry, what is the impact. So, many different things can be done with this and with this we can also look at some of the energy sectors as well as the manufacturing sector. So, the impact of energy intensity, we can also, all of this can be done in, this was done in money units, we can also do it in hybrid units where some years, some of the terms, some of the sectors are represented in energy terms and the other sectors are represented in money terms and we will look at some of these examples and the applications in the next module.