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Module 1: Leontief's Input/Output Analysis

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Country Input/Output Analysis

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So, we continue with this module where we are going to go ahead with input-output analysis. In the previous module we looked at the basics of input-output analysis, we looked at Leontief's initial formulation, we also said that in an economy when we talk of several different sectors, an output of one sector is used in the other sectors and it is also then used to meet the final demand. When we look at this, this and we sum up for each sector we get a set of interactions between the different sectors and this is represented, through the matrices that we create. We then for the input-output we said that the requirement of letting us say steel for automobiles depends on the total output of the automobiles and this requirement, this correlation is assumed to be linear. So, with linear constants when we have the direct coefficients, we create a matrix equation. It is a set of linear equations in the n variables and with that, we can see that if the final demand for once our sector increases, what happens to the rest of this. So, we then talked about two sets of coefficients, the direct coefficient, which is the direct requirement, let us say of steel for agriculture or steel for electricity or agricultural output for chemicals and then we have the total which is direct plus indirect and we did this by then creating the matrix creating the Leontief index inverse matrix, and then that relates to the final demand. So that any change in final demand results in a corresponding requirement or a change in the output of the different sectors. When we created that matrix, we saw that the diagonal elements of the matrix are greater than 1 which is sort of intuitive. Because, if we need a certain amount of final demand for steel, because of that final demand for steel to produce that we will need other chemicals, we will need electricity, for that chemicals and that electricity again we need a certain amount of steel. So, when we couple that up, we will see that for the diagonal elements will all be greater than 1. Now to take this forward, just to illustrate this from the book by Miller and Blair on input-output analysis, I would like to just show you some examples of aggregate for a country input-output tables and what they mean. (Refer Slide Time: 3:13) So, this is the US, this is the A matrix which we talked about this is A ij for the US and this is a 7 sector 7 by 7 matrix if you can see. If you look at this, let me just get the laser point. You can see for agriculture, from 1 to 1 is the agricultural products being used for agriculture. Then agricultural products being used for mining for construction for manufacturing, for trade, transportation, utilities, for services and others and similarly mining to this manufacturing to a variety of things and services going to a variety of things and so, you can see all of these are between 0 and 1 and this is the total. (Refer Slide Time: 4:18) Now, when we take this A matrix, we can write down for this matrix, we can calculate I minus A and then take the inverse of that and that gives you the matrix which we are talking of, this is the inverse matrix that we are looking at and this is the L matrix. So, this is the matrix that we calculate. If you look at, this is the Leontief inverse that we are talking of. Now you see the diagonal elements, agriculture is 1.26, which means that an increase in the final demand of agriculture by 1 unit results in a net overall requirement of increasing agricultural product by 1.26. Because an increase of agriculture requires all other inputs from the other sectors, which again, in turn, requires the amounts from agriculture. So you can see all the diagonal elements 1.26, 1.07, 1.0047, 1.34, 1.008, 1.41, 1.03, all the diagonal elements are greater than 1. All the off-diagonal elements are less than 1 they are between 0 and 1. And so this is the L matrix. We could then take the L matrix and see what happens when you if you change the final demands. So, the assumption in the input-output method is that these coefficients are static, and these coefficients remain constant. (Refer Slide Time: 5:54) Now just to give you an idea of this, we could also represent… See all of these, when we talked about these input-output tables, these were all represented in monetary units. It was also possible that we can talk in terms of the physical units in terms of bushels and tons. So, if you are looking at, let us say corn or agriculture in bushels of corn, and if you are looking at let us say oil, tons of oil. If we had an example, where this is a physical quantity, we said that 75 bushels of corn are used in the agricultural sector, 250 is used in the manufacturing sector and 175 is the final demand. So, the total demand is 500. Similarly, 40 tons are used in agriculture, 20 tons used here 340 tons and 400 tons. So if you look at this then if we had a price in dollars per physical unit that means dollars 2 per bushel and dollars 5 per ton, the n we could multiply each of these units, 75 into 2, 250 into 2 and so on so that you can get it in money terms and then what we get is this is the conventional matrix that we had used for the input-output analysis, then we can do the normal analysis in terms of the Leontief inverse and make the calculations, we can get the coefficients and these will be all in the monetary terms. We could also go back and if you see this, this 150, 200, 500, 100 that we got after multiplying this, if we change the revised physical units of measures to reflect the price, then this becomes, this is the matrix that we got can be converted into physical terms. That means we now have this as 150, 500, 350, 1000. (Refer Slide Time: 8:11) In physical terms, now, this is in rupees or dollars in this example it is when dollars this is, 1 dollar is the cost of half a bushel per unit price. So, physically this represents half a bushel and that is a physical and this one if you look at it in tons 1700, 2000 and this is 1 fifth ton. So, we can move between physical and money terms and there are certain cases where we can also look at this in terms of the hybrid unit, where you have both physical as well as monetary terms. (Refer Slide Time: 9:06) Now, we can have on this site, if you remember in the column, the last row in the column are the different payment sectors, one of them could be wages and if we look at this, this is the wages 650, 1400, 1100, 3150 and if you look at this in terms of the output, you will see that the wages or the labour or the employment is 650 by 1000. (Refer Slide Time: 9:53) And so that unit is, we can get a coefficient which is 650 by 100 which is 0.65 and 1400 by 2000 which is 0.7. And these are representing the employment factors or the employment index per unit of the money that we are spending in that sector. And this could be useful for instance, if we were thinking in terms of, instead of coal we go for renewables and we go for photovoltaics, we can see the growth in the two different sectors. We can have an employment factor in terms of ratios, and then see how many jobs are being created, how many jobs are being lost. And so that that is an interesting way in which we can look at that. (Refer Slide Time: 10:39) So, that we could take, for instance, the ratios of this and then use that to then calculate what is the amount of labour under different conditions. We can come to make a composite index of energy. For instance, in the case of US, we saw the Input-Output matrix for a particular year, we can also draw as we have seen, in the initial lectures, we saw the Sankey or the energy balance diagram for the world and India and these diagrams will represent the relative proportions of the different fields and the flows in different sectors and this can be then converted into… (Refer Slide Time: 11:30) So, we could have the input-output in terms of some sectors, the energy sectors and non-energy sectors. So, we could create a hybrid input-output table where your transaction matrix has energy units and money. So, then what happens is that when I have a transaction from energy to another sector, it will be in terms of the value add which is provided by steel. The steel, cement or the other industrial sectors chemicals all of them are put in terms of millions of rupees or millions of dollars and the energy could be in megajoules, pita joules or kilowatt-hours. In the case of in the Miller and Blair example, they have talked of it in terms of BTU and dollars, British thermal Unit. So, when we have this kind of this is called a hybrid input-output framework. Please remember this is equivalent to the same thing we can take the hybrid, the energy terms multiplied by the price and then convert it into the conventional input-output table that we had seen earlier, which would be everything in money terms. And then we can see the amount of electricity which is in money given to the industry sector. In the case of a hybrid system, where you have energy, we can look at the energy use per million rupees of steel produced. And so, this is as long, as we are consistent in terms of the units we can otherwise go ahead and do the same kind of example. So, to just give you an example, this is again the example from the textbook. (Refer Slide Time: 13:50) And there are two things here, there is some output of some products which are called widgets and there is energy and then there is a final demand, final demand we saw, FI and then we have the total output. So, in this case, if you see 10 million dollars of widgets being used for the widgets, for making the widget and 10 million dollars of widgets being used for the 20 million dollars for the energy sector and 70 is the final demand for the widget. So, total when we add it up, this is 100 million dollars and in this case, it is 30, this is 40 and this is 50, this is 120. So, in quad BTU this is given in terms of this can the same row can also be represented. Now this is million dollars and this one is in quad BTU in energy units, this will be 60, 80, 100, 240. So, if you see this is equivalent to a price in terms of 30 by 60, the price is 0.5 million dollars per quad BTU. And one could operate this with the money terms, do the calculations, after we get the final results, use this factor to get it into the energy term. So, we can move seamlessly between energy and money. Of course, another way is sometimes you operate with a hybrid input-output framework, but we just have to remember this that these coefficients will have then units. In the case, in the normal case, the aids are all ratios, which are in terms of between 0 and 1 and so then that becomes an easy way of doing this. (Refer Slide Time: 16:40) So, this is in terms of the essence, we then have the following matrices, the normal matrix that we talked Zi plus f is equal to x. This was our financial one and instead of this now, we also have the Ei, that is the energy plus the energy demand is equal to the total g and then that could be how we could write this. So, q is a vector of energy deliveries to the total final demand and g is the factor of the total energy consumption. So, we could operate it this way or we could operate it in the normal input-output with the money terms and then make the calculation. (Refer Slide Time: 17:28) So, just to give you, if you look at you can look at the textbook by Miller and Blair, there are several examples of this. So, for instance, there are this three-sector, three sectors and one automobile sector. So, you have crude oil, refined petroleum, electric power, and then you have the crude oil is going for refined petroleum sector then it goes, some of it goes to the electric power sector, there is no final demand for the crude oil, you add it up that comes to 10 million US dollars. Refined petroleum, some of it is being used in the crude oil sector and some of it, of course, is going into this and so on, when you add it up, this is the total. And then electric power electricity going into each of these sectors and they have, there is a final demand for automobiles and then this is the total output. (Refer Slide Time: 18:27) And one could then convert this in terms of the price. And you can get them in terms of BTU, this is the kind of matrix which will then come. So, it is dividing those money units by the prices. And please remember, in a situation, prices of energy to different sectors may be different and that can be also configured into this framework. (Refer Slide Time: 18:58) So that is the situation in terms of looking at the examples, different kinds of examples where we take these different sectors, the energy sector and the automobile sector and then convert it into this. (Refer Slide Time: 19:13) We also, I showed you earlier the input-output table from the textbook on the Input-Output analysis and similar kind of input-output table is shown here, which is now a hybrid unit and this is hybrid unit has transactions millions of dollars for the non-energy sector and in quads or 10 raised to 15 BTU for the energy sector. So, you can see coal mining oil, natural gas, petroleum utilities, gas utilities, all of these will be in BTU. The chemicals agriculture, mining, transport and communication rest of economy are all going to be in the money terms and then we can make the if we know the prices, we can convert it into a money term aspect and then do. (Refer Slide Time: 20:16) And so, we saw last time, the numbers in terms of direct coefficients for 2003. Please remember that as the economy changes, you will find that the coefficients also will change. And so, when we talk about input-output analysis, if you are taking fixed coefficients that will be valid only for short term kinds of calculations. If you are looking at long term calculations and if the structure changes, it is quite likely that there will be very significant changes. Even when we compare, you can take this table with the values and compare it with the 2003 coefficients and you will find that there are some changes in some of these coefficients and over a longer period, you will see that these coefficients change quite significantly. For instance, the energy use for the industry may decrease if there have been significant improvements in energy efficiency and so that those are that is how there are coefficients. (Refer Slide Time: 21:30) And if you look at the total coefficients, now this is after we take that same, if you take this 2006 matrix that we had got I minus A, take the inverse of that, that will give you the Leontief inverse. And you will find then that these coefficients, we talked about the diagonal coefficients being greater than 1 and you can remember. If you remember the earlier in 2003, this value was lower than this, this is now almost 1.6 times, and so on. So, this gives you an idea of how you can use this and then also that these things change. And that is just to show you another set of data, this is the 97 data. And you can see when you take 97, 2003 and 2006, you can see quite clearly there are reasonable differences in all of this. (Refer Slide Time: 22:16) And so, we will take an example before we take that example, let me talk to you about how this can be used, to assess the impact of different kinds of possibilities for a particular sector. So, this is one of the papers, one of the research work done by one of our PhD students and you can see this paper, you can look it up in the energy journal. It is an integrated modelling framework for energy economy and emissions modelling and this is a case study for India. (Refer Slide Time: 22:50) Integrated Modelling framework So, in this if you see the approach that we had was, we essentially looked at the emissions intensity, the emissions intensity is the emission per unit of GDP. And we broke up the emission intensity into the difference in terms of the energy intensity of the GDP and the sectoral contribution to the GDP. So, typically what happens is in any, in any country, the GDP comes from a whole set of different sectors. (Refer Slide Time: 23:33) So, if you look at it, typically the most important sectors are industry, services and agriculture and over a long period, if you look at India, for instance, over the last 10-20 years, you will see that the share of agriculture in the GDP has been declining. Share of services has been increasing, the share of the industry more or less remain constant, slight increases slight decreases. So, when we look at this, what has happened is that the share of services in the total GDP has been much higher than, has grown and as compared to the industry and industry share has declined a bit. Now, if you look at the energy requirement for industry and the high energy-intensive industries, that energy requirement is much higher per million rupees of value add as compared to something in the services sector and the services sector, at most your need something with the energy for the air conditioning and space cooling. But in industry, we are looking at manufacturing and transformations and so that is much more energy-intensive. (Refer Slide Time: 25:04) Integrated Modelling framework So, we can the first thing is we did a decomposition analysis to see what is the share of what is the breakup of the share of the sectoral contribution and the how much of the energy intensity improvements and then we got ranges for these parameters. This is from the past and from that we started with a particular base year and then made projections for the target year. So, when we looked at the projection, we projected different possible scenarios for India in terms of industrial growth, services growth and agricultural growth. And based on that, we got, we took an input-output model with some coefficients and then, saw when we looked at this with the kind of investments required we also built a detailed model for the power sector. And for this kind of requirement we estimated what is the electricity demand, then saw what kind of new capacities have to be added, we try to do an optimization model under different scenarios. And using that we estimated what is the total demand for goods and services and then ran an input-output method to model to see what will happen in different sectors and this then gives us an idea to see, we then saw what is the impact of different household classes and the income and income distributions and if you remember earlier we talked about equality and inequality in incomes and we talked about the GINI coefficients. So, after looking at this kind of investments in the energy sector, and whether how much is the government and private investment, based on that we try to see what will be the investment in the other sectors and as a result of that, we try to see the impact on the income and income distribution. So, this is the method, it is a set of coupled models. There is an optimization model of the power sector, there is an input of model and then there is a decomposition analysis and different scenarios. (Refer Slide Time: 27:06) So, under each of these scenarios, we first identify different drivers, we took a high services scenario, high industry scenario, and then we looked at the additional investment, either if it is, if the investment which is made, have been proportional cutbacks from each of the sectors, or the additional investment from cutbacks in the welfare sector and then in the power sector, we ran 2 scenarios where there is no restriction on emissions and we go for the minimum cost or if there are restrictions on emissions, and then the initial investments may be higher. (Refer Slide Time: 27:45) And based on that we could see under these scenarios, what happens in terms of the growth rates and the per capita income, and interestingly we can also see the difference in the GINI coefficient. So, for instance, in this case, in the case when we have more restrictions on emissions, we see that it results in a slightly higher inequality and this is just, the numbers are not that important, you can look at the details in the paper, but basically, it gives you an illustration of how to input-output analysis can be used to answer what-if questions about the impacts of policy. (Refer Slide Time: 28:27) So, that is the idea of how this input-output analysis can be used at the aggregate level or the energy sector. Now, let us take one simple example and try to solve and we have already seen in both the, in the last module as well as in this module how to do the calculations. (Refer Slide Time: 00:24) Now, let us do this for the, for this example. So, there are two sectors given here, the agriculture and the manufacturing sector. (Refer Slide Time: 00:38) Agriculture, manufacturing and so agriculture, manufacturing and let us say that we are talking of this in terms of money terms, in million rupees and what has it, the partial table has been given to you of transactions. So, the questions the, unit is million or million Indian rupees. We are considering a two-sector economy with an input-output table as shown for 2017.We are asked to fill in the blanks in the input-output table, compute the A matrix and the L matrix. (Refer Slide Time: 01:21) Then we are supposed to consider two cases, one is where the agriculture final demand increases by 200 million rupees in 2018, while the final demand for manufacturing remains constant. So, if agriculture increases and manufacture remains constant, the second one is where agricultural final demand remains constant while manufacturing demand increases by 200 million rupees, we want to compare the two cases in terms of the input-output tables. Is the total output of the economy the same in both the cases and then we also want to ask in concept, how can we use the input-output table to compute the impact of employment of two different options? (Refer Slide Time: 02:09) So, let us do this example. Please try this, it is fairly simple. It is related to whatever we have done so far. So, we had this is 300, 500, 800, 200, 400, 1500 and then you have the payment sector and then the total. It is given to you, this the value is given, payment sector outside. This is fairly straightforward we have already seen this. We can sum this up 300 plus 500, 800 plus 800, 1600 this will be the total output here. Here, 200 plus 400, 600 plus 1500, 2100 million tons. Now, we know that the column when we are looking at agriculture, agriculture is being used for agriculture, and these are transactions here. So, the total payments in terms of wages, profits in everything which is there must be such that this total output is the same. So, this total output here will be 1600, total output here will be 2100. As we subtract, we can take 1600 minus 500 and that will give us 1100. Similarly, when we look at this, it is going to be 2100 minus 900, it is 1200. Then when we add this up, this is 8000, 2300 plus 1000, 3300. Let us add this up, 2300 plus 1000, 3300. Now, this two has to add up and that is clear. So, 2100, 1600, 3700 plus 3300 is 7000. 7000 million tons is the total output of the economy. (Refer Slide Time: 04:54) And now let us see the question which has been asked is to see what happens if we change if this increase, if the, let us see with the question says that agricultural final demand increases by 200 million, final demand for manufacturing remains constant. Before that, we are asked to fill in the blanks which we have done. Compute the A matrix and the L matrix. (Refer Slide Time: 05:05) So, A matrix is straightforward. Let us look at the A matrix, A matrix is going to be 300 by 1600. This is 500 by 2100, this is 200 by 1600, 400 by 2100. So, this comes out to be 0.1875 and this is 0.2381, I will just round it off. This is 0.125. This is the A matrix. Remember the F matrix is 800 and 1500. (Refer Slide Time: 06:57) So, when we look at this A matrix, we can now calculate I minus A, I minus A becomes 1 minus 0.1875 is 0.8125, 1 minus point. So, this is minus 0.2381, this is minus 0.125, 1 minus 0.1905, this is 0.8095, this is I minus A. We can take the inverse of this and you can do this. I am not going to show you all the steps, you just with the rounded-up values, you will find that this is turning out to be 1.29, 0.38, 0.2. It is rounding off; this is almost similar. So, this is your I minus A inverse and very interestingly these are the diagonal elements and when you now multiply this, by now the value off, we will change. new is going to be 800 will become 1000 and this remains as 1500. We can multiply this and what you will find is Xnew, you can calculate this multiply this with this and add this and then you get, you will get 1858 and 2340. Remember the total output last time which we had this was, earlier it was 1600 and 2100. So, both of them have increased and have increased by different amounts. So, this is increased by 258 and this is increased by 220. Of course, the increase in agriculture is the percentage increase in agriculture is higher. (Refer Slide Time: 10:06) Using this we can then make the final table that we had. And you will see that now, agriculture manufacturing, agriculture manufacturing. So, we have the final values which we calculated 1858, 2340. The direct coefficients will remain the same, we can just multiply by the direct coefficients to get the ese values and you can cross-check that and round it off, you will get 348 and here you get 509, that s 0.1875 into the value that we had and then this is 232, 408 final demand that we had was 1000, 800, 2000 and this is 1500. When you add this up you should get 1509 plus 348, you get the same value that we had 1858 and here also 19 and 2,1540, 1500 and this comes to, this should have been the value of Xnew here when you multiply it is 2140, 2340 it is 2140. So, we have a slight increase in the value of the, manufacturing output from 2100 to 2140 but significant increase in this. So, this is 2140. Now when we look at the payments sector, we get this as the subtraction, take this as 1858, 2140 this one will remain constant. The remaining part of 1000. So, then this is 2000 and 1500, 3500 and these values we will get as 1277 and this is 1223. When we add this up, this comes out to be 3600 and when we add this total up, we will get a total of 75, 58, 48, 98, 7598. Now just compare this with what we had earlier. This was 7000, 3300, 2100, 1600. This has now become 7598, if you look at the overall output growth, this is 7598 by 7000. It is less than 10 per cent, you can, we can calculate the amount 598 by 7000 that is the percentage growth. (Refer Slide Time: 13:44) So, let us look at section b, where we now keep agricultural final demand remaining constant while manufacturing demand increases by 200 million rupees. So, the question is whether an increase of 200 million rupees in the agricultural final demand. That is what we saw last time and instead of that, we keep that constant and manufacturing increases by 200. (Refer Slide Time: 14:09) So, now the final demand that we are looking at is going to be 800 million rupees for agriculture and industry increases from 1500 to 1700. So, we can take this pre-multiply by the inverse that we had and that is going to be 0.20, 1.29 multiplied by 800, 1700. This will give us the value of Xnew and you can multiply 1.29 into 800, plus 0.38 into 1700 and you will see this comes out to be 1676 is the total final output of the agricultural sector. And for the industrial sector, this will be 0.2 into 800, plus 1.29 into 1700, this will come out to be 2358. This is now Xnew.