 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, zin plus fi and xn would be zn1, zn2, znj, znn plus fn ok. So, this can be written in matrix form whatever we have written so far this can be written in matrix form. So, that we have these following matrix is x is x1, x2 and so on to xn, z matrix is z11 to z1n and so on i1 to zin 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 1111 all the i1 value. So, this is the way in which we can write this we can also see this is the equations which are there and this is the values. 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 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 entire 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 zij pi xj 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. So, for instance if you are looking at aluminium being used for aircraft production. So, this will be aluminium input by aircraft output 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 value of aluminium bought by the aircraft producers in the last year in the year which we are looking at divided by the value of aircraft production. Now, can we say anything about aij? So, aij has to be between 0 and 1 it cannot be negative it is a physical amount of quantity that is required cannot be greater than 1 because finally, the total value add that is there in that sector has to be has to be a combination of all the value adds of different components and since all this 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 production operates under constant returns to scale. So, we can now write this as if we look at this matrix a11, a12, 1n zij if you remember the zij's which we had we can write the expression let us let us start from the scratch let us 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 zi plus fi we can write the zij as we said is going to be a combination of a into x. So, we are going to have x is equal to the zij z i plus f is what we had and this is going to be nothing but a into your x. So, this is ax 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 we 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. So, essentially what we had is we started off 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 a 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 lij. So, we can write this as x is equal to x 1 is equal to l 1 1 f 1 plus l 1 2 f 2 l 1 f n and so on x i is equal to l i 1 f 1 similarly x n. So, we have to calculate in each case the Leontief inverse and so 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 where determinant of a is a d minus b c and of course determinant of a should not be equal to 0 with 2 by 2 is something we can do by hand but if we want to calculate this for 3 by 3 4 by 4 5 by 5 n by n we can use any we can use matlab or you can use excel and you can do the matrix inversion and get the Leontief. 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 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 labor government services and imports and this will add up to the total 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. These are the balance equations that we already talked of for the 2 sectors that is x is equal to x 1 plus x 2 plus this is the balance equation for the row and the balance equation for the column and with this you will get this kind of calculation. So, if we look at these sectors L is the labor services employment and all other value added and M are imports 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 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 the 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. 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 the 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 and 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 the total final demand is going to be 150 plus 500 plus sorry this is 350 you can see this is 350. So, this total 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 demands or 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 outlays which are XI total outlays must balance out. So, this must be equal to a 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 AIJs. So, if we look at A11, A11 is what is the amount per unit of agriculture what is the amount of agriculture used. 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 the sector itself. A12 is the percentage of the A12 is the percentage of the agriculture which is used in the, 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. These are the technical coefficients we have the A matrix which is 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 change where the agricultural output decreased and both of these the agricultural output suppose 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. So, in doing this first thing which we can do is calculate I minus A if you remember this is A. So, 1 minus 0.15 is 0.85 and here is 0 minus 0.25. So, it is minus 0.25 this is 0 minus 0.2. So, it is 0.21 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 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 essentially 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. So, we can now take if we want to calculate the value of x nu we can just take l into f nu and this is 1.2541, 0.3300, 0.2640 you multiply the two matrices 600, 1500 we get 1247, 0.5218, 41.58 and what are the z nu, z nu can be the new inter industry transactions will just be a into x nu and if you do this you will find that z nu is 187.13460.40249.5 and then if we look at this then we can we can get the new final output input output table and that will now be agriculture manufacturing 187.13460.4, fi is 600, 1247. We could round it off also this is 249.592 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 8110.89 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, 326089.10 let us look at what it was earlier and then we can compare these two. So, you see 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 agricultural sector in order 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 reduced by 200 units, but the total output has also reduced but not in the same amount it has reduced proportionately reduced to 184 watt and when we look at the addition of this the earlier the total output of the economy the two 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, there are many different things that 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 some of the terms some of the sectors are represented in energy terms and the other sectors are represented in money terms. We will look at some of these examples and the applications in the next module.