 module 128 in production analysis we are going to study the Euler theorem that is also sometimes called the Euler theorem. This theorem was basically provided a mathematician a Swiss mathematician whose name was Lenard Euler or basically this provides us a mathematical relationship that applies to various homogeneous function and that homogeneous function can be homogeneous consumption function that can be any homogeneous production function. So, any function or any mathematical relationship having the degree t when the t is equal to 1 that will be the homogeneous there that we can apply. Now, coming to the point that we use this Euler theorem in microeconomics and particularly in production and its need has come forward. Actually, when we see that all the inputs that we have in production have a particular break even point and that profit maximization or output maximization point, when we see that the break even point requires that all the factors of production are equal to the payment of their marginal productivity. That means the factor which is contributing to the output is paid according to that. So, because it is very genuine and it is very, very imperative means a factor that will have more share in the output that should be having more wage or that should have more rent. If it is like this, it was break even point that if we look at it, it means that all the marginal productivity is that if they are paid according to that, it means that the output will be totally exhausted. When we look at this concept, the theorem of product exhaustion has come forward and then we have a problem. Is that adding up problem that we are checking, that product is actually and equally properly exhausted or there can be certain left out amount in the form of inventory or surplus or this will require more and we will have the deficit of the product. So, looking at this concept and problem, when we have this problem, the name of this problem that was called adding up problem or product exhaustion theorem. Now, to solve this problem, our economist that is called Philip Wichstedt and to solve this problem, the concept used was the concept that was provided in the form of the Euler theorem. This product exhaustion theorem was basically solved with three methods and one of them is the Euler theorem. Now, if we go into the detail of this, the mathematical proposition that we have, it says that the production function is if it is homogeneous of degree t and if it is homogeneous of degree t, it means that homogeneous means concentrate returns to scale and homogeneous of degree t in the form of t is equal to 1. So, we can say homogeneous of degree 1 production function So, the first derivatives related to input means the marginal productivities, if we look at it, we solve it. So, these marginal productivities that we have, of various inputs, they properly solve our product exhaustion theorem in this form that there is no deficit there and there is no surplus there. And this is only possible when the derivatives of various inputs are equal to each other. And that is why when we solve it, we solve it after that parameter. So, we go into its example and we use the elasticity of substitution equation that returns to scale Now, it returns to scale equation which is change in output due to change in scaling of input means if we solve it and it gives, now if this scaling has the power t and we solve it and when we come to the next and we know that it is equal to the level t and if this t is equal to 1, then means that scaling or our production function is homogeneous of degree 1. So, its further if we, due to inputs, if we solve the change, then also scaling when we solve the derivatives, now the change in this scaling S, Z1 and S, Z2 where Z1 and Z2 are the two inputs, we can put capital or labour in this place, we can put K1, K2, L1 and L2, so when this scaling means change due to, we change in total factor due to scaling, we solve the derivatives and when we solve it further here, then we equate these derivatives with the derivative of right side equation which comes to us means t and S power t minus 1 function into Z, this one and this was equal to this other one, so we say that this is our production function, that is degree t and the marginal productivities of both of them are equating with each other and since marginal productivities equate in this way that the changes due to them were coming into the equation on the input side, the same changes were coming into the output side, so then we can say that due to the scaling, if we use product exhaustion theory, then it enables us that the change we have made in the input side is the same as we have in the output side and nothing is left unused and waste in the economy. Now, the result that we see is called the Euler theorem and production function, when the lead and homogenous function is used, it will be adding up property and this is adding up property which we named as product exhaustion theorem. Now, if we look at the output, it is basically the summation of marginal productivities of the inputs and marginal productivities are further multiplied by their level of use. So, here two things will come forward, it means one marginal productivity of either capital and labour or Z1 or Z2 and their level of use. Now, this level of use means that basically, what was the internal ratio of the inputs? So, it depends on these two things and if we come to its significance, then what does it mean? When we talk about marginal productivities, actually, we see these marginal productivities, so labour's marginal productivity, we always assign value to it, so value of marginal product of labour that will be called wage and likewise, value of marginal product of capital that will be called rent. So, whenever these marginal productivities are equated with their price, then we can say that the firm that is maximizing profit is able to reach the break even point because of this Euler theorem. And here, we can also go to that point and see that our break even point always provides us a point where marginal revenue is equal to the marginal cost. Now, here, marginal revenue and marginal cost, if we look at it, the cost is basically derived from the input side and revenue that is derived from the output side. So, the scaling of the input is just going to equate the same degree level into the output.