 Module 136, in production analysis we are going to study transcendental logarithm production function. Previously we have studied various types of the production function, the very simple linear production function, the fixed proportion, Lyuntov production function, the Cobb-Douglas production function and CES mean the constant elasticity of substitution. And they all have gradually improved the capacity of the economist to explain the production function in the form of the technical and mathematics. But at the same time there was certain trouble that as we have already studied the new classical approach of the production function that is based on the three stages of the production function. And when it was applied to the actual data sometime it was very difficult for the economist that we were not able to explain these three stages. And if I will explain in a very simple form, if I tell you here, so till now the production function that we have done, when we talk about the change of the inputs, we are mostly moving from one unit to the other unit to the third unit. But it is not possible that in real life, in many things, if we say that the labor unit has been shifted from five to six units, or if we say that one input was already ten, now it has put fifteen units, where the actual form can be the same, which can be the in-between of those two. And it was not possible in our calculation because it required a little more precision. And similarly, since when the production function started, that era, in 1940, 50, and 60, there was mostly start of industrial production and there was also the agricultural production function. So, the units that we had, they were in the very big form. Meaning, we had the forms or we had the forms. And the unit of production of labor, if you look at it, now we mostly divide it in the form of hours or days divide. So, because if we say that we gave half labor, then it is difficult to explain. So, now the labor productivity is mostly expressed in the form of hours. Meaning, there is two hours of labor, there is four hours of labor. So, looking at all these factors, the problems that were there, some economists, such as Halter, Hawke, and Carter, and Hawking, they were working. And keeping in view their work, they little bit improve or slightly change the existing cob-dogless production function in a way that they introduced the base of the log in the existing cob-dogless production function. And when that function, that was introduced by Harter in 1957, that base E was added and that base E was added. In a form that it was given a power and that power was the function of the amount of the input that was used. So, in a very simple manner, we have to explain this output, is the function of A technology. And single input case, if this is only single, we can say now this x has the power of this A, mean it is again the marginal productivity or you can say that this is the contribution of this input x towards the total output y. Previously, we have used the notation of Q and again we can utilized like this that Q is the function of A. And here if x is replaced by a labor, that will be L, the function of alpha. And now the addition will be this E base, that E base has this power gamma and multiply by this value of that input. Here it is x and likewise here x, we can have this in the form of the L. So, in certain books, you can find this type of the production function or like this type of the production function, but they both will be expressing the same manner. And at the same time, we have studied that there is a very small number of the production function, that they will be having single input case. So, mostly production function, they have more than one or more than even the two inputs. When this input is explained with the two input case, we have modified in this manner that now the output is equal to technology A, x 1 input, x 2 input and their respective elasticity of production. And again now the base E will be utilizing again the share of this input x 1 and x 2. And if I will write in the form of the Q, this will be equal to A. And in the form of the capital and labor, I can write. If I can write here this capital, it is with the power of alpha and labor with the power of beta and this E. And now it will have this power rho multiply by this K plus again rho multiply by L. So, this R will this base of the log will be having now the proportion of this input 1 and input 2 and summing of this will be their respective power. Now, as we know that the marginal physical productivity of the single input version case using the composite function, we can have this the marginal physical productivity is the change in the output with respect to input. So, we can again have the derivative of our previous production function and that production function we have put like this. If it was y is equal to A, we can utilize this and here we have taken its first derivative. So, this will be its marginal physical productivity. So, average physical productivity is equal to y by x as it is always equal to the ratio of total output divide by total input. So, it will be equal to average physical productivity and the elasticity of production for the single input if we have to write that will be the ratio of MPP divide by average physical productivity. So, when we calculate this elasticity of production utilizing mean this will be the case of the marginal physical productivity and then divided by this y and x. So, elasticity of production will be depending upon only and only on the amount of that input that has been used. So, when we will now utilize the case of the two input case again we will be having the same strap that now the contribution of except one input there will be the utilization of the two inputs. Thank you.