 module 107 and we are going to study the chapter of production analysis and the topic is production function. What is meant by a production function? If we generally say it is just a relationship, relationship that has to just join certain inputs and those inputs will be utilized to have certain output. But if in a technical term we will say then we can say that it is a mathematical or the technical relationship of combining various inputs to have a very certain specific output. And this production function that has to be utilized, it has to utilize various aspects of the theory of the firm. And now if we coming to that point we have to decide what are the basic two main aspects that has to be decided or that has to be kept in front of when we have to utilize the production function. The one thing that the pure economics whenever we are going to study it means we are having the limited resources. So whenever we are dealing with the aspect of any type of the production we have to keep in our mind that our entrepreneur has two, whatever the aspects, he has many resources but at a certain level his resources are limited. And when the resources are limited, the entrepreneur, the producer, the owner, what will be the biggest desire of that is that he or she has to select the least cost combination of the resources or the least cost method or the least cost technology which can produce its given level of output. So in the production function, our various inputs, we provide this opportunity that we exchange their quantities in their various systems to see which method or which combination is available with the least cost best output and which can be produced further in the market and can be sold. And in other words, we say that if there are given resources, then with that given resources we can produce the maximum level of output. Now coming to the next point, if we see that if we see all these aspects, then the main avenue in front of us is that there can be the production possibility set. That is one combination, that is the second combination and that is the third combination. These all will be the various production possibility sets that has to be utilized. Now here if we see that the production function basically is transforming what is transformation of large or transformation of diverse, transformation of various nature, number of the inputs in a number of the outputs. Now if we see here, then we can say that there are, if two only inputs, like we are taking an example here in the second equation, that if we can say in the example that the simplest form of production is in which we say that we are taking only two inputs, Z1 and Z2. Or we can say that we can take Z1, Z2, we can take labor 1, labor 2, we can take capital 1, capital 2, we can also take labor and capital. And initially the output produced by this, we make it a vector and we say that this Z vector is the function and this output is giving us Z1 and Z2 and non-negative means there should be certain production. Now we have the main thing that this Z1 and Z2 combination, is the maximum available or the optimum level of the output? So now we have to decide that the firm is going to produce, but either the firm is producing the optimal level of the output or the optimal level, the maximum level that has to be attained with the utilization of these resources. And otherwise, something less can happen to it. Now in these two or three stages, when we see, we will decide that this level of the output that is called y, it starts with zero and we have to take it as positive. Maximum it can go to this level and this maximum level, this is the optimal or the maximum level. If this production is equal to that actual level, then we will say that the firm is having efficient level of the production or we can say that firm that is producing efficiently. Now in this way, if we see, if I see a firm in this graph, they are utilizing the Z1 input and on the other side they are utilizing the firm Z2. Now while utilizing them, if they are producing an output, this is the output. And in the same way, the various combinations are optimal for us, so we have a boundary which we call as the optimal or the maximum output. So now the firm can have either this maximum level of output or in between here, here, here or here. Now if we see, this inefficient level which we are showing, so inefficient level and here maximum or efficient level, if we have any firm, then we can further study that why the firm is not coming from this point to this and this. What are the factors? Is it just an utilization of inputs issue or organizational matters? Or is it the issue of different combinations? So we can further study those factors. Another thing that we have to see here is that in the production function, there are some essential points for utilization of inputs which we have to keep in mind. And if we look at those essential points, the first aspect is that the first essentiality is that that nothing can be produced without the non-utilization of any input. Means, whenever we produce something, then it is necessary to use any input. So if we say, we have to say that function of 0 and 0 means the both two inputs, it cannot give any output, it will be 0. But along with this, if we use any input, we say yes, there is a function in which we have taken one input as 0. And the second input, suppose we say we have taken a labor, but the output is 0 again, why? Because whatever type of input was required, that is not provided. So it means some time a very specific level of input that is required for that particular level of production, even if that is not provided, then production will not be possible. So if we look at this, this equation, it gives us the concept of essentiality. And this equation, it gives us the concept of essential, strict essentiality. So this can, we can utilize, means function of 0 and capital A is there. And at the same time, we can say function 0 of anything input like Z2 if it is utilized in the other form. Now if we look at this, in this whole production process, we have two aspects of differentiation, meaning twice. And this twice differentiation, the next aspects, if we look at it, this will explain our marginal productivity relative to input properly. And that marginal input or rate of change in output with respect to input when we look at it, it will tell us further aspects that this change in output with respect to one input is not only due to one input, rather we have to include some other factors of production also. And when we have to deal with the change in only one input, we have to assume the other factors of production as factors paribus or constant.