 Module 129 in production analysis and we are going to study technologies with distinct outputs and inputs. Actually in real life we can see that there are very little or the small number of production functions that can be produced or utilized or expressed with only one or the two inputs. Mostly production functions are those that require more than two or the three inputs. A simple example if we want to take, so we take two examples, if we take one from agriculture, means there is a production on any form, so if we see that there is wheat production, so it requires multiple number of the inputs, it requires seed, it requires farmers practice in the form of you can say the labor, it requires pesticides or it can require water, it requires any type of the what you can say fertilizer and even fertilizer can be of the various types. Likewise if we take the example of any industrial production whatever there will be a type of output it will definitely be requiring. I mean if there is an example of the textile we say it means we require the labor, we require the energy, we require the resources in the form of capital, we require certain dyes, we require certain stitching, we require certain chemical, so mostly production function in the real life are those that include more than two or the three inputs, so keeping in view those real life practices we can say mostly production function they include n number of the inputs. So here if we say that there is a q level of the output that is being produced by mostly production function including mean there is a function of x 1, x 2, x 3 and x 4, x 5 and up to n, so here now n is the multiple or the many number of the input case. If this type of the production function is available to us then we can generalize our previous concept of elasticity of production or returns to scale even to this level of the production function. In our returns to scale example we have come up with the one conclusion that if the inputs they are being scaled up by any factor t then the output will also be scaled up by the same level of the scaling that can be t or greater than t or less than t. So if we look at this example we have seen that the same level of scaling means constant returns to scale or greater than one will be increasing returns to scale or less than one will be decreasing returns to scale. So when we have studied those two input scaling cases so we can apply them here as well. Now when we do multiple input cases then there will be only one difference that is as we said earlier that z 1 and z 2 are inputs and they are multiplied by their scalar t. So we said t is e 1, t is e 2 but now we are having more than 1, 2, 3, 4, 5 many inputs. So now x 1 multiplied by t, x 2 multiplied by t, x 3 multiplied by t. So whatever the series of the inputs we all are utilizing by the same constant factor t and that scalar value will increase that all inputs or the production function by the same factor and that t if of the same degree will be having the influence or impact in the form of the increased output by the same level then it will be constant returns to scale and in the same way now if our output increase compared to the increase of the input t then we will say that increasing returns and in the same way if it is less than that it can also be expressed as decreasing returns to scale. So in multiple input cases there will be some differences but mostly concepts of economics can be explained in the same way. Just like we used Euler theorem for two inputs, the same way we can utilize the Euler theorem for multiple inputs in the same way that means the marginal productivity of all the inputs, if we look at all these inputs, the marginal productivity of all these inputs when we sum up then definitely they will be equal to the total change in the output or the product exhaustion theorem that is the multiple inputs in the same way. Now coming to the next, if we look at all the inputs, we were multiplying t or k and k is equal to 1 then this is our constant returns to scale and in the same way if it is greater than 1 or less than then we can do increasing returns to scale or decreasing returns to scale but in this we have a crucial aspect which is that when we were doing two or three inputs then we assumed that the scaling that is constant for all inputs means the actual concept of scaling is that all the inputs of total production function are the same amount that will either increase or decrease and in the real world situation for us it becomes difficult to apply it in economics no doubt mathematically we can multiply it in the same way but if we go in the real world situation then you can see that in many things if we count as an example I had already put the agriculture in front of you then in the production of wheat it is possible that a farmer who was putting 40 kg seed in the field instead of putting 40 kg seed, we can say that he started putting 60 kg seed and if he was putting 1 bag of fertilizer first then he started putting 2 bags means he is doubling it but it is not possible that he says that as much as I have one bag of lands, I have to double it definitely because it will require some IPR or more cost is involved and in the same way sometimes we have a managerial skills issue and if we look at it we say that if any farm is there it is mostly a boss, now that boss or managerial can make its decision now if he has to enhance the production then he can enhance the inputs but the boss, the manager or the producer is also a part of the management team and if we are going to double the same then there might be a conflict between those instead of increasing the production it will start decreasing and in the same way we can say that the farmer who took the first example of agriculture that he might have had one bag of land to increase his production then he says that he should double his land now he takes two bags of land but production in that way and he also doubles all the inputs in the same amount he is adding as much water as he was putting the unit on one bag and he is doubling it in the same amount so now he expects that if the first farmer I had had 40 months then now I have 80 months from two fields but the possible farmer is getting 100 out of 80 and this also might be 60 out of 80 the reason being all the factors are constant but we have not included the fertility of that land meaning that was, that is not explicit if we say that this is an intrinsic or a latent quality and sometimes it is also possible that a particular weather condition and both the space of land are at a distance so one's weather and the other's are different so these are certain factors which if we say in real life then we say that that is not possible that the concept of scaling if we are in the case of multiple inputs then mathematically then maybe we can put all the rules in the same form and explain them and prove them but in the real life situation sometimes the concept of scaling for all types of inputs sometimes it becomes very difficult to put them on and that is why the law of diminishing productivity is in the same way if we call it diminishing returns to scale then we can utilize it in the same way because actually in terms of diminishing returns to scale because somewhere we will have some factors of production in that production function which we cannot scale which we have to keep fixed like we have already talked about the manager or the land or a farm which cannot duplicate its production unit so likewise there will be some aspect which will not be possible to scale in the actual form so the multiple input case mathematically and technically is going to follow all the rules that we have studied for two input cases and we have applied that as Euler theorem we have applied the rule of isoquant we have looked at the point of tangency we have looked at returns to scale they all will be applicable but in the real life situation sometimes it is difficult to follow all these rules thank you