 Module 131, in production analysis we are going to study fixed coefficient production function. As we have already studied that production function can include various type of the inputs and those inputs they have the various types and at the same time various natures. There is a possibility that 2 or the 3 inputs they can be substituted with each other and there is a possibility that they cannot be substituted to each other. So, due to this property of substitution we have various types of production function and out of those there is one type that is called fixed coefficient production function. When we say fixed coefficient it means it is that type of a production function that requires that the output it will be produced by a fixed proportion of certain inputs. So, this production function it requires that the inputs they will not be substituted to each other, but they will be utilized in a various minimal level or you can say a various fixed level of that that will be required from one to another. That combination can be very unique means there is a possibility that we have stated that 1 unit of labour plus 3 units of capital if this is the combination. So, whatever the situation will be scaling of that type can be utilized, but that minimal level of combination has to be remained fixed throughout the process of the production function. And the key economic feature is related to that is mostly an entrepreneur or a producer they have a very valid reason for switching of the inputs keeping in view the availability or the non availability or particularly when they see that in the market certain amount of input is not available or if available it is costly mean due to the relative changes in the prices. So, due to that price change the respective budget line or the respective cost line it affects. Due to that entrepreneur they have the ability to switch the inputs in a way that now they come up with certain solution that they produce the same amount of the output now, but with various combination of the inputs. But this fixed coefficient production function it necessitates that we cannot substitute the inputs despite of the fact that we are facing the change in the relative prices of the factors of production. So, now coming to the other part that we will draw a diagram and we say that these inputs that what we are utilizing in this fixed proportion they are basically the complement to each other means this production is not possible until and unless we are going to maintain that complementarity or that combination. So, that production is not possible in the sense if we say that if we have to utilize in the production that the pen with the ink. So, that pen cannot write something if we have not utilized the ink and likewise there can be certain other utilization also in the industry or something else that they has to be utilized in a very particular and unique manner that they are complement to each other and without that they cannot be utilized. Here one very simple example I can give you that if we look at it we talk about a nut and bolt that whenever somewhere this is a very small portion in the production manufacturing when we have to make the furniture of the wood we use a lot of electronic things so you will know that on one side the secret is to close it, there will be a volleys or bolts so that is the peer they are the complement to each other without that they cannot be used likewise in our stitching if we see a simple button is used which the two parts until you come and cross it will not do its function of closing. So, these are the aspects that if we take a lot of things from a very small portion, even if they go to a very large level, so that complementarity or the combination has to be maintained. Now, if we use this in a mathematical form, then the way we used to write in the production function earlier we said that Q amount of output is the function of capital and the labour and where capital and labour they have their particular elasticity of production that is given with the alpha and beta. But in this we have a very unique aspect in which we are attaching a word that is called minimum and that minimum it has a very unique feature in the sense that that combination is capital or labour that has to be maintained throughout the production function whenever we are going to scale it up. So, this minimum operator is that particular unique feature of the fixed proportion and this smaller or the minimum amount has to be maintained throughout. Now, if we say that we have their elasticity of production whatever it is, but if we double the amount of capital and double the amount of capital, then we accept that maybe this will enhance our production. So, like production functions, here our production will not be enhanced at all rather in doubling we have invested the initial amount of capital on that value or we have invested that will be total waste. If we look at this in a graphic form, then this shows if we look here that we are showing a production function in this form. Basically, this is not shown in the format, this only this vertex shows its combination and this vertex's combination shows that only this combination is required that minimum level of capital and labour that give equal to the q amount of output or any amount which we will enhance here on its right, but that will be totally waste and likewise, if we enhance its vertical capital here, then that will not give any type of the incremental value in our output, but only and only production is possible at this vertex. So, if we have to scale it up, then we have to scale it up in equal amount, here we have to enhance it in one unit, which in other production functions, our production increase is possible, it will not be like this. Therefore, when we make its different forms of isoquants, we see that this is the first isoquant, this is the second one and this is the third one. So, and we join this vertex point here and this vertex point here shows the points where production is possible and here if we see this, then our region shows this or this, if we say this, then we can say that this is our expansion path, in which if we have to enhance the output, then only and only this is our path when we can enhance our output in fixed coefficients. Thank you.