 module 133 and we are going to study linear production function. A production function that expresses the change in the output for one unit change in the input in the at manner that for every unit change in input there is a constant change in the output that will be the linear or what we can say for one unit change in input whatever will be the change in the output that remains constant or the same. In other words when we explain that the output level q is the function of capital and labour and here capital and labour they have very peculiar marginal productivities of alpha and beta in the sense that we are going to say that the output is equal to alpha multiplied by l plus beta multiplied by the k. So, here alpha and beta are particularly marginal productivities of labour and capital. When we express this type of the production function we can say that we are having the isoquants in the form that they are in the straight lines that has to be expressed in a manner that whenever we are going to switch capital and labour in between they are having the same rate of technical substitution or the rate of technical substitution between capital and labour it is constant throughout that line. In these type of the production function we see that mostly there will be similar slope throughout and that straight line it can be of various nature it can be of less slope or the other, but whatever it will be maintained it will remain constant. And when we explain in the form of the scalability or when we have to explain in the form of the returns to scale. So, when we multiply the inputs with any factor that is either k or either t we are very expressively explained that the output is also going to increase by the same factor t. So, that production function also exhibits constant returns to scale property. So, when various isoquants of a production function they are parallel to each other then we can express the slope of that isoquant in the form that will be equal to beta divided by alpha and as the beta is related to the marginal productivity of the capital that is mostly expressed on the y axis. So, here it will be explained in the form of minus beta by alpha. So, such isoquant map is mostly expressed in the form of the straight line, but actually in reality it is very difficult that we are having this type of the production function in real life. We can mathematically explain it, we can graphically explain it, but in actual life it is very difficult because in such a production function only two inputs and their substitutability are also in this form that only they both can be substituted with each other. It is very difficult to be in real life. Now, if we look at it because the rate of technical substitution constant is along the straight line and because the rate of technical substitution is the main factor under which we calculate the elasticity of substitution and when we calculate the elasticity of substitution. So, because the rate of technical substitution that we have in this constant is the denominator in this production function if we look at it then we have 0 in this form. So, when that constant will be divided by 0. So, there will be the product that will be the infinity. Now, if we explain it graphically. So, it means there are one two or the three various isoquants and these isoquants all are exhibiting linear production function in the form that the slope of this line is equal to minus beta by alpha and whatever the formation is here that is expressed in the way and if we have to explain the expansion path. So, it is possible that if we have to check the optimal point here or here or here then the expansion path will be equal to this if expansion optimal points are here and here then the expansion path will be equal at this and if our optimal points are here. So, we can have our expansion path at this. So, there is possibility that this type of the production function they may express various types of the expansion path, but they all expansion path will be originating from the this origin and they will be expressing the constant returns to scale.