 Module 135 in production analysis we are going to explain constant elasticity of substitution production function. This production function was given by Aero in 1961 and it mostly exhibits the same function that are explained through the Cobb-Douglas production function, but it has certain more capacity to explain the other forms of the elasticity of substitution as well. Likewise again we are going to explain that the output is a function of capital and labour and now the capital and labour they have their respective elasticity of production not only in the form of their power rather here we have added in this form that the k has the power of rho and labour again has the power of the rho, but now the whole function is have the collective power of gamma divided by rho and how it is going to have various implications that is explained in this slide. As we know that the degree of returns to scale mostly in the Cobb-Douglas production function was in the three forms mean it was constant returns to scale, increasing returns to scale and decreasing returns to scale. And now depending upon the values of this elasticity of input that alpha and beta that here are explained in a different form, we again can utilize the same if all these inputs now they are multiplied by a factor t the same t will be added up in the output and the total output will have the returns to scale. But now we have to decide that this change in the inputs of capital and labour multiplied by t either they will have the similar change in the output or greater than or less than 1. In this equation we can see this respective elasticities are in the form of the rho. So if this rho and this rho because this is explained at the three places if it is less than or equal to 1 then this function will be equal to constant returns to scale and likewise it may have the value when we can say that it is not equal to 1. And if this rho is that it is in the form of numerator it is gamma. So if this gamma is have the value greater than 0 means it should be positive. So this function will closely resemble to constant elasticity of substitution of utility function. So when this gamma will be greater than 1 means this numerator will be greater than 1 it will be increasing returns to scale and when this gamma it will be less than 1 it will have the decreasing returns to scale. And when this will be equal to 1 we can expect that it will be constant returns to scale. To simplify this explanation we have one another narration that this elasticity of substitution and this elasticity of substitution again is equal to the ratio of the capital to labor divided by change in rate of technical substitution. So it is equal to 1 divided by 1 minus rho and this rho that we have already put here in this form. So to simplify now we can say that this linear fixed proportion function or the Cobb-Douglas they all cross bond to this case only and only by the change of this rho. So if this rho will be equal to 1 or this rho will be equal to infinity or rho will be equal to 0 these 3 they will have the different cross bonding case related to our previous all 3. So when this rho will be equal to 1 that will be equal to the linear production function when this rho will be equal to infinity it will be fixed proportion and when this will be equal to 0 we can have the Cobb-Douglas production function. So keeping in view the more insight that mostly the rate of technical substitution and the substitutability it depends upon one input or the ratio of the input. So to emphasize this this CES function it uses the distributional weight and that distributional weight mostly if we say that this alpha and alpha should be positive mean it should be greater than 1 and it should be less than 1 because alpha plus beta they both have come to the unit. So when that alpha is multiplied by this k and that 1 minus alpha is again multiplied by this l. So it means now this is the cross bonding contributing share of labour and capital that is added up in this equation just to give the relative significance of the input. So there is a possibility that the capital is having here 0.7. So if there will be 0.7 definitely 1 minus 0.7 means now labour will have the contribution only by the 30 percent. So in this equation not only their respective shares rather their respective productivities both are entered in the equation. Now we can explain that what are the various types of production function shapes that are mostly explained by the elasticity of substitution as we have seen in our previous that when elasticity of substitution that is equal to 1 divided by 1 minus rho and when rho will be equal to 1 then it will have this straight line or unitary elastic production function that is the linear production functions. And when we will have again this elasticity of substitution by the same Farcoula and where this rho will be equal to 0 that will be the case of the normal Cobb-Douglas production function that exhibit the convex shape production functions. And again the third case when this elasticity of substitution will be equal to infinity and that will be explaining when this rho will having the value of infinity we will have the respective production function of this fixed proportion when production is only possible at their vertex. So we will have this fixed proportion production function and this will be the Cobb-Douglas. And one another factor that I have explained only here the generalized shape the three shapes can also be explained in the form of the increasing rate of return or the decreasing rate of return means if this will be the case when rho is equal to 1 and the one case normal the other when we can have the decreasing rate of return. So when decreasing rate of return the difference between these isoquants will be increasing it means now we will requiring more amounts of inputs to have the level of the output. And if it will be increasing rate of return shape will be same but the difference will be less in the next steps like this and when the shape is like this that rho is equal to 0 it means constant elastic function can have like this Cobb-Douglas shape normally and when there will be increasing rate of return gradually for the next it will have the smaller difference to attain and if there will be this will be increasing and when there will be decreasing rate of return then the next it will have the more difference it means now the more will be required. And likewise for this shape increasing they will be further away and for increasing rate of return the next that will be closer to this that small units of inputs will be required to have the same level of the output or the increasing level of the output. So this CES constant elasticity substitution function it not only provides all shapes of the production function that we can explain in one form depending upon it is various rates of substitution and elasticity of substitution and production elasticity of it is inputs and it can also exhibit the three rates of returns to scale thank you.