 Module 134 in production analysis we are going to study Cobb-Douglas production function. Generally Cobb-Douglas production function is the one that is mostly utilized in the analysis of the production. Likewise the other production function it exhibits many properties that has to be explained in very much detail in various coming sessions. But here we see related to the elasticity of substitution. So any production function that has elasticity of substitution equal to 1 that will be called the Cobb-Douglas production function. And in the Cobb-Douglas production function the isoquants they are usually the normal of the convex shape that we mostly explain in our all types of the analysis. And in other words we can say that mostly the shape of the isoquant in the Cobb-Douglas production function is similar to the indifference curve that we have explained in our consumer analysis. And the mathematical form of the Cobb-Douglas production function is just equal that mean quantity is the function of capital and labor. And that function is equal to A for technology, K for capital and alpha labor. But with the chain that here alpha and beta that are the respective output or the production elasticity of the input they are put at the power of that capital and labor. So here production function will be equal to A K for its output elasticity of production means alpha and for labor its own elasticity of production beta. Now coming to the next slide we explain that these two elasticities of production of the respective inputs that are alpha and beta they are in the manner that alpha plus beta it will be equal to 1. So when it will be equal to 1 we will say that the Cobb-Douglas production function it exhibits the constant returns to scale and these returns to scale we will explain in the next slide. And at the same time if we take the same formula of our returns to scale that if the inputs they are multiplied by a unit t or a scalar value we expect that the same scalar value will be exhibited to be multiplied by our output. So in this manner this alpha plus beta it will also be summed up for the power of that function. And so alpha plus beta if will equal to be 1 here also it will be the constant returns to scale and if this power of alpha plus beta will be greater than 1 then we will expect that it is increasing returns to scale and if this alpha plus beta is less than 1 then we will say that it is decreasing returns to scale case. So in this manner the Cobb-Douglas production function has the property that it can exhibit the three all returns to scale constant increasing and the decreasing. So if we have to explain that the elasticity of substitution equal to 1 this we will explain and here the Cobb-Douglas production function has also proved that in very other applications of the production it is linear in the logarithm means it is very simple to add up all the these factors. When we convert this Cobb-Douglas production in the form of the log we just convert this q into form of natural log of q and this part of technology into natural log and this alpha natural log K and beta natural log L. So in this way the effects of technology effects of capital and effects of labor they can be decomposed and then further summed up to have their contribution in the total output and here the constant this alpha this beta and they both they give the capital input and the labor input ratio and they are basically the elasticities of these inputs. Now coming to that elasticity of substitution is equal to 1 we can derive in the manner. So now coming to that as we know that elasticity of substitution when we explain through theta it is basically the ratio of change in capital and labor ratio divided by their respective change in rate of technical substitution and here we can say this rate of technical substitution is basically is change in capital due to change in labor and that gives equal to marginal physical productivity of labor divide by marginal physical productivity of capital. So when we explain this and we utilize with respect to the Cobb-Douglas production function where Cobb-Douglas production function is given by q equal to a k power alpha and l power beta and here rate of technical substitution will be equal to MPL by MPK. So when we derive MPL we will take first derivative of this function with respect to l that will be equal to this beta a k power alpha and then l beta minus 1 and now it will be divided by marginal physical productivity of capital for which we will take the first derivative of again this function. But now with respect to k and it will be again equal to this alpha a k power alpha minus 1 and then l power beta. So now when we simplify we can say this beta by alpha and this a will be cancelled by this and now this k power alpha into k power alpha plus 1 mean this k alpha minus 1 we will take above and l power beta and then again l beta plus 1 this will be taken downward. So this will be crossed by this and the resultant will be beta by alpha into this k by l. So this rate of technical substitution here it will just come up to this and when we have to now calculate this elasticity it will be equal to k by l change in k by l divided by change in RTS. So when we will substitute its values it will just come equal to 1 and now here we have read that elasticity of substitution will be substituted for this value and this value and it will give us the unit that is equal to 1 and now coming again to the same that we can utilize the same equation function in the form of the logarithm if we say. So again we can say that when q is equal to a k power alpha and labour power beta. So the Cobb-Douglas production function when given in this form so we can also write in the form of the log where rate of technical substitution is given in the form of log like this that it is equal to natural log of beta by alpha plus natural log of k by l. So in this way now when we write our elasticity of substitution it will be equal to the change in the natural log of k by l and then change in like this. So again it will give us the result of 1. So this is the proof that Cobb-Douglas production function has elasticity of substitution equal to 1 and with the various changes of this alpha plus beta we can have the three various types of the returns to scale. Thank you.