 Module 132, in production analysis we are going to study the Lyuntov technology. Actually this topic is very much related with the fixed proportion of production function that is called our production function that is utilizing the inputs in the fixed proportion. So here is one production function of any type of the good that is expressed in a way that q is the function of capital and labor and in the sense that it is equal to capital plus labor plus 2 into under root k and l. So mathematically if we are going to explain this equation we say that this is a very special case or the classification function named for the Russian American that economist that was Vasily Lyuntov. The function clearly exhibits the constant returns to scale. So when we say the constant returns to scale definitely it will be homogeneous of the degree 1. So when it will be homogeneous of the degree 1 it has to be expected that it will follow the rate of what we say that the product exhaustion theorem or the at the same time Euler theorem and the marginal productivities of all the inputs they will exhaust the total output. And when we utilize the same production function and we explain the marginal productivities function in the this form by taking the derivative. We take the marginal productivity of this total function with respect to capital that comes equal to 1 plus into capital by labor ratio it is power minus 0.5 and the same is for the marginal productivity of labor. It means marginal productivity of capital and marginal productivity of labor they are equal to each other and they are positive and at the same time diminishing returns. So this is that expected that we have to expect from the product exhaustion theorem and the Euler. And at the same time the rate of technical substitution here depends only on the ratio of the two inputs that is capital and the labor and all things that are utilized here now means this scaling of the t has nothing to do with the marginal productivities. So rate of technical substitution it is equal to the marginal productivity of labor divided by the marginal productivity of capital and when we divide this it is showing us that there is an isoquant that have the generally as usual the convex shape that we have expressed even in many other forms. But this marginal productivities are positive and diminishing and this explain that due to this if we say sometimes there is the fixed proportion case there can be, but it can express some other shapes also. So the fixed proportion case and the Luntov they are mostly the utilized in the mirror image of the each other, but if the special case will be different than the other it may have the different value and the different behavior there.