In the theory of homogeneous functions, there is a special, quite famous theorem, which was proven by mathematician Euler in the end of the 18th century. It is called Euler's theorem, and I'll provide the rigorous statement. Let F, a function of n variables be continuous differential function, and it is homogeneous of degree m, then it satisfies equation. This equation, by the way, is called Euler's equation, the sum consisting of n terms. By the way, this part of the theorem is quite easily proven, but the converse statement is also true, we'll talk about it in a minute, and it's more difficult to prove it, still quite possible. So the converse statement is as follows. So given a function f of n variables, which is continuously differentiable, if it satisfies in its domain, which is the positive orthant. If it satisfies this equation, Euler's equation, then it is homogeneous of degree m. M is the factor here. We'll employ the converse theorem to this. Actually, both parts are being summed up in one statement which has the name of Euler. Let's consider a minimization of costs problem when the production function is homogeneous of degree m. So let's suppose we have a production function which is continuously differentiable, so this is the production function, and it is homogeneous of degree m. We set cost minimization problem. As always, and as it happens almost always, the constraint, inequality constraint, for this minimization problem will be binding and it can be written as an equation. Also, let's suppose that all factors are employed in positive quantities. Now what we can see, we can form Lagrangian. Lagrangian. We apply- this is not a Kuhn-Tucker problem. We apply a simple Lagrange method. But the order of the terms here within the brackets is important because we believe that this constraint was initially inequality. So then, here, we need to subtract y minus f of x, and then we get a system of first order conditions in the form of- okay. So, i ranges from 1 to n. Now let us recall one of the envelopes theorems. In this theorem, we're interested in finding derivative of the value function which is the cost, total cost function in our problem with respect to the output dy. What did we get then? We've got Lambda star. So check with the previous lectures and find out. What I'm aiming at, since we know it's given that the function is homogeneous and there is some degree of homogeneity, then we can employ the Euler's equation then. Right? So this employment will give us the form of this equation when we substitute the partial derivatives of the production function using first-order conditions. What I mean is as follows. Since we know for sure that Lambda is a positive number, otherwise, the price will be zero, then we can solve for marginal product, and we get a w_i over Lambda from here. Now, it's quite convenient to write just below this equation. We can substitute into our left-hand side these derivatives. What do we get? One over Lambda goes out, and here, we have w_i x_i. By the way, we're operating at the optimal bundle point. So we can write asterisks everywhere. Here we have mf but the value of the production function is described by y. So we'll have my. What does we get here as a sum? As a sum, we get a total cost function. So this can be written in the form of 1 over TC prime, the derivative with respect to y, following the envelope theorem result. Here we have TC over y and here we have my. Here we are dealing, probably that's the first time in the series of lectures, with a differential equation. Luckily, it's so easy and it depends to the easiest type of first-order differential equation, which is called separable. That means that we can perform since TC prime is differential of the function over dy and that can be used here, so we can write before integrating. Here we get 1 over m integral and here we have dy over y, both to see as a variable is positive takes positive values and also y is the output takes positive values. So immediately, we can write expressions after integration. So anti-derivative of 1 over x is a logarithmic function, so this is how we get lnTC as a function of y equals 1 over m lny, and as always, we need to attach a constant of integration which will be probably Gamma, some number. Now we need to exponentiate this equation. That means raising the base of natural logarithms e, a special number, to the left-hand side and to the right hand side. After that, we get in the left TCy and here we have e to the power of Gamma, y to the power 1 over m. All we have to do, we need to replace this awkwardly-looking constant into a different constant and that's how we get and I'll put it as the final result, interesting result that if a production function is a homogeneous function, then the total cost function based on this production function is a power function, c times y to the power of 1 over m. Once specific example concerns m equals 1, remember, if m equals 1, we're dealing with a constant returns to scale function and then TC is simply linear, c times y. We'll know in microeconomic theory this is quite often case of production functions. So we always rely on the fact that marginal cost function is fixed as well as the average cost is also fixed, and it all follows from the homogeneity of the production function.