We proceed with the envelope theorems. There are at least two theorems which are called envelope, and one is about unconstrained optimization, another one is about constrained and we start with the unconstrained firstly. We have an objective function, a function of many variables but also it depends on some scalar parameter alpha. So we are considering a problem of unconstrained either maximization or minimization, and alpha as I said is scalar parameter so it belongs to some interval. As for the function f, this function is continuously differentiable in all the variables. Now, let's suppose we have found the solution of this optimization problem. To be more concrete, let's suppose the function has been maximized. So then, the maximum point depends on alpha. Now we'll also suggest that this vector function of alpha is still continuous differentiable on the interval I. Later on we'll find out how to guarantee this condition. For the time being, let's suppose they found maximum point is continuously differentiable and now we'll substitute x star into the function itself, having a function of alpha alone and we'll call it value function. So by definition I'll be using phi letter, this value function is defined exactly like this. Now the question is, how to find derivative of this value function with respect to parameter. The envelope theorem is exactly the theoretical statement which simplifies the answer to this question. We form it as a theorem. I will need just one formula. So under condition that a function is continuously differentiable, that x star is also continuous differentiable, this derivative can be formed by partial differentiation or the objective function with respect to alpha and later on we substitute into the argument of the function the maximum point. We'll provide the proof. The proof is quite simple. Actually, to find this derivative means just to apply the chain rule to the composite function. This is our composite function, so then becomes. Don't forget that x star is the maximum point and that means that all first order derivatives of the objective function turns zero at this point. So what's left? All this terms under the summation symbol disappear and we get just one term where we substitute x star point. Now, there are a lot of applications to the microeconomic theory of this particular theorem. I will just provide one. Let's suppose we have a perfectly competitive firm whose cost function is c(y) function which is continuously differentiable, and we are considering a profit maximization problem where p is the price of the production per unit and this is the revenue, y is the output. So the firm maximizes its profits. Maximization takes place with respect to the output. Now the question is how to find the derivative of the profit function with respect to the price. Without knowing the envelope theorem, the problem is solved through some steps. Firstly, you need to find the supply function of the firm by differentiation of the pie function and having the first or the conditions. After that, actually you have to apply implicit function theorem in order to find y as the function of price p and later, you have to substitute into the pie function or the supply function and only then you are able to differentiate with respect to the price. But the application or the envelope theorem makes it much easier because all we need to do, we need to take this formula and we need to simply differentiate with respect to the parameter which is p. Now this pie star function which is called in microeconomic theory a profit function, is exactly the value function. All we need to do, we need to take the formula and differentiate with respect to p then we get the answer. It's y star the supply which is positive and that means that with the growth of the price, the profit function which is the maximum profit attainable at a given price is also positive. The derivative is positive so it's growing.