Now we recall that we have the constraint and when we substitute x* into the constraint, we get equation being satisfied identically. Let us differentiate this equation with respect to α. What do we get then, this is also 0. Then let us take this equation, multiply it by λ* and add to the previous one, so once again we're taking this equation and multiply by λ* and then add to this formula, to the previous derivative. We can easily see that under the summation symbol, all these derivatives, with respect to α, are multiplied this time by a combination of two terms, df over dx, plus λ* multiplied by dg over dx, and that makes the derivative of the Lagrangian function. So, all in all we get the formula. But since x* is either maximum point or a minimum point, at this point the derivatives of the Lagrangian turn 0 so they are all zeros. This term completely disappears and what's left, dl over df. Now, let us apply this envelope theorem to a particular problem of utility maximization. We are considering a utility function over xy, which is maximized subject to a budget constraint. I is the income. Let's suppose that x and y takes on negative values. Moreover, I will consider a case that there are no corner solutions that are only internal solution to this problem, meaning that let's suppose that x and y are both positive. Both values, all their goods which this consumer buys and consumes. Now the question is, what will be the value function? The value function, in this particular case for this particular problem, is called an indirect utility function and its definition is as follows. I will be using v as a notation for the indirect to the function. This function depends on the prices and the income because the demand on goods depends on prices and income. Now, we need to answer the question, what will be the derivative of this indirect utility function with respect to the income? We can think of this derivative as a sensitivity indicator, how the welfare of the consumer changes with the change in income. Can we employ this particular envelope theorem? Yes, we can. The only problem is that we have three parameters here instead of one; we have prices and we have income, but nothing will be changed in the state of this problem if we have more parameters. Here in this particular derivative, we keep prices fixed. We change only the income. Now, according to the theorem, we need to use the Lagrangian. Lagrangian is this function and we need to differentiate the Lagrangian with respect to the income. What do we get then, we get λ*. So, it's interesting that we can find the economic meaning of the Lagrange multiplier. It tells us how the indirect utility function will change with the small changes in income. Well, and it can be shown that this λ* is a positive number. So, with the increase in income the welfare or the consumer will rise. We can ask similar questions, but this time we're interested in the sensitivity with respect to prices. Once again, taking the Lagrangian and differentiation with respect to the price of x will give us a negative result. So negative, and the same true for another derivative. And we have many more applications of these envelope theorems.