So, first let's check that these point where x1 is consumed in positive amount but x2 is not consumed at all can be a corner solution. So, that will be our corner solution point number one. So, for that point we have x1* equals I over P and x2* is 0. So then, we can substitute, since x1 is positive from the complimentary slackness condition, we get that this should be 0, x2 is also zero. Then we get immediately λ* is 1 over P, which is good, this is a positive number because price is positive, of course. Then, we need to check whether this is fulfilled. It's interesting, that provides us the range of income values for which a consumer, in his consumption, follows this pattern just spending money on buying good number one. So, according to this inequality we have I over P, minus one over P, that's our multiplier, should be non-positive. So, then we have a smaller income. Finally, this point. This point cannot be the maximization point, clearly, but we need to check that still. So, at this point x2* is I, x1* is zero, then we get immediately from these two values, if we substitute that λ is zero, but that's impossible. We've seen it when we begin solving the problem. Remember, we were considering the case where the maximization point lies below the budget line and that situation wasn't possible at all. So, we have finished analyzing the first order conditions. Now, we have found for the values of income greater than 1. Well, we can still call it inner solution because this is not a corner solution then, right? For small values of income described here, we have this particular corner solution. How can we be sure that we've got the maximization points? Once again, we referred to Weierstrass theorem. Since the budget constraint is a closed bounded set. So, Weierstrass theorem is applicable, the utility function is continuous. So, it takes the greatest value at some point here, and we have found just one critical point or the Lagrangian. So, that means that we have found exactly the optimal bundle or the consumer. Now, we start another problem. This will be a problem of minimization of costs. So, we have a production function. Let me write it here. This is the output described by a letter Y. This is our production function. Now, we are minimizing costs. This is a two factor production and that should be minimized. Subject two. The minimal output level is Y, so the constraint is written in the form of the value of the production function, should exceed or at least be the same as y and x1, x2, takes non-negative values. We haven't analyzed yet the Kuhn-Tuckent conditions for minimization problems. We may, but actually, as we know, we can always replace a minimization problem with a maximization problem. It's done by simply reversing the sign of the objective function. So, if we take this condition on minimization and replace by minus a 100x1 minus x2, should be maximized. Then, we get an equivalent problem and we can fill in, we can form the Lagrangian function, for this time for the maximization problem. So, how will it look like? L, Kuhn-Tucker is Lagrangian for maximization problem, you start with objective function written there. What now would take plus, we take λ and what will be written here? Let's see. For maximization problems, we subtract from the bigger term the smaller term that becomes like this, and it's quite time to write down the first order conditions using Kuhn-Tucker's conditions. So, following their format. So, we have, since no space left I will write the complementary slackness condition in the form of. I have skipped NDCQ checking for the time being but I'll return to it later. Now, λ takes on a non-negative values, also I'm rewriting the constraint and the signs of the factors employed here. Now, check NDCQ. In order to check NDCQ, we need to visualize then set the constraints set under which we maximize this function. This is actually a separate problem because we need to find out what will be the shape of the isoquant. We get an isoquant when we rewrite inequality sign into equality sign. Then, in the space of factors we have a curve and this curve looks like that. It can be done by, this is single-variable calculus part of the problem, but clearly you can solve for x2 as a function of x1 and you get a P solve the hyperbola curve. So, either we have a solution which belongs to this isoquant in between the endpoints or we get some endpoint. All in all, we have a gradient which is not 0, and that means NDCQ holds no concerns about that. Now, we proceed with solving this system of inequalities, equalities. It will require some time to do.
