Let us consider a particular problem on utility maximization. Utility function is maximized subject to the budget constraint. The price of x1 goods will be P dollars per unit. The second good will cost just one dollar. So, the price is one. It's not greater then the income and clearly these goods should be consumed in non-negative quantities. So, this is the problem we need to solve and it has two parameters, P the price of the first good per unit and the income I. We'll start with NDCQ as always. Let's draw the budget constraint set. The boundary of this triangle consists of three sides. At each side, we're not talking about vertices here. Only one constraint holds as a binding constraint and the length of this gradient of the constraint is one, that means that NDCQ holds and also for the vertices including even the zero point. When we fill in the Jacobian matrix, each time we have a matrix of the full rank which is two. So, NDCQ holds as the conclusion. We can proceed with the Lagrangian. That will be a Kuhn-Tucker Lagrangian of course. It won't include non-multiplies associated with the non-negativity constraints. So, the Lagrangian will have just two terms. Now, we start filling in the first order conditions. Now, if we differentiate with respect to lambda, we get the constraint itself and complimentary slackness, looks like that. Now, we start proceeding the same path. First of all, looking for the inner solutions. So step one, assuming that, or rather I will be interested whether there are any solutions which lie below the budget line. Assuming that one minus is greater than zero, we get lambda zero, should equal zero. But that doesn't suit the first inequality at all because if lambda is zero, and X2 take non-negative values, how come it can be non-positive. So, impossible. The conclusion is, then we conclude, then should be I. But, all that we know already that the solution lies somewhere within this linear segment. Still we have three possibilities. Step two, checking whether X1 optimal value and X2 are positive. How to check that. This is an assumption. If we assume that, then we get immediately this time a system of equations, not inequalities and it will look like that. Why is that? Because if X1 is positive then that follows from the complementary slackness conditions. Both of them, the first and the second and we add also the budget constraint. Now, we need to find lambda. Don't forget we always check that the found value lambda star should be non-negative. This is a must. So, let's substitute the values of X2 and X1 into the budget constraint, then we get P lambda plus p lambda minus one equals I. That makes lambda star equal, and we see this condition is met because this is always positive, which is good. Now, let us find the values of X2 and X1. What do we get then? I'll write here. So, X1 we can find, if this is just lambda star according to the equation. What about X2 star? X2 star is different because we take lambda multiply by P and we subtract one and that becomes. So, for some low values or the income, it may be negative but that doesn't suit the condition of non-negativity. So, we claim that given the income the conclusion. So, when do we get this solution? Both goods are consumed in positive amounts. Given I greater than one, we get both X1 star positive and X2 star positive. So, this is a solution which lies somewhere in between excluding the end points or this segment. But now it's time to check whether we can get corner solutions. So, these points are called corner solutions and it will be done in step three.