 This module pertains to the method of analysis and in the method of analysis our main topic is the local and global solutions. Before going to this topic we have to see that what are the various topics for the analysis available for us in the subject of economics. There is not a single method of analysis there can be mathematical statistical econometrics comparative analysis the graphical analysis historical analysis and case studies and at the end we can have the game theory and these all methods they utilize one technique the basic technique that will be called the optimization. The optimization is the combination of the two words the one the optimus and this optimus derived from the Latin having the meaning of the best among all or the most favorable thing. Here in the students I would like to tell you that from which our Urdu language is Arabic and Farsi so when you look at the many words of English then when we look at the root word or the language then it will be either Greek which will be on the other side or Latin or Greek or Latin or Greek so that is why you are not sure that every time the Latin or Greek word is the source of it. Optimus had an other word with it that was the opus which means work, work or work related and if we look at it optimus means the thing which is the work and is the best among the things of that work and omnius for all so all the things which will be the best solution or the best thing for us that will be the optimal or the optimus and optimization technique is based on the fact that when we have to select the best thing and if we look at its significance in economics then the basic work of it is that it is our work in any thing because the work of economics is related to resources which are scarce resources so optimization will help us to reduce the cost to save the resources to improve the efficiency means to get the best solution with the same scarce resources and help us to get the best solution with accuracy and to help us to make the best decision. Now if we look at some elements of optimization the first of those elements is we have choice variables and defining those choice variables is one of the first things that optimization will tell us to define which choice variable can be different. A consumer who is going to buy anything will have a choice variable that I have to maximise my utility. In the same way, it is a firm whose choice variable can be that I have to increase the output of my production and many times it can be that for the same production it is that I have to create that production in less and less expenses so its choice variable reduction will be in cost. When we decide our choice variable then our second main part will be objective function and objective function is basically a mathematical relationship in which we decide that we have a choice variable on one side and what value do we give that variable and we have to optimise it or maximise it or minimize it. And to achieve this objective function we have available resources. We have supply, capital, labour, or we have money so we have available constraints. And by utilising all these we will decide the final method and we will see how our optimization technique works. We can also draw that optimization through graphs. We can use calculus, we can use linear algebra, we can use optimization. Now how will we get the solution of optimization technique? So the basic solution if we call it technical then we say that the choice variables we take we make them vectors of values. Actually choice variable if we have any if we have then we say to quantify it or in other words we have to get it in the form of number or in the form of measurement or in the form of number. So that is why we say that we have to decide its values and those values will be in the feasible set or available resources. And then we will see that for that objective function it will be at the minimum or maximum stage. When we talk about the minimum and maximum then basically we will have two important things. One is called global solution and the other is called local solution. First I will talk about local solution which is located below. We say that local solution is the one that fulfills all our conditions while staying at that point where that objective function which we want to get or solve. We take the value of it which is nearby which means we call it to some extent and the nearby we call it neighborhood or it is a particular range. In that range it is the most optimal or best point or its value. But when we talk about global then we will say that global will be the point that instead of our nearby all the points of that whole analysis will be the best among all that will be global. It will be local for its nearby or neighborhood but while looking at the rest of the locals the best local will be called global solution. Now if we look at it then we have the main questions which come for optimization so if we look at it there is an existence of whether the solution that we are looking for is really available. It is also made in that system because if it is possible to solve some equations then it is not solvable. Similarly if you have read the math then we will see that in the math many times we have a matrix like this which does not have a solution. So in this we basically have that question in which we say that is that objective function is solvable or not and if it is then we have to look at it as a local and global point. After doing that the local or global point is unique it is not repeated it is very important then the interior solution and its location when we talk about the interior solution then it is on the boundary or within the system because if it is on the boundary then its repercussions come up. With the help of this graph if we look at these things then I would like to help out for you that if we draw in this then we have drawn a variable on the x axis and in the same way we have drawn a function of a variable which is on the y axis and when we look at the different combinations of these two which one of them is coming to this point and after that this curve is drawn and goes down then it comes up then it is present here so if we see then we have a point in this graph so if I look at these nearby points which I am updating then this becomes its neighborhood so in this neighborhood this local maximum will be one in the same way if we note these points and this will bring all its neighborhoods then in this neighborhood this point will also be a local maximum but when we compare these two then from these two this point is also local maximum compared to our neighborhood and compared to this it is higher point so if we say while doing these two then we will say that this will make it a global solution and on this global solution if we say that this graph is instead of coming down in this way if it goes up then when it goes up then we will see that if you go towards infinity because it is not coming down so if a point goes to a place like this then that is called the boundary point so while looking at these aspects we will have to see in our models or methods now if we see in this so if we have to calculate one point here then when we do any point at this place then this will be its minimum point in the same way when this above point will be then this local maximum will be called and in this way this will be its maximum point if we see some more graphs in it then this will only look like this because we draw a lot of graphs in our economics in that if I draw in front of you that our cycle can be like this in which you see that it is increasing every time so if I see that this is a minimum then this is a minimum this is a minimum this is a maximum this is a maximum this is a maximum and this is a maximum but from these four maximums if we decide then where is this line coming from this is higher this is next higher and this is above so because the highest range is this so this point will be called the global maximum and like this if we have to talk about the minimum then we can also decide that many times in many of our points which we see from the minimum that if I draw it in this way then this is also minimum this is also minimum but this is the most minimum so this point will be called global minimum and in the same way this point that will be the global maximum so any such point if we see if I do this curve on a side line in which we see so I will say this point on one side if we draw this point then this is the turning point of the two contours means when there is the turning of the slope when we draw this point we call the saddle point and whenever we have to solve the local or global either minima or maxima we have to calculate the slope of this saddle point with respect to either this active axis and we see that from that one unit change what has changed in the slope in this or not so the saddle point will be that point on which we say that when the rate of change becomes zero