 So, we have looked at the resources. We have also looked at a model by which we can identify how the non-renewable resources would get utilized, we have just finished the hotling model. We now switch tracks, we are going to now move into environmental economics and we are going to look at how do we trade off between energy development and environment. Before we do this, we need to understand some basics of demand and we will talk about preferences and utility. So, we will introduce some simple concepts where we talk about preferences and utility. Now, in general, in life, we are always encountering choices and we have, in these choices, we always have preferences. So, in when we talk in terms of economics, we would like to quantify these preferences. So, we would like to rank the preferences between bundles of goods and services. So, to simplify things, we can even look at two goods X and Y and we are talking of goods but it could be goods and bads that means it could be pollution, it could be garbage. But in general, let us look at if we are talking of two goods X and Y, how do we choose between those goods and then we want to understand consumer choices and preferences. This is a whole field in itself but we are just going to look at the basics of it and use it and quantify it and then use it to see how we can make trade offs between different choices. So, we look at it, start with first looking at individual choices and then we will come to when we aggregate these individual choices into a societal choice. So, for simplicity, let us look at two goods X and Y. These goods could be wine and cheese that is the example which is taken in the book by Colstad on environmental economics, pizzas and movies, you will see if you look at different kinds of lectures which are available, people are talking of pizzas and movies and sometimes when you have a fixed amount of money which you want to spend and you want to enjoy a movie or you want to see the movies or you want to have pizza or you want to do both and we look at that. And in a similar fashion, we take these two goods as things that we can relate to but we can translate that finally to resources, energy use and environment. In the case of preference relation, we have some properties. The first property is that there is completeness. That means each individual knows his or her preference between a different set of goods. At most one can say that we are indifferent, these two are equivalent. Otherwise, I prefer this to that or I prefer the other one and so on. So, it is not that it will be left ambiguous, that is in terms of completeness of the choices. Then the second is that there is a transitivity that we have, that means the preferences are rational. For instance, we create a suppose we say that you prefer, suppose I prefer apples to oranges. So, this symbol which I say is apple is preferred to orange and suppose orange is preferred to bananas. If the preference is rational, then it will be transitive. That means if A is preferred to B and B is preferred to C, then A is preferred to C which means that apple would be preferred to banana. Then the, so this is the second part, we talked of completeness, we talked of transitivity. The third assumption for preference relation, the property is non-sociation and this is not always a correct assumption in real cases, but for the economic calculation, we say that more is better. That means if you have, if you watch two movies, it is preferred to watching one movie. If you are having two pizzas better than one pizza and so on, so that the idea is that it is monotonicity, there is no satiation. In actual practice, you know that after a point, you stop, you do not have any hunger and you would not want to. So, there would be satiation, but for all the, in the economic utilities context, we talk of non-satiation. So, in the preference relations which we have, now we have the following possibilities. We had this sign where we talked about x preferred to y. This is a strict preference, that means x is always preferred to y and the weak preference is where x is preferred to or equivalent to y. This is a weak preference and x, this is an indifference relation which means that the consumer feels x and y are equivalent and this is called indifference. So with this, we would like to introduce this concept of, we talked about transitivity and non-satiation. We would like to look at these preferences. So for instance, if we look at the, if you are talking of three points, in the case of A, we have, you have one movie and two pizzas, here we have two movies and one pizza and C is where you have two movies and two pizzas. As per the non-satiation, C would be preferred to B and C would be preferred to A. Between A and B, we have to then decide, it will depend on the individual and suppose both are equivalent for the individual, then we can say that A and B are, we are indifferent to this. We can join a set of all points which are equivalent to us and that is called the indifference curve. Of course, here we are talking of discrete values but in general, we can look at a continuous kind of curve and this is the indifference curve is a set of consumption bundles that the consumer think are equally good. She or he are indifferent to the consumption bundles. That means if you have a consumption bundle x indifferent to y, this set of these points is provided by an indifference curve and this x could have a whole set of different goods and y could have another set of different goods. Of course, we are looking at initially just looking at, to simplify things, we are looking at two different goods and the combinations of that. So for instance, in that example that we talked of in pizzas and movies, if you said that 2, 1 and 1, 2 are equivalent, then we have an indifference curve here and this with 2, 2 is on a different curve and the utility or the value that I get from this is higher than the value that I get from this. So this is based on the fact that there is non-satiation, this is the, so the utility is a mathematical way to represent the value that the consumer assigns to a particular good or a bundle of goods and services and this, the indifferent curve represents a locus of constant utility and so this is utility is equal to constant, this is and the utility increases in this direction. So this is the basic way in which we represent consumer choices. And let us look at some of the characteristics of these indifference curves. So let us look at this whether or not it is possible for two indifference curves to intersect. Based on our initial principles, if you said, all the points on this indifference curve will have the same amount of utility and all points on this indifference curve will have the same utility. So which means that if we take the point which is intersection, intersecting that is x and compare that with y, these would have the same value of utility. On the other hand, if we compare that with z, z will also have the same value of utility as x, but if you look at these two points, y based on non-satiation, the value of utility at y is greater than that at z which means that we have violated this. So the indifference curves will not intersect, they will be, they will be, in general they would be parallel to each other, they will not intersect. So this is not possible. The other thing is that is it possible, let us look at this. It is not possible to have an upward-sloping indifference curve. Again this goes based on the fact that we have the non-satiation. Also it is not possible to have a thick indifference curve. So the way in which we will have, in general the utility shape of the indifference curve would be such that it would be convex, which means that instead of having a combination of two goods and services will have higher utility than the individual amounts of goods and services. So this is the characteristic of convexity. Let us now define what is utility. Utility is an economic term referring to the total satisfaction received from consuming a good or service. In many cases we try to quantify some of these things by money but in general we are looking at it as an expression by which we look at the satisfaction received from consuming a good or service. So what is the utility that we have by consuming an apple or consuming an orange and we said this is dependent on the preferences, this is dependent on the individual choice and when we have this and we can quantify it then we can look at two different individuals, their choices and look at the utilities which are obtained. So in general the utility is a function of the amount of consumption, it is a mathematical representation of a preference relation. There are many different possible utility functions and in the tutorial we will give you these, you can cross check whether these are convex and whether they can represent the utility functions and you can see these are some of the functions which are there and which have been used in literature in mapping some of the utility. What are the properties of the utility functions? So the first one is independence, the utility is dependent only on the choice between those goods, completeness we said that yes it maps the entire range of choices, transitivity we saw the transitivity of the preferences and now transitivity is there for the utility function continuity and it is an increasing function with u dash x greater than 0. The other point which is to be remembered is that in general in principle we consider them as ordinal not cardinal that means we rank our preferences. We say that an apple is better than an orange or is preferred to an orange but we do not say we would not like to say that an apple is 50% more than that orange. So we do not convert it into that kind of thing but in many cases when you look at the kind of calculations is done this assumption is often relaxed. So with the result that you create something like a utility function we want to also now calculate the change in utility per unit of additional goods. So that means we can look at what is del u by del x and if you look at the change in utility per unit of additional good the question is should this increase, decrease or remain constant. Think about any context and you can understand how this will change and there is this law of diminishing marginal utility and that if you see so for instance this is this example which is given by Serrano and Feldman where we are looking at there is a rich tourist who is stranded in a desert it is hypothetical context and there is a monopolist who has bottles of water and the tourist has 100 units of money available with her and we see that for the really thirsty so for the first bottle of water you are willing to pay a significant amount that means 25 units for the second bottle you are willing to pay less for the third bottle even less and so on. Even till the point so basically what happens is for every additional good that we have we are actually the utility of that good diminishes. So del u by del x actually decreases as you go ahead and so this is the law of diminishing marginal utility and we can also then see that what happens if we talk about two goods and we have a utility function of these goods we have this curve where we have shown x or let us put this as y and x and you have u as a function of x y we have u as a function of x y we want that we have a fixed amount of resources that means for x there is a price of x and there is a price of y and there is a final like budget that we have. So we can say that we want to maximize the utility that we have maximize u as a function of x y subject to the total amount that we are willing to spend on both these goods. So you may have a budget on each of these we can say that let us say p x into x p y into y less than or equal to b. So we want to maximize u x y subject to the budget the total expenditure on x and y should be less than the budget that we have. If we take that then we can create again the Lagrangian which will be u x y plus lambda b minus p x x. So if we look at this we can differentiate this what would we get we get del u by del x minus lambda p x equal to 0 by del y del u by del y lambda p y is equal to 0. So we can write this as lambda is equal to del u by del x by p x lambda is also equal to del u by del y by p y. If we equate this what would we get we get and you can see this essentially what you will get is del u by del x by del u by del y is p x by p y and this is essentially giving us the marginal rate of substitution and the marginal rate of substitution is the how much of x which we marginal rate of substitution when we look at if we move on the utility curve and we have a change where the utility remains constant then this will get minus delta x 2 by delta x 1 and this will be marginal utility of 1 by marginal utility of 2 this is going to be equal to del u by del x 1 by del u by del x 2. So this is the marginal rate of substitution in the equation in the utility curve that we had look at x y then we had the utility curve what we found with respect to the budget constraint is that you would have a line which is tangential to the utility curve which is maximized and with the slope which is the ratio of the prices and this will give us the optimum point and then with this now we can see that what happens if the prices change what happens with the utility changes and in the tutorials will take a few examples where we can look at this. You can look at of course in the case of indifference curve we can also look at indifference between environment and resources and this is what we will go through. There are a number of different resources that you can use for this section and you can look at any of these there is a book by Colstad and there is some lecture notes in the open coursework in MIT and then the lecture notes in Serrano and Feldman and this will give you an idea of preferences and utility. We will next look at when we want to take these preferences and we want to aggregate these preferences and we want to look at societal choices and societal choices between environment and energy and resources and economics and we will link this all up in the next section. Before we do that we are going to look at the philosophical basis by which we make decisions. So we will take a look at the different kinds of ways in which we can take philosophical decisions related to the environment what are the kind of choices which are there then we will look at the Kenneth Arrow's theory of social choice.