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Claim My Discount!We have already looked at fossil fuels and renewable resources. Today in today’s module we are going to start looking at resource economics. So, we will start with some of the earliest studies and the earliest research which was done on non-renewable or fossil resource economics. (Refer Slide Time: 00:47) So, we start with way back in the 1930s Herald Hotelling, was an economist who was concerned about resources, published a paper and we will upload the original paper for you to look at. But in the introduction of the paper, he talks about contemplation of the worlds disappearing supplies of minerals, forests and other exhaustible assets has led to demands for regulation of their exploitation. And the feeling that these products are now too cheap for the good of human future generations that they have been selfishly exploited at too rapid a rate and that in consequence of their excessive cheapness they are being produced and consumed wastefully has given rise to the conservation movements. So this was the initiation of his study, remember this is the 1930s an era where we were thinking that we have abundant fossil fuels, abundant material and the idea is to try and use them at a fast pace so that the world develops and the economy is developed and this was the first paper which talked about a fundamental theory to look at how much should be produced in a particular year given the fact that the overall resources are finite and that is also called in literature as the mine managers problems, so we will start with his analysis and then we will build on that and see how to work that. (Refer Slide Time: 02:22) So, this is the paper which was published in the economics of, this is the paper title was Economics of Exhaustible Resources, it was published in 1931. (Refer Slide Time: 02:38) Πt ¿ Pt qt−C qt Pt=MARKET PRICE OF RESOURCE qt=OUTPUTOF RESOURCE∈PERIOD C=MARGINALCOST OF RESOURCE (ASSUMEDCONSTANT ) And it starts with the idea that we have a function in the Tth year the profit that a mine manager has is Pt where Pt is the market price of the resource, let us say coal or oil market price of the resource per unit of the course and qt is the production or the output of that resource in the Tth year, output of resource in the Tth year, in period t. So, this will be Pt into Qt minus, minus the cost that means there is a marginal cost of the resource for extraction of the resource there is an amount that we have to pay to extract the resource marginal cost of the resource, which in our simplest model we will assume that to be constant, assumed to be constant. So, from the Pt qt which is the revenue we subtract minus C into qt, this is the profit in the Tth year. Now for a mine, the decisions that we have to take care how much should we produce in each of the years and then you are going to have a production till the time when the entire resource gets utilized and that is where it gets exhausted. (Refer Slide Time: 05:01) So if we look at this we are looking at the mine owner having a fixed stock of homogeneous resource let us call that Q max and that is the total amount of coal or oil whatever we have in the mind and we are saying that the fixed cost of extraction which is C, so we are trying to get the discounted value of the profits, so we want to maximize T equal to 0 to T we want to maximize the profit and this profit remember just we discuss in the economics portion when we looked at a project. Each year we will have a profit so you will have like profit Pie 1, Pie 2 and so on till the resource gets utilized. So each one will be the different values which you have will be defined as Pie t divided by, so will take the discounted sum of this, so we will take this as, we are maximizing this, subject to the constraint t is equal to 0 to t qt is equal to Q max. Naturally, this constraint in the when you get the maximum profit the entire resource will be utilized, so the decision is we have a total amount of resource we want to decide how much should be mine each year till the resource gets over. And in each year based on this, we are getting a certain amount of profit that is discount, that is summed up and we get the net present value of all the profits that is what we want to maximize. So, this is a classic optimization problem where we can see the time horizon T is exogenously determine we have to find out what is that time horizon T. we can form the Lagrange and then differentiate it to get the optimum. (Refer Slide Time: 07:38) Σt =0 T πt (1+d)t MAXΣt =0 T ¿ ¿ ¿ Σt =0 T qt=QMAX So, the problem which is being defined is simply sigma t is equal to t Piet by 1 plus d raise to t, d is the discount rate and in Hoteling’s original formulation he talked about this as the interest rate but we now know that when we are looking at this it is the discount rate that we should be talking about. So, this Piet we can write this down in this form t is equal to 1, 0 to t Pt qt minus C qt 1 plus d raise to t, subject to sigma t to 0 to T qt is equal to Q max. This is an equality constraint and we are maximizing an objective function subject to an equality constraint we can use the method Lagrange multipliers, we can create the Lagrangian and then differentiate it concerning the variables. (Refer Slide Time: 09:11) So, the Lagrange that we will get will be, Lagrange will be sigma t is equal to 0 to t, Pt qt minus C qt by 1 plus d raise to t, this is your original objective function, plus lambda there is only one constraint lambda. You associate a Lagrange multiplier concerning the constraint and we take that constraint as Q max minus sigma qt this is the Lagrangian that we have constructed. Remember this is the objective function, this is the constraint which we have multiplied by the Lagrange multiplier and added it to the modified objective function. Now we can differentiate this concerning the variables. What are the decision variables, what are the variables that we have? The variables are qt that means q1, q2, q3 and so on till it gets depleted, so this we can differentiate till Lagrangian by del q and this will come to Pt minus C and then when you differentiate this Q max is constant, this will have a term minus lambda is equal to 0. We have equations so we will have this as the equations that we have you will have the dell by dell qt we can essentially, we will have this as t is equal to 0, 1, 2 and so on to t. So, if you look at this what will be, there is the lambda will be Pt minus C by 1 plus d raise to t that is the Lagrange multiplier and this is constant across the time intervals. (Refer Slide Time: 11:59) λ = P0−C= P1−C (1+d) = Pt−C (1+d)2 .... . Pt=C+λ (1+d)t Rt +1−Rt Rt = Δ Rt Rt =d Rt=Pt –C So, if we write this down we will find that P0 minus C by 1 plus d raise to 0 which is 1 is equal to P1 minus C by 1 plus d equal to P t minus C by 1 plus d square and so on. So, if you look at this, this is essentially what we get is Pt is constant plus lambda into 1 plus d raise to t. So, the price is equal to the marginal cost of extraction plus a user cost which increases with the time. So, if we look at what is this difference between Pt minus C which is the profit that we are getting if we take return Rt plus 1 minus Rt by Rt we will get this as per unit profit that we are getting or return is increasing has to increase year by year, the revenue in time t, this is Rt is Pt minus C. Revenue per unit in time t has to increase on a year to year basis at the discount rates, so that is what we have shown. That would mean that in general what will happen is that the price, if the c is the constant price, will increase with time and the rate of increase would be at the rate of the discount rate. (Refer Slide Time: 14:15) So, if you look at it in terms of a graph, what we will get is, you will get that, if this is the marginal cost which is C which is constant and we start with price and time till the time when we will actually, T when it becomes so costly that qt becomes equal to 0. Now, this can be put in the form of what is known as a demand curve but before we see the demands curve we will look at what the result which Hotelling put in his paper. (Refer Slide Time: 15:06) And the result shows that the market price of a resource net the extraction cost must rise at a rate equal to the rate of interest and this is what was there in his paper but we know subsequently that when we are looking at this we are talking of the market price of a resource, net its extraction cost will be rising at the discount rate and that is when we talk of an exhaustible resource we will try to get the optimum strategy is to see that the price minus the extraction cost divided by 1 plus d raise to t is constant across different time intervals and that gives us an optimal extraction strategy. Now, let us move forward and look at what we understand as a demand curve. (Refer Slide Time: 16:10) So, typically what happens is for any commodity when we look at price and quantity, typically what happens is that as the price of quantity increases the quantity demand decreases, so you, we have something like this which is, this is a something called a demand curve, this is a linear demand curve and typically what we will do is we will say quantity is a function of the price that is typically what we expect, this is a sort of linear correlation. Now, one of the things when we want to understand in economics when we talk in terms of the demand we define something called the elasticity. (Refer Slide Time: 17:15) Price Elasticity ɳ=|d qt d pt pt | And it is called the price you would have studied this in your basic economics, we talk about elasticity and we will talk in terms of price elasticity of demand and that typically is that if we say that suppose the price of a commodity or a good increase by 1 per cent what happens, what is the percentage of demand for a unit percentage change in the price? So, the way we define this is we define an elasticity which will be the delta dqt by qt divided by dPt by Pt, so this is in general what will happen is as the price increases the quantity demanded will be decreasing, so this could, this would generally be a negative term. Of course, in the elasticity, we may take the elasticity and we may take the absolute value of the elasticity, so this is what we can define as the elasticity of price. And of course, remember that as prices increase in the short term depending upon the kind of good and service sometimes if prices increase for instance the prices of onions in seasons sometimes increase very drastically. Sometimes you can substitute onions by something else at some point in some cases for instance if you are looking at using a certain amount of electricity and the price increases, in the short run you may still require that electricity and you have no option but to pay that price. So, then the demand may be considered you might have a relatively a demand where there is it is inelastic, so even if the price increases it may not change much. On the other hand, you might have a possible substitute and you might be able to instead of electricity you may be able to use some other source of energy. So then there would be an elasticity, so there in all these cases we have, we define two terms. One is the short-term elasticity and then there is a long term elasticity. Normally long-term elasticity is higher because you can put investments so that you can have different kinds of substitutes and there are different studies by which we estimate these long terms and short-term elasticity. (Refer Slide Time: 20:09) qt=f (pt) Inverse Demand Curve Pt=a−b qt a>0 b>0 ɳt=|d qt Pt d ptqt | d pt d qt =−b ¿ a−bqt −b qt =−a b qt +1 So, we will take the demand curve that we have and then define this in term of an inverse. So, we said demand curve is where you have qt as a function of Pt, we will talk about an inverse demand curve and here will say Pt is a minus bqt, so if you look at the figure here you will see that this corresponds and we are saying a greater than 0, b greater than 0 then you have a straight line where you as there is a certain point at which at a particular Pt where the Pt goes very high then the qt is equal to 0 and so on. So, this is the kind of inverse demand curve, let us now calculate for this what is the elasticity of this demand curve, we want to calculate what is delta t dqt by dpt and to pt by qt, so if you see this, this is going to be, DPT by dqt is equal to minus b and so the elasticity is we look at dpt by dqt, pt by minus dqt and pt is a minus bqt minus pqt. So, you will see that essentially the, at different points the elasticity will change at different points on this curve. So, you look at this and you will see that this is minus a by b qt plus 1, 1 minus a by bqt and so this is the kind of calculation that you can do. (Refer Slide Time: 23:10) You will see that in these kinds of the curve what will happen is that you will be able to calculate and you will have the values will be different at different points, it is not a constant elasticity curve. There is we can also look at an equation where we talk of a different kind of curve where you have a constant elasticity curve and then in the constant elasticity curve you will see that we can look at this, this is the kind of curve which we will have. (Refer Slide Time: 23:52) In this case, Pt we can write down an expression for Pt which is a by qt raise to b this is another inverse demand curve where essentially. So, at t tends to infinity the qt tends to 0. So, in general, this is a constant and we look at, let us look at the calculating the elasticity of this curve, so if we look at the Pt by dqt this is going to be a minus b qt raise to minus b minus 1 and if we look at the elasticity whi we had, if we look at the elasticity this is going to be now we are going to calculate this as Pt by qt into dpt by dqt which is we have a minus b qt raise to minus b minus 1. And if we see this becomes qt raise to minus 1 plus 1, 0. So, this qt raises to minus b and Pt is nothing but a by qt raise to b, so this is minus ab qt raise to minus b, this cancels and so you get minus 1 by b. If it is the modulus value then this is 1 by b. So, if you see this a and b are constants so this curve essentially represents a constant elasticity case. So we have in the last module we have just seen the basics of hotelling’s model in which we saw the wide mangers problems reduces to a situation where we essentially keep the profit in different intervals divided by the discount, the discounted profit in different years should be constant and we also saw that when we talk in terms of the demand curve, we can look at different kinds of elasticity. And we will now take the result that we had got where we said that the price of the commodity net the extraction cost, should increase at the discount rate for different time horizons, we will take that and derive for a given demand curve which is known, we would like to see how long will the resource last. So, will consider a situation where there is a competitive mining industry which has a known facing will first start with taking a linear inverse demand curve and then we will take constant elasticity demand curve and we will derive how much is the time for which a given resource will last in a mine. So, that is the problem that we are looking at and as we saw this is a kind of inverse demand curve that we are focusing on, we would like to see first of all we start with a competitive mining industry which is facing a linear inverse demand curve. (Refer Slide Time: 02:05) Let us say that when Pt is a minus bqt, so at that is given as the demand curve. Now at the time when qt will be equal to 0, we talked about, so we were looking at a situation when we will have the qt will be equal to 0 at some time interval. And this is at a time where essentially we will have the price when qt is equal to 0, the price Pt let us take linear when qt is equal to 0, Pt turns out to be a, so let us call this time interval as t when the entire this is where we talk of Pt and qt at the time when the qt is equal to 0 that pt becomes equal to a this is a and this is at a time when over a period when the resource has got completely utilized it is like we have something like there is a superabundant substitute. So that no one continues to use this let us say coal, you have something which is much bet preferred and then this is and then there are no additional resources. Now the question for us is to determine this t, what is the date of exhaustion t and we would like to determine it using the fact that the, it is a competitive economy and the mine manger is trying to maximize the profit. So we have, we know that the optimal strategy is where Pt will be P0 into 1 plus d raise to t, for this please remember that we have taken a constant extraction cost and we have subtracted, so we can take essentially we are taking P0 minus C, we are neglecting the extraction cost or we taking that the extraction cost is constant. So having said this, if this is the equation that we have got we can now substitute and get Pt is a, is equal to P0 1 plus d raise to t. (Refer Slide Time: 05:29) If that is the case then from this equation we can substitute and we will get P0, a is known P0 is 1 a divided by 1 plus d raise to t. We can substitute back in that equation so that we get Pt and any time interval will be P0 into 1 plus d raise to t and substitute P0, so we get an into 1 plus d raise to capital T, capital T is a constant which we do not know which we want to find out and this is at any time interval t is equal to 1, 2, 3, 4 etc. So, then this can be written as a 1 plus d raise to t minus capital T, this is now our expression for Pt, we also know from the equation that Pt is a minus b qt this was our original inverse demand curve, so now we can equate these two and what do we get? We will get that bqt is equal to bqt will be equal to an into a minus, that’s the what we equate this, that means b qt is equal to a 1 minus 1 plus d raise t minus T that means qt we have now got an expression for qt. The generic expression for qt in terms of known coefficients a, b and d only unknown is capital T, now what we can do is that we will take that the total sum t equal to 0 to T minus 1, why do we say T minus 1? Because q capital T is equal to 0, so production is there from the 0th year to T minus 1 here, qt this will be equal to the total reserve which is R0, or in the earlier case what we have said q max and this is we can write the sigma notation t is equal to 0 to T minus 1 a by b into 1 minus 1 plus d raise to t minus T. Now look at this we have got one equation we know R0, we know a, we know b, we know d the only unknown that we have is capital T, we have a sigma notation, this is a geometric progression, we can derive an expression for capital T in terms of these coefficients a, b and R0. In which case we have completely solved the problem we found the time for which the resource will last when it gets exhausted and we can substitute back and then we can get essentially qt as a function of time or Pt as a function of time and was our objectives. So, let us just do the simple calculations which are the, so when we look at this we get. (Refer Slide Tim10:00) We open out the bracket and you will get a by b into sigma t is equal to 0 to t minus 1 there are t terms, so this will be a by b into t minus a by b into sigma t is equal to 0 to t minus 1, 1 plus d raise to t minus t. Now, this is geometric progression let us just write this down in a separate way, let us write this as S is equal to sigma t is equal to 0 to T minus 1, 1 plus d raise to t minus T. Let us expand it, let us write it down, this is going to be the t is equal to 0 this will become 1 plus d raise to minus t, so that is capital T and then it will be 1 plus d raise to t minus 1 and it gets the denominator changes till you get 1 plus t. There are t terms and you can see that it goes 1 plus d raise to T, 1 plus d raise to T minus 1, T minus 2 and so on till 1 plus T. So if you take this as one equation which is S, I can just take S by 1 plus d, this will be 1 plus d raise to T plus 1 to and then we can take 1 minus 2. We can just do 1 minus 2 and if you do this you will find that we get S minus S by 1 plus d and if you see in this case what remains is that we are going to have the all of these terms will get cancel you will get 1 by 1 plus d over here minus 1 by 1 plus d raise to d plus 1 these are the 2 terms which will be remaining. So this is going to be 1 by 1 plus d minus 1 by 1 plus d raise to t plus 1. So, we can take this as with simple algebra, this takes 1 by 1plus d common this is just you could have just used the geometry progression formula but we just in one step we just derive it and this turns out to be Sd by 1 plus d equal to 1 by 1 plus d now 1 plus d not equal to 0, so we can cancel this and we get S is equal to 1 by d into 1 minus 1 by 1 plus d raise to t. (Refer Slide Time: 14:28) So, we can substitute back in the equation that we had earlier which was R0 is equal to a by b capital T minus a by b into S and S is we just derived the S as 1 by d into 1 minus 1 plus d raise to T. So, now we can simplify this, we can take b on b by, a by b is common here, I can take it on the other side so that I get b R0 by an as T minus 1 by d into 1 minus 1 plus d raise to T. Now you want to solve for t so we can just write this now as T is equal to take the terms on the other side, we have now solved for capital T which is what we wanted to do, we wanted to find the date of exhaustion which is unknown, we do this in terms of the coefficients a, b, R0 and d, a, b, R0 and d are known please remember this is an equation where you have T on both sides, so you can do it iteratively, you can start by assuming a certain value of T get the next value and then so on till we get a converge value. And so with the result that we right now we can now determine T and we will do a tutorial example where we will plug in the values and then see how this can be done. So, once we do this then we know the T, once we know T we now know the price trajectory, as well as we, know theqt, so we know Pt as a function of T and we know Qt as a function of T. (Refer Slide Time: 17:05) And if you just see for a linear model typically for a this is an example from Conrad book on resource economics and we will also solve particular example with some values so that we can make our calculations. This is a linear inverse demand curve and corresponding to this you see that the price starts from a certain value and the price as a function of time increases at the discount rate. (Refer Slide Time: 17:35) And when we look at the qt versus the time we will see that the qt starts from some value and it decreases till about the 20th year where it becomes equal to 0, so this is an example to show you how the whole mine model problem is done for a competitive market.
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