 So, we have in the last module, we have just seen the basics of hotelings model in which we saw that the buying managers problem 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 elasticities. 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, we 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 a 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 the kind of inverse demand curve that we are focusing on. We would like to see first of all we start with the competitive mining industry which is facing a linear inverse demand curve. Let us say that when Pt is A minus Bqt, so 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 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 of time when the resource has got completely utilized, it is like we have something like there is a super abundant substitute, super abundant substitute so that no one continues to use this, let us say coal may have something which is much bad 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 mind manager 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 raised 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 are 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 raised to t. 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 raised to t. We can substitute back in that equation, so that we get Pt in any time interval will be P0 into 1 plus d raised to t and substitute P0. So, we get a into 1 plus d raised 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 raised 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 b qt is equal to, b qt will be equal to a into a minus, that is the one we equate this. That means b qt is equal to a 1 minus 1 plus d raised to t minus t. That means qt, we have now got an expression for qt. 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 1th year qt. This will be equal to the total reserve which is r 0 or in the earlier case what we had said was 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 raised to t minus t. Now, look at this we have got one equation, we know r 0, 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 r 0. 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 that was our objective. So, let us do the simple calculations which are there. So, when we look at this we get 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 raised to t minus t. Now, this is a 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 raised 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 raised to minus t. So, that is capital T and then it will be 1 plus d raised 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 raised to t, 1 plus d raised to t minus 1, t minus 2 and so on till 1 plus d. 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 raised to t plus 1, 2 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 it remains is that we are going to have the, all of these terms will get cancelled. You will get 1 by 1 plus d over here minus 1 by 1 plus t raised to t plus 1. These are the two terms which will be remaining. So, this is going to be 1 by 1 plus d minus 1 by 1 plus d raised to t plus 1. So, we can take this as it is simple algebra. We can just take 1 by 1 plus d common. This is actually just, you could have just used the geometric progression formula, but we just in one step we just derive it. And this turns out to be s d by 1 plus d equal to 1 by 1 plus t. Now, 1 plus d not equal to 0. So, we can cancel this and we get s is equal to, we get s is equal to 1 by d into 1 minus 1 by 1 plus t raised to t. So, we can substitute back in the equation that we had earlier which was r 0 is equal to a by b capital T minus a by b into s. And s is, we just derived s as 1 by d into 1 minus 1 plus d raised 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 r 0 by a as t minus 1 by d into 1 minus 1 plus t raised to t. Now, we want to actually 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, r 0 and d. a, b, r 0 and d are known. Please remember this is an equation where you have t on both sides. So, you can do it in an iterative fashion. You can start by assuming a certain value of t, get the next value and then so on till we get a converged value. And so, with the result that we now, 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 the qt. So, we know pt as a function of t and we know qt as a function of t. And if you just see for a linear model, typically for a, this is an example from Conrad's book on resource economics and we will also solve one particular example with some values so that we can make our own 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. And when we look at the qt versus time you will see that the qt starts from some value and decreases till about the 20th year where it becomes equal to 0. So, this is an example to show you how the whole mind model problem is done for a competitive market. Now, what happens if instead of the linear inverse demand curve, we had a different demand curve which is we had the constant elasticity demand curve. Let us derive for that. pt is equal to a by qt raised to b. So, in this case what would happen is you would see that typically if we have this curve you will find that the qt would not become 0 and qt would tend to 0 only as it goes to infinity. So, what we can do is we have the same term which we saw in the earlier Hotelling's analysis where we said pt will be 1 plus d raised to t p0. So, in this case we do not have a choke price or a price which where we find that that will be equal to a in this earlier case where qt was equal to 0, vpt was equal to a. We just have this expression and we can use this expression and substitute back and we will get qt we can write in this pt is a by qt raised to b. So, qt raised to b is qt raised to b is going to be a by 1 plus d raised to t p0 and what we need to do is we need to find out. So, qt if I write this can be written as a by 1 plus d raised to t p0 raised to 1 by b. So, what we know now is this is an expression that we have now for qt we know a, b and d we do not know p0 and in order to find p0 we can use the value fact that sigma t is equal to 0 to infinity qt will be equal to r0 or sigma t is equal to 0 to infinity we plug in this value our problem what we want to do is we want to determine this value of p0. In the earlier case we had the value of p0. So, we want to find p0 in terms of the known coefficients a, b and r0 and d and now when you look at this you will see that this is basically again this is a geometric progression when we look at this sigma if we take this out you will see that this is we can take a by p0 raised to 1 by b and you get sigma t is equal to 0 to infinity. So, this is actually a geometric progression where we have this is of the form you know if we have a geometric progression and we are multiplying the constant factor the multiplicant that we are doing is 1 plus d raised to t by b. So, 1 plus 1 by 1 plus d raised to 1 by b and as the limit as we are looking at the term where this is of course, going to be less than 1 in the limit as we take it to infinity we take this sum to infinity this is going to be equal to the initial term divided by 1 plus 1 minus r. So, this sum is going to be equal to so, the what we get is that expression that we got a by p0 raised to 1 by b into 1 by 1 minus r and r is nothing but 1 plus d raised to 1 by b and this term will be equal to r0 you can convince yourself that this is true. So, what this would mean is that we can take this as a by p0 raised to 1 by b and we can take this as we can multiply by 1 plus d raised to 1 by b. So, that you get 1 plus d raised to 1 by b 1 plus d raised to 1 by b minus 1 equal to r0. So, we can now simplify and write this in terms of the basically we can get we want to what is our objective we know d we know b we know a we need to know p0. So, if you look at this and you want to take this as a by p0 we want to find p0 p0 by a p0 by a raised to 1 by b is what we have here and this will be equal to 1 plus d raised to 1 by b raised to r0 into this is going to come in a 0 will come in here and this will come there. So, you get r0 1 plus d raised to 1 by b minus r0. So, when we simplify this we will get an expression which is going to be nothing but p0 a into 1 plus d raised to b. So, once we know p0 we know that pt is p0 1 plus d raised to t we know what is pt and we if you remember we have already calculated what was qt qt was. So, we can calculate at any point of time qt what about the time of exhaustion capital t in this particular case since there is a constant elasticity it will go asymptotically to 0 it will never become equal to 0. So, we can just take the point where a certain amount 90 percent of the resource is used or 80 percent but it will never get completely used because the price is going to keep increasing but you will always use a certain amount of it and this is the constant elasticity case. So, let us just take stock what we have done is we have solved the mine managers problem and we first started with Hotelling's original paper and we said that when we talk about a non-renewable resource we want and if the we are looking at a mine manager who is trying to maximize the profit given the constraint in terms of the known total resource which is there in the mine we want to decide how much to mine at different points of time and based on that we saw that that the quantity which will be mined will be such that the price will increase at the rate of the discount rate and of course this is assuming a constant extraction cost. We can have more complicated models where if the extraction costs are also a function of time and one can have various additional complications but the principle by which we will do the calculation will be exactly the same and with that we found that there will be a point at which the price increases to the point where there is a super abundant substitute and the mine gets completely exhausted and that was the choke point we used that and then said that suppose we know we have a perfect competition case and we know what is the demand curve and we know the inverse demand curve we showed that for a linear inverse demand curve we can calculate what is the time for which the resource the mine would last and this is under perfect competition. In the second case we said that suppose we have another situation where there is a constant elasticity demand curve in the constant elasticity demand curve we never get to a choke point but the quantity the quantity keeps decreasing and beyond a particular point and this will go asymptotically to the price will keep increasing and the quantity goes asymptotically to 0 and in such a case we identified all the parameters so that we could get qt as a function of time and we got pt also as a function of time. So, this is the constant elasticity price trend and constant elasticity qt trend you can see that beyond the point after a certain point of time it goes asymptotically it is not exactly 0 but it still keeps getting going down and this is we had derived this where we are looking at the competitive extraction and price path there is no choke point but we get the price and this is how we said that we can derive this and we as t tends to infinity qt tends to 0 pt tends to infinity and assuming an exhaustion we got this and we got the expression for the p0. So, with this we have now derived all the expressions for the price trajectory and the quantity trajectory for perfect competition. Now, the question is what if there is a monopoly what if the mind manager controls all the minds and the mind manager can then will the strategy remain the same or will it be different we will look at this in the next module.