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. (Refer Slide Time: 18:08) Let us derive for that. Pt is equal to a by qt raise 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 raise 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 an in this earlier case where qt is equal to 0, Pt 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 raise to b. So, qt raise to b is going to be a by 1 plus d raise 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 raise to t P0 raise to 1 by b, so what we know is this is an expression that we have now for qt we know a, b, and d we do not know P0. (Refer Slide Time: 20:33) And to find P0 we can use the 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 plugin 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 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 raise to 1 by b and you get sigma t is equal to 0 to infinity. (Refer Slide Time: 22:32) So, this is 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 multiplicand that we are doing is 1 plus d raise t by b, so 1 plus 1 by 1 plus d raise to 1 by b and as the limit as we are looking at 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 divide by 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 raise to 1 by b into 1 by 1 minus r and the r is nothing but 1 plus d raise 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 raise to 1 by b and we can take this as we can multiply by 1 plus d raise to 1 by b so that you get 1 plus d raise to 1 by b 1 plus d raise to 1 by b minus 1 equal to R0. So, we can now simplify and write this in terms of the basics 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 raise to 1 by b is what we have here and this will be equal to 1 plus d raise to 1 by braise to R0 into this is going to come in a0 will come in here and this will come there. So, you get R0 1 plus d raise to 1 by b minus R0. (Refer Slide Time: 26:09) So, when we simplify this will get an expression which is going to be nothing but P0 an into 1 plus d raise to b, so once we know P0 we know that Pt is P0 1 plus d raise to t we know what it 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. Now, 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 per cent of the resource is used or 80 per cent 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 we are looking at a mine manager whose 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 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 cost is also a function of time and one can have various additional complications. But the principle by which we will do the calculation will be the same and with that we found that there wi be a point at which the price increases to the point where there is a superabundant 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 show 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, mean the constant elasticity demand curve we never get to a choke point but the quantity keeps decreasing and beyond the particular point 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. (Refer Slide Time: 30:31) And constant elasticity qt trend you can see that beyond a point after a certain point of time it goes asymptotically it is not exactly 0 but its keep getting going down. (Refer Slide Time: 30:43) 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. (Refer Slide Time: 30:53) And this is how we say that we can derive this and as t tends to infinity qt tends to 0, Pt tends to infinity and assuming 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 mine manager controls all the mines? And the mine manager can then, will the strategy remain the same or will it be different? We will look at this in the next module. We continue looking at Non-renewable Resource Economics and in the previous module, we have looked at a situation where there is perfect competition and we found out what is the optimal strategy for the mine manager. We found out what is the price trajectory, we found out how much quantity would be taken in different years and the time for which the resource will last. In this condition what happens is the demand or the inverse demand curve is given and any company or an owner of a mine has no way of influencing that demand. Let us now look at the question of what happens if all the mines are controlled by one company or one individual? Does that mean what happens if there is a monopoly? And of course, as you would expect the rules would be changed because then the monopolists can influence the total quantity which is being released and because the total quantity being released is being influenced the price would change. And so the monopolists have a way to dictate a price and can then decide so, in this case, the optimization changes, the monopolist tries to maximize the revenue and this in a similar fashion like the analysis we have done in the last section where we said that the cost is constant and we can take the price minus the cost or we can neglect the cost. (Refer Slide Time: 02:03) So the revenue that we will have Rt will be Pt into qt, now in actual practise, we have seen this case of a monopolist affecting the prices, in some cases, it may be one individual which is a monopoly or it could be a cartel of producers, for instance, Opaque is a cartel of oil-producing and exporting countries and in the 1970s Opaque decided that it was going to control the quantity the oil that it is going to release. And with the result you could see a sudden spurt in the oil prices, this is called the oil shock and that is the point at which all countries started looking at energy independence, looking at energy efficiency and this was the start of the whole movement to look at energy conservation and energy efficiency. So, we would like to now look at from a monopolist point of view if you have the revenue and we want to maximize the discounted sum of the revenue Rt by 1 plus d raise to t, t is equal to 0 to T. So, it is very similar to the earlier situation that we had, only thing is that in the case of Rt the monopolist can influence the quantity that is being released overall and hence is also able to influence the price and this will be subject to the constrain that sigma qt t is equal to 0 to T is equal to the total reserve we had which is R0. (Refer Slide Time: 04:10) So when we take this we can take the Lagrange which is very similar to the last analysis that we did but R0 minus sigma t is equal to 0 to T qt and this Lagrange divided by then differentiated concerning del qt set this equal to 0, what we get is we will get the, we are differentiating the total revenue that we have concerning qt. So, what we will get is the marginal revenue, we can differentiate this and we will get delta Rt by del qt 1 plus d raise to t minus lambda is equal to 0. So, essentially what we get is delta Rt by del qt is also known as the marginal revenue, that means the revenue per unit of q and the what we would get then is that the lambda value is going to be equal to marginal revenue divided by 1 plus d raise to t. So, this is that in each time interval just like we had in the earlier case we had the price increasing at the discount rate. (Refer Slide Time: 06:17) Now we are having the marginal revenue. Marginal revenue 1 by 1 plus d and so on marginal revenue t by 1 plus t raise to t. Now let us take a situation where we have linear inverse demand curve, so we have Pt is a minus b qt so then Rt becomes a qt minus b qt square, so del Rt by del qt is a minus 2 but. So, once we plug this in the value of lambda which we get is a minus 2 b qt by 1 plus d raise to t. (Refer Slide Time: 07:35) And we said that MRt increase, so M R marginal revenue in time horizon t will be equal to marginal revenue 0 first year into 1 plus d raise to t. Now let us consider the linear inverse demand curve and take the situation when the resource is completely exhausted. When the resource is completely exhausted at that point Qt capital T will be equal to 0. At this point what will be happening will be the marginal revenue which we have must equal to the price per unit that will be equal to the price and that is when the Monopolist will not want to produce any more the marginal revenue will be equal to the price then that is equal to we said a minus b qt. So this is going to be equal to a, so a is going to be equal to MR0 1 plus d raise to t and then we can substitute this in this expression so that we get MR t is equal to this is capital T. When it gets exhausted capital T this is going to be a by 1 plus d raise to capital T multiplied by 1 plus d, so this is marginal revenue is going to be 1 plus d raise to t minus T. (Refer Slide Time: 09:26) And we have already derived that the marginal revenue for the linear inverse demand curve case is a minus 2 bqt, so we can put this as a minus we can equate these 2 terms bqt is a 1 plus d raise to t minus T. We can now get from this we can put this as 2 bqt is equal to an into 1 minus, looks very similar to the competition case but with a difference, we have now this is qt is a by 2b into 1 minus 1 plus d raise to t minus t. If you remember you can look back at the earlier derivation that we had done. In the case of the this is for the monopoly and for perfect competition for a competitive market we got qt is equal to a by b. So, if you look at this of course in the case of the monopoly the capital value the value of exhaustion the number of years T would be different but in general what you would find this that the monopolist would, in a particular year release less amount of q so that the price increases and the overall revenue increases with the result that as we would expect the resource is going to last for a longer period under a monopolist case. (Refer Slide Time: 11:46) So, if we look at this we would expect qualitatively a curve like this where you have qt and t this is for if this is the shape of our competition, the competitive market then the monopolist would be this is how the monopoly would look like. So, you can look at the book by Conrad on the nonrenewable resource, on resource economics and there is a chapter on renewable resource economics which shows some of this. If we then take this the same thing you can see the this is the actual this is for a this is a plot which his shown from Conrad which shows similar kind of trend for a particular example. (Refer Slide Time: 12:56) So now what we would like to do is we would like to look at this take that expression and just like we did for the competition case we would like to derive how much time the resource is going to last for. (Refer Slide Time: 13:09) So similarly we take t is equal to 0 to T minus 1, remember that in the last year q capital T is equal to 0 so that will not be added q t will be equal to 0 to T minus 1 a by 2 b into 1 minus 1 plus d raise to, and if we use this in the same fashion as we did derive for the competition case this becomes a geometric progression and finally, we get an expression which is we get 2 bR0 this sum will be equal to R0. So, 2 b R0 by a is T minus 1 by d 1 minus 1 plus d raise to t and the final expression that we get is t is equal to 2 bR0 by a plus 1 by d 1 minus 1 plus d raise to T. This is for the time for a monopoly and you would remember that we have a similar expression when we had the competition and only difference was that in this case, this was b R0 by a and so what you would find is that the time taken would increase and now the question is of course thus that means that a monopoly is better from a resource point of view? From a resource point of view, the monopolists conserve the resource because the monopolists are looking at the overall maximization of revenue but in the process given the discount rate that is where the population and the consumers are exposed to much higher prices and because of that the overall utility of society is less under a monopolist case even though the resources gets conserved for a longer time. So now let us do one thing let us take the same whatever we have learnt for competition and monopoly let us now solve one particular example a numerical which is there in your tutorial sheet. (Refer Slide Time: 16:26) I will just show you this number and this is the tutorial sheet. This shows that we have a tutorial problem the inverse demand function for fossil fuel is given to you as Pt is equal to 1 minus 0.1 qt which means that a is equal to 1, b is equal to 0.1 and we have also the value of discount rate R0 is given to you as 75, R0 is 75 and d is 5 per cent which is nothing but 0.05. So, the first part of the question is, what is the price of elasticity of demand for this function and qt is equal to 5 units, so when qt is equal to 5, let us just substitute qt is equal to 5, so what is the value of Pt? Just 1 minus 0.1 into 5 this is 0.5, so the answer is 0.5. So, the differentiate this del Pt by del qt this is minus 1, so if we look at the elasticity that is going to be del qt by del Pt into Pt by qt. Which is this is del qt by del Pt this is 1 by minus 0.1 Pt we said is 0.5 and qt is 5. So, you will find that the elasticity is minus 1 which implies that if we have a 1 per cent increase in the price there will be a 1 per cent decrease in the quantity and that is the elasticity, so we have solved the first part of the question. The second part b says to determine the time value of extraction for a mining industry under pure competition. (Refer Slide Time: 19:10) So, when we talk about solving this for a pure competition we will have this as we have Pt will be for pure competition this will be Pt into 1 plus d raise to t, so 1 point d is 0.05, 1.05 t. Nowt is equal to T, qt is equal to 0 and Pt is equal to a which is 1. So, 1 is equal to P0 1.05 raise to T we can substitute for P0 so that we get Pt is equal to 1.05 raise to t minus T. So, you remember the formula that we had derived for the time that this will last, we want to this is the only unknown is capital T we need to determine capital T then we can plug it back and we can get the equations for Pt and qt in which case we would determine the time path of extraction. So, once we do this we check b R0 by a plus 1 by d into 1 minus 1 plus d raise to T. Just substitute the values this is going to be 0.1 into 75 by 1 plus 1 by 0.05 into 1 minus 1 by 1.05 raise to T. (Refer Slide Time: 20:56) So, if we simplify this we will get T is equal to 7.5 plus 20 into 1 minus 1 by 1.05 raise to T. Now when we look at this we will this is as we said this is an equation where we have to iteratively solve for T. So, let us assume a certain value of T, let us say suppose T is equal to 10 years we can substitute T and get T is equal to 7.5 plus 20 into 1 minus 1 by 1.05 raise to 10. You can plug in this values and you will see you get T comes out to be 15.2. Now we take 15.2 as a starting point and solve t get the next value of T then you get T is equal to 18.0 and then the next iteration we get 19.2 you can solve this and check 19.7, 19.8 and it converges to about 19 points, you get T approximately 19.94. We can round this off to about 20 years. So, we solve this part C when does the resource get exhausted. The resource gets exhausted in at 20 years and then what happens is that if we now substitute back we get the Pt which we had already solve we got this as Pt is equal to 1.05 into t divide by 1.05 raise to 20 and if you see this value of 1.05 raise to 20 turns out to, so you can get this so we got an expression for Pt, we also now can substitute this and get the expression for qt and with that, we will get essentially the value of qt in different years. Now let us look at for the same situation the part d, would the time path of extraction, so once we have this we can plot it for different years and we have got the plot of Pt versus time and q t versus time. Now the question is would the time path of extraction for a monopolistic mining industry be different. So, if you look at this as we have seen earlier what happens in a monopoly is that you can affect the quantity supplied and hence the price and because of that you release less then it perfect competition you should have less quantity mine and with the result that we expected to last longer. (Refer Slide Time: 24:12)If we take this if you remember we had derived now for the monopolists that this is going to be 2 b R0 by a plus 1 by d 1 minus 1 plus d raise to T more or fewer things the equation looks very similar expect instead of 7.5 this is now 15. So, once we do this we will get a different converge solution. So if we start with T is equal to 10, you will get T is equal to 15 plus 20 into 1 minus 1.05 raise to 10 and you get the next value becomes T is equal to 22.7 and as we go ahead you will find that it converges to about 30.5 years. So t approximately 31 years in the first case we found T is 20 years and in this case, it is 31 years. So, qualitatively we realize that essentially the monopolists want to maximize revenue and because of that, we produce less from the mine in the initial years as compared to the competition case and with the result that overall the revenue increases. Now the question is that what happens we also saw repeat this with d is equal to 10 per cent if the discount rate is higher what would happen? If the discount rate is higher it means that we are counting future cash flows and discounting it by a larger amount. So, we would prefer to have a profit or a revenue today as compared to in the future with the result that what would happen is that we would actually mine qt at the initial period would be higher and then the mine would get exhausted in a shorter period. You can repeat this on your own and cross-check. So, with this we have completed the portion on non-renewable resource economics we have of course done this with a simplistic set of assumption, you can relax all of these assumptions you could have a situation where the cost of extraction change, you could have a situation where there are different kinds of demand inverse demand curve and but this gives us a way in which from first principles we can identify how an optimal mine manager would think and what is how the resources would be used subject to the fact that of course the total amount of resources are finite.
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