 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 mind 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 mind has no way of influencing that demand. Let us now look at the question, what happens if all the minds are controlled by one company or one individual, that means what happens if there is a monopoly and of course as you would expect the rules would be changed because then the monopolist can actually 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 monopolist has 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 costs are constant and we can take the price minus the cost or we can neglect the cost. So the revenue that we will have RT will be PT into QT. Now in actual practice 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, OPEC is a cartel of oil producing and exporting countries and in the 1970s OPEC decided that it was going to control the quantity of oil that it was 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 and 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 raised 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 is able to influence the quantity that is being released overall and hence is also able to influence the price and this will be subject to the constraint that sigma u t, t is equal to 0 to t is equal to the total reserve that we had which is r 0. So when we take this we can take the Lagrangian which will be very similar to the last analysis that we did but r 0 minus sigma t is equal to 0 to t q t and this Lagrangian divided by the differentiated with respect to del q t set this equal to 0. What we get is we will get the, we are differentiating the total revenue that we have with respect to q t. So what we will get is the marginal revenue, we can differentiate this and we will get delta RT by del q t 1 plus d raised to t minus lambda is equal to 0. So essentially what we get is delta RT by del q t is also known as the marginal revenue that means the revenue per unit of q and what we would get then is that the lambda value is going to be equal to marginal revenue divided by 1 plus t raised to t. So this is that in each time interval just like we had in the earlier case we had the price increasing we had the discount rate now we are having the marginal revenue, marginal revenue 1 by 1 plus t and so on marginal revenue t by 1 plus t raised to t. Now let us take a situation where we have a linear inverse demand curve so we have p t is a minus b q t so then RT becomes a q t minus b q t square right. So del RT by del q t is a minus 2 b q t. So once we plug this in the value of lambda which we get is a minus 2 b q t by 1 plus d raised to t and we said that MRT increases so MR marginal revenue in time horizon t will be equal to marginal revenue in 0 first year into 1 plus d raised 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 q t 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 anymore the marginal revenue will be equal to the price and that is equal to we said a minus b q t so this is going to be equal to a. So a is going to be equal to MR0 1 plus d raised to t and then we can substitute this in this expression so that we get MRT is equal to this is capital T when it gets exhausted capital T this is going to be a by 1 plus d raised to capital T multiplied by 1 plus t. So this is marginal revenue is going to be 1 plus d raised to t minus t and we have already derived that the marginal revenue for the linear inverse demand curve case is a minus 2 b q t. So we can put this as a minus we can equate these two terms b q t is a 1 plus d raised to t minus t. We can now get from this we can put this as 2 b q t is equal to a e into 1 minus so looks very similar to the computation case but with a difference we have now this is q t is a by 2 b into 1 minus 1 plus d raised 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 q t is equal to a by b. So if you look at this of course in the case of the monopoly the capital the value of exhaustion the number of years t would be different but in general what you would find is 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. So if we look at this we would expect qualitatively a curve like this where you have q t and t this is for if this is the shape or competition competitive market then the monopolist would be. So you can look at the book by Conrad on renewable resource on resource economics and there is a chapter on non-readable resource economics which shows some of these if we then take this the same thing can you can see the this is the actual this is for a this is a plot which is shown from Conrad which shows similar kind of trend for a particular example. 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. So in a similar fashion 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 need 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 derived for the competition case this becomes a geometric progression and finally we get an expression which is like we get 2 b r 0 this sum will be equal to r 0. So 2 b r 0 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 b r 0 by a plus 1 by d 1 minus 1 plus d raise to t this is for the time for the 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 r by b r 0 by a and so what you would find is that the time taken would increase and now the question is of course does that mean that a monopoly is better from a resource point of view from a resource point of view the monopolist conserves the resource because the monopolist is looking at the overall maximization of revenue but in the process given the discount rate that is there 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 get conserved for a longer time. So now let us do one thing let us take the same whatever we have learnt for competition and for monopoly let us now solve one particular example a numerical which is there in your tutorial sheet 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 a 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 we have also the value of discount rate R0 is given to you as 75 R0 is 75 and D is 5 percent 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 when 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 is 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 percent increase in the price there will be a 1 percent decrease in the quantity and that is the elasticity. So we solve the first part of the question the second part B says determine the time value of extraction for a mining industry under pure competition. So when we talk about solving this for pure competition we will have this as we have Pt will be for pure competition this will be Pt into 1 plus d raised to t. So 1 point d is 0.05 1.05 t. Now we know at t 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 raised to t we can substitute for P0 so that we get Pt is equal to 1.05 raised 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 Br0 by a plus 1 by d into 1 minus 1 plus d raised 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 raised to t. So if we simplify this we will get t is equal to 7.5 plus 20 into 1 minus 1 by 1.05 raised 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 0.05 raised 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 to get the next value of t and 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 19.4. 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 that p t which we had already solved we got this as p t is equal to 1.05 into t divided by 1.05 raised to 20. And if you see this value of 1.05 raised to 20 turns out so you can get this. So we got an expression for p t we also now can substitute this and get the expression for q t and with that we will get essentially the value of q t 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 p t versus time and q t versus time. Now the question is with 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 are able to affect the quantity supplied and hence the price. And because of that you release less than in perfect competition it is better for you to have less quantity mined and with the result that we expected to last longer. If we take this if you remember we had derived now for the monopolist that this is going to be 2 b r 0 by a plus 1 by d 1 minus 1 plus d raised to t more or less things the equation looks very similar except instead of 7.5 this is now 15. So once we do this we will get obviously 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 raised 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 monopolist wants 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 percent. 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 periods would be higher and then the mine would get exhausted in a shorter time 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 simplistic set of assumptions. You can relax all of these assumptions. You could have a situation where the cost of extraction change. You could have situation where there are different kinds of demand inverse demand curves 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 the way in which the resources would be used subject to the fact that of course the total amount of resources are finite.