Question

Recall the simple two-period model we covered during class: when the demand curve is stable over time and the marginal cost (of extraction) is assumed to be constant, the value of the marginal user cost was shown to rise over time at the rate ?? in an efficient allocation. With the aid of appropriate diagrams, one can generalize the results to loner time periods:

1) Discuss what happens to the efficient allocation path when we extend the time horizon from 2 periods to ?? periods while retaining all the other assumptions?

2) Discuss what happens to the efficient allocation when we further assume a substitute renewable resource is available at constant marginal cost? (This case, for example, could describe the efficient allocation of oil or natural gas with a solar or wind substitute).

3) Discuss what happens to the efficient allocation path when we further assume the marginal cost of extraction rises with the cumulative amount extracted? (This is commonly the case, for example, with minerals, where the higher-grade ores are extracted first, followed by an increasing reliance on lower-grade ones).

Answer 1

1) In this two period model the demand is assumed to be stable,marginal cost of extraction constant and the marginal user cost rising over time.Now,an efficient allocation in any time period implies an extraction level of the resource taking into account the marginal cost of extraction,the resource price and the marginal user cost in each of the time periods.In other words,based on the profit maximizing principle of production,a extraction level where the price level of the resource is equal to the sum total of marginal extraction cost and the marginal user cost would provide efficient/optimal allocation of the resource.

Now,since the current extraction of the resource depletes the resource availability for the future time periods the marginal user cost which is the present value of the foregone opportunities of resource availability for the future will always be increasing at every following time periods.Hence,the discount rate of the resources will also increase in future time periods.Therefore,in every future time period the resource price will also rise to match the increasing marginal user cost to hold the efficient allocation condition in every future time period(Notice here that the marginal cost of extraction is constsnt).Thus,in every future time period the efficient allocation level of the resource can be expressed as:-

Resource Price(P)=Marginal Cost of Extraction(MC)+Marginal User Cost(MUC)[Where both P and MUC will keep increasing in every time period to hold this optimaility/efficiency condition]

2) If any substitute renewable resource is discovered to replace the current resurce in use then expectedly its marginal user cost or MUC will not increase anymore in future as now users will shoft to the substitute resource and hence the depletion rate of the current resource will decrease(discount rate of the current resource will also drop).Concequently the resource price P will also be stable over time as now MUC will be stable as well over time(it can remain stable or decrease if the extraction becomes minimal due to the advent of the new resource).

Now from the production perspective,the producers will only switch to the renewable resource only if the marginal willingness to pay by the users for the new resource is atleast equal or higher than the marginal cost of extraction.Other wise,it would not be profitable to switch.Furthermore,at the point of transition between the existing and the substitute resource,the marginal cost of extractions for both the resources have to be equal so that the producers will have no issue switching to the new resource.Again,as mentioned in part 1),the producer will continue to extract the level at which the Resource price P=Marginal Cost of Extraction MC+Marginal User Cost MUC for the substitute resource to maintain the efficient/optimal allocation condition.

3) Assuming all other conditions as same,now if the marginal cost of extraction increases with further cumulative extractions,the overall extraction level will decrease if the price remains the same and the sacrifice made by users in future time periods also decreases due to reduced depletion rate presently.As the marginal extraction cost keeps increasing in future it will be equal to the overall marginal cost(indicating there are no extra costs aside from marginal extraction cost) at the transition point from the existing to the substitute resource which denotes the effective/optimal allocation of the substitute resource in case of switching.Again,the idea is that the producers will switch due to increasing marginal cost of the existing resource if the resource price does not rise to compensate the increasing marginal extraction cost.But switching only occurs when the total marginal cost including extraction cost of the substitute resource and any additional cost do not exceed the marginal extraction cost of the current resource.