Econ 380 Problem Set 1 Answer Sheet

Transcription

1 Econ 38 Problem Set Answer Sheet. Consider the two period model discussed in class. Assume that the inverse demand equations f oil differ across the two periods, with: MB = 6 q MB = 8 q where q denotes the amount of oil consumed in period and q denotes the amount consumed in period. The higher demand f oil in period might result, f example, from an increased population. On the other hand, suppose that the marginal cost equations f extraction remain fixed across the two periods with: MC = 2 + q and MC = 2 + q. If there are a total of 2 barrels of oil available f the two periods and the discount rate is 6.25% (i.e., r =.625): a. How much oil would each time period wish to have, igning the other time period? That is, what is the statically efficient amount of oil to consume in each period? Each period will want to continue to extract oil as long as the marginal benefit from doing so is at least as great as the marginal cost of the extraction. That is, they will continue to extract as long as MBt MCt ( t =,) and they will stop extracting once MBt = MCt ( t =,). After that point, any additional oil extraction costs me (at the margin) than it benefits users (at the margin). q F Period, then, we have that denotes the statically efficient quantity of oil f period, where solves: q MB 6 q = 2 + q 4 = 2q q = MC = 2. Similarly, f period, we have that the statically efficient quantity of oil,, solves. q

2 MB 8 q = 2 + q 6 = 2q q = MC = 3. In reality, since there are only 2 barrels of oil available, period would have to settle f 2 barrels of oil. Graphically, Period s problem is to find the intersection of the marginal benefits and marginal cost curves depicted in the graph below. Note that, once again, we find the statically efficient amount of oil f period would be to have q = 2. MC, MB 8 6 MC 4 2 MB 2 3 q Similarly, we can solve graphically f the statically efficient quantity f period as indicated below, yielding q = 3

3 MC, MB 8 6 MC 4 MB q b. How much would be allocated to period and how much would be allocated to period in a dynamically efficient allocation? Does the quantity used go up down over time? How does this compare to what we found in the example in class and what is the source of the differences (if any) in the pattern of consumption over time? As the answer to the previous question indicates, we have a scarcity problem. The two periods jointly want me oil than there is available. Indeed, f period alone, they would want me oil than there is available. In der to allocate this scarce resource, we have to balance the competing demand f the oil. Dynamic efficiency seeks to maximize the present value of the total net benefits received from the use of the oil. As we saw in class, dynamic efficiency is achieved when: [ ] [ PV MNB = PV MNB ]; i.e., when the net gains that each period would receive from having just a little me oil are the same. If this is not the case, we should shift some oil to the higher valued time period and increase the total net benefits. F the current problem, we know that: MNB = MB MC = 6 q 2 + = 4 2q Since these marginal net benefits occur today, there is no discounting necessary and [ ] PV MNB = MNB = 4 2 q.

4 Similarly, f period, we know that: MNB = MB MC = 8 q 2 + = 6 2 q. In this case, since we have to wait a whole period to receive these benefits, we need to take into consideration the cost of that waiting using discounting, so that: [ ] = PV MNB MNB + r 6 2q = = q = q. Setting these two equal yields: 4 2q = q, where the double asterisks are used to distinguish the dynamically efficient solutions. These second piece of infmation we have is that we have only 2 barrels of oil available f use in the two periods. Since both periods want me than is available, we will want to use up all of the oil and we have: q + q = 2 q = 2 q. Substituting this into our efficiency criteria, we have: 4 2q = q = q q = Given the period s allocation, period simply gets what is left, so that: q = Notice that, unlike the problem in class, we actually allocate me to period than to period. This is due largely to the fact that period receives greater marginal benefit from (i.e., they have me demand f) the oil than period at any given quantity level. F the problem in class, the two periods had the same demand curve. The fact that the discount rate is smaller in

5 this problem than in the problem discussed is class has some impact as well, though this is min. We could also have solved this problem graphically, using the two-winged graphs from class. In this case, the width of the graph should be 2, since there are 2 barrels of oil available. As the figure below indicates, we end up with much the same solution as we found mathematically above PV [ MNB ] q q = q q = c. What would be the efficient prices in the two periods? The price in each period needs to be high enough to encourage the individuals in that period to consume exactly the efficient quantity. An individual will want to purchase oil as long as their marginal benefits from the oil exceeds the price. They will stop consuming oil once their marginal benefits exactly equals the price. Thus, we want: P = MB q = 6 q = = Similarly, f period, we have: P = MB q = 8 q = =

6 Notice that the price is higher in period than in period. d. What would be the marginal user's cost in each period? The marginal user s cost measures the marginal net benefits fegone to the period in question due to scarcity. Thus, f period we have: MUC = MNB q = 4 2q = = ( ) Similarly, f period : MUC = MNB q = 6 2q ( ) = = 39.. Notice that the marginal user s cost is 6.25% higher in period than in period. e. How would your answer in parts b, c, and d change if it was known that OPEC planned to reduce their production of oil in period, raising the anticipated marginal costs in period to: MC = 5 + q. How would the actions of OPEC impact the dynamically efficient allocation of oil over time? Are the effects isolated to period (during which the costs are higher)? In answering this part of the question, we essentially have to go through the same steps as we did in parts a, b, and c above, but not using the new cost infmation in period. The big change occurs in terms of the MNB f period. Specifically, we now have: MNB = MB MC = 8 q 5 + = 3 2 q. In present value terms, then, we have:

7 [ ] = PV MNB MNB + r 3 2q = = q = q. Our efficiency criterion then becomes: 4 2q = q. Substituting in our resource constraint, we then find that: 4 2q = q = q q = As a result, q = Now, even though the second period has a greater demand f oil than the first period, this is offset by the fact that OPEC has increase the price of oil. Now me oil is consumed in period than in period. Both periods are impacted by the change. The prices become P = 6 q = and P = 8 q = OPEC s actions actually results in lower prices today, because it becomes me efficient to consume the oil today in der to avoid the higher costs in period. The only way to achieve this higher level of consumption in period is to lower price. Price in increased in period in der to reduce the consumption further in that period. The marginal user s costs become MUC = 3 2q = MUC = 4 2q = and 2. TRUE/FALSE/UNCERTAIN - EXPLAIN: Indicate whether the following statement is true false uncertain. Uncertain means that it can be either true false, depending upon the circumstances. F any statement thought to be true false, state why it is true false. F any statement thought to be uncertain, identify the circumstances under which the statement would be true and false. a. Given a fixed resource stock (e.g., the number of barrels of oil available) in our simple two period model, dynamic efficiency will required that the resource stock be exhausted during those two periods, with a higher percentage of the resource extracted in the first period relative to the second period. This statement is false f two reasons. First, it is not necessarily the case that all of the resource be exhausted in the two time periods. I gave the example in class in which the statically efficient quantity of oil demanded was 8 barrels of oil f each period. If we had 2 barrels of oil, it

8 would be efficient to consume only 6 barrels total (i.e., 8 barrels in each period) and leave the remaining 4 barrels in the ground. Because we can extract the oil, it does not mean we have to. If the marginal cost of doing so exceeds the marginal benefit, then clearly we should not do so and instead leave some oil unused. Second, it is not necessarily the case that we want to have a higher percentage of the resource extracted in the first period relative to the second period. In fact, in the first problem set of this homewk set, we saw that it was efficient to extract me in period than in period because the marginal net benefits from the oil were higher in period than in period.