Homework Assignment #2: Answer Sheet

Transcription

1 Econ 427 Energy Economics and Energy Security Professor Ickes Spring 2013 Homework Assignment #2: Answer Sheet 1. Consider an exhaustible resource problem with demand given by =100.Letthecost of extraction be given by = 10, and suppose that the rate of interest, = 10. Suppose that the total reserve of the resource is = 153. Assume that Hotelling s Rule is satisfied. (a) Let be the final year of production. If exactly one ton of the resource is extracted in period, during how many years will extraction take place (using a spreadsheet is fine here)? brief answer We know that =100 1 =99. Wealsoknowthat 1 10 = = =80 909, so =91,and 1 =9,and P 1 =0 =1+9=10 Continue this process until P = 153, and we have the answer. So: Figure 1: (b) What is the price of the resource in period 1? What is output in period 1? Draw the path price and the extraction path. brief answer We see from the table that the initial price is 60 and initial output is So that path price and the extraction path look like: Figure 2: Price and Output Path (c) Suppose that = 354. If everything else is unchanged, how do your answers change? brief answer Now we have greater total reserves so we must produce longer. We just need P =354 Nothing else is changed, so we now have:we now produce for 11 periods, initial price is lower and initial output is higher. 1

2 Figure 3: 2. Suppose that demand for an exhaustible resource is given by =,where 0are constants. Draw this demand curve. Suppose that extraction costs are constant at rate,. The economy is competitive and the interest rate is given at. (a) If the initial price level is 0, and if Hotelling s rule holds, then how does relate to 0? As what happens to. brief answer We know that = 0 (1+ ). We also know that as then as there is no choke price. The demand curve looks like p q Figure 4: Demand Curve for =,with =1 =1 (b) Using the expression you have from part (a) and the demand curve, derive an expression for in terms of the parameters. From this expression can you determine what happens to as? brief answer We have = for we have: and = 0 (1 + ),so = " = 0 (1 + ),sosolving # 1 (1) (1 + ) 0 As the denominator gets bigger and bigger so 0, which means that (c) If initial reserves are given at 0, and if reserves are exhausted as,whatisthe relationship between total extraction and reservesl? brief answer We know that total production equals reserves, so we sum up (1) and thus P h i =0 1 (1+ ) 0 = 0. You could solve this expression for 0 but I did not ask you to do that. 2

3 pt(1+r)^t parameters time quantity Tq r p a b Figure 5: (d) Using a spreadsheet, assume that =1 = 01 = 05 =0 and 0 =75 Can you figure out what 0 has to be (within a couple of decimal points)? What would happen to the price path if = 01? Explain. brief answer Just use expression (1) with the parameters given, start with period 0, let keep increasing and total production. Try different values for 0 and notice that = 0 (1 05). See what value of 0 works so that total production approaches 75 I got this when I tried 714.If the interest rate was lower then the price will grow at a slower rate. The present value of production in the future is now greater than when = 05. The initial price would have to be higher to lower initial quantity. I get an initial price of 822 and 0 =7 1. (e) Suppose 0 =200,and = 05. What would be your new estimate of 0? Can you explain why the initial price changes that way when reserves rise? brief answer If we have = 05 but higher reserves then we must have a lower price so that current production is greater. That way we can approach the higher quantity of reserves. I get an initial price of about 648 and quantity 76. The price rises at the same rate in both cases, so if the price did not fall output would not be high enough to approach the higher level of reserves. 3. Return to problem (1, yes problem 1) but assume now that the producer is a monopolist. If everything else is unchanged, re-do problem 1. brief answer The key here is to note that total revenue, = 100 2, so = Then we reason exactly as in problem 1 but we note that for the monopolist we have = +1. With =1in the last period we have 1+ =98,so 1 = =90 0, and we just keep proceeding accordingly. Since we have the 1+ 1 same reserves, we get (a) Is the monopolist s initial price higher or lower than in the competitive economy? What about the time to completion? brief answer Comparing this table with that from problem 1, we see that the initial quantity is lower (26 versus 40) and the initial price is higher. Production takes longer till exhaustion. 3

4 MR Q MR C (MR c)/(1+r) TQ periods Figure 6: tity 25 a n u20 Q Monopolist Competitive Figure 7: (b) Does the monopolist or the competitive economy conserve more? Explain. brief answer The monopolist produces for a longer period of time. But this does not raise welfare since the competitive solution re-produces the path the social planner would produce. Marginal benefit is too high in the initial periods with the lower production the monopolist chooses. 4. Consider the basic Hotelling model of exhaustible resources. Assume a competitive economy with many producers, a fixed cost of extraction,, and a choke price,. The rate of interest is given at rate. What happens to the extraction path of the resource (the plot of output,,againsttime)if: (a) the rate of interest falls. brief answer If falls, it is all of a sudden better to keep a dollar s worth of oil in the ground than a dollar in the bank. So oil production falls. This causes 0 to rise and 0 to fall. The Hotelling Rule requires that net rent grows at the rate of interest which is now lower. So clearly the time to exhaustion must rise. If price starts lower than before, and if it grows slower than before, it must take longer to reach Economically, the present value of future production has increased, so we should shift extraction towards the future (see figure 8). (b) the demand for the resource increases suddenly. brief answer If the choke price remains unchanged this means that the demand curve becomes flatter greater demand at any price below In the case of the inverse 4

5 q t q 0 initial optimal extraction path q 0 new optimal extraction path T T ' t Figure 8: Shift in the Extraction Path demand curve used in class, =, this means that falls. If the price path did not change we would extract more very period and total production would exceed. So 0 must rise to dampen down the quantity demanded. Since prices still grow at the rate it follows that 0 must also rise. If not, then we would reach before exhaustion. You can also see this from the expression we derived in class for output: = [1 (1 + ) ],soif falls is higher. But this means that we must reach exhaustion at a lower, so the new extraction path must have higher 0 and lower. (c) the choke price falls brief answer This means that is lower. If the price path were unchanged we would end up with extra oil which cannot be optimal. So 0 must fall and 0 must increase. The time to exhaustion must also fall, since reaches the new lower in less periods. So 0 rises and falls. (d) a tax on the sales (gross revenue) of the resource, per barrel is imposed on producers. brief answer I said a tax on revenues not rents. If I sell a barrel I in period, Inow keep (1 ). The producer is now equating (1 ) = (1 ) +1 (2) 1+ thetaxisclearlynotneutral(asitwouldbeifthetaxwereonrents,thenwecould cancel out the (1 ) terms). The impact of the tax is to lower the present value of current profits relative to future profits, as the left-hand side of (2) fall by more than the RHS. So the producer wants to produce less in the current period. So 0 falls. Since production is moved to the future the time to exhaustion must rise, since prices still rise at the rate of interest. (e) a new discovery of oil takes place that doubles reserves. brief answer If reserves now equal 2, then production must rise. But if the choke price is unchanged and prices grow at the rate of interest, then 0 must rise and must increase as well h so that the total amount of production Σ =0 1 must rise. But we know that = i 1 (1 + ) then the sum of production is given by: Σ =0 1 = Σ =0 1 [1 (1 + ) ]= 5

6 so if doubles, and is given, the only thing that can rise is. 6