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PS3 Soln

# PS3 Soln - EEP 101/Econ 125 GSI Diana Lee and Biswo Poudel...

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Unformatted text preview: EEP 101/Econ 125 GSI: Diana Lee and Biswo Poudel Spring 2010 Problem set 3 Due Thursday, April 15th at lecture Late problem sets will not be accepted. 1. Natural Resources Suppose lolli—trees grow in Lolli—land with a growth rate given by G (st) = st— 18—620 where s is the stock of lolli—trees and t is measured in years. Lolli—trees produce lollipops and are therefore highly desired by the candy—loving residents of Lolli—land. One harvested lolli—tree sells for a market price of 5 dollars. The is a ﬁxed cost of harvesting lolli—trees equal to 100 dollars, but variable costs decrease with the size of the stock, at the rate of —s so that the total cost of harvesting is given by T0 = 100 — 3. Finally, Lolli-land is in a boom period with a relatively high interest rate of 5%. (a) Suppose Chucky, the mayor of Lolli—land, realizes that lolli—trees are on the brink of extinction due to a series of severe droughts over the past few years. There are currently only 10 lolli—trees left in Lolli—land. Chucky decides to enforce a ban on any lolli—tree harvesting and allow lolli—trees to return to their natural steady state. Find the growth rate G and stock 3 that would be the long run result of Chucky’s policy. The natural steady state occurs when the stock is at its carrying capacity. This is because at that point, there is zero net growth and thus the stock remains the same size from year to year. To ﬁnd this, we set the growth equation equal to zero and ﬁnd the maximum stock that makes it zero. 2 s—ﬁszo 3:0,100 Thus, at the carrying capacity, 3 = 100 and G = 0. 1 (W Calculate the number of years t that it would take for the lolli-tree orchard to reach its carrying capacity. (You can round the growth rate for each year to the nearest integer.) To do this question, you should consider the growth of the orchard each year and iterate until the stock grows until 100. Time Beginning stock st Growth: C(st) 2 st — % Ending stock: 8,; + C(st) 0 10 10—-%ﬁ=:9 10+9=19 1 19 1 ——%i==15 19—15=34 2 34 34—-%%==22 34—22256 3 56 56——%g?==25 56+25=81 4 81 81-— gig ==15 81——15==96 5 96 96—-§i9==4 96—42100 Thus, at the end of 6 years (counting year zero), the orchard will reach its carrying capacity at 100 trees. Suppose now that Angelica runs a ﬁrm interested in harvesting lolli—trees. Find the optimal number of trees that she would sell each period. Set up the proﬁt maximization. Proﬁt is given by: «=5Q—ﬁQ—ww0—g Taking the derivative we have g=5u—§Q+1=0 5—§+1=0 62% 3:60 The number of trees she will sell is given by the harvest, so we need to plug back into the growth equation. Gmm=6o—%=24 Angelica wants to harvest every period because she can do so sustainably and make a proﬁt. She tells Chucky that she will go ahead and do so unless he pays her enough now to compensate her for all her proﬁt both now and in the future (into eternity). Calculate the amount that Chucky must pay Angelica to meet her demand. Let’s calculate the proﬁt she would receive each year if she did what was optimal. 7r=5*24—(100—60)=120—40280 We can use the formula for an annuity (same as perpetuity in this class) where NPV=§. Here x is our yearly proﬁt. NPV— 0—835 — —1600 After calculating this amount, Chucky decides that he is not willing to pay such a large lump sum to maintain his desired level of lolli—trees. He decides instead that he is willing to allow the stock of lolli—trees to decrease to 90% of its carrying capacity and thus allow Angelica to harvest some trees each period. He does so by levying a tax on Angelica. Calculate the tax per harvested tree required for Angelica to maintain a steady state at 90% of the carrying capacity. (Hint: a tax per tree effectively lowers the price received per tree and will thus change the proﬁt maximization equation.) Again, let’s set up the proﬁt equation while putting in a tax. 52 7r— — 5(5 — W)_ (100 — s) — t(s — W) 7? = (5 — t)(s — ﬁzo)— (100 — 8) Taking the derivative we have s=(5—t><1—a)+1=o Recall that Chucky is using the tax to change Angelica’s optimal stock to 90. So we can plug in s=90 to solve for the tax that will do so. (5—t)( —%)+1=0 (5 — t)(50 — 90) +50 2 0 (5 — t)(—40) + 50 = 0 40t—200+50=0 40t= 150 t* = % =3.75 Phil and Lil have invented a new technology that has a ﬁxed cost of 200 dollars but changes the marginal variable costs to —23. Determine whether or not Angelica will adopt the new technology if she decides based solely on the proﬁts for one period. Recall that using a new technology may change the optimal stock that Angelica chooses, so we must ﬁrst calculate the new optimal stock. The ﬁxed cost of 200 replaces the old ﬁxed cost of 100. Set up the proﬁt maximization. Proﬁt is given by: 7r— — 5(3 — F20)_ (200— 23) Taking the derivative we have d—g=15(1——)+2=0 5 — — :+ 2— — 0 7— _ s— — 70 C(70) :70—49:21 7r=5*21—(200—2*70)=105—60=45 wold = 80 > mew = 45, so Angelica will not adopt the new technology. (g) Interested in pushing their product, Phil and Lil try to convince Angelica that she should adopt the technology because its beneﬁts extend to future periods in addition to the current period. In particular, the ﬁxed cost only has to be paid once but the change in variable costs apply to every period. Assuming that Angelica can continue to harvest every period into eternity, calculate the maximum ﬁxed cost that Phil and Lil can charge for their technology and still ensure that Angelica will adopt. (Hint: Use the difference in present values of the proﬁt streams for the old and new technologies.) The maximum ﬁxed cost they can charge is the difference in long—run proﬁts between the new and old technology. I calculate the case here where the ﬁxed cost is only paid once for both the new and old technology but gave you credit if you assumed otherwise and did the calculations correctly. NPV(7Told,nofc) — 100 = NPV(7rnew,nofc) — F0 7l—oldﬂwfc : 80 ‘l’ 100 = 180 7rn6’w,nofc : 45 ‘l‘ 200 : 245 NPV(7rold,nofc) = ﬂ = 3600 .05 NPV(7rnew,nofc)% = 4900 Plugging into the formula above, 3600 — 100 = 4900 — F0 F0 2 1400 2. Short Answer Questions (a) The above ﬁgure illustrates the price trajectory of a non-renewable resource under two difference interest rates. For the curve associated with TI, explain why the price trajectory has the shape that it does. With non—renewable resources, price represents scarcity. The upward curve is associated with a non-renewabe resource where there is no recycling or new dis- coveries over time. With mining, the resource becomes more scarce over time, thus causing the price to rise. The exponential shape of the curve is the result of Hotelling’s rule, which states that the price must rise exactly with the interest rate for the most efﬁcient use of the resource. From the graph above, determine whether r1 > r2 or r1 < r2. Explain. r2 > T1. This is because higher interest rates mean that the present is worth more than the future (since the NPV of the future is less with a higher interest rate). Thus, under the higher interest rate, more will be harvested now (relative to the lower interest rate), resulting in a lower price since there is more available. With more harvest now, though, the stock decreases more quickly and thus the price rises more quickly. In ﬁsheries, we sometimes get what is called a backwards bending supply curve. This is because, after a certain point, to increase the catch of ﬁsh, we end up overﬁshing and depleting the stock of ﬁsh. This thus makes catching ﬁsh harder and harder, which explain why the backwards bending portion shows an increase in costs with less and less quantity. With the given supply curve, it is possible for a monopoly to produce a higher market quantity than the competitive equilibrium? If so, graphically construct such a case. If not, explain why not. Yes, it is possible. See graph below. Note, this occurs because the monopoly only cares about maximizing proﬁt. When we are maximizing proﬁt, he is taking into account the trade—off between price and quantity. By lowering its quantity, it can raise the price, but when it lowers its quantity, it will sell less. Normally, it is worth lowering the quantity because the gain in price more than makes up for it. In this case, though, the monopolist actually increases its revenue by increasing its quantity. Because of the backward bending shape of the curve, it can increase its quantity by a large portion without lower the price very much at all. ...
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