Chapter 23. The Economics of Resources. Chapter Outline. Chapter Summary

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1 Chapter 23 The Ecoomics of Resources Chapter Outlie Itroductio Sectio 23. Growth Models for Biological Populatios Sectio 23.2 How Log Ca a Noreewable Resource Last? Sectio 23.3 Sustaiig Reewable Resources Sectio 23.4 The Ecoomics of Harvestig Resources Sectio 23.5 Dyamical Systems ad Chaos Chapter Summary Geometric growth models for biological populatios ted to use the atural rate of icrease (the differece betwee the birth ad death rates) to represet the growth rate of a populatio. Ufortuately, birth ad death rates fail to remai costat over time. Noetheless, such models ca be used i short-term plaig. Sice huma populatios rely o resources to sustai them, it is importat to be able to predict future amouts of both reewable ad oreewable resources. The maagemet of reewable resources poses a very iterestig problem: determie how much of our resource we ca harvest each year ad still allow the resource to repleish itself. I other words, what is the maximum harvest our resource ca sustai? Fidig this value requires kowig the size of ext year s populatio give this year s populatio. This iformatio is usually provided by a reproductio curve for the resource. This problem is a importat oe i forest ad fishery maagemet. Not surprisigly, ecoomic factors play a role i the maagemet of resources. Harvestig will ot take place if it is ot profitable. Furthermore, the resource may be elimiated etirely if it is more profitable to ivest the proceeds elsewhere tha to sustai the resource over time. I dyamical systems the state of a system at ay time depeds upo its state at previous times. Certai dyamical systems, such as weather, are chaotic, i the sese that the evolutio of the system is sesitive to iitial coditios. That is, a small chage i the iitial coditios may result i large chages at some future time. Biological populatio growth ca sometimes exhibit chaotic tedecies. 43

2 432 Chapter 23 Skill Objectives. Determie a coutry s projected populatio i a give umber of years whe its curret populatio ad its projected growth rate are give. 2. Explai why a sustaiable yield policy is eeded for harvestable resources. 3. Give the curret geeratio populatio size, use its reproductio curve to estimate the ext geeratio s populatio. 4. Iterpret the meaig of the lie y = x i relatio to the reproductio curve. 5. Approximate from a reproductio curve the projected sustaiable yield for a harvestable resource whe the curret geeratio populatio is give. 6. Estimate from the reproductio curve the maximum sustaiable yield for a harvestable resource. Teachig Tips. It s somewhat surprisig to fid that studets with a algebra backgroud ofte do t relate the value of a fuctio to the height of its graph; cosequetly, the process of iterpretig a reproductio curve may require some preparatio i terms of discussig fuctios. Oce that has y value bee doe, you might cosider havig studets read the ext geeratio populatio for various curret geeratio populatios ( x values) before attemptig to determie the sustaiable yield. 2. Whe begiig to approach the cocept of sustaiable yield, some studets eed a review of the lie y = x i terms of idetity fuctio. Uderstadig that it represets a equality betwee the curret populatio ad the ext-geeratio populatio sets the stage for fidig the sustaiable yield. 3. The otatio f ( x) x ca sometimes be uderstood more clearly i terms of the physical subtractio of legths of lie segmets If you use a overhead projector, drawig the vertical segmets f ( x ) ad x i differet colors o the graph of the reproductio curve may help emphasize this relatioship. Drawig their differece i yet a third color ad followig this chagig differece alog the curve may help studets visualize the maximizatio problem. 4. This chapter offers may possibilities for extra-credit projects. If your geographic area has a specific reewable resource idustry such as lumber i the Pacific Northwest or fishig i the Northeast ad the Gulf of Mexico, studets ca cotact local agecies ad compaies to fid out their policy o reewig the atural resources. It would the be iterestig to report these fidig back to the class. Research Paper Studets should fid researchig the topic of fractals very iterestig. They may wish to ivestigate the lives ad cotributios of mathematicias such as Beoit Madelbrot (Polish) or Gasto Julia (Frech). Other studets may wish to focus o the complex shapes geerated by computers or fractals i ature. I all cases, studets ca fid a wealth of iformatio o the Iteret Spreadsheet Project To do this project, go to This spreadsheet project is desiged to explore the upredictable ad chaotic aspects of the logistics model, icludig the effects of roudig.

3 The Ecoomic of Resources 433 Collaborative Learig Idustry Awareess As a icebreaker, ask the studets to form groups ad determie if they are aware of idustries that deal i reewable resources, ad how they go about guarateeig that the resource will ot be elimiated. Oe obvious example is the paper idustry, which plats ew trees to replace those it cuts dow, thereby repleishig its stock. The fishig idustry is more difficult to maage, sice idividual fisherme have o cotrol over the size of what their competitors catch. I this case, govermet itervetio or agreemet amog the fisherme may be eeded to maitai a stable populatio ad regular harvests of fish.

4 434 Chapter 23 Solutios Skills Check:. b 2. a 3. c 4. b 5. a 6. a 7. c 8. b 9. c 0. c. b 2. a 3. c 4. a 5. c 6. b 7. b 8. a 9. a 20. c Exercises:. I late summer populatio i mid-2025 is as follows. 8 8 populatio i mid growth rate = billio 5.59 billio 3. populatio i mid-2025 is as follows billio 8 8 populatio i mid growth rate = billio 5.49 billio 5. The populatio of Africa would be 925 (.024) 8 = 48 millio, almost 00% greater tha, or twice as large, as Europe s populatio. 6. (a) 7. (a) years years years years.3 8. (a) 206 The demad will be ot 2.0 but 2.2 times as much i 206 as i 99, of which busiess will accout for almost two-thirds rather tha half. 9. (a) populatio i mid-2025 is as follows. 8 8 ( populatio i mid-2007) ( + growth rate) = 6.593( ) billio 8.3 billio populatio i mid-2050 is as follows populatio i mid growth rate = billio.5 billio No chage i growth rate, o chage i death rates, o global catastrophes, etc.

5 The Ecoomic of Resources (a) For 2025 More-developed coutries.26( ) 8 billio.238 billio Less-developed coutries (excludig Chia) 4.056( ) 8 billio billio Chia.32( ) 8 billio.47 billio Sum For billio More-developed coutries.26( ) 43 billio.269 billio Less-developed coutries (excludig Chia) 4.056( ) 43 billio billio Chia.32( ) 43 billio.709 billio Sum There is ot much differece. Aswers will vary..73 billio. (a) The static reserve will be years. The expoetial reserve will be (c) Aswers will vary ( 77.9 ) years. l (a) The static reserve will be years The expoetial reserve will be l[ ] (c) Aswers will vary. 42 years 3. (a) The static reserve will be 00 years. We are seekig the expoetial reserve. This will be + 00( 0.025) 5 years. l (c) [ ] years l , years l (a) The static reserve will be 00 years. We are seekig the expoetial reserve. This will be + 00( 0.025) 65 years. l (c) [ ] years l , years l 0.025

6 436 Chapter (a) [ ] years l [ ] ( ) [ ] l l0 = = l 0.0 l 0.99 l 0.99 This theoretically would imply forever! 6. (a) The static reserve will be 0,000 years. We are seekig the expoetial reserve. This will 0,000( 0.035) be + 70 years l [ ] + 5, years l (c) Sice (.035) the static reserve will be years. (d) Aswers will vary tos plats 800 years 5.5 tos/plat/year 30 lb/plat/day, which is ureasoable l % l % 30 l % 2. After the first year, the populatio stays at , 20, 0, 0, , 8.2, 6.6, 7.6, 8.4, 9.5, 2.0, 7.3, 8.6, We must have f ( x ) = x, or 4( 0.05 ) =, or x = 5. x 26. Usig x f ( x ) x ( x ) 4x 0.05 x = x. The oly solutios are x = 0 ad with x = 5 we have the followig (rouded). 5,.3, 4.8,.6, 4.6,.8, 4.5,.9, 4.4, Usig x f ( x ) x ( x ) with x = 0 we have the followig (rouded). 0, 5.0,.3, 4.8,.6, 4.6,.8, 4.5,.9, 4.4 The populatio is oscillatig but slowly covergig to Those are the oly values. 29. We must have f ( x ) = x, or 40 3( 0.05 ) =, or x = x 30. Aswers will vary x 0.05 x = x. The oly solutios are x = 0 ad

7 The Ecoomic of Resources The red dashed lie idicates the same size populatio ext year as this year; where it itersects the blue curve is the equilibrium populatio size. x+ x = 3x 0.05x x = 2x 0.5 x, ad we wat to maximize this quatity. By graphig, symmetry of a parabola, or (for the istructor) by calculus, 20 the maximum occurs at x = 6.7, for which the yield is 3 x 2 x The graph shows the fuctio Usig x f ( x ) x ( x ) with x = we have the followig (rouded)., 2.0, 2.6, 2.9, 3., 3.2, 3.3, 3.3, 3.3, Usig x f ( x ) x ( x ) with x = 5 we have the followig (rouded). 5, 6.6, 8.2, 9.8,., 2.0, 2.6, 3.0, 3., The populatio sizes are, 5.0, 3.7, 4.9, 4.9, 5.0, 4.8, 4.3, 3.0, 9.5 ad the followig year the populatio is wiped out. 36. Usig x f ( x ) x ( x ) with x = we have the followig (rouded)..0,.5, 2., 3.0, 4.2, 5.6, 7.2, 8.9, 0.4,.5 Thus, after 0 years harvestig may resume. 37. About 5 millio pouds. Maximum sustaiable yield is about 35 millio pouds for a iitial populatio of 25 millio pouds. 38. (a) About 6 MSY 7, for a iitial populatio of approximately (a) The last etry show for the first sequece is the fourth etry of the secod sequece, so the first jois the secod ad they the both ed up goig through the same cycle (loop) of umbers over ad over. 39, 78, 56, ad we have joied the secod sequece. However, a iitial 00 stays 00 forever; ad ay other iitial umber edig i 0 jois the loop sequece 20, 40, 80, 60, 20,.... (c) Regardless of the origial umber, after the secod push of the key we have a umber divisible by 4, ad all subsequet umbers are divisible by 4. There are 25 such umbers betwee 00 ad 99. You ca verify that a iitial umber either jois the self-loop 00 (the oly such umbers are 00, 50, ad 25); jois the loop 20, 40, 80, 60, 20,... (the oly such are the multiples of 5 other tha 00, 50 ad 25); or jois the big loop of the other 20 multiples of Aswers will vary. 4. (a) 33, 9, 82, 68, 00,,,.... The sequece stabilizes at. Aswers will vary. (c) That would trivialize the exercise! (d) For simplicity, limit cosideratio to 3-digit umbers. The the largest value of f for ay digit umber is = 243. For umbers betwee ad 243, the largest value of f is = 63. Thus, if we iterate f over ad over say 64 times startig with ay umber betwee ad 63, we must evetually repeat a umber, sice there are oly 63 potetially differet results. Ad oce a umber repeats, we have a cycle. Thus, applyig f to ay 3-digit umber evetually produces a cycle. How may differet cycles are there? That we leave you to work out. Hits: ) There are t very may cycles. 2) There is symmetry i the problem, i that some pairs of umbers give the same f 68 = f 86. result; for example,

438 Chapter 23 42. (a), 4, 2,,.... 3, 40, 20, 0, 5, 6, 8, 4, 2,..... (c) 2, 6, 3, 0, 5, 6, 8, 4, 2,,... 43. (a) 0.0397, 0.540773, 0.545072626,.288978, 0.75942, 0.5978202,.39379, 0.056275776, 0.

8 438 Chapter (a), 4, 2,,.... 3, 40, 20, 0, 5, 6, 8, 4, 2,..... (c) 2, 6, 3, 0, 5, 6, 8, 4, 2,, (a) , , , , , ,.39379, , , , , , , , , , , , , ,.32383, , , , , , , , , (c) 0.722,.32448, , , , , , , , (a) Uless r = 0 (which would t be a very dyamic system), the oly equilibrium poits are x = 0 ad x =. For a logistic model with λ 0, the oly equilibrium poits are x = 0 ad x = (carryig capacity). 45. Period 2 begis at λ = 3, period 4 at , period 8 at 3.544, period 3 at , ad chaotic behavior osets at about See Word Search Solutio

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