Survey of Math Chapter 23: The Economics of Resources Page 1

Transcription

1 Chapte 23: The Economics of Resouces Page 1 Intoduction Populations gow and decease ove time, and thee ae diffeent mathematical models fo this. Some common models ae the following thee ae many othes! Aithmetic Gowth Hee, the gowth is by a fixed amount each time peiod. So we have gowth ate = P = = constant. This is the same as simple inteest, and leads to a staight line fo the population, since the slope of the population line (ise/un) is always the same. Geometic Gowth Hee, the gowth is by a fixed pecentage of the cuent population each time peiod. So we have gowth ate = P = P. This is the same as compound inteest, and leads to a cuving line fo the population, since the slope of the population line (ise/un) is always inceasing. Logistic Gowth Both the pevious gowth models have a flaw they would indicate that the population can gow without estaint, becoming vey lage. But that isn t what happens in natue the population is constained by the esouces

2 Chapte 23: The Economics of Resouces Page 2 available to it and often thee is a maximum population that can be suppoted. This maximum population is called the caying capacity M, and logistic gowth incopoates this into the gowth ate. ( gowth ate = P = P 1 P ). M Fo logistic gowth, the gowth is initially inceasing, then eventually deceases and eaches a maximum at M. If the initial population is above M, the population will decease to the caying capacity. Compaing these diffeent models, we see that they ae all vey simila fo shot times, but the long time behaviou is quite diffeent. Moe advanced models will also contain the idea of extinction, and if a population falls below a cetain amount they will eventually become extinct.

3 Chapte 23: The Economics of Resouces Page 3 The Rule of 70: Population Doubling Fo estimating how long it takes fo a population to double, one can use the Rule of 70: If a county s population continues to gow at a constant ate of % pe yea, then it will double in size evey 70/ yeas. Since population gowth and savings/boowing ae closely elated, you should not be supised to lean that thee is a simila ule (The Rule of 72) fo estimating an investments doubling time. This ule aises fom the geometic gowth model, with i = /m = since the populations ae measued each yea. Since geometic gowth is eally just compound inteest, we have New Population = Old Population(1 + i) n 2 Old Population = Old Population(1 + ) n 2 = (1 + ) n ln(2) = ln((1 + ) n ) ln(2) = n ln(1 + ) n = ln(2) ln(1 + ) Fo a small numbe, ln(1 + ) (check it out on a calculato), and ln 2 = So we have n 0.70 = 70 % whee is the pecentage. as the time it takes fo the population to double. Example Execise 6 Using the ule of 70, when will the populations of the following counties have doubled? (a) Afica, 925 Million, gowth ate of 2.4% in (b) USA, 301 million, gowth ate of 0.8% in Solution: Afica time to double = 70 = USA time to double = 70 = = 29 yeas. = 87.5 yeas. Example Wisconsin s electicity demand in 1991 was about half fom business use and half fom esidential use, and inceased at 2.8% pe yea. When will the demand be twice what it was in 1991? time to double = 70 = = 25 yeas = 2016.

4 Chapte 23: The Economics of Resouces Page 4 How Long will Nonenewable Resouces Last Resouces come in two basic types: enewable esouces and nonenewable esouces. Once nonenewable esouces ae used up, it takes a long time fo the ecosystem to geneate moe, if moe can be geneated at all. Gasoline and coal ae examples of nonenewable esouces. How long nonenewable esouces last is obviously a vey impotant question, and so modeling this situation is citical. We have two basic types of models: If the ate of use of the esouce is constant, then we can calculate the static eseve. If the ate of use is inceasing (which typically happens as the population inceases), then a bette model would be to calculate the exponential eseve Static Reseve If we use U units pe yea and have a eseve of S units of the esouce, then we have enough esouces to last S U yeas. Exponential Reseve If we use U 1 units in the fist yea, and the usage inceases at % each yea, and have a eseve of S units of the esouce, then the poblem is moe complex. But it is still just a compound inteest poblem at heat, so we can answe it. We can modify the savings fomula, with i = /m = to answe the question of when we un out of the esouce. [ ] (1 + i) n 1 A = d i [ ] (1 + ) n 1 S = U We need to solve this fo n, the time it takes to deplete the esouce. (1 + ) n 1 = S U (1 + ) n 1 = S U (1 + ) n = 1 + S U ln((1 + ) n ) = ln (1 + SU ) n ln(1 + ) = ln (1 + SU ) n = ln ( 1 + S U ) ln(1 + ) This gives the numbe of yeas that it will take to deplete the esouce.

5 Chapte 23: The Economics of Resouces Page 5 Example In 2004, the natual gas esouces totaled tillion cubic feet, while annual consumption was 90 tillion cubic feet pe yea. Wold consumption is pojected to incease by 2.2% pe yea though (a) What is the static eseve in 2004? (b) What is the exponential eseve in 2004? Solutions: (a) Static eseve = S/U = /90 = 67.5 yeas (b) Exponential eseve = ln ( 1 + S U ) ln(1 + ) Sustaining Renewable Resouces = ln ( ) ln(1.022) = 41.8 yeas. A enewable esouce is one that eplenishes itself (any fom of wildlife, tees, bamboo, con, etc). The question with enewable esouces is not when will they un out, but how much can we havest and still leave enough of the population to egeneate itself. This is paticulaly impotant fo wild populations of animals which ae havested especially fish. Repoduction Cuves A epoduction cuve is a mathematical model that gives the population size in the next yea if the population size this yea is known: Figue 1: Population cuve that elates the population in a cuent geneation x to the population in the next geneation by the fomula y = f(x). The dashed line is y = x which epesents no population change fom yea to yea. The basic shape of this cuve is no accident. It ises on the left since thee ae enough esouces fo the population to incease, and falls on the ight since the population is too lage fo the available esouces and must decline.

6 Chapte 23: The Economics of Resouces Page 6 The details of the shape of the epoduction cuve depend on the paticula population being studied, but the basic shape is always the same. The epoduction cuve is usually used in the following manne. Vetical lines epesent moving fom one yea to the next. The amount of the vetical line between the line y = x and the epoduction cuve epesent the natual incease o natual decease of the population fom one yea to the next. An equilibium population is a population which does not change fom yea to yea. An equilibium population occus whee the epoduction cuve intesects the line y = x. Figue 2: Left: Population stats at 0.2. In following yeas, the population is 0.725, , The geen lines epesent population gowth (above y = x) o decease (below y = x) fom yea to yea. Right: Equilibium population. Sustained Yield Havesting of a Population A simple model of havesting a population would include the idea that geate effot would esult in geate havest. The havest line could theefoe be a staight line with steepe slope indicating geate havest. It looks like the y = x line we used, but it has a diffeent slope to indicate diffeent levels of havesting. A sustained yield havesting policy is a policy that if continued indefinitely will maintain the same yield. We ae most inteested in the maximum sustainable yield, which would maintain the population at at a constant fixed amount, and havest the maximum each yea (if we havest less than the maximum, the population would incease fom yea-to-yea). This is descibed in Fig. 3. An Altenate Way of Showing The Population Dynamics: Cobweb Diagams Anothe way of pesenting this infomation is shown in Fig. 4. The hoizontal lines that move towads the staight line y = x ae used to eset to the next yea (and theefoe epesent the natual incease o natual decease of the population, since they ae the same length as the geen lines on the left).

7 Chapte 23: The Economics of Resouces Page 7 Figue 3: Left: Population cuve. Cente: Population Gain cuve, f(x) x. The dashed line shows the population gain going fom 0.2 to 0.725, which is = Right: The maximum sustainable yield would maintain the population at 0.42 yea-to-yea, with a maximum sustainable yield of Chaos in Biological Populations Figue 4: Left: Population cuve. Right: Cobweb diagam. One definition of chaos is that knowing the cuent population does not allow you to pedict the futue population. Of couse, mathematically we can pedict futue populations, so a bette definition of chaos would be a system fo which small changes in initial conditions lead to lage diffeences late. This is eally what pedictability is all about the pedictions fo a system without chaos would be oughly the same if we changed ou stating position by a small amount. This isn t tue fo chaotic systems, and much of natue is chaotic.

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