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Chapter 23: The Economics of Resources Lesson Plan Growth Models for Biological Population How Long Can a Nonrenewable Resource Last? Sustaining Renewable Resources The Economics of Harvesting Resources Dynamical Systems and Chaos 1 Mathematical Literacy in Today’s World, 7th ed. For All Practical Purposes © 2006, W.H. Freeman and Company

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Chapter 23: The Economics of Resources Growth Models for Biological Population 2 Growth Models Geometric growth model is used to make rough estimates about sizes of human populations. Birth, death, and migration rates rarely remain constant for long, so projections must be made with care. Using the model for short-term projections may be useful. Rate of Natural Increase The annual birth rate minus the annual death rate.

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Chapter 23: The Economics of Resources Growth Models for Biological Population 3 Predicting the U.S. Population The U.S. population is increasing at an average growth rate of 1.0% per year to an expected total of 301 million by mid-2007. What is the anticipated population mid- 2011? Answer: Apply the compound interest formula, A = P(1 + r) n with P = 301 million, r = 0.01, n = 4 A = P (1 + r) n = 301 million (1 + 0.01) 4 = 301,000,000 (1.01) 4 = 313,000,000 Population in 2011 is rounded off and estimated to be 313 million. Using Compound Interest Formula for Population Growth – A = P (1 + r) n Where: A = Amount owed after interest is added (future population amount after growth rate) P = Principal amount (initial population) r = Interest (growth) rate n = Years where r (growth) rate is applied ~

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Chapter 23: The Economics of Resources Growth Models for Biological Population 4 Comparing Growth Rates in Different Nations Rate of increase of U.S. was estimated at 1.0% The rates of increase in most developing nations are much higher than in industrialized nations. Nigeria, Africa’s most populous country, has an estimated growth rate of 2.8%, and will grow from 135 million in mid-2007 to 231 million in mid-2030. This is an increase of more than two-thirds within this period. These projections raise concern over providing sufficient food and resources for all people. Also, population structure (subgroups of the population) may change in poorer countries. It is likely the life expectancy of poorer nations will increase, resulting in the proportion of the older population to increase. In poorer countries, the proportion of the population over 60 years of age will be 20% by 2050, compared with 8% now.

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Chapter 23: The Economics of Resources Growth Models for Biological Population 5 Limitations on Growth Population growth is eventually constrained by the availability of resources such as food, shelter, and psychological and social “space.” Carrying capacity of the environment is the term for the maximum population size that can be supported by the available resources. Logistic Model A particular population model that begins with near-geometric growth but then tapers off toward a limiting population (the carrying capacity). The logistic model reduces the annual increase r P by a factor of how close the population size P is to the carrying capacity M: Growth rate P ′ = r P ( 1 – ) = r P ( 1 – ) population size carrying capacity P M

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Chapter 23: The Economics of Resources How Long Can a Nonrenewable Resource Last? 6 Nonrenewable Resource A resource that does not tend to replenish itself and of which only a fixed supply S is available. Important examples include gasoline, coal, and natural gas. Example: How long will the supply of a resource last? As long as the rate of use of the resource remains constant; say for example, we use a constant U units per year, then the supply will last S/U years. Example: U.S. recoverable coal reserves will last 250 years. For most resources, its consumption or rate of use tends to increase with population and with a higher standard of living. To determine how long the supply will last, we can manipulate the savings formula (similar to making regular withdrawals [with interest] from a fixed supply of the nonrenewable resources). The terms we use to describe this calculation are referred to as the static reserve and the exponential reserve.

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Chapter 23: The Economics of Resources How Long Can a Nonrenewable Resource Last? 7 Static Reserve Static reserve is how long the supply S will last at a particular constant annual rate of use U. The length of time that a static reserve will last is S/U years. Exponential Reserve Exponential reserve is how long the supply S will last at an initial rate of the use U that is increasing by a proportion r each year. This length of time (the number of years, n) is determined by evaluating the following expression: ln [1 + (S/U) r ] ln (1 + r ) years Where ln is the natural logarithm and can be evaluated easily on a calculator. n =

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Chapter 23: The Economics of Resources Sustaining Renewable Resources 8 Renewable Natural Resource A resource that tends to replenish itself (fish, wildlife, and forests). We would like to know how much we can harvest and still allow for the resource to replenish itself. We concentrate on the subpopulation with commercial value. We measure the population size as its biomass. Biomass is the physical mass of the population and is written in common units of equal value. Examples of biomass can be measuring fish in pounds rather than in number of fish and in forests counting the number of board feet of usable timber rather than the number of trees. Reproduction Curves A curve that shows population size in the next year plotted against population size in the current year. We use this curve to predict next year’s population size (biomass) based on this year’s size.

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Chapter 23: The Economics of Resources Sustaining Renewable Resources 9 Reproduction Curves For each x-value (the point on the horizontal axis that represents this year’s population), a corresponding y-value exists that is the height of the curve at that x- value. The y-value is a function of x f (x) and is the prediction of what next year’s population will be, which takes into account the following: Continuing members Addition of new members Minus losses due to death and other factors

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Chapter 23: The Economics of Resources Sustaining Renewable Resources 10 Equilibrium Population Size Equilibrium population size does not change from year to year. The harvest yield depends on both the population x (of fish, lumber, etc.) and the amount of effort to harvest the population. The population size x e is the equilibrium population size, for which the population one year later is the same, or f (x e ) = x e.

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Chapter 23: The Economics of Resources Sustaining Renewable Resources 11 Sustained-Yield Harvesting Policy A policy that is continued indefinitely will maintain the same yield. For a sustainable yield, the same amount is harvested every year and the population remaining after each harvest is the same. To achieve this stability, the harvest must exactly equal the natural increase each year, the length of the green vertical line. Maximum Sustainable Yield The maximum sustainable yield is obtained by selecting an x- value whose colored vertical line is as long as possible, marked x M.

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Chapter 23: The Economics of Resources The Economics of Harvesting Resources 12 Harvesting Resources We assume that the price p received is the same for each harvested unit and does not depend on the size of our harvest. We assume that our operation is a small part of the total market and does not affect the overall supply and hence price. Example: Cattle Ranching We assume that the cost of raising and bringing a steer to market is the same for every steer and does not depend on how many steers we bring to market (that is, there is no volume discount). As long as the selling price per unit is higher than the harvest cost per unit, we make a profit.

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Chapter 23: The Economics of Resources The Economics of Harvesting Resources 13 Harvesting Resources Example: Fishing and Logging We assume that the cost of harvesting a unit of the population decreases as the size of the population increases—this is the principle of economy of scale (similar to volume discounts). For example, the same fishing effort may yield more fish when the fish just happen to be more abundant. Also, the logger’s costs per tree are less when the trees are clumped together. Above a certain population size where the unit cost curve intersects the unit price curve, a profit is possible.

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Chapter 23: The Economics of Resources Dynamical Systems of Chaos 14 Dynamical System Various dynamical systems provide useful models for physical, biological, chemical, financial, and other phenomena. The system’s state depends only on its states at previous times. Mathematical Chaos Chaos is generally confusion, unpredictability, and apparent randomness. Mathematicians and other scientists use the word to describe systems whose behavior over time is inherently unpredictable. Three main features of mathematical chaos: Determinism – Future behavior of the system is completely determined by its present state, its past history, and known laws; chance is not involved. Complex behavior Sensitivity to initial conditions – A small change now can make a big difference later. This is known as the butterfly effect.

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Chapter 23: The Economics of Resources Dynamical Systems of Chaos 15 Mathematical Chaos Butterfly Effect A small change in initial conditions of a system can make an enormous difference later on. Iterated Function Systems (IFS) A sequence of elements (numbers or geometric objects) in which the next element is produced from the previous one according to a function rule. It just means that we take an initial value, apply a function to it, then repeat it over and over. This is exactly what we did earlier, geometrically, with reproduction curves for population.

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