chapter 23: the economics of resources for all practical ......chapter 23: the economics of...

Chapter 23: The Economics of Resources

Lesson Plan

Growth Models for Biological

Population

How Long Can a Nonrenewable

Resource Last?

Radioactive Decay

Sustaining Renewable Resources

The Economics of Harvesting

Resources

Dynamical Systems and Chaos

Mathematical Literacy in Today’s World, 9th ed.

For All Practical Purposes

© 2013 W. H. Freeman and Company


Growth Models for Biological Population

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.



Predicting the U.S. Population The U.S. population increased at

an average growth rate of 0.95% per year to 310.5 million at the beginning of 2011. What is the anticipated population at the beginning of 2015?

Answer: Apply the compound interest formula, A = P(1 + r)n with P = 310.5 million, r = 0.0095, and n = 4.

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

4. 2015 310,500,000 1 0.0095

310,500,000 1.0385

322,000,000

Pop in



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

population size PP = rP 1 - = rP 1 -

carrying capacity M


How Long Can a Nonrenewable Resource Last?

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.


How Long Can a Nonrenewable Resource Last?

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:

Where ln is the natural logarithm and can be evaluated easily on a calculator.

years

ln 1 + S/U rn =ln 1 + r


Radioactive Decay

Radioactive materials are characterized by exponential

decay (geometric growth with a negative rate of growth)

Decay Constant

The decay constant, λ, for a substance decaying exponentially is the proportion of the substance that decays per unit of time.

For an original amount P, the amount A remaining after t time

units is:

Half-life

The half-life of a substance is the amount of time that it would

take for one-half of the quantity to decay.

-λtA= Pe

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Radioactive Decay

Decay Constant and Half-life Relationship

For a substance with half-life H and decay constant λ, the following relationship holds:

Example: The half-life of carbon-14 is 5730 years. Find the decay constant λ.

Solution:

ln2 0.693 Hλ

4

ln2

5730 ln2

ln2 0.6931.209 10 .0001209 / yr

5730 5730

Hλ

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.



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



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 xe is the equilibrium population size, for which the population one year later is the same, or

f (xe) = xe



Sustained-Yield Harvesting Policy A policy that if 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 xM.


The Economics of Harvesting Resources

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 (i.e., 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.


The Economics of Harvesting Resources

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.



Dynamical Systems

Useful in modeling systems (such as populations)

Depends largely on previous states

A dynamical system is a mathematical model for a system whose state evolves with time and whose future states depend deterministically on its present and past states.

A system is deterministic if its changes through time depend only on natural and mathematical laws and are not substantially affected by chance or free will (e.g., The path of a golf putt).

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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.

A dynamical system exhibits chaos if it is:

Orbitally dense―any state is near one that eventually will recur

Transitive―from any state you can eventually get close to any other.

Sensitive―a small change in the initial state can produce widely diverging results later. This is known as the butterfly effect.



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.

chapter 23: the economics of resources for all practical ......chapter 23: the economics of...

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