Analyzing Accumulated Change Integrals in Action 6.1 -Perpetual Accumulation and Improper Integrals Improper Integral Definite integrals have specific numbers for both the upper limit and the lower limit. In this section, what happens to the accumulation of change when one or both of the limits of the integral are infinite is considered. Improper integrals play a role in economics and statistics as well as in other fields of study. 1
Improper Integral Evaluations An improper integral of the form is evaluated by applying a limit: Replace with a variable, N. Evaluate the limit of the integral as N increases without bound, provided the limit exists. Evaluating Improper Integrals where F is an anti-derivative of f. Improper integrals show up in situations in which quantities are being evaluated over an indefinitely long interval. 2
Divergence and Convergence If the limit of an improper integral exists, the improper integral converges. Sometimes the limit of an improper integral does not exist. (The limit increases or decreases without bound.) In this case, improper integral diverges. EXAMPLE 3
Example EXAMPLE: 2, 6, 8, 12, 18, 20 4
Analyzing Accumulated Change Integrals in Action 6.2 - Streams in Business and Biology Income Stream and Flow Rates An income stream is a regular flow of money that is generated by a business or an investment. The rate of flow is a function Rthat varies according to time t. Constant %allocated + profit Linear % allocated (profit ± increment * t) Exponential % allocated (1+ % increase) Exponential % allocated (1 - % decrease) 5
Future Value of a Continuous Income Stream Suppose that an income stream flows continuously into an interest-bearing account at the rate of R(t) dollars per year, where t is measured in years and the account earns interest at 100r% compounded continuously. The future value of the account at the end of T years is Flow Rates and Rates of Change The flow rate for a continuous income stream is not the same thing as the rate of change of that stream. Using the Fundamental Theorem of Calculus, the rate-ofchange function for future value is: The function F (t)= R(x) e r(t-t) gives the rate of change (after t years) of the future value (in T years) of an income stream whose income is flowing continuously in at a rate of per year. The rate-of-change function, rather than the flow rate of the income stream,, is graphed when illustrating future value as the area of a region beneath a rate-of-change function. 6
Present Value of a Continuous Stream The present value of a continuous income stream is the amount P that would need to be invested at the present time so that it would grow to a specified future value under stated conditions. Suppose that an income stream flows continuously into an interest-bearing account at the rate of R(t) dollars per year, where t is measured in years, and that the account earns interest at the annual rate of 100r% compounded continuously. The present valueof the account is Present Value of a Continuous Income Stream (when the future value is known) 7
Perpetual Income Streams When there is no specific end date to an income stream, the stream is considered to flow in perpetuity. For example, when a company leases land rights from its parent corporation to harvest bio-mass from that land for the life of the company, the parent corporation might, in turn, create a fund with the lease money that allows it to purchase more land at some indefinite time. Because there is no definite end date for the income stream, it can be considered to flow in perpetuity. In the case of a perpetual income stream, the end time T is considered to be infinite and the present value of such a stream is calculated as the improper integral Example : 2, 4, 8 8
Streams in Biology Biology and other fields involve situations similar to income streams. An example of this is the growth of populations of animals. Functions that model biological streams where new individuals are added to the population and the rate of survival of the individuals is known are referred to as survival and renewal functions. Future Value of a Biological Stream The future value (in byears) of a biological stream with initial population size P, survival rate 100s% and renewal rate r(t), where tis the number of years, is + b 9
EXAMPLE: 22 Analyzing Accumulated Change Integrals in Action 6.3 -Calculus in Economics Demand and Elasticity 10
Demand function or Demand curve Understanding consumer demand is important in economics, management, and marketing. The amount of a good or service that consumers buy can be considered as a function of the price they have to pay. This function is known as a demand function or demand curve. Demand Functions Demand is affected by several factors, including utility(usefulness), necessity, the availability of substitutes, and buyer income. However, when all other factors are held constant, the quantity demanded can be considered as a function of market price. 11
Demand Function A function giving the expected quantity of a commodity purchased at a specified market price is referred to as a demand function or demand schedule. The demand function Drelates the input variable p (price per unit) with the output variable q = D(p) (quantity). The graph of a demand function is referred to as a demand curve. The Law of Demand All other factors being constant, as the price of a commodity increases, the market will react by demanding less and, as the price of a commodity decreases, the market will react by demanding more. 12
Why does my economics text show demand as a function of quantity? In economic theory, many functions (such as cost, revenue, and profit) are graphed with quantity on the horizontal axis. The demand curve is likewise drawn with quantity on the horizontal axis. Reading the economics text carefully can reveal whether price or quantity is considered to be the input variable. If quantity is considered to be the input variable, the economics text is dealing with the inverse demand function. If price is considered to be the input variable, the economics text is simply drawing the demand function on a reversed set of axes to keep quantity on the horizontal axis. Consumer Expenditure Consumer expenditure is the price per unit of a commodity times the quantity purchased by consumers. Assuming that demand is satisfied, the quantity purchased is the same as the quantity in demand. This expenditure is represented graphically as the area of a rectangle under the demand curve. 13
For a commodity with demand function D and price per unit p, when market price is fixed at p 0, the quantity demanded is written q 0 = D(p 0 ) and Consumer expenditure = p 0. q 0 Consumer Surplus Consumer surplus is the amount that consumers are willing and able to spend but do not actually spend because the market price was fixed at p0. This is the amount that consumers have in excess from not having to spend as much as they were willing and able. 14
Consumer Willingness and Ability to Spend When consumers demand a certain quantity of a commodity, they are willing and able to pay more (as an aggregate) than they actually spend at the correlating market price. Consumer willingness and ability to spend is the total amount of money consumers have available and are willing to pay to obtain a certain quantity of a commodity. Consumer Willingness and Ability to Spend For a continuous demand function D, the amount that consumers are willing and able to spend for a certain quantity of a commodity is given by Consumer Expenditure Consumer Surplus 15
Consumer Willingness and Ability to Spend where p 0 is the market price at which q 0 units are in demand, and p max is the price above which consumers will purchase none of the commodity. (If the demand function approaches but does not cross the input axis, the integral is improper with an upper limit of.) Example: 12, 14, 24 16
Price Elasticity of Demand Elasticity is a measure of the responsiveness of a function s output to a change in its input variable. Because the measures of the quantities of output and input are normally very different, elasticity uses a ratio of percentage rates of change to compare relative changes. For a commodity with differentiable demand function D and price per unit p, the price elasticity of demand is Demand is elastic when ƞ > 1 and inelastic when ƞ < 1. Demand is at unit elasticity when ƞ = 1. Price elasticity of demand is normally negative Example: 28 17
Analyzing Accumulated Change Integrals in Action 6.4 -Calculus in Economics Supply and Equilibrium Calculus in Economics Supply and Equilibrium Supply Function A function giving the expected quantity of a commodity supplied at a specified market price is referred to as a supply function or supply schedule. The supply function S relates the input variable p(price per unit) with the output variable q = S(p) (quantity). The graph of a supply function is referred to as a supply curve. 18
Shutdown The shutdown price, p s, is the lowest market price that producers are willing and able to accept to supply any quantity of a certain commodity. The shutdown point is the point (p s, S(p s ))on the supply curve that marks the conditions (market price and quantity)under which the production of a commodity will shut down. Producer Revenue Producer revenue is the price per unit of a commodity times the quantity supplied to the market (assuming the entire quantity is also purchased at market price). Producer revenue is represented graphically as the area of a rectangle with one corner at the origin (0, 0) and the opposite corner on the supply curve at point (p 0, q 0 ) where p 0 is the market price. Producer revenue = p 0. q 0 19
Producer Willingness and Ability to Receive For a continuous supply function S, the minimum amount producers are willing and able to receive for a certain quantity of a commodity is given by where p 0 is the market price at which q 0 units are supplied, and p s is the shutdown price.(if there is no shutdown price, p s =0.) Producer Surplus The amount that producers receive in excess of the minimum amount that they require to supply a certain quantity of a commodity is known as producer surplus. Producer surplus benefits society as a whole because producers are able to use surplus to develop better or different products, invest in other economic sectors, or pass it on to their employees as money or added benefits. 20
Producer Surplus Producer surplus is calculated as producer revenue minus producer willingness and ability to receive: Producer revenue Producer willingness to pay Producer surplus = Equilibrium and Social Gain The market price and quantity (p*, q*) at which supply equals demand is called the equilibrium point. At the equilibrium price p*, the quantity demanded by consumers coincides with the quantity supplied by producers. This quantity is q*. Society benefits when consumers and/or producers have surplus funds. 21
total social gain When the market price of a product is the equilibrium price for that product, the total benefit to society is the consumers surplus plus the producer surplus. This amount is known as the total social gain. EXAMPLE: 10, 16, 22, 24 22