The marginal propensity to consume is the derivative of a consumption function Cxwith ( )

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1 Marginal Propensity to Consume/Save The marginal propensity to consume is the increase (or decrease) in consumption that an economy eperiences when income increases (or decreases). The marginal propensity to consume is a measure of what consumers will do when they have additional income. Suppose the marginal propensity to consume is Then for every dollar of increase in income, consumers will spend 75 cents and save 25 cents. The marginal propensity to consume is the derivative of a consumption function Cwith ( ) respect to income and is denoted dc d. The flip side of the marginal propensity to consume is the marginal propensity to save. Consider that there are only two possibilities with respect to the additional income the consumer can either save it or spend it. We can epress a propensity to save function as S( ) where represents income. Then S( ) C( ). ds dc Take the derivative of both sides of the equation S( ) C( ), so 1. Once you d d know the propensity to consume, you can find the propensity to save, and vice versa. Eample 9: Suppose the consumption function for a country s economy can be modeled by 1.07 C ( ) , where C ( ) and are both measured in billions of dollars. Find the marginal propensity to consume and the marginal propensity to save when 7. Note that 7 in this problem represents $7,000,000,000. If we substitute 6 instead of 7 into the derivative, we are actually subtracting $1 billion. The difference between the final answer using $7,000,000,000 and using $6,999,999,999 is negligible, so in the case of marginal propensity to save/consume, we do not reduce the number of interest by 1 before substituting. Lesson 9 Marginal Analysis 6

2 Poppers 1-3 Suppose a company s weekly cost incurred in producing of its products is given by the function and the weekly demand for the product is given by p.8 200, where p denotes the unit price. 1. Find the marginal cost function a b c d Find the average cost function a b c d Find the revenue function. a b c d Lesson 9 Marginal Analysis 7

3 Math 1314 Lesson 10 Elasticity of Demand Suppose you owned a small business and needed to make some decisions about the pricing of your products. It would be helpful to know what effect a small change in price would have on the demand for your product. If a price change will have no real change on demand for the product, it might make good sense to raise the price. However, if a price increase will cause a big drop in demand, then it may not be a good idea to raise prices. There is a measure of the responsiveness of demand for product or service to a change in its price: elasticity of demand. This is defined as percentage change in demand percentage change in price. To develop this formula, we ll start by solving our demand function for, so that we have a function f( p). Then we have a demand function in terms of price. If we increase the price by h dollars, then the price is p h and the quantity demanded is f ( p h). f ( p h) f( p) The percentage change in demand is 100 and the percentage change in price is f( p) 100 h. p If we compute the ratio given above, we have 100 f ( p h) f( p) f( p) h 100 p. We can simplify this to p f( p h) f( p) f( p) h. For small values of h, f( p h) f( p) h f ( p), so we have p f ( p). f ( p) This quantity is almost always negative, and it would be much easier to work with a positive quantity. So the negative of this ratio is the elasticity of demand. Elasticity of Demand p f ( p) E( p), where p is price and f ( p) is the demand function and is differentiable at f ( p) p. Lesson 10 Elasticity of Demand 1

4 Revenue responds to elasticity in the following manner: If demand is elastic at p, then An increase in unit price will cause revenue to decrease or A decrease in unit price will cause revenue to increase If demand is unitary at p, then An increase/decrease in unit price will cause the revenue to stay about the same. If demand is inelastic at p, then An increase in the unit price will cause revenue to increase A decrease in unit price will cause revenue to decrease. We have these generalizations about elasticity of demand: Demand is said to be elastic if E( p) 1. Demand is said to be unitary if E( p) 1. Demand is said to be inelastic if E( p) 1. So, if demand is elastic, then the change in revenue and the change in price will move in opposite directions. If demand is inelastic, then the change in revenue and the change in price will move in the same direction. Eample 1: Find E p for the demand function 2p 15 0 and determine if demand is elastic, inelastic or unitary when p 4. Lesson 10 Elasticity of Demand 2

5 Eample 2: Suppose the demand function for a product is given by p This function gives the unit price in dollars when units are demanded. a. Find the elasticity of demand. b. Find E (100) and interpret the results. c. Find E (300) and interpret the results. d. If the unit price is $100, will raising the price result in an increase in revenues or a decrease in revenues? e. If the unit price is $300, will raising the price result in an increase in revenues or a decrease in revenues? Lesson 10 Elasticity of Demand 3

6 What else can E( p ) tell you? Eample 3: If revenue? 1 E( p) when p 250, what effect will a 1% increase in price have on 2 Eample 4: If revenue? 3 E( p) when p 250, what effect will a 1% increase in price have on 2 Popper 5: If demand is elastic at p, then a decrease in unit price will cause revenue to increase. a. True b. False Lesson 10 Elasticity of Demand 4

7 Math 1314 Lesson 11: Eponential Functions as Mathematical Models or Eponential models can be written in two forms: f a e g a b. In the first model, the base of the eponent is the number e, which is approimately In the second model, the base of the eponent is a positive number other than 1. In both cases, the variable is located in the eponent, and that s why these are called eponential models. Eponential functions can be either increasing or decreasing. b The function f a e is an eponential growth function and it s increasing. The function f a e b is an eponential decay function and it s decreasing. In both equations the variable a is called the initial amount and b is called the growth or decay constant, depending on the type of function. g a b, the function is an eponential growth function if b 1 and is an eponential decay function if 0 b 1. For a function of the form We can also compute the rate at which an eponential function is increasing or decreasing. We ll do this by finding a numerical derivative. We can use the regression feature to find an eponential equation for data that s given. Popper 28 is D Eample 1: Identify each function as a growth function or a decay function. Find the initial value. Calculate f (30) and f '(30) t a. f( t) 12.8e b. g ( ) 38.6(1.0489) Eample 2: Suppose the points (0, 10000) and (2, 10940) lie on the graph of a function. Find b the equation of the function in the form f a e using GeoGebra, assuming that the function is eponential. Begin by entering the two given points in the spreadsheet, then make a list. Lesson 11 Eponential Functions as Mathematical Models 1

8 Uninhibited Eponential Growth Some common eponential applications model uninhibited eponential growth. This means that there is no upper limit on the value of the function. It can simply keep growing and growing. Problems of this type include population growth problems and growth of investment assets. Eample 3: A biologist wants to study the growth of a certain strain of bacteria. She starts with a culture containing 25,000 bacteria. After three hours, the number of bacteria has grown to 63,000. Assume the population grows eponentially and the growth is uninhibited. a. Find the equation of the function in the form State the two points given in the problem. b f a e using GGB. Enter the two points in the spreadsheet, then make a list. b. How many bacteria will be present in the culture 6 hours after she started her study? Popper 13: Given the following function, state the initial value for the function a b Lesson 11 Eponential Functions as Mathematical Models 2

9 Eample 4: The sales from company ABC for the years are given below. Year profits in millions of dollars Rescale the data so that = 0 corresponds to Begin by making a list. a. Find an eponential regression model for the data. b. Find the rate at which the company's sales were changing in Eponential Decay Eample 5: At the beginning of a study, there are 50 grams of a substance present. After 17 days, there are 38.7 grams remaining. Assume the substance decays eponentially. a. State the two points given in the problem. Enter the two points in the spreadsheet and make a list. b. Find an eponential regression model. c. What will be the rate of decay on day 40 of the study? Eponential decay problems frequently involve the half-life of a substance. The half-life of a substance is the time it takes to reduce the amount of the substance by one-half. Lesson 11 Eponential Functions as Mathematical Models 3