Marginal Propensity to Consume/Save The marginal propensity to consume is the increase (or decrease) in consumption that an economy experiences 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 0.75. Then for every dollar of increase in income, consumers will spend 75 cents and save 5 cents. The marginal propensity to consume is the derivative of a consumption function C( x) with respect to income and is denoted dc dx. 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 express a propensity to save function as S( x) where x represents income. Then S( x) = x C( x). ds dc Take the derivative of both sides of the equation S( x) = x C( x), so = 1. Once you dx dx know the propensity to consume, you can find the propensity to save, and vice versa. Example 9: Suppose the consumption function for a country s economy can be modeled by 1.07 C( x) = 0.749x + 34.759, where C( x) and x are both measured in billions of dollars. Find the marginal propensity to consume and the marginal propensity to save when x = 7. Note that x = 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 7
Math 1314 Test Review Lessons 8 3 1. Given f ( x) = x x. A. Find any zeros of f. B. Find any local (relative) extrema of f. C. Find f '( 0.5) and f ''( 0.5). Given f ( x) = x e + 3x x 1. A. Find any zeros of f. B. Find any extremum of f. 1314 Test Review 1
3. The following table of values gives a company s annual profits in millions of dollars. Rescale the data so that the year 003 corresponds to x = 0. Create a list of points. Year 003 004 005 006 007 008 Profits (in millions of dollars) 31.3 3.7 31.8 33.7 35.9 36.1 A. Find the cubic regression model for the data. B. Find the R value for the cubic regression model. C. Use the cubic regression model to predict the company's profits in 010. D. Find the exponential regression model for this data. 4. The graph of f ( x ) is shown below. A. x 4 B. + x 4 C. x 4 1314 Test Review
5. Suppose x 5, x 1 f x x x ( ) = 8, 1 < 3 + > x 1, x Determine, if they exist, A. B. C. + x 1 x 1 x 1 6. 4x lim x 1 x 5 7. x + 3 lim x 4 x 8 x + 5x + 6 8. lim x x + 9. 10x x lim x 3 4 x 10. 3 lim x + 5x 7x 1 x 7 4 + x x 1314 Test Review 3
3 x 1 11. lim x x 7x Enter the function into GGB. Look at the graph to determine your answer. 1. The graph of f ( x ) is shown below. Which of the following statements is true? I. x exists and is equal to 3. II. x 5 exists and is equal to 3. III. IV. x 6 x does not exist. does not exist; there is a hole where x =. V. does not exists; there is unbounded behavior as x approaches 4. x 4 1314 Test Review 4
13. The graph of f ( x ) is shown below. Which of the following statements is true? I. The function is continuous at x = 3. II. The function is discontinuous at x = 3 because x 3 does not exist. III. The function is discontinuous at x = 3 because f(3) does not exist. IV. The function is discontinuous at x = 3 because even though f(3) exists and exists, the two quantities are not equal. x 3 14. Find the first and second derivative: f x = x x + x + x 4 3 ( ) 5 3 8 7 1 4 3x 5 x 15. Let f ( x) = x ln( x 1) + e A. Find the slope of the tangent line at x = 3. B. Write the equation of the tangent line at the given point. 1314 Test Review 5
16. Find the average rate of change of f ( x) = 0.8x 0.11x on the interval f ( x + h) f ( x) [ 1.5, 4 ]. Recall: = average rate of change/difference quotient h 17. The model gives the number of bacteria in a culture t hours after an experiment begins. What will be the bacteria population 6 hours after the experiment begins? 18. A country s gross domestic product (GDP) in billions of dollars, t years from now, is projected to be for 0 t 5. What will be the rate of change of the country s GDP years from now? 19. A ball is thrown upwards from the roof of a building at time t = 0. The height of the ball in feet is given by, where t is measured in seconds. Find the velocity of the ball after 3 seconds. 0. Suppose a manufacturer has monthly fixed costs of $50,000 and production costs of $4 for each item produced. The item sells for $40. Assume all functions are linear. State the: A. cost function. B. revenue function. C( x) = cx + F R( x) = sx c = cost/unit; F = fixed costs s = selling price C. profit function. P( x) = R( x) C( x) 1314 Test Review 6
D. Find the break-even point. Recall: R( x) = C( x) 1. Cost data and demand data for a company's best-selling product are given in the tables below. Create two lists. Quantity produced 1,000,000 3,000 4,000 Total cost $13,400 $14,00 $14,900 $15,400 Quantity demanded 1,000,000 3,000 4,000 Price in dollars $10.75 $10.15 $9.85 $9.70 A. Find linear regression model for cost. B. Find the linear regression model for demand. Then find the revenue function. Linear Demand Equation: Linear Revenue Equation: Recall: R( x) = px D. Use the linear cost and revenue function to find the number of items that must be sold to break even on that product. Round your answer to the nearest unit. 1314 Test Review 7
. Suppose that a company has determined that the demand equation for its product is 5x + 3p 30 = 0 where p is the price of the product in dollars when x of the product are demanded (x is given in thousands). The supply equation is given by 5x 30 p + 45 = 0, where x is the number of units that the company will make available in the marketplace at p dollars per unit. Find the equilibrium quantity and price. The following formulas will be provided with Test. It will be a link. f ( x + h) f ( x) f ( b) f ( a) = h b a C( x) = cx + F R( x) = sx or R( x) = xp P( x) = R( x) C( x) 1314 Test Review 8