Marginal Analysis

Math 1314 Lesson 9: Marginal Analysis Marginal Cost Suppose a business owner is operating a plant that manufactures a certain product at a known level. Sometimes the business owner will want to know how much it costs to produce one more unit of this product. The cost of producing this additional item is called the marginal cost. Example 1: Suppose the total cost in dollars per week by ABC Corporation for producing its best-selling product is given by C ( x) = 10,000 + 3000 x − 0.4 x 2 . Find the actual cost of producing the 101st item. The cost of producing the 101st item can be found by computing the average rate of change, that C (101) − C (100 ) C ( x + h) − C ( x) is by computing where x = 100 and h = 1. = 101 − 100 ( x + h) − x c.. ( 101) 101 - - c_( too) 100 \ The right hand side of this equation is the formula for average rate of change of the cost function. This will give us the actual cost of producing the next item. However, it is often inconvenient to use. For this reason, marginal cost is usually approximated by the instantaneous rate of change of the total cost function evaluated at the specific point of interest. That is to say, we’ll find the derivative and substitute in our point of interest. Example 2: Suppose the total cost in dollars per week by ABC Corporation for producing its best-selling product is given by C ( x) = 10,000 + 3000 x − 0.4 x 2 . Find C ' (100) and interpret the results. Note that the answers for examples 1 and 2 are very close. This shows you why we can work with the derivative of the cost function rather than the average rate of change. The derivative will be much easier for us to work with. So, we’ll define the marginal cost function as the derivative of the total cost function. You will find that by a marginal function, we mean the derivative of the function. So, the marginal cost function is the derivative of the cost function, the marginal revenue function is the derivative of the revenue function, etc. Lesson 9 – Marginal Analysis 1 Example 3: A company produces noise-canceling headphones. Management of the company has determined that the total daily cost of producing x headsets can be modeled by the function C ( x) = 0.0001x 3 − 0.03 x 2 + 135 x + 15,000 . a. Find the marginal cost function. b. Use the marginal cost function to approximate the actual cost of producing the 25th headset. Average Cost and Marginal Average Cost Suppose C (x ) is the total cost function for producing x units of a certain product. If we divide this function by the number of units produced, x, we get the average cost function. We denote C ( x) this function by C (x ) . Then we can express the average cost function as C ( x ) = . The x derivative of the average cost function is called the marginal average cost. We’ll use the marginal average cost function solely to determine if the average cost function is increasing or if it is decreasing. If C′(a) > 0 , then the average cost function is increasing when x = a . If C′(a) < 0 , then the average cost function is decreasing when x = a . Example 4: A company produces office furniture. Its management estimates that the total annual cost for producing x of its top selling executive desks is given by the function C ( x ) = 400 x + 500,000 . a. Find the average cost function, and the average cost of producing 3000 desks? c. Find the marginal average cost function. Lesson 9 – Marginal Analysis 2 _______ d. Find the C '(2587) e. What happens to C (x ) when x is very large? Marginal Revenue We are often interested in revenue functions, as well. The basic formula for a revenue function is given by R ( x ) = px where x is the number of units sold and p is the price per unit. Often p is given in terms of a demand function in terms of x, which we can then substitute into R (x ) . The derivative of R (x ) is called the marginal revenue function. We will use the derivative of the revenue function as an approximation of the actual revenues realized on the sale of the additional item. Example 5: A company estimates that the demand for its product can be expressed as p = −.04 x + 800 , where p denotes the unit price and x denotes the quantity demanded. a. Find the revenue function. b. Find the marginal revenue function. c. Use the marginal revenue function to approximate the actual revenue realized on the sale of the 4003rd item. * Lesson 9 – Marginal Analysis 3 Example 6: The table below gives pricing information about a company’s best-selling product. Use the data to: Price when x items are demanded 525 505 491 484 475 A. Create a list. Then find the linear demand function. Command: Answer: C. Use the marginal revenue function to approximate the actual revenue realized on the sale of the 1101st item. Marginal Profit The profit function can be expressed as P ( x ) = R ( x ) − C ( x ) , where R (x ) is the revenue function and C (x ) is the cost function. As before, we will find the marginal function by taking the derivative of the function, so the marginal profit function is the derivative of P (x ) . This will give us a good approximation of the profit realized on the sale of the additional unit of the product. - Example 7: A company estimates that the cost to produce x of its products is given by the function C ( x) = 0.000003 x 3 − 0.08 x 2 + 500 x + 250,000 and the demand function is given by p = 600 − 0.8 x . a. Find the profit function. b. Find the marginal profit function. c. Use the marginal profit function to compute the actual profit realized on the sale of the 53rd unit. Example 8: The tables below give the costs associated with the production of x items and pricing information when x items are demanded. Assume that the cost data is exponential and the demand data is linear. x 200 400 600 800 1000 C:::... Cost in dollars of producing x items 134,350 145,480 157,750 173,400 185,000 200 400 600 800 1000 Price when x items are demanded 525 505 491 484 475 Begin by creating two lists in GGB. A. Find the exponential cost function. B. Find the linear demand function. ~~ Answer: Suppose a company’s weekly cost incurred in producing x of its products is given by the function 0.02 0.2 10 800 and the weekly demand for the product is given by p = −0.8x + 200 , where p denotes the unit price. Poppers 1-3 \J 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 25 cents. The marginal propensity to consume is the derivative of a consumption function C ( x ) with respect to income and is denoted 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 ) . know the propensity to consume, you can find the propensity to save, and vice versa. Take the derivative of both sides of the equation S ( x ) = x − C ( x ) , so Example 9: Suppose the consumption function for a country’s economy can be modeled by C ( x) = 0.749 x1.07 + 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