Chapter 3.4 Notes-Marginal Analysis and Economics. (1) Cost Functions

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1 Chapter 3.4 Notes-Marginal Analysis and Economics (1) Cost Functions (2) Revenue Functions (3) Profit Functions (4) Elasticity of Demand Marginal analysis is the study of the rate of change of economic quantities, i.e. the increase or decrease in the rate of GDP or total cost production. It is extremely useful in real life applications. (1) Cost Functions DEFN: The cost (sometimes known as the marginal cost) is the actual cost in producing an additional unit of a certain product given that the company is already at a certain level or operation. If you want to know how much it cost to produce the a T H unit after have already produced a 1 units then the actual cost is: C(a) C(a 1) where C(x) is the cost function. Note that in words we have: C(a) is the total amount it cost to produce a units, C(a 1) is the total amount it cost to produce a 1 units, SO C(a) C(a 1) is how much it cost to produce the a T H unit (given you have already produced a 1). DEFN: The marginal cost function is the derivative of the total cost function. The derivative (marginal cost function) gives an estimation to the actual cost. If you want to approximate how much it cost to produce the a T H unit after have already produced a 1 units then the approximate cost is: C (a 1) where C(x) is the cost function. **NOTICE: you plug in a 1 into the marginal cost function. This is because we are approximating the actual cost (which is a slope) using the derivative of the point before: C(a) C(a 1) = C(a) C(a 1) a (a 1) C (a 1) 1

2 Example 1. Let C(x) be the cost of producing x items. C(x) = x x 2 a) Find the marginal cost function. C (x) = x. b) Find the marginal cost when producing x = 99 items C (99) = (99) = ***What does this tell us? It cost about $3.86 to produce the 100 th Furby assuming 99 of them were already produced. c) Find the actual cost in producing the 100 th item. C(100) C(99) = $3.88. DEFN: The average cost function, denoted C(x), is C(x) x where C(x) is the cost function. DEFN: The marginal average cost function is the derivative of the average cost function. It measures the rate of change of the average cost function with respect to the number of units produced, C (x). Example 2. Let C(x) be the cost function of producing x computers. a) Find the average cost function. C(x) = x C(x) = C(x) x = 7500 x b) Find the marginal cost function. C (x) = ( ) 7500 x + 20 = 7500 x 2 2

3 (2) Revenue Functions: DEFN: If a company give the demand equation p = f(x), then the revenue function is given by Note: R(x) is selling price quantity. R(x) = p x = x f(x). DEFN: The revenue (sometimes known as the marginal revenue) gives the actual revenue realized from the sale of an additional unit given that the sales are already at a certain level. If you want to know how much you make selling the a T H unit after have already sold a 1 units then the actual revenue is: R(a) R(a 1) where R(x) is the revenue function. Note that in words we have: R(a) is the total amount made from selling a units, R(a 1) is the total amount made from selling a 1 units, SO R(a) R(a 1) is how much you make from selling the a T H unit (given you have already sold a 1). DEFN: The marginal revenue function is the derivative of the revenue function, R (x). derivative (marginal revenue function) gives an estimation to the actual revenue. The If you want to approximate how much you make selling the a T H unit after have already sold a 1 units then the approximate revenue is: R (a 1) where R(x) is the cost function. **NOTICE: you plug in a 1 into the marginal revenue function. This is because we are approximating the actual revenue (which is a slope) using the derivative of the point before: R(a) R(a 1) = R(a) R(a 1) a (a 1) R (a 1) Example 3. Let f(x) = p be the demand equation of chocolate bars, that is, x = number of chocolate bars and p = price in dollars. p = 0.01x a) Find the revenue function b) Find the marginal revenue function. R(x) = p x = ( 0.01x + 20)x = 0.01x x. R (x) = 0.02x

4 c) Compute R (50). Interpret your results. R (50) = 0.02(50) + 20 = 19 That says that you make approximately $19 selling the 51 st chocolate bar (assuming you have already sold 50). (3) Profit Functions As we have seen, the profit function is given by P (x) = R(x) C(x) where R(x) is the revenue function and C(x) is the cost function. If you want to know your profit from making and selling the a T H unit after have already made and sold a 1 units then the actual profit is: where P (x) is the revenue function. P (a) P (a 1) DEFN: The marginal profit function is the derivative of the profit function, P (x). The derivative (marginal profit function) gives an estimation to the actual profit or loss realized. If you want to approximate how much your profit is from making and selling the a T H unit after have already made and sold a 1 units then the approximate profit is: where P (x) is the cost function. P (a 1) Example 4. The monthly profit of yoga pants is given by P (x) = 2x x a) What is the actual profit when selling the 101 st item assuming that 100 have already been sold? P (101) P (100) = b) Compute the marginal profit when x = 100. Interpret your results. P (x) = 4x = P (100) = This is saying that your profit in making and selling the 101 th yoga pant (assuming you have already made and sold 100) is $2, which is a lot for one pair of yoga pants... maybe they re from Lululemon or something... 4

5 (4) Elasticity of Demand Economists use elasticity of demand to analyze the demand function. DEFN: If f is a differentiable demand function defined by x =, then the elasticity of demand at price p is given by DEFN: The demand is said to be 1. elastic if E(p) > 1, 2. unitary if E(p) = 1 and 3. inelastic if E(p) < 1. Theorem: 1. If demand is elastic, E(p) > 1, then in price = in revenue and in price = in revenue. 2. If demand is unitary, E(p) = 1, then a change in unit price will cause the revenue to stay about the same. 3. If demand is inelastic, E(p) < 1, then in price = in revenue and in price = in revenue We can think of elastic, inelastic, and unitary in some examples: 1. A good example of an elastic good is Coke and Pepsi. If Coke increases their prices by too much, a lot of soda drinkers will switch to the lower priced Pepsi and Coke s revenue will decrease. 2. If I had a son and gave him $20 to buy candy, he will spend $20 on candy, no matter the price of the candy (aka, he ll go to the clerk in the store and say: please give me the amount of candy I can afford for $20). In this case, his demand has unit elasticity: no matter the price, he will spend the $20. If the price goes up by 10%, he will just buy 10% fewer pieces of candy and hence, the revenue remains about the same. 3. A good example of an inelastic commodity is gas. No matter how high gas prices are, the quantity demanded won t decrease because people still need gas and so, an increase in price will give an increase in revenue. 5

6 Example 5. The demand equation for KU basketball tickets is given by x = 201 3p for 0 p 67 where p is price in dollars and x is number of tickets. (a) Compute the elasticity of demand when p = 4, p = 134 3, and p = 47. Since, we need to know f (p) where = x = (201 3p) 1/2. f (p) = 1 2 (201 3p)1/2 3 = p. Then Now, = p p 201 3p = 3p 2(201 3p). E(4) = = inelastic at p = 4. ( ) 134 E = 1 = unitary at p = E(47) = = elastic at p = 47. (b) Without using part (a), find the price p that makes the elasticity of demand unitary. To be unitary we need E(p) = 1. So = 3p 402 = 1 = 3p = 2(201 3p) = 3p = 402 6p = 9p = 402 = p = 2(201 3p) 9 6