MAT 125 - Applied Calculus 3.4 – Marginal Functions in Economics 4/24/2017 MAT 125 – Applied Calculus 3.4 – Marginal Functions in Economics 3.4 – Marginal Functions in Economics

Today’s Class We will be learning the following concepts today: MAT 125 - Applied Calculus 4/24/2017 Today’s Class We will be learning the following concepts today: Cost Functions Average Cost Functions Revenue Functions Relative Rate of Change Elasticity of Demand 3.4 – Marginal Functions in Economics Dr. Erickson 3.4 – Marginal Functions in Economics

Introduction Marginal Analysis is the study of the rate of change of economic quantities. An economist is not merely concerned with the value of an economy’s gross domestic product (GDP) at a given time, but is equally concerned with the rate at which it is growing or declining. 3.4 – Marginal Functions in Economics Dr. Erickson

Definitions Marginal Cost The actual cost incurred in producing an additional unit of a certain commodity given that a plant is already at a certain level of operation is called marginal cost. Marginal cost is approximated by the rate of change of the total cost function evaluated at the appropriate point. 3.4 – Marginal Functions in Economics Dr. Erickson

Definitions Marginal Cost Function The marginal cost function is the derivative of the corresponding total cost function. In other words, if C is a total cost function, then the marginal cost function is defined to be its derivative C. NOTE: The adjective marginal is synonymous with derivative of. 3.4 – Marginal Functions in Economics Dr. Erickson

Example 1 3.4 – Marginal Functions in Economics Dr. Erickson

Definitions Average Cost The average cost of producing x units of the commodity is obtained by dividing the total production cost by the number of units produced. 3.4 – Marginal Functions in Economics Dr. Erickson

Average Cost Function 3.4 – Marginal Functions in Economics Dr. Erickson

Definitions Marginal Average Cost Function The derivative 𝐶 ′(𝑥) of the average cost function, called the marginal average cost function, measures the rate of change of the average cost function with respect to the number of units produced. 3.4 – Marginal Functions in Economics Dr. Erickson

Example 2 3.4 – Marginal Functions in Economics Dr. Erickson

Revenue Functions Recall that a revenue function R(x) gives the revenue realized by a company from the sale of x units of a certain commodity. If the company charges p dollars per unit and this price is related to the quantity x of the commodity demanded, then p = f (x) and R(x) = px = xf (x) 3.4 – Marginal Functions in Economics Dr. Erickson

Definitions Marginal Revenue The marginal revenue gives the actual revenue realized from the sale of an additional unit of the commodity given that sales are already at a certain level. Marginal Revenue Function We define the marginal revenue function to be R(x), where R is the revenue function. 3.4 – Marginal Functions in Economics Dr. Erickson

Definitions Profit Function The profit function P is given by MAT 125 - Applied Calculus 4/24/2017 Definitions Profit Function The profit function P is given by P(x) = R(x) – C(x) where R and C are the revenue and cost functions and x is the number of units of a commodity produced and sold. Marginal Profit Function The marginal profit function P(x) measures the rate of change of the profit function P and provides us with a good approximation of the actual profit or loss realized from the sales of the (x + 1)st unit of the commodity assuming the xth unit as been sold. 3.4 – Marginal Functions in Economics Dr. Erickson 3.4 – Marginal Functions in Economics

Example 3 3.4 – Marginal Functions in Economics Dr. Erickson

Relative Rate of Change 3.4 – Marginal Functions in Economics Dr. Erickson

Example 4 Find the percentage rate of change of f at the given value of x. 3.4 – Marginal Functions in Economics Dr. Erickson

Elasticity of Demand 3.4 – Marginal Functions in Economics Dr. Erickson

Elasticity of Demand Elastic – E(p) > 1 If the demand is elastic at p, then either a small increase in the unit price will cause the revenue to decrease or a small decrease in the unit price will cause the revenue to increase. What does this mean? If demand is elastic, then the change in revenue and the change in unit price will move in opposite directions. 3.4 – Marginal Functions in Economics Dr. Erickson

Elasticity of Demand Inelastic – E(p) < 1 If the demand is inelastic at p, then either a small increase in the unit price will cause the revenue to increase or a small decrease in the unit price will cause the revenue to decrease. What does this mean? If demand is inelastic, then the change in revenue and the change in unit price will move in the same direction. 3.4 – Marginal Functions in Economics Dr. Erickson

Elasticity of Demand Unitary – E(p) = 1 If the demand is unitary at p, then a small increase or decrease in the unit price will cause the revenue to stay about the same. 3.4 – Marginal Functions in Economics Dr. Erickson

Example 5 For the demand equation, compute the elasticity of demand and determine whether the demand is elastic, unitary, or inelastic at the indicated price. 3.4 – Marginal Functions in Economics Dr. Erickson

Example 6 3.4 – Marginal Functions in Economics Dr. Erickson

Example 7 3.4 – Marginal Functions in Economics Dr. Erickson

Next Class We will discuss the following concepts: MAT 125 - Applied Calculus 4/24/2017 Next Class We will discuss the following concepts: Higher-Order Derivatives Please read through Section 3.5 – Higher-Order Derivatives in your text book before next class. 3.4 – Marginal Functions in Economics Dr. Erickson 3.4 – Marginal Functions in Economics