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3.4 Marginal Functions in Economics Marginal Analysis • 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. – A manufacturer is not only interested in the total cost of corresponding to a certain level of production of a commodity, but also is interested in the rate of change of the total cost with respect to the level of production. Supply • In a competitive market, a relationship exists between the unit price of a commodity and the commodity’s availability in the market. • In general, an increase in the commodity’s unit price induces the producer to increase the supply of the commodity. • The higher unit price, the more the producer is willing to produce. Supply Equation • The equation that expresses the relation between the unit price and the quantity supplied is called a supply equation defined by p = f ( x ) . • In general, p = f ( x ) increases as x increases. Demand • In a free-market economy, consumer demand for a particular commodity depends on the commodity’s unit price. • A demand equation p = f ( x ) expresses the relationship between the unit price p and the quantity demanded x. • In general, p = f ( x ) decreases as x increases. • The more you want to buy, the unit price should be less. Cost Functions • The total cost is the cost of operating a business. Usually includes fixed costs and variable costs. • The cost function C(x) is a function of the total cost of operating a business. • The actual cost incurred in producing an additional unit of a certain commodity given that a plant is already at a level of operation is called the marginal cost. Rate of Change of Cost Function Suppose the total cost in dollars incurred each week by Polaraire for manufacturing x refrigerators is given by the total cost function C ( x ) = 8000 + 200 x − 0.2 x 2 (0 ≤ x ≤ 400) a. What is the actual cost incurred for manufacturing the 251st refrigerator? b. Find the rate of change of the total cost function with respect to x when x = 250 . a. the actual cost incurred for manufacturing the 251st refrigerator is C (251) − C (250) [ ] − [8000 + 200(250) − 0.2(250) ] = 8000 + 200(251) − 0.2(251) 2 2 = 45,599.8 − 45,500 = 99.80 b. The rate of change is given by the derivative C ' ( x ) = 200 − 0.4 x Thus, when the level of production is 250 refrigerators, the rate of change of the total cost is C ' (250) = 200 − 0.4(250) = 100 • Observe that we can rewrite C (251) − C (250 ) C (251) − C (250) = 1 C (250 + 1) − C (250 ) C (250 + h ) − C (250) = = 1 h • The definition of derivative tells us that C (250 + h ) − C (250 ) C ' (250) = lim h →0 h • Thus, the derivative C' ( x ) is a good approximation of the average rate of change of the function C ( x ) . Marginal Cost Function • The marginal cost function is defined to be the derivative of the corresponding total cost function. • If C ( x ) is the cost function, then C' ( x ) is its marginal cost function. • The adjective marginal is synonymous with derivative of. Revenue Functions • 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, then R ( x ) = px . • The demand function p = f ( x ) tells the relationship between p and x. Thus, R ( x ) = xf ( x ) Marginal Revenue Functions • The marginal revenue function gives the actual revenue realized from the sale of an additional unit of the commodity given that sales are already at a certain level. • We define the marginal revenue function to be R' ( x ) . Profit Functions • The profit function 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. • The marginal profit function P' ( x ) measures the rate of change of the profit function and provides us with a good approximation of the actual profit or loss realized from the sale of the additional unit of the commodity. Average Cost Function • The average cost of producing units of the commodity is obtained by dividing the total production cost by the number of units produced. • The average cost function is denoted by C ( x ) and defined by C (x ) x • The marginal average cost function C' ( x ) measures the rate of change of the average cost. The weekly demand for the Pulser 25 color LED television is p = 600 − 0.05 x (0 ≤ x ≤ 12,000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing the Pulser 25 is given by 3 2 ( ) C x = 0.000002 x − 0.03x + 400 x + 80,000 where C(x) denotes the total cost incurred in producing x sets. a. Find the revenue function R and the profit function P. R ( x ) = px = (600 − 0.05 x )x = 600 x − 0.05 x 2 P ( x ) = R ( x ) − C ( x ) = (600 x − 0.05 x 2 ) − (0.000002 x − 0.03x + 400 x + 80,000) 3 2 = −0.000002 x − 0.02 x + 200 x − 80,000 3 2 b. Find the marginal cost function, the marginal revenue function, and the marginal profit function. C ' ( x ) = 0.000006 x − 0.06 x + 400 2 R' ( x ) = 600 − 0.1x P' ( x ) = −0.000006 x − 0.04 x + 200 2 c. Compute C ' (2000) , R' (2000 ) , and P' (2000 ) and interpret your results. • • • Since C ' (2000 ) = 304 , the cost to manufacture the 2001st LED TV is approximately 304. Since R ' (2000 ) = 400 , the revenue increased by manufacturing the 2001st LED TV is approximately 400. Since P ' (2000 ) = 96 , the profit for manufacturing and selling the 2001st LED TV is approximately 96. Elasticity of Demand • Question: when you produce more commodities, do you actually get the more revenue? • It is convenient to write the demand function f in the form x = f ( p ) ; that is, we will think of the quantity demanded of a certain commodity as a function of its unit price. • Usually, when the unit price of a commodity increases, the quantity demanded decreases Suppose the unit price of a commodity is increased by h dollars from p dollars to p+ h dollars. Then the quantity demanded drops from f (p) units to f(p + h) units. The percentage change in the unit price is h (100) p and the corresponding percentage change in the quantity demanded is ⎡ f ( p + h ) − f ( p )⎤ (100) ⎢ ⎥ f ( p) ⎣ ⎦ Percentage change in the quantity demanded Percentage change in the unit price ⎡ f ( p + h ) − f ( p )⎤ (100) ⎢ ⎥ f ( p) ⎣ ⎦ = h (100) p p f ( p + h) − f ( p) = f ( p) h Elasticity of Demand f ( p + h) − f ( p) ≈ f ' ( p) h • Since previous ratio as , we can write the pf ' ( p ) E( p) = − f ( p) called the elasticity of demand at price p. • We will see in section 4.1 that f ' ( p ) < 0 since f is decreasing. Because economists would rather work with a positive value, we put a negative sign. Consider the demand equation p = −0.02 x + 400 (0 ≤ x ≤ 20,000) which describes the relationship between the unit price in dollars and the quantity demanded x of the Acrosonic model F loudspeaker systems. Find the elasticity of demand E(p). pf ' ( p ) p (− 50) = E( p) = − f ( p ) − 50 p + 20,000 p = 400 − p • When p = 100 1 , we have E (100) = . This result 3 tells us that when the unit price is set at $100 per speaker, an increase of 1% in the unit price will cause a decrease of approximately 0.33% in the quantity demanded. • When p = 300 , we have E (300 ) = 3 . This result tells us that when the unit price is set at $100 per speaker, an increase of 1% in the unit price will cause a decrease of approximately 0.33% in the quantity demanded. Since the revenue is R ( p ) = px = pf ( p ) , the marginal revenue function is R' ( p ) = f ( p ) + pf ' ( p ) ⎡ pf ' ( p )⎤ = f ( p )⎢1 + = f ( p )[1 − E ( p )] ⎥ f ( p) ⎦ ⎣ 1 • Since E (100 ) = < 1 , we have R ' (100 ) > 0 . If we 3 increase the unit price, the revenue increases. • Since E (300) = 3 > 1 , we have R ' (300 ) < 0 . If we decrease the unit price, the revenue decreases. • If the demand is elastic at p [E(p)>1], then an increase in the unit price will cause the revenue to decrease, whereas a decrease in the unit price will cause the revenue to increase. • If the demand is inelastic at p [E(p)>1], then an increase in the unit price will cause the revenue to increase, whereas a decrease in the unit price will cause the revenue to decrease. • If the demand is unitary at p [E(p)>1], then an increase in the unit price will cause the revenue to stay about the same. Consider the demand equation p = −0.01x 2 − 0.2 x + 8 (0 ≤ x ≤ 20,000) and the quantity demanded each week is 15. If we increase the unit price a little bit, what will happen to our revenue? 1.2 dx dx 1 = −0.02 x − 0.2 ⇒ = dp dp − 0.02 x When x=15, p=2.75, f(p)=x=15, and f ' ( p ) = ( 2.75)(− 4 ) 11 pf ' ( p ) = E( p) = − =− 15 15 f ( p) dx = −4 dp