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PROFIT, REVENUE & COST: AN APPROACH USING DIFFERENTIAL CALCULUS MARGINAL COST • Marginal Cost Analysis studies how to estimate economic values (such as profit, revenue and cost) change when the input increases by 1 unit. • While marginal analysis is an accurate approximation of how these quantities change when the input increases by 1, • You can also calculate the exact change. • The increase in Total Cost may in certain cases be approximated to be a function of quantity. MARGINAL ANALYSIS • To perform marginal analysis on • either profit • Revenue • Cost • Find the derivative function for the one quantity out of these three that you are estimating for. • The derivatives of these quantities are called • marginal profit function • marginal revenue function • marginal cost function NOTATIONS • If profit is given by P(x), • then the marginal profit function is given by P′(x) • If revenue is given by R(x), • then the marginal revenue function is given by R′(x) • If cost is given by C(x), • then the marginal cost function is given by C′(x) MARGINAL ANALYSIS • The Use of Marginal Analysis • an estimate of how much profit, revenue and/or cost changes when the nth unit is produced or sold. Here, we use derivatives. • the exact amount of how much profit, revenue and/or cost changes. Here, we use the original function. • Marginal analysis can either estimate (get close to), or get the real quantity, that adding 1 unit results in. • Both approaches require knowledge of functions. ESTIMATE • To get an estimate of how much profit, revenue and/or cost are changing for the nth unit, we need to find the marginal function and plug one less than n (or, number of units − 1) into the marginal function. • To estimate how profit, revenue and/or cost are changing when the nth unit is produced or sold, plug in (n−1) into the marginal function (derivative) • Remember: The derivative of a function is the slope of the function. In essence, we are looking for the slope of f’(n-1) EXACT CHANGE • Calculating exactly how the quantity changes (instead of estimating) uses the original function instead of the derivative. • To find the exact change in profit, revenue or cost after producing or selling the nth unit, we need to evaluate the original function at n and subtract the original function evaluated at n−1 • To calculate the exact change in profit, revenue or cost for the nth unit, calculate f(n)−f(n−1), where f(n) is the original function • f(n) = f(n) – f(n-1) • if you are asked to calculate the exact cost of producing the 20th unit, you need to plug in both 20 and 19 into the original function, and subtract the latter from the former • f(n) = f(20) – f(20-1) SAMPLE PROBLEM • For a company that sells kids' toys, • the total cost of producing x is given by the function 𝐶 𝑥 = 2350 + 80𝑥 − 0.04𝑥 2 • and that all x toys are sold when the price (small letter p) is equal to 𝑝 𝑥 = −2𝑥 + 35 ESTIMATE THE MARGINAL COST OF PRODUCING THE 6TH UNIT • For a company that sells kids' toys, • the total cost of producing x is given by the function 𝐶 𝑥 = 2350 + 80𝑥 − 0.04𝑥 2 • and that all x toys are sold when the price (small letter p) is equal to 𝑝 𝑥 = −2𝑥 + 35 • Find the marginal cost function, differentiate the total cost function. 𝐶′ 𝑥 = 80 − 0.08𝑥 • Find the marginal cost of producing the 6th unit. Plug in one less than the x that was given into the marginal function. 𝐶′(5) = 80 − 0.08 5 = 79.6 • The marginal cost of producing the 6th unit is $79.60 CALCULATE THE ACTUAL COST OF PRODUCING THE 6TH UNIT • For a company that sells kids' toys, • the total cost of producing x is given by the function 𝐶 𝑥 = 2350 + 80𝑥 − 0.04𝑥 2 • and that all x toys are sold when the price (small letter p) is equal to 𝑝 𝑥 = −2𝑥 + 35 • we calculate the cost of producing 6 units and subtract the cost of producing 5 units. The result of this must be the cost of producing unit 6. • Marginal function is only used to estimate. • To find actual amounts, use the original profit, revenue and/or cost function. 𝐶 6 −𝐶 5 𝐶 6 − 𝐶 5 = 2,828.56 − 2749 𝐶 6 − 𝐶 5 = $79.56 ESTIMATE THE MARGINAL REVENUE FROM SELLING THE 6TH UNIT 𝐶 𝑥 = 2350 + 80𝑥 − 0.04𝑥 2 𝑝 𝑥 = −2𝑥 + 35 However, we were not given a revenue function in the problem. But…… Revenue = Quantity * Price Revenue Function : 𝑅 𝑥 = 𝑥 ∗ 𝑝(𝑥) 𝑅 𝑥 = 𝑥 ∗ 𝑝 𝑥 = 𝑥 ∗ −2𝑥 + 35 = −2𝑥 2 + 35𝑥 • Find the marginal revenue function, differentiate the revenue function. 𝑅′ 𝑥 = −4𝑥 + 35 • to estimate the revenue the from selling the 6th, we plug in 5 (one less) into the marginal revenue function 𝑅′ 5 = −4(5) + 35 = 15 • The estimated revenue of selling the 6th unit is $15 CALCULATE THE ACTUAL REVENUE OF SELLING THE 6TH UNIT 𝐶 𝑥 = 2350 + 80𝑥 − 0.04𝑥 2 𝑝 𝑥 = −2𝑥 + 35 𝑅 𝑥 = 𝑥 ∗ 𝑝 𝑥 = 𝑥 ∗ −2𝑥 + 35 = −2𝑥 2 + 35𝑥 • find the total revenue of selling the first 6 units and subtract the revenue from selling the first 5 units. • The difference will be the revenue produced by the 6th unit. • To find actual amounts, use the original revenue function. 𝑅 6 −𝑅 5 𝑅 6 − 𝑅 5 = (138 − 125) 𝑅 6 − 𝑅 5 = $13 the actual revenue of selling the 6th unit is $13, and our estimate was of $15. FIND THE MARGINAL PROFIT FUNCTION • • • • • To find the marginal profit function, find the profit function first. The profit function was not given in the original problem. How is profit calculated? Profit (capital P) is Revenue minus Cost. The Profit function is just the Revenue function minus the Cost function. 𝑝 𝑥 = −2𝑥 + 35 𝑝𝑟𝑖𝑐𝑒 (𝑠𝑚𝑎𝑙𝑙 𝑝) 𝐶 𝑥 = 2350 + 80𝑥 − 0.04𝑥 2 𝑅 𝑥 = 𝑥 ∗ 𝑝 𝑥 = 𝑥 ∗ −2𝑥 + 35 = −2𝑥 2 + 35𝑥 𝑃 𝑥 = 𝑅 𝑥 − 𝐶 𝑥 = −2𝑥 2 + 35𝑥 − (2350 + 80𝑥 − 0.04𝑥 2 ) 𝑃 𝑥 = 𝑅 𝑥 − 𝐶 𝑥 = −1.96𝑥 2 − 45𝑥 − 2350 FIND THE MARGINAL PROFIT FUNCTION • • • • • To find the marginal profit function, find the profit function first. The profit function was not given in the original problem. How is profit calculated? Profit (capital P) is Revenue minus Cost. The Profit function is just the Revenue function minus the Cost function. 𝑃 𝑥 = −1.96𝑥 2 − 45𝑥 − 2350 𝑃′ 𝑥 = −3.92𝑥 − 45 NOTE: • • • • Do not confuse the profit function with the price function. The price function is usually written as p(x) while the profit function is the uppercase version, P(x). In summary, big P is for Profit!