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Table m5 1 the demand curve p 10 q price kg 10 9 8 7

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TABLE M.5-1 The Demand Curve P = 10 – Q Price ($/kg) 10 9 8 7 6 5 4 3 2 1 0 Quantity (kg) 0 1 2 3 4 5 6 7 8 9 10 Total Revenue ($) 0 9 16 21 24 25 24 21 16 9 0 Marginal Revenue ($/kg) +9 +7 +5 +3 +1 –1 –3 –5 –7 –9 The Marginal Revenue (or ∆TR/ ∆Q ) curve may be derived in three ways. For those with calculus, it is simply the slope or derivative of the TR curve at any value of Q : dTR/d Q = 10 – 2 Q. Those without calculus should glance at Module 10, which contains some basic rules for calculating derivatives, including the one used here. Yet it is also possible to derive it from Table M.5-1. Note that the values for Marginal Revenue are located at the midpoints of the relevant values of Q. For example, the value of MR as we go from Q = 2 to Q = 3 (or vice-versa) is equal to +$5/kg, and is situated at Q = 2.5 kg, since it is the change in TR in moving between 2 and 3 kg. Note also that the MR declines by 2 for each increase of Q by 1 unit: its slope is –2. Its equation is therefore MR = 10 – 2 Q. The third way of calculating it is to use “The Rules.” The Rules apply to any related linear average and marginal curves and the corresponding total function. “THE RULES” 1 . From Average to Marginal Curve : “Same intercept, twice the slope.” [If Average Revenue = P = a + bQ , then MR = a + 2 bQ . In our example, if P = AR = 10 – Q , then MR = 10 – 2 Q .] 2 . From Marginal to Average Curve : “Same intercept, half the slope.” [If MR = a + bQ , then P = AR = a + 0.5bQ. In our example, if MR = 10 – 2 Q, then P = AR = 10 – Q .] 3 . From Average to Total Curve : “Average times Q = Total.” [If Average Revenue = P = a + bQ , then TR = aQ + bQ 2 . In our example, if P = AR = 10 – Q , then TR = 10 Q Q 2 .] 4 . From Total to Average Curve : “Total divided by Q = Average.” [If Total Revenue = PQ = aQ + bQ 2 , then AR = a + bQ . In our example, if TR = 10 Q Q 2 , then P = AR = 10 – Q .] 5 . From Total to Marginal and From Marginal to Total Curve : Either use calculus or use a 2- step procedure: Total Average Marginal or Marginal Average Total.
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While “The Rules” have been expressed in terms of Demand and Marginal Revenue curves, they apply equally as well for any of the sets of functions in Section 1.1 of this Module. The only set that poses any problems is the Total Cost/ Marginal Cost one. The reason is that Fixed Costs complicate the situation slightly. The needed adjustments to “The Rules” in this case are covered in your text, on page 349, footnote 14, and you will get some practice in the Exercises. M5-4 MATH MODULE 5: TOTAL, AVERAGE, AND MARGINAL FUNCTIONS TR ($) TR = P Q D = AR P = AR, MR ($/kg) MR Q (kg) 25 0 0 5 5 5 10 10 10
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MATH MODULE 5: TOTAL, AVERAGE, AND MARGINAL FUNCTIONS M5-5 2. Exercises 1. For each of the following cases, provide the Total Revenue, Average Revenue, and Marginal Revenue equations and give the value for each of the 3 equations when Q = 10 tonnes: (a) P = AR = 30 – Q (b) TR = 10 Q – 0.1 Q 2 (c) MR = 30 – 6 Q (d) MR = 40 – 4 Q (e) TR = 2 Q – 0.2 Q 2 (f) P
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