[Marginal Revenue Function] Find the marginal revenue (MR)

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[Marginal Revenue Function] Find the marginal revenue (MR) functions for each of the following demand functions and evaluate them at Q = 4 and Q = 10. (a) Demand Function Q = 36 – 2P. (b) Demand Function 44 – 4P – Q = 0

The Correct Answer and Explanation is:

To find the marginal revenue (MR) function, we need to first express the revenue function (R) in terms of quantity (Q), and then differentiate it with respect to Q. Here’s how we can proceed for each case:

(a) Demand Function: Q = 36 – 2P

  1. Step 1: Express the demand function in terms of price (P). Rearranging the equation, we get: Q=36−2P  ⟹  P=36−Q2Q = 36 – 2P \implies P = \frac{36 – Q}{2}Q=36−2P⟹P=236−Q​
  2. Step 2: Find the revenue function, RRR. Revenue is given by: R=P×QR = P \times QR=P×Q Substituting the expression for PPP into the revenue formula: R=(36−Q2)×Q=(36−Q)Q2=36Q−Q22R = \left( \frac{36 – Q}{2} \right) \times Q = \frac{(36 – Q)Q}{2} = \frac{36Q – Q^2}{2}R=(236−Q​)×Q=2(36−Q)Q​=236Q−Q2​
  3. Step 3: Differentiate the revenue function with respect to QQQ to find the marginal revenue function: MR=dRdQ=ddQ(36Q−Q22)MR = \frac{dR}{dQ} = \frac{d}{dQ} \left( \frac{36Q – Q^2}{2} \right)MR=dQdR​=dQd​(236Q−Q2​) Using basic differentiation: MR=12(36−2Q)=18−QMR = \frac{1}{2} \left( 36 – 2Q \right) = 18 – QMR=21​(36−2Q)=18−Q
  4. Step 4: Evaluate the MR function at Q=4Q = 4Q=4 and Q=10Q = 10Q=10:
    • When Q=4Q = 4Q=4: MR(4)=18−4=14MR(4) = 18 – 4 = 14MR(4)=18−4=14
    • When Q=10Q = 10Q=10: MR(10)=18−10=8MR(10) = 18 – 10 = 8MR(10)=18−10=8

Thus, the marginal revenue function is MR=18−QMR = 18 – QMR=18−Q, and at Q=4Q = 4Q=4, MR=14MR = 14MR=14, and at Q=10Q = 10Q=10, MR=8MR = 8MR=8.

(b) Demand Function: 44 – 4P – Q = 0

  1. Step 1: Express the demand function in terms of price (P). Rearranging the equation: 44−4P−Q=0  ⟹  4P=44−Q  ⟹  P=44−Q444 – 4P – Q = 0 \implies 4P = 44 – Q \implies P = \frac{44 – Q}{4}44−4P−Q=0⟹4P=44−Q⟹P=444−Q​
  2. Step 2: Find the revenue function, RRR. Revenue is: R=P×QR = P \times QR=P×Q Substituting for PPP: R=(44−Q4)×Q=(44−Q)Q4=44Q−Q24R = \left( \frac{44 – Q}{4} \right) \times Q = \frac{(44 – Q)Q}{4} = \frac{44Q – Q^2}{4}R=(444−Q​)×Q=4(44−Q)Q​=444Q−Q2​
  3. Step 3: Differentiate the revenue function with respect to QQQ to find the marginal revenue: MR=dRdQ=ddQ(44Q−Q24)MR = \frac{dR}{dQ} = \frac{d}{dQ} \left( \frac{44Q – Q^2}{4} \right)MR=dQdR​=dQd​(444Q−Q2​) Using basic differentiation: MR=14(44−2Q)=11−Q2MR = \frac{1}{4} \left( 44 – 2Q \right) = 11 – \frac{Q}{2}MR=41​(44−2Q)=11−2Q​
  4. Step 4: Evaluate the MR function at Q=4Q = 4Q=4 and Q=10Q = 10Q=10:
    • When Q=4Q = 4Q=4: MR(4)=11−42=11−2=9MR(4) = 11 – \frac{4}{2} = 11 – 2 = 9MR(4)=11−24​=11−2=9
    • When Q=10Q = 10Q=10: MR(10)=11−102=11−5=6MR(10) = 11 – \frac{10}{2} = 11 – 5 = 6MR(10)=11−210​=11−5=6

Thus, the marginal revenue function is MR=11−Q2MR = 11 – \frac{Q}{2}MR=11−2Q​, and at Q=4Q = 4Q=4, MR=9MR = 9MR=9, and at Q=10Q = 10Q=10, MR=6MR = 6MR=6.

Summary:

  • For demand function Q=36−2PQ = 36 – 2PQ=36−2P, the MR function is MR=18−QMR = 18 – QMR=18−Q, with values at Q=4Q = 4Q=4 and Q=10Q = 10Q=10 as 14 and 8, respectively.
  • For demand function 44−4P−Q=044 – 4P – Q = 044−4P−Q=0, the MR function is MR=11−Q2MR = 11 – \frac{Q}{2}MR=11−2Q​, with values at Q=4Q = 4Q=4 and Q=10Q = 10Q=10 as 9 and 6, respectively.

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