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Suppose a bar in a remote mining town is the only bar within one hundred kilome...
Jun 10, 2024
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Solution by Steps
step 1
The cost function is given by C(Q)=80+4QC(Q) = 80 + 4Q
step 2
The market demand function is Q=400−40pQ = 400 - 40p
step 3
To find the revenue, we use R=pQR = pQ. Substituting Q=400−40pQ = 400 - 40p into the revenue function, we get R=p(400−40p)=400p−40p2R = p(400 - 40p) = 400p - 40p^2
step 4
The profit function is π=R−C\pi = R - C. Substituting R=400p−40p2R = 400p - 40p^2 and C=80+4QC = 80 + 4Q into the profit function, we get π=(400p−40p2)−(80+4(400−40p))\pi = (400p - 40p^2) - (80 + 4(400 - 40p))
step 5
Simplifying the profit function: π=400p−40p2−80−1600+160p=240p−40p2−1680\pi = 400p - 40p^2 - 80 - 1600 + 160p = 240p - 40p^2 - 1680
step 6
To maximize profit, we take the derivative of the profit function with respect to pp and set it to zero: dπdp=240−80p=0\frac{d\pi}{dp} = 240 - 80p = 0
step 7
Solving for pp, we get p=3p = 3
step 8
Substituting p=3p = 3 back into the demand function Q=400−40pQ = 400 - 40p, we get Q=400−40(3)=280Q = 400 - 40(3) = 280
step 9
Substituting p=3p = 3 and Q=280Q = 280 back into the profit function, we get π=240(3)−40(3)2−1680=720−360−1680=−1320\pi = 240(3) - 40(3)^2 - 1680 = 720 - 360 - 1680 = -1320
Part (b): Find the cover charge AA per person and the price per drink pp to maximize profits.
step 1
The new profit function with the cover charge is π=20A+(400−40p)p−(80+4(400−40p))\pi = 20A + (400 - 40p)p - (80 + 4(400 - 40p))
step 2
Simplifying the profit function: π=20A+400p−40p2−80−1600+160p=20A+560p−40p2−1680\pi = 20A + 400p - 40p^2 - 80 - 1600 + 160p = 20A + 560p - 40p^2 - 1680
step 3
To maximize profit, we take the derivative of the profit function with respect to pp and set it to zero: dπdp=560−80p=0\frac{d\pi}{dp} = 560 - 80p = 0
step 4
Solving for pp, we get p=7p = 7
step 5
Substituting p=7p = 7 back into the demand function Q=400−40pQ = 400 - 40p, we get Q=400−40(7)=120Q = 400 - 40(7) = 120
step 6
Substituting p=7p = 7 and Q=120Q = 120 back into the profit function, we get π=20A+560(7)−40(7)2−1680=20A+3920−1960−1680=20A+280\pi = 20A + 560(7) - 40(7)^2 - 1680 = 20A + 3920 - 1960 - 1680 = 20A + 280
step 7
To maximize profit, we set the profit function equal to the maximum profit: 20A+280=020A + 280 = 0
step 8
Solving for AA, we get A=−14A = -14
Answer
The bar's short run profit is -1320.
The cover charge per person is -14, and the price per drink is 7.
Key Concept
Profit maximization in a monopoly
Explanation
To find the profit-maximizing price and quantity, we set the derivative of the profit function to zero and solve for the price. The cover charge is determined by setting the profit function equal to the maximum profit.
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