17. Florida Power & Light is the sole provider of electricity in 32 counties of
ID: 1190793 • Letter: 1
Question
17. Florida Power & Light is the sole provider of electricity in 32 counties of southern Florida. To meet the monthly demand for electricity in these counties, which is given by the inverse demand function P = 1,200 - 4Q, the utility company has set up two electric generating facilities: Q1 kilowatts are produced at facility 1, and Q2 kilowatts are produced at facility 2 (so Q = Q1 + Q2). The costs of producing electricity at each facility are given by C1(Q1) = 8,000 + 6Q12 and C2(Q2) = 6,000 + 3Q22, respectively. Determine the profit-maximizing amounts of electricity to produce at the two facilities, the optimal price, and the utility company’s profits.
Explanation / Answer
P = 1200 - 4Q = 1200 - 4Q1 - 4Q2 [Since Q = Q1 + Q2]
C1 = 8000 + 6Q12
C2 = 6000 + 3Q22
Profit is maximized when MR1 = MC1 & MR2 = MC2
TR1 = P x Q1 = 1200Q1 - 4Q12 - 4Q1Q2
MR1 = dTR1 / dQ1 = 1200 - 8Q1 - 4Q2
MC1 = dC1 / dQ1 = 12Q1
Equating MR1 with MC1,
1200 - 8Q1 - 4Q2 = 12Q1
20Q1 + 4Q2 = 1200
5Q1 + Q2 = 300 ...... (1)
Again, TR2 = P x Q2 = 1200Q2 - 4Q1Q2 - 4Q22
MR2 = dTR2 / dQ2 = 1200 - 4Q1 - 8Q2
MC2 = dC2 / dQ2 = 6Q2
Equating MR2 with MC2:
1200 - 4Q1 - 8Q2 = 6Q2
4Q1 + 14Q2 = 1200
2Q1 + 7Q2 = 600 ...... (2)
Equilibrium price & quantity are found by solving equations (1) & (2):
5Q1 + Q2 = 300 ...... (1)
Multiplying by 7,
35Q1 + 7Q2 = 2100 ..... (3)
2Q1 + 7Q2 = 600 ...... (2)
(3) - (2) gives:
33Q1 = 1500
Q1 = 1500 / 33 = 45.45
Q2 = 300 - 5Q1 [From (1)]
= 300 - (5 x 45.45) = 72.75
So, Q = Q1 + Q2 = 45.45 + 72.75 = 118.2
P = 1200 - 4Q = 1200 - (4 x 118.2) = 727.2
Therefore,
C1 = 8000 + (6 x 45.45 x 45.45) = 20,150
C2 = 6000 + (3 x 72.75 x 72.75) = 21,878
Total cost, C = C1 + C2 = 20,150 + 21,878 = 42,028
Total revenue, R = TR1 + TR2 = (1200Q1 - 4Q12 - 4Q1Q2) + (1200Q2 - 4Q1Q2 - 4Q22)
= 1200 x (Q1 + Q2) - 8Q1Q2 - 4 x (Q12 + Q22)
= (1200 x 118.2) - (8 x 45.45 x 72.75) - 4 x [(45.45 x 45.45) + (72.75 x 72.75)]
= 141,840 - 26,452 - 29,433
= 85,955
So, Profit = R - C
= 85,955 - 42,028
= 43,927
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