The San Francisco power company faces a total cost function of TC= (1/2)Q 2 + 10
ID: 1117625 • Letter: T
Question
The San Francisco power company faces a total cost function of
TC= (1/2)Q2 + 100,000
where prices are in cents/kilowatt-hour and quantities are in millions of kilowatt-hours. Th demand function of San Francisco residents for off-peak hours is:
D0: P=4-Q or D0': Q=4-P
The demand function of San Francisco residents for peak hours is:
Dp: P=8-Q or Dp': Q=8-P
Calculate (a) the short-run marginal cost function (SMC); (b) the equilibrium price (P0) and quantity demanded (Q0 ) during off-peak hours; (c) the equilibrium price (Pp) and quantity demanded (Qp) during peak hours; and (d) the total demand for electricity (Q0+Qp)
Explanation / Answer
(a) TC = (Q2 / 2) + 100,000
SMC = dTC / dQ = 2Q / 2 = Q
(b) Equilibrium is obtained when Marginal revenue (MRo) equals SMC.
Qo = 4 - Po
Po = 4 - Qo
Total revenue (TRo) = Po x Qo = 4Qo - Qo2
MRo = dTRo / dQo = 4 - 2Qo
Equating with SMC,
4 - 2Qo = Qo
3Qo = 4
Qo = 1.33
Po = 4 - 1.33 = 2.67
(c) Equilibrium is obtained when Marginal revenue (MRp) equals SMC.
Qp = 8 - Pp
Pp = 8 - Qp
Total revenue (TRp) = Pp x Qp = 8Qp - Qp2
MRp = dTRp / dQp = 8 - 2Qp
Equating with SMC,
8 - 2Qp = Qp
3Qp = 8
Qp = 2.67
Pp = 8 - 2.67 = 5.33
(d)
Total demand = Qo + Qp = 1.33 + 2.67 = 4
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