Question

demand during peak hour is Qp=20-(1/2)Pp and demand for off peak hour is Qo=10-(1/2)Po, quantity measured in kilowatt hours. Peak and off peak each take up half the day. the variable cost per unit of output per period is $4.00. capital costs, sunk and cannot be adjusted, are $2.00 per unit of capacity per day. if peak and off peak pricing is allowed, what is the peak price? what is the peak quantity of electricity demanded? what is the off peak price for electricity? what is the off peak quantity of electricity demanded? what is the optimal capacity?

Answer 1

The demand function during peak hours is given as Qp=20-1/2Pp or Qp=20-0.5Pp or Qp-20=-0.5Pp or 40-Qp/0.5=Pp where Qp and Pp represent the quantity demanded of output and output price during the peak hours.

The total revenue during the peak hours or TRp can be denoted as Pp*Qp=(40-Qp/0.5)*Qp=40Qp-Qp^2/0.5

The marginal revenue obtained by the firm during the peak hours or MRp=dTRp/dQp=40-4Qp

The variable cost per unit of output per period is $4.00 and an additional $2 has to be paid per unit of capacity per day. Hence, the total marginal cost of output produced in this case or MC=($4+$2)=$6.

Based on the profit-maximizing principle of a price discriminatory firm, the firm will maximize profit during the peak hours by producing the output level at which MRp is equal to MC.

Therefore, based on the profit-maximizing principle of the price discriminatory firm, we can state:-

40-4Qp=6

-4Qp=-40+6

-4Qp=-34

Qp=-34/-4

Qp=8.5

Hence, the profit-maximizing output or quantity demanded during the peak hours is 8.5 units of output, in this case.

Now, plugging the profit-maximizing output or quantity demanded into the demand function during peak hours, we get:-

Qp=20-0.5Pp

8.5=20-0.5Pp

8.5-20=-0.5Pp

-11.5=-0.5Pp

-11.5/-0.5=Pp

23=Pp

Therefore, the profit-maximizing output price charged by the firm during the peak hours would be $23 per unit of output.

The demand function during off-peak hours is given as Qo=10-1/2Po or Qo=10-0.5Po or Qo-10=-0.5Po or 20-Qo/0.5=Po where Qo and Po represent the quantity demanded of output and output price during the off-peak hours.

The total revenue during the off-peak hours or TRo can be denoted as Po*Qo=(20-Qo/0.5)*Qo=20Qo-Qo^2/0.5

The marginal revenue obtained by the firm during the peak hours or MRo=dTRo/dQo=20-4Qo

The total marginal cost of output produced or MC is $6.

Therefore, based on the profit-maximizing principle of any price discriminatory firm, we can state:-

20-4Qo=6

-4Qo=-20+6

-4Qo=-14

Qo=-14/-4

Qo=3.5

Hence, the profit-maximizing output produced by the firm or the quantity demanded during the off-peak hours is 3.5 units of output.

Now, plugging the profit-maximizing output or quantity demanded into the demand function during the off-peak hours, we can derive:-

Qo=10-0.5Po

3.5=10-0.5Po

3.5-10=-0.5Po

-6.5=-0.5Po

-6.5/-0.5=Po

13=Po

Thus, the profit-maximizing price charged per unit of output by the firm during the off-peak period is $13 per unit of output.

The firm will achieve the optimum scale or level of output which corresponds to the per-unit output price which is equal to the MC in the case of both peak and off-peak periods.

Therefore, based on the condition of optimal capacity in the case of the peak period, we can state:-

40-Qp/0.5=6

-Qp/0.5=-40+6

-Qp/0.5=-34

-Qp=-17

Qp=17

Hence, the optimal unit of output or capacity during the peak period would be 17 units of output produced by the firm.

Again, based on the condition of optimal capacity in the case of the off-peak period, we can obtain:-

20-Qo/0.5=6

-Qo/0.5=-20+6

-Qo/0.5=-14

-Qo=-28

Qo=28

The optimal output produced by the firm during the off-peak period would be 28 units of output, in this case.