In this section, we try to compare the social welfare under a guaranteed power price scenario and that in the benchmark model.
4.1 Benchmark model
In benchmark model, we assume that the public power plant is a monopolist for supplying the power, i.e., Qm = x, where the superscript m represents the monopoly case. The social welfare function in this case is composed by the consumer surplus, i.e., (b/2)Qm2, and the producer surplus, i.e., m = (p cx)Qm. The social planner sets the optimal power price to maximize the social welfare function, i.e., pm = cx. We find that the rule of marginal cost pricing will cause the highest social welfare. Given pm = cx, the consumer surplus is
, the producer surplus is 0, and the maximized social welfare is
4.2 A guaranteed power price scenario
In this subsection, we assume that the public power plant is a monopoly to sell the power in the market, and it not only generates the power by itself but it also purchases power form the private power plant by a guaranteed purchasing price. We employ the backward induction to obtain the optimal solution in this scenario.
4.2.1 The case of cy > cx
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The decision making for the private power plant, i.e., Stackelberg leader, at stage 3 as follows:
Equation (4) means that if the guaranteed price makes the profit of the private power plant is positive (negative), i.e., w cy (<) 0, then the private power will (not) sell all (any) power to the public power plant. According to Equation (2), the public power plant chooses the output to maximize the profit as follows:
x = b the profit function of the public power plant, we find 1/w < 0. This result tells us the optimal guaranteed purchasing power for the public power plant is
w = cy. (6) Refer back to Equation (4) and we realize that the private power plant will sell all power i.e., y = y to the public power plant under the scenario of guaranteed purchasing price.
At stage 1, the social planner decides the optimal power price to maximize the social welfare. Because of w = cy, the social welfare function in Equation (3) can be reduced as follows:
W = (b/2)Q2 + 1, (7) where Q = x + y. By the first order condition of social welfare function, we obtain the optimal quantity of power production for the public power plant is
x = b c a x
y. (8) The optimal market total output, the power price, and the social welfare are
11 planner still adopts the rule of marginal cost pricing to maximize the social planner.
4.2.2 The case of cy < cx
Following the same step in the case of cy > cx, we can obtain the optimal solution of this scenario as follows:
Q** = where “**” represents the case of cy < cx. We arrange the optimal solution in various scenarios in Table 1.
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Table 1. The optimal solution in various scenarios
Monopoly
Table 1 shows that it causes a maximized social welfare if the social planner adopts the rule of marginal cost pricing on matter in the monopoly case or in the case of guaranteed purchasing price with cy > cx or cy < cx. Hence, we have the proposition 1 as follows:
Proposition 1 No matter what the market structure is, the rule of marginal cost pricing always maximizes the social welfare.
We also find that the system of guaranteed purchasing price does not change the consumer surplus. The reason is that the public power plant still is a monopoly to sell the power under the system of guaranteed purchasing price. The rule of marginal cost pricing makes the profit of public power plant in the case of monopoly be zero. But in the system of guaranteed purchasing price, the profit of public power plant is likely to be negative (positive) if the guaranteed purchasing price is higher (lower) than its own electricity generation cost. And the public power plant must use the marginal cost of private power plant as a guaranteed purchasing price in order to maximize its own profit. As a result, the profit of private power plant is always zero. Hence, the
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size of social welfare depends on the size of public power plant. We conclude some important findings in the proposition 2 as follows:
Proposition 2 Under the rule of marginal cost pricing
(i) No matter what the market structure is, the sizes of consumer surplus are the same;
(ii) The guaranteed purchasing price set by public power plant is just as the marginal cost of private power plant;
(iii)The system of guaranteed purchasing price is likely to cause a positive or negative profit for public power plant;
(iv) The system of guaranteed purchasing price is likely to cause a low social welfare if this system causes a profit loss for public power plant.
4.3 Some discussions
It is obviously that a decrease in the marginal cost of public power plant will cause an increase in consumer surplus no matter in which kind of the market structure.
This result is that the public power plant is a monopoly in selling power market.
We next concern the effect of change in marginal costs of private power plant and public power plant on the social welfare. For getting the answer, we show some comparative statistic results as follows:
cy public power plant increases if cy < cx, or the loss of public power plant decreases if cy
> cx, when the marginal cost of private power plant decreases. More importantly, a
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decrease in the marginal cost of public power plant may cause the social welfare to decrease when the purchased amount of power by the public power plant is too much, i.e., y > (a cx)/b. Although a decrease in marginal cost of public power plant is advantage to consumer surplus, the public power plant purchases too much power that price is fixed on cy is disadvantage to producer surplus. Given any purchased amount of power y, if an increase size of consumer surplus can’t cover a decrease size of producer surplus, then it will cause the social welfare to decrease. This result is concluded in proposition 3 as follows:
Proposition 3 Under the system of guaranteed purchasing price, the marginal cost of
public power plant decreases, and the public power plant purchases power too much, it will cause the social welfare to decrease.