*5.2. Speculative Trading*

Interestingly, the date 1 supply spikes at price *p*1 = *a*. The optimistic normal suppliers have the incentive to sell electricity at date 1, but their marginal cost is *a*. Therefore, they choose to sell electricity only if *p*1 exceeds *a*.

If the speculative trade is prohibited, the market structure become very simple. Suppliers cannot buy electricity at *t* = 1; that is, *xb* must be 0. Therefore, normal suppliers' profit maximization at *t* = 1 changes as follows:

$$\begin{aligned} \max\_{\mathbf{x}\_1, \mathbf{x}^b} \ (p\_1 - a)\mathbf{x}\_1 + E\_i[v(\mathbf{x}\_1)] \\ \text{s.t.} \ x\_1 \le X, \ p\_1 \mathbf{x}^b \le I, \ x\_1 \ge 0 \end{aligned} \tag{41}$$

*x*1 is the selling volume, and *<sup>v</sup>*(*<sup>x</sup>*1) is the value function for *t* = 2.

$$\begin{aligned} v(\mathbf{x}\_1, \mathbf{x}^b) = \text{Max}\_{\mathbf{x}\_2} \ (p\_2 - a)\mathbf{x}\_2\\ \text{s.t.} \ \mathbf{x}\_2 \le X - \mathbf{x}\_1, \ \mathbf{x}\_2 \ge 0 \end{aligned} \tag{42}$$

The renewable suppliers face the same problem. Thus, the market without speculative trading equilibrium is as follows:

$$d\_1(p\_1) = F\_{p\_1}(p\_1)R\_1 \qquad \text{ if } p\_1 \le a \tag{43}$$

$$F\_a(a)R\_1 < d\_1(p\_1) < F\_a(a)R\_1 + F\_a(a)X \qquad \text{if } p\_1 = a \tag{44}$$

$$d\_1(p\_1) = F\_{p1}(p\_1)R\_1 + F\_{p1}(p\_1)X \qquad \text{if } p\_1 > a \tag{45}$$

$$p\_2 = 0 \qquad \text{if } d\_2(0) + d\_1(p\_1) \le \mathcal{R}\_1 + \mathcal{R}\_2 \tag{46}$$

$$0 < p\_2 < a \qquad \text{if } d\_2(0) + d\_1(p\_1) > R\_1 + R\_2 \text{ and } d\_2(a) + d\_1(p\_1) \le X^b + R\_1 + R\_2 \tag{47}$$

$$p\_2 = a \qquad \text{if } d\_2(a) + d\_1(p\_1) > R\_1 + R\_2 \text{ and } d\_2(a) + d\_1(p\_1) \le X + R\_1 + R\_2 \tag{48}$$

$$p\_2 > a \qquad \text{if } d\_2(a) + d\_1(p\_1) > X + R\_1 + R\_2 \tag{49}$$

Because suppliers cannot buy the electricity, the demand on date 1 is only *d*1(*p*1).

Because optimistic suppliers expect that renewable supplies are low and electricity prices at *t* = 2 is high, they buy date 1 electricity for the purpose of making profits through resale. The date 1 price *p*1 is affected by the elasticity of the demand. Even if the consumers' demand *d*1(*p*1) is inelastic with regard to the price, total demand is not inelastic with regard to the price.

**Lemma 1.** *The total demand becomes more elastic with regard to the price through speculative trading at t* = 1*.*

**Proof.** In the speculative trading model, the date 1 total demand is elastic.

$$D\_1(p\_1) = d\_1(p\_1) + (1 - F\_{\mathbb{P}1}(p\_1)) \frac{I}{p\_1} \tag{50}$$

Even if the consumers' demand *d*1(*p*1) is perfectly inelastic with regard to the price, the total demand is not inelastic with regard to the price.

This simple lemma implies that the trade of the suppliers has a role in establishing stability in the electricity market. If speculative trade is prohibited, the price *p*1 jumps from 0 to *a*, with demand for *d*1 shifting (see Figure 5).

**Figure 5.** Date 1 market equilibrium without speculative trading.

Governments may often worry about consumers' surplus in the speculative electricity market. Interestingly, consumers' surplus is not always lower than that of the market without resale.
