*5.1. Price Relation*

The market at date 2 is simpler than the market at date 1. However, the date 2 price is influenced by the date 1 price *p*1. The next lemma shows the relation between *p*1 and *p*2.

The market equilibrium at date 2 is simpler than the date 1 equilibrium. As noted, total supply *<sup>S</sup>*2(*p*2) is expressed as follows:

$$0 \le S\_2(0) \le R\_1 + R\_2 + X^b - d\_1(p\_1) \qquad \text{if } p\_2 = 0 \tag{36}$$

$$S\_2(p\_2) = R\_1 + R\_2 + X^b - d\_1(p\_1) \qquad \text{ if } 0 \le p\_2 \le a \tag{37}$$

$$R\_1 + R\_2 + X^b - d\_1(p\_1) \le \mathcal{S}\_2(p\_2) \le R\_1 + R\_2 + X - d\_1(p\_1) \qquad \text{if } p\_2 \ge a \tag{38}$$

*X<sup>b</sup>* electricity units are sold by normal suppliers at date 1. Therefore, based on the date 1 equilibrium, the following can be expressed:

$$X^b = F\_{p\_1}(p\_1)X\tag{39}$$

Therefore, *X<sup>b</sup>* is an increasing function of *p*1. Moreover, *d*1(*p*1) is a decreasing function of *p*1. Thus, if *p*1 gets higher (and other conditions at date 2 remain equal), total supply moves towards the right. As a result, *p*2 decreases (see Figure 4).

We can examine Assumption 2 by this price relation. Higher *p*1 implies lower *p*2 for all suppliers; therefore, *p*1 ≤ *p*1 for all *Ei*[*p*2].

$$F\_{p\_1}(E\_i[p\_2]) \le F\_{p\_1'}(E\_i[p\_2])\tag{40}$$

This implies that Assumption 2 is satisfied by this equilibrium.

**Figure 4.** Date 2 market equilibrium with higher *p*1.
