*4.2. Model Equilibrium*

The market equilibrium is determined by the intersection between the supply and demand. We can solve this model by conducting backward induction.

The equilibrium at *t* = 2 matches the normal electricity noted in Section 3.

To explain the equilibrium, we note the total electricity sold at date 1 as *Xb*.

$$X^b = \int\_i \mathbf{x}\_i^b d\mathbf{i} \tag{13}$$

The market equilibrium is determined with some price *p*1 (see Figure 2).

**Figure 2.** Date 2 market equilibrium.

Total demand *<sup>D</sup>*2(*p*2) is expressed as follows:

$$D\_2(p\_2) = d\_2(p\_2) \tag{14}$$

Total supply *<sup>S</sup>*2(*p*2) is expressed as follows:

$$0 \le S\_2(0) \le R\_1 + R\_2 + X^b - d\_1(p\_1) \qquad \text{if } p\_2 = 0 \tag{15}$$

$$S\_2(p\_2) = R\_1 + R\_2 + X^b - d\_1(p\_1) \qquad \text{if } 0 \le p\_2 \le a \tag{16}$$

$$R\_1 + R\_2 + X^b - d\_1(p\_1) \le S\_2(p\_2) \le R\_1 + R\_2 + X - d\_1(p\_1) \qquad \text{if } p\_2 \ge a \tag{17}$$

The market equilibrium is determined through the intersection of the demand and supply functions:

$$D\_2(p\_2) = \mathbb{S}\_2(p\_2) \tag{18}$$

The electricity price at *t* = 2, *p*2 is expressed as follows:

$$p\_2 = 0 \qquad \text{if } d\_2(0) + d\_1(p\_1) \le R\_1 + R\_2 + X^b \tag{19}$$

$$0 < p\_2 < a \qquad \text{if } d\_2(0) + d\_1(p\_1) > X^b + R\_1 + R\_2 \text{ and } d\_2(a) + d\_1(p\_1) \le X^b + R\_1 + R\_2 \tag{20}$$

$$p\_2 = a \qquad \text{if } d\_2(a) + d\_1(p\_1) > X^b + R\_1 + R\_2 \text{ and } d\_2(a) + d\_1(p\_1) \le X + R\_1 + R\_2 \tag{21}$$

$$p\_2 > a \qquad \text{if } d\_2(a) + d\_1(p\_1) > X + R\_1 + R\_2 \tag{22}$$

*p*2 depends on the renewable supply *R*2 and the price at *t* = 1, *p*1.

The market equilibrium is determined by the intersection of the demand and supply functions:

$$D\_2(p\_2) = \mathcal{S}\_2(p\_2) \tag{23}$$

The belief of *p*2 depends on the belief of *R*2 and *p*1.

Let *Fp*1 (*Ei*[*p*2]) be the distribution of the expected price at *t* = 2 at price *p*1. For solving the equilibrium, we need the following assumption.

**Assumption 1.** *If p* ≤ *p, for all Ei*[*p*2]*, Fp*(*Ei*[*p*2]) ≤ *Fp*(*Ei*[*p*2])*.* *Energies* **2019**, *12*, 2946

Although this is an assumption, we check whether this assumption is satisfied through the market equilibrium.

At *t* = 1, the population of suppliers who have beliefs *p*1 > *Ei*[*p*2] is 1 − *Fp*1 (*p*1).

Therefore, at *t* = 1, if *p*1 ≤ *a*, 1 − *Fp*1 (*p*1) normal suppliers buy electricity, and *Fp*1 (*p*1) renewable suppliers sell electricity.

*p*1 > *a*, 1 − *Fp*1 (*p*1) normal suppliers buy electricity, and *Fp*1 (*p*1) renewable and normal suppliers sell electricity.

Total demand *<sup>D</sup>*1(*p*1) is expressed as follows:

$$D\_1(p\_1) = d\_1(p\_1) + (1 - F\_{p\_1}(p\_1))\frac{I}{p\_1} \tag{24}$$

An analysis of Assumption 2 showed that the demand function is decreasing: *limp*1→0*D*1(*p*1) = <sup>∞</sup>, and *limp*1<sup>→</sup> ∞ *<sup>D</sup>*1(*p*1) = *limp*1<sup>→</sup> ∞*d*1(∞) = 0. Total supply *<sup>S</sup>*2(*p*2) is expressed as follows:

$$F\_{p\_1}(p\_1)R\_1 \qquad \text{ if } p\_1 \le 0 \tag{25}$$

$$F\_a(a) \le S\_1(a) \le F\_a(a)(R\_1 + X) \qquad \text{if } p\_1 = a \tag{26}$$

$$F\_{p\_1}(p\_1)(R\_1 + X) \qquad \text{ if } p\_2 \ge a \tag{27}$$

An analysis of Assumption 2 showed that the supply function is increasing in *p*1, and *S*(0) = 0 *limp*1<sup>→</sup> ∞*S*1(*p*1) = *R*1 + *X*. The market equilibrium is determined by the intersection of the demand and supply functions (see Figure 3):

$$D\_1(p\_1) = S\_1(p\_1) \tag{28}$$

The electricity price at *t* = 1, *p*1 is expressed as follows:

$$d\_1(p\_1) + (1 - F\_{p\_1}(p\_1))\frac{I}{p\_1} = F\_{p\_1}(p\_1)R\_1 \qquad \text{if } p\_1 \le a \tag{29}$$

$$F\_a(a)R\_1 < d\_1(p\_1) + (1 - F\_a(a))\frac{I}{p\_1} < F\_a(a)R\_1 + F\_a(a)X \qquad\qquad\text{if } p\_1 = a \tag{30}$$

$$d\_1(p\_1) + (1 - F\_{p\_1}(p\_1))\frac{I}{p\_1} = F\_{p\_1}(p\_1)R\_1 + F\_{p\_1}(p\_1)X \qquad \text{if } p\_1 > a \tag{31}$$

*<sup>D</sup>*1(*p*1) is decreasing, and *<sup>S</sup>*1(*p*1) is increasing.

$$\lim\_{p\_1 \to 0} D\_1(p\_1) = \infty \tag{32}$$

$$\lim\_{p\_1 \to \infty} D\_1(p\_1) = \lim\_{p\_1 \to \infty} d\_1(p\_1) = 0 \tag{33}$$

$$\lim\_{p\_1 \to 0} S(p\_1) = S(0) = 0\tag{34}$$

$$\lim\_{p\_1 \to \infty} \mathcal{S}\_1(p\_1) = R\_1 + X \tag{35}$$

**Figure 3.** Date 1 market equilibrium.

## **5. Results and Discussion**

This model has a mixed structure; it includes aspects of the heterogeneous belief model as well as the electricity market. Therefore, it includes many special features that are not available in other models. As noted, the shape of supply curve at date 1 is one of them. This shape is derived from the electricity market model. Electricity suppliers have ladder cost-related functions, and their strategy depends on the shape of that functions. We analyzed three major points of this model: price relation, speculative trading effect, and belief effect.
