*5.3. Heterogeneous Belief*

The market equilibrium is heavily influenced by the suppliers' beliefs. It is essential to compare some measures of optimistic beliefs.

#### **Assumption 2.** *Gp*1 (*p*2) > *Fp*1 (*p*2) *for all p*1, *p*<sup>2</sup>*.*

This means that, with distribution *Fp*1 , for all *p*1, the population that expects to encounter a date 2 price that is higher than *p*2, which is larger than that with *Gp*1 . Therefore, *Fp*1 is a more optimistic distribution than *Gp*1 .

**Lemma 2.** *The market price p*1 *is higher under Fp*1 *compared to that under Gp*1 *.*

**Proof.** This implies *<sup>D</sup>*1(*p*) under *G*(*d*) is lower than that under *F*(*d*) for all *d*. Similarly, *<sup>S</sup>*1(*p*) is higher than that under *<sup>F</sup>*(*d*). Then, the date 1 price *p* is lower than that under *<sup>F</sup>*(*d*). This is because the date 2 price *p*2 is expressed as follows:

$$d\_1(p\_1) + (1 - F\_{\mathbb{P}1}(p\_1))\frac{I}{p\_1} = F\_{\mathbb{P}1}(p\_1)R\_1 \qquad \qquad \text{if } p\_1 \le a \tag{51}$$

*Energies* **2019**, *12*, 2946

$$F\_{\mathfrak{a}}(a)R\_1 < d\_1(p\_1) + (1 - F\_{\mathfrak{a}}(a))\frac{1}{a} < F\_{\mathfrak{a}}(a)R\_1 + F\_{\mathfrak{a}}(a)X \qquad \qquad \text{if } p\_1 = a \tag{52}$$

$$(d\_1(p\_1) + (1 - F\_{p\_1}(p\_1)) \frac{I}{p\_1} = F\_{p\_1}(p\_1)R\_1 + F\_{p\_1}(p\_1)X \qquad \text{if } p\_1 > a \tag{53}$$

Based on the assumption, for all, *p*1 is expressed as follows:

$$d\_1(p\_1) + (1 - F\_{p\_1}(p\_1))\frac{I}{p\_1} \ge d\_1(p\_1) + (1 - G\_{p\_1}(p\_1))\frac{I}{p\_1} \tag{54}$$

$$F\_{p\_1}(p\_1)R\_1 \le G\_{p\_1}(p\_1)R\_1\tag{55}$$

$$F\_{p\_1}(p\_1)R\_1 + F\_{p\_1}(p\_1)X \le G\_{p\_1}(p\_1)R\_1 + G\_{p\_1}(p\_1)X \tag{56}$$

Therefore, the total demand under *Fp*1 is larger than that under *Gp*1 , and the total supply under *Fp*1 is less than that under *Gp*1 . As a result, the market price increases under *Fp*1 (*p*2).

Because of heterogeneous beliefs, the price of the electricity is higher compared to the no-resale market price. Firms have the incentive to buy the electricity for the speculative resale, and the consumer suffers the higher prices.

This model involves a heterogeneous bubble. Heterogeneous beliefs among investors create such bubbles. In such models, investors' beliefs differ because they have different prior belief distributions. Agents' heterogenous beliefs can occur because of many factors. For example, overconfidence about the precision of signals among investors can lead to different prior distributions (with lower variance) regarding the signals' noise term. Investors without common prior beliefs can agree to disagree even after they share all their information with each other. In the heterogeneous beliefs model with short-sale constraints, the asset price can result in the creation of bubbles. Optimistic agents buy the asset, and the price rises. Under the conditions of a short-sale constraint, pessimistic traders cannot make use of the high asset prices (Miller (1977) [19], Harrison et al. (1978) [20]). In a dynamic model, the asset price can even exceed the valuation of the most optimistic investor's expectation regarding the economy. In the model, firms with pessimistic beliefs about the demand on the next day sell the electricity immediately; this implies that the supply on the next day increases. Therefore, a pessimistic belief distribution, such as G(x), is very beneficial for the consumers' surplus.

As noted in Section 4.1, higher *p*1 implies lower *p*2 in this model. Therefore, the next lemma shows some negative correlations between the expected price and the realized price.

#### **Lemma 3.** *The market price p*2 *is lower under Fp*1 *than that under Gp*1 *.*

Therefore, if the suppliers' belief is optimistic (that is, they expect higher *<sup>E</sup>*[*p*2]), *p*2 tends to be lower. This can be interpreted as a heterogeneous belief bubble, and this bubble burst on date 2.

#### **6. Policy Implications and Conclusions**
