3.2.1. Firm

(1) R&D investment of the firm

As a typical emerging industry, the fuel ethanol industry is still faced with the urgen<sup>t</sup> need for continuous improvement of its related technologies. Therefore, one of the stylized facts is that almost all firms in the fuel ethanol industry have research and development (R&D) investments. We assume that the firm's R&D investment consists of two parts. The first part is the fixed amount of R&D investment. Whether the company is profitable or not, it will invest in R&D. The second part is that when the firm has a positive profit, a fixed proportion of profit will be invested in R&D. Thus, R&D investment can be expressed as:

$$R\_{i,t} = \text{Max} \{ rd + \sigma \cdot \pi\_{i,t}, rd \} \tag{1}$$

where *rd* denotes the fixed amount of R&D investment; σ denotes the fixed proportion of profit invested in R&D, and π*it* denotes firm's profit which is defined by Equation (A7) in the Appendix A.

(2) Entry of firms

We assume that a firm's entry decision is influenced by the industry's profit to cost ratio. Let ϕ(*x*) = Φ · exp(−ϕ · *<sup>x</sup>*), where ϕ and Φ are positive constants, and Φ ∈ (0, 1] is given a distribution function *ps*, *s* = 1,2...,l; then, the number of latecomers in each period can be expressed by the following equation:

$$\gamma\_s = \begin{cases} 0 & \text{with probability } \psi(\mathbf{x})\\ \text{s with probability } p\_s \cdot (1 - \psi(\mathbf{x})) & \end{cases} \tag{2}$$

where ψ(*x*) = ϕ(max[<sup>Γ</sup>*t*, <sup>0</sup>]).

In Equation (2), if Φ = 1 is set; then, ϕ(0) = 1, which represents when the incumbent firm loses money, and no new firms enter this industry. That is to say, Φ = 1 indicates that the firm is completely rational. If Φ < 1 is set, even if the incumbent firms have losses, there will still be latecomers. In other words, this model can satisfy the theoretical hypothesis of rationality or incomplete rationality by making di fferent assumptions.

This study assumes that the initial size and technical e fficiency of the latecomers are equal to the average level of the whole industry.

(3) Adjustment rules of the firm

During each period, the firm can determine the optimal output, *si*,*t*, and the corresponding demand of feedstock, *mi*,*t*, according to the feedstock price and product price. Due to the matching relationship between the feedstock input and fixed assets, the required asset size should be *mi*,*<sup>t</sup>*/<sup>α</sup>. If the firm's own asset scale *Fi*,*t*−<sup>1</sup> is smaller than *mi*,*<sup>t</sup>*/<sup>α</sup>, then the firm's asset scale expands to *Fi*,*<sup>t</sup>* = *mi*,*<sup>t</sup>*/<sup>α</sup>, and the corresponding firm's capacity utilization ratio is η*i*,*<sup>t</sup>* = 1. Otherwise, if the firm's own asset scale *Fi*,*t*−<sup>1</sup> is larger than *mi*,*<sup>t</sup>*/<sup>α</sup>, the firm's asset scale remains unchanged, that is *Fi*,*<sup>t</sup>* = *Fi*,*t*−1, and the capacity utilization rate is η*i*,*<sup>t</sup>* = *mi*,*<sup>t</sup>*/(<sup>α</sup> · *Fi*,*<sup>t</sup>*).

(4) Exit rules of the firm

We assume that when the firm has losses for several consecutive periods, the firm will withdraw production.
