Patents and P2: Innovation and Technology Adoption for Environmental Improvements

Abstract

In this paper, we identify factors that influence adoption of two types of environmental innovations, environmental patents and pollution prevention (P2) activities, and then measure the resulting influence of each on the pollution profile of the firm. We find that environmental patenting is most strongly driven by the technological capacity of the firm, including prior environmental patenting and prior P2 adoptions. While P2 activities are also influenced by prior adoptions of P2, other factors play an important role, including environmental innovation opportunities, the regulatory environment, and firm-specific characteristics. In terms of environmental outcomes, we find that both environmental technologies reduce pollution. Due to strong knowledge accumulation effects of environmental patents, the long-term impact of an environmental patent is stronger and longer lasting than the long-term impact of a P2 practice.

Appendix

The empirical results allow us to specify our general expressions in (1) and (2) such that that patents are a function of 1-year lagged patent flow, 2-year lagged patent stock, and 2-year lagged P2 stock (Table 2, Model VI); while P2 is explained by 1-year lagged P2 flow and 1-year lagged patent flow (Table 3, Model I). Pollution is affected by contemporaneous P2 and patents (Table 4, Model II). Denoting current (lagged) EnvPatents as ${P}_{it} \left({P}_{it-1}\right)$, 2-year lagged EnvPatent Stock as ${P}_{it-2}^{stock}$, current (lagged) Total P2 as ${S}_{it}\left({ S}_{it-1}\right)$, 2-year lagged Total P2 Stock as ${S}_{it-2}^{Stock}$, and lagged Total Releases as ${Q}_{it-1}$, we can express these models as follows:

$${P}_{it}=exp\left({a}_{P}+{b}_{P}{Q}_{it-1} +{c}_{P2}{S}_{it-2}^{stock}+{d}_{P1}{P}_{it-1}+{d}_{P2}{P}_{it-2}^{stock}+{{\varvec{e}}}_{{\varvec{P}}}{{\varvec{Z}}}_{{\varvec{it}}}^{{\varvec{P}}}+{{\varvec{\eta}}}_{{\varvec{i}}}^{{\varvec{P}}}+{{\varvec{v}}}_{{\varvec{t}}}+{\varepsilon }_{it}^{P}\right)$$

(A.1)

$${S}_{it}=exp\left({a}_{S} +{b}_{S}{Q}_{it-1} + {c}_{S}{S}_{it-1} +{d}_{S}{P}_{it-1} +{{\varvec{e}}}_{{\varvec{S}}}{{\varvec{Z}}}_{{\varvec{it}}}^{{\varvec{S}}}+{{\varvec{\eta}}}_{{\varvec{i}}}^{{\varvec{S}}}+{{\varvec{v}}}_{{\varvec{t}}}+{\varepsilon }_{it}^{S}\right)$$

(A.2)

$${Q}_{it}={a}_{Q}+{b}_{Q}{Q}_{it-1}+{c}_{Q}{S}_{it}+{d}_{Q}{P}_{it} +{{\varvec{e}}}_{{\varvec{Q}}}{{\varvec{Z}}}_{{\varvec{it}}}^{{\varvec{Q}}}+{\eta }_{i}^{Q}+{{\varvec{v}}}_{{\varvec{t}}}+{\varepsilon }_{it}^{Q}$$

(A.3)

In the derivations that follow, we simplify the notation when calculating marginal effects (as elasticities) using count models. For example, we denote $\frac{\partial {P}_{t+1}}{\partial {P}_{t}}=\frac{\partial log\left[E({P}_{t+1}|Q,K,Z)\right]}{\partial {P}_{t}}$, and $\frac{\partial {S}_{t+1}}{\partial {S}_{t}}=\frac{\partial log\left[E({S}_{t+1}|Q,K,Z)\right]}{\partial {S}_{t}}$ .  We use elasticities implied by estimates in Tables 2 and 3.

To calculate the effect of an extra environmental patent adopted at time t on pollution τ periods hence, we take the derivative of Eq. (3′) with respect to ${P}_{t}$. We note that pollution at any period is a function of lagged pollution which is in itself a function of ${P}_{t}$ and ${S}_{t}$, and of ${P}_{t+\tau }$ and ${S}_{t+\tau }$, that are themselves functions of ${P}_{t}$ and ${S}_{t}$.

$$\begin{aligned} \frac{{dQ_{t + \tau } }}{{dP_{t} }} & = \underbrace {{\frac{{\partial Q_{t + \tau } }}{{\partial Q_{t + \tau - 1} }}\left[ {\frac{{\partial Q_{t + \tau - 1} }}{{\partial Q_{t + \tau - 2} }} \ldots \frac{{\partial Q_{t} }}{{\partial P_{t} }}} \right]}}_{pollution\;path\;dependence} \\ & \quad + \underbrace {{\frac{{\partial Q_{t + \tau } }}{{\partial P_{t + \tau } }}\left[ {\overbrace {{\left( {\frac{{\partial P_{t + \tau } }}{{\partial P_{t + \tau - 1} }}\frac{{\partial P_{t + \tau - 1} }}{{\partial P_{t + \tau - 2} }} \ldots \frac{{\partial P_{t + 1} }}{{\partial P_{t} }}} \right)}}^{via\;lagged\;patent} + \overbrace {{\left( {\frac{{\partial P_{t + \tau } }}{{\partial P_{t + \tau - 2}^{stock} }}\frac{{\partial P_{t + \tau - 2}^{stock} }}{{\partial P_{t} }}} \right)}}^{via\;patent\;stock} + \overbrace {{\left( {\frac{{\partial P_{t + \tau } }}{{\partial P_{t + \tau - 1} }}\frac{{\partial P_{t + \tau - 1} }}{{\partial P_{t + \tau - 2} }} \ldots \frac{{\partial P_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial P_{t} }}} \right)}}^{adjustment\;factor}} \right]}}_{pollution\;change\;due\;to\;patent\;knowledge\;accumulation\;from\;extra\;patent} \\ & \quad + \underbrace {{\frac{{\partial Q_{t + \tau } }}{{\partial S_{t + \tau } }}\left[ {\overbrace {{\left( {\frac{{\partial S_{t + \tau } }}{{\partial S_{t + \tau - 1} }}\frac{{\partial S_{t + \tau - 1} }}{{\partial S_{t + \tau - 2} }} \ldots \frac{{\partial S_{t + 1} }}{{\partial P_{t} }}} \right)}}^{via\;lagged\;P2} + \overbrace {{\left( {\frac{{\partial S_{t + \tau } }}{{\partial S_{t + \tau - 1} }}\frac{{\partial S_{t + \tau - 1} }}{{\partial S_{t + \tau - 2} }} \ldots \frac{{\partial S_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial P_{t} }}} \right)}}^{adjustment\;factor}} \right]}}_{pollution\;change\;due\;to\;P2\;knowledge\;accumulation\;from\;extra\;patent} \\ \end{aligned}$$

The first term illustrates how the trajectory of pollution is lower due to patenting. The second term is the same-knowledge accumulation effect that results from most recent new patent and further lagged patent knowledge stock. The third term is the cross-knowledge accumulation effect because past patenting affects new P2 which in turn affects pollution. Note that for the second and third terms the learning effects include an adjustment factor arising from how the effect of an initial patent on pollution can encourage or discourage future patenting and future P2. Because patenting occurring in the initial period can lower pollution immediately, $\frac{\partial {Q}_{t}}{\partial {P}_{t}}<0$, the adjustment terms will reinforce (mitigate) the knowledge accumulation effects if lower pollution in the adoption year will encourage (discourage) subsequent patenting and P2, i.e., $\frac{\partial {P}_{t+1}}{\partial {Q}_{t}}<0;\; \frac{\partial {S}_{t+1}}{\partial {Q}_{t}}<0$ ($\frac{\partial {P}_{t+1}}{\partial {Q}_{t}}>0;\; \frac{\partial {S}_{t+1}}{\partial {Q}_{t}}>0$). All these effects will be felt 2 years after adoption because of the 2-year lagged effect of patent stock on new patent flow).

In the year of adoption, τ = 0, the only effect is through the contemporaneous impact which is the focus of Sect. 5.2, $\frac{d{Q}_{t}}{d{P}_{t}}={d}_{Q}$.

One year after adoption at time t + 1, the effect of patenting at time t is:

$$\begin{aligned} \frac{{dQ_{t + 1} }}{{dP_{t} }} & = \frac{{\partial Q_{t + 1} }}{{\partial Q_{t} }}\left[ {\frac{{\partial Q_{t} }}{{\partial P_{t} }}} \right] + \frac{{\partial Q_{t + 1} }}{{\partial P_{t + 1} }}\left[ {\left( {\frac{{\partial P_{t + 1} }}{{\partial P_{t} }}} \right) + \left( {\frac{{\partial P_{t + 1} }}{{\partial P_{t - 1}^{stock} }}\frac{{\partial P_{t - 1}^{stock} }}{{\partial P_{t - 1} }}} \right) + \left( {\frac{{\partial P_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial P_{t} }}} \right)} \right] \\ & \quad + \frac{{\partial Q_{t + 1} }}{{\partial S_{t + 1} }}\left[ {\left( {\frac{{\partial S_{t + 1} }}{{\partial P_{t} }}} \right) + \left( {\frac{{\partial S_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial P_{t} }}} \right)} \right] \\ \end{aligned}$$

Since we are interested in a new patent adopted at time t, not any earlier, several terms are zero. By substituting for all coefficients from (1′), (2′) and (3′):

$$\frac{{dQ_{t + 1} }}{{dP_{t} }} = b_{Q} d_{Q} + d_{Q} \left[ {d_{P1} + 0 + 0} \right] + c_{Q} \left[ {d_{S} + b_{S} d_{Q} } \right] = b_{Q} d_{Q} + d_{Q} d_{P1} + { }c_{Q} \left[ {d_{S} + b_{S} d_{Q} } \right]$$

(A.4)

Two years after adoption at time t + 2, the effect of patenting at time t is derived similarly, noting that new patenting at time t + 2 now also influenced by the effect of 2-year lagged patent stock, ${P}_{t}^{stock}$ and the adjustment terms are non-zero.

$$\begin{aligned} \frac{{dQ_{t + 2} }}{{dP_{t} }} & = \frac{{\partial Q_{t + 2} }}{{\partial Q_{t - 1} }}\left[ {\frac{{\partial Q_{t - 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial P_{t} }}} \right] \\ & \quad + \frac{{\partial Q_{t + 2} }}{{\partial P_{t + 2} }}\left[ {\left( {\frac{{\partial P_{t + 2} }}{{\partial P_{t + 1} }}\frac{{\partial P_{t + 1} }}{{\partial P_{t} }}} \right) + \left( {\frac{{\partial P_{t + 2} }}{{\partial P_{t}^{stock} }}\frac{{\partial P_{t}^{stock} }}{{\partial P_{t} }}} \right) + \left( {\frac{{\partial P_{t + 2} }}{{\partial P_{t + 1} }}\frac{{\partial P_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial P_{t} }}} \right)} \right] \\ & \quad + \frac{{\partial Q_{t + 2} }}{{\partial S_{t + 2} }}\left[ {\left( {\frac{{\partial S_{t + 2} }}{{\partial S_{t + 1} }}\frac{{\partial S_{t + 1} }}{{\partial P_{t} }}} \right) + \left( {\frac{{\partial S_{t + 2} }}{{\partial S_{t + 1} }}\frac{{\partial S_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t + 1} }}{{\partial P_{t} }}} \right)} \right] \\ \end{aligned}$$

By substituting for all coefficients from (1′), (2′) and (3′) and noting that depreciated cumulated stock of patents can be expressed as ${P}_{it}^{stock}=\sum_{\tau }^{t}{\delta }^{\tau }{P}_{t-\tau }$:

$$\frac{{dQ_{t + 2} }}{{dP_{t} }} = b_{Q}^{2} d_{Q} + d_{Q} \left[ {d_{P1}^{2} + d_{P2} \delta^{0} + d_{P1} b_{P} d_{Q} } \right] + { }c_{Q} \left[ {c_{S} d_{S} + c_{S} b_{S} d_{Q} } \right]$$

(A.5)

Generalizing (A.1.2) for all years $\tau \ge 1$, we have

$$\frac{{dQ_{t + \tau } }}{{dP_{t} }} = b_{Q}^{\tau } d_{Q} + d_{Q} \left[ {d_{P1}^{\tau } + d_{P2} \delta^{\tau - 2} + d_{p1}^{\tau - 1} b_{P} d_{Q} } \right] + { }c_{Q} \left[ {c_{S}^{\tau - 1} d_{S} + c_{S}^{\tau - 1} b_{S} d_{Q} } \right]$$

(A.6)

After multiplying (8) by $\widehat{P}$, which represents what one extra patent is as a proportion to mean patent, we derive Eq. (4) in Sect. 5.3.

The effect of an extra P2 adopted at time t on future emissions can be analyzed similarly. To calculate the effect of an extra P2 adopted at time t on pollution τ periods hence, we take the derivative of Eq. (3′) with respect to ${S}_{t}$, recognizing that pollution at any period is a function of lagged pollution which is in itself a function of ${P}_{t}$ and ${S}_{t}$, and of ${P}_{t+\tau }$ and ${S}_{t+\tau }$, that are themselves functions of ${P}_{t}$ and ${S}_{t}$.

$$\begin{aligned} \frac{{dQ_{t + \tau } }}{{dS_{t} }} & = \underbrace {{\frac{{\partial Q_{t + \tau } }}{{\partial Q_{t + \tau - 1} }}\left[ {\frac{{\partial Q_{t + \tau - 1} }}{{\partial Q_{t + \tau - 2} }} \ldots \frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right]}}_{pollution\;path\;dependence} \\ & \quad + \underbrace {{\frac{{\partial Q_{t + \tau } }}{{\partial S_{t + \tau } }}\left[ {\overbrace {{\left( {\frac{{\partial S_{t + \tau } }}{{\partial S_{t + \tau - 1} }}\frac{{\partial S_{t + \tau - 1} }}{{\partial S_{t + \tau - 2} }} \ldots \frac{{\partial S_{t + 1} }}{{\partial S_{t} }}} \right)}}^{via\;lagged\;P2} + \overbrace {{\left( {\frac{{\partial S_{t + \tau } }}{{\partial S_{t + \tau - 1} }}\frac{{\partial S_{t + \tau - 1} }}{{\partial S_{t + \tau - 2} }} \ldots \frac{{\partial S_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)}}^{adjustment\;factor}} \right]}}_{pollution\;change\;due\;to\;P2 \;knowledge\;accumulation\;from\;extra\;P2} \\ & \quad + \underbrace {{\frac{{\partial Q_{t + \tau } }}{{\partial P_{t + \tau } }}\left[ {\overbrace {{\left( {\frac{{\partial P_{t + \tau } }}{{\partial P_{t + \tau - 1} }}\frac{{\partial P_{t + \tau - 1} }}{{\partial P_{t + \tau - 2} }} \ldots \frac{{\partial P_{t + 2} }}{{\partial S_{t}^{stock} }}\frac{{\partial S_{t}^{stock} }}{{\partial S_{t} }}} \right)}}^{via\;lagged\;patent} + \overbrace {{\left( {\frac{{\partial P_{t + \tau } }}{{\partial P_{t + \tau - 2}^{stock} }}\frac{{\partial P_{t + \tau - 2}^{stock} }}{{\partial P_{t + 2} }}\frac{{\partial P_{t + 2} }}{{\partial S_{t}^{stock} }}\frac{{\partial S_{t}^{stock} }}{{\partial S_{t} }}} \right)}}^{via\;patent\;stock} + \overbrace {{\left( {\frac{{\partial P_{t + \tau } }}{{\partial P_{t + \tau - 1} }}\frac{{\partial P_{t + \tau - 1} }}{{\partial P_{t + \tau - 2} }} \ldots \frac{{\partial P_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)}}^{adjustment\;factor}} \right]}}_{pollution\;change\;due\;to\;patent\;knowledge\;accumulation\;from\;extra\;P2} \\ \end{aligned}$$

All expressions are analogous to the expression for patents. The first term illustrates how the trajectory of pollution is lower due to an additional P2 practice. The second term is the same-knowledge accumulation effect that results from most recent new P2. The third term is the cross-knowledge accumulation effect because P2 affects patenting which then affects pollution. There are again adjustment factors in the second and third terms arising from how the effect of an initial P2 on pollution can encourage or discourage future P2 and future patenting. Because P2 occurring in the initial period lowers pollution immediately, $\frac{\partial {Q}_{t}}{\partial {S}_{t}}<0$, the adjustment terms will reinforce (mitigate) the knowledge accumulation effects if lower pollution in the adoption year will encourage (discourage) subsequent patenting and P2, i.e., $\frac{\partial {P}_{t+1}}{\partial {Q}_{t}}<0;\; \frac{\partial {S}_{t+1}}{\partial {Q}_{t}}<0$ ($\frac{\partial {P}_{t+1}}{\partial {Q}_{t}}>0;\; \frac{\partial {S}_{t+1}}{\partial {Q}_{t}}>0$). All these effects will be felt 4 years after adoption because of the 2-year lagged effect of patent stock on new patent flow,  the 2-year lagged effect of P2 stock on new P2 flow, and the 2-year lagged effect of P2 stock on patents.

Differentiating Eq. (3′) with respect to ${S}_{t}$, on the year of adoption, the only effect we can see is the immediate effect, $\frac{d{Q}_{t}}{d{S}_{t}}={c}_{Q}$.

One year after adoption at time t + 1, the effect of an extra P2 adopted at time t is:

$$\begin{aligned} \frac{{dQ_{t + 1} }}{{dS_{t} }} & = \frac{{\partial Q_{t + 1} }}{{\partial Q_{t} }}\left[ {\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right] + \frac{{\partial Q_{t + 1} }}{{\partial S_{t + 1} }}\left[ {\left( {\frac{{\partial S_{t + 1} }}{{\partial S_{t} }}} \right) + \left( {\frac{{\partial S_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)} \right] \\ & \quad + \frac{{\partial Q_{t + 1} }}{{\partial P_{t + 1} }}\left[ {\left( {\frac{{\partial P_{t + 1} }}{{\partial S_{t - 1}^{stock} }}\frac{{\partial S_{t - 1}^{stock} }}{{\partial S_{t - 1} }}} \right) + \left( {\frac{{\partial P_{t + 1} }}{{\partial P_{t - 1}^{stock} }}\frac{{\partial P_{t - 1}^{stock} }}{{\partial P_{t - 1} }}\frac{{\partial P_{t - 1} }}{{\partial S_{t - 3}^{stock} }}\frac{{\partial S_{t - 3}^{stock} }}{{\partial S_{t - 3} }}} \right)} \right. \\ & \quad \left. { + \left( {\frac{{\partial P_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)} \right] \\ \end{aligned}$$

Substituting the coefficients from (1′), (2′), the expression at t + 1 is:

$$\begin{aligned} \frac{{dQ_{t + 1} }}{{dS_{t} }} & = b_{Q} c_{Q} + c_{Q} \left[ {c_{S} + b_{S} c_{Q} } \right] + d_{Q} \left[ {0 + 0 + b_{P} c_{Q} } \right] \\ & = b_{Q} c_{Q} + c_{Q} \left[ {c_{S} + b_{S} c_{Q} } \right] + d_{Q} b_{P} c_{Q} \\ \end{aligned}$$

(A.7)

Two and three years after adoption, the general expression can be extended to incorporate longer pollution path dependence effects and longer adjustment in the P2 knowledge accumulation term. We are also able to capture the first expression in the third term. The second expression in the third bracket will still be zero because we are interested in an extra P2 at time t and not any earlier.

$$\begin{aligned} \frac{{dQ_{t + 2} }}{{dS_{t} }} & = \frac{{\partial Q_{t + 2} }}{{\partial Q_{t} }}\left[ {\frac{{\partial Q_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right] + \frac{{\partial Q_{t + 2} }}{{\partial S_{t + 2} }}\left[ {\left( {\frac{{\partial S_{t + 2} }}{{\partial S_{t + 1} }}\frac{{\partial S_{t + 1} }}{{\partial S_{t} }}} \right) + \left( {\frac{{\partial S_{t + 2} }}{{\partial S_{t + 1} }}\frac{{\partial S_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)} \right] \\ \quad + { }\frac{{\partial Q_{t + 2} }}{{\partial P_{t + 2} }}\left[ {\left( {\frac{{\partial P_{t + 2} }}{{\partial S_{t}^{stock} }}\frac{{\partial S_{t}^{stock} }}{{\partial S_{t} }}} \right) + \left( {\frac{{\partial P_{t + 2} }}{{\partial P_{t}^{stock} }}\frac{{\partial P_{t}^{stock} }}{{\partial P_{t} }}\frac{{\partial P_{t} }}{{\partial S_{t - 2}^{stock} }}\frac{{\partial S_{t - 2}^{stock} }}{{\partial S_{t - 2} }}} \right) + \left( {\frac{{\partial P_{t + 2} }}{{\partial P_{t + 1} }}\frac{{\partial P_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)} \right] \\ \frac{{dQ_{t + 3} }}{{dS_{t} }} & = \frac{{\partial Q_{t + 3} }}{{\partial Q_{t + 2} }}\left[ {\frac{{\partial Q_{t + 2} }}{{\partial Q_{t + 1} }}\frac{{\partial Q_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right] + \frac{{\partial Q_{t + 3} }}{{\partial S_{t + 3} }}\left[ {\left( {\frac{{\partial S_{t + 3} }}{{\partial S_{t + 2} }}\frac{{\partial S_{t + 2} }}{{\partial S_{t + 1} }}\frac{{\partial S_{t + 1} }}{{\partial S_{t} }}} \right) + \left( {\frac{{\partial S_{t + 3} }}{{\partial S_{t + 2} }}\frac{{\partial S_{t + 2} }}{{\partial S_{t + 1} }}\frac{{\partial S_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)} \right] \\ & \quad + \frac{{\partial Q_{t + 3} }}{{\partial P_{t + 3} }}\left[ {\left( {\frac{{\partial P_{t + 3} }}{{\partial P_{t + 2} }}\frac{{\partial P_{t + 2} }}{{\partial S_{t}^{stock} }}\frac{{\partial S_{t}^{stock} }}{{\partial S_{t} }}} \right) + \left( {\frac{{\partial P_{t + 3} }}{{\partial P_{t + 1}^{stock} }}\frac{{\partial P_{t + 1}^{stock} }}{{\partial P_{t + 1} }}\frac{{\partial P_{t + 1} }}{{\partial S_{t - 1}^{stock} }}\frac{{\partial S_{t - 1}^{stock} }}{{\partial S_{t - 1} }}} \right)} \right. \\ & \quad \left. { + \left( {\frac{{\partial P_{t + 3} }}{{\partial P_{t + 2} }}\frac{{\partial P_{t + 2} }}{{\partial P_{t + 1} }}\frac{{\partial P_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)} \right] \\ \end{aligned}$$

Substituting the coefficients from (1′), (2′) and noting that depreciated cumulated stock of P2 can be expressed as ${S}_{it}^{stock}=\sum_{\tau }^{t}{\delta }^{\tau }{S}_{t-\tau }$:

$$\frac{{dQ_{t + 2} }}{{dS_{t} }} = b_{Q}^{2} c_{Q} + c_{Q} \left[ {c_{S}^{2} + c_{S} b_{S} c_{Q} } \right] + d_{Q} \left[ {c_{P2} \delta^{0} + 0 + d_{P} b_{P} c_{Q} } \right]$$

(A.8)

$$\frac{{dQ_{t + 3} }}{{dS_{t} }} = b_{Q}^{3} c_{Q} + c_{Q} \left[ {c_{S}^{3} + c_{S}^{2} b_{S} c_{Q} } \right] + d_{Q} \left[ {d_{P1} c_{P2} \delta^{0} + 0 + d_{P}^{2} b_{P} c_{Q} } \right]$$

(A.9)

Four years after an extra P2 adoption, all terms are non-zero:

$$\begin{aligned} \frac{{dQ_{t + 4} }}{{dS_{t} }} & = \frac{{\partial Q_{t + 4} }}{{\partial Q_{t + 3} }}\left[ {\frac{{\partial Q_{t + 3} }}{{\partial Q_{t + 2} }}\frac{{\partial Q_{t + 2} }}{{\partial Q_{t + 1} }}\frac{{\partial Q_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right] + \frac{{\partial Q_{t + 4} }}{{\partial S_{t + 4} }}\left[ {\left( {\frac{{\partial S_{t + 4} }}{{\partial S_{t + 3} }}\frac{{\partial S_{t + 3} }}{{\partial S_{t + 2} }}\frac{{\partial S_{t + 2} }}{{\partial S_{t + 1} }}\frac{{\partial S_{t + 1} }}{{\partial S_{t} }}} \right)} \right. \\ & \quad \left. { + \left( {\frac{{\partial S_{t + 4} }}{{\partial S_{t + 3} }}\frac{{\partial S_{t + 3} }}{{\partial S_{t + 2} }}\frac{{\partial S_{t + 2} }}{{\partial S_{t + 1} }}\frac{{\partial S_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)} \right] + { }\frac{{\partial Q_{t + 4} }}{{\partial P_{t + 4} }} \\ & \quad \left[ {\begin{array}{*{20}c} {\left( {\frac{{\partial P_{t + 4} }}{{\partial P_{t + 3} }}\frac{{\partial P_{t + 3} }}{{\partial P_{t + 2} }}\frac{{\partial P_{t + 2} }}{{\partial S_{t}^{stock} }}\frac{{\partial S_{t}^{stock} }}{{\partial S_{t} }}} \right) + \left( {\frac{{\partial P_{t + 4} }}{{\partial P_{t + 2}^{stock} }}\frac{{\partial P_{t + 2}^{stock} }}{{\partial P_{t + 2} }}\frac{{\partial P_{t + 2} }}{{\partial S_{t}^{stock} }}\frac{{\partial S_{t}^{stock} }}{{\partial S_{t} }}} \right)} \\ { + \left( {\frac{{\partial P_{t + 4} }}{{\partial P_{t + 3} }}\frac{{\partial P_{t + 3} }}{{\partial P_{t + 2} }}\frac{{\partial P_{t + 2} }}{{\partial P_{t + 1} }}\frac{{\partial P_{t + 1} }}{{\partial Q_{t} }}\frac{{\partial Q_{t} }}{{\partial S_{t} }}} \right)} \\ \end{array} } \right] \\ \end{aligned}$$

Substituting the coefficients from (1′), (2′) and noting that ${S}_{it}^{stock}=\sum_{\tau }^{t}{\delta }^{\tau }{S}_{t-\tau }$:

$$\frac{{dQ_{t + 4} }}{{dS_{t} }} = b_{Q}^{4} c_{Q} + c_{Q} \left[ {c_{S}^{4} + c_{S}^{3} b_{S} c_{Q} } \right] + d_{Q} \left[ {d_{P1}^{2} c_{P2} \delta^{0} + d_{P2} \delta^{0} c_{P2} \delta^{0} + d_{P}^{3} b_{P} c_{Q} } \right]$$

(A.10)

Generalizing for all years $\tau \ge 1$, and simplifying

$$\frac{{dQ_{t + 4} }}{{dS_{t} }} = b_{Q}^{\tau } c_{Q} + c_{Q} \left[ {c_{S}^{\tau } + c_{S}^{\tau - 1} b_{S} c_{Q} } \right] + d_{Q} \left[ {d_{P1}^{\tau - 2} c_{P2} + d_{P2} \delta^{\tau - 4} c_{P2} + d_{P1}^{\tau - 1} b_{P} c_{Q} } \right]$$

(A.11)

After multiplying (13) by $\widehat{S}$, which represents what one extra P2 is as a proportion to mean P2, we derive Eq. (5) in Sect. 5.3.