## 4.2 Research and Development

One of the main features of the WITCH model is the characterisation of endogenous technical change. Albeit difficult to model, technological innovation is key to the decoupling of economic activity from environmental degradation, and the ability to induce it using appropriate policy instruments is essential for a successful climate agreement, as highlighted also in the Bali Action Plan.

Both innovation and diffusion processes are modelled. We distinguish dedicated R&D investments for enhancing energy efficiency from investments aimed at facilitating the competitiveness of innovative low carbon technologies (backstops) in both the electric and non-electric sectors. The returns to R&D investment depend on the stock of previously accumulated knowledge. Higher knowledge stock facilitates generation of new, energy saving innovations. In addition, international spillovers of knowledge are accounted for to mimic the flow of ideas and knowledge across countries.

### 4.2.1 Knowledge stock in backstop technologies, electric batteries, and energy efficiency

Stocks of knowledge are defined for two backstop technologies, batteries for EDVs, and overall energy efficiency improvements. At each point in time, new ideas are produced using a Cobb-Douglas combination between domestic investments in innovation, $$I_{rd}$$, the existing stock of knowledge, RD_{rd} and the knowledge of other countries, $$SPILL$$. That is, for four sectors $$rd_{el\_back, nel\_back, RDEN, battery}$$ we have a knowledge stock, which accumulates with the perpetual rule and with the contribution of international knowledge spillovers, $$SPILL$$, in the respective sector $$RD$$

$RD_{rd}(t+1,n) = RD_{rd}(t,n) (1-\delta_{rd})^{\Delta_{t}} + \Delta_{t} \times a \times I_{rd}(t,n)^b \times RD_{rd}(t,n)^c \times SPILL_{rd}(t,n)^d$ The coefficients of the knowledge production function are sector dependent, and are calibrated based on (Bosetti et al. 2008) and (Popp 2004). The parameter a is a simple scale parameter, based on an original value of a=0.3022, converted from billion USD to trillions computed as 0.3022*1000^(b+c+d-1) = 0.041925858 for energy eficiency, while it is equal to one for the other sectors, where the exponentes add up to one. The parameter d=0.15 implies that a 1% increase in spillovers increases output of domestic ideas by 0.15%. The return to one’s own investment (b) is slightly higher (b=0.18) for energy eficiency while it contributes the remaining 85% for the ogher sectors. Only for energy efficienct, the additional knowledge produced depends also on the existing knowledge stock (c=0.38>0) since Popp (2004, JEEM) estimates 0.53 for the total effect from knowledge stock without spillovers. Thus deducting the efect from spillovers, we use 0.53-0.15=0.38 for the vakye of $$c$$ for energy efficiency, while this effect is zero for the other sectors.

Parameter elback nelback battery en
a 1 1 1 0.0419258
b 0.85 0.85 0.85 0.18
c 0 0 0 0.3840625
d 0.15 0.15 0.15 0.15

The contribution of foreign knowledge through spillovers to the production of new domestic ideas depends on the interaction between two terms: the first describes the absorptive capacity whereas the second captures the distance from the technology frontier, which is represented by the stock of knowledge in OECD countries.

$SPILL_{rd}(t,n) = \frac{RD_{rd}(t,n)}{\sum_{n\in \text{OECD}} RD_{rd}(t,n)} \times \left( \sum_{n\in \text{OECD}} RD_{rd}(t,n) - RD_{rd}(t,n) \right)$

The knowledge stock dedicated for energy efficiency is combined with energy supply and autonomous energy efficiency improvement to form energy services. Energy services are then used as an input in production of final good.

In backstop technologies, the knowledge stock is used to lower installation costs, $$SC$$, which are determined by a two-factor learning curve.

### 4.2.2 Two-Factor Learning Curve

In two-factor learning curves (see e.g. (Klaassen et al. 2005) and (Söderholm and Sundqvist 2007)), investment costs decrease as a result of the accumulation of knowledge (learning-by-researching) or experience (learning-by-doing). The accumulation of knowledge is produced by investments in research and development, as discussed above, while the stock of experience is proxied with global cumulated installed capacity, $$wcum$$ (full global technology spillover is assumed). The two-factor learning curve takes the following form:

$\frac{SC_{j}(t,n)}{SC_{j}(0,n)} = \left( \frac{RD_{j}(t,n)}{RD_{j}(0,n)} \right)^{-lbr\_factor} \left( \frac{wcum_{j}(t,n)}{wcum_{j}(0,n)} \right)^{-lbd\_factor}$

where $$lbr\_factor$$ and $$lbd\_factor$$ measure the strength of the learning effect. They relate to the corresponding learning rates, $$lbr\_rate$$ and $$lbd\_rate$$, which measure the rate at which unit costs decrease for each doubling of the knowledge or capacity stock, through the following relationship:

${lbd\_rate}= 1 - 2^{-lbd\_factor}$

The equation is written for learning-by-doing, but the same applies to learning-by-researching, obviously.

### 4.2.3 One-Factor Learning Curve

The cost evolution of wind (onshore and offshore) and solar (PV and CSP) technologies follows a technical change framework as well, but in this case only learning-by-doing is taken into account. Thus, investment costs decrease according to the progressive technology deployment (global cumulative capacity), while no dedicated R&D investments are considered.

$\frac{SC_{j}(t,n)}{SC_{j}(0,n)} = \left( \frac{wcum_{j}(t,n)}{wcum_{j}(0,n)} \right)^{-lbd\_factor}$

On the contrary, the cost evolution of vehicle batteries follows a one-factor learning curve based on learning-by-researching:

$\frac{SC_{j}(t,n)}{SC_{j}(0,n)} = \left( \frac{RD_{j}(t,n)}{RD_{j}(0,n)} \right)^{-lbr\_factor}$

### References

Bosetti, Valentina, Carlo Carraro, Emanuele Massetti, and Massimo Tavoni. 2008. “International Energy R&D Spillovers and the Economics of Greenhouse Gas Atmospheric Stabilization.” Energy Economics 30 (6): 2912–29. https://doi.org/http://dx.doi.org/10.1016/j.eneco.2008.04.008.

Popp, David. 2004. “ENTICE: Endogenous Technological Change in the Dice Model of Global Warming.” Journal of Environmental Economics and Management 48 (1): 742–68. https://doi.org/http://dx.doi.org/10.1016/j.jeem.2003.09.002.

Klaassen, Ger, Asami Miketa, Katarina Larsen, and Thomas Sundqvist. 2005. “The Impact of R&D on Innovation for Wind Energy in Denmark, Germany and the United Kingdom.” Ecological Economics 54 (2): 227–40.

Söderholm, Patrik, and Thomas Sundqvist. 2007. “Empirical Challenges in the Use of Learning Curves for Assessing the Economic Prospects of Renewable Energy Technologies.” Renewable Energy 32 (15): 2559–78.