Silviase/jgov_v1.4_full_ann huggingface dataset:
$$ \frac{\partial Y\left(t\right)}{\partial HKL\ \left(t\right)}\frac{\partial HKL\ \left(t\right)}{\partial I_{HKL}\ \left(t-1\right)}+\frac{I_{HKL}\ \left(t\right)}{I_{HKL}\ \left(t-1\right)}=\beta\ \frac{c\left(t\right)}{c\left(t-1\right)} $$

Proof: See Appendix B.

From the two conditions above we can easily derive:

$$ \frac{I_{HKL}\left(t\right)}{I_{HKL}\left(t-1\right)}=\frac{I_{HE}\left(t\right)}{\underbrace{I_{HE}\left(t-1\right)}_{A}}+\frac{\partial Y\left(t\right)}{\underbrace{\partial HE\left(t\right)}^{}\frac{\partial HE\left(t\right)}{\partial I_{HE}\left(t-1\right)}}-\frac{\partial Y\left(t\right)}{\underbrace{\partial HKL\left(t\right)}^{}\frac{\partial HKL\left(t\right)}{\partial I_{HKL}\left(t-1\right)}} $$

Equation (10) allows us to identify the different channels through which mitigation policy affects the rate of growth of investments in Non-Energy R&D. First, from equation (10) we can rule out any “direct” crowding out effect by Energy R&D. The positive sign of the first term on the RHS (A) shows instead that an increase in Energy R&D investments has a direct positive effect on the growth rate of investments in Non-Energy R&D. However, the effect of an increase in Energy R&D investments also depends on the magnitude of the change of the marginal product of Energy R&D investments and of the marginal product of Non-Energy R&D investments, the second (B) and the third (C) term on the RHS of equation (10), respectively.

To isolate the total effect of an increase in Energy R&D, we have performed an exercise in which, in the BaU scenario, we impose an Energy R&D expenditure path equal to the equilibrium path in the stabilization scenario. This exercise is designed to identify the amount of non-energy R&D that would be displaced by an exogenous increase in energy R&D, with virtually no price effect. We must emphasize that, in running our test, we do not make any additional restriction to the other choice variables in the model whose equilibrium path is thus adjusted to the new assumptions in the R&D sector. Simulation results show that Non-Energy R&D investments respond positively to an increase in Energy R&D, revealing a (mild) degree of complementarity between the two knowledge stocks.

In addition, using equation (10) and numerical results from simulations in the stabilization scenario, we can provide an explanation for the reduction of the rate of investments in Energy R&D that depends on the structure of the economy, as portrayed in the model, and not on the “crowding out” hypothesis usually proposed in the literature. The decrease in the rate of investments in Non-Energy R&D can indeed be explained by the effect of mitigation policy on the marginal product of Energy
R&D investments and on the marginal product of Non-Energy R&D investments, the second $ (B) $ and the third term $ (C) $ on the RHS of equation (10), respectively.

As to the marginal product of investments in Energy R&D, since the mitigation policy increases investments to achieve higher energy efficiency and induces a substitution from Energy Services (ES) to Capital-Labor services (KLS), the equilibrium path determined in the stabilization scenario shows that $ \partial HE/\partial I_{HE} $ is always lower than in the BaU scenario and $ \partial Y/\partial HE $ is instead always higher because of a higher KLS/ES ratio. The combination of the two effects results in a contraction of B in the first 60 years (due to the strong effort in Energy R&D investments) and in an expansion in the last 40 years of the century (due to a marked shift away from energy in the outer nest), with respect to the BaU scenario.

As to the marginal product of investment in Non-Energy R&D, C on the RHS, its absolute value is always higher than in the BaU scenario because both $ \partial HKL/\partial I_{HKL} $ and $ \partial Y/\partial HKL $ increase. The first term increases because of the reduction of Non-Energy R&D investments, the latter increases because, as discussed in Section 4, our model features energy-biased technical change – i.e. higher investments in Non-Energy R&D trigger higher demand of energy services (ES). Therefore, when an ambitious climate policy is implemented, energy demand can be reduced by slowing down the pace of accumulation of the pollution augmenting Non-Energy knowledge stock, thus increasing its marginal product.

### 5.4 Induced Technical Change and the Cost of Climate Policy

In this section we compare the cost of the stabilization policy under alternative assumptions about induced technical change (ITC). We consider four alternative scenarios.

In the first one, we assume that it is not possible to change the Non-Energy knowledge stock and we exogenously set it equal to its equilibrium BaU level. In the second scenario, we maintain the assumption that Non-Energy knowledge stock is exogenously set at its BaU level, and we make the additional assumption that investments in Energy R&D partly crowd out Non-energy R&D (we assume a 50% crowding out as in Popp, 2004). In the third scenario, we assume that it is not possible to adjust the Energy knowledge stock, which is exogenously set equal to its BaU level. In the fourth scenario, we assume that both knowledge stocks are exogenously set at their BaU level.

For each scenario, we compare the effect of the stabilization policy on the price of carbon permits and on discounted climate policy costs, measured as the ratio between net discounted GWP losses and the
net discounted GWP in the BaU scenario. Both indicators are widely used to assess the implications of alternative scenarios of ITC on climate policy costs.

Table 9. Percentage change of permit prices under alternative scenarios

2002	2022	2042	2062	2082	2102
NoITC_HKL	0.31%	0.36%	0.63%	1.14%	1.50%	0.12%
Crowding-out	-0.31%	0.28%	0.52%	0.90%	1.26%	0.11%
NoITC_He	0.62%	0.69%	0.99%	1.49%	1.84%	0.15%
NoITC	0.00%	1.09%	1.61%	2.49%	3.34%	0.24%

As to the price of emission permits, we find that ITC always lowers the equilibrium profile of carbon prices, as shown in Table 9. It is interesting to notice that omitting ITC in the Energy Sector leads to a higher overestimate of carbon prices than in the scenario where Non-Energy ITC is excluded.

Finally, let us focus on global GWP losses. The most interesting conclusion is as follows. If pollution-augmenting technical change is omitted, the cost of stabilizing $ CO_{2} $ concentrations decreases. Therefore, by focusing only on energy R&D, the cost of climate policy is underestimated (from 3.9% to 3.6% of global GWP in our model).

Therefore, neglecting ITC tends to overestimate the total cost of climate policy. However, if only Energy R&D is endogenized and can be induced by climate policy, the total cost is underestimated.

A sensitivity analysis of the main results reported in this section is contained in Appendix C. Our sensitivity analysis, performed with respect to the values of the main elasticities of substitution assumed in this report, confirms our results.

### 6. CONCLUSIONS

To improve our understanding of the effects of climate policy on technological change, we have introduced directed technical change in WITCH. In this new version of the model, R&D expenditures, and therefore knowledge accumulation, are factor-specific and can be directed either towards increasing energy-efficiency or towards increasing the productivity of Non-Energy inputs, namely capital and labor.
By explicitly modeling the equilibrium dynamics of two R&D capital stocks, we avoid making implicit assumptions on energy R&D crowding-out, and we can study how mitigation policies change the direction and the magnitude of technical change.

Then, we analyzed the implications of a climate policy whose target is to stabilize $ CO_{2} $ concentrations in the atmosphere at 450 ppmv at the end of the century. Simulation results show that this policy induces an increase of Energy R&D investments, as expected. More specifically, the stabilization target requires enhanced energy efficiency and thus switches the ratio of Energy R&D to Non-Energy R&D investments from a declining path to a rising one.

Therefore, contrary to findings in Goulder and Schneider (1999) and Sue Wing (2003), we obtain that climate policy strongly re-directs investments to Energy R&D, and to the energy sector in general. However, the contraction in Non-Energy R&D is greater than the substantial increase in Energy R&D. Therefore, the equilibrium total R&D expenditure is smaller in the stabilization scenario than in the BaU scenario. This latter result was also in Goulder and Schneider (1999) and Sue Wing (2003).

We also find that there is no direct competition between the Energy and Non-Energy R&D (no crowding out), despite the contraction of Non-Energy R&D investments. The decline of this latter variable can be explained as follows. First, a major cause of the Non-Energy R&D investments reduction is output contraction induced by the stabilization policy, which in turn induces a lower demand of capital-labor services and a lower investment in Non-Energy R&D. Second, technical change is mainly capital-labor augmenting and energy-biased in our baseline. This implies that Non-Energy R&D investments are pollution-biased and are therefore strongly discouraged under climate policy.

Therefore, our results suggest that a greater expenditure on energy R&D under a stabilization policy might not lead to a lesser effort on other types of R&D, and thus to negative macroeconomic consequences. Output contraction is the reason of lower total R&D investments, and not vice-versa. Finally, sensitivity analysis on key elasticities of substitution has shown that our results are robust to a wide range of key elasticities' values.

Our results challenge the widespread intuition that a low-carbon world would be a world with a higher rate of technological innovation. By using the WITCH model, we have shown that R&D efforts to increase energy efficiency and to de-carbonize the energy sector will increase enormously if a stringent climate policy is implemented. However, since they represent a small fraction of overall R&D expenditure, this expansion of knowledge creation in the energy sector might be not sufficient.
to compensate for the contraction of R&D expenditure in other sectors, induced by a lower level of economic activity.

Finally, it is clear that when energy and other inputs are complements – as suggested by many empirical studies – any capital-labor augmenting technical change is fundamentally a pollution-biased technical change and is thus discouraged by price signals induced by climate policy.

The consequence for studies that aim at assessing the economic cost of climate policy is as follows. The role of technical change to achieve GHG concentration stabilization targets is crucial. If a study neglects the impact of climate stabilization on technical change, it will overestimate the cost of climate policy, as clearly shown in Edenhofer (2006). However, if a study considers the impact of climate policy only on energy-related and carbon-free technical change, the cost of climate policy will be underestimated. A proper assessment of the cost of climate policy must take into account the contrasting effects induced on both Energy and Non Energy R&D and related consequences. $ ^{9} $
## Appendix A: Model Equations and List of Variables

In this Appendix, we reproduce the main equations of the WITCH model. For a full description of the model please refer to Bosetti, Massetti and Tavoni (2007). The list of variables is reported at the end of this Appendix. In each region, indexed by n, a social planner maximises the following utility function:

$$ W(n)=\sum_{t}U\left[C(n,t),L(n,t)\right]R(t)=\sum_{t}L(n,t)\Big\{\log\big[c(n,t)\big]\Big\}R(t), $$

where t represents 5-year time spans and the pure time preference discount factor is given by:

$$ R(t)=\prod_{\nu=0}^{t}\left[1+\rho(\nu)\right]^{-5}, $$

where the pure rate of time preference $ \rho(v) $ is assumed to decline over time. Moreover, $ c(n,t)=\frac{C(n,t)}{L(n,t)} $ is per capita consumption.

Economic module

The budget constraint defines consumption as net output less investments:

$$ \begin{aligned}C(n,t)=&Y(n,t)-I_{C}(n,t)-I_{R\&D,EN}(n,t)-I_{R\&D,KL}(n,t)\\&-\sum_{j}I_{R\&D,j}(n,t)-\sum_{j}I_{j}(n,t)-\sum_{j}O\&M_{j}(n,t)\end{aligned} $$

Where j is an index identifying different energy technologies. Output is produced via a nested CES function that combines a capital-labor aggregate and energy; capital and labor are obtained from a CES function. The climate damage $ \Omega $ reduces gross output. To obtain net output we also subtract the fuel costs f and the costs of CCS:

$$ \begin{align*}Y\big(n,t\big)=&\frac{TFP\big(n,t\big)\Big[\alpha_{Y}\left(n\right)\cdot KLS^{\rho_{Y}}+\left(1-\alpha_{Y}\left(n\right)\right)\cdot ES\big(n,t\big)^{\rho_{Y}}\Big]^{1/\rho_{Y}}}{\Omega\big(n,t\big)}\\&-\sum_{f}\Big(P_{f}\big(n,t\big)X_{f,extr}\big(n,t\big)+P_{f}^{int}\big(t\big)X_{f,netimp}\big(n,t\big)\Big)\end{align*}. $$

Total factor productivity $ TFP(n,t) $ evolves exogenously with time. Energy services are an aggregate of energy and a stock of knowledge combined with a CES function:

$$ E S\big(n,t\big)=\big[\alpha_{H E}\big(n\big)H E(n,t)^{\rho_{E S}}+\alpha_{E N}\big(n\big)E N(n,t)^{\rho_{E S}}\big]^{1/\rho_{E S}}. $$

Energy is a combination of electric and non-electric energy:

$$ EN\big(n,t\big)=\left[\alpha_{EL}EL(n,t)^{\rho_{EN}}+\alpha_{NEL}NEL(n,t)^{\rho_{EN}}\right]^{1/\rho_{EN}}. $$

Each factor is further decomposed into several sub-components. Factors are aggregated using CES, linear and Leontief production functions. Capital-labor services are obtained aggregating a capital-labor input and a knowledge stock with a CES function:

$$ KLS\big(n,t\big)=\Big[\alpha_{HKL}\big(n\big)HKL(n,t)^{\rho_{KLS}}+\alpha_{KL}\big(n\big)KL(n,t)^{\rho_{KLS}}\Big]^{1/\rho_{KL}} $$
The capital-labor input is a CES combination of capital and labor. Labor is assumed to be equal to population and evolves exogenously.

$$ KL\big(n,t\big)=\Big[\alpha_{K}\big(n\big)K_{C}(n,t)^{\rho_{KL}}+\alpha_{L}\big(n\big)L(n,t)^{\rho_{KL}}\Big]^{1/\rho_{KL}} $$

Final good capital accumulates following the standard perpetual rule:

$$ K_{C}\big(n,t+1\big)\;=\;K_{C}\big(n,t\big)\big(1-\delta_{C}\big)+I_{C}\left(n,t\right) $$

New ideas which contribute to the stock of energy knowledge, $ Z_{HE}(n,t) $ , are produced using R&D investments, $ I_{R\&D,EN}(n,t) $ , together with the previously cumulated knowledge stock $ HE(n,t) $ :

$$ Z_{_{H E}}\big(n,t\big)\;=\;a\;I_{_{H E}}(n,t)^{b}\;H E(n,t)^{c}\;. $$

Similarly, new ideas in the non-energy sector are generated as follows:

$$ Z_{H K L}\big(n,t\big)=f I_{H K L}(n,t)^{g}H K L(n,t)^{h} $$

The two knowledge stocks evolve as follows:

$$ \begin{aligned}&HE(n,t+1)=HE\left(n,t\right)(1-\delta)+Z_{HE}\left(n,t\right)\\&HKL(n,t+1)=HKL(n,t)(1-\delta)+Z_{HKL}\left(n,t\right)\\ \end{aligned} $$

For illustrative purposes, we show how electricity is produced via capital, operation and maintenance and resource use through a zero-elasticity Leontief aggregate:

$$ E L_{j}\big(n,t\big)=\min\big\{\mu_{n,j}K_{j}\big(n,t\big);\tau_{n,j}O\&M_{j}\big(n,t\big);\varsigma_{j}X_{j,E L}\big(n,t\big)\big\}. $$

Capital for electricity generation technologies accumulates as follows:

$$ K_{j}\big(n,t+1\big)=K_{j}(n,t)\Big(1-\delta_{j}\Big)+\frac{I_{j}(n,t)}{S C_{j}(n,t)}, $$

where, for selected technologies, the new capital investment cost $ SC(n,t) $ decreases with the world cumulated installed capacity by means of Learning-by-Doing:

$$ S C_{j}\big(n,t\big)=B_{j}(n)\sum_{t}\sum_{n}K_{j}\big(n,t\big)^{-\log_{2}P R_{j}}. $$

Operation and maintenance is treated as an investment that fully depreciates every year. The resources employed in electricity production are subtracted from output in equation A3 and A4. Their prices are calculated endogenously using a reduced-form cost function that allows for non-linearity in both the depletion effect and in the rate of extraction:

$$ P_{f}\big(n,t\big)=\chi_{f}\left(n\right)+\pi_{f}\big(n\big)\left[Q_{f}\big(n,t-1\big)\big/\overline{Q}_{f}\big(n,t\big)\right]^{\psi_{f}\left(n\right)} $$

where $ Q_{f} $ is cumulative extraction of fuel f:

$$ Q_{f}\big(n,t-1\big)=Q_{f}\big(n,0\big)+\sum_{s=0}^{t-1}X_{f,e x t r}\big(n,s\big). $$

Each country covers consumption of fuel f, $ X_{f}(n,t) $ , by either domestic extraction or imports, $ X_{f,\text{netimp}}(n,t) $ or by a combination of both. If the country is a net exporter, $ X_{f,\text{netimp}}(n,t) $ is negative.
$$ X_{f}\big(n,t\big)=X_{f,e x t r}\big(n,t\big)+X_{f,n e t i m p}\big(n,t\big) $$

## Climate Module

GHGs emissions from combustion of fossil fuels are derived by applying stoichiometric coefficients to the total amount of fossil fuels minus the amount of $ CO_{2} $ sequestered:

$$ C O_{2}\big(n,t\big)=\sum_{f}\omega_{f,C O_{2}}X_{f}\big(n,t\big)-C C S\big(n,t\big). $$

When a cap on emission (CAP) is included we have an additional equation, constraining emissions, given the possibility to sell and buy permits:

$$ CO_{2}\big(n,t\big)=CAP(n,t)+NIP(n,t) $$

In addition, carbon permits revenues/expenses enter the budget constraint:

$$ C\big(n,t\big)=Y\big(n,t\big)-I_{C}\big(n,t\big)-I_{R\&D,EN}\big(n,t\big)-I_{R\&D,KL}\big(n,t\big) $$

$$ -\sum_{j}I_{R\&D,j}\big(n,t\big)-\sum_{j}I_{j}\big(n,t\big)-\sum_{j}O\&M_{j}\big(n,t\big)-p\big(t\big)NIP\big(n,t\big) $$

The damage function impacting output varies with global temperature:

$$ \Omega(n,t)=\frac{1}{1+\left(\theta_{1,n}T(t)+\theta_{2,n}T(t)^{2}\right)}\cdot $$

Temperature increases through augmented radiating forcing $ F(t) $ :

$$ T(t+1)=T(t)+\sigma_{1}\Big\{F(t+1)-\lambda T(t)-\sigma_{2}\big[T(t)-T_{LO}(t)\big]\Big\} $$

which in turn depends on $ CO_{2} $ concentrations:

$$ F(t)=\eta\Big\{\log\Big\lfloor M_{A T}(t)/M_{A T}^{P I}\Big\rfloor-\log(2)\Big\}+O(t), $$

caused by emissions from fuel combustion and land use change:

$$ M_{A T}(t+1)=\sum_{n}\Bigl[C O_{2}\bigl(n,t\bigr)+L U_{j}(t)\Bigr]+\phi_{11}M_{A T}(t)+\phi_{21}M_{U P}(t), $$

$$ M_{U P}(t+1)=\phi_{22}M_{U P}(t)+\phi_{12}M_{A T}(t)+\phi_{32}M_{L O}(t)\;, $$

$$ M_{L O}(t+1)=\phi_{33}M_{L O}(t)+\phi_{23}M_{U P}(t)\;. $$

Model variables are denoted with the following symbols:

$$ W=welfare $$

U = instantaneous utility

$$ C=\mathbf{c o n s u m p t i o n} $$

$$ c=\mathrm{p e r-c a p i t a c o n s u m p t i o n} $$

$$ L = \text{population} $$

$$ R = \operatorname{discount factor} $$
Y = production
$ I_{c} $ =investment in final good
$ I_{R\&D, EN} $ =investment in energy R&D
$ I_{R\&D, KL} $ =investment in non-energy R&D
$ I_{j} $ =investment in technology j
O&M=investment in operation and maintenance
TFP=total factor productivity
$ K_{c} $ =final good stock of capital
ES=energy services
KLS=capital-labor services
KL=capital-labor aggregate
$ Z_{HE} $ =flow of new energy knowledge
$ Z_{HKL} $ =flow of new non-energy knowledge
$ \Omega $ =damage
$ P_{j} $ = fossil fuel prices
$ X_{j} $ = fuel resources
$ P_{CCS} $ = price of CCS
$ CCS=CO_{2} $ =sequestered
HE=energy knowledge
EN=energy
EL=electric energy
NEL=non-electric energy
$ K_{j} $ = stock of capital of technology j
$ SC_{j} $ =investment cost
$ CO_{2} $ =emissions from combustion of fossil fuels
NIP = Net import of carbon permits
p = Price of carbon permits
$ M_{AT} $ = atmospheric CO $ _{2} $ concentrations
LU = land-use carbon emissions
$ M_{UP} $ = upper oceans/biosphere CO $ _{2} $ concentrations
$ M_{LO} $ = lower oceans CO $ _{2} $ concentrations
F = radiative forcing
T= temperature level
## Appendix B: Simplified WITCH model and Proof of Proposition 1

In this Appendix, we use a simplified version of the WITCH model, in which we omit some features such as labor inputs, operation and maintenance costs and the damage cost of climate change, to prove Proposition 1. We focus our analysis on the outer nest of the model, thus also omitting the detailed description of the energy sector. For clarity's sake, let us skip the regional index. The list of variables is reported in the previous page. In the simplified WITCH model, a social planner maximizes the following utility function:

$$ W=\sum_{t}\left\{\log\left[c(t)\right]\right\}\beta^{t} $$

subject to:

$$ c\big(t\big)=Y\big(t\big)+I_{_{C}}\left(t\right)+I_{_{E}}\big(t\big)+I_{_{H E}}\big(t\big)+I_{_{H K L}}\big(t\big)+p_{_{f}}X_{_{f}}\big(t\big) $$

$$ K_{C}\left(t+1\right)=\left(1-\delta_{C}\right)K_{C}\left(t\right)+I_{C}\left(t\right) $$

$$ EN(t+1)=E\Big(I_{EN}\left(t\right),X_{_{f}}\left(t\right)\Big) $$

$$ HE(t+1)=HE(t)+I_{_{HE}}(t)^{b}HE(t)^{c} $$

$$ HKL(t+1)=HKL(t)+I_{HKL}(t)^{g}HKL(t)^{h} $$

$$ Y\big(t\big)=T F P\big(t\big)\Big[K L S\big(t\big)^{\rho_{Y}}+E S\big(t\big)^{\rho_{Y}}\Big]^{1/\rho_{Y}} $$

$$ E S\big(t\big)=\Big[H E(t)^{\rho_{E S}}+E N(t)^{\rho_{E S}}\Big]^{1/\rho_{E S}} $$

$$ KLS\big(t\big)=\Big[HKL(t)^{\rho_{KLS}}+KL(t)^{\rho_{KLS}}\Big]^{1/\rho_{KLS}} $$

Let L be the Lagrangian :

$$ L=\sum_{t=0}^{T}\frac{u(c_{t})}{\left(1+\rho\right)^{t}}+\sum_{t=0}^{T}\phi_{t}^{1}\Big(Y(t)-c(t)-I_{E}(t)-I_{H E}(t)-I_{H I L}(t)-p_{f}X_{f}(t)+(1-\delta)K_{c}(t)-K_{c}(t+1)\Big) $$

$$ +\sum_{t=0}^{T}\phi_{t}^{2}\Big(E\big(I_{E N}\big(t\big),X_{f}\big(t\big)\big)-E N(t+1)\Big)+\sum_{t=0}^{T}\phi_{t}^{3}\left(H E(t)+I_{H E}\left(t\right)^{b}H E(t)^{c}-H E(t+1)\right) $$

$$ +\sum_{t=0}^{T}\phi_{t}^{4}\left(HKL(t)+I_{HKL}(t)^{\varepsilon}HKL(t)^{h}-HKL(t+1)\right) $$

The associated first order conditions are:

$$ \frac{\partial L}{\partial c(t)}=\frac{\beta^{t}}{c(t)}-\phi_{t}^{1}=0 $$

$$ \frac{\partial L}{\partial I_{_{E}}(t)}=\phi_{_{t}}^{2}\frac{\partial\big(E(t)\big)}{\partial I_{_{E}}(t)}-\phi_{_{t}}^{1}=0 $$

$$ \frac{\partial L}{\partial I_{_{H E}}\big(t\big)}=\phi_{t}^{3}b I_{_{H E}}\big(t\big)^{b-1}H E^{c}\big(t\big)-\phi_{t}^{1}=0 $$

$$ \frac{\partial L}{\partial I_{H K L}\big(t\big)}=\phi_{t}^{4}g I_{H K L}\big(t\big)^{g-1}H E^{h}\big(t\big)-\phi_{t}^{1}=0 $$

$$ \frac{\partial L}{\partial X_{_{f}}\left(t\right)}=\phi_{_{t}}^{^{2}}\frac{\partial\left(E\left(t\right)\right)}{\partial X_{_{f}}\left(t\right)}-\phi_{_{t}}^{^{1}}p_{_{f}}=0 $$

$$ \frac{\partial L}{\partial K_{C}\left(t\right)}=\phi_{t}^{1}\frac{\partial Y\left(t\right)}{\partial K_{C}\left(t\right)}+\left(1-\delta\right)\phi_{t}^{1}-\phi_{t-1}^{1}=0 $$