*3.3. Optimal Decisions in Policy T*

In this subsection, models are formulated and the optimal decision is derived in Policy T. A similar type of subsidy policy is discussed in the forward SC setting by several researchers [59]. For example, to strengthen sustainable innovation, \$400 million was allotted to fund the R&D of energy technologies as a part of the American Recovery and Reinvestment Act of 2009 [48]. The demand function in this policy becomes *D<sup>T</sup> <sup>i</sup>* = *<sup>a</sup>* − *<sup>p</sup><sup>T</sup> <sup>i</sup>* + *βθ<sup>T</sup> <sup>i</sup>* , and the corresponding profit functions of the manufacturer and retailer; and SW remains similar with previous subsections, and they are obtained as follows:

$$
\pi\_{ri}^T(p\_i^T) = (p\_i^T - w\_i^T) D\_i^T \tag{7}
$$

$$\pi\_{\rm mi}^T(w\_i^T, \theta\_i^T, \tau\_i^T) = (w\_i^T - \lambda\_1 \theta\_i^T) D\_i^T + X \tau\_i^T D\_i^T - \kappa \tau\_i^{T^2} - (1 - \mu\_i^T) \lambda\_2 \theta\_i^{T^2},\tag{8}$$

$$
\pi\_{\rm gi}^T(\mu\_i^T) = \pi\_{ri}^T + \pi\_{mi}^T + \frac{D\_i^{T^2}}{2} - \mu\_i^T \lambda\_2 \theta\_i^{T^2}.\tag{9}
$$

Note that the manufacturer receives a subsidy on the total R&D investment, not on per unit product *λ*1*θ<sup>T</sup> <sup>i</sup>* , *<sup>i</sup>* = *<sup>m</sup>*,*r*. If *<sup>λ</sup>*<sup>1</sup> = 0 and *<sup>μ</sup><sup>T</sup> <sup>i</sup>* = 0, the profit functions become similar to Ghosh and Saha [18], as well as Song and Gao [19], where the authors examined the optimal decision for a forward SC setting where the government does not provide any subsidy. Similar to previous subsections, we derive the optimal decision for Policy T and omit the detailed derivation. Lemma 5,6 characterize the optimal decisions under the MS and RS games, respectively.

**Lemma 5.** *Optimal decision in Scenario MT are obtained as follows:*

*μT <sup>m</sup>* = <sup>6</sup>*<sup>κ</sup> N*<sup>5</sup> *; w<sup>T</sup> <sup>m</sup>* <sup>=</sup> *<sup>N</sup>*5*YZκ*+*N*4(*aN*2+4*cmκ*)*λ*<sup>2</sup> <sup>Δ</sup>3*<sup>m</sup> ; <sup>p</sup><sup>T</sup> <sup>m</sup>* <sup>=</sup> *<sup>N</sup>*5*YZκ*+*N*4(*aN*3+2*cmκ*)*λ*<sup>2</sup> <sup>Δ</sup>3*<sup>m</sup> ; <sup>θ</sup><sup>T</sup> <sup>m</sup>* <sup>=</sup> (*a*−*cm*)*N*5*Z<sup>κ</sup>* <sup>Δ</sup>3*<sup>m</sup> ; <sup>τ</sup><sup>T</sup> <sup>m</sup>* = (*a*−*cm*)*N*4*Xλ*<sup>2</sup> <sup>Δ</sup>3*<sup>m</sup> ; <sup>π</sup><sup>T</sup> mm* <sup>=</sup> (*a*−*cm*)2*N*4*κλ*<sup>2</sup> <sup>Δ</sup>3*<sup>m</sup> ; <sup>π</sup><sup>T</sup> rm* <sup>=</sup> <sup>4</sup>(*a*−*cm*)2*N*<sup>4</sup> <sup>2</sup>*κ*2*λ*<sup>2</sup> 2 <sup>Δ</sup>3*<sup>m</sup> ; <sup>π</sup><sup>T</sup> gm* <sup>=</sup> (*a*−*cm*)2*N*5*κλ*<sup>2</sup> <sup>Δ</sup>3*<sup>m</sup> ; <sup>Q</sup><sup>T</sup> <sup>m</sup>* <sup>=</sup> <sup>2</sup>(*a*−*cm*)*N*4*κλ*<sup>2</sup> <sup>Δ</sup>3*<sup>m</sup> , where* Δ3*<sup>m</sup>* = *N*<sup>4</sup> <sup>2</sup>*λ*<sup>2</sup> <sup>−</sup> *<sup>N</sup>*5*Zκ.*

**Lemma 6.** *Optimal decision in Scenario RT are obtained as follows: μT <sup>r</sup>* <sup>=</sup> *<sup>N</sup>*3*λ*2−*Z*2*<sup>κ</sup>* <sup>2</sup>(*N*2+*κ*)*λ*2)*; <sup>w</sup><sup>T</sup> <sup>r</sup>* <sup>=</sup> *<sup>N</sup>*2(*aN*1+*cm <sup>N</sup>*3)*λ*2+*Zκ*((*aN*1−*cm*(3*N*2+2*κ*))*β*+(*aN*4+*cm <sup>N</sup>*2)*λ*1) <sup>2</sup>Δ3*<sup>r</sup> ; <sup>p</sup><sup>T</sup> <sup>r</sup>* = *N*2((*a*+*cm*)*κ*+*aN*2)*λ*2−*Zκ*(*a*((*Z*+*β*)*κ*−*N*3*λ*1)+*cm*(*N*2*β*+*κλ*1) <sup>Δ</sup>3*<sup>r</sup> ; <sup>θ</sup><sup>T</sup> <sup>r</sup>* <sup>=</sup> (*a*−*cm*)*Zκ*(*N*2+*κ*) <sup>Δ</sup>3*<sup>r</sup> ; <sup>τ</sup><sup>T</sup> <sup>r</sup>* <sup>=</sup> (*a*−*cm*)*X*(*N*2*λ*2+*Z*2*κ*) <sup>2</sup>Δ3*<sup>r</sup> ; π<sup>T</sup> mr* <sup>=</sup> (*a*−*cm*)2*κ*(*N*2*λ*2+*Z*2*κ*) <sup>4</sup>Δ3*<sup>r</sup> ; <sup>π</sup><sup>T</sup> rr* <sup>=</sup> (*a*−*cm*)2*κ*(*N*2*λ*2+*Z*2*κ*) <sup>2</sup>Δ3*<sup>r</sup> ; <sup>π</sup><sup>T</sup> gr* <sup>=</sup> (*a*−*cm*)2*κ*((3*N*2+2*κ*)*λ*2+*Z*2*κ*) <sup>4</sup>Δ3*<sup>r</sup> ; Q<sup>T</sup> <sup>r</sup>* <sup>=</sup> (*a*−*cm*)*κ*(*N*2*λ*2+*Z*2*κ*) <sup>Δ</sup>3*<sup>r</sup> , where* <sup>Δ</sup>3*<sup>r</sup>* = *<sup>N</sup>*<sup>2</sup> <sup>2</sup>*λ*<sup>2</sup> <sup>−</sup> *<sup>N</sup>*3*Z*2*κ.*

The concavity of profit functions for CLSC members and SW in Scenarios MT and RT is ensured by condition Δ3*<sup>m</sup>* > 0 and Δ3*<sup>r</sup>* > 0, respectively. Similar to Policy RE, optimal decisions differ according to the power of CLSC members, and increasing market potential does not effect the subsidy rates. By comparing optimal decisions, the following theorem is proposed.
