Inventory planning and tax incentives for charitable giving

Abstract

Many of America’s top corporate donors share a common feature: the bulk of their giving is in the form of in-kind products, not cash. This phenomenon is not a coincidence but rather closely tied to the tax code creating such a preference due to an enhanced deduction for inventory donations. We examine a model of inventory choice under uncertainty and demonstrate that enhanced tax deductions not only promote giving, they also notably influence inventory planning and accelerate learning of customer demand. The results confirm that enhanced deductions can be used to promote pro-social behaviors such as boosting charitable giving, aligning inventories with consumer needs, and alleviating supply chain shortages. The results also demonstrate the potential risks of excessive tax preferences for inventory donations, including inflated retail prices and additional environmental waste.

Appendix

1.1 Proof of Proposition 1

Using backward induction, we first solve for the firm’s period 2 stocking decisions. Given $\psi$ and ${Q}_{1}$, in the event ${S}_{1}<{Q}_{1}$, ${D=S}_{1}$. In this case, the firm sets ${Q}_{2}^{l*}({Q}_{1};\psi ) =D$ since $e<c\left[1-\tau \right]/\tau$ and $p>c$ implies there is no reason to produce more or less than demand, respectively. When ${S}_{1}={Q}_{1}$, the firm learns only that $D\ge {Q}_{1}$. In this case, the firm’s period 2 problem is in (1) and, using Leibniz’s rule, the first-order condition with respect to ${Q}_{2}^{h}({Q}_{1};\psi )$ is:

$$\frac{\left[1-\tau \right]\left[p-c\right]\left[\psi \left(1-p\right)-{Q}_{2}^{h}({Q}_{1};\psi )\right]-\left[c(1-\tau )-\tau e\right]\left[{Q}_{2}^{h}({Q}_{1};\psi )-{Q}_{1}\right]}{\psi [1-p]-{Q}_{1}}=0.$$

Solving the above first-order condition yields the following:

$${Q}_{2}^{h*}({Q}_{1};\psi )={Q}_{1}+\frac{\left[1-\tau \right]\left[p-c\right]\left[\psi (1-p)-{Q}_{1}\right]}{\left[1-\tau \right]p-\tau e}.$$

(4)

Using ${Q}_{2}^{l*}({Q}_{1};\psi )=D$, ${Q}_{2}^{h*}({Q}_{1};\psi )$ in (4), and noting ${Q}_{1}$ can be contingent on $\psi$, the firm’s period 1 problem is in (2). Solving the first-order condition of this problem yields ${Q}_{1}^{*}\left(\psi \right)$ in Proposition 1. Substituting this in (4) yields ${Q}_{2}^{h*}({Q}_{1}^{*}\left(\psi \right);\psi )={Q}_{2}^{h*}\left(\psi \right)$, completing the proof of the proposition.

1.2 Proof of Proposition 2

• (i) The firm learns demand at the end of period 1 if and only if $D<{Q}_{1}^{*}\left(\psi \right)$. Thus the ex ante probability of learning demand is ${\rho }^{*}={{E}_{\psi }[Q}_{1}^{*}\left(\psi \right)/(\psi [1-p])]$. Substituting for ${Q}_{1}^{*}\left(\psi \right)$ from Proposition 1 and noting that ${Q}_{1}^{*}\left(\psi \right)/(\psi [1-p])$ is free of $\psi$ yields the value of ${\rho }^{*}$.

• (ii) The firm’s expected charitable contributions in equilibrium, ${\Omega }^{*}$, equal:

$${\Omega }^{*}={E}_{\psi }\left[{\int }_{0}^{{Q}_{1}^{*}\left(\psi \right)}\left[{Q}_{1}^{*}\left(\psi \right)-D\right]\frac{1}{\psi \left[1-p\right]}dD+{\int }_{{Q}_{1}^{*}\left(\psi \right)}^{{Q}_{2}^{h*}\left(\psi \right)}\left[{Q}_{2}^{h*}\left(\psi \right)-D\right]\frac{1}{\psi \left[1-p\right]}dD\right].$$

(5)

Substituting ${Q}_{1}^{*}\left(\psi \right)$ and ${Q}_{2}^{h*}\left(\psi \right)$ from Proposition 1 in (5) and replacing ${Q}_{1}^{*}\left(\psi \right)$ by ${\rho }^{*}\psi [1-p]$ yields the expression for ${\Omega }^{*}$.

• (iii) The expected consumer surplus in equilibrium, ${\Phi }^{*}=[1-p]{E}_{\psi }[{S}_{1}+{S}_{2}]/2$, equals:

$$\begin{array}{c}\Phi^\ast=\frac{1-p}2E_\psi\left[\int_0^{Q_1^\ast\left(\psi\right)}\left[D+D\right]\frac1{\psi\left[1-p\right]}dD+\right.\int_{Q_1^\ast\left(\psi\right)}^{Q_2^{h\ast}\left(\psi\right)}\left[Q_1^\ast\left(\psi\right)+D\right]\frac1{\psi\left[1-p\right]}dD+\\\left.\int_{Q_2^{h\ast}\left(\psi\right)}^{\psi\left[1-p\right]}\left[Q_1^\ast\left(\psi\right)+Q_2^{h\ast}\left(\psi\right)\right]\frac1{\psi\left[1-p\right]}dD\right].\end{array}$$

(6)

Substituting ${Q}_{1}^{*}\left(\psi \right)$ and ${Q}_{2}^{h*}\left(\psi \right)$ from Proposition 1 in (6) and replacing ${Q}_{1}^{*}\left(\psi \right)$ by ${\rho }^{*}\psi [1-p]$ yields the expression for ${\Phi }^{*}$.

• (iv) The firm’s expected production in equilibrium, ${\Lambda }^{*}$, equals:

$${\Lambda }^{*}={E}_{\psi }\left[{\int }_{0}^{{Q}_{1}^{*}\left(\psi \right)}\left[{Q}_{1}^{*}\left(\psi \right)+D\right]\frac{1}{\psi \left[1-p\right]}dD+{\int }_{{Q}_{1}^{*}\left(\psi \right)}^{\psi [1-p] }\left[{Q}_{1}^{*}\left(\psi \right)+{Q}_{2}^{h*}\left(\psi \right)\right]\frac{1}{\psi \left[1-p\right]}dD\right].$$

(7)

Substituting ${Q}_{1}^{*}\left(\psi \right)$ and ${Q}_{2}^{h*}\left(\psi \right)$ from Proposition 1 in (7) and replacing ${Q}_{1}^{*}\left(\psi \right)$ by ${\rho }^{*}\psi [1-p]$ yields the expression for ${\Lambda }^{*}$.

1.3 Proof of Proposition 3

The following derivatives of the equilibrium quantities noted in Proposition 1 are convenient in establishing the comparative statics in Proposition 3.

$$\begin{array}{c}\frac{\partial {Q}_{1}^{*}\left(\psi \right)}{\partial e}=\frac{{{[Q}_{1}^{*}\left(\psi \right)]}^{2}\tau \{{[p(1-\tau )-\tau e]}^{2}+{[c(1-\tau )-\tau e]}^{2}\} }{\psi \left[1-p\right]\left[p-c\right]\left[1-\tau \right]{\left[\left(p+c\right)\left(1-\tau \right)-2\tau e\right]}^{2}}>0, {\text{and}}\\ \frac{\partial {Q}_{2}^{h*}\left(\psi \right)}{\partial e}=\frac{[{{Q}_{2}^{h*}\left(\psi \right)]}^{2}\tau [c(1-\tau )-\tau e][(2p+c)(1-\tau )-3\tau e]}{\psi \left[1-p\right]\left[p-c\right]\left[1-\tau \right]{\left[\left(p+2c\right)\left(1-\tau \right)-3\tau e\right]}^{2}}>0.\end{array}$$

(8)

Let $e=\gamma Z$, where $Z=Min\left\{c,[p-c]/2\right\}>0$.

• (i) Using ${{{\rho }^{*}=E}_{\psi }[Q}_{1}^{*}\left(\psi \right)/(\psi [1-p])]$ and (8), it follows that:

   

$$\frac{\partial {\rho }^{*}}{\partial \gamma }=Z{\frac{\partial {\rho }^{*}}{\partial e}=ZE_{\psi }}\left[\frac{1}{\psi \left[1-p\right]}\frac{\partial {Q}_{1}^{*}\left(\psi \right)}{\partial e}\right]>0.$$

• (ii) Using ${\Omega }^{*}$ from (5), it follows that:

$$\frac{\partial {\Omega }^{*}}{\partial \gamma }=Z\frac{\partial {\Omega }^{*}}{\partial e}=Z{E}_{\psi }\left[\left(\frac{2{Q}_{1}^{*}\left(\psi \right)-{Q}_{2}^{h*}\left(\psi \right)}{\psi \left[1-p\right]}\right)\frac{\partial {Q}_{1}^{*}\left(\psi \right)}{\partial e}+\left(\frac{{Q}_{2}^{h*}\left(\psi \right)-{Q}_{1}^{*}\left(\psi \right)}{\psi \left[1-p\right]}\right)\frac{\partial {Q}_{2}^{h*}\left(\psi \right)}{\partial e}\right].$$

The inventory levels in Proposition 1 are ordered as $2{Q}_{1}^{*}\left(\psi \right)>{Q}_{2}^{h*}\left(\psi \right)>{Q}_{1}^{*}\left(\psi \right)$. This, in conjunction with (8), implies $\partial {\Omega }^{*}/\partial \gamma >0$.

• (iii) Using ${\Phi }^{*}$ from (6), it follows that:

$$\frac{\partial {\Phi }^{*}}{\partial \gamma }=Z\frac{\partial {\Phi }^{*}}{\partial e}=\frac{Z[1-p]}{2}{E}_{\psi }\left[\left(1-\frac{{Q}_{1}^{*}\left(\psi \right)}{\psi \left[1-p\right]}\right)\frac{\partial {Q}_{1}^{*}\left(\psi \right)}{\partial e}+\left(1-\frac{{Q}_{2}^{h*}\left(\psi \right)}{\psi \left[1-p\right]}\right)\frac{\partial {Q}_{2}^{h*}\left(\psi \right)}{\partial e}\right].$$

With $\psi \left[1-p\right]>{Q}_{2}^{h*}\left(\psi \right)>{Q}_{1}^{*}\left(\psi \right)$ and (8), it follows that $\partial {\Phi }^{*}/\partial \gamma >0$.

• (iv) Using ${\Lambda }^{*}$ from (7), it follows that:

$$\frac{\partial {\Lambda }^{*}}{\partial \gamma }=Z\frac{\partial {\Lambda }^{*}}{\partial e}=Z{E}_{\psi }\left[\left(1-\frac{{{Q}_{2}^{h*}\left(\psi \right)-Q}_{1}^{*}\left(\psi \right)}{\psi \left[1-p\right]}\right)\frac{\partial {Q}_{1}^{*}\left(\psi \right)}{\partial e}+\left(1-\frac{{Q}_{1}^{*}\left(\psi \right)}{\psi \left[1-p\right]}\right)\frac{\partial {Q}_{2}^{h*}\left(\psi \right)}{\partial e}\right].$$

With $\psi \left[1-p\right]>{Q}_{2}^{h*}\left(\psi \right)>{Q}_{1}^{*}\left(\psi \right)$ and (8), it follows that $\partial {\Lambda }^{*}/\partial \gamma >0$.

1.4 Proof of Proposition 4

Define the firm’s expected profit expression in (3) by $\Pi (p,e(p)).$ Consider $e(p)=\gamma Min\{c,[{p}^{*}-c]/2\}=\gamma c$. Thus, at the optimal price ${p}^{*}$, the firm’s expected profit is $\Pi ({p}^{*},\gamma c).$ Differentiating the first-order condition of the firm’s pricing problem, that is, $\partial\Pi /\partial {p}^{*}=0$, with respect to $\gamma$ yields $d{p}^{*}/d\gamma =-({\partial }^{2}\Pi /\partial {p}^{*}\partial \gamma )/({\partial }^{2}\Pi /\partial {p}^{*2}).$ At the interior maximum ${\partial }^{2}\Pi /\partial {p}^{*2}<0$, and hence the sign of $d{p}^{*}/d\gamma$ is the same as the sign of ${\partial }^{2}\Pi /\partial {p}^{*}\partial \gamma$. This cross partial is presented below:

$$\frac{{\partial }^{2}\Pi ({p}^{*},\gamma c)}{\partial {p}^{*}\partial \gamma }=\frac{\psi c({p}^{*}-c){(1-\tau )}^{2}\tau A}{2{[{{p}^{*}}^{2}{(1-\tau )}^{2}+c{p}^{*}(1-\tau )(1-\tau -3\gamma \tau )+{c}^{2}((2+\gamma )\tau +({\gamma }^{2}-\gamma -1){\tau }^{2}-1)]}^{3}},$$

where

$$\begin{array}{lll}A=2c^4-p^{\ast5}{(1-\tau)}^4-14c^4\gamma{(1-\tau)}^3\tau+30c^4\gamma^2{(1-\tau)}^2\tau^2-28c^4\gamma^3(1-\tau)\tau^3\\\quad+10c^4\gamma^4\tau^4+cp^{\ast4}{(1-\tau)}^3(-4+4\tau+9\gamma\tau)-2c^4(2-\tau)\tau(2-(2-\tau)\tau)\\\quad+p^{\ast3}(2c(1-{\tau)}^3-2c(1+\gamma){(1-\tau)}^3\tau-2c^2\gamma{(1-\tau)}^2\tau(-7(1-\tau)+12\gamma\tau))\\\quad+c^5(2-2(2-\tau)\tau+\gamma\tau(-6+(6+5\gamma)\tau))((2+\gamma)\tau+(\gamma^2-\gamma-1)\tau^2-1)\\\quad+p^{\ast2}(-6c^2\gamma{(1-\tau)}^2\tau+6c^2\gamma(1+\gamma){(1-\tau)}^2\tau^2+c^3(1-\tau)(-3+9(1+\gamma)\tau\\\quad-9(1+\gamma(2+3\gamma))\tau^2+(3+\gamma(9+\gamma(27+31\gamma)))\tau^3))+p^\ast(10c^4\gamma^3(1-\tau)\tau^3\\\quad-15c^4\gamma^4\tau^4-6c^3(1-\tau)(-1+\tau+\gamma\tau)(1+\tau(-2+\tau+2\gamma(-1+\tau+\gamma\tau))))\end{array}$$

  

Given $\partial\Pi /\partial {p}^{*}=0$, tedious computations confirm $A\le 0$. With the remaining term in the numerator and the denominator of ${\partial }^{2}\Pi /\partial {p}^{*}\partial \gamma$ each positive, it follows that ${\partial }^{2}\Pi /\partial {p}^{*}\partial \gamma \le 0$. Hence $d{p}^{*}/d\gamma \le 0$ in the $e=\gamma c$ case. Next consider $e(p)=\gamma Min\{c,[{p}^{*}-c]/2\}=\gamma [{p}^{*}-c]/2$. In this case, the cross partial derivative is as below:

$$\frac{{\partial }^{2}\Pi ({p}^{*},\gamma [{p}^{*}-c]/2)}{\partial {p}^{*}\partial \gamma }=\frac{\psi {({p}^{*}-c)}^{2}{(1-\tau )}^{2}\tau B}{{[{{p}^{*}}^{2}(4-4(2-\tau )\tau +\gamma \tau (-6+(6+\gamma )\tau ))+{c}^{2}(-4+\tau (8-4\tau +\gamma (-2+(2+\gamma )\tau )))+2c{p}^{*}(2-\tau (4(1-\gamma )-(2-4\gamma -{\gamma }^{2})\tau ))]}^{2}},$$

where

$$\begin{array}{l}B=\lbrack-2p^{\ast5}(4-8(1+\gamma)\tau+(4+\gamma(8+5\gamma))\tau^2)(4-4(2-\tau)\tau+\gamma\tau(-6+(6+\gamma)\tau))+\\c^5(8-8(2-\tau)\tau+\gamma\tau(12+(-12+5\gamma)\tau))(-4+\tau(8-4\tau+\gamma(-2+(2+\gamma)\tau)))+\\c^4\gamma\tau(32+\tau(36\gamma{(1-\tau)}^2+22\gamma^2(1-\tau)\tau+5\gamma^3\tau^2-32(3-(3-\tau)\tau)))+p^{\ast4}((4-8(1+\\\gamma)\tau+(4+\gamma(8+5\gamma))\tau^2)(4-4(2-\tau)\tau+\gamma\tau(-6+(6+\gamma)\tau))+c(-112+\tau(120\gamma(1-\\{\tau)}^3+144\gamma^2{(1-\tau)}^2\tau-242\gamma^3(1-\tau)\tau^2+45\gamma^4\tau^3-112(-2+\tau)(2-(2-\tau)\tau))))+\\p^{\ast2}(2c^3(-24+\tau(-76\gamma{(1-\tau)}^3-174\gamma^2{(1-\tau)}^2\tau-56\gamma^3(1-\tau)\tau^2+35\gamma^4\tau^3+24(2-\\\tau)(2-(2-\tau)\tau)))+6c^2(8+\tau(4\gamma{(1-\tau)}^3-26\gamma^2{(1-\tau)}^2\tau-8\gamma^3(1-\tau)\tau^2+5\gamma^4\tau^3-\\8(2-\tau)(2-(2-\tau)\tau))))+p^{\ast3}(-4c(4-8(1+\gamma)\tau+(4+\gamma(8+5\gamma))\tau^2)(-5+\tau(10-\\3\gamma+(-5+\gamma(3+\gamma))\tau))-4c^2(12+\tau(40\gamma{(1-\tau)}^3-57\gamma^2{(1-\tau)}^2\tau-67\gamma^3(1-\tau)\tau^2+\\20\gamma^4\tau^3-12(2-\tau)(2-(2-\tau)\tau))))+p^\ast(96c^3+4c^3\tau(28\gamma{(1-\tau)}^3+15\gamma^2{(1-\tau)}^2\tau-\\7\gamma^3(1-\tau)\tau^2-5\gamma^4\tau^3-24(2-\tau)(2-(2-\tau)\tau))+2c^4(16+\tau(72\gamma{(1-\tau)}^3+78\gamma^2(1\\{-\tau)}^2\tau+4\gamma^3(1-\tau)\tau^2-15\gamma^4\tau^3-16(2-\tau)(2-(2-\tau)\tau))))\rbrack/\lbrack p^{\ast2}(4-4(2-\tau)\tau+\\\gamma\tau(-6+(6+\gamma)\tau))+c^2(-4+\tau(8-4\tau+\gamma(-2+(2+\gamma)\tau)))+2cp^\ast(2-\tau(4(1-\gamma)-\\(2-4\gamma-\gamma^2)\tau))\rbrack.\end{array}$$

  

Given $\partial\Pi /\partial {p}^{*}=0$, tedious computations confirm $B\ge 0$. With the remaining term in the numerator and the denominator of ${\partial }^{2}\Pi /\partial {p}^{*}\partial \gamma$ each positive, it follows that ${\partial }^{2}\Pi /\partial {p}^{*}\partial \gamma \ge 0$. Hence $d{p}^{*}/d\gamma \ge 0$ in the $e=\gamma [{p}^{*}-c]/2$ case.

1.5 Proof of Proposition 5

The socially optimal period 2 stocking decisions are determined as follows. When ${S}_{1}<{Q}_{1}$, demand is known and ${Q}_{2}^{l\dagger}\left(\psi \right)=D$ since $\kappa >\beta$ and $p>\kappa$ implies there is no societal reason to produce more or less than demand, respectively. When ${S}_{1}={Q}_{1}$, the socially optimal ${Q}_{2}^{h\dagger}({Q}_{1};\psi )$ is obtained by solving:

$$\begin{array}{c}\begin{array}{c}Max\\Q_2^h(Q_1;\psi)\end{array}\int_{Q_1}^{Q_2^h(Q_1;\psi)}\left[\left(\frac{1+p}2\right)D-\kappa Q_2^h\left(Q_1;\psi\right)+\beta(Q_2^h\left(Q_1;\psi\right)-D)\right]\frac1{\psi\lbrack1-p\rbrack-Q_1}dD\\+\int_{Q_2^h(Q_1;\psi)}^{\psi\lbrack1-p\rbrack}\left[\left(\frac{1+p}2\right)Q_2^h\left(Q_1;\psi\right)-\kappa Q_2^h\left(Q_1;\psi\right)\right]\frac1{\psi\lbrack1-p\rbrack-Q_1}dD.\end{array}$$

(9)

The first order condition of (9) yields:

$${Q}_{2}^{h\dagger}({Q}_{1};\psi )={Q}_{1}+\frac{\left[\left(\frac{1+p}{2}\right)-\kappa \right]\left[\psi \left(1-p\right)-{Q}_{1}\right]}{\left[\frac{1+p}{2}\right]-\beta }.$$

(10)

Stepping back, the socially optimal stocking level in period 1 solves:

$$\begin{array}{c}\begin{array}{c}Max\\ {Q}_{1}\left(\psi \right)\end{array}{\int }_{0}^{{Q}_{1}\left(\psi \right)}\left[\begin{array}{c}\left(\frac{1+p}{2}\right)D-\kappa {Q}_{1}\left(\psi \right)+\beta [{Q}_{1}\left(\psi \right)-D)]+\\ \left(\frac{1+p}{2}\right)D-\kappa D\end{array}\right]\frac{1}{\psi [1-p]}dD+\\ {\int }_{{Q}_{1}\left(\psi \right)}^{{Q}_{2}^{h\dagger}\left({Q}_{1}\left(\psi \right);\psi \right)}\left[\begin{array}{c}\left(\frac{1+p}{2}\right){Q}_{1}\left(\psi \right)-\kappa {Q}_{1}\left(\psi \right)+\\ \left(\frac{1+p}{2}\right)D-\kappa {Q}_{2}^{h\dagger}\left({Q}_{1}\left(\psi \right);\psi \right)+\beta [{Q}_{2}^{h\dagger}\left({Q}_{1}\left(\psi \right);\psi \right)-D)]\end{array}\right]\frac{1}{\psi [1-p]}dD+\\ {\int }_{{Q}_{2}^{h\dagger}\left({Q}_{1}\left(\psi \right);\psi \right)}^{\psi [1-p]}\left[\begin{array}{c}\left(\frac{1+p}{2}\right){Q}_{1}\left(\psi \right)-\kappa {Q}_{1}\left(\psi \right)+\\ \left(\frac{1+p}{2}\right){Q}_{2}^{h\dagger}\left({Q}_{1}\left(\psi \right);\psi \right)-\kappa {Q}_{2}^{h\dagger}\left({Q}_{1}\left(\psi \right);\psi \right)\end{array}\right]\frac{1}{\psi [1-p]}dD.\end{array}$$

(11)

The first-order condition of (11) yields ${Q}_{1}^{\dagger}\left(\psi \right)$. Substituting this in (10) yields ${Q}_{2}^{h\dagger}({Q}_{1}^{\dagger}\left(\psi \right);\psi )={Q}_{2}^{h\dagger}\left(\psi \right)$, completing the proof of part (i). Using the social planner’s preferred inventory levels in Proposition 5(i) and the firm’s preferred stocking choices in Proposition 1, part (ii) follows immediately since ${\left.{Q}_{1}^{*}\left(\psi \right)\right|}_{e={e}^{*}}={Q}_{1}^{\dagger}\left(\psi \right)$ and ${\left.{Q}_{2}^{h*}\left(\psi \right)\right|}_{e={e}^{*}}={Q}_{2}^{h\dagger}\left(\psi \right)$, where ${e}^{*}$ is free of $\psi$. Finally, ${e}^{*}>0$ is equivalent to the condition $\kappa <{\kappa }^{*}$, proving the tax preferences for donations.

1.6 Proof of Proposition 6

The firm’s profits in (1) and (2) are reduced by the additional amount of $\left[1-\tau \right]\eta \left[{Q}_{t}-D\right]$ in the event ${Q}_{t}>D$, and an additional amount of $\left[1-\tau \right]\lambda \left[D-{Q}_{t}\right]$ in the event ${Q}_{t}<D$. With this change, the firm’s optimal quantities are derived using precisely the same steps as in the proof of Proposition 1.

1.7 Proof of Proposition 7

The proof to determine the firm’s optimal (interior) stocking levels in the general distribution case using the first-order conditions follows the same steps as the proof of Proposition 1. In particular, the firm’s period 2 problem is as in (1) with $1/[\psi (1-p)-{Q}_{1}]$ in the two integrands replaced by the general conditional distribution $f\left(D;\psi ,p\right)/[1-F\left({Q}_{1};\psi ,p\right)]$; in addition, the upper limit of the integral is the upper bound of the general distribution. Again, ${Q}_{2}^{l*}({Q}_{1};\psi )=D$. Using Leibniz’s rule, the first-order condition for ${Q}_{2}^{h*}({Q}_{1};\psi )$ is:

$$\frac{1}{1-F\left({Q}_{1};\psi ,p\right)}\left(\begin{array}{c}\left[1-\tau \right]\left[p-c\right]\left[1-F\left({Q}_{2}^{h}\left({Q}_{1};\psi \right);\psi ,p\right)\right]-\\ \left[(1-\tau )c-\tau e\right]\left[F\left({Q}_{2}^{h}({Q}_{1};\psi );\psi ,p\right)-F\left({Q}_{1};\psi ,p\right)\right]\end{array}\right)=0.$$

(12)

From (12), it follows that ${Q}_{2}^{h*}({Q}_{1};\psi )$ satisfies:

$$F\left({Q}_{2}^{h*}\left({Q}_{1};\psi \right);\psi ,p\right)=F\left({Q}_{1};\psi ,p\right)+\frac{\left[1-\tau \right]\left[p-c\right]\left[1-F\left({Q}_{1};\psi ,p\right)\right]}{\left[1-\tau \right]p-\tau e}.$$

(13)

The firm’s period 1 problem is in (2). With ${Q}_{2}^{l*}\left({Q}_{1}\left(\psi \right);\psi \right)=D$, the first-order condition of this problem with respect to ${Q}_{1}\left(\psi \right)$ yields:

$$\begin{array}{c}\left[(1-\tau )c-\tau e\right]\left[\begin{array}{c}\left[{Q}_{2}^{h*}\left({Q}_{1}\left(\psi \right);\psi \right)-{Q}_{1}\left(\psi \right)\right]f\left({Q}_{1}\left(\psi \right);\psi ,p\right)-\\ F\left({Q}_{1}\left(\psi \right);\psi ,p\right)-\left[F\left({Q}_{2}^{h*}\left({Q}_{1}\left(\psi \right);\psi \right);\psi ,p\right)-F\left({Q}_{1};\psi ,p\right)\right]\frac{d{Q}_{2}^{h*}\left({Q}_{1}\left(\psi \right);\psi \right)}{d{Q}_{1}\left(\psi \right)}\end{array}\right]+\\ \left[1-\tau \right]\left[p-c\right]\left[1-F\left({Q}_{1}\left(\psi \right);\psi ,p\right)+\left[1-F\left({Q}_{2}^{h*}\left({Q}_{1}\left(\psi \right);\psi \right);\psi ,p\right)\right]\frac{d{Q}_{2}^{h*}\left({Q}_{1}\left(\psi \right);\psi \right)}{d{Q}_{1}\left(\psi \right)}\right]=0.\end{array}$$

(14)

Substituting for $F\left({Q}_{2}^{h*}\left({Q}_{1}\left(\psi \right);\psi \right);\psi ,p\right)$ from (13) into (14) and solving for ${Q}_{2}^{h*}\left({Q}_{1}\left(\psi \right);\psi \right)$ yields:

$${Q}_{2}^{h*}\left({Q}_{1}\left(\psi \right);\psi \right)={Q}_{1}\left(\psi \right)+\frac{F\left({Q}_{1}\left(\psi \right);\psi ,p\right)}{f\left({Q}_{1}\left(\psi \right);\psi ,p\right)}-\frac{\left[1-\tau \right]\left[p-c\right]\left[1-F\left({Q}_{1}\left(\psi \right);\psi ,p\right)\right]}{\left[(1-\tau )c-\tau e\right]f\left({Q}_{1}\left(\psi \right);\psi ,p\right)}.$$

(15)

From (13) and (15), it follows that ${Q}_{1}^{*}\left(\psi \right)$ satisfies:

$$\begin{array}{c}F\left({Q}_{1}^{*}\left(\psi \right);\psi ,p\right)+\frac{\left[1-\tau \right]\left[p-c\right]\left[1-F\left({Q}_{1}^{*}\left(\psi \right);\psi ,p\right)\right]}{\left[1-\tau \right]p-\tau e}=\\ F\left({Q}_{1}^{*}\left(\psi \right)+\frac{F\left({Q}_{1}^{*}\left(\psi \right);\psi ,p\right)}{f\left({Q}_{1}^{*}\left(\psi \right);\psi ,p\right)}-\frac{\left[1-\tau \right]\left[p-c\right]\left[1-F\left({Q}_{1}^{*}\left(\psi \right);\psi ,p\right)\right]}{\left[(1-\tau )c-\tau e\right]f\left({Q}_{1}^{*}\left(\psi \right);\psi ,p\right)};\psi ,p\right).\end{array}$$

(16)

(15), (16) with ${Q}_{1}\left(\psi \right)={Q}_{1}^{*}\left(\psi \right)$, and ${Q}_{2}^{l*}\left(\psi \right)=D$ correspond to the stocking levels.