Leasing or selling? The channel choice of durable goods manufacturer considering consumers’ capital constraint

Abstract

In this paper, we explore how a monopoly manufacturer chooses the market strategies and decides the optimal price to obtain maximum profit. We divide the market into the C-type market and the N-type market, and analyze the profitability of the monopoly manufacturer who takes the pure-selling, pure-leasing and hybrid strategy respectively, considering consumers’ capital constraint and the life span of the durable goods in an indefinite time horizon model. (1) We find that a larger proportion of the consumers with capital constraint has a more significant impact on the prices and it could slow down the development of the rental market. When the scale of the group attains up to a threshold level, it would greatly influence the customers’ demand and their marketing strategy, and encourages more manufacturers to take the leasing strategy. (2) In the Hybrid Strategy, we see explicit growth in the overall profit with both the leasing channel and the selling channel working together, although the former outperforms the latter. The suppressed selling channel, in fact, has to lower the price to keep the market coverage, with an independent market structure. (3) Finally, we find that a leasing agent may help the manufacturer at first and then become a tough competitor. These findings provide new insights for the operation of large construction machinery manufacturing companies. subject classification numbers as needed.

Appendix

1.1 Appendix: Proofs

1.1.1 Proof of lemma 1

From the constraint, we have

$Q_{ns}=Nq_{ns}=1-\frac{p_{s}}{V_{s}}(P_{s}=Np_{s})$

Substitute them into the profit function

$\Pi _{S}=(1-m)(P_{s}-c)\cdot q_{ns}$

After solving the first order conditions we obtain the only interior solution

$P^{S}_{s}=\frac{NV_{s}+c}{2}$

Substituting it back, we obtain the optimal quantity and profit:

$Q_{ns}=\frac{NV_{s}-c}{2NV_{s}}$;

$\Pi _{S}=\frac{(1-m)(NV_{s}-c)^{2}}{4N^{2}V_{s}}$ $\square$

1.1.2 Proof of lemma 2

From the constraint, we have

$Q_{nl}=Nq_{nl}=1-\frac{P_{l}}{V_{l}}$

$Q_{c}=Nq_{c}=\omega -\frac{P_{l}}{V_{l}}$

Substitute them into the profit function

$\begin{aligned} \Pi _{L} = & (1 - m)[(P_{l} - c) \cdot q_{{nl}} + (P_{l} - c_{m} ) \cdot (Q_{{nl}} - q_{{nl}} )] \\ & + m[(P_{l} - c) \cdot q_{c} + (P_{l} - c_{m} ) \cdot (Q_{c} - q_{c} )] \\ \end{aligned}$

And the function becomes

$\begin{array}{rcl} \Pi _{L}&{}=&{}\frac{(1-m)}{N}[(P_{l}-c)\cdot (1-\frac{P_{l}}{V_{l}}) +(P_{l}-c_{m})\cdot (N-1)\cdot (1-\frac{P_{l}}{V_{l}})]\\ &{}~&{}+\frac{m}{N}[(P_{l}-c)\cdot (\omega -\frac{P_{l}}{V_{l}})+(P_{l}-c_{m})(N-1)(\omega -\frac{P_{l}}{V_{l}})] \end{array}$

After solving the first order conditions we obtain the only interior solution

$P^{L}_{l}=\frac{V_{l}}{2}(1-m+m\omega )+\frac{c+(N-1)c_{m}}{2N}$

Substituting it back, we obtain the optimal quantity and profit:

$Q^{L}_{nl}=\frac{1+m-m\omega }{2}-\frac{c+(N-1)c_{m}}{2NV_{l}}$

$Q^{L}_{c}=\frac{2\omega -1+m-m\omega }{2}-\frac{c+(N-1)c_{m}}{2NV_{l}}$

$\Pi _{L}=\frac{(NV_{l}(1-m+m\omega )-c-(N-1)c_{m})^{2}}{4N^{2}V_{l}}$ $\square$

1.1.3 Proof of lemma 3

From the constraint, we have

$Q_{ns}=Nq_{ns}=\frac{P_{l}-p_{s}}{V_{l}-V_{s}}-\frac{p_{s}}{V_{s}}$

$Q_{nl}=Nq_{nl}=1-\frac{P_{l}-p_{s}}{V_{l}-V_{s}}$

$Q_{c}=Nq_{c}=\omega -\frac{P_{l}}{V_{l}}$

Substitute them into the profit function

$\begin{aligned} \Pi _{H} = &\, (1 - m)[(P_{s} - c) \cdot q_{{ns}} + (P_{l} - c) \cdot q_{{nl}} + (P_{l} - c_{m} ) \cdot (Q_{{nl}} - q_{{nl}} )] \\ & + m[(P_{l} - c) \cdot q_{c} + (P_{l} - c_{m} ) \cdot (Q_{c} - q_{c} )] \\ \end{aligned}$

And the function becomes

$$\begin{aligned} \begin{array}{rcl} \Pi _{H}&{}=&{}\frac{(1-m)}{N}[(P_{s}-c)(\frac{P_{l}-p_{s}}{V_{l}-V_{s}} -\frac{p_{s}}{V_{s}})+(P_{l}-c)(1-\frac{P_{l}-p_{s}}{V_{l}-V_{s}})\\&&+\, (P_{l}-c_{m})(N-1)(1-\frac{P_{l}-p_{s}}{V_{l}-V_{s}})]\\ &&+\,\frac{m}{N}[(P_{l}-c)(\omega -\frac{P_{l}}{V_{l}})+(P_{l} -c_{m})(N-1)(\omega -\frac{P_{l}}{V_{l}})] \end{array} \end{aligned}$$

We use $p_{s}$ instead of $P_{s}$ to simplify the calculation. After solving the first order conditions we obtain the only interior solution

$P^{H}_{l}=\frac{V_{l}}{2}(1-m+m\omega )+\frac{c+(N-1)c_{m}}{2N}$;

$P^{H}_{s}=\frac{c+(1-m+m\omega )NV_{s}}{2N}(P^{H}_{s}=Np^{H}_{s})$.

Then we check the second order conditions

$\frac{\partial ^{2}\Pi _{H}}{\partial p^{2}_{s}}=\frac{-2(1-m)V_{l}}{V_{s}(V_{l}-V_{s})};~~~$ $\frac{\partial ^{2}\Pi _{H}}{\partial p_{s}\partial P_{l}}=\frac{2(1-m)}{V_{l}-V_{s}};~~~$ $\frac{\partial ^{2}\Pi _{H}}{\partial P^{2}_{l}}=\frac{-2(V_{l}-mV_{s})}{V_{s}(V_{l}-V_{s})}$.

The Hessian is semi-definitive. Therefore, it’s the optimal solution. Substituting it back, we obtain the optimal quantity and profit:

$Q^{H}_{ns}=\frac{(N-1)c_{m}}{2N(V_{l}-V_{s})}-\frac{c}{2NV_{s}}$

$Q^{H}_{nl}=1-\frac{N(1-m+m\omega )(V_{l}-V_{s})+(N-1)c_{m}}{2N(V_{l}-V_{s})}$

$Q^{H}_{c}=\omega -\frac{(1-m+m\omega )V_{l}N+c+(N-1)c_{m}}{2NV_{l}}$

$\Pi _{H}=\frac{(NV_{l}(1-m+m\omega )-c-(N-1)c_{m})^{2}}{4N^{2}}+ \frac{(1-m)[c(V_{l}-V_{s})-(N-1)c_{m}V_{s}]^{2}}{4N^{2}V_{l}V_{s}(V_{l}-V_{s})}$ $\square$

1.1.4 Proof of proposition 1

We set function

$$\begin{aligned} \begin{array}{rcl} F(m) &{} = &{} \Pi _{L}-\Pi _{S}\\ &{}=&{}\frac{(NV_{l}(1-m+m\omega )-c-(N-1)c_{m})^{2}}{4N^{2}V_{l}}-\frac{(1-m)(NV_{s}-c)^{2}}{4N^{2}V_{s}}\\ &{} = &{} \frac{V_{l}(1-\omega )^{2}}{4}m^{2}+\frac{(NV_{s}-c)^{2}-2 NV_{s}(1-\omega )(NV_{l}-c-(N-1)c_{m})}{4N^{2}V_{s}}m\\&&+\,\frac{V_{s}(NV_{l}-c-(N-1)c_{m})^{2}-V_{l}(NV_{s}-c)^{2}}{4N^{2}V_{l}V_{s}} \end{array} \end{aligned}$$

$f(V_{s},V_{l})=\frac{V_{s}(NV_{l}-c-(N-1)c_{m})^{2} -V_{l}(NV_{s}-c)^{2}}{4N^{2}V_{l}V_{s}}$

(a)$F^{''}(m)=\frac{V_{L}(1-\omega )^{2}}{2}>0$, $F'(m)$ is strictly increasing in m.

(b)If $F'(0)=\frac{(NV_{s}-c)^{2}-2 NV_{s}(1-\omega )(NV_{l}-c-(N-1)c_{m})}{4N^{2}V_{s}}>O$, then it is equivalent to $\omega >\omega _{1}=1-\frac{(NV_{s}-c)^{2}}{2NV_{s}(NV_{l}-c-(N-1)c_{m})}$

(c1)Therefore, when the C-type market satisfies the following: $\omega >\omega _{1}$, we can see that F(m) is strictly increasing in m.

(c2)Otherwise, F(m) initially decreases, and turns to increase after $m>m_{1}$, when $F'(m)=0$, $m_{1}=\frac{2NV_{s}(1-\omega )(NV_{l}-c-(N-1)c_{m})-(NV_{s}-c)^{2}}{2N^{2}V_{s}V_{l}(1-\omega )^{2}}$.

Then we figure out the number of zeroes of F(m). $F(1)=\frac{(NV_{l}\omega -c-(N-1)c_{m})^{2}}{4N^{2}V_{l}}$.

(a1) So if $\frac{V_{s}(NV_{l}-c-(N-1)c_{m})^{2}-V_{l}(NV_{s}-c)}{4N^{2}V_{l}V_{s}}<0$. It’s easy to verify that there is only one zero point (No matter whether it is in the situation c1 or c2).

(a2) Then we assume $F(0)>0$ and $F'(0)<0$. There are two zero points. We set them as $V_{s1}$ and $V_{s2}$, respectively.

So,$\Delta =b^{2}-4ac =[\frac{(NV_{s}-c)^{2}-2NV_{s}(1-\omega )(NV_{l}-c-(N-1)c_{m})}{4N^{2}V_{s}}]^{2}-\frac{(m^{2}(1-\omega )^{2}V_{l})(V_{s}(NV_{l} -c-(N-1)c_{m})^{2}-V_{l}(NV_{s}-c)^{2})}{4N^{2}V_{l}V_{s}}$ $\Delta >0$ is equivalent to $(NV_{s}-c)^{2}-4NV_{s}(1-\omega )(\omega NV_{l}-c-(N-1)c_{m})>0$. And there is $V_{s2}$ in [0,1], $\Delta <0$ is equivalent to $(NV_{s}-c)^{2}-4NV_{s}(1-\omega )(\omega NV_{l}-c-(N-1)c_{m})<0$. There is a point $V_{s1}$ which satisfies the $F(0)=0$ .

$\begin{array}{lll} \frac{\partial \Pi _{S}}{\partial m} &{} = &{}\frac{-(NV_{s}-c)^{2}}{4N^{2}V_{s}}<0;\\ \frac{\partial \Pi _{L}}{\partial m} &{} = &{}\frac{2(\omega -1)(NV_{l}(1-m+m\omega )-c-(N-1)c_{m})^{2}}{4N^{2}V_{s}}<0. \end{array}$ $\square$

1.1.5 Corollary 1

The leasing company’s total demand can be written as $Q^{L}=\frac{1-m+m\omega }{2}+\frac{c+(N-1)c_{m}}{2NV_{l}}$. The derivatives of profit, price and demand with respect to $c_{m}$ can be seen as following.

$\begin{array}{rcl} 1-m &{}+&{} m\omega = 1+(\omega -1)m;\\ \frac{\partial \Pi _{L}}{\partial c_{m}}&{} = &{} \frac{-2(N-1)(NV_{l}(1-m+m\omega )-c-(N-1)c_{m})}{4N^{2}V_{l}}<0;\\ \frac{\partial Q^{L}}{\partial c_{m}}&{} = &{} \frac{N-1}{2NV_{l}}>0;\\ \frac{\partial p^{L}_{l}}{\partial c_{m}}&{} = &{} \frac{N-1}{2N}>0. \end{array}$ $\square$

1.1.6 Proof of proposition 2

The derivatives of profit of leasing company with respect to N can be seen as

$\frac{\partial \Pi _{L}}{\partial N}=\frac{(c-c_{m})[NV_{l}(1-m+m\omega )-c-(N-1)c_{m}]}{2N^{3}V_{l}}>0$

As for selling companies, we have $\frac{\partial \Pi _{S}}{\partial N}=\frac{(1-m)[1-\delta ^{N}-c(1-\delta )]}{4N^{2}(1-\delta )} \{\frac{-N\ln \delta \cdot \delta ^{N}[1-\delta ^{N}+c(1-\delta )] +c(1-\delta )(1-\delta ^{N})}{(1-\delta ^{N})^{2}}- (\frac{1-\delta ^{N}}{1-\delta ^{N}})^{2}\}(NV_{s}=\frac{1-\delta ^{N}}{1-\delta })$

$\begin{array}{rcl} g(N)&{}=&{}\frac{-N\ln \delta \cdot \delta ^{N}[1-\delta ^{N}+c(1-\delta )]+c(1-\delta )(1-\delta ^{N})}{(1-\delta ^{N})^{2}}-(\frac{1-\delta ^{N}}{1-\delta ^{N}})^{2}~[\delta \in (0,1)]\\ g'(N)&{}=&{}\frac{-N\ln ^{2}\delta \cdot \delta ^{N}[1-\delta ^{N}+c(1-\delta )]-\ln \delta \cdot \delta ^{N}(1-\delta ^{N})}{(1-\delta ^{N})^{3}}\\ &{}<&{}\frac{-N\ln ^{2}\delta \cdot \delta ^{N}(1-\delta ^{N})-\ln \delta \cdot \delta ^{N}(1-\delta ^{N})}{(1-\delta ^{N})^{3}}\\ &{}=&{}\frac{(-\ln \delta \cdot \delta ^{N})(1-\delta ^{N})(N\ln \delta +1-\delta ^{N})}{(1-\delta ^{N})^{3}}\\ &{}<&{}0 \end{array}$

Finally, we have $g(1)=\frac{-\ln \delta \cdot \delta }{1-\delta }(1+c)+c-1>0$($\delta$ is approximately equal to product quantity 1), and ${\lim \limits _{N\rightarrow \infty }} g(N)=c(1-\delta )-1<0$.

Therefore, there is a unique $N^{*}$ that satisfies $N<N^{*}$, then $\frac{\partial \Pi _{S}}{\partial N}>0$, and $\Pi _{S}$ increases monotonously. That is to say, there is a unique optimal product life span for selling companies. $\square$

1.1.7 Proof of proposition 4

The derivatives of profit, prices and demand with respect to m can be seen as

$\frac{\partial \Pi _{H}}{\partial m}=\frac{(NV_{l}(1-m+m\omega )-c-(N-1)c_{m})(\omega -1) V_{l}}{2N}-\frac{[c(V_{l}-V_{s})-(N-1)c_{m}V_{s}]^{2}}{4N^{2}V_{s}V_{l}(V_{l}-V_{s})};$

$\frac{\partial P^{H}_{l}}{\partial m}=\frac{V_{l}}{2}(\omega -1)<0$; $\frac{\partial P^{H}_{s}}{\partial m}=\frac{(\omega -1)NV_{S}}{2}<0$.

$\frac{\partial Q^{H}_{ns}}{\partial m}=0$; $\frac{\partial Q^{H}_{nl}}{\partial m}=\frac{\partial Q^{H}_{c}}{\partial m}=\frac{1-\omega }{2}>0$.

The results above can be interpreted into Proposition 4. $\square$

1.1.8 Corollary 2

The derivatives of prices and demand with respect to life span N can be written as

$\begin{array}{rcl} \frac{\partial P^{H}_{l}}{\partial N} &{} = &{}-\frac{c-c_{m}}{2N^{2}}<0;\\ \frac{\partial P^{H}_{s}}{\partial N} &{} = &{}(1-m+m\omega )V_{s}>0;\\ \frac{\partial p^{H}_{s}}{\partial N} &{} = &{}-\frac{c}{2N^{2}}<0;\\ \frac{\partial Q^{H}_{nl}}{\partial N} &{} = &{}\frac{c_{m}}{2N^{2}(V_{l}-V_{s})}>0;\\ \frac{\partial Q^{H}_{c}}{\partial N} &{} = &{}\frac{c-c_{m}}{2N^{2}V_{l}}>0. \end{array}$ The results above can be interpreted into Corollary 2. $\square$

1.1.9 Proof of lemma 4

From the constraint, we have

$\begin{array}{lcl} Q_{ns}&{}=&{}Nq_{ns}=1-\frac{p_{s}}{V_{s}}(P_{s}=Np_{s}=\frac{NV_{s}+c}{2});\\ Q_{A}&{}=&{}Nq_{A}=\omega -\frac{P_{A}}{V_{l}}.\\ \\ \end{array}$

(a) Substitute them into the profit function

$\Pi ^{a}_{A}=m[(P_{A}-P_{s})q_{A}+(P_{A}-c_{m})(Q_{A}-q_{A})]$

And the function becomes

$\Pi ^{a}_{A}=\frac{m}{N}[(P_{A}-P_{s})(\omega -\frac{P_{A}}{V_{l}}) +m(P_{A}-c_{m})(N-1)(\omega -\frac{P_{A}}{V_{l}})]$

After solving the first order conditions we obtain the only interior solution

$P^{a}_{A}=\frac{2\omega NV_{l}+NV_{s}+c+2(N-1)c_{m}}{4N}$

Substituting it back, we obtain the optimal quantity and profit of agents and the manufacturer:

$\begin{array}{cll} Q^{a}_{A} &{}=&{} \frac{2\omega NV_{l}-[NV_{s}+c+2(N-1)c_{m}]}{4NV_{l}};\\ \Pi ^{a}_{M}&{}=&{}\frac{(1-m)(NV_{s}-c)^{2}}{4N^{2}V_{s}}+\frac{m(NV_{s}-c) [2\omega NV_{l}-NV_{s}-c-2(N-1)c_{m}]}{8N^{2}V_{l}};\\ \Pi ^{a}_{A}&{}=&{} \frac{m}{16N^{2}V_{l}}[2\omega NV_{l}-NV_{s}-c-2(N-1)c_{m}]^{2}.\\ \\ \end{array}$

(b) We set the agents’ wholesale price as $\alpha P_{s}$ and think the N-type consumers include the agents.

Substitute the constraints into the profit function:

$\Pi ^{b}_{A}=m[(P_{A}-\alpha P_{s})q_{A}+(P_{A}-c_{m})(Q_{A}-q_{A})]$

And the function becomes

$\Pi ^{b}_{A}=\frac{m}{N}[(P_{A}-\alpha P_{s})(\omega -\frac{P_{A}}{V_{l}})+m(P_{A}-c_{m}) (N-1)(\omega -\frac{P_{A}}{V_{l}})]$

After solving the first order conditions we obtain the only interior solution

$P^{b}_{A}=\frac{2\omega NV_{l}+\alpha NV_{s}+\alpha c+2(N-1)c_{m}}{4N}$

Substituting it back, we obtain the optimal quantity and profit of agents and manufacturers:

$\begin{array}{rcl} Q^{b}_{A} &{}=&{} \frac{2\omega NV_{l}-[\alpha NV_{s}+\alpha c+2(N-1)c_{m}]}{4NV_{l}};\\ \Pi ^{b}_{M}&{}=&{}\frac{(1-m)(NV_{s}-c)^{2}}{4N^{2}V_{s}}+\frac{(1-m)(\alpha NV_{s}+\alpha c-2c)[2\omega NV_{l}-\alpha NV_{s}-\alpha c-2(N-1)c_{m}]}{8N^{2}V_{l}};\\ \Pi ^{b}_{A}&{}=&{} \frac{m}{16N^{2}V_{l}}[2\omega NV_{l}-\alpha NV_{s}-\alpha c-2(N-1)c_{m}]^{2}. \end{array}$ $\square$

1.1.10 Proof of Lemma 5

(a) As the manufacturer is the Stackelberg leader. We first figure out the profit of the agent. Similar to the Hybrid Strategy, we have the constraint

$\begin{array}{lcl} Q_{ns}&{}=&{}Nq_{ns}=\frac{P_{A}-p_{s}}{V_{l}-V_{s}}-\frac{p_{s}}{V_{s}};\\ Q_{nA}&{}=&{}Nq_{nA}=1-\frac{P_{A}-p_{s}}{V_{l}-V_{s}};\\ Q_{A}&{}=&{}Nq_{A}=\omega -\frac{P_{A}}{V_{l}}. \end{array}$

Substitute them into the profit function

$\begin{aligned} \Pi _{A}^{1} = &\, (1 - m)[(P_{A} - P_{s} )q_{{nA}} + (P_{A} - c_{m} )(Q_{{nA}} - q_{{nA}} )] \\ & + m[(P_{A} - P_{s} )q_{A} + (P_{A} - c_{m} )(Q_{A} - q_{A} )] \\ \end{aligned}$

After solving the first order conditions we obtain the only interior solution

$P^{1}_{A}=\frac{2V_{l}-mV_{l}-mV_{s}}{2N(V_{l}-mV_{s})}P_{s}+ \frac{V_{l}(V_{l}-V_{s})B}{2(V_{l}-mV_{s})}+\frac{(N-1)c_{m}}{2N}(B=1-m+m\omega )$;

Substituting it back to profit function of the manufacturer, we obtain the optimal price of manufacturer is

$P^{1}_{s}=\frac{c}{2}+\frac{NV_{s}V_{l}(2V_{l} -mV_{l}-mV_{s})B-mV_{s}(V_{l}-mV_{s})(N-1)c_{m}}{2(2(1-m)V^{2}_{l}+m^{2}V_{s}V_{l}-m^{2}V^{2}_{s})}.$

(b) We set $\alpha P_{s}$, $\alpha \in (0,1]$ as the agent’s wholesale price and think the N-Type consumers include the agents. Then,

$\begin{array}{ccl} P^{2}_{A}&{}=&{}\frac{(1-m+\alpha )V_{l}-m\alpha V_{s}}{2N(V_{l}-mV_{s})} P_{s}+\frac{V_{l}(V_{l}-V_{s})B}{2(V_{l}-mV_{s})}+\frac{(N-1)c_{m}}{2N};\\ P^{2}_{s}&{}=&{}\frac{\alpha (1+\omega )}{2E}+\frac{[(1-m+\alpha )V_{l}-m \alpha V_{s}]V_{s}-2V_{l}(V_{s}-mV_{s})}{4NV_{l}V_{l}(V_{l}-mV_{s})E} c\\&&+\frac{(V_{l}-2\alpha V_{l}+\alpha V_{s})}{4E(V_{l}-mV_{s})}B+ \frac{(V_{l}-2\alpha V_{l}+\alpha V_{s})(N-1)c_{m}}{4 E NV_{l}(V_{l}-V_{s})}.\\ \\ (E&{}=&{}\frac{V_{l}-\alpha V_{s}}{NV_{s}(V_{l}-V_{s})}+\frac{(V_{l} -2\alpha V_{l}+\alpha V_{s})[(1-m+\alpha )V_{l}-m\alpha V_{s}]}{2NV_{l}(V_{l}-V_{s})(V_{l}-mV_{s})})\\ \end{array}$ $\square$