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Investment strategies of duopoly firms with asymmetric time-to-build under a jump-diffusion model

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Abstract

This paper employs a two-factor jump-diffusion model to investigate the optimal investment timing and capacity choice of the duopoly firms in the presence of uncertain and asymmetric time-to-build. By assuming that both the market demand and investment cost follow the jump-diffusion process, we show that the impacts of uncertainty of time-to-build on duopoly firms’ the optimal investment decisions depend on the directions of jumps in demand and investment cost. Moreover, the asymmetry of time-to-build makes it possible for the dominated firm to preempt the market successfully and becomes the leader. The leader’s capacity level increases with the dominated firm’s time-to-build and the follower’s decreases, even if the dominated firm is the leader. We also apply numerical simulation to compare the main results between two-factor diffusion model and two-factor jump-diffusion model.

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Funding

This research was funded by the Key Project of Education Science Planning in Heilongjiang Province for 2021 (No. 477) and Open topic project of Think tank (No. ZKKF2022208).

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Authors and Affiliations

  1. Department of Management Science and Engineering, Harbin Institute of Technology, Xidazhi, Harbin, 150001, Heilongjiang, China

    Yanyun Liu & Baiqing Sun

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  1. Yanyun Liu
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  2. Baiqing Sun
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Correspondence to Baiqing Sun.

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Appendix A Linear demand function

Appendix A Linear demand function

1.1 Proof of Theorem 1

Given the current demand-to-cost ratio \(\theta \), maximizing (10) with respect to \(Q_M\), we can yield the optimal capacity as follows:

$$\begin{aligned} Q_{M}^{*}(\theta )=\underset{Q_{M}\ge 0}{\arg \max } g_{M}(\theta ,Q_{M})=\frac{1}{2\eta }\left( 1-\frac{\xi +b_{X}}{\xi +b_{Y}} \frac{1}{\theta }\right) \end{aligned}$$
(A1)

Following Wu and Hu (2022), the firm’s value under the new state variable, \(v_{M}(\theta )\) in (9), satisfies the following HJB equation

$$\begin{aligned} r v_{M}(\theta )&=\frac{1}{2}\left( \sigma _{X}^{2}+\sigma _{Y}^{2}\right) \theta ^{2}v_{M}^{''}(\theta )+(\mu _{X}-\mu _{Y})\theta v_{M}^{'}(\theta )+(\mu _{Y}-\lambda _{X}-\lambda _{Y})v_{M}(\theta )\nonumber \\&\quad +\lambda _{X} E\left[ v_{M}\left( \theta (1+U^{X})\right) \right] +\lambda _{Y} E\left[ (1+U^{Y}) v_{M}\left( \frac{\theta }{1+U^{Y}}\right) \right] \end{aligned}$$
(A2)

subject to

$$\begin{aligned}{} & {} v_{M}(0)=0 \end{aligned}$$
(A3)
$$\begin{aligned}{} & {} v_{M}\left( \theta \right) \mid _{\theta =\theta _{M}^{*}}=g_{M}\left( \theta , Q_{M}^{*}(\theta )\right) \mid _{\theta =\theta _{M}^{*}} \end{aligned}$$
(A4)
$$\begin{aligned}{} & {} \frac{\partial v_{M}(\theta )}{\partial \theta }\mid _{\theta =\theta _{M}^{*}}=\frac{\partial g_{M}(\theta ,Q_{M}^{*}(\theta )}{\partial \theta }\mid _{\theta =\theta _{M}^{*}} \end{aligned}$$
(A5)

Meanwhile, the general solution of (A2) with the initial condition (A3) takes the form

$$\begin{aligned} v_{M}(\theta )=A_{M}\theta ^{\beta _1} \end{aligned}$$
(A6)

where \(\beta _{1}\) is the positive root (\(>1\)) of (13) (e.g., Nunes and Pimentel 2017). Substituting (A6)(10) into (A4)(A5) gives

$$\begin{aligned} \theta _{M}^{*}(Q_{M})= & {} \frac{\beta _{1}}{\beta _{1}-1}\frac{b_{X}(\xi +b_{X})}{b_{Y}(\xi +b_{Y})(1-\eta Q_{M})} \end{aligned}$$
(A7)
$$\begin{aligned} A_{M}= & {} \frac{\xi }{(\beta _{1}^2-1)\eta b_{Y}(\xi +b_{Y})}(\theta _{M}^{*})^{-\beta _{1}} \end{aligned}$$
(A8)

After solving the system of Eqs. (A1) and (A8) we get the results in Theorem 1.

1.2 Duopoly market

First of all, we investigate the duopoly firms’ optimal investment decisions when the market roles are exogenous. To this end, we first formalize the firm’s value function, and consequently by maximizing the value function at the moment of investing, the optimal capacity level can be derived. By solving the HJB equation the value function satisfies, the optimal investment threshold can be obtained. According to this train of thought, we discuss the follower’s optimal investment decisions when its investment timing is earlier and later than the manufacturing timing of the leader in Appendix A.2.1 and  A.2.2. In Appendix A.2.3, we proceed to explore the leader’s investment decision. By taking the preemptive incentive into consideration, we further study the equilibrium investment strategies of duopoly enterprises under competition in Appendix A.2.4.

1.2.1 Proof of Theorem 2

Given the current demand-to-cost ratio \(\theta \) and the leader’s capacity level \(Q_L^i\), the type j follower’s profit flow obtained at the investment timing when investing with capacity \(Q_{F1}^{j}\) can be evaluated as

$$\begin{aligned} g_{F1}^{j}\left( \theta ,Q_{L}^{i},Q_{F1}^{j}\right)&=\frac{1}{Y}G_{F1}^{j}\left( X,Y,Q_{L}^{i},Q_{F1}^{j}\right) \nonumber \\&=\xi _{j}Q_{F1}^{j}\left[ \frac{\left( 1-\eta (Q_{L}^{i}+Q_{F1}^{j})\right) \theta }{b_X(\xi _{j}+b_{X})}-\frac{1}{b_Y(\xi _{j}+b_{Y})}\right] \end{aligned}$$
(A9)

maximizing (A9) with respect to \(Q_{F1}^j\), the optimal capacity level can be obtained as

$$\begin{aligned} Q_{F1}^{j*}(\theta ,Q_{L}^{i})=\frac{1}{2\eta }\left( 1-\eta Q_{L}^{i}-\frac{\xi _{j}+b_{X}}{\xi _{j}+b_{Y}} \frac{1}{\theta }\right) \end{aligned}$$
(A10)

Before investing, the follower’s value, \(v_{F1}^{j}(\theta )\), equals to the value of the investment option the firm holds, which is

$$\begin{aligned} v_{F1}^{j}(\theta )=A_{F1}^{j}\theta ^{\beta _1} \end{aligned}$$
(A11)

Substituting (A9) (A10) (A11) into the corresponding value matching and smooth pasting conditions yields

$$\begin{aligned} \theta _{F1}^{j*}(Q_{L}^{i})= & {} \frac{(\beta _{1}+1)b_X(\xi _{j}+b_{X})}{(\beta _{1}-1)b_Y(\xi _{j}+b_{Y})(1-\eta Q_{L}^{i})} \end{aligned}$$
(A12)
$$\begin{aligned} A_{F1}^{j}(Q_{L}^{i})= & {} g_{F1}^{j}(Q_{L}^{i})(\theta _{F1}^{j*}(Q_{L}^{i}))^{-\beta _{1}} \end{aligned}$$
(A13)

Substituting (A12) into (A10) we can conclude the follower’s optimal investment strategy after the leader’s product entering the market in Theorem 2.

1.2.2 Proof of Theorem 3

Given the current demand-to-cost ratio \(\theta \) and the leader’s capacity level \(Q_L^i\), assuming that the type j follower invests with capacity \(Q_{F0}^{j}\), then the expected profit flow the follower obtained at the moment of investment can be evaluated as

$$\begin{aligned} g_{F0}^{j}(\theta ,Q_{L}^{i},Q_{F0}^{j})&=\frac{Q_{F0}^{j}\xi _{i}\xi _{j}\left( 1-\eta (Q_{L}^{i}+Q_{F0}^{j})\right) \theta }{b_X(\xi _{j}+b_{X})(\xi _{i}+\xi _{j}+b_{X})}-\frac{Q_{F0}^{j}\xi _{j}}{b_Y(\xi _{j}+b_{Y})}\nonumber \\&\quad +\frac{Q_{F0}^{j}\xi _{j}\theta }{b_X(\xi _{i}+\xi _{j}+b_{X})}\left[ (1-\eta Q_{F 0}^{j})-\frac{\xi _{i}\eta Q_{L}^{i}}{\xi _{i}+b_{X}}\right] \end{aligned}$$
(A14)

maximizing (A14) with respect to \(Q_{F0}^j\), the follower’s optimal capacity level can be obtained as described in (39).

In the meanwhile, before investing, the firm’s value, \(v_{F0}^{j}(\theta )\), should satisfy the following HJB equation:

$$\begin{aligned}&\frac{1}{2}(\sigma _{X}^{2}+\sigma _{Y}^{2})\theta ^{2}v_{F0}^{j''}(\theta )+(\mu _{X}-\mu _{Y})\theta v_{F0}^{j'}(\theta )-(r-\mu _{Y}+\lambda _{X}+\lambda _{Y})v_{F0}^{j}(\theta )\nonumber \\&\quad +\lambda _{X}E\left[ v_{F0}^{j}\left( (1+U^{X})\theta \right) \right] +\lambda _{Y} E\left[ (1+U^{Y})v_{F0}^{j}\left( \frac{\theta }{1+U^{Y}}\right) \right] \nonumber \\&\quad +\xi _{i}(v_{F1}^{j}(\theta )-v_{F0}^{j}(\theta ))=0 \end{aligned}$$
(A15)

where \(v_{F1}^{j}(\theta )\) is defined by (A11) through (A13). Unlike before, a general solution of (A15) with the initial condition is

$$\begin{aligned} v_{F0}^{j}(\theta )=v_{F1}^{j}(\theta )+A_{F0}^{j}\theta ^{\gamma _i} \end{aligned}$$
(A16)

where \(\gamma _i\) is the positive (\(>1\)) of the following equation

$$\begin{aligned}&\frac{1}{2}(\sigma _{X}^{2}+\sigma _{Y}^{2})\gamma (\gamma -1)+(\mu _{X}-\mu _{Y}) \gamma +\lambda _{X}E\left[ (1+U^{X})^{\gamma }\right] +\lambda _{Y}E\left[ (1+U^{Y})^{1-\gamma }\right] \nonumber \\&\quad -(r+\lambda _{i}-\mu _{Y}+\lambda _{X}+\lambda _{Y})=0 \end{aligned}$$
(A17)

Substituting (A14), (A16) and (39) into the boundary conditions, we can derive successively the value function, optimal investment threshold as well as capacity level of the follower given in Theorem 3.

1.2.3 Proof of Theorem 4

Given the current demand-to-cost ratio \(\theta \) and the follower’s investment strategies \(\left\{ \theta _{F0}^{j},\theta _{F1}^{j}\right\} , \left\{ Q_{F0}^{j}, Q_{F1}^{j}\right\} \), the expected profit flow the type i leader obtains when invests with capacity level \(Q_{L}^{i}\) at the moment of investment can be evaluated as follows:

$$\begin{aligned}&g_{L}^{i}(\theta ,Q_{L}^{i})=\frac{\xi _{i}(1-\eta Q_{L}^{i})Q_{L}^{i}\theta }{b_X(\xi _{i}+b_{X})}-\frac{\xi _{i} Q_{L}^{i}}{b_Y(\xi _{i}+b_{Y})}\nonumber \\&\quad -\left( \frac{\theta }{\theta _{F0}^{j*}(Q_{L}^{i})}\right) ^{\gamma _{i}}\frac{\xi _{i}\xi _{j}\eta Q_{L}^{i}Q_{F0}^{j*}(Q_{L}^{i})\theta _{F0}^{j*}(Q_{L}^{i})}{b_X(\xi _{i}+\xi _{j}+b_X)}\left( \frac{1}{\xi _{i}+b_{X}}+\frac{1}{\xi _{j}+b_{X}}\right) \nonumber \\&\quad -\left[ \left( \frac{\theta }{\theta _{F1}^{j*}(Q_{L}^{i})}\right) ^{\beta _{1}}-\left( \frac{\theta }{\theta _{F0}^{j*}(Q_{L}^{i})}\right) ^{\gamma _{i}} \left( \frac{\theta _{F0}^{j*}(Q_{L}^{i})}{\theta _{F1}^{j*}(Q_{L}^{i})}\right) ^{\beta _{1}}\right] \frac{\xi _{j}\eta Q_{L}^{i}Q_{F1}^{j*}(Q_{L}^{i})\theta _{F1}^{j*}(Q_{L}^{i})}{b_X(\xi _{j}+b_{X})} \end{aligned}$$
(A18)

maximizing (A18) with respect to \(Q_L^i\), the optimal capacity level can be obtained as

$$\begin{aligned} Q_{L}^{i*}(\theta )=\underset{Q_{L}^{i}\ge 0}{\arg \max }g_{L}^{i}(\theta , Q_{L}^{i}) \end{aligned}$$
(A19)

Meanwhile, the value of the leader before investing, \(v_{L}^{i}(\theta )\), equals to the value of the investment option the firm holds, which is

$$\begin{aligned} v_{L}^{i}(\theta )=A_{L}^{i} \theta ^{\beta _{1}} \end{aligned}$$
(A20)

Substituting (A18) and (A20) into the boundary conditions, we can obtain successively the value function of the leader, optimal investment threshold as well as the corresponding optimal capacity given in Theorem 4.

1.2.4 Proof of Theorem 5

Firstly, following Huisman and Kort (2015), we assume that both firms have incentives to preempt the market, that is, \(\theta _{P}^{i}\ne \emptyset \) for \(i\in \{A,B\}\), indicating that there exists at least a \(\theta \) such that \(V_{P}^{i}(\theta )\ge V_{F0}^{i}(\theta , Q_{L}^{j*}(\theta ))\) holds. With this assumption, two possible scenarios are be considered.

If \(\theta _{P}^{i*}\le \theta _{P}^{j*}\le \theta _{L}^{i*}\) for \(i\ne j\) and \(i,j\in \{A,B\}\), the type i firm succeeds in the preemption game and optimally invests at \(\theta _{P}^{j*}-\varepsilon \), which is infinitely close to \(\theta _{P}^{j*}\). Therefore, the investment is made at \(\theta _{P}^{j*}\) as \(\varepsilon \) approaches 0 and the capacity choice is \(Q_{P}^{i*}=Q_{L}^{i*}(\theta _{P}^{j*})\). Meanwhile, failure in the preemptive game leads to the type j firm implements the investment decision as a follower, whose optimal investment decisions are shown in Theorems 2 and 3. Otherwise, if \(\theta _{P}^{i*}\le \theta _{L}^{i*}\le \theta _{P}^{j*}\) for \(i\ne j\), since the type j firm will not invest before the demand-to-cost ratio reaches \(\theta _{P}^{j*}\), the type i firm’s optimal investment should be made at \(\theta _{L}^{i*}\) instead of \(\theta _{P}^{i*}\). Consequently, the firms’ investment strategies follow those in Theorem 2 through 4.

Secondly, we assume that only one firm has an incentive to preempt the market. Without loss of generality, we assume that \(\theta _{P}^{i}\ne \emptyset \) and \(\theta _{P}^{j}=\emptyset \) for \(i\ne j\) and \(i,j\in \{A,B\}\), indicating that there exists at least a \(\theta \) such that \(V_{P}^{i}(\theta )\ge V_{F0}^{i}(\theta ,Q_{L}^{j*}(\theta ))\) holds, whereas \(V_{P}^{j}(\theta )< V_{F0}^{j}(\theta ,Q_{L}^{j*}(\theta ))\) for all \(\theta \). In this case, both firms’ investment strategies follow those in Theorem 2 through 4.

1.3 Proof of Theorem 6

Suppose that we have already assigned the type i and j firm as the market leader and follower respectively. First of all, with the assumption that the leader’s product has already entered into the market, we can examine social planner’s optimal investment decision by solving the following problem:

$$\begin{aligned}&sw_{F1}^{ij}(\theta )=\frac{1}{Y}SW_{F1}^{ij}(X,Y)=\frac{1}{Y}\underset{T_{F1}^{j}\ge 0,Q_{F1}^{j}\ge 0}{\max }E^{(X,Y)}\left[ \int _{0}^{{\overline{T}}_{F1}^{j}}e^{-rt}\frac{X_{t} Q_{L}^{i}(2-\eta Q_{L}^{i})}{2}\right. \nonumber \\&\quad \left. dt+\int _{{\overline{T}}_{F1}^{j}}^{+\infty }e^{-rt}\left( \frac{X_{t}(Q_{L}^{i}+Q_{F1}^{j})\left( 2-\eta (Q_{L}^{i}+Q_{F1}^{j})\right) }{2}- Q_{F1}^{j}Y_{t}\right) dt\right] \end{aligned}$$
(A21)

Combining the initial condition and boundary conditions, we can straightforward calculate the optimal investment threshold \(\theta _{F1}^{j**}(Q_{L}^{i})\) and capacity level \(Q_{F1}^{j**}(Q_{L}^{i})\) of the type j follower in view of welfare-maximizing, as shown in (62) and (63).

Secondly, we examine social planner’s optimal investment decision with the assumption that the leader’s product hasn’t entered into the market yet by solving

$$\begin{aligned}&sw_{F0}^{ij}(\theta )=\frac{1}{Y}\max _{T_{F0}^j,Q_{F0}^j\ge 0}E^{(X, Y)}\left[ 1_{\left\{ {\overline{T}}_L^i<T_{F0}^j\right\} }e^{-r{\overline{T}}_L^i}SW_{F1}^{ij}(X_{{\overline{T}}_L^i},Y_{{\overline{T}}_L^i})+1_{\left\{ T_{F 0}^j \le {\overline{T}}_L^i<{\overline{T}}_{F0}^j\right\} }\right. \nonumber \\&\left\{ \int _{{\overline{T}}_L^i}^{{\overline{T}}_{F0}^j}e^{-rt}\frac{X_tQ_L^i\left( 2-\eta Q_L^i\right) }{2} dt+\int _{{\overline{T}}_{F 0}^j}^{+\infty } e^{-rt}\left[ \frac{X_t(Q_L^i+Q_{F 0}^j)\left( 2-\eta (Q_L^i+Q_{F 0}^j)\right) }{2}\right. \right. \nonumber \\&\quad -\left. \left. Q_{F0}^jY_t\right] dt\right\} +1_{\left\{ {\overline{T}}_L^i\ge {\overline{T}}_{F0}^j\right\} }\left\{ \int _{{\overline{T}}_{F 0}^j}^{{\overline{T}}_L^i} e^{-rt}\frac{X_tQ_{F0}^j(2-\eta Q_{F0}^j)}{2}dt\right. \nonumber \\&\quad \left. \left. +\int _{{\overline{T}}_L^i}^{+\infty }e^{-rt}\left[ \frac{X_t(Q_L^i+Q_{F0}^j)\left( 2-\eta (Q_L^i+Q_{F0}^j)\right) }{2}-Q_{F0}^j Y_t\right] dt\right\} \right] \end{aligned}$$
(A22)

Similar to Appendix A.2.2, the profit flow the type j follower can obtain at the investment timing in view of social welfare-maximization can be written as follows:

$$\begin{aligned}&\frac{\xi _{i}\theta }{2b_X(\xi _{i}+\xi _{j}+b_{X})}\left[ Q_{L}^{i}(2-\eta Q_{L}^{i})+\frac{Q_{F 0}^{j}(2-\eta Q_{F0}^{j}-2\eta Q_{L}^{i})\xi _{j}}{\xi _{j}+b_{X}}\right] \nonumber \\&\quad +\frac{\xi _{j}\theta }{2b_X(\xi _{i}+\xi _{j}+b_{X})}\left[ Q_{F0}^{j}(2-\eta Q_{F0}^{j})+\frac{Q_{L}^{i}(2-\eta Q_{L}^{i}-2\eta Q_{F0}^{j})\xi _{i}}{\xi _{i}+b_{X}}\right] -\frac{\xi _{i}Q_{F0}^{j}}{\xi _{i}+b_{Y}} \end{aligned}$$
(A23)

Maximizing (A23) with respect to \(Q_{F0}^j\) yields the optimal capacity level. Combining the value-matching and smooth-pasting conditions, we can get the optimal investment strategies of the type j follower in light of welfare-maximizing, as shown in (64) and (65).

Finally, the social planner needs to deal with the type i leader’s investment problem as follows:

$$\begin{aligned}&sw_{L}^{ij}(\theta )=\frac{1}{Y}\max _{T_{L}^{i},Q_{L}^{i}\ge 0}E^{(X,Y)}\left[ 1_{\left\{ {\overline{T}}_{L}^{i}<T_{F0}^{j**}\right\} }\left\{ \int _{{\overline{T}}_{L}^{i}}^ {{\overline{T}}_{F1}^{j**}}e^{-rt}Q_{L}^{i}\left( \frac{X_{t}(2-\eta Q_{L}^{i})}{2}-Q_{L}^{i}Y_{t}\right) \right. \right. \end{aligned}$$
(A24)
$$\begin{aligned}&\quad \left. \left. dt+\int _{{\overline{T}}_{F1}^{j**}}^{\infty }e^{-rt}\left( \frac{(Q_{L}^{i}+Q_{F1}^{j**}(Q_{L}^{i})) \left( 2-\eta (Q_{L}^{i}+Q_{F1}^{j**}(Q_{L}^{i}))\right) X_{t}}{2}-Q_{F1}^{j**}(Q_{L}^{i})Y_{t}\right) dt\right\} \right. \nonumber \\&\quad +\left. 1_{\left\{ T_{F0}^{j**}\le {\overline{T}}_{L}^{i}<{\overline{T}}_{F0}^{j**}\right\} } \left\{ \int _{{\overline{T}}_{L}^{i}}^{{\overline{T}}_{F0}^{j**}}e^{-rt}Q_{L}^{i}\left( \frac{X_{t}(2-\eta Q_{L}^{i})}{2}- Q_{L}^{i}Y_{t}\right) dt\right. \right. \nonumber \\&\quad \left. \left. +\int _{{\overline{T}}_{F0}^{j**}}^{\infty }e^{-rt}\left( \frac{(Q_{L}^{i}+Q_{F0}^{j**}(Q_{L}^{i}))\left( 2-\eta (Q_{L}^{i} +Q_{F0}^{j**}(Q_{L}^{i}))\right) X_{t}}{2}-Q_{F0}^{j**}(Q_{L}^{i})Y_{t}\right) dt\right\} \right. \nonumber \\&\quad \left. +1_{\left\{ {\overline{T}}_{L}^{i}\ge {\overline{T}}_{F0}^{j**}\right\} }\left\{ \int _{{\overline{T}}_{F0}^{j**}}^{{\overline{T}}_{L}^{i}} e^{-rt}\frac{X_{t}Q_{F0}^{j**}(Q_{L}^{i})\left( 2-\eta Q_{F0}^{j**}(Q_{L}^{i})\right) }{2}dt\right. \right. \nonumber \\&\quad \left. \left. +\int _{{\overline{T}}_{L}^{i}}^{+\infty }e^{-rt}\left( \frac{X_{t}(Q_{L}^{i}+Q_{F0}^{j**}(Q_{L}^{i}))\left( 2-\eta (Q_{L}^{i}+Q_{F 0}^{j**}(Q_{L}^{i}))\right) }{2}-Q_{F0}^{j**}(Q_{L}^{i})Y_{t}\right) dt\right\} \right] \end{aligned}$$
(A25)

After a series of complicated calculations, we evaluate (A25) as (67), and consequently, the optimal investment strategies can be obtain as shown in (66).

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Liu, Y., Sun, B. Investment strategies of duopoly firms with asymmetric time-to-build under a jump-diffusion model. Math Meth Oper Res 98, 377–410 (2023). https://doi.org/10.1007/s00186-023-00833-0

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