Optimal strategy for corporate international investment and consumption problem with stochastic hyperbolic discounting
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Long, Jun; Zeng, Sanyun; Gupta, Brij; Zhang, Jindan; Nedjah, Nadia Article Optimal strategy for corporate international investment and consumption problem with stochastic hyperbolic discounting Journal of Innovation & Knowledge (JIK) Provided in Cooperation with: Elsevier Suggested Citation: Long, Jun; Zeng, Sanyun; Gupta, Brij; Zhang, Jindan; Nedjah, Nadia (2024) : Optimal strategy for corporate international investment and consumption problem with stochastic hyperbolic discounting, Journal of Innovation & Knowledge (JIK), ISSN 2444-569X, Elsevier, Amsterdam, Vol. 9, Iss. 1, pp. 1-12, https://doi.org/10.1016/j.jik.2023.100459 This Version is available at: https://hdl.handle.net/10419/327365 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/
Optimal Strategy for Corporate International Investment and Consumption Problem with Stochastic Hyperbolic Discounting Jun Long a , Sanyun Zeng b, *, Brij B. Gupta c,d,e,f,g , Jindan Zhang h , Nadia Nedjah i a School of Information Technology & Engineering, Guangzhou College of Commerce, Guangzhou 511363, PR China b School of Accounting, Guangdong University of Foreign Studies, Guangzhou 510006, PR China c Department of Computer Science and Information Engineering, Asia University, Taichung 413, Taiwan d Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea e Symbiosis Centre for Information Technology (SCIT), Symbiosis International University, Pune, India f Department of Electrical and Computer Engineering, Lebanese American University, Beirut 1102, Lebanon g Center for Interdisciplinary Research, University of Petroleum and Energy Studies (UPES), Dehradun, India h Xianyang Vocational Technical College, Xianyang, Shanxi, China i Department of Electronics Engineering and Telecommunications of the Engineering Faculty, State University of Rio de Janeiro, Rio de Janeiro, Brazil ARTICLE INFO Article History: Received 10 March 2023 Accepted 30 December 2023 Available online 12 January 2024 ABSTRACT This study investigates the dynamics of a stochastic hyperbolic discounting model in a continuous-time framework to address the complexities associated with the corporate international investment consumption problem (CIICP). By formulating a dynamic programming equation based on the principles of dynamic programming, we seek to untangle the intricacies of CIICP by incorporating log utility. Our research provides valuable insights into the economic ramifications of the proposed model and presents a comprehensive analysis of numerical sensitivities. This investigation not only contributes to a deeper understanding of the CIICP phenomenon but also contributes to more informed decision-making within the field. © 2024 The Authors. Published by Elsevier España, S.L.U. on behalf of Journal of Innovation & Knowledge. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: Corporate international investment and consumption problem (CIICP) Stochastic hyperbolic discounting time-inconsistent Dynamic programming equation (DPE) JEL Classification: C610 G150 Introduction Hyperbolic discounting refers to a phenomenon wherein individuals tend to place greater value on immediate benefits compared with future benefits, resulting in a time-inconsistent preference structure that can be categorized into naive or sophisticated. Naive individuals are unaware of their changing preferences over time, leading to inconsistent planning that is not executed practically. In contrast, sophisticated individuals are aware of their time-inconsis- tent preferences and choose an optimal time-consistent plan. In the context of international investment and consumption for a corporation, this study hypothesizes a continuous-time model with stochastic hyperbolic discounting. Specifically, we propose that corporate managers prefer short-term benefits over future benefits, and this preference structure may have implications for the corporation’s investment and consumption decisions, warranting further investigation. The majority of economic decisions involve a tradeoff between present and future returns, which essentially gives them a time-span- ning nature. Discounted utility theory has become the main framework for evaluating intertemporal choices in economics (see, Hajdini, 2012;Mankiw, 2020;Portney & Weyant, 2013). The discount function, which is an essential component of the theory of decision making, values delayed returns in the present. In problems with finite horizons, individuals typically discount short- and long-term utility functions applying the same constant discount rate of their preferences over time. However, experimental studies on time preferences have demonstrated that the time-consistent hypothesis is unrealistic (Ainslie, 1992;Loewenstein & Prelec, 1992;Thaler, 1981) since individuals behave more patiently when both rewards are far removed from the present and more impatiently when both rewards approach This research is supported by grants from: The Key Project of National Natural Science Funds (No. 71231008), Guangdong Province Philosophy and Social Sciences Planning Project (No. GD23CGL28), and Guangdong Province General University Humanities and Social Sciences Research Characteristic Innovation Project (No. 2018WTSCX035). * Corresponding author. E-mail address: [email protected] (S. Zeng). https://doi.org/10.1016/j.jik.2023.100459 2444-569X/© 2024 The Authors. Published by Elsevier España, S.L.U. on behalf of Journal of Innovation & Knowledge. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Journal of Innovation & Knowledge 9 (2024) 100459 Journal of Innovation &Knowledge https://www.journals.elsevier.com/journal-of-innovation-and-knowledge
the current time. Laibson (1997) proposed a quasi-hyperbolic discount function in which the discount rate decreases over time to capture the phenomenon of time-varying impatience. This type of preference is referred to as present-biased preferences by Palacios- Huerta and P erez-Kakabadse (2013). Time-inconsistent preferences have been extensively investigated by economists, with a large body of literature exploring hyperbolic discounting models (Ainslie & Haslam, 1992;Ainslie & Herrnstein, 1981;DellaVigna & Malmendier, 2006;Kirby & Herrnstein, 1995;Loewenstein & Prelec, 1992;McClure et al., 2004;Myerson & Green, 1995;Thaler & Shefrin, 1981). These models have been applied to a wide range of economic problems, including Barro (1999),DellaVigna and Malmendier (2004),Grenadier and Wang (2007),O’Donoghue and Rabin (1999), and Palacios- Huerta and P erez-Kakabadse (2013), among others. However, these models have inherent limitations, as with the assumption of consistent preferences over time and the difficulty of estimating discount rates accurately. Further research is needed to address these limitations and improve our understanding of time-inconsistent preferences. A surge in research activities regarding corporate international investment has occurred over the years, exploring various aspects of this phenomenon from different perspectives. Choi (1989) examined the evolution of the theory of corporate international investment, focusing on its financial dimension. Bellalah and Wu (2009) proposed a model of corporate investment among international concerns that incorporates stochastic and deterministic elements. The model considers the impact of domestic and foreign prices on firms’outputs and inputs, along with exchange rate risk and competitiveness between firms. Zhang and Zhang (2012) developed an optimal corporate investment model that accounts for daytime and nighttime differences. Overall, these studies have contributed to our understanding of the factors that influence international investment in corporations; however, some limitations remain. For instance, Choi’s (1989) analysis primarily focused on the financial aspects of international investment, neglecting other significant factors such as political and cultural considerations. Bellalah and Wu’s (2009) model assumed that firms are rational and profit-maximizing, which might not always be the case in reality. Zhang and Zhang’s (2012) model is based on the assumption that daytime and nighttime differences have a significant impact on investment decisions, which might not be true in all contexts. Therefore, future research must address these limitations and provide a more comprehensive understanding of corporations’international investment. The literature on firms’international investment and consumption has included several studies. Wu and Zhang (2006) examined the optimal corporate portfolio and consumption choice problem, in which investors could invest their wealth in bonds (bank accounts) and real projects owned by production. Huang and Wu (2011) focused on the optimal international portfolio and consumption choices for corporations, in which investors could invest their wealth in domestic bonds (bank accounts) or physical projects abroad. Huang and Zhang (2013) proposed a model for firms’optimal investment and consumption that considered the effects of inflation and differences in market opening and closing. Long and Zeng (2016) investigated an international investment problem for corporations, proposing time-consistent strategies with a mean-variance criterion. Bellalah et al. (2016) conducted a study on corporate international investment and analyzed the impact of information cost, short selling, and taxation on management decision making, finding that these factors have a significant impact on corporate investment decisions. Bellalah and Zhang (2017) investigated the impact of firms’international investment decisions around market closure on exchange rate risk, incomplete information, and short selling restrictions, determining that these factors had a significant impact on investment. Choi et al. (2018) examined the predictions of real option theory, confirming that it accurately predicts investment decisions. Daszkiewicz (2019) explored the internationalization patterns of family high-tech firms and found that such firms tend to internationalize slowly and cautiously. Glodowska et al. (2019) investigated multinational enterprises’risk mitigation strategies in emerging markets, finding that these strategies are essential for successful internationalization. Zhu and Sardana (2020) investigated the moderating effect of internationalization motives on Polish firms’multinationalization−performance relationship, determining that such motives had a significant impact on the relationship. Barlozewski and Trapczyzski (2021) examined the impact of foreign exchange rate risk on the expected returns generated by South African investors’portfolios, demonstrating that this risk has a significant impact on returns. Djemo et al. (2021) examined how small family firms responded to the COVID-19 crisis, finding that such firms adapted and responded to the crisis in various ways. Marjanski & Sulkowski, 2021 examined how new multinational firms structured their foreign direct investment location portfolios and found that such decisions have a significant impact on performance outcomes. Grieco (2021) provided a comprehensive overview of the theories and evidence on foreign investment and development, while Song (2021) examined the conditions under which multinational corporations adjust production volume among subsidiaries. Koren and Vodopyanova (2022) focused on methods to increase transnational corporations’investment attractiveness, and Li et al. (2022) constructed a proposed mathematical model to study the optimal investment problem. Finally, Nguyen (2023) investigated the relationship between financial constraints and firm productivity in Vietnamese manufacturing industries. Despite the significant progress made in the field of corporate investment and consumption decisions, some limitations remain in the previous literature. Specifically, the majority of studies have primarily focused on corporations’optimal investment and consumption choices, while neglecting the potential impact of external factors such as political and economic risks. Furthermore, some studies have solely concentrated on a particular type of investment, such as bonds or physical projects, and have not explored other investment options. To address these limitations, future research should expand on the approaches of previous research by considering a broader range of investment options and external factors to provide more comprehensive insights into firms’international investment and consumption decisions. As noted, some studies have only focused on specific factors and did not consider other important factors that may affect investment decisions. Additionally, some studies have only focused on specific types of firms or industries, which may limit the generalizability of the findings. Finally, some studies have relied on selfreported data, which could be subject to bias and may not accurately reflect corporations’actual investment decisions. Therefore, future research must incorporate a more comprehensive set of factors and a broader range of firms and industries to establish a more complete understanding of corporate international investment decision making. In doing so, scholars can gain a deeper understanding of the complex decision making processes that firms engage in when making international investment and consumption decisions and provide more accurate and reliable recommendations for practitioners and policymakers. As the scope and complexity of problem-solving expands, demand for central processing unit (CPU) resources also rises. To optimize CPU resources, Jararweh et al. (2019) proposed the use of parallelization and vectorization techniques to enhance the performance of the Needleman−Wunsch algorithm; a widely used sequence alignment algorithm in bioinformatics with high computational complexity. The authors suggested distributing the workload to multiple processors using parallelization techniques and optimizing the use of CPU resources using vectorization techniques. They implemented the parallelized and vectorized Needleman−Wunsch algorithm, testing it on a set of DNA sequence data. The results demonstrated that their algorithm significantly improved performance compared with traditional J. Long, S. Zeng, B.B. Gupta et al. Journal of Innovation & Knowledge 9 (2024) 100459 2
algorithms. This technology has significant implications for future research in this field. We combine stochastic hyperbolic discounting (SHD) and a corporate international investment and consumption problem (CIICP) to investigate their implications for understanding and solving contemporary economic and financial issues, providing valuable insights for corporate investment and consumption decisions. Our CIICP is constructed referencing Bellalah and Wu (2009), and the SHD method was proposed by Zou et al. (2014). We obtain some interesting results by combining CIICP and SHD. This study makes several notable contributions. First, dynamic programming is applied to formulate a dynamic programming equation (DPE) that describes the optimal corporate international investment strategy and consumption choice for sophisticated individuals. Second, we provide explicit solutions for logarithmic utility. Third, we conduct a sensitivity analysis to examine the volatility parameter of the optimal solution. The results of this study provide valuable insights for CIICP that are of considerable significance for corporate investment and consumption decisions. The remainder of the paper is organized as follows. Section 2 presents the CIICP market setting and introduces the concept of SHD. Section 3 details the DPE for the problems of international investment and consumption with complex individual corporations. In Section 4, we obtain an explicit solution for log utility. Section 5 provides simulation results to illustrate the impact of volatility parameters on foreign production markets and exchange rates. We present some conclusive and suggested extensions in Section 6. Finally, the Appendix presents the model’s technical details. Model hypotheses In this section, we present the basic model for the CIICP framework and describe SHD referencing Zou et al. (2014). Let ðV;F;F;PÞdenote a filtered probability space, where Vis a probability space, P is a measurable probability, Ftrepresents the information available up to time t, and the filtration F¼f Ftg;t2½0;T satisfies the usual conditions (i.e., fFtg;t2½0;Tis right-continuous and P-complete). Here, Trepresents the terminal horizon. We assume that all stochastic processes and random variables are defined in the filtered probability space ðV;F;F;PÞ. Additionally, let B1 t;B2 t, and B3 t be three mutually dependent, one-dimensional Brownian motions with correlation coefficients r12;r13, and r23 2½1;1, respectively, which represent uncertain external sources in the market. The industrial market model In this section, we construct a market referencing Bellalah and Wu (2009), considering a “two-country”firm for which domestic ðRtÞand foreign ðR tÞstatic cash flows are denoted as follows: Rt¼ð1kÞðPtCtÞQt;Qt¼Pb t; R t¼ð1kÞðP tC tÞQ t;Q t¼ðP tÞb;ð1Þ where Pt;Ct(domestic), P tand C t(abroad, marked with *) are the output and input production prices. Qtand Q trepresent the amount of production output, band bdenote positive constants, etrepresents the exchange rate, which is uncertain, and kand kare the tax rates in home and foreign countries. Choi (1989) assumed the quantity of output (Qt) to be certain and did not consider the tax. Similar to Choi (1989) and Bellalah and Wu (2009), we assume that the dynamic input and output prices of products and the exchange rate can be expressed in the following form: dPt Pt ¼mpdt þsdB1 t;dCt Ct ¼mcdt þsdB1 t; dP t P t ¼mpdt þsdB2 t;dCt Ct ¼mcdt þsdB1 t; ð2Þ and det et ¼medt þsedB3 t;ð3Þ where the terms mp;mp;mc;mc, and meare constant-bounded and represent the rates that are expected instantaneously for various variables. The terms s;sand serepresent the coefficients of instantaneous fluctuations. The initial values of the above variables are P0;C0; P 0;C 0, and e0, respectively. The cash flow generated from the domestic market ðRtÞis expressed as follows: dRt Rt ¼fðtÞdt þðbþ1ÞsdB1 t;ð4Þ with fðtÞ¼1 2ðbþ1Þ2s2þðbþ1Þmp1 2s2 þðmpmcÞC0emct P0emptC0emct:ð5Þ Similarly, the cash flow ðRtÞin the foreign market that has not yet been remitted is denoted as follows: dRt Rt ¼fðtÞdt þðbþ1ÞsdB2 t;ð6Þ where fðtÞ¼1 2ðbþ1Þ2ðsÞ2þðbþ1Þmp1 2ðsÞ2 þðmpmcÞC 0emct P 0emptC 0emct:ð7Þ From R¼etRtand Eqs. (3) and (6), the variation in cash flow from abroad ðR tÞis obtained as follows: dR t R t ¼½fðtÞþmedt þðbþ1ÞsdB2 tþsedB3 t;ð8Þ where fðtÞ¼fðtÞþðbþ1Þsser23:ð9Þ However, a sunk cost is paid by investors who require extra returns to compensate for the resources with which they engage in the search to obtain information. Information cost has an incremental return dimension, and it is included in the descriptions of the dynamics of different variables and added to the drift. Thus, the actual change in domestic cash flow should be indicated as follows: dRt Rt ¼ftðÞþλR ½dt þbþ1ðÞsdB1 t; R0¼1kðÞPb 0P0C0 ðÞ: ð10Þ where λRrepresents the rate of information cost in the domestic market. The real changes in the exchange market are expressed as follows: det et ¼½meþλedt þsedB3 t; eð0Þ¼e0: ð11Þ where λerepresents the rate of information cost in the exchange market. Then, from R¼etRtand Eqs. (8) and (11), the change in cash flow at a specific time coming from the foreign market can be easily obtained as follows: dR t R t ¼½fðtÞþλRþmeþλedt þðbþ1ÞsdB2 tþsedB3 t; R 0¼ð1kÞPb 0ðP 0C 0Þ: ð12Þ J. Long, S. Zeng, B.B. Gupta et al. Journal of Innovation & Knowledge 9 (2024) 100459 3
Wealth process Let Wtrepresent corporate wealth and ptdenote the proportion of corporate investment abroad at time t0. 1 ptis the proportion of home market investment. Referencing the corporate consumption introduced by Sumner (2008), in addition to decision making on investment, the manager can also make consumption decisions to satisfy shareholder demands. Eqs. (10) and (12) indicate that corporate wealth can be expressed as follows: dWt¼ptfðtÞþλRþmeþλe ðÞWtþð1ptÞfðtÞþλR ðÞWtct ½dt þð1ptÞðbþ1ÞWtsdB1 t þptðbþ1ÞWtsdB2 t þptWtsedB3 t; ð13Þ where ctis corporate consumption. We assume a positive initial corporate wealth (W0), and pð¢Þis a measurable process that is almost surely locally bounded. SHD We use the framework of Merton (1969) to introduce the stochastic hyperbolic discount function from Harris and Laibson (2013) to examine time-inconsistent individuals’consumption and investment. Referencing Harris and Laibson (2013), the discount function Dðt;sÞis expressed as follows: Dðt;sÞ¼ erðstÞ;s2½t;tþtÞ; herðstÞ;s2½tþt;þ1Þ; ð14Þ where ½tþt;þ1Þis the forward interval and ½t;tþtÞis the closest interval. We assume that the closest interval ðtÞhas an exponentially distributeddurationwithparameterλ, and the stationarity assumption is satisfied for the stochastic hyperbolic discount function Dðt;sÞ(i.e., Dðt ;tþsÞ¼Dð0;sÞ). The expected length of the closest interval is E½t¼1 λ. Alowerλindicates a longer expected duration. The duration of the closest interval is 1as λ¼0, and the discount function degenerates to a constant exponential discount, and the problem represents Merton’s classical case; however, when λ!1, the closest interval becomes zero, and the discount function is transformed into the following: Dðt;sÞ¼ 1;s¼t; herðstÞ;s2½t;þ1Þ; ð15Þ Ajumpoccursats¼t. The parameter hð0<h1Þshows the present deviation in the degree of preference. A smaller hindicates a larger present deviation. When h¼1, the present and future intervals are not distinguished and the discount function Dðt;sÞonce again degenerates to an exponential discount function with a discount rate of r, and this implies that the problem reduces to Merton’s classical case. CIICP optimization model This study simultaneously considers firms’investment and consumption to maximize the utility of wealth by determining the investment strategy and consumption choice. A mature corporate investor with temporally inconsistent preferences is required to allow for future selves in the current decision. The following optimization problems must be solved by complex individual firms for finite and infinite planning periods with SHD to obtain optimal and timeconsistent consumption and portfolio policies, respectively: maxfct;ptgERtþt tertUðctÞdt þhRT tþtertUðctÞdt þherTBðT;WTÞ hi s:t:dWt¼ptfðtÞþλRþmeþλe ðÞWtþð1ptÞfðtÞþλR ðÞWtct ½dt þð1ptÞðbþ1ÞWtsdB1 t þptðbþ1ÞWtsdB2 t þptWtsedB3 t; ð16Þ and maxfct;ptgERtþt tertUðctÞdt þhR1 tþtertUðctÞdt hi s:t:dWt¼ptfðtÞþλRþmeþλe ðÞWtþð1ptÞfðtÞþλR ðÞWtct ½dt þð1ptÞðbþ1ÞWtsdB1 t þptðbþ1ÞWtsdB2 t þptWtsedB3 t; ð17Þ where Uð¢Þis the corporate consumption utility in the investment period, Bð¢Þis the corporate terminal wealth utility, ris the manager’s time preference in the recent period, hrepresents the diminished rate of future discounting, and trepresents the length of time of the current phase. In the next subsection, we first derive the problem for complex individuals, and then extend the analysis to yield DPE that are valid for infinite level problems. The DPE of the finite horizon problem Our analytical approach is based on Zou et al. (2014), and has also been used by Karp (2007). Specifically, a continuous-time problem is converted into a discrete-time problem, which we solve using backward induction, obtaining the optimal action of the future self first, and then we obtain the DPE for a finite horizon problem for mature individuals by iterating and passing the problem to the limit of continuous time. Denoted by Vðt;WtÞthe value function of Eq. (16) is as follows: Vðt;WtÞ¼ maxfct;ptg EZtþt t ertUðctÞdt þhZT tþt ertUðctÞdt þherTBðT;WTÞ ;ð18Þ with VðT;WTÞ¼BðT;WTÞ. We assume that Vðt;WtÞis of classC1. Similar to Zou et al. (2014), we deduce the DPE to replace the Hamilton-Jacobi-Bellman (HJB) equation as follows: rVVtK¼maxfct;ptgUVWcþVWWptfðtÞþλRþmeþλe ðÞ½ f þð1ptÞfðtÞþλR ðÞ þ1 2VWW W2½ð1ptÞ2ðbþ1Þ2s2þp2 tðbþ1Þ2ðsÞ2 þp2 tðseÞ2þ2ð1ptÞptðbþ1Þðbþ1Þssr12 þ2ð1ptÞptðbþ1Þsser13 þ2p2 tðbþ1Þsser23g; ð19Þ where Kðt;WtÞ¼ZT t eðλþrÞðstÞUðc sÞdt;ð20Þ and c tsatisfies Eq. (19). Therefore, the DPE for Eq. (16) is Eqs. (19) and (20), for which the boundary conditions for complicated individuals are as follows: VðT;WTÞ¼BðT;WTÞ:ð21Þ To simplify, we denote the following: D1 u¼fðtÞþλR; D2 u¼fðtÞþλRþmeþλe; D1 d¼ðbþ1Þ2s2; D2 d¼2ðbþ1Þ2s2þ2ðbþ1Þðbþ1Þssr12 þ2ðbþ1Þsser13; D3 d¼ðbþ1Þ2s2þðbþ1Þ2ðsÞ2þðseÞ2 2ðbþ1Þðbþ1Þssr12 2ðbþ1Þsser13 2ðbþ1Þsser23; ð22Þ J. Long, S. Zeng, B.B. Gupta et al. Journal of Innovation & Knowledge 9 (2024) 100459 4
then Eq. (19) is equivalent to the following: rVVtK¼maxfct;ptgUVWcþVWWD1 uþ1 2VWW W2D1 d þpVWWðD2 uD1 uÞþ1 2VWW W2D2 d þ1 2p2VWW W2D3 dg:ð23Þ By the first-order conditions, we obtain the optimal solution for Eq. (16) as follows: Uðc tÞ¼VW; p t¼ VW VWW W D2 uD1 u D3 d 1 2 D2 d D3 d : 8 > < > :ð24Þ Remark 1. (1) The term VWW W VWis the counterpart to the Pratt−Arrow measure (Pratt, 1976) with respect to risk aversion. This measure reflects how risk-averse a company is toward domestic and foreign investments. (2) The term D3 drefers to the variability of investment strategy. (3) The first term, VW VWW W D2 uD1 u D3 d in p t, is referred to as “speculative demand”or aggressive demand and is contingent on risk aversion measures. (4) The second term, 1 2 D2 d D3 d ,inp tis referred to as hedging demand, which covers the demand for foreign investments on a risk hedging basis and is not related to the manager’s risk attitude. Remark 2. Eq. (22) shows that D2 uD1 u¼fðtÞfðtÞþmeþðλRλeλRÞ. Information costs are included in the investment strategy, and this influence is represented by λRλeλR. This expression indicates the information costs involved in cost advantage seeking, wherein managers will not agree to invest abroad until they have obtained some useful information regarding the foreign and exchange markets. Without this information, managers are more inclined to make investments in the local market. The DPE of the infinite horizon problem The value function defines Vðt;WtÞas in Eq. (17) as follows: Vðt;WtÞ¼ maxfct;ptgEZtþt t ertUðctÞdt þhZ1 tþt ertUðctÞdt :ð25Þ Following Zou et al. (2014), we the DPE can be obtained as follows: dVtK¼maxfct;ptgUVWcþVWWD1 uþ1 2VWW W2D1 d þpVWWðD2 uD1 uÞþ1 2VWW W2D2 d þ1 2p2VWW W2D3 dg:ð26Þ where Kt;Wt ðÞ¼λ1hðÞ R1 teλþrðÞsUc s ds:ð27Þ We further assume limT!1herðTtÞBðT;WTÞ¼0. As a consequence, a transversal condition with the boundary condition Eq. (21) becomes the following: limt!1herTVðt;WtÞ¼0:ð28Þ For this reason, the HJB of Eq. (17) is subject to Eq. (13), which is Eq. (26) for sophisticated individuals, and the horizontal condition is Eq. (28). The explicit solution for CIICP with log utility In this section, we consider the CIICP with log utility with the following: UðctÞ¼lnct;BðT;WTÞ¼&lnWT:ð29Þ We present the following lemma to solve the optimal problem with log utility. Lemma 1. The solution for the following ODE: _ atrat¼Pt;ð30Þ can be characterized as follows: at¼aTerðTtÞZT t erstðÞ Psds;ð31Þ where aTis a given scalar, and Ptis a given function. The proof is presented in Appendix A. From Eq. (23), we have the following: Theorem 1. Eq. (16) is optimally solved as follows: c t¼Wt at ; p t¼1 D3 d ftðÞftðÞþλRþmeþλeλR1 2D2 d ; ð32Þ and the sophisticated t-agent’s value function is as follows: Vðt;WtÞ¼atlnWtþbt;ð33Þ where at;btsatisfies the following: at¼erðTtÞ&h r 1h λþreðλþrÞðTtÞþλhþr rðλþrÞ; bt¼RT terðstÞPsds; ð34Þ with Pt¼lnatD1 u1 2D1 dþ1 2D3 d D2 uD1 u1 2D2 d 2 "# atþ1 þλð1hÞRT teðλþrÞðstÞRs tG0ðvÞ 1 av dv lnas ds; G0ðtÞ¼ D2 uD1 u D3 d 1 2 D2 d D3 d ! ðD2 uD1 uÞþD1 u 1 2D1 dþD2 uD1 u D3 d 1 2 D2 d D3 d ! aD2 dþ1 2 D2 uD1 u D3 d 1 2 D2 d D3 d ! D3 d "# ; ð35Þ and D1 u;D2 u;D1 d;D2 d;and D3 dare given in Eq. (23). The proof is presented in Appendix B. Remark 3. We find that the optimal investment strategy does not determine the wealth process (Wp t), but the optimal consumption choice depends on the wealth process (Wp t). Results of simulation and parameter sensitivity In this section, we examine the impact of parameters with respect to firms’optimal investment strategies and consumption choices, and supply several numerical examples demonstrating the impact. J. Long, S. Zeng, B.B. Gupta et al. Journal of Innovation & Knowledge 9 (2024) 100459 5
For the description below, in the absence of any other indication, the fundamental parameters are provided as follows: b¼2;b¼2;mp¼0:08;mp¼0:06;mc¼0:02;mc¼0:03; s¼0:2;s¼0:15;se¼0:05;r12 ¼0:8;r13 ¼0:6;r23 ¼0:4; me¼0:004;λR¼0:001;λR¼0:002;λe¼0:003;P0¼6¼P 0; C0¼2¼C 0: (The data are obtained from Bellalah & Wu, 2009). (1) Based on Theorem 1, we can obtain the portfolio changing over time. In Fig. 1, (i) letting λR¼0:001;λR¼0;and λe¼0, we obtain the curve that describes the relationship between the portfolio and information cost in the home market. (ii) Letting λR¼0;λR¼0:002;and λe¼0, or letting λR¼0;λR¼0;and λe¼0:003;we obtain a similar curve that describes the relationship between the portfolio and information cost in the home or exchange market. (iii) Letting λR¼0;λR ¼0;and λe¼0, we obtain the curve that demonstrates the portfolio without any information cost. (2) In Theorem 1, we analyze the sensitivity of information cost rates on optimal investment strategies. According to Theorem 1,@p @λR <0;@p @λR>0;@p @λe >0 and @p @me >0, which means that @p @λR <0, indicating that the manager should invest less money in the foreign market when the home market’s information cost rate (λR) increases and other parameters are fixed. @p @λR>0 indicates that more funds should be allocated to the foreign market when other parameters are fixed, and as the information cost rate in the foreign market rises, λRincreases. If managers do not want to spend much on information cost in the foreign market, they choose to invest in the home market. Therefore, information has a central influence on foreign investment and can explain the home bias in international finance. @p @λe >0 means that more information cost in the exchange market indicates a higher proportion of the total budget is allocated to foreign investment. @p @me >0. This suggests that when the exchange rate (me) increases and other parameters are fixed, the manager should allocate more money to foreign market investment. It is crucial to determine the relationship between the portfolio and the cost of a single piece of information. We first set λR¼ 0;λR¼0;λe¼0;me¼0 for every case. To not show the curves superposed, we let t¼1;2;3;and 4. In Fig. 2, (i) letting t¼1, we obtain the curve that demonstrates the portfolio changing with λR. When λR¼0:009, the strategy for achieving the best in the foreign market is p¼0:9735. In this case, the highest proportion of corporate wealth is invested in the foreign market. (ii) Letting t¼2, we obtain the curve that demonstrates the portfolio changing with λe. When λe¼0:006, the strategy to achieve the best in the foreign market is p¼0:9149. (iii) Letting t¼3, we obtain the curve that demonstrates the portfolio changing with me.Whenme¼0:01, for the CRRA case, the strategy for achieving the best in the foreign market isp¼0:9344. (iv) Letting t¼4, we obtain the curve that demonstrates the portfolio changing with λR.WhenλR¼0:001, for the CRRA case, the strategy for achieving the best in the foreign market is p¼0:729, indicating that 72.9 % of the corporate wealth is made available for investment in the domestic market. To compare with the research results from Bellalah and Wu (2009), this study uses the authors’investment proportion in foreign markets and our own, respectively. When λR¼0:01, their result is x= 0.9073, while our result is about 0.8; when me¼0:01, their result is x= 0.9508, while our result is about 0.8; when λe¼0:001, their result is x= 0.7769, while our result is about 0.45; and when λR¼0:0095, their result is x= 0.4943, while our result is about 0.25. Considering SHD, the results of this study are generally lower than those of Bellalah and Wu (2009), which indicates that SHD has a significant impact on investment decision making, which is consistent with the hypothesis of time-consistent preferences being unrealistic. (3) From Theorem 1, we next examine the influence of λand Ton the consumption−wealth ratio. (i) The expected duration of the nearest interval goes to zero as λ!1, and in the case of the stochastic discount function, is changed to Eq. (15) as a deterministic Fig. 1. Portfolio changes over time. J. Long, S. Zeng, B.B. Gupta et al. Journal of Innovation & Knowledge 9 (2024) 100459 6
jump function that jumps at t.IfT<1,byEq. (34), we obtain the following: at¼erðTtÞ&h r þh rðλ!1Þ; which leads to c t Wt ¼1 erðTtÞ&h r þh r : (ii) For an infinite planning horizon, i.e., Eq. (17),byEq. (34) and letting T!1, we have at¼λhþr rðλþrÞ, and the consumption−wealth ratio can be written as follows: c t Wt ¼rðλþrÞ λhþr: (4) We analyze the sensitivities of parameters λ;h, and time ton the consumption−wealth ratio. Note that for any x>0;ex>1þx, and taking the partial derivatives of atw.r.t parameter λ, we can obtain the following: @at @λ¼ 1h ðλþrÞ2eðλþrÞðTtÞþ1h λþrðTtÞeðλþrÞðTtÞ1h ðλþrÞ2 ¼1h ðλþrÞ2eðλþrÞðTtÞ1þðλþrÞðTtÞeðλþrÞðTtÞ hi <0; which means the following: @ct=Wt @λ¼@ct=Wt @at @at @λ¼ 1 a2 t @at @λ>0: By taking the derivative of atin relation to the parameter h,we obtain the following: @at @h¼1 reðTtÞþ1 r11 λþr ¼1 r1eðTtÞ hi þ1 λþreðλþrÞðTtÞ1 r : Taking the partial derivatives of the above equation in relation to time t, we obtain the following: Fig. 2. Consumption−wealth ratio c t /W t changes over time t. J. Long, S. Zeng, B.B. Gupta et al. Journal of Innovation & Knowledge 9 (2024) 100459 7
@ @t @at @h ¼eðTtÞ1eλðTtÞ hi <0; which shows that @at @his decreasing with respect to time t. Next, we will observe whether @at @his negative by its value at t=T. Noting that the parameter r2½0;1, we obtain the following: @at @ht¼T ¼1 λþr11 r <0:ð36Þ Thus, we determine that @at @h<0, which means the following: @ct=Wt @λ¼@ct=Wt @at @at @λ¼ 1 a2 t @at @λ<0: Therefore, the consumption−wealth ratio c t=Wtis decreasing with respect toh. According to Eq. (34), we obtain the following: @at @t¼r&h r erðTtÞð1hÞeðλþrÞðTtÞ ¼1 1herðTtÞr&h 1heλðTtÞ : If r&h 1h1, then @at @t0, which means @ @t ct Wt ¼@ @t1 at ¼1 a2 t @at @t0. If r&h 1h<eλT, then @at @t<0, which means @ @t ct Wt >0. Otherwise, if eλT<r&h 1h<1, then when t>Tþ1 λln r&h 1h;@at @t<0, or else @at @t0. Furthermore, λ¼0orh¼1 leads to the following: atλ¼0 ¼erðTtÞ&h r 1h rerðTtÞþ1 r ¼1 r1þð&r1ÞerðTtÞ hi ; ath¼1 ¼erðTtÞ&1 r þλþr rðλþrÞ ¼1 r1þð&r1ÞerðTtÞ hi ; ð37Þ which means that atjλ0or h¼1¼1 r½1þð&r1ÞerðTtÞ. Therefore, the problem reduces to Merton’s classical case yielding the following: ct Wt ¼r 1þð&r1ÞerðTtÞ: By Eq. (32) and for λ>0;0<h<1, we obtain the following: atλ>0;0<h<1 ;>atλ¼0 ;atλ>0;0<h<1 >ath¼1 ; which indicates that the consumption−wealth ratio in this study is larger than Merton’s. In Fig. 2(i), letting z1¼1 and h1¼0:3, we plot the curves with λ1¼0;0:5;1, and 10. The results show that the consumption−wealth ratio is increasing with respect to parameter λ. In Fig. 2(ii), letting λ1¼1 and h1¼0:3, we plot the curves for z1¼ 1;2;3 and 4. The results show that the influence of zon the consumption−wealth ratio is significant in the later period. In Fig. 2(iii), letting λ2¼1 and &1¼1, we plot the curves with h1 ¼0;0:05;0:1 and 1. The results show that the influence of hon the consumption−wealth ratio is significant in the early stage. (5) In Theorem 1,weobtainthefigures demonstrating the optimal consumption−wealth ratio ( ct Wt) changes with the parameters λand &. In Fig. 3(i), using three different values of &with 0.01, 0.1 and 0.8, the results show that the consumption−wealth ratio ( ct Wt) is scaling up relative to parameter λ. In Fig. 3(ii), using three different values of &with 1, 10 and 100, the results show that the consumption−wealth ratio ( ct Wt) is also scaling up relative to parameter λ. In Fig. 4(i), taking three different values of λwith 0, 1 and 10, the results show that the consumption−wealth ratio ( ct Wt) is reduced in relation to parameter &. In Fig. 4(ii), taking three different values of &with 1, 10 and 100, the results show that the consumption−wealth ratio ( ct Wt) is also reduced in relation to parameter &. Conclusion This study builds upon the CIICP theory and introduces the concept of SHD based on the prior research conducted by Zou et al. (2014) and Bellalah and Wu (2009). By taking into account the evolving preferences of sophisticated individuals, our study enables the formulation of consumption and investment strategies in a consistent manner over time. We employ a recursive approach to derive explicit solutions for corporate international investment strategy and consumption choices, considering log utility and encompassing both infinite and finite planning periods. Fig. 3. Consumption−wealth ratio c t /W t changes with λ. J. Long, S. Zeng, B.B. Gupta et al. Journal of Innovation & Knowledge 9 (2024) 100459 8