Determinant Factors of FDI Spillovers – What Do We Really Know?

Empirical evidence about FDI spillovers to domestic firms has provided mixed results. This global evaluation has recently been complemented with the analysis of the factors that determine the existence, dimension and sign of FDI spillovers. We survey the arguments that support these factors and analyze the empirical evidence already produced. FDI spillovers depend on many factors, frequently with an indeterminate effect. Absorptive capacity of domestic firms and regions are a precondition for incorporating the benefits of FDI spillovers. Concerning the remaining factors, the results suggest opposite effects or, in some cases, are still insufficient to legitimate decisive conclusions.

Introduction
Foreign Direct Investment (FDI) is considered, in most countries, to be an important component of their development strategy, and policies are accordingly designed to stimulate inward flows. An important motivation for this interest is the possible existence of FDI spillovers, a concept that embodies the fact that MNEs (multinational enterprises) own technology 2 which can be transmitted to domestic firms and thereby raise their productivity level. The spread of productivity spillovers is thus a matter of externalities being transmitted from established foreign producers to domestic ones. Since the pioneering study of Caves (1974), the occurrence of FDI spillovers has been widely investigated. However, empirical evidence, as surveyed for instance by Meyer (2003) or Görg and Greenaway (2004), has provided mixed results.
Recently, there has been an effort to increase the knowledge about the factors that determine the existence, sign and magnitude of FDI spillovers. Perhaps the most important lesson to be learned from the existing studies is that we need to complement the "global evaluation" of whether FDI spillovers exist or not with a detailed analysis of "the different circumstances and policies of countries, industries and firms that promote or obstruct spillovers" (Lipsey, 2002, p. 32). A substantial body of work along these lines can already be extracted from the literature on the subject.
While previous surveys on this topic focused on the "global evaluation" of FDI spillovers, this paper provides a comprehensive description of the determinant factors of this phenomenon, both in terms of the arguments that have been proposed so far and the empirical evidence already produced.
The remainder of the paper is organized as follows: section 2 presents the channels through which domestic firms may benefit from FDI spillovers; section 3 analyzes the factors that determine these spillovers; section 4 focuses on the empirical evidence on this question; and section 5 concludes. 1

Channels of technological diffusion
FDI spillovers can occur through five main channels: demonstration/imitation, labor mobility, exports, competition and backward and forward linkages with domestic firms.
Demonstration (by MNEs)/imitation (by domestic firms) is probably the most evident spillover channel (Das, 1987;Wang and Blomström, 1992). The introduction of a new technology into a given market may be too expensive and risky for a domestic firm to undertake due to the costs inherent in acquiring its knowledge and the uncertainty of the results that may be obtained. If a technology is used successfully by a MNE, this will encourage domestic firms to adopt it. Obviously, the relevance of this effect increases with the similarity of the good produced by the two types of firms .
The second channel is related to the possibility of domestic firms hiring workers who, having previously worked for a MNE, know about the technology and are able to implement it in the domestic firm (Fosfuri et al., 2001;Glass and Saggi, 2002). 3 Nevertheless, it is important to stress a possible negative impact arising through this channel, as MNEs may attract the best workers from domestic firms by offering higher wages (Sinani and Meyer, 2004). The influence of this labor mobility on the efficiency of local firms is, however, difficult to evaluate as it involves tracking the workers in order to investigate their impact on the productivity of other workers (Saggi, 2002). It is not surprising, therefore, that there is a shortage of detailed studies in relation to this particular aspect.
Exports are a third channel through which the presence of MNEs may benefit domestic firms (Aitken et al., 1997;. Several studies have highlighted the positive impact of MNEs on the export capacity of domestic firms (Rhee, 1990;Aitken et al., 1997;Kokko et al., 2001). Among other aspects, export activity involves costs associated with the establishment of distribution networks, transport infrastructures or knowledge of consumers' tastes in foreign markets , which MNEs are better able to afford. By following the export processes of foreign firms (through imitation or, in specific circumstances, through collaboration), domestic firms may reduce the costs of entry into the foreign market. 4 The gains obtained in this way may have favorable repercussions on the productive efficiency of domestic firms.
The increased competition induced by MNEs is a fourth channel of FDI spillovers (Wang and Blomström, 1992;Markusen and Venables, 1999). Competition in the 2 domestic economy between MNEs and domestic firms is an incentive for the latter to make a more efficient use of existing resources and technology or even to adopt new technologies; it may also restrict the market power of domestic firms. 5 However, the efficiency of domestic firms may also be negatively affected through this channel, as the presence of MNEs may imply significant losses of their market shares, forcing them to operate on a less efficient scale, with a consequent increase in their average costs (Harrison, 1994;Aitken and Harrison, 1999).
A final channel concerns the relationships that domestic firms establish in local markets with MNEs as their suppliers (backward linkages) or customers of intermediate inputs (forward linkages) 6 as pointed out, for instance, by Lall (1980), and formalized by Rodríguez-Clare (1996), Markusen and Venables (1999) or Lin and Saggi (2004).
Let us firstly consider the case of backward linkages. With increasing returns to scale, the presence of MNEs may benefit domestic suppliers if it increases the demand for local inputs. In their attempts to assure a certain quality pattern, MNEs may also benefit domestic suppliers in several ways: providing technical support for the improvement of the quality of goods or for the introduction of innovations, through labor training for instance; providing support for the creation of productive infrastructures and for the acquisition of raw materials, as well as support at the organizational and management levels, among other aspects (Lall, 1980). We should also consider the possible increase in the efficiency of domestic firms brought about by the competition among them to become MNE suppliers. Furthermore, Matouschek (1999) considers that the benefits for domestic suppliers resulting from the presence of MNEs may be extended to other domestic firms that produce end-user consumer goods.
As far as the channel of forward linkages is concerned, the most evident link consists in the MNEs' supply of a higher quality inputs and/or at a lower price to domestic producers of end-user consumer goods (Markusen and Venables, 1999). Nevertheless, it is not possible to exclude the fact that the upgrade of production quality may lead to an increase in prices. If domestic firms do not have the capacity to benefit from this upgrade of quality, they will suffer the negative effects associated with increased costs (Javorcik, 2004b).
This short summary of the spillover channels clearly shows the existence of several, and frequently opposite, effects, making it difficult to formulate a clear expectation as to their global impact. Besides, "it is (…) difficult to distinguish one from 3 the other since the mechanism of technology spillovers from FDI is complex and often interdependent" (Kinoshita, 2001, p. 5).

Determinant factors of FDI spillovers -the main arguments
Recently, empirical studies have shown that the existence, sign and magnitude of FDI spillovers to domestic firms depend on a multiplicity of factors related to the characteristics of the MNEs and of foreign investment, as well as on the characteristics of the host countries, sectors and firms. In this section, we summarize the several factors that have so far been taken into consideration.
The factor that has been analyzed in most detail is the absorptive capacity of domestic firms, together with the influence of the technological gap between foreign and domestic firms. Using the definition of Narula and Marin (2003), "absorptive capacity includes the ability to internalize knowledge created by others and modifying it to fit their own specific applications, processes and routines" (Narula and Marin, 2003, p. 23). It is maintained that there must exist some technological gap between the two groups of firms for spillovers to occur. If the technological gap is too low, MNEs will transmit few benefits to the domestic firms (Kokko, 1994). According to some authors -following Gerschenkron (1962) -the magnitude of FDI spillovers will increase with the technological gap, as it increases the opportunities for domestic firms to obtain higher levels of efficiency via imitation of foreign technology (Findlay, 1978;Wang and Blomström, 1992). However, the gap cannot be too high, as the domestic firm would be unable to absorb the MNEs' technological advantage. The argument is that technology diffusion is not an automatic and direct effect deriving from the existence of a stock of knowledge at other firms: it also requires the recipient to have the capacity to absorb and adopt such technology (Lapan and Bardhan, 1973;Wang and Blomström, 1992;Perez, 1997, Kinoshita, 2001. The lesson to be drawn from these arguments is that domestic firms must have a moderate technological gap vis-à-vis MNEs in order to benefit from the higher technology associated with MNEs. The concept of absorptive capacity has been expressed not only at the microeconomic level, but also at the macroeconomic level. It has usually been associated with the level of development of a particular country (Borensztein et al., 1998;Xu, 2000) and particularly with its human capital stock. Moreover, Blomström et al. (1994) and 4 Kokko and Blomström (1995) show that MNEs use more advanced technology in countries and sectors that have a higher share of skilled labor.
Nevertheless, the concept of "absorptive capacity" comprises other factors, which we could label as "support infrastructures". For instance, Hermes and Lensink (2003) argue that a developed financial system favors the occurrence of FDI spillovers, as it reduces the risks inherent in the investment made by domestic firms to imitate the MNEs' technologies or to upgrade the qualification of their employees. 7 The relationship between the development level of the host country and the magnitude of FDI spillovers has also been established through two additional arguments.
Firstly, in the context of the labor mobility channel, a lower spillover level should occur in less developed countries. On average, MNEs pay higher wages than domestic firms, among other reasons in order to avoid labor turnover (Lipsey and Sjöholm, 2004). In less developed countries, this wage differential is usually higher, making more difficult the transfer of workers from MNEs to domestic firms. Secondly, it is considered less likely that less developed countries (with a lower absorptive capacity) will attract MNEs that have strong linkages with local suppliers and customers (Rodríguez-Clare, 1996).
Another factor that may affect the occurrence of spillovers is related to the export capacity of domestic firms. It has been argued that exporting firms already face significant competitive pressure in the foreign market and that MNEs operating in the domestic market are not expected to create relevant additional pressures (Blomström and Sjöholm, 1999). One may also argue that, as a firm increases its exporting capacity, the relevance of the domestic market decreases and the positive effects associated with the competition from MNEs become less important. In view of this, FDI spillovers will be more evident in the case of non-exporting domestic firms. In contrast to this relationship, it is suggested that domestic firms already exposed to foreign competition will probably have a greater capacity not only to absorb foreign technology but also, as emphasized by  and Schoors and van der Tol (2002), for countering the competition provided by MNEs in the local market, precluding this way the negative effect through the competition channel.
The size of domestic firms has also been linked to their capacity for obtaining the benefits associated with the presence of MNEs. Small firms (in terms of employment or production) may be less apt to compete with MNEs, suffering more significant losses (Aitken and Harrison, 1999). Furthermore, such firms may not have a sufficient 5 production scale to imitate some of the technologies introduced by MNEs. We therefore expect larger firms to benefit more from the presence of foreign companies.
It has also been maintained that spillovers have a circumscribed geographical dimension or, at least, that they decrease with distance (Audretsch and Feldman, 1996;Audretsch, 1998). Several reasons have been suggested for this fact (Girma and Wakelin, 2001;Girma, 2003;Torlak, 2004). For example, labor turnover is considered to be regionally confined. The existence of demonstration effects is also limited in space.
Finally, due to transport costs, backward linkages are expected to occur mainly at a regional level. However, in the context of the model proposed by Fosfuri et al. (2001), spillovers increase if the domestic firm and the MNE do not directly compete in the same region. In that case, the MNE is less affected by the transfer of the trained worker (who knows the MNE's technology) to the domestic firm and it becomes easier for the latter to hire that worker.
Is FDI from different countries equally prone to generate spillovers to domestic firms? The different sources of FDI can be expressed through several factors such as culture, language, levels of protection and the sectoral structures of FDI, among other aspects. For instance, in a study for the Indian economy, Banga (2003) argues that Japan invests in more standardized sectors whilst the USA preferentially invests in capitalintensive sectors. Considering the characteristics of the Indian economy, the expectation is that Japanese FDI produces more spillovers. Rodríguez-Clare (1996) argues that backward linkages depend positively on transport costs (and, hence, probably on distance) between the home country of the MNE and the host country, as well as on cultural, social and legal differences. If they are high enough, the MNE may have an incentive to buy inputs in the host country. Nevertheless, we can also consider that differences in cultures and languages may limit the domestic firms' capacity for assimilating the new technology, and therefore the net impact of these elements is ambiguous.
FDI spillovers to domestic firms are also influenced by the entry mode of FDI.
The argument is as follows: when the MNE enters through a merger or acquisition, technological transfer occurs gradually, restricting, or at least delaying, spillovers. On the contrary, when FDI occurs through greenfield investment, the introduction of the new technology is instantaneous (Braconier et al., 2001). In opposition to this effect, if FDI takes place through a merger or acquisition, the MNE, due to its pre-integration in the local economy, is expected to establish wider inter-sectoral linkages with domestic firms 6 than when entry occurs through greenfield investment (Scott-Kennel and Enderwick, 2001;Belderbos et al., 2001). In considering 272 Japanese MNEs operating in 24 countries, Belderbos et al. (2001) confirm this hypothesis.
Another determinant factor of FDI spillovers is the degree of foreign ownership of investment projects (Blomström and Sjöholm, 1999;Louri, 2001, 2002;Javorcik and Spatareanu, 2003). Minority ownership reduces the incentive for the parent firm to transfer more advanced technology to its affiliate due to its reduced control over the management. In view of this, the technology transferred increases with the degree of foreign ownership, making it more possible for spillovers to occur (Ramachandran, 1993).
However, one can also assume that a larger share of local ownership allows for easier access to foreign technology. Furthermore, affiliates with a greater degree of local participation can be expected to create more inter-sectoral linkages with the local economy, as was confirmed by Toth and Semjen (1999) in the Hungarian case.
According to Kokko et al. (2001), trade policy influences the technological characteristics of inward FDI (particularly in developing countries) and, consequently, the level of FDI spillovers. In countries with an inward-oriented trade policy, MNEs are focused on the local market. In order to succeed in that market, MNEs may be led to use technologies that are not available to domestic firms (or are only weakly developed), thus creating a vast potential for the existence of demonstration and learning effects. With an outward-oriented trade policy, MNEs will base their advantage mainly on their international distribution and marketing networks. Although in this latter case FDI spillovers may be created through exports, they can be expected to be more relevant in the first case (Kokko et al., 2001) not only on account of the above-mentioned reason, but also because of the greater competition with domestic firms. 8 It has also been argued that MNEs that are more focused on the local market establish greater inter-sectoral relationships with domestic firms, increasing the possibility of spillovers (Reuber, 1973;Altenburg, 2000). Nevertheless, we should note that an outward-oriented regime will probably attract a greater volume of FDI since the size of the domestic market ceases to act as a constraint (Bhagwati, 1978). 9 Still in the context of the same determinant factor, it has been observed that if MNEs produce for the foreign market and domestic firms produce for the local one, the possibility of spillovers via imitation will be diminished if goods produced for the local market use different production processes to goods produced for export due to differences in quality (or other characteristics), as emphasized by Javorcik (2004b). However, if the 7 requirements imposed by MNEs serving foreign markets are greater, more significant adjustments may be induced in local suppliers and the potential for spillovers will be increased.
Intellectual property rights are another important factor, not only increasing the probability of MNEs investing in a given country (Lee and Mansfield, 1996), but also increasing the likelihood of the occurrence of spillovers, since, when there is weak protection, there will be a tendency to attract mainly low-technology FDI (Javorcik, 2004a) and MNEs will tend to opt for fully-owned investment projects (Sherwood, 1990). 10 Javorcik (2004a) also suggests that a weak protection of intellectual property rights will induce MNEs to opt for investment projects centered preferentially on distribution and not on local production. All these factors work against the emergence of spillovers. Nevertheless, intellectual property rights can be considered an additional cost for those who imitate, and, consequently, they will be seen as a restriction on the potential benefits for domestic firms. In the context of a model in which MNEs choose between exports and FDI, Markusen (2001) concludes that the optimal solution for the domestic economy is a level of intellectual property rights equal to the minimum amount needed to guarantee entry.
Two other factors that determine the existence of FDI spillovers can be inferred from the model of Fosfuri et al. (2001). The first concerns the type of training received by workers at MNEs. If the worker receives training in a more firm-specific technology, local firms have less advantage in obtaining that technology as it is more costly to adapt it to their own production process. The second is connected with the duration of the work contract and/or the existence of restrictions on labor mobility. Both aspects may limit the transfer of workers from MNEs to domestic firms, and, as such, the occurrence of spillovers through the labor mobility channel.
In the model proposed by Wang and Blomström (1992), if MNEs face stronger competition in the local market, they will be forced to use more advanced technology in order to assure their market share. This is empirically confirmed by Blomström et al. (1994), in a study for the Mexican economy. In such case, spillovers can be expected to increase with competition in the local market. However, the high level of competition may also lead MNEs to protect their technological advantage in a more active way, as, for instance, in the model of Fosfuri et al. (2001).
As we mentioned in Section 2, in the context of the model proposed by Matouschek (1999), downstream firms can also take advantage of the presence of a MNE, 8 due to upstream improvements induced by its entry into the market. However, this effect will only occur when the MNE chooses a supplier arrangement that leads to a net increase in the domestic production of the input. Another element obviously affecting the possible emergence of inter-sectoral spillovers is an intensive use by MNEs of intermediate inputs, as this is a critical condition for the occurrence of spillovers through backward linkages (Rodríguez-Clare, 1996).
What motivates a MNE when it decides to locate abroad also affects the existence of FDI spillovers. The motivation implicit in most studies in this literature is based on the argument of traditional FDI theories that "when firms establish affiliates abroad and become multinational (…) they bring with them some amount of proprietary technology that constitutes their firm-specific advantage and allows them to compete successfully with local firms who have the superior knowledge of local markets, consumer preferences and business practices" (Blomström and Sjöholm, 1999, p. 915-916). FDI is thus technology sourcing, according to this view. Nevertheless, there has been empirical evidence showing that FDI may instead be technology exploiting, in the sense that it is attracted by the wish to gain access to the technological advantages of the host country (Cantwell, 1989;Kogut and Chang, 1991;Neven and Siotis, 1996). Fosfuri and Motta (1999) consider this last possibility in theoretical terms. In their model, the affiliate of a MNE, by locating abroad close to technologically-advanced local firms, may benefit from spillovers induced by the latter (which may then be transferred to the parent firm). There are therefore good reasons to consider that FDI spillovers will be greater when the traditional motivation for investing abroad prevails, i.e. when FDI is technology sourcing (Driffield and Love, 2003).
The "value" of the technology (in a broad sense that incorporates the innovative level of that technology) is a final determinant factor (and, perhaps, the most obvious one) of FDI spillovers. On the one hand, it incites domestic firms to try to accede to the technology but, on the other hand, it motivates MNEs to protect it (Blomström et al., 2000). If we accept that MNEs that are more recently established in the host economy make use of more advanced technology, we can argue, as in Karpaty and Lundberg (2004), that their impact will be negatively related to the length of time elapsing since their entry into the local market. 9

Determinant factors of FDI spillovers -the empirical evidence
In view of the contrasting arguments in the case of certain factors presented in the previous section, empirical analysis becomes increasingly important for clarifying the determinant factors of FDI spillovers. In this section, we consider the empirical evidence produced in this area.
As the knowledge content of the spillover effect is inherently an abstract concept and thus not directly measurable, the approach usually adopted in the empirical literature consists in capturing the spillover effect in the framework of an econometric analysis in which labor productivity (or total factor productivity) of domestic firms is regressed on a number of covariates assumed to have an effect on productivity, including the presence of foreign firms. In the present context, we take into account a range of studies which divide the sample in order to evaluate if the impact of the foreign presence on the domestic productivity differs according to the specific factor that determines that partition.
As mentioned in section 3, the determinant factors of FDI spillovers that have been most widely investigated are absorptive capacity (both at firm and country level) and the influence of the technological gap. It is therefore not surprising that this topic is the most profusely analyzed in empirical terms.
The importance of absorptive capacity emerges as a solid conclusion in most studies on this subject. Kinoshita (2001) uses R&D investment as a proxy for absorptive capacity, as it is considered that this increases the capacity of domestic firms to imitate new technologies (Cohen and Levinthal, 1989;Griffith et al., 2000Griffith et al., , 2003. With statistical information for the Czech Republic, he confirms that domestic firms only benefit from the presence of MNEs when they perform R&D actively, i.e. when they develop the ability to imitate new technologies. R&D activity and FDI thus appear to be complementary in their effect on the productivity of domestic firms. Keller and Yeaple (2003) analyze the case of the USA and conclude that only firms operating in high technology sectors -which invest more in R&D -benefit from positive FDI spillovers. In a sequence of studies for the Indian case, Kanturia (2000Kanturia ( , 2001Kanturia ( , 2002 separates "scientific" from "non-scientific" sectors and notes that, besides there being a distinct impact in the case of the group of sectors taken into account, as far as the "scientific sectors" are concerned, there is a positive impact in the case of firms with a higher absorptive capacity. The importance of absorptive capacity is also stressed by a vast group of additional studies, including those undertaken by Schoors and van der Tol (2002), , Girma (2003), and Karpaty and Lundberg (2004). 11 The study of Damijan et al. (2003) is a partial exception to this evidence, as they obtain a positive relation in the case of Hungary and Slovakia, but a negative one in the cases of Estonia and Latvia.
It has also been shown that the particular characteristics and development level of the region/country are relevant for the occurrence of the spillover phenomenon. According to Ponomareva (2000) and Yudaeva et al. (2003), who used data for Russia, FDI spillovers depend positively on the education level of the region considered. In turn, Sgard (2001) analyzes the existence of a distinct impact of FDI in spatial terms in the Hungarian case.
He considers a region between Budapest and the border with Austria -which is more developed than other parts of the country and closer to the EU border -and another region comprising the remaining area towards the border with Yugoslavia, Romania and Ukraine.
Although there is a positive impact in both regions, it is stronger in the first one. Imbriani and Reganati (1999) develop a similar investigation, considering three large regions in the context of the Italian economy. The results show that only the North-West region (the most dynamic region and the one where, for example, a large part of the R&D activity developed by large firms is concentrated) benefits from the foreign presence. This last group of studies emphasizes an important message: although FDI may work as a convergence mechanism at a national level if it produces significant gains in efficiency for domestic firms, it can also increase domestic inequalities at a regional level. This is an important result that certainly justifies further investigation.
In spite of the strong evidence concerning the relevance of absorptive capacity at both levels of the analysis (micro and macro), "the full potential of the concept of absorptive capacity is yet to be exploited. Future research ought to explore the concept in more detail to assert what contributes to a strong absorptive capacity on the firm as well on the national level" (Meyer, 2003, p. 22). For instance, Alfaro et al. (2003) show the importance of a developed financial system as a condition for obtaining benefits from the presence of MNEs.
Let us now evaluate the impact of the technological gap between domestic firms and MNEs. Kokko (1994) represents a pioneering contribution in this area. In order to investigate the influence of the technological characteristics of the sectors on the extent of spillovers, he considers three variables: the level of technological complexity (proxied by the amount of patent fees per employee in different industries), the average capital intensity of MNEs and the technological gap (evaluated by the difference in labor productivity between domestic firms and MNEs). The results suggest that an increase in 11 technological complexity and capital intensity makes the occurrence of FDI spillovers less likely, but that the influence of the technological gap is neutral. However, Kokko (1994) also concludes that wide technological gaps, together with large foreign market shares, generate a less favorable situation for the emergence of spillovers since, in this case, MNEs may operate in "enclaves", without connection with domestic firms.
Other studies do not allow to generalize the neutrality of the technological gap as drawn by Kokko (1994). In Sjöholm (1999a), the results differ in terms of the dependent variable used, thereby precluding a clarifying conclusion. Using data for Uruguay, Kokko et al. (1996) distinguish between domestic establishments with low and high technological gaps vis-à-vis MNEs operating in the same sector and obtain evidence of positive spillovers only in the first group. The same result emerges from the studies of Cantwell (1989), Tsou and Liu (1998), Imbriani and Reganati (1999), Girma and Wakelin (2000) and Kanturia (1998). Flôres et al. (2002) and Proença et al. (2002) try to identify, for the Portuguese case, the range in terms of productivity within which spillovers are maximized. The results of Flôres et al. (2002) suggest that spillovers are maximized when the average level of domestic productivity is between 50% and 80% of the corresponding productivity level of foreign firms. 12 Proença et al. (2002) perform a similar exercise, obtaining a range between 60% and 95%. This non-coincidence of results may be due mainly to the different proxies used for the technological gap. Narula and Marin (2003) analyze the case of Argentina by including the two above-mentioned topics -absorptive capacity and technological gap. They conclude that, regardless of the technology gap, the influence of the foreign presence at the sectoral level is not significant when absorptive capacity is not taken into account. When a proxy for the absorptive capacity of domestic firms is included -more specifically, investment in new equipment oriented to product/process innovation or investment in training activitiesthey observe positive spillovers to those domestic firms that have an absorptive capacity but not to the remaining ones, which reinforces the relevance of this determinant factor. Blomström and Sjöholm (1999) investigate whether the effect of the presence of MNEs differs according to whether domestic firms export or instead are oriented towards the domestic market. Using statistical information for Indonesia, they detect a positive effect (significant at the 1% level) in the case of non-exporting firms while, by contrast, the variable is not significant when exporting firms are considered. 13 Following the same line of analysis, Ponomareva (2000) includes a dummy variable equal to one if the domestic firm belongs to a sector exporting over 30% of its production and notes, in confirmation of the results obtained by Blomström and Sjöholm (1999), that the spillover effect is higher for non-exporting firms (or those with a low level of exports). However, Sinani and Meyer (2004) fail to find any remarkable difference concerning the impact on the two groups of domestic firms. Schoors and van der Tol (2002) split their sample into three groups: "closed" sectors (exporting less than 1/3 of their production), "open" sectors (exporting between 1/3 and 2/3 of their production) and "very open" sectors (exporting more than 2/3 of their production) and find that positive intra-sectoral spillovers only  Aitken and Harrison (1999) analyze the impact of the firms' size on the existence of FDI spillovers. They distinguish between firms with more or less than 50 workers (on average, throughout the period analyzed) and conclude that the impact on the efficiency of domestic firms of the foreign presence at the sectoral level is negative in both cases, but only significant for the smaller firms. This result confirms the idea that such firms have a lower capacity for obtaining positive effects from the presence of MNEs and are less suited to face competition from MNEs. Nevertheless, Dimelis and Louri (2001) find evidence to the opposite. In their study, only small domestic firms (those with less than 50 workers) benefit from positive spillovers. Considering FDI from Japan and from the rest of the world (mainly Europe), Girma and Wakelin (2001) also conclude that small-sized firms are the ones which benefit more from FDI spillovers. However, in the case of FDI from the USA, the impact is not significant for either small or large domestic firms.
Finally, Sinani and Meyer (2004) note that only small-sized domestic firms (with less than 50 workers) and medium-sized domestic firms (employing between 50 and 100 workers) benefit from FDI spillovers, the effect being greater in the first case. The impact is not significant when larger firms are considered. In short, the evidence concerning this determinant factor is inconclusive.
The interaction between the size of the domestic firms and absorptive capacity is emphasized by Girma and Wakelin (2001). They conclude that large and highly-skilled domestic firms do not benefit from foreign presence because they are "probably the 13 nearest to foreign multinationals in terms of technology and market share, and may already operate at the technological frontier" (Girma and Wakelin, 2001, p. 17). However, the group of firms that gain most from foreign presence consists of small firms with a high proportion of skilled labor.
A different topic is to analyze whether FDI spillovers have a local/regional dimension. This is one of the factors that enjoys a wider empirical evaluation but, once again, the results are differentiated. Sjöholm (1999b), Aitken and Harrison (1999) and Yudaeva et al. (2003) do not confirm this geographically circumscribed dimension. The first of these studies considers the case of Indonesia and concludes that the variable that measures intra-sectoral spillovers has a positive coefficient at the national level but a negative one when the evaluation is confined to a regional level. A similar result is obtained by Aitken and Harrison (1999) for the case of Venezuela and by Yudaeva et al. (2003) with data for Russia. However, Ponomareva (2000), also analyzing the Russian economy, confirms the hypothesis of a regional dimension in the spillover effect. Torlak (2004) follows the same line of research using statistical information for five countries. The existence of positive spillovers at the regional level is detected in the cases of the Czech Republic and Poland. Nevertheless, when the so-called agglomeration effect is controlled (by taking as an additional independent variable, the total number of firms in the region) the positive influence at the regional level only holds firm in the case of the Czech Republic and a negative effect is even detected in the Bulgarian case. Girma and Wakelin (2001) and Girma (2003) arrive at an opposite conclusion. The former detect a positive impact on the productivity of domestic firms caused by the foreign presence in the same region, both in the sector defined at a 4-digit level (intra-sectoral spillovers) and in the one defined at a 2-digit level (inter-sectoral spillovers) 14 , although the effect of the foreign presence outside the region is found to be non-significant. The latter study also concludes favorably as to the existence of a regional dimension in the spillover effect, using data for the UK.
A distinct question is evaluated by Haskel et al. (2002), Harris and Robinson (2002) and Konings (2001) concerning, in the two first cases, the UK and, in the third one, Poland, Romania and Bulgaria. Instead of capturing the foreign presence with the MNEs' share in the sector and region, these studies consider the MNEs' presence in the region as a whole (without sectoral disaggregation). Konings (2001) does not confirm the expected positive impact, with a negative sign even being detected in the Polish case (albeit only significant at the 10% level). The paper by Harris and Robinson (2002) estimates the 14 impact of the MNE's presence on each sector taken individually. Analyzing 20 sectors, they find that the impact is non-significant in 13, negative in four and positive in three.
Finally, Haskel et al. (2002) try alternative specifications with different lags for the foreign presence variable and, in general, they obtain non-significant results. As a general rule, therefore, this empirical evidence is not in favor of a positive impact. However, as Girma (2003) has pointed out, this evaluation relates to the agglomeration effect and not to intra-sectoral spillovers.
Another factor that may determine the magnitude of the spillover effect is related to the nationality of FDI. Banga (2003) concludes that Japanese FDI is more susceptible to create spillovers to Indian domestic firms than FDI from the USA. Haskel et al. (2002) detects evidence of positive spillovers associated with FDI from the USA and France (greater in the French case), a non-significant effect in the case of German MNEs and a negative one when Japanese FDI is considered. Using data for the Swedish economy, Karpaty and Lundberg (2004) distinguish between FDI from the USA, Japan and the rest of the world and, in spite of the fact that FDI spillovers are always significant, the greatest effect occurs in the Japanese case. The main lesson to be drawn from this group of studies is that the nationality of FDI is important for the existence of FDI spillovers. Hu and Jefferson (2002) investigate this aspect using evidence for the electronic and textile sectors in China. They examine whether there are relevant differences between the impact of FDI from Macao, Hong-Kong and Taiwan in comparison with FDI from OECD countries. The results show that only FDI from OECD countries has a significant -and negative -effect on the performance of local firms, which they relate to the higher technological level of firms from OECD countries and the consequent stronger competition on the local market.
Some studies have evaluated the influence on the spillover effect of the degree of foreign ownership of the affiliates of MNEs obtaining mixed results. Blomström and Sjöholm (1999) do not find a significant impact of this determinant factor. This result differs from that obtained by Louri (2001, 2002). Using three alternative variables to measure the foreign presence -sales, employment and capital - Dimelis and Louri (2002) discover that the impact of MNEs with minority foreign ownership is clearly greater. Through quantile regressions, this study concludes that majority foreign ownership only influences the 25% more efficient domestic firms, while the impact of minority foreign ownership is positive at any efficiency level of domestic firms. Javorcik and Spatareanu (2003) in turn, working with data for Romania, concluded that, in the case 15 of intra-sectoral spillovers, the impact is positive only when there is no local participation.
However, as far as inter-sectoral spillovers through backward linkages are concerned, they find a positive effect in the case of firms with local participation and a negative one in the opposite case. This last result is corroborated by Javorcik (2004b). Kokko et al. (2001) examine the role of trade policy on the dimension of FDI spillovers. Considering the Uruguayan case, they separate the MNEs which were established there in the period before 1973 (characterized by an import substitution regime) and those which were established after that period (in a more open regime). They obtain a positive coefficient for the variable relating to the presence of MNEs established in the former period and a negative one in the case of the MNEs established in the subsequent period. Therefore, inward-oriented MNEs appears to generate a more favorable impact on the productivity of local firms. Nevertheless, when they analyze, with a probit model, the impact of the foreign presence on the probability of the local firms beginning to export, a positive impact is only detected in the case of MNEs established during the outward-oriented period, which suggests that gains may be obtained through the export channel. Javorcik (2004b), using data for Lithuania, finds some (weak) evidence to suggest that FDI projects oriented towards the domestic market generate more spillovers. This aspect is also analyzed by Li et al. (2001), using Chinese data, in a study that distinguishes between the impact on domestic firms caused by the foreign presence and the impact resulting from competition with MNEs. It produces two interesting conclusions. Firstly, the negative effect of competition for domestic firms is restricted to those sectors where FDI is preferentially oriented towards the domestic market. Secondly, as far as the effect directly related with the existence of foreign presence is concerned, domestic firms only benefit, in terms of increased efficiency, in the case of export-oriented FDI. To sum up, once again, the conclusions from this set of studies do not legitimate an unequivocal interpretation.
The impact of different competition levels on the magnitude of the spillover effect is analyzed by Sjöholm (1999a) and his results suggest that a higher level of competition increases the possibility of FDI spillovers.
Another factor that has been analyzed concerns the MNEs' motivation for investing abroad. The analysis of this factor is, however, faced with an important difficulty: how to measure this motivation? Driffield and Love (2003) proxy it with the R&D intensity differential between the host country and the home country, at the sectoral level. The assumption is that, if this differential is positive, FDI is technology exploiting 16 and, by contrast, if the differential is negative, FDI is technology sourcing. It has been argued that the latter type of FDI is likely to produce greater productivity spillovers to domestic firms, as mentioned in section 3. Driffield and Love (2003) check this assumption by considering FDI inflows in the UK from 13 countries. They obtain a positive sign in the case of FDI with the traditional motivation (technology sourcing) and a negative sign in the case of technology exploiting FDI. Both results are highly significant (at 1% level). Girma (2003) performs a similar exercise, but with an analysis at the firm level, and broadly confirms these results.
Finally, using data for the Swedish economy, Karpaty and Lundberg (2004) investigate whether FDI spillovers depend on the length of time elapsing since the MNEs entered the local market. The evidence shows that only MNEs established after 1990 (and therefore probably using more advanced technology) generate positive spillovers.

Final Remarks
In this paper, we have shown that FDI spillovers to domestic firms depend on a multiplicity of factors associated with the entry mode of FDI, as well as with the characteristics of the MNEs and of the recipient economies, sectors and firms. These factors often work in opposite directions making the overall effect difficult to establish.
As far as the empirical evidence is concerned, it is insufficient to allow us to draw definitive conclusions for most of the factors. The most robust empirical result relates to the importance of the absorptive capacity of domestic firms, which appears to be a fundamental precondition for enabling them to capture these indirect benefits from FDI.
Furthermore, empirical evidence also shows that FDI spillovers appear to be greater in more developed regions. This last result allows us to consider that, although FDI spillovers may accelerate convergence at the national level by increasing the productivity of domestic firms, they may also contribute towards increased inequalities within each country. The evidence concerning the remaining factors is, in some cases, inconsistent or, in other cases, still too scarce to allow for unequivocal conclusions.
The research into this subject is far from being exhausted, but it already gives interesting insights into the mixed evidence resulting from the empirical literature as regards the existence of FDI spillovers. The main challenge now is to deepen the analysis of their determinant factors along the lines we have suggested in this survey, including the way in which they interact, in both theoretical and empirical terms. 17


Hereditary Portfolio Optimization with Taxes and Fixed Plus Proportional Transaction Costs—Part I

This is the first of the two companion papers which treat an infinite time horizon hereditary portfolio optimization problem in a market that consists of one savings account and one stock account. Within the solvency region, the investor is allowed to consume from the savings account and can make transactions between the two assets subject to paying capital gain taxes as well as a fixed plus proportional transaction cost. The investor is to seek an optimal consumption-trading strategy in order to maximize the expected utility from the total discounted consumption. The portfolio optimization problem is formulated as an infinite dimensional stochastic classical-impulse control problem. The quasi-variational HJB inequality (QVHJBI) for the value function is derived in this paper. The second paper contains the verification theorem for the optimal strategy. It is also shown there that the value function is a viscosity solution of the QVHJBI.

Introduction
This is the first of the two companion papers (see [1] for the second paper) which treat an infinite time horizon hereditary portfolio optimization problem in a financial market that consists of one savings account and one stock account. It is assumed that the savings account compounds continuously with a constant interest rate r > 0 and the unit price process, {S(t), t ≥ 0}, of the underlying stock follows a nonlinear stochastic hereditary differential equation (see (2.5)) with an infinite but fading memory. The main purpose of the stock account is to keep track of the inventories, (i.e., the time instants and the base prices at which shares were purchased or short-sold) of the underlying stock for purpose of calculating the capital gain taxes, and so forth. In the stock price dynamics, we assume that both f (S t ) (the mean rate of return) and g(S t ) (the volatility coefficient) depend on the entire history of stock prices S t over the time interval (−∞,t] instead of just the current stock price S(t) at time t ≥ 0 alone. Within the solvency region κ (to be defined in (2.29)) and under the requirements of paying fixed plus proportional transaction costs and capital gain taxes, the investor is allowed to consume from his savings account in accordance with a consumption rate process C = {C(t), t ≥ 0} and can make transactions between his savings and stock accounts according to a trading strategy -= {(τ(i), ζ(i)), i = 1,2,...}, where τ(i), i = 0,1,2,... denotes the sequence of transaction times and ζ(i) stands for quantities of the transaction at time τ(i) (see Definitions 2.4 and 2.5).
The investor will follow the following set of consumption, transaction, and taxation rules (Rules 1-6). Note that an action of the investor in the market is called a transaction if it involves trading of shares of the stock such as buying and selling. Rule 1. At the time of each transaction, the investor has to pay a transaction cost that consists of a fixed cost κ > 0 and a proportional transaction cost with the cost rate of μ ≥ 0 for both selling and buying shares of the stock. All the purchases and sales of any number of stock shares will be considered one transaction if they are executed at the same time instant and therefore incur only one fixed fee κ > 0 (in addition to a proportional transaction cost).

Rule 2.
Within the solvency region κ , the investor is allowed to consume and to borrow money from his savings account for stock purchases. He can also sell and/or buy back at the current price shares of the stock he bought and/or short-sold at a previous time.
Rule 3. The proceeds for the sales of the stock minus the transaction costs and capital gain taxes will be deposited in his savings account and the purchases of stock shares together with the associated transaction costs and capital gain taxes (if short shares of the stock are bought back at a profit) will be financed from his savings account.
Rule 4. Without loss of generality, it is assumed that the interest income in the savings account is tax-free by using the effective interest rate r > 0, where the effective interest rate equals the interest rate paid by the bank minus the tax rate for the interest income.
Rule 5. At the time of a transaction (say t ≥ 0), the investor is required to pay a capital gain tax (resp., be paid as a capital-loss credit) in the amount that is proportional to the amount of profit (resp., loss). A sale of stock shares is said to result in a profit if the current stock price S(t) is higher than the base price B(t) of the stock and it is a loss otherwise. The base price B(t) is defined to be the price at which the stock shares were previously bought or short-sold, that is, B(t) = S(t − τ(t)) where τ(t) > 0 is the time duration for which those shares (long or short) have been held at time t. The investor will also pay capital gain taxes (resp., be paid as capital-loss credits) for the amount of profit (resp., loss) by short-selling shares of the stock and then buying back the shares at a lower (resp., higher) price at a later time. The tax will be paid (or the credit will be given) at the buying back time. Throughout the end, a negative amount of tax will be interpreted as a capital loss credit. The capital gain tax and capital loss credit rates are assumed to be the same as β > 0 for simplicity. Therefore, if |m| (m > 0 stands for buying and m < 0 stands for selling) shares of the stock are traded at the current price S(t) at the base B(t) = S(t − τ(t)), then Mou-Hsiung Chang 3 the amount of tax due at the transaction time is given by |m|β S(t) − S t − τ(t) .
(1.1) Rule 6. The tax and/or credit will not exceed all other gross proceeds and/or total costs of the stock shares, that is, where m ∈ denotes the number of shares of the stock traded with m ≥ 0 being the number of shares purchased and m < 0 being the number of shares sold.
Under the above assumptions and Rules 1-6, the investor's objective is to seek an optimal consumption-trading strategy (C * ,- * ) in order to maximize the expected utility from the total discounted consumption over the infinite time horizon, where δ > 0 represents the discount rate and 0 < γ < 1 represents the investor's risk aversion factor. Due to the fixed plus proportional transaction costs and the hereditary nature of the stock dynamics and inventories, the problem will be formulated as a combination of a classical control (for consumptions) problem and an impulse control (for the transactions) problem in infinite dimensions. A classical-impulse control problem in finite dimensions is treated in [2]. In this paper a quasi-variational Hamilton-Jocobi-Bellman inequality (QVHJBI) for the value function together with its boundary conditions is derived. The second paper (see [1]) establishes the verification theorem for the optimal investment trading strategy. In there, it is also shown that the value function is a viscosity solution of the QVHJBI (see (QVHJBI ( * )) in Section 4.3.4). Due to the complexity of the analysis involved, the uniqueness result and finite-dimensional approximations for the viscosity solution of (QVHJBI ( * )) will be treated separately in a future paper.
In recent years, there has been extensive amount of research on the optimal consumption-trading problems with proportional transaction costs (see, e.g., [3][4][5][6], and references contained therein) and fixed plus proportional transaction costs (see, e.g., [7]) within the geometric Brownian motion financial market. In all these papers, the objective has been to maximize the expected utility from the total discounted or averaged consumption over the infinite time horizon without considering the issues of capital gain taxes (resp., capital loss credits) when stock shares are sold at a profit (resp., loss). In different contents, the issues of capital gain taxes have been studied in [8][9][10][11][12][13][14][15], and references contained therein. In particular, [9,10] considered the effect of capital gain taxes and capital loss credits on capital market equilibrium without consumption and transaction costs. These two papers illustrated that under some conditions, it may be more profitable to cut one's losses short and never to realize a gain because of capital loss credits and capital gain taxes as some conventional wisdom will suggest. In [8] the optimal transaction time problem with proportional transaction costs and capital-gain taxes was considered in order to maximize the long-run growth rate of the investment (or the so-called Kelley criterion), that is, where V (t) is the value of the investment measured at time t > 0. This paper is quite different from ours in that the unit price of the stock is described by a geometric Brownian motion, and all shares of the stock owned by the investor are to be sold at a chosen transaction time and all of its proceeds from the sale are to be used to purchase new shares of the stock immediately after the sale without consumption. Fortunately due to the nature of the geometric Brownian motion market, the authors of that paper were able to obtain some explicit results.
In recent years, the interest in stock price dynamics described by stochastic delay equations has increased tremendously (see, e.g., [16,17]). To the best of the author's knowledge, this is the first paper that treats the optimal consumption-trading problem in which the hereditary nature of the stock price dynamics and the issue of capital gain taxes are taken into consideration. Due to drastically different nature of the problem and the techniques involved, the hereditary portfolio optimization problem with taxes and proportional transaction costs (i.e., κ = 0 and μ,ν > 0) remains to be solved. This paper is organized as follows. The description of the stock price dynamics, the admissible consumption-trading strategies, and the formulation of the hereditary portfolio optimization problem are given in Section 2. In Section 3, the properties of the controlled state process are further explored and corresponding infinite-dimensional Markovian solution of the price dynamics is investigated. Section 4 contains the derivations of the QVHJBI together with its boundary conditions (QVHJBI ( * )) using a Bellmantype dynamic programming principle.
The verification theorem for the optimal consumption-trading strategy and the proof that the value function is a viscosity solution of the (QVHJBI ( * )) are contained in the second paper [1].

The hereditary portfolio optimization problem
Throughout the end, we use the following convention.

Convention 2.
If t ≥ 0 and φ : → is a measurable function, define Under Conditions 1-2, it can be shown that ρ is essentially bounded and strictly positive on (−∞,0]. Furthermore, 3) The following are two examples of ρ : (−∞,0] → [0,∞) that satisfy Conditions 1 and 2: ρ (−∞,0) (or simply × L 2 ρ for short) be the history space of the stock price dynamics, where L 2 ρ is the class of ρ-weighted Hilbert space of measurable functions φ : For t ∈ (−∞,∞), let S(t) denote the unit price of the stock at time t. It is assumed that the unit stock price process {S(t), t ∈ (−∞,∞)} satisfies the following stochastic hereditary differential equation with an infinite but fading memory: (2.5) In the above equation, the process {W (t), t ≥ 0} is one-dimensional standard Brownian motion defined on a complete filtered probability space (Ω,Ᏺ,P;F), where F = {Ᏺ(t), t ≥ 0} is the P-augmented natural filtration generated by the Brownian motion {W (t), t ≥ 0}. Note that f (S t ) and g(S t ) in (2.5) represent, respectively, the mean growth rate and the volatility rate of the stock price at time t ≥ 0. Note that the stock is said to have a hereditary price structure with infinite but fading memory because both the drift term S(t) f (S t ) and the diffusion term S(t)g(S t ) in the right-hand side of (2.5) explicitly depend on the entire past history prices (S(t),S t ) ∈ × L 2 ρ in a weighted fashion by the function ρ satisfying Conditions 1-2.
Note that we have used the following notation in the above: It is assumed for simplicity and to guarantee the existence and uniqueness of a strong solution S(t), t ≥ 0, that the initial price function (S(0),S 0 ) = (ψ(0),ψ) ∈ + × L 2 ρ,+ is given and the functions f ,g : L 2 ρ → [0,∞) are continuous, and satisfy the following Lipschitz and linear growth conditions (see, e.g., [18][19][20][21][22] for the theory of stochastic functional differential equations with an infinite or a bounded memory).
Assumption 2.2 (Lipschitz condition). There exists a constant c 2 > 0 such that Assumption 2.3. There exist positive constants α and σ such that Note that the lower bound of the mean rate of return f in Assumption 2.3 is imposed to make sure that the stock account has a higher mean growth rate than the interest rate r > 0 for the savings account. Otherwise, it will be more profitable and less risky for the investor to put all his money in the savings account for the purpose of optimizing the expected utility from the total consumption.
Although the modeling of stock prices is still under intensive investigations, it is not the intention of this paper to address the validity of the model stock price dynamics treated in this paper but to illustrate the hereditary optimization problem that is explicitly dependent upon the entire past history of the stock prices for computing capital gain taxes or capital loss credits. The term "hereditary portfolio optimization" is therefore coined in this paper for the first time. We, however, mention here that stochastic hereditary equation similar to (2.5) was first used to model the behavior of elastic material with infinite memory and that stochastic functional differential equations with bounded memory have been used to model stock price dynamics in option pricing problems (see [16,17]).
It can be shown that, for each initial historical price function (ψ(0),ψ) ∈ + × L 2 ρ,+ , the price process {S(t), t ≥ 0} is a positive, continuous, and F-adapted process defined on (Ω,Ᏺ,P;F) but it is not Markovian with respect to any filtration that makes sense. For this reason, we frequently consider the corresponding + × L 2 ρ,+ -valued process {(S(t), S t ), t ≥ 0} instead of the real-valued process {S(t), t ≥ 0}. However, following approaches similar to that of [20,Section 3], it can be shown under Conditions 1-2 and Assumptions 2.
We also note here that, since security exchanges have only existed in a finite past, it is realistic but not technically required to assume that the initial historical price function Mou-Hsiung Chang 7 (ψ(0),ψ) has the property that The stock inventory space. The space of stock inventories, N, will be the space of bounded functions ξ : (−∞,0] → of the following form: where {n(−k), k = 0,1,2,...} is a sequence in with n(−k) = 0 for all but finitely many k, and Note that the function ξ : (−∞,0] → defined above denotes the inventory of the investor's stock account. In particular, when As illustrated in Sections 2.3 and 2.5, N is the space in which the investor's stock inventory lives. The assumption that n(−k) = 0 for all but finitely many k implies that the investor can only have finitely many open positions in his stock account. However, the number of open positions may increase from time to time. Note that the investor is said to have an open long (resp., short) position at time τ if he still owns (resp., owes) all or part of the stock shares that were originally purchased (resp., short-sold) at a previous time τ. The only way to close a position is to sell what he owns and buy back what he owes.
If η : → is a bounded function of the form then for each t ≥ 0, we define, using Convention 2, the function η t : (−∞,0] → by In this case, ρ,+ be the investor's initial portfolio immediately prior to t = 0. That is, the investor starts with x ∈ dollars in his savings account, the initial stock inventory, (2.20) and the initial profile of historical stock prices (ψ(0),ψ) ∈ + × L 2 ρ,+ , where n(−k) > 0 (resp., n(−k) < 0) represents an open long (resp., short) position at τ(−k). Within the solvency region κ (see (2.29)), the investor is allowed to consume from his savings account and can make transactions between his savings and stock accounts under Rules 1-6 and according to a consumption-trading strategy π = (C,-) defined below.
..} is a trading strategy with τ(i), i = 1,2,..., being a sequence of trading times that are G-stopping times such that and for each i = 0,1,..., is an N-valued Ᏻ(τ(i))-measurable random vector (instead of a random variable in ) that represents the trading quantities at the trading time τ(i). In the above, m(i) > 0 (resp., m(i) < 0) is the number of stock shares newly purchased (resp., short-sold) at the current time τ(i) and at the current price of S(τ(i)) and, for k = 1,2,..., m(i − k) > 0 (resp., m(i − k) < 0) is the number of stock shares bought back (resp., sold) at the current time τ(i) and at the current price of S(τ(i)) in his open short (resp., long) position at the previous time τ(i − k) and at the base price of S(τ(i − k)).
For each stock inventory ξ of the form expressed (2.13), Rules 1-6 also dictate that the investor can purchase or short sell new shares and/or buy back (resp., sell) all or part of Mou-Hsiung Chang 9 what he owes (resp., owns). Therefore, the trading quantity {m(−k), k = 0,1,...} must satisfy the constraint set (ξ) ⊂ N defined by (2.24)

Solvency region.
Throughout the end of this paper, the investor's state space S is taken to be S = × N × + × L 2 ρ,+ . An element (x,ξ,ψ(0),ψ) ∈ S is called a portfolio, where x ∈ is investor's holding in his savings account, ξ is the investor's stock inventory, and (ψ(0),ψ) ∈ + × L 2 ρ,+ is the profile of historical stock prices. Define the function H κ : S → as follows: where G κ : S → is the liquidating function defined by (2.26) In the right-hand side of the above expression, x − κ = the amount in his savings account after deducting the fixed transaction cost κ; (2.27) and for each k = 0,1,..., The solvency region κ of the portfolio optimization problem is defined as Note that within the solvency region κ , there are positions that cannot be closed at all, namely, those (x,ξ,ψ(0),ψ) ∈ κ such that This is due to the insufficiency of funds to pay for the transaction costs and/or taxes, and so forth. Observe that the solvency region κ is an unbounded and nonconvex subset of the state space S. The boundary ∂ κ will be described in detail in Section 4.3.

Portfolio dynamics and admissible strategies.
At time t ≥ 0, the investor's portfolio in the financial market will be denoted by the quadruplet (X(t),N t ,S(t),S t ), where X(t) denotes the investor's holdings in his savings account, N t ∈ N is the inventory of his stock account, and (S(t),S t ) describes the profile of the unit prices of the stock over the past history (−∞,t] as described in Section 2.1. Given the initial portfolio and applying a consumption-trading strategy π = (C,-) (see Definition 2.4), the portfolio dynamics of {Z(t) = (X(t),N t ,S(t),S t ), t ≥ 0} can then be described as follows. Firstly, the savings account holding {X(t), t ≥ 0} satisfies the following differential equation between the trading times: (2.32) and the following jumped quantity at the trading time τ(i): As a reminder, m(i) > 0 (resp., m(i) < 0) means buying (resp., selling) new stock shares at τ(i) and m(i − k) > 0 (resp., m(i − k) < 0) means buying back (resp., selling) some or all of what he owed (resp., owned).
Mou-Hsiung Chang 11 Secondly, the inventory of the investor's stock account at time t ≥ 0, N t ∈ N, does not change between the trading times and can be expressed as the following equation: It has the following jumped quantity at the trading time τ(i): Thirdly, since the investor is small, the unit stock price process {S(t), t ≥ 0} will not be in anyway affected by the investor's action in the market and is again described as in (2.5).
Mou-Hsiung Chang 13 Let ( × L 2 ρ ) * be the space of bounded linear functionals (or the topological dual of the space × L 2 ρ ) equipped with the operator norm · * defined by Note that ( × L 2 ρ ) * can be identified with × L 2 ρ by the well-known Riesz representation theorem.
Let ( × L 2 ρ ) † be the space of bounded bilinear functionals Φ : In this case, DΦ(φ(0),φ) ∈ ( × L 2 ρ ) * is called the (first-order) Fréchet derivative of Φ at (φ(0),φ) ∈ × L 2 ρ . The function Φ is said to be continuously Fréchet differentiable if its Fréchet derivative DΦ : and where In this case, the bounded bilinear functional ρ . The second-order Fréchet derivative D 2 Φ is said to be globally Lipschitz on × L 2 ρ if there exists a constant K > 0 such that Assuming all the partial and/or Frechet derivatives of the following exist, the actions of the first-order Fréchet derivative DΦ(φ(0),φ) and the second-order Fréchet D 2 Φ(φ(0),φ) can be expressed as where ∂ φ(0) Φ and ∂ 2 φ(0) Φ are the first-and second-order partial derivatives of Φ with respect to its first variable φ(0) ∈ , D φ Φ and D 2 φ Φ are the first-and second-order Fréchet derivatives with respect to its second variable φ ∈ L 2 ρ , ∂ φ(0) D φ Φ is the second-order derivative first with respect to φ in the Fréchet sense and then with respect to φ(0), and so forth.
Let C 2,2 ( × L 2 ρ ) be the space of functions Φ : × L 2 ρ → that are twice continuously differentiable with respect to both its first and second variables. The space of Φ ∈ C 2,2 ( × L 2 ρ ) with D 2 Φ being globally Lipschitz will be denoted by C 2,2 lip ( × L 2 ρ ).
It will be shown in the proof of Theorem 3.5, however, that any tame function of the above form can be approximated by a sequence of quasi-tame functions that are in Ᏸ(Γ).
We have the following Ito formula in case Φ ∈ × L 2 ρ is a quasi-tame function in the sense defined above.
Proof. The Ito formula for a quasi-tame function Φ : × L 2 ([−h,0]) → for the × L 2 ([−h,0]) solution process {(x(t), x t ), t ≥ 0} of a stochastic function differential equation with a bounded delay h > 0 is obtained in an unpublished dissertation by Arriojas [18] (the same result can also be obtained from [21,22] with some modifications). The same arguments can be easily extended to the infinite memory stochastic hereditary differential equation (2.5) considered in this paper. To avoid further lengthening the paper, we omit the proof here.
Throughout the end of this proof, we define for each φ ∈ L 2 ρ and each k = 1,2,... the function φ (k) (−θ;h) by  (3.26) and c > 0 is the constant chosen so that and by the Lebesque dominating convergence theorem, we have Therefore, for any finite G-stopping time τ, we have from Theorem 3.4 and sample path convergence property of the Ito integrals (see [23,24]) that  This proves the theorem.

Dynkin's formula for the controlled state process.
Combining the above results in this section and results for general jumped processes (see [25,26]), we have the following Dynkin formula for the controlled (by the admissible strategy π) κ -valued state process {Z(t) = (X(t),N t ,S(t),S t ), t ≥ 0}: and A and Γ are as defined in (3.12) and (3.15).

The quasi-variational HJB inequality
The main objective of this section is to derive the dynamic programming equation for the value function in form of an infinite-dimensional quasi variational Hamilton-Jacobi-Bellman (HJB) inequality (or QVHJBI) (see (QVHJBI ( * )) in Section 4.3.4).

4.1.
The dynamic programming principle. The following Bellman-type dynamic programming principle (DPP) was established in [6] and still holds true in our problem by combining it with that obtained in [27][28][29] for optimal classical control of stochastic functional differential equations with a bounded memory. For the sake of saving space, we take the following result as the starting point without proof for deriving our dynamic principle equation.

Derivation of the QVHJBI.
In this section, we will derive the Hamilton-Jacobi-Bellman (HJB) quasi-variational inequality (see (QVHJBI ( * )) in Section 4.3.4) based on the dynamic programming principle described in Proposition 4.1. We emphasize here that it is not our intension to rigorously verify every step involved in the derivations since the rigorous verification is to be done in [1], the continuation of this paper. To derive (QVHJBI ( * )) in Section 4.3.4, we consider the effects on the value function when there is consumption but no transaction and when there is transaction but no consumption.
Mou-Hsiung Chang 21 In this case, V κ (X(t),N t ,S(t),S t ) = V κ (X(t−),N t− ,S(t),S t ) for all t ≥ 0, since there is no jump transaction. Assume that the value function V κ : κ → + is sufficiently smooth. From Proposition 4.1 and (4.4), we have This shows that since the maximum of the above expression is achieved at Note that the Fréchet differential operators A and Γ are defined in (3.12) and (3.15), respectively.

22)
ᏹ κ Φ is as given in (4.8), and the operators A and Γ are given as follows.

Boundary values of the QVHJBI.
4.3.1. The solvency region for κ = 0 and μ > 0. When there is no fixed transaction cost (i.e., κ = 0 and μ > 0), the solvency region can be written as where G 0 is the liquidating function given in (2.26) with κ = 0. This is because In this case, all shares of the stock owned or owed can be liquidated due the absence of a fixed transaction cost κ = 0.   The interface (intersection) between ∂ +,I,1 κ and ∂ +,I,2 κ is denoted by  Mou-Hsiung Chang 25 In this case, ∂ −,ℵ κ = ∅ (the empty set),

Decomposition of
On the other hand, if I = ∅ (the empty set), that is, n(−i) ≥ 0 for all i ∈ ℵ, then (4.31)

Boundary conditions for the value function.
Let us now examine the conditions of the value function V κ : κ → + on the boundary ∂ κ of the solvency region κ defined in (4.25)-(4.26). We make the following observations regarding the behavior of the value function V κ on the boundary ∂ κ .  To guarantee that (X(t),N t ,S(t),S t ) ∈ 0 , we require that Applying Theorem 3.5 to the process {e −rt G 0 (X(t),N t ,S(t),S t ), t ≥ 0} we obtain for every almost surely finite G-stopping time τ, where X(t) and N t are given in (2.32)-(2.36) with κ = 0. Taking into the account of (2.5) and (2.32)-(2.36) and substituting them into the function G 0 , we have G 0 X(t),N t ,S(t),S t = G 0 X(t−),N t− ,S(t),S t . (4.37) Intuitively, this is also because of the invariance of liquidated value of the assets without increase of stock value. Hence (4.36) becomes the following by grouping the terms n(Q(t) − i) according to i ∈ I and i / ∈ I: (4.38) Mou-Hsiung Chang 27 Now fix the first exit time τ ( τ is a G-stopping time) defined by (4.39) We can integrate (4.38) from 0 to τ, keeping in mind that (x,ξ,ψ(0),ψ) ∈ ∂ I,1 0 (or equivalently, G 0 (x,ξ,ψ(0),ψ) = 0), to obtain (4.40) Now use the facts that 0 < μ + β < 1, C(t) ≥ 0, α ≥ f (S t ) > r > 0, n(−i) < 0 for i ∈ I and n(−i) ≥ 0 for i / ∈ I and Rule 6 to obtain the following inequality: Now define the following process: Then by the Girsanov transformation (see [23,24]), { W(t), t ≥ 0} is a Brownian motion defined on a new probability space (Ω,Ᏺ, P;F), where P and P are equivalent probability measures, and hence Since G 0 (X(t),N t ,S(t),S t ) ≥ 0 for all t ≥ 0, this implies that τ = 0 a.s., that is, We need to determine the conditions under which the exit time occurred. Let k be the index of the shares of the stock where the state process violated the condition for the stopping time τ. In other words, if k ∈ I, then We will examine both cases separately.  Mou-Hsiung Chang 29 We have established that X( τ),N τ ,S( τ),S τ ∈ ∂ I,1 0 , (4.50) and this is inconsistent with (4.51) Therefore, we know n(Q( τ − k)) = 0. we see that n(Q( τ) − k) = 0. We conclude from both cases that (X( τ),N τ ) = (0,{0}). This means that the only admissible strategy is to bring the portfolio from (x,ξ,ψ(0),ψ) to (0,0,ψ(0),ψ) by an appropriate amount of the transaction specified in the lemma. This proves the lemma.
We conclude from some simple observations and Theorem 4.5 the following.
Boundary condition (ii). On ∂ I,1 κ for I ⊂ ℵ, then the investor should not consume but buy back n(−i) shares for i ∈ I and sell n(−i) shares for i ∈ I c of the stock in order to bring his portfolio to {0} × {0} × + × L 2 ρ,+ after paying transaction costs and capital gains taxes, and so forth. In other words, bring his portfolio from the position (x,ξ,ψ(0),ψ) ∈ ∂ I,1 κ to (0, 0,ψ(0),ψ) by the quantity that satisfies (4.57)-(4.58). In this case, the value function V κ : ∂ I,1 κ → + satisfies the following equation: (4.61) Note that this is a restatement of Theorem 4.5.

Mou-Hsiung Chang 31
Boundary condition (iii). On ∂ +,I,2 κ for I ⊂ ℵ, the only optimal strategy is to make no transaction but to consume optimally according to the optimal consumption rate function c * (x,ξ,ψ(0),ψ,) = (∂V κ /∂x) 1/(γ−1) (x,ξ,ψ(0),ψ) which is obtained via c * x,ξ,ψ(0),ψ = argmax c≥0 ᏸ c V κ x,ξ,ψ(0),ψ + c γ γ , (4.62) where ᏸ c is the differential operator defined by This is because the cash in his savings account is not sufficient to buy back any shares of the stock but to consume optimally. In this case, the value function V κ : ∂ +,I,2 κ → + satisfies the following equation provided that it is smooth enough: Boundary condition (iv). On ∂ −,I,2 κ , the only admissible consumption-investment strategy is to do no consumption and no transaction but to let the stock price grow as in the boundary condition (i).