Individual Comparative Advantage and Human Capital Investment under Uncertainty

Transcription

1 Individual Comarative Advantage and Human Caital Investment under Uncertainty Toshihiro Ichida Waseda University July 3, 0 Abstract Secialization and the division of labor are the sources of high roductivity in modern society. When worker skills are multi-dimensional, workers may face a choice between general versus seci c human caital investment. Given that individual agents face uncertainty in the relative outut rice, what are the otimal strategies for heterogeneous individual agents in human caital investment? In the absence of insurance markets, general investment gives an otion value for changes in the environment. We analyze a model in which workers are born heterogeneous and are endowed with two-dimensional skills in di erent sectors, to determine if incentives exist for workers to invest in skills in which they had originally excelled or struggled. We nd that some workers choose to invest in their weaker skill, via seci c human caital investment, rovided that the scale of the risk is big and that the arameter of relative risk aversion is greater than one. We nd that there exist agents whose otimal human caital investment decisions reverse their ex ante comarative advantages ex ost. JEL Classi cation: D8; F6; J3; J4; J6 Keywords: a self-selection model of occuational choice with multidimensional heterogeneity, individual comarative advantage, choice under uncertainty, intersectoral labor mobility, and seci c and general human caital investments.

2 Introduction This aer analyzes the theoretical imlications of multi-dimensional heterogeneity of agent-skills and of interactions between the innate and acquired comarative advantages of individual agents. While many existing models deal with one-dimensaional human caital skills when they analyze investment decisions, the model in this aer looks at multi-dimensional skill investments. Several studies have looked at the comarative advantages of individual agents that come from multi-dimensional heterogeneity, but none examined the interaction between ex ante and ex ost di erences in agent s comarative advantages. This aer introduces the innate heterogeneity of agents with multi-dimensional skills and considers the directional incentives for individual agents to invest in their human caital skills: general skill investments versus seci c skill investments. The model looks at the distinction between general and seci c human caital investment as in Becker (993), although not in exactly the similar manner. While general skills are useful for many other emloyers, both inside and outside their current industry, seci c skills in this aer are not only useful skills for the current emloyer but also for emloyers in the same sector. Therefore, in this aer, seci c skills denote sector-seci c skills. This aer adots multi-dimensional heterogeneity and follows the regular job-assignment models based on individual comarative advantages. The model in this aer starts from the Roy (95) model from labor economics (as with Ohnsorge and Tre er 007 and Costinot and Vogel 00), but asks a di erent question: What are the directional incentives of human caital investment for individual agents given the terms of trade uncertainty? When the heterogeneity of human caital skills is multi-dimensional, the investment decision involves not only how much to invest but also which skill (direction) to enhance. For examle, if each agent is endowed with two tyes of skills such as mathematics and music, then the question is which school (say, MIT to enhance math skills or Juilliard to enhance musical skills) the agent should enter in addition to the regular human-caital-investment question of how long he or she should attend the chosen school. If an agent chooses between somewhat secialized schools, then he or she is deciding to make a secialized human caital investment. Instead, an agent might go to a school (say, Harvard to enhance both math and music skills) where he or she can do a double major in music and math. In this case, an agent conducts a generalized human caital investment. This aer asks what tye of agents will choose secialized investment or generalized investment and exlores the relationshi between the innate (ex ante) comarative advantages and the

3 acquired or ost-education (ost-investment) comarative advantages. The analysis will be conducted under uncertainty because, without it, all agents will make seci c investments based on his or her innate comarative advantage. With uncertainty, many agents will enhance their comarative advantage skills regardless of their degree of risk aversion. This occurs if the agents have very strong comarative advantage in one tye of skill. However, some risk averse agents may make general skill investments if their comarative advantage is not articularly strong in either tye. Under certain conditions, we nd that some agents make seci c skill investments in the direction of their innately weak comarative advantages. In articular, we can nd some agents whose otimal human caital investment decisions reverse their ex ante comarative advantages ex ost. The acquired comarative advantage may well be di erent from the innate comarative advantage. To introduce uncertainty in a simle manner, I adot the assumtion of small oen economy where outut rices are exogenous in the model. Several aers have looked at the roblem of human caital investment with international trade, including Findlay and Kierzkowski (983) and Falvey, Greenaway, and Silva (00). However, these aers dealt with one-dimensional skill. So there was no question about secialization versus generalization in human caital skill investments. In this aer, by working with multidimensional skill model, I will try to work on the dichotomy of secialized skill investments versus general skill investments within the set u of oen economy. In articular, this aer o ers the rst model to link innate multidimensional heterogeneity of individuals with the directional (seci c versus general) incentive to invest in their human caital skills. 3 This aer o ers an analysis of directional incentives of human caital investment by individual agents. When worker skills are multi-dimensional, the direction of investment becomes a choice roblem. Each individual may secialize in one tye of skill or may generalize in multile skills. Both the secialization and generalization of skills have costs and bene ts. Several revious studies have looked at these issues, but all of them (with which the This aer does not address the question of how long agents choose to go to school. Please refer to Becker (993) for this choice roblem. In a recent aer, Bougheas and Riezman (007) looked at the determinants of trade in relation to the distribution of human caital. There, even if the countries are the same in the aggregate endowments of human caital, the di erence in the roerties of distribution of human caital exlains the di erent atterns of trade. 3 Note that, in traditional labor literature, seci c human caital investment usually means rm-seci c skills rather than industry- or sector-seci c. In this aer, it is the latter. 3

4 author is familiar) started from identical agents who conduct either seci c or general investments. No revious study looked at the case of the innate heterogeneity of individual agents with multi-dimensional skills. When there are increasing returns on human caital investment, Sherwin Rosen (983) showed that secialized investments tend to revail in a model without uncertainty. The imortance of secialization has also long been widely recognized. (See, for examle, the very rst chater in Wealth of Nations by Adam Smith 776.) The division of labor and secialization are the key sources of higher roductivity in the modern world. The division of labor may be the result of endowed di erences in individual comarative advantages as in Rosen (978). Secialization, however, may also be a result of secialized human caital investment as in Rosen (983). It is now widely recognized that individual agents have an incentive to secialize when there is no uncertainty about which sector individuals are emloyed. [Kevin M. Murhy (986),. 37.] When the world is uncertain, however, ex-ante seci c human caital investment may be a risky strategy because secialization makes individual agents in exible as factors of roduction ex ost. To borrow the exressions from Gene M. Grossman and Carl Shairo (98), an investment in secialization is like reducing the degree of intersectoral mobility in a subsequent eriod. A general (as oosed to seci c) human caital investment is a form of self-insurance if insurance markets are absent. (See, for examle, Ehrlich and Becker 97.) A general investment brings the bene t of greater exibility in resonding to economic change. Murhy (986, section 5.) also looked at a case where there is uncertainty about which sector will become favorable. He also concluded that less secialization exoses workers to less risk. Both Grossman and Shairo (98) and Murhy (986) examined the individuals incentives to make general versus seci c human caital investment when every agent is identical before conducting an investment in human caital skill(s). So, what haens to these results if we start from a situation in which agents are heterogeneous in their innate comarative advantage before making the human caital investment? In this aer, I aim to analyze the incentives of individual agents to invest in human caital skills when the role of endowed di erences is imortant. By looking at the directional incentives to invest in human caital skills for the self-selection model of occuational choice, I am able to address the following (new) set of research questions: What is the role of endowed di erences of individuals in human caital investment? Do eole with heterogeneous skills secialize or generalize their innate skills when the world is uncertain? What kind of eole do invest in their innately strong (comara- 4

5 tive advantage) skills? Do eole secialize in what they were relatively good at when they were born? If so, under what conditions? Do agents invest their time (and money) to enhance their skills in a socially ine cient way? Do they go to schools in which they will learn something they are innately oor at? Will there be the same number of job-switching individuals if we allow the dynamic develoment of human caital? All of these questions were unanswered by revious work in this eld since they assumed identical agents before the decision of human caital investment. Although there is a large volume of literature on human caital investment, to my knowledge there is no revious work that starts from multi-dimensional heterogeneity. I introduce such a model in the next section. I nd that, in general, many workers tend to secialize in their strengths even under uncertainty. This is true esecially for the workers who have a very strong innate comarative advantage in one sector. For examle, if you are born a good singer and you did not do well in high school, then you will choose to go to a school of music to enhance your strong innate skills in terms of comarative advantage. For workers who are born almost equally good in both sectors, the roblem becomes very di cult. Some workers try to invest in general skills so that they can choose their jobs in a exible manner once the uncertainty resolves. Others may secialize in their innately strong or weak skills deending on their attitude toward risk. If workers are very risk averse, we nd that some workers decide to make secialized investments in their weak skills (in the direction of their comarative disadvantages) rather than in their strong ones (in the direction of their comarative advantages). This haens to those who have skills whose levels are similar enough so that their degree of comarative advantage is not very strong. For examle, consider a erson A who has high talent in music but also has a high IQ. This erson might choose to give u music to go to MIT even if his talent in music is actually the stronger of his skills. On the other hand, suose there is another erson, B, who has similar skills as erson A. Suose also that B s music skills are not as good as A s and that B s IQ is slightly higher than A s. Therefore, if there is no uncertainty, A chooses to be a musician and B gives u the idea of being a musician to focus on academic study. However, given the setu of the model with uncertainty, A goes to MIT by giving u becoming a musician and B goes to Juilliard to be a musician! So the reversal of training occurs for those who are in certain classes of skills in this model. In the end, we may see some cases where a erson s comarative advantage reverse from ex ante to ex ost. Person A (resectively, B) has a comarative advantage in music (resec- 5

6 tively, academic skills) ex ante, but after the training A (resectively, B) has a comarative advantage in academic skills (resectively, music). This reversal result is quite new as far as we know. The remaining art of the aer is organized as follows: the next section develos the basic model of human caital investment with two dimensional heterogeneity of innate skills. The nal section summarizes the results and suggests some ossible extensions. The Model We consider a simle two-eriod and two-sector model of a small oen economy that faces exogenously given international outut rices. In eriod 0, no roduction takes lace, agents are endowed with multi-dimensional heterogeneous skills, and they invest in their human caital to enhance their innate skills. Uncertainty about the terms of trade will be realized in eriod and agents choose their occuations and engage in roduction. Multidimensional skills are embodied in an individual agent and can only be sold as a ackage. Therefore, we assume that an individual agent can take only one job at a time. Outut markets for sectors X and Y are assumed to be cometitive, both internationally and domestically. In making the investment decision in eriod 0, each agent is assumed to have rational exectations concerning the rices that will revail in eriod. The economy consists of a continuum of self-emloyed agents j J, each of whom is endowed with an individual-seci c occuational skill vector ( j ; j ) jointly distributed over a unit square [0; ] [0; ] R in eriod 0. 4 Let t and F t (; ) denote the sace and the joint distribution function of human caital skills for each eriod t = 0;. Thus, we know that 0 = [0; ] [0; ]. Let f t (; ) > 0 denote the joint density function for F t (; ), and assume that f t is integrable over any artition of the human caital skill sace t for t = 0;. Agents are rice takers in the outut markets. Each comonent of the skill vector ( j ; j ) reresents an sector-seci c human caital skill; their magnitudes measure the caabilities of the agent j in the roduction of X and Y in e ciency units. Once the terms of trade become known in eriod, each agent decides either to roduce X using, or Y using. Each individual undertakes only one occuation at a time because human caital skills are assumed to 4 Tye sace does not have to be a unit square. Here for simlicity, a unit square tye sace is assumed. 6

7 be embodied in human beings. [Murhy (986, Sec.II)] Each comonent of the skill vector ( j ; j ) is indivisible and non-transferable. The size of the comonents of individual skill vectors in each eriod ( j ; j ) t is rivate information for j, but its aggregate distribution F t is ublicly known. To summarize, the timing of the model is as follows:. In eriod 0, the tye of each individual ( j ; j ) [0; ] [0; ] = 0 is given.. An individual choose to invest in human caital skills for either sector X or Y, or both. This individual decision will create the new skill sace and the new distribution of skills F (; ) for eriod. 3. In eriod, a new relative rice P = (P X ; P Y ) is realized. 4. An individual chooses occuation and roduce. Let us now state the assumtions about human caital investment and uncertainty.. Key Assumtions of the Model Let us now state the assumtions about how individual agents can invest in their skills in eriod 0. There are two kinds of human caital investment: () General Human Caital Investment and () Sector Seci c Human Caital Investment. Because there are two directions in sector seci c investments, we can categorize ossible investments into three investment tyes. A: [Human Caital Investment] An individual with the skill vector (; ) can choose among the three tyes of human caital investment: I HC fs X ; G; S Y g which are listed as:. S X = Seci c Investment in Sector X: (; ) =) (; ). G = General Investment in both sectors: (; ) =) (; ) 3. S Y = Seci c Investment in Sector Y: (; ) =) (; ) where and reresent arameters such that > > : () The fact that both arameters are larger than imlies that human caital investment, regardless of seci city or generality, is e ective. The size of 7

8 investment e ciency for seci c investment is assumed to be larger than that for general investment. Otherwise, everyone will invest in general skills alone because seci c skill investment will be dominated by general investment in either direction. It is assumed that the seci c investments in two sectors share the same arameter. This means that the e ect of the seci c investment is symmetric between sector X and Y. It is also assumed that only one tye of general investment can occur. General investment enhances skills in both sectors in a symmetric manner. Both comonents will be multilied by the same. These symmetry assumtions are made to simlify the analysis. Another eculiarity about investment technology is that the e ect of investments is roortional to the individual s original strength (innate caabilities) in each sector. If your innate skill in sector X roduction is very large, then your ost investment skill in the same sector will be roortionally large. This assumtion is aroriate if all agents in the economy are considered to be young. For examle, young Michael Jordan can be trained to be a suerstar because of his excetional innate talent as a basketball layer. The e ect of training Jordan is much larger than the e ect of training a mediocre layer. The assumtion of roortional e ectiveness of human caital skill investment would create a theoretical roblem if we were to start thinking about an economy with matured (and old) agents who already have invested in their skills and have little margin for additional skill develoment. However, we focus on this articular case of roortionality in this aer. Comared to more general investment frontier deicted in Murhy (986) and Grossman and Shairo (98), this aer s seci cation of human caital investment given by the above assumtion is somewhat restrictive by not allowing a range of intermediate cases (such as 40% on X and 60% on Y, etc.) in general investment. This restrictive assumtion is made for simlicity. The fact that is larger than means an imlicit assumtion about some form of increasing returns from secialization in human caital investment. Because is larger than, we can conclude that the tye sace in eriod should be [0; ] [0; ]. Now let us assume that all consumers have identical and homothetic references that can be reresented by a utility function u(x; y) for ositive consumtion amounts for each roduct: x > 0 and y > 0. When we denote the income of the individuals as M > 0 and the vector of outut rices as P (P X ; P Y ), then the indirect utility function for homothetic references 8

9 can be written as the following searable function: V (P; M) = C(P ) v(m) () We know, by Deaton and Muellbauer (980), that the rice-index function C(P ) is linear homogeneous and concave in P. The art v(m) is an increasing function of income M of an individual. If the original utility function u(x; y) is homogeneous of degree one, then v(m) = M should hold. Therefore, if the agents are risk neutral, the indirect utility can be written as V (P; M) = C(P ) M (3) and if the agents are risk averse, then v(m) in the equation () is strictly concave in M. [Jack Hirshleifer and John G. Riley, 99,. 3.] Thus, we could write v 0 > 0 and v 00 < 0. For now, let us leave the second one as v 00 0, in order to include both cases of risk neutrality and risk aversion. Now let us introduce uncertainty in eriod. Since we assume that we live in a small oen economy, every individual agent takes terms of trade (outut rices) as given. Suose there are two states of nature, as follows: A. [Uncertainty] Uncertainty about the terms of trade in eriod takes the following symmetric form: P = (; ) with robability (; ) with robability (4) where is a ositive arameter larger than. We concentrate on a case with the same robability for the two states of nature. When the state of nature is such that P = (; ), roducers in sector X will bene t in eriod because P X will be more exensive relative to P Y. When P = (; ) occurs, roducers in sector Y will bene t because P Y will be more exensive relative to P X. As we have seen in the regular selfselection models based on individual comarative advantage, the realization of articular terms of trade may induce some workers to take di erent jobs once the uncertainty is resolved. (See, for examle, Sattinger 975.) Thus, some agents will choose to work in a favorable sector while others may stay in the sector at which they were innately good. Let us further assume that the demand condition is symmetric in sectors X and Y. In the equation of indirect utility (), C(P ) reresents a consumer rice index (CPI). 9

10 A3. [Equal CPI] The consumer rice index (CPI) in eriod is the same for the di erent states of the world, namely holds in the neighborhood. C(; ) = C(; ) (5) This assumtion will enable us to comare welfare by directly looking at v(m) in (), the utility art only from income in eriod without worrying about substitution e ects in consumtion. (See Aendix for an exlanation.) This way, irresective of the outcome of uncertain terms of trade, we can comare the economic welfare of individual agents only by looking at v(m). Assume also that no income-insurance market exists. (See Grossman and Shairo [98] for a comarison between self-insurance versus insurance markets.) A4. [No Insurance] The market for income insurance is absent. This may be because there is no market for skills er se. Or it may be because information asymmetry 5 revents insurance rms from oerating ro tably. According to an article in The Economist (March 6, 994), the caital markets for human caital investment may be imerfect. The article reads as follows: For instance, borrowing to nance an investment in human caital may be di cult because would-be trainees lack collateral, or because the costs of administration and collection make such loans unattractive to rivate lenders. In any case, we focus on the case without insurance. Given the investment made in eriod 0 and given the resolved terms of trade, individual agents choose their occuations and start roducing either X or Y in eriod. How do individual agents decide in which sector to work? To determine this, let s rst introduce a constant-returns-to-scale roduction function for each sector: x = NX (6) y = N Y where N X and N Y are the total sum of (e ective) skill levels emloyed in each sector. 6 Given the realization of terms of trade, P, the skill sace in 5 Skill levels are rivate information about the individual agents and the insurance comany cannot know them. 6 We could add coe cients in the roduction functions such as x = N X, but we omit them for the sake of simlicity. 0

11 eriod,, can be artitioned into two: () X (P ) grou of roducers of X and () Y (P ) grou of roducers of Y. Then, N X and N Y can be written as follows: ( NX = R X (P ) df (; ) N Y = R Y (P ) df (7) (; ) All individual agents earn wages that are equal to the value of their marginal roduct. Therefore, when the terms of trade is given as P = (P X ; P Y ), if an individual with a vector of skills (; ) who engages in roduction in sector X, then that individual earns a wage of P X, while the same individual earns P Y if he or she engages in roduction in sector Y. So whether an individual (; ) works in sector X or Y deends on the direction of inequality in P X? P Y : (8) Thus, an individual who had made human caital investment I HC fs X ; G; S Y g in eriod 0 and faces terms of trade P (P X ; P Y ) f(; ); (; )g in eriod will earn the income M(I HC ; P ). The ossible combinations can be written as follows: 8 M(S X ; (; )) = max [ ; ] M(G; (; )) = max [ ; ] >< M(S Y ; (; )) = max [ ; ] (9) M(S X ; (; )) = max [ ; ] M(G; (; )) = max [ ; ] >: M(S Y ; (; )) = max [ ; ] Given this information about ossible income in eriod, we can now consider the exected utility of an individual (; ) who conducted human caital investment I HC fs X ; G; S Y g in eriod 0. Since the lottery about terms of trade in eriod is given by (4), exected utility U(I HC ) for each investment attern is written as 8 < : U(S X ) = f v[m(s X; (; ))] + v[m(s X; (; ))]g C(P ) U(G) = f v[m(g; (; ))] + v[m(g; (; ))]g C(P ) U(S Y ) = f v[m(s Y ; (; ))] + v[m(s Y ; (; ))]g C(P ) (0) where C(P ) C(; ) = C(; ) is de ned as such from (5). Thus, to comare the sizes among (0), we only need to comare the inside of the curly brackets {} because we can think of C(P ) as an exogenous arameter. We will come back to this oint later.

12 Now, to leave sace for analysis, I would like to make additional innocuous assumtion about the size of risk. Let us consider the case with large risk in this aer. A5. [Risk Size] We focus on our analysis for the case of >. The size of the rice arameter is larger than the size of a arameter for seci c human caital investment. The reason for this assumtion is a matter of simlicity. By assuming otherwise, the analysis is very similar excet for the combination of arameters and cases. For those who are interested in the other case of <, lease consult my Ph.D. Dissertation, Ichida (004, Ch. 4), for the case of riskneutral agents.. Incentives for Each Individual Agent With the assumtions given by A.-A5. above, let us now consider the incentives for each individual agent. What tye of human caital investment do agents choose? What jobs do agents take once the uncertainty is resolved? If all agents were identical before the investment decision as in Grossman and Shairo (98) and in Murhy (986), then in equilibrium all investment choices must give the same exected utility. Otherwise, every agent will ick the best alternative that gives the highest exected utility level. The income of all agents in eriod must also be equalized across sectors for identicalagents-case. That is how the allocation of jobs would be done. However, this shall not be the case for the model in this aer because agents are born to be di erent. In fact, agents are heterogeneous in both absolute and relative skill levels in both eriods. Deending on the relative sizes of skills in the vector (; ), the largest income that is chosen among those in (9) may di er. Therefore, let us break down the initial tye sace 0 = [0; ] (in eriod 0) into the following eight artitions by ossible occuational choice with

13 di erent investment choices and with di erent risk outcomes. 8 K : (; ) [0; ] \ f 0 < < g K : (; ) [0; ] \ f < < g K 3 : (; ) [0; ] >< \ f < < g K 4 : (; ) [0; ] \ f < < g K 5 : (; ) [0; ] \ f < < g K 6 : (; ) [0; ] \ f < < g K 7 : (; ) [0; ] >: \ f < < g K 8 : (; ) [0; ] \ f < < g () I will exlain the meaning of these artitions in words.. Think about the case where an individual conducted S X (seci c investment in sector X skill) in eriod 0. We think of the case of M(S X ; (; )) or M(S X ; (; )). If the terms of trade turn out to be favorable to sector X, i.e. (; ) is realized, 7 then agents in S 7 i= K i will roduce in sector X while those in K 8 will roduce in sector Y. If the terms of trade turn out to be favorable to sector Y, i.e. (; ) is realized, then agents in S 3 i= K i will roduce in sector X while those in S 8 i=4 K i will roduce in sector Y.. Think about the case where an individual conducted G (general investment) in eriod 0. We think of the case of M(G; (; )) or M(G; (; )). If the terms of trade turn out to be favorable to sector X, i.e. (; ) is realized, then agents in S 6 i= K i will roduce in sector X while those in S 8 i=7 K i will roduce in sector Y. If the terms of trade turn out to be favorable to sector Y, i.e. (; ) is realized, then agents in S i= K i will roduce in sector X while those in S 8 i=3 K i will roduce in sector Y. 3. Think about a case in which an individual conducted S Y (a seci c investment in sector Y skill) in eriod 0. We think of the case of M(S Y ; (; )) or M(S Y ; (; )). If the terms of trade turn out to be favorable to sector X, i.e. (; ) is realized, 8 then agents in S 5 i= K i will roduce in sector X while those in S 8 i=6 K i will roduce in sector Y. If the terms of trade turn out to be favorable to sector Y, i.e. (; ) is S realized, then agents in K will roduce in sector X while those in 8 i= K i will roduce in sector Y. 7 This also means that the individual guessed correctly. 8 This also means that the individual guessed wrong. 3

14 Given the above examination of individual decisions about which sector agents work in eriod, we want to think about the investment incentives of individuals in each artition by using backward induction. However, the kind of investment each individual chooses deends on the agent s attitude toward risk..3 Exected Utility and Attitude toward Risk Now, because the exected utility function can reresent the same reference u to the monotonic linear transformation, 9 we can de ne the exected utility for the agents in artition K i who conducted investment I HC as follows: for all i f; ; ::; 8g, a new von-neumann Morgenstern exected utility function V (K i ; I HC ) is de ned by V (K i ; I HC ) U(IHC ) C(P ) : () (;)Ki This can be done because C(P ) C(; ) = C(; ) is an exogenous arameter, as we noted before. For the agents in di erent artitions in (), we are able to calculate the von-neumann Morgenstern exected utility function V (K i ; I HC ) by combining the equations of income (9) and the equations of the exected utility (0). The v(m) art of the equations (0) can be thought of as the Bernoulli utility function. When v 00 = 0, we can say that agents are risk neutral and when v 00 < 0, we can say that agents are strictly risk averse. Here I would like to introduce seci c functional forms for the Bernoulli utility function. Let it be the Constant Relative Risk Aversion (CRRA), written as ( M v(m) = for 6= ln M for = where the size of reresents the coe cient of relative risk aversion. To simlify the analysis, I focus on some seci c cases in this aer. Three cases of di erent attitude toward risk will be analyzed: () RN = risk neutral case, () RA = risk aversion with a coe cient of relative risk aversion is, and (3) RA = risk aversion with a coe cient of relative risk aversion is. Let us start with the risk neutral case. RN When the agents are risk neutral, the Bernoulli utility function takes the linear form: v(m) = M. 9 See Pro.6B in age 73 of Mas-Colell et al. (995). 4

15 The case for risk neutrality (RN) is straightforward. We adot simly the linear function without coe cients for the Bernoulli utility. The case for risk averse is more comlicated because there are many ways to be risk averse. To make it easier to obtain the analytical results, let us focus on the following two cases of risk aversion. RA One tye of risk averse agents has a Bernoulli utility function in the form of v(m) = ln M. The arameter for relative risk aversion for this log form is constant at. The rst tye of risk aversion RA utilizes log utility form. This utility has a constant elasticity for any income level. RA Another tye of risk averse agent has a Bernoulli utility function in the form of v(m) = M. The arameter for relative risk aversion for this form is. The second tye of risk aversion RA utilizes the negative inverse form of utility. The arameter for relative risk aversion is greater than. Therefore, the degree of risk aversion is the strongest among these three cases..4 The results for individual agents Let us state the main result for the RN case as Theorem. Theorem When the assumtions given by A.-A5. hold in the model with the RN Bernoulli utility function v(m) = M, the following three situations will occur deending on the size of the arameter value of. (i) All agents in the economy will make seci c investments in the direction of their innate comarative advantage; agents (; ) 0 with < will make S X (Seci c investment in sector X) and agents (; ) 0 with > will invest S Y (Seci c investment in sector Y ), if the arameter of general investment is smaller than a articular threshold value, or < + : (ii) There will be some agents who will make general investments G. In articular, agents (; ) 0 with < < 5

16 will invest G (General investment in both sectors), agents (; ) 0 with 0 < < will invest S X, and agents (; ) 0 with < < will invest S Y, if the arameter of general investment is within a articular threshold values, or + < < ( + ) + : (iii) There will be some agents who will make general investments G. In articular, agents (; ) 0 with + < < + will invest G (General investment in both sectors), agents (; ) 0 with + 0 < < will invest S X, and agents (; ) 0 with + < < will invest S Y, if the arameter of general investment is larger than a articular threshold value, or ( + ) + < : The roof is in the later section. In the case of risk neutrality, seci c investments are made only to strengthen the agent s innate comarative advantage. When the arameter of general investment is smaller than a articular threshold value, no general investment occurs. If the arameter is larger, then the general investment occurs for agents with an intermediate level of comarative advantage. Risk neutral agents will never make seci c investments in the oosite direction, namely, the direction of the agent s innate comarative disadvantage. Let us state the main result for the RA case with log utility. 6

17 Theorem When the assumtions given by A.-A5. hold in the model with the RA Bernoulli utility function v(m) = ln M, the following two situations will occur deending on the size of the arameter value of. (i) All agents in the economy will make seci c investments, but the direction of investments vary deending on the strength of their innate comarative advantage; agents (; ) 0 with 8 >< >: 0 < < () invest S X < < () indi erent between S X and S Y < < () invest S Y if the arameter of general investment is smaller than a articular threshold value, or < : (ii) There will be some agents who will make general investments G. In articular, agents (; ) 0 with < < will invest G (General investment in both sectors), agents (; ) 0 with 0 < < will invest S X, and agents (; ) 0 with < < will invest S Y, if the arameter of general investment is larger than a articular threshold value, or < : As with the risk neutral case, when the arameter of general investment is larger than a articular threshold value, then the general investment occurs for agents with an intermediate level of comarative advantage. When the arameter of general investment is smaller, however, the direction of seci c investments is quite di erent. Those with a very strong innate comarative advantage make seci c investments in their innately strong skills. 7

18 Those with an intermediate comarative advantage are indi erent when it comes to conducting seci c investments in sector X and sector Y. This is a striking result. For those who are closer to the 45-degree line, the exected utility values from S X and S Y are the same. This is because of the exibility of agents who are in the range of < < (3) about which sector they work in after the realization of the terms of trade. For those in artition (3), they work in sector X (resectively, Y) if the terms of trade are favorable for workers in X (resectively, Y). Let us state the main result for the RA case as Thorem 3. Theorem 3 When the assumtions given by A.-A5. hold in the model with the RA Bernoulli utility function v(m) = M, the following two situations will occur deending on the size of the arameter value of. (i) All agents in the economy will make seci c investments, but the direction of investments vary deending on the strength of their innate comarative advantage; agents (; ) 0 with 8 >< >: 0 < < + () invest S X + < < () invest S Y < < ( + ) () invest S X ( + ) < < () invest S Y if the arameter of general investment is smaller than a articular threshold value, or < + : (ii) There will be some agents who will make general investments G. In articular, agents (; ) 0 with 8 0 < < + () invest S X >< + < < ( ) () invest S Y ( ) ( ) < < () invest G ( ) >: < < ( + ) () invest S X ( + ) < < () invest S Y 8

19 if the arameter of general investment is larger than a articular threshold value, or + < : When the degree of risk aversion is very strong for the agents, we observe a very interesting henomenon. Some agents secialize in their innately weak skills. When agents are born with very strong comarative advantages, then they try to develo their strong skills regardless of the uncertainty about future terms of trade. As the relative ability of the agents in two sectors get closer, their individual incentive to invest in their human caital skills becomes something like ersonal insurance against a ossible misfortune a ecting their innately strong skills. The roofs to the Theorems -3 are given below..5 Analysis for Each Partition Let us start analyzing the incentives to invest in human caital skills for agents in each artition. Deending on the realized outcome of uncertain terms of trade, the choice of occuation di ers among di erent artitions. Therefore, the realized income may di er. Based on the exected choice of occuation and exected income level, the individuals choose to decide to invest in seci c or general human caital investment..5. Partition K and K 8 : Consider agents in artition K. All agents in this artition will work in sector X in eriod regardless of the realized terms of trade. Their exected utility can be written as follows: 8 < : V (K ; S X ) = v() + v() V (K ; G) = v() + v() : (4) V (K ; S Y ) = v() + v() In this case, it is straight forward to show V (K ; S X ) > V (K ; G) > V (K ; S Y ) (5) regardless of the shaes of Bernoulli utility function. Irresective of the attitude toward risks, i.e. RN, RA, or RA, the Bernoulli utility functions are increasing monotonic functions, i.e. v 0 > 0. We know that > > and > > for > > > because of the assumtions of 9

20 the model. (5) shows that all agents in artition K will make a seci c investment in sector X. The same logic hold true for the agents in artition K 8 and they invest S Y..5. Partition K and K 7 : Next, consider agents in artition K. They work in sector X unless they have invested in S Y and the terms of trade turn out to be unfavorable to sector X, i.e. P = (; ). Thus, their exected utility can be written as 8 < V (K ; S X ) = v() + v() V (K ; G) = : v() + v() : (6) V (K ; S Y ) = v() + v() From the analysis for K, we know that V (K ; S X ) > V (K ; G) (7) holds always true since V (K ; S X ) = V (K ; S X ) and V (K ; G) = V (K ; G) can be seen in (6). So we are left to comare the size between V (K ; S X ) and V (K ; S Y ). We should note that artition K belongs to the region < < : (8) We rst look at the RN case: v(m) = M. By looking at fv (K ; S X ) V (K ; S Y )g = ( + ) (9) we can conclude that the exression (9) is ositive because + = + > together with (8). Second, look at the case for RA: v(m) = ln M. The exression fv (K ; S X ) V (K ; S Y )g = ln ln + ln ln (0) is ositive because < always hold for agents in (8). Third, look at the RA case: v(m) = M. Let s look at the following exression: fv (K ; S X ) V (K ; S Y )g = 0 ( + ) ()

21 Because it is easy to verify that < + () and, by noting the inequality (8), we can say that exression () is always ositive for agents in K. So we can conclude that V (K ; S X ) > V (K ; S Y ) (3) for all three cases. Therefore, we can conclude that in artition K, the seci c investment for sector X, S X, is always chosen for all RN, RA and RA cases. The same logic hold true for agents in artition K 7 and they invest S Y..5.3 Partition K 3 and K 6 : Now consider agents in artition K 3 who are in the region: < < : (4) They work in sector Y when the terms of trade are (; ) only if they had invested either G or S Y. Otherwise, they work in sector X. Therefore, their exected utility can be written as 8 < : V (K 3 ; S X ) = v() + v() V (K 3 ; G) = v() + v() V (K 3 ; S Y ) = v() + v() : (5) Note that V (K 3 ; S X ) = V (K ; S X ) and V (K 3 ; S Y ) = V (K ; S Y ), but V (K 3 ; G) 6= V (K ; G). To conserve sace, let denote reference over the human caital investment I HC fs X ; G; S Y g for a articular artition. If it is clear from the context, assume that the following notation can be used for any i f; :::; 8g: I HC I HC0 for agents in K i () V (K i ; I HC ) > V (K i ; I HC0 ): By using the notation, we look at the three cases for agents in K 3 and K 6.

22 RN case for K 3 and K 6 : Let us rst look at the RN case: v(m) = M. By looking at fv (K 3 ; S X ) V (K 3 ; S Y )g = ( + ) (6) we know this is the same as (9). By looking at the following exression: + ( )( ) = ; we can conclude that < + holds true since > and >. Then we can say that S X S Y agents in K 3. Now we need to comare S X with G. Look at (7) for RN fv (K 3 ; S X ) V (K 3 ; G)g = ( + ) : (8) The sign of (8) deends on the direction of the following inequality:? + Considering the condition for artition (4), < + always hold if holds or < + <, < ( + ) + ( + ) +, S X G: (9) When the arameter is larger, then there exist regions in which the reverse may occur. > ( ( + ) +, SX G for < < + + G S X for < < (30) The same logic can exlain symmetrically the agents in K 6.

23 RA case for K 3 and K 6 : Look at the case for RA: v(m) = ln M. We can conclude that S X S Y for RA agents in K 3 because fv (K 3 ; S X ) V (K 3 ; S Y )g = ln + ln ln ln (3) is ositive for <. If we now comare S X and G, there are two cases deending on the arameter values for and. Look at fv (K 3 ; S X ) V (K 3 ; G)g = ln + ln ln ln ln (3) and we nd that the sign of this deends on 7 hence, the direction of inequality in 7. We can easily see that for the left side of the arrow in (33) imlies > =) S X G (33) < : Otherwise, the region in K 3 is divided into two in the following manner: ( < SX G for, < < G S X for < < (34) Some workers in K 3 choose to make a seci c investment in X while others make general investments. The same logic can exlain symmetrically the agents in K 6. RA case for K 3 and K 6 : Now let us look at the case for RA: v(m) = M. To rovide the integrated analysis, we ostone the analysis for the RA case for agents in K 3 and K 6 to the section under the title: RA case: K 4 and K Partition K 4 and K 5 : Now consider agents in artition K 4 who are in the region: < < : (35) 3

24 These are the most exible workers. So all of them work in sector X (resectively, Y) when the terms of trade is (; ) (resectively, (; )), irresective of their investment decisions. Therefore, their exected utility can be written as 8 < V (K 4 ; S X ) = v() + v() V (K 4 ; G) = : v() + v() : (36) V (K 4 ; S Y ) = v() + v() Note that V (K 4 ; G) = V (K 3 ; G) and V (K 4 ; S Y ) = V (K 3 ; S Y ), but V (K 4 ; S X ) 6= V (K 3 ; S X ). RN case for K 4 and K 5 : Let us rst look at the RN case: v(m) = M. By looking at fv (K 4 ; S X ) V (K 4 ; S Y )g = ( )( ) which is ositive because of (35) and > 0 and >, we can conclude that S X S Y for agents in K 4. Now we have to comare S X with G. Look at fv (K 4 ; S X ) V (K 4 ; G)g = [( ) ( )] (37) The sign of (37) deends on the direction of the following inequality:? : There are three ossible arameter saces:. All agents in K 4 choose S X if <, < + (38) because (35) and (38) imly > :. Some agents choose S X and others choose G deending on the location if < <, + ( + ) < < + 4

25 holds. Therefore, ( 0 < < () invest S X < < () invest G should hold. 3. All agents in K 4 choose S X if < because (35) and (39) imly for all agents in K 4. >, ( + ) + < (39) The last case must combine with the analysis on K 3. The same logic can exlain symmetrically the agents in K 5. RA case for K 4 and K 5 : Let us rst look at the RA case: v(m) = ln M. By looking at fv (K 4 ; S X ) V (K 4 ; S Y )g = 0; we must conclude that S X S Y for all agents in K 4. To comare with general investments, look at fv (K 4 ; S X ) V (K 4 ; G)g = ln ln whose sign deends on the direction of inequality in? : If > holds, then we should know that S X S Y G for all agents in K 4. Therefore, we can say that S X S Y, < < : But if < holds, then we should know that < holds. Therefore, we know that G S X S Y for all agents in K 4. This last case must combine with the analysis of K 3. The same logic can exlain symmetrically about the agents in K 5. 5

26 RA case for K 4 and K 5 : Let us look at the RA case: v(m) = M. Here we want to analyze artitions K 3 and K 4 together. The case for artitions K 5 and K 6 is similar, so we omit the analysis. We rst make it clear that agents belong to the regions: ( K3 : < < and K 4 : < <. We also list here the conditions for the reference of agents for investment tyes. Because V (K 4 ; G) = V (K 3 ; G) and V (K 4 ; S Y ) = V (K 3 ; S Y ) holds, the relationshi between G and S Y is the same between K 3 and K 4. By checking the sign of the exression fv (K i ; S Y ) V (K i ; G)g = ( ) ( ) for i = 3; 4; we can conclude that ( > ( ), G S Y for K 3 and K 4 < ( ), S Y G for K 3 and K 4 (40) Now start looking at artition K 3, to comare the two seci c investments, we must look at the sign of the following exression: fv (K 3 ; S X ) V (K 3 ; S Y )g = ( + ) (4) and we can conclude that ( > +, S Y S X for K 3 < +, S X S Y for K 3 : (4) If we comare S X and G, we must check the sign of the following: fv (K 3 ; S X ) V (K 3 ; G)g = ( + ) (43) and we can conclude that ( > +, G S Y for K 3 < +, S : (44) Y G for K 3 Look at artition K 4. Let us rst comare the two seci c investments. Look at ( )( ) fv (K 4 ; S X ) V (K 4 ; S Y )g = 6

27 which is always ositive because > 0, > and > for all agents in K 4. Therefore, we can conclude that S Y S X for all agents in K 4 : (45) If we comare S X and G for artition K 4, we must check the sign of the following: fv (K 4 ; S X ) V (K 4 ; G)g = ( ) ( ) and we can conclude that ( > ( ), S X G for K 4 ( ) <, G S X for K 4 : (46) To reare the further analysis, we now claim the following two results: Lemma The following relationshi holds true. < ( ) < (47) Proof. The left side of (47) is obvious because >. To rove the right side, ( ) <, ( ) < ( ) which is true for < by assumtion. The second result is given here. Lemma The following relationshi holds true. + < + + < ( + ) < + + (48) Proof. From the left side, because >, it is obvious that Next, look at the mid inequality, + + < + + < + + :, ( )( ) < 0 + 7

28 which is true because > and >. From the right side, ( + ) < +, ( )( ) < 0 + which is also true. This concludes the roof. Now, given the relative size in (48), we can analyze ve cases deending on the size of the arameter : Case : < + Case : + < < + + Case 3: + + < < + Case 4: Case 5: (+) + + < < (+) + <.5.5 Case : < +. When < + and 8 >< holds, the following relationshi holds true: < + < < (49) >: ( ) < < < ( ) + : (50) This is because < + + automatically imly < + and we can derive a few results: 8 >< < +, < + < + ( ) +, < < + >: < +, ( ) < < ( ) and < +, Together with (47), (49) and (50) can be shown. If we summarize all conditions (40)-(46), we can state the following results for the case < +. 8 >< >: < < +, S X S Y G + < <, S Y S X G < <, S Y S X G 8

29 .5.6 Case : When < < < + +. holds, then the following relationshi holds true: < + < < + < (5) and ( ( ) < < ( ) Because < + results: 8 >< + >: : automatically imly < +, we can derive a few + <, + < < + ( ) +, < +, < < + ( ) < < ( ) Together with (47), (5) can be shown. If we summarize all conditions (40)-(46), we can state the following results for the case + < < >< < < +, S X S Y G + >: < <, S Y S X G < <, S Y S X G.5.7 Case 3: + + < < +. When + + < < + holds, then the following relationshi holds true: < + < + < < ( ) < < ( ) (5) We can derive a few results: ( + + <, < +, + < ( ) < ( ) < < ( ) We also have to check if + + (53) 9

30 cannot occur. Suose it does. Then we will encounter a contradiction because there must exist agents within + ( ) (54) whose reference can be reresented by the following three: S Y G S X S Y S X G (55) which violates the transitivity of the reference. First, S Y G because all agents in (54) must satisfy < ( ) since (5) and (53). S X S Y because of the right side in (54) and whose reference can also be reresented by S X G because of the left side of (54). This leads to (55). If we summarize all conditions (40)-(46), we can state the following results for the case 3: 8 >< < < +, S X S Y G + < < +, S Y S X G ( ) + < < >:, S Y G S X ( ) < <, S Y S X G.5.8 Case 4: When + < < (+) + + < < (+) +. holds, then the following relationshi holds true: < + < + < < ( ) < < ( ) (56) Because + + < automatically imlies + 8 >< >: + + <, + <, ( ) < (+), + + < 30 <, we can derive a few results: ( ) < ( ) < < < ( ) (57)

31 In addition to the above, we must check if + + (58) cannot occur. Suose it does. Then we will encounter a contradiction because there must exist agents within + ( ) (59) whose references can be reresented as follows: S Y G S X S Y S X G (60) which violates the transitivity of the reference. First, S Y G because all agents in (59) must satisfy < ( ) since (56). S X S Y because of the right side in (59) and whose reference can also be reresented by S X G because of the left side of (59). This leads to (60). If we summarize all conditions (40)-(46), we can state the following results for case 4. 8 >< < < +, S X S Y G + < < +, S Y S X G + >: < < ( ), S Y G S X ( ) < <, G S Y S X.5.9 Case 5: (+) + <. When (+) + < holds, then the following relationshi holds true: < + < + < ( ) < ( ) < < (6) 3