Interaction of Human and Physical Capital in a Model of Endogenous Growth

Transcription

1 Economics of Planning 28: , Kluwer Academic Publishers. Printed in the Netherlands. Interaction of Human and Physical Capital in a Model of Endogenous Growth JOB GRACA, SAQIB JAFAREY "~ and APOSTOLIS PHIL1PPOPOULOS Department of Economics, University of Essex, Wivenhoe Park, Colchester, Essex, C04 3SQ, U.K. Abstract. A model of development is presented where growth is initially driven by physical capital accumulation, as in the neoclassical model. After a critical level of physical capital is reached, the economy 'takes off' and enters a stage of sustained growth driven by human capital accumulation. The link between these two stages is provided by the assumption that private incentives for human capital accumulation increase with the average levels of human and physical capital. At the early stages of development, these incentives are low so the level of human capital stays stagnant until sufficient physical capital is accumulated. Other results are that some economies may reach a steady state of physical capital before a 'take-off' is possible. This is especially likely for economies in which agents have low savings propensities. Such economies remain stuck in a no-growth equilibrium forever. Economies that do grow may experience endogenous cycles if the return to investment in human capital is sufficiently increasing in the level of physical capital. 1. Introduction This paper presents a model of endogenous growth where human and physical capital interact, not just in the production technology, but in the technology for augmenting human capital as well. The return on time spent in augmenting human capital is assumed to be an increasing function of the economy-wide levels of both physical and human capital. This formulation generates dynamic equilibria which are consistent with both 'poverty traps' and 'self-sustaining' growth. It also generates outcomes in which economies grow out of poverty traps due to the initial accumulation of physical capital and then embark on the path of self-sustaining growth with increases in both human and physical capital. The 'new' growth theory, by emphasizing the role of human capital, has yielded considerable insight into the possibility of self-sustaining growth. This outcome sat uncomfortably with the predictions of the neoclassical theory of the 1960s. The main reason for this was theoretical: the neoclassical theory had ignored human capital, focusing solely on physical capital, and could thus reconcile the principle of diminishing returns to individual factors with the possibility of sustained growth only by appeal to an exogenous process of technical change. 1 By contrast, the new theory assumes that in addition to physical capital, improvements in the quality of labour, i.e. human capital, can be endogenously produced. Further, it assumes that marginal improvements in human capital are not subject To whom correspondence should be addressed

2 94 JOB GRACA ET AL. to decreasing returns in the level of human capital, while the production of output exhibits constant returns to physical and human capital taken together. In the long run, these assumptions make possible a balanced growth equilibrium, where endogenous human capital accumulation, rather than exogenous technical change, drive the process of growth. 2 Thus, the new growth theory has drawn attention to human capital as an important factor of production and has attributed long-run growth to the accumulation of this factor. At the same time, a branch of this theory has shown that, under plausible modifications, models of human capital can generate 'poverty trap' equilibria in which no accumulation of human capital and no growth takes place. Examples of poverty-trap equilibria are in papers by Becker etal. (1990), Ehrlich and Lui (1991), Galor and Zeira (1993) and Azariadis and Drazen (1990). The first two papers study the interaction between numbers of children and schooling per child in the context of endogenous fertility models. The latter two papers examine the effects of nonconvexities in the technology for augmenting human capital. While the frameworks are different, a common thread runs through these papers - economies that start off with a low level of human capital stagnate while economies that start off with sufficiently high levels of human capital grow. With the exception of Becker et al., none of these papers considers the implications of initial levels of physical capital for growth. 3 The available evidence, however, suggests that both human and physical capital are positive determinants of growth in developing economies. 4 Two contributions are especially relevant. In his analysis of the sources of growth in the U.S. economy during 1909 to 1957, Denison found a reversal in the relative importance of human to physical capital. Over the period , improvements in the educational composition of the workforce contributed 12% while increases in physical capital contributed 26% towards accounting for the growth in real national income over this period. Over the period , 23% of the growth in real national income was contributed by education and only 15% by physical capital. 5 Thus, there was a reversal in the relative magnitudes of these contributions. 6 A recent empirical study by De Long and Summers (1991) also argues for the importance of physical capital. Distinguishing between equipment and other types of capital (e.g. structures), the authors find a significant positive relationship between cross-country growth rates and equipment investment over the period Indeed, their results indicate a far greater importance of this type of capital for growth than has been previously recognised, either in the theoretical or the empirical literature. They reconcile their results with previous findings by proposing that equipment, as opposed to other forms of capital, can potentially generate externalities which promote growth, something which cannot be captured by studies based on aggregate concepts of capital. Our model incorporates the insights discussed above, albeit in a stylized fashion. An externality from physical capital to the educational sector is crucial for the results. This externality allows for equilibria in which physical capital formation

3 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 95 drives the initial phase of growth, which we term as neoclassical growth (because, by itself, this phase cannot go on forever), while human capital begins to accumulate at a later stage, and drives the economy into sustained growth with increases in both factors. Other results from our model are: First, self-sustaining growth can arise with sufficiently high levels of either human or physical capital (thus, a high level of human c~tpital is not a prerequisite for a 'take-off' although human capital accumulation drives growth in the long run). Second, savings propensities matter for whether an economy takes off or not. Third, if the externality from physical capital to the human capital accumulation technology is strong enough, endogenous cycles can occur along the balanced growth path. Section 2 presents the model. Section 3 characterizes three types of equilibria within the framework of a specific example. Section 4 solves for these equilibria. Section 5 discusses some comparative statics regarding capital mobility and savings. Finally, Section 6 contains concluding remarks. 2. Model 2.1. ENVIRONMENT Time is discrete and denoted by t -- 0, 1, 2,... The economy consists of a series of two-period lived, overlapping generations. A member of generation t is born in period t and lives till period t + 1. The population is constant at 2N, so each member of generation t is associated with one successor, born in period t + 1. Individuals of a given generation are identical in all respects. The representative individual of generation t derives utility only from her own consumption and not from the consumption or utility of her successor; thus, there is no bequest motive. Production is carried out by firms, using capital and labour. Firms acquire capital by borrowing the savings of private agents, and labour by hiring workers in return for a wage. Labour services consist of effective hours of labour which depend not only upon labour-time, but also the skill level, or human capital, of the worker. Thus, the hourly wage depends upon the skill level of the worker. Members of each generation start life with a certain level of human capital which they costlessly inherit from their respective predecessors (who are old by then). They are also endowed with a fixed amount of non-leisure time (normalized to unity) in both periods of their lives. When young, they divide this time between employment and education; the latter activity allows them to augment their inherited level of human capital. They also divide their employment income between consumption and savings to be carried over to old age. Upon becoming old, they work full-time for a wage, as there are no further incentives to augment human capital. Young individuals augment their human capital through a technology which generates increments to human capital as functions of the fraction of time spent in education, the individual's initial human capital, the average level of human capital

4 96 JOB GRACAETAL. in that period and a measure of the average level of physical capital (to be defined below). The augmented level of human capital becomes available to individuals when they become old. Note that since individuals are identical, average quantities equal individual quantities. However, the notion that returns to investment in human capital depend on the average levels of human and physical capital captures the external nature of these effects. Each individual takes the average levels of these variables as given and does not take into account the effect of their own actions upon these averages. The following elaborates on the model discussed above PRODUCTION Each period, a single good is produced by a constant returns to scale technology using two factors: capital and labour. Capital depreciates fully each period. The production function is assumed to be Cobb-Douglas: where Y~ is the level of output at time t, Kt is the capital stock at time t and Lt is the effective labour supply at time t. The latter is given by Lt = ntxt where nt is total hours of labour supplied at time t and xt denotes the common skill level of individuals working at time t. Note that the skill level xt is the same for young and old people alive at the same time t. Each old person supplies one unit of labour and each young person supplies 1 - rt units, where rt is the amount of time spent in education during period t. Since there is an equal number, N, of young and old people at any time t, nt can be written as nt = (2- rt)n. Therefore, equation (2) can be expressed as (1) (2) Lt = (2 - Tt)xtN. Output can be written in intensive form as Yt = kilt (3) (4) where yt = Yt/Lt and kt = Kt/Lt. Since private returns to scale are constant, firms earn zero profits in competitive equilibrium. Therefore, factor prices are given by rt = /3kt ~-1 wt = (1 - ~)kt ~ (5a) (5b)

5 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 97 where rt is the rate of return on capital and wt is the wage rate at time t. Note that the gross rate of return on capital equals the net rate of return because of the assumption of full depreciation. Since only young agents save and accumulate capital, we shall define per-capita quantities in terms of the population of young agents only. Let yt and kt denote output per young-saver and capital per young-saver respectively. In any period, these are given by Yt/N and Kt/N respectively. The relationship between per young-saver and intensive quantities is given by /It = (2 - "ct)xtyt / t = (2- Tt)xtkt. (6a) (6b) 2.3. HUMAN CAPITAL ACCUMULATION The skill level of the ith individual worker at time t is denoted by x~, while the average skill is denoted by xt. Although these quantities will be equal in equilibrium, individuals take the average skill level as given in private decision making. Each young agent faces the following technology in deciding the amount of time to devote to education (denoted by -r{) Xt+xi ~. X{(1 --}- ")/(xt, kt )T{) (7) where "/(xt, kt) represents the marginal efficiency of time spent in augmenting human capital. 7(.) depends upon the average levels of both human and physical capital. 7 Specifically, we assume that 7(.) is: (i) increasing in each argument; (ii) concave; (iii) bounded above by a maximum value 7"-8 The assumption that 3'(.) is an increasing function of the average level of human capital has been used previously by Azariadis and Drazen (1990). Its analytical purpose is to allow for the existence of multiple steady state equilibria. This assumption may be justified by appeal to 'peer group' effects. The assumption that 7(.) is also increasing in the average level of physical capital requires somewhat greater comment. While it may be more plausible to have 3'(.) depend upon private purchases of physical capital for educational purposes, it is the presence of the external effect in our model that generates 'take-off' equilibria. 9 Therefore, we neglect the role of private physical capital in education in favour of examining the purely external effect of social physical capital. In order to justify this external effect, it is helpful to appeal to more precise concepts of both physical and human capital. First of all, if capital may be distinguished by vintage, then a higher level of physical capital may be interpreted as embodying more modern techniques. Similarly, if we interpret investment in human capital as the attainment of higher education which develops engineering and management skills, then clearly physical capital, as laboratories, computes, etc., plays an important role in such education. 1

6 98 JOB GRACA ET AL. To a great extent, this role may be manifested through private purchases of capital inputs by educational institutions, something which we admittedly are not considering. However, externalities may arise through two sources: (i) geographical proximity to modern technology may allow for 'hands-on' experience (e.g. internships) which cannot be duplicated in the classroom; (ii) access to and interaction with a modem industrial sector may facilitate the adoption of training programmes (e.g. in computer-assisted design) that complement modem techniques of production UTILITY MAXIMIZATION A typical individual born in period t has a time-additive utility function Ut = Oq ln(c~) + ce2 ln(c~+l) (8) where c~ is the consumption of generation t when young and ct +l is the consumption of generation t when old. The ith young agent at time t supplies (1 - r~) labour hours (which implies an effective labour supply of (1 - r~)x{ for employment, and r~ labour hours for education). At time t + 1, her human capital is increased to xt+ li and this is devoted full-time to income-earning activity. Thus, her budget constraint in each period is given by (1 - r~)wtx~ = cyt + st (9a) i 0 Wt+lXt+ 1 + rt+lst = ct+ 1 (9b) where st is savings carried over by a young person at time t. Capital per young saver kt+l equals st. The intensive measure of capital is related to st by st = (2 - ~+l)xt+lkt+l. (10) The maximization problem facing a young agent can be expressed as max o/1 ln((1 - r~)wtx{ - st) + o~2 in(wt+l (1 --}- "y(xt, kt)r~)x~ + rt+lat) 7"~ ~8t subject to r~ _> 0.12 In stating the maximization problem, (7) has been substituted into (9b), and (9a) and (9b) into (8). The first-order conditions with respect to st and r~ are, respectively, o /1Ct+l a2cty _ rt+l (lla) -'~2 " i c~+"17(xt, kt)wt+lx, - e---l1 i 0). (1 lb) gywtxt<o (= 0ifr~ >

7 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 99 (The superscripts denoting individual agents and the generations to which they belong were relevant for keeping track of private decision variables while taking the first-order conditions. These shall henceforth be dropped on the grounds that individual and average quantities will be equal in equilibrium.) Combining (1 la) and (1 lb) gives the following inequality ")'(.)Wt+l ~ rt+l" (12) wt Condition (12) states that the yield on investment in human capital must equal that on physical capital in any interior maximum, i.e. for any % > 0. If (12) holds as a strict inequality, that implies that no training takes place in period t, i.e. Tt = O. Equilibrium paths are characterized by (1 la) and (12) (which determine the optimal choices of st and "rt), along with (5a) and (5b) (which determine wt, wt+l and rt), (9a) and (9b) (which determine ct y and ct +l). Taking into account (5) and (9), and using (10) to eliminate st, the optimality conditions may be written as /3k > 7(xt, kt) (13a) kt+l /3kG1 = [(2 = +,t)22 [(1 -/3)(1 - "rt)k~ -(2- Tt+l)(1 + "yttt)kt+l] (13a) where = OZl/O~ 2 and 3't = 7(xt, kt). Equations (13a) and (13b) constitute a set of two equations and two unknowns (Tt and kt+l). If (13a) holds as an inequality, then Tt = 0 and (13b) alone determines kt+l as a function of kt and the parameters/3 and. Equation (7) completes the system by describing the evolution of the stock of human capital. Equation (13b) may be solved for kt+l as a function of kt, Tt and Tt+l. Let denote this function; thus, kt+l = (kt, Tt, Tt+l). 13 It can be shown that is increasing in kt and "rt+l, and decreasing in Tt. Thus (13b) represents a downward sloping locus in ('rt, kt+l) space, i.e. for fixed values of kt and "rt+l. Equation (13a) does not depend upon Tt, therefore it is a horizontal line in (Tt, kt+l) space. An interior equilibrium occurs at time t if (13a) and (13b) intersect at a positive value of Tt (see Figure la). However, if they intersect at a negative value of %, then Tt = 0 (see Figure lb). 3. Equilibrium paths Depending on initial values of human and physical capital, three of the possible types of equilibria are (i) underdevelopment equilibria in which the level ofhuman capital does not change over time while the level of physical capital reaches a steady state value; (ii) sustained growth equilibria in which the level of human capital grows steadily; (iii) take-off equilibria in which the level of human capital stays constant over some initial period of time but then begins to grow as in the sustained growth equilibria. 14

8 100 JOB GRACA ET AL. kt+l kt+l (13~) Fig. la. Equilibrium with r~ > 0. "rt 1 I 1 I ( to,, r,, r~+~) kl+t (13~) k~+l (k,,t,,r,+l) Fig. lb. Equilibrium with ~-t = UNDERDEVELOPMENT EQUILIBRIA An underdevelopment equilibrium (UDE) is characterized by "rt = 0 for all t _> 0. In this equilibrium (13a) holds as an inequality for all t >_ 0. The paths of physical and human capital accumulation are given by: fl(1 - fl) kt ~ (14a) kt+l = +(~+2)fl

9 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 101 kt+t -I Fig. 2. ko Dynamics of the capital stock in the UDE. k~ xt = xo. (14b) Equation (14a) is derived from (13b) after setting rt = rt+l = 0. It defines an upward sloping, concave locus in (kt, kt+l) space. The unique fixed point of this locus is denoted by k which represents the steady state level of physical capital in a UDE (see Figure 2). The analytic expression for k is = /3(1 -/3) 2/3 + (1 -+-/3)' (15) From any given value of k0, there is monotonic convergence to k. The level of per-capita income may be higher or lower at this steady state, depending upon the starting value of human capital, but there is no growth in per-capita income once is reached. The defining feature of the UDE is that (13a) holds as an inequality at all points of time. In the steady state of the UDE, (13a) reduces to /3 > ~l--fl -- which holds for any xo _< g, where g is a critical value such that the above barely holds as an equality. Thus, as we shall prove below, any economy that starts off with xo _< g and k0 _< k remains in the UDE.

10 102 JOB GRACA ET AL. k* (16a) Fig. 3. L I l l T* (]~b) The existence of a balanced growth path SUSTAINED GROWTH EQUILIBRIA A sustained growth equilibrium (SGE) is characterized by (13a) holding as an equality and by the fact that 7-t > 0 for all t. Equations (13a), (13b) and (7) jointly determine the evolution of the variables r~, kt+l and Xt+l. The long-run behaviour of SGE is characterized by a balanced growth path associated with steady-state values of kt and 7-t (denoted by k* and 7-* respectively) and a constant growth rate for xt (denoted by v). Along the balanced growth path the equations characterizing equilibrium reduce to 9 (16a) (1 *) 1-/3 -- ~* (k*) 1-~ 1 (1-3)(1-7-*) 3 ~b [ [(2-7-*)3 + (1-3)](1 + 7*r*) (2-7-*)/~+(1-/3)J x +l = (1 + 7*7-*)xt (16b) (16c) where 7" is the upper bound on 7. Equations (16a) and (16b) jointly determine k* and 7-*. Equation (16a) determines k* independently of 7-*. Equation (16b) describes a locus in (7-*, k*) space. The LHS of (16b) is increasing in k* while the RHS is decreasing in both 7-* and k*. Thus, (16b) describes a downward-sloping locus in (7-*, k*) space. Figure 3 illustrates the balanced growth equilibrium, where both (16a) and (16b) hold. Figure 3 shows that k* < k, i.e. capital intensity is lower in the SGE than in the UDE. This is because the solution for k* from (16b), at the point where 7-* equals zero, must equal the solution for k from (14a). Thus, for the case 7-* > 0, the

11 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 103 equals zero, must equal the solution for k from (14a). Thus, for the case r* > 0, the solution k* must be smaller than k. This of course does not imply that per-capita stocks of physical capital are lower in the SGE than in the UDE. The reason that the ratio of per-capita physical capital to per-capita human capital is lower in the SGE has to do with the fact that human capital is much higher, not that physical capital is lower, in the SGE than in the UDE. The above analysis of long-run equilibria in a SGE is identical to that in Azariadis and Drazen (1990). However, the dynamics of SGE in our model may differ from those in the latter paper. Azariadis and Drazen show that the balanced growth path is locally saddle-point stable. This need not be the case in our model, where 3`(.) is an increasing function of kt. If the elasticity of 3` (.) to changes in kt is large enough, the balanced growth path in our model may display oscillations (this is analysed in the next section) TAKE-OFF EQUILIBRIA An equilibrium in which an economy experiences a transition from the growth path associated with an underdevelopment equilibrium (UDE) to a sustained growth equilibrium (SGE) is defined to be a take-off equilibrium (TOE). A TOE is characterized by (14a), (14b) for t E [0, T- 1]; and by (13a) holding as an equality, along with (13b) and (7) for t E [T, oo), Thus, rt = 0 for t E [0, T - 1] and rt > 0 at time T. The jump in the value oft takes place because 3'(.) increases sufficiently by t = T to induce young members of generation t to start investing in education. In the following section, we shall characterize the set of equilibria, and show how they arise from the initial combinations of physical and human capital. 4. Characterization of equilibrium We shall now prove that for the economy described in Section 2 there is a threshold 3`t (denoted by "~) such that if 3`t < "Y for all t, then the economy remains in a UDE for all t. We shall also prove that if 3`t exceeds -~ at some time t, then this economy must be either in a TOE or a SGE; If it is in the TOE, it may begin to accumulate human capital before % crosses the threshold "~. LEMMA 1. For any t with kt+l given by q3(kt, O, O) and for any kt, (13b) implies that /3kt~ -- A( a constant) = ~b(1 + [3) +2/3 kt+l 1 - Proof. with "rt = "rt+l = 0, (13b) becomes (1 +/3)kt~+l ~kt~+l 1 = V5 (1 - e - 2kt+l"

12 104 JOB GRACA ET AL. Multiplying both sides of the above equation by ] ~t/k~tq_l, setting ~] ~t/kt+l and inverting both sides gives the following equation. (1) A 1 +/3) (1 +/3) which after straightforward manipulation leads to = A A = (1 +fl) +2fl [] 1- ~ PROPOSITION 1. Let ~/ = A. Then 7t for all t is a necessary and sufficient condition for a UDE. Proof. (Sufficiency) Let 7t <_ "Y - A for all t. Assume initially that ~-t = 0 for all t so that xt = xo. Then, from Lemma 1, the LHS of (13a) is equal to A along the entire path followed by the variable kt as it converges to k. If A _> 7(x0, kt) over the entire path for kt, then our initial assumption that rt = 0 for all t is consistent with the equilibrium. (Necessity). Suppose that ~/t at some time t. Then, Tt = 0 cannot be an equilibrium irrespective of future values of T. To see this, first consider the case where Tt+l = 0. If "rt = 0 as well, then, from Lemma 1, the LHS of (13a) equals "~. But the RHS equals 7t by assumption. Equation (13a) is violated, implying that rt = 0 cannot be an equilibrium. Now consider the case ~'t+l > 0. IfT-t = 0 as well, then since kt+l is an increasing function of Tt+l, the LHS of (13a) will be even lower than in the case where rt+l = 0. Thus, the LHS of (13a) is less while the RHS equals 7t >_ 7, implying that (13a) will be violated even more strongly. Therefore, regardless of the value of "rt+l, ~'t = 0 cannot be an equilibrium when 7t [] The intuition behind the necessity proof is as follows. When 7t the lefthand side of (13a) is too small when evaluated at a point where "rt = 0. Equality in (13a) is achieved by increasing Tt, which given (13b) implies a fall in kt+l. This causes an increase in the left-hand side of (13a) leading to the required equality. We next look at the conditions under which an economy that starts off with no growth in human capital takes off at some future date. Proposition 1 has already established that a precondition for take-off is that 7t > "~ at some date t. As argued above, such a situation requires that "rt > 0 in order for (13a) to be-satisfied. However, a take-off can occur even before 7t crosses the critical To see this, consider an economy that is growing in physical capital but not in human capital. Along this growth path, 7t rises due to physical capital accumulation. Suppose that if the economy remains on this path, 7t crosses the critical value for the first time at some date s', i.e. %, Then, "rs, > 0. However, this will raise savings at date s' - 1, and therefore ks, will rise (as implied by the (.) function above). This will imply that the LHS of (13a) will be less than A, its value in a

13 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 105 UDE. In that case, if %,_ 1 was very close to ~, then the inequality required by (13a) may be reversed, requiring T~,-1 > 0. Thus, human capital may begin to accumulate even before "/t crosses ~. The following Lemma allows us to establish another critical value of "7 below which an anticipated future increase in T does not induce current investment in human capital. LEMMA 2. For any t, with kt+l given by (kt, O, 1) and for any kt, (13b) implies that t3k3t -- A' (a constant) =_ +13 kt+a 1 - fl Proof. (Follows the steps outlined in the proof of Lemma 1, with rt = O, "rt+ 1 = 1). [] Note that A' < A for any positive/3. Let '} - A'; therefore "~ While 7t _< ") implies that rt = 0 if rt+l = 0, 7t <_ /implies that ~-t = 0 even if'rt+l is positive; in particular, even if Tt+l = 1. At one extreme, consider an economy in which 7t _< /at date t. The discussion following Lemma 2 implies that no growth in human capital will take place at date t, regardless of the value of ~-t+l. At the other extreme, if 7t > 5', then as argued in the proof of Proposition 1, *t > 0, again regardless of "rt+l. However, if ~ _> 7t > ~, then the value of 7t depends upon the value of Tt+I. Two cases may be distinguished. If-yt over all future periods, then current ~-t will remain zero since all future values of r will also be zero. However, if at some future date s', %, > ~, then future values of Tt will not remain zero. This may feed back to cause Tt+l and consequently rt to turn positive. This result is summarized in the following Proposition. PROPOSITION 2. Given s' > s > O, if vt <_ 5"/for t 6 [0, s), < Vt <_ ~ for t E [s, s') and ~/t > ~/at t = s', then there is some date T E [s, s'] such that "it = 0 for t < T, and 7T > O. [] 4.1. INITIAL CONDITIONS AND EQUILIBRIUM PATHS We now turn to characterizing UDE and TOE on the basis of initial values of the state variables z0 and k0. Define ~?t - x (-~, kt) as the level of xt that satisfies the equation -y (x (~/, kt ), kt) The properties of 7(.) imply that, for x(.) is a downward sloping and convex function of kt. Lemmas 3 and 4 establish regions for (x0, k0) that lead to UDE, while Lemmas 5 and 6 establish regions for (x0, k0) that lead to TOE.

14 106 JOB GRACA ET AL. LEMMA 3. Let ~ = x('~, k) where ~/ is the critical value and-k is the steady state level of physical capital in the UDE. Then, for any ko <_ -k, xo < ~ implies that the stock of physical capital increases towards -k and the economy converges to (xo, k) in a UDE. Proof Since 7(~, k) = "~ and xo _< ~, k0 < k, 7(xo, ko) < "~. Thus, the economy grows towards the steady state level k of physical capital with no growth in human capital. As kt rises do does 7, but since x0 < 5, 7(xo, k) < "~.?(.) never crosses the threshold level required for growth in human capital to take place, and the economy remains in the UDE. [] LEMMA 4. For any ko > -k, there exists a critical value :~o - z('~, ko), such that xo < xo implies that the stock of physical capital falls towards k and the economy converges to (x0, k) in a UDE. Proof Since xo _< xo, 7(xo, ko) _< ~. as kt falls with no change in xt, 7t also falls. Thus, the inequality is strengthened and ~/t _< "~ for all t along this path. [] LEMMA 5. For any ko < k, if xo lies in the interval (~, :~o], then 7(.) will cross the critical value zy at some finite date t. Proof Note that since x(~, ko) is a downward sloping function of ko, ko < implies that ko >_ ~. Since x0 < 5:o, 7(x0, ko) < "~. However, as kt grows towards k, 7t will rise. Even with no change in xt along the way, xo > ~ would imply that 7(xo, k) > "Y. If xt begins to grow along the way, the above inequality would be even stronger. Thus, at some point, 7t has to cross the critical value -~. [] For the following lemma, define ~:t - x(,~, kt), where "~ is as defined in the previous section. LEMMA 6. For any ko < -k, if ~:o > ~ and xo lies in the interval (~, 5:o], then: (i) 7(.) will cross the critical value ;y at some future date t, and (ii) the economy is in a TOE, i.e. shows zero growth in human capital initially, butpositive growth at a future date. Proof (i) Since "~ > ~, then k0 > x0 and the fact that xo lies in the interval (~, 5:o] implies that it also lies in the interval (~, ko]. Hence, from Lemma 5, 7t will cross ~ at some future date. (ii) xo < xo implies that 7(xo, ko) < "~. Thus, TO = 0, and there is no growth in xt initially. The growth path from (x0, ko) satisfies the conditions for Proposition 2. Thus the economy experiences a 'take-off'. [] Finally, note that if x0 > ~0 the economy will experience growth in human capital from the very start. This is similar to the threshold property required for growth in Azariadis and Drazen (1990). However, in this case the threshold level is a decreasing function of the initial capital stock. Figure 4 shows how the space for initial conditions (xo, ko) may be divided up into regions for UDE, TOE and SGE respectively. 15

15 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 107 x SGE \ Fig. 4. Initial conditions for the three types of equilibria. 7 Fig, 5a. Dynamics DYNAMICS Figure 5a shows the dynamics of these equilibria, while Figure 5b compares equilibria in our model with those in Azariadis and Drazen (1990). Any path that starts off in a UDE converges monotonically along a horizontal ray towards k. A path that starts off in the TOE region grows horizontally until it crosses ~; after this, it may enter the transition to a SGE at any point until it hits

16 108 JOB GRACA ET AL. I I I Fig. 5b. Dynamics in the Azariadis-Drazen model. 5~. A path that starts off in the SGE follows an upward pointing arrow, indicating growth in human capital, and converges towards the steady state level k* of physical capital. This convergence will be locally monotonic if e'~k' i.e. the elasticity of 7t with respect to kt in a neighbourhood of 7", is small. However, if this is not the case, oscillations may result. This is captured by Proposition 3 below (the proof of which is in the Appendix). Let "Yk = 07t/Okt and 7k =- ")'k kt/7t. Let 7~ --- 7k evaluated at 7" and k, = k* respectively and 7k = 7~(k*/7"). PROPOSITION 3. If e~k <-/3 (resp. >/3), SGE growth paths display monotonic convergence (resp. display oscillations) in the neighbourhood of the steady state Note that/3 represents capital's share in total output. Empirical estimates of this parameter tend to be around 0.2 to 0.4. Thus, locally monotonic convergence of SGE growth paths requires that the proportional response of 7(.) to changes in physical capital be less than about 20%. Offhand, this appears to be a plausible restriction. One possibility that our analysis has not accounted for is that growth paths in the SGE region may hit the boundary of the UDE region and lead to stagnation in human capital. This is especially true for paths that begin to the right of "~ but below ~. Such paths lead to declining levels of physical capital, evan as the level of human capital rises. The effect of this combination on 7(.) is ambiguous. It is possible that 7(.) may fall below ~ along such paths. The assumption that 7~, the

17 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 109 3; SGE 5: ~ ~ slope = -w/v I "-L"<. Fig. 6. The state space for the Section 4.3 example. sensitivity of 7(.) to kt, is small may rule this out. Note that this is similar to the condition required for locally monotonic convergence of SGE growth paths AN EXAMPLE OF 7(xt, kt) In order to illustrate the results of the above sections, we shall assume that 7(x, k) follows 7(x, k) = 7* - b. exp -(vx+~ k) where b, v and w are all non-negative constants. It is easy to verify that 7 : R+ x R+ ~ [7* - b, 7*] is everywhere continuous and differentiable, increasing at a decreasing rate in each argument and that it is weakly concave throughout. Note that b = 0 implies no externality from human or physical capital to 7(.), while v = 0 (resp. w = 0) implies that the externality depends upon physical capital (resp. human capital) alone. For a fixed value of7t = 7, the equation for x(7, kt) becomes ln(b) - In(-/* - 7) w xt = - --" kt (17) 72 1) which defines level curves in (x, k) space for given values of 7(.)- Figure 6 illustrates these curves and shows how the regions for UDE, TOE and SGE may be determined for this example. The level curves associated with the 7(.) function are straight lines. These level curves get flatter as w/v, which measures the external effect from physical capital

18 110 JOB GRACA ET AL. SGE TOE / slo,... /v Fig. 7. The effect of steep level curves upon the state space. relative to that from human capital, gets smaller. As w approaches zero, the level curves become horizontal as in Azariadis and Drazen (1991]. At the other end, as w approaches infinity, the level curves become vertical. Given k, the steady state level of physical capital in a UDE, there is a critical value of w such that UDE get eliminated altogether. 16 This means that starting from any positive x0, an economy will eventually increase 7(.) sufficiently to begin accumulating human capital. This critical value is derived from (17) by equating the RHS to zero at O't = "~ and kt = -k: W c = ln(b) - - k If w _> we, the level curve associated with "~ intersects the horizontal axis to the left of k (see Figure 7). Thus, all growth paths hit 5' before they hit k. 5. Comparative statics 5.1. INTERNATIONAL CAPITAL MOBILITY The preceding analysis was confined to closed economies. However, a simple extension to the open economy framework may proceed as follows. Consider a small economy, hitherto closed to trade with the world, that is growing along a horizontal line in the TOE region of Figure 4a. Opening this economy up to trade in a world dominated by 'developed' countries, i.e. countries that are growing along a SGE with physical capital equal to k* will prevent this economy from increasing its capital beyond k* (note that the return on capital is greater at k* than at k).

19 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 111 X X* TOEto UDE~ Fig. 8. k* The effect of capital mobility upon the state space. I I I If this economy cannot attain the critical value ~ before it hits K*, it will remain in a UDE forever. In other words, the scope for an economy remaining in a UDE will increase at the expense of the scope for an economy to experience a take-off. This is because, given two steady states k and k*, with k > k*, the boundary between the UDE and TOE regions shifts up from ~ to x* -- x(5/, k*), so that any economy starting off with x0 < x* and k0 < k* will never raise "Yt sufficiently to induce self-sustaining growth. This is shown in Figure 8. The shaded region in that diagram represents a set of initial conditions that belong to the TOE region in the case of a closed economy, but switch to the UDE region in the presence of capital mobility. The international mobility of physical capital does not imply the equalization of growth rates of per-capita output in our model, unlike in Rebelo (1992). Returns to capital do equalize, but not per-capita incomes, either in levels or in growth rates. 17 The reason is that 'residents' of SGE economies experience growth in per-capita savings, and therefore per-capita stocks of physical capital, at a rate equal to the growth rate of their individual human capital. 'Residents' of UDE economies end up with stagnant levels of individual human capital; thus, their per-capita savings and stocks of physical capital also stop growing. Although interest rates equalize along with the intensive measure of physical capital, per-capita quantities remain divergent. Therefore, the international mobility of capital can have adverse implications for the distribution of incomes across countries. Of course, if both labour and

20 112 JOB GRACA ET AL. 1: TOE to UDE III: SGE to TOE Fig. 9. / The effect of savings behaviour upon the state space. physical capital were internationally mobile, human capital growth rates could equalize, allowing convergence of other per capita variables as well SAVINGS BEHAVIOUR Returning to the closed economy framework, consider the effect of a decrease in savings propensities. In the context of the example of Section 4, this can be achieved through assuming that the relative preference for young-age consumption, given by ~ oq/o~2 increases. The steady state level of capital in the UDE, given by (15), is repeated below. Clearly, an increase in causes k to fall. = /3(1 -/3) 2/3 + (1 +/3)" Further, given the proof of Lernma 1 and Proposition 1, the critical value "~ depends positively upon : ~= (1 +/3) + 2/3 1-/3 An increase in the value of causes the set of initial conditions that lead to UDE to expand at the expense of both TOE and SGE, while initial conditions that lead to TOE expand at the expense of SGE. Thus, the likelihood of UDE increases, of SGE decreases while that of TOE may increase or decrease. All in all, the change reduces the set of growth equilibria.

21 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 113 These shifts are depicted in Figure 9. k~ and "~' denote the new values of and ~ respectively, and ~ denotes the new boundary between the TOE and UDE regions. The implication is that two economies that differ in savings propensities can grow differently even if they start with the same initial conditions. 6. Conclusions The model presented in this paper shows the implications of allowing the technology for human capital accumulation to depend upon the level of physical capital. Long-run growth takes place through constant returns to factors that can be accumulated. These factors are human and physical capital, and the production technology displays constant returns jointly to these two factors. The engine of growth in our model is human capital accumulation. Allowing an interaction of human and physical capital at the level of the human capital technology has allowed us to characterize a richer set of equilibria than in previous papers. Thus, not only do we have the two long-run equilibria of 'poverty trap' models, but we also have the possibility of some economies escaping what appears to be a poverty trap through an endogenous mechanism. The dependence of the human capital technology on physical capital also reemphasizes the role of physical capital, at least for driving the early stages of growth. In our model, unlike other 'poverty trap' models, an economy with sufficiently high physical capital may escape the poverty trap, even if its initial human capital is low. This has both positive and normative implications: the positive implications are reflected in the take-off equilibria, where a period of simple physical capital accumulation precedes a take off into sustained growth. The normative implications are that exogenous increases in the physical capital stock (perhaps through foreign aid) may have the indirect effect of raising private incentives for human capital accumulation, thus allowing an otherwise stagnating economy to cross the threshold for sustained growth. These considerations may provide some justification for capital controls in developing countries. Because of the external nature of the effect that physical capital has on human capital accumulation in our model, private decision makers will underinvest in physical capital. In the presence of capital mobility, this underinvestment may result in underdevelopment equilibria in countries that may have grown if physical capital were immobile across countries. Acknowledgements The authors would like to thank S. Lahiri, seminar participants at the Federal Reserve Bank of Cleveland and participants at the ESRC Development Study

22 114 JOB GRACA ET AL. Group Conference (University of Leicester, March 1994), especially J. Knight and P. Levine, for comments. All errors are our own. Appendix: Proof of Proposition 3 In this Appendix, we shall present the analysis of local dynamics around the SGE steady state (k*, 7.*). The first-order conditions (13a) and (13b) are totally differentiated, with all relationships evaluated at the steady state values, k*, 7"* and 7*. This yields the following system: Idkt+l A1 B1 B2 j L dt.t j dt.t+l ] = [ 0 ] [dkt ] (A.1) where dkt, dt.t, etc. represent small deviations from the steady state values. At the steady state, (13a) and (13b) imply: /3(k*) ~-1 = 7* (a.2a) ( 1 ] (1 -/3)(1-7.*)(k*) ~-I 2-7.*+,1+~-7.*. = 1+7"7.* (A.2b) Taking account of (A.2a) and (A.2b), the coefficients A1 and B2 may be written as A /37"-'71"k* [ 127' ] B2 = (1 -/3)(k*) ~-l (1 + )(1 +7*7.*)23. The coefficient B1 is not reported because it does not affect the local dynamics of the system. Because the system (A. 1) is block-diagonal, it has two, distinct real roots, given by /~1 = A1 A2 = B2. Monotonicity requires both roots to be positive. If this fails, then oscillations occur, although the system could remain saddlepoint stable, i.e. converge to the balanced growth equilibrium along a unique path. Since we have one predetermined variable (k) and one jump variable (7.), saddlepoint stability requires the magnitude of one root to be larger and the other to be smaller than unity. The unstable root implies (given k0) a unique choice of 7.o such that explosive paths are ruled out. LEMMA A. 1:.~2 ---~ B2 _~ 1. Proof. t32 may be multiplied and divided by (1 - T*). Therefore, B2 = (1-7.*)(1-/3)(k*) ~-1 (1-+ )(1 + 7"7.*)2(1-7.*) '

23 INTERACTION OF HUMAN AND PHYSICAL CAPITAL 115 B2 may be rewritten as: (1-7-*)(1 -/3)(k*) fl-1 B2 (1 + 7*7-*) 1 +7" (1 + %b)(1 + 7*r*)(1-7-*)" Using (A.2b), and rearranging, B2 becomes 2 + ~b(1 +/3-1) _ (1 q- ~/3)7-* B2 = (a " 1 1+'7"7-* 1-7-* The above expression consists of three terms, each of which may be shown to be no less than unity. The second and third terms obviously satisfy this, given that 0 < ~-* < 1. The first term may be broken into 2 + ~(1 +/3-1) _ 7-*. (1 +%b) Since 0 <_ 7-* _< 1, the above expression will be no less than unity if 2 + ~'b(1 +/3-1) >2 (1 -.'. 2+%b(1+/3-1)_>2+2~b.'. 1+/3-1 >2 which holds since/3 < 1. Thus, A2 = B2 _> 1. [] From the derivation of A1 above, it is a straightforward matter to see that A1 = A~ _> (resp. _<)0 as/3 > (resp. <_)'y~(k*/7*). Proposition 3 follows by denoting e.rk - 7k (k /7 ). Thus, the second root is positive (resp. negative) if the elasticity of 7(.) with respect to k, evaluated at the steady state value of 7(.) is not too large (resp. large enough). Finally, note that if A1 >_ 0, then its magnitude is no greater than unity, since/3 < 1. This implies the saddlepoint stability of the balanced growth equilibrium. Notes 1 Some economists, notably Haavelmo (1954), Schultz (1960), Arrow (1962) and Uzawa (1965) recognized the importance of human capital for economic growth. However, their insights exerted little influence on the vast body of literature that grew around the Solow (1956) growth model. 2 Human capital as a factor of production has been used both to endogenize technical change (Romer, 1990) and to introduce nondecreasing returns overall without imputing them to a single reproducible factor (see Sala-i-Martin, 1990). 3 Becker et al. (1990) derive the rather paradoxical result that an economy starting off with low human capital and low physical capital is more likely to grow than one with low human capital and high physical capital. This is because they assume that the level of physical capital has a

24 116 JOB GRACA ET AL. negative impact upon the incentives to invest in human capital. Our paper presents an alternative setup and reaches the opposite conclusion. 4 Thus, Barro (1992) finds that growth rates for the period across a cross-section of countries are positively related to 1960 measures of human capital and also to the ratio of physical investment to GDP. A recent World Bank report [1993] also argues for positive effects from schooling as well as physical capital formation upon the growth of the East Asian economies. 5 See Denison (1962). Denison's method involved assuming a constant productivity premium for educated versus uneducated workers, taking account only of changes in the level of educational attainment by the workforce. Had the productivity premium increased between the first and second periods, the increase in the relative contribution of educational composition would have been even larger. 6 In a similar vein, it is claimed by some historians of the Industrial Revolution that, at least in Great Britain, large-scale declines in literacy followed the onset of the factory system rather than preceding it, implying that education was not an important factor in the early stages of the 'Revolution'. However, this remains a controversial matter (see West, 1985). 7 The specification for 3'(.) is the main point of departure of our model from other models of human capital accumulation. For example, Lucas (1988) follows Uzawa (1965) in assuming that 3` is constant, while Azariadis and Drazen (1990) assume that it is an increasing function of the average level of human capital alone. At the same time, Caballe and Santos (1993) and King and Rebelo (1990) allow, as we do, physical capital to affect the technology for human capital. However, they treat physical capital only as a private input into the educational sector, while we consider only its external effects. 8 3' may be defined as 7" = lim 7(xt,k*) ~t~oo where k* is the steady state value of k~ in a sustained growth equilibrium. 9 Altematively, we could have assumed an externality from some other variable, such as output per capita, to human capital accumulation. So long as this variable could grow during periods when there is no increase in human capital, it would induce take-offs during subsequent periods. 10 Lucas (1993, p. 257) offers a similar justification for including physical capital in the human capital equation. 11 Note the distinction between the use of computers by students in e.g. philosophy, and the recognition of a particularly innovative application of computers that increases the productive potential of students in, e.g. architecture. Thus, under our maintained hypothesis, educational institutions in less advanced countries may not offer curricula at the 'frontier' of each field, not just because of the private cost of offering such curricula, but also because they outpace the technical level at which production is carried out. 12 In principle, ~- should also be constrained to be less than unity. However, the utility function of our example satisfies Inada conditions for each period's consumption. Spending all the available time in education would reduce young-age consumption to zero while leaving old-age consumption positive. This consumption stream cannot be optimal under Inada conditions. 13 The explicit solution of (13b) is too messy and adds little insight. Hence, it is not reported here. 14 Other possible equilibria are: (i) human capital grows over some initial interval of time but stops growing later on as the economy enters an underdevelopment equilibrium; (ii) human capital accumulation cycles between intervals of positive and zero accumulation. While acknowledging such possibilities, we shall not analyse them in this paper. 15 Note that in Figure 3 the steady states k and k* could be determined as functions of the underlying technology and taste parameters, k is derived in (15), while the solution for k* is too unwieldy to report. 16 This outcome results from the fact that the level curves of 3`(.) touch both axes. If this were not the case, then sufficiently low initial values of either type of capital would always lead to UDE.

25 INTERACTION OF HUMAN AND PHYSICAL CAPITAL Other models in which the openness of an economy may have adverse implications for its growth rate are Stokey (1991a,b) and Young (1991). However, our result is more drastic in the sense that capital mobility may affect the type of equilibrium in which an economy finds itself. 18 Alternatively, if the 'world' stock of physical capital rather than the domestic capital stock exerted a positive externality on domestic education, growth rates might again equalize. However, given the justifications outlined in Section 2 for the inclusion of this externality, geographical proximity or at least unimpeded labour mobility would be a requirement for physical capital to have beneficial effects on human capital. Thus, assuming that the externality 'stops at the border' appears to be plausible. References Arrow, K. (1961), 'The economic implications of learning by doing', Review of Economic Studies, 29, Azariadis, C. and Drazen, A. (1990), 'Threshold externalities in economic development', Quarterly Journal of Economics, 105, B arro, R. (1991), 'Economic growth in a cross section of countries', Quarterly Journal of Economics, 106, Becker, G., Murphy, K. and Tamura, R. (1990), 'Human capital, fertility and economic growth, Journal of Political Economy, 98, S 13-$37. Caballe, J. and Santos, M. (1993), 'On endogenous growth with physical and human capital', Journal of Political Economy, 101, De Long, J.B. and Summers, L. (1991), 'Equipment investment and economic growth', Quarterly Journal of Economics, 106, Denison, E. (1962), 'Education, economic growth and gaps in information', Journal of Political Economy, 70, $124-S128. Ehrlich, I. and Lui, E (1991), 'International trade, longevity, and economic growth', Journal of Political Economy, 99, Galor, O. and Zeira, J. (1993), 'Income distribution and macroeconomics', Review of Economic Studies, 60, Haavelmo, T. (1954), A Study in the Theory of Economic Evolution, North-Holland, Amsterdam. King, R. and Rebelo, S. (1990), 'Public policy and economic growth: Developing neoclassical implications', Journal of Political Economy, 98, S126-S150. Lucas, R. (1988), 'On the mechanics of economic development', Journal of Monetary Economics, 22, Lucas, R. (1993), 'Making a miracle', Econometrica, 61, Quah, D. (1993), 'Empirical cross section dynamics in economic growth', European Economic Review, 37, Quah, D. (1994), 'Convergence empirics across economies with (some) capital mobility', London School of Economics Mimeo. Rebelo, S. (1992), 'Growth in open economies', Centre for Economic Policy Research Discussion Paper No Romer, E (1990), 'Endogenous technical change', Journal of Political Economy, 98, $71-S 102. Sala-i-Martin, X. (1990), 'Lecture notes on economic growth (II): five prototype models of endogenous growth', NBER Working Paper No Schultz, T. (1960), 'Capital formation by education', Journal of Political Economy, 68, Solow, R.M. (1956), 'A contribution to the theory of economic growth', Quarterly Journal of Economics, 70, Stokey, N. (1991a), 'Human capital, product quality, and growth', Quarterly Journal of Economics, Stokey, N. (1991b), 'The volume and composition of trade between rich and poor countries', Review of Economic Studies, 58, Summers, R. and Heston, A. (1991), 'The Penn World Table (Mark 5): an expanded set of international comparisons, ', Quarterly Journal of Economics, 106,

26 118 JOB GRACAETAL. Uzawa, H. (1965), 'Optimum technical change in an aggregative model of economic growth', International Economic Review, 6, Young, A. (1991), 'Learning by doing and the dynamic effects of international trade', Quarterly Journal of Economics, 1006, West, E.G. (1985), 'Literacy and the Industrial Revolution', in Mokyr J. (ed.), The Economics of the Industrial Revolution, George Allen and Unwin, London, pp World Bank (1993), The East Asian Miracle: Economic Growth and Public Policy, Oxford University Press, New York.