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Public education under capital mobility
Jean-Marie Viaenea and Itzhak Zilcha
, 
, b
a Erasmus University and Tinbergen Institute, Rotterdam, Netherlands
b The Eitan Berglas School of Economics, Tel-Aviv University, Ramat Aviv, 69978 Israel
Received 28 February 2000; accepted 26 February 2001 Available online 20 May 2002.
The paper considers a two-country model of overlapping generations economies with intergenerational transfers motivated by altruism and investment in human capital. We examine in a non-stationary competitive equilibrium the optimal provision of education with and without capital market integration. First, we explore how regimes of education provision––public, private or mixed––arise and how they affect the dynamics of autarkic economies. Second, we study the effects of capital market integration, in equilibrium, on the optimal provision of education. Third, we show that capital market integration enhances government intervention in the provision of public education (to improve the welfare of its constituents) and consider various solutions to such a competition.
Author Keywords: Altruism; Education; Growth; Human capital; Capital market integration
JEL classification codes: D9; E2; F2; J2
Trends manifested during the 1990s suggest a worldwide acceleration in the flows of foreign direct and portfolio investments. International production has become a significant element in the world economy and substantial flows of foreign investments to emerging markets is a recent phenomenon dating only from the beginning of last decade. This would not have been possible if it were not for the ongoing integration of international capital markets (see, e.g., UNCTAD, 1997). The increased mobility of capital coincided with the growing recognition that economies have come to revolve around the production and the use of knowledge. With the continuous upskilling of jobs, investment in education has become a high priority of many developed and developing countries. But does capital mobility enhance public investment in education? This paper seeks to study how international capital market integration (CMI) affects investment in education by households and government.
This paper integrates few main features in the recent literature on endogenous growth. Investments in human capital are used as an engine for growth (see, e.g., Lucas, 1988; Azariadis and Drazen, 1990; van Marrewijk, 1999). Public education is provided by governments, although private provision of education exists, in order to foster growth (see, e.g., Glomm and Ravikumar, 1992; Eckstein and Zilcha, 1994). Various aspects of the role of capital market integration in enhancing economic activity have been studied (see, e.g., Barro et al., 1995; Buiter, 1981; Dellas and de Vries, 1995; Leiderman and Razin, 1994; Rivera-Batiz and Romer, 1991; Ruffin and Yoon, 1993; Ruffin, 1985; Stokey, 1996). Our model in a steady state can be viewed as an AK-type endogenous growth model where A is proportional to the long-run capital–labor ratio. However, we shall consider mainly the transition path and non-stationary competitive equilibria.
The theoretical literature in public economics and international trade has devoted a significant amount of attention to issues concerning tax competition between governments and the provision of public goods. In a model with international capital mobility, Buiter and Kletzer (1995) study the issue of education and domestic capital market imperfection. In a closed economy, Lin (1998) examines an OLG model in which the young generations invest in their own education. Gradstein and Justman (1995) consider an economy where individuals (over-) invest in their own human capital accumulation for the purpose of attracting foreign capital. In a trade model, Wilson (1987) obtains that a tax on mobile capital can cause an inefficient distribution of public goods across regions because of the positive and negative externalities linked to investment inflows and outflows. In our framework, local governments finance public education by taxing labor earnings, the immobile factor, and therefore allow for the implementation of an efficient zoning policy. Also, individuals do not invest in their own human capital. With compulsory schooling in mind, it seems that the acts of training and the allocation of resources are not fully decided by the young generations.
The specific model we analyze is as follows. We consider a two-country economy with overlapping generations, identical households (in each generation) in both countries. Parents care about their offspring's income, hence we observe intergenerational resource transfers to the young (gift-bequest motive) and investment in her/his human capital. Education to the young generation is provided by both public schooling and parents. Governments tax labor income to finance the costs of public education, while parents may use some of their free time to enhance their child's human capital. As capital market integration affects wages and interest rates differently in different countries, the bequest transfers and the relative size of investments in education are expected to change across countries. Equilibrium levels of physical capital, effective labor and output will, therefore, differ between the integrated economies.
Our model allows us to deal with several important issues. First, though policy-makers wish to contain overall public spending, they make sure that `sufficient' resources are devoted to education. In this regard, it is important to determine the `optimal' level of real resources put into public education subject to government budget balance.1 Second, any increase in a country's investment in education raises this country's marginal return to physical capital, and thus attracts capital inflows. Capital market integration enhances government intervention in the provision of public education in order to improve the welfare of its current population (or voters). Therefore, the question how capital market integration affects the education policy, or coordination, between governments assumes importance. While we are aware of previous research on international tax competition (see, e.g., Mendoza and Tesar, 1998) we have not seen any discussion in the literature of such a conflict between governments.
We consider first the endogenous growth process in an autarkic competitive equilibrium under various regimes of education provision. The effects of capital market integration on the `optimal provision' of public education for the capital-exporting country (`domestic') and the capital-importing country (`foreign') are then studied. We find that, following CMI, the allocation of output between the two countries (in each date) depends upon the relative stocks of human capital. Each government has therefore an incentive to enhance the formation of human capital in order to increase the well-being of its constituents. The only tool it can use in this competition is the level of public education. We consider a two-stage game in which governments first select levels of public education and then, after the formation of human capital in each country, the two integrated economies follow a competitive equilibrium path. Various solutions to such a competition are considered. We show that the `optimal' public education is the same in the case where governments agree on a cooperative solution or a Nash bargaining solution. Moreover, this Pareto optimal provision is the same as that in the autarky case. In contrast, in the Nash equilibrium between the two governments, we obtain a non-Pareto optimal level. This implies that international coordination might be necessary to avoid inefficiencies that arise from overinvestment, or underinvestment, in public education.2
The remainder of the paper is organized as follows. Section 2 presents the OLG model with altruistic representative agents and characterizes the autarky equilibria. Section 3 studies the effects of capital market integration on the optimal provision of education in a two-country model. It also examines the education provision competition between governments in the integrated economy. Section 4 discusses the robustness of the results and concludes the paper. Most proofs are relegated to the appendix to facilitate the reading.
Consider an overlapping generations economy with identical agents in each generation. Consumers are economically active during two periods––a working period followed by a retirement period. At the end of the first period, every individual gives birth to one offspring. Denote by Gt the individuals born at the outset of period t and refer to them as generation t. The analysis starts at t=0, where the G-1 individuals live the retirement period, consuming their savings.
In this economy parents derive utility from the future income of their child. This motivates the intergenerational transfers in the forms of human capital and physical capital. The levels of human and physical capital transfers together with the relevant interest rate and wage determine the offspring's total income. Intergenerational transfers are carried out in two ways: (a) Improving the earning capability of the offspring via education; (b) the `joy of giving' or the `gift-bequest motive'. Denote by bt the transfer of physical capital to his/her child and denote by et the effort, measured in time, invested in educating this offspring.
The human capital of the representative individual in Gt+1, denoted ht+1, depends upon et and the parent's level of human capital ht. Moreover, we assume that the public sector invests in educating the young generation; this public education expenditure is fully financed by a proportional tax on wage income. Each individual is endowed with two units of time. Labor supply is assumed to be inelastic and equal to one unit of time. The other unit of time is allocated between time devoted to private education, denoted by et, and leisure (or home production), denoted 1-et. Let etg be the public investment (measured in time) in each child. Although individuals in each generation Gt are identical3 to provide public education, a fraction of the work force is devoted to this assignment. Let us assume a continuum of individuals in each Gt, then a proportion (1-etg) of this population is engaged in production while a proportion etg are `teachers', i.e., engaged in public education, working 1 unit of time in teaching activity. Moreover, we assume for simplicity that `teachers' are chosen at random from the population.
The mechanism of transfer of human capital to the younger generation and the evolution of this process has attracted a lot of attention in the economic literature during the last two decades (see, e.g., Becker and Tomes, 1986; Lucas, 1988; Azariadis and Drazen, 1990; Jovanovich and Nyarko, 1995; Orazem and Tesfatsion, 1997). It is clear that the human capital level of the younger generation is affected, significantly, by the direct investment in education, the environment (represented here by the average human capital of the teachers, which is the same as the average human capital of the older generation) and the human capital of the parents. 4 Since education takes place mostly by people the human capital of the `educators' should be taken into account. Thus some kind of weighing of the human capital of the corresponding `instructor' (a parent or a teacher) with the `time' spent on that type of `education' makes sense. Although the human capital production process is complex, to simplify our analysis we assume the following evolution process. For some constants 
1 and 2 we have
| (1) |
where is the average human capital of the `teachers' which, by our assumption, is equal to the average human capital of generation t. We take
1 and 2 to be constants (larger than 1) that represent the efficiency of the process which generates human capital: 1 is affected by the home environment while 2 is affected by the schooling system, neighborhood, facilities, etc. It is possible to generalize the process to allow for decreasing (or increasing) returns to scale. However, such a generalization will `weaken' our results and complicate our analysis. Also, the perfect substitution between public and private inputs can be relaxed, to some extent, without affecting most of the results, but with much more complex proofs.
Lifetime preferences of the individuals are assumed to be given by a Cobb–Douglas utility function
ut=c1t 1 c2t2 yt+13 [1-et]4, | (2) |
where i are known parameters and i>0 for i=1,2,3,4; c1t and c2t denote, respectively, consumption in first and second period of the individual's life; yt+1 is the income of the offspring and (1-et) represents leisure (or some home-produced good).5 Let bt-1 be the intergenerational transfer made to each `young' individual at the outset of his working period, rt and wt be the interest rate and the wage rate in period t respectively, then the lifetime income for individuals in Gt is given by
yt=(1+rt)bt-1+(1- t)wtht, t=0,1,..., | (3) |
where the tax at rate t on wage earning, determined by the government, finances the public education at level etg. We do not assume taxation of intergenerational transfers since such a tax does not exist in most countries, and its effect on our subsequent results is insignificant. Since the human capital of all individuals of generation t is the same, for each t, the government budget constraint is
| twtht=wthtetg, t=0,1,..., | (4) |
which implies that etg=t.
Production in this economy is carried out by competitive firms that use labor and capital to produce a single commodity. This commodity serves for consumption as well as input in the production process. There is full depreciation of the physical capital. The aggregate human capital in date t (not including the human capital devoted to public education) is an input in the production process. In particular we take the aggregate production function to be
| qt=F(kt,(1-etg)ht), | (5) |
where kt is the capital stock and (1-etg)ht=(1-t)ht is the effective human capital used in the production process. F(· ,·) is assumed to exhibit constant returns to scale, it is strictly increasing, concave, continuously differentiable and satisfies Fk(0,(1-t)ht)=
, Fh(kt,0)=, F(0,(1-t)ht)=F(kt,0)=0.
Production at each date t is carried out by competitive firms which borrow capital at date t-1 and hire labor services at date t. Thus factors' prices are given, in competitive equilibrium, by the corresponding marginal products. Since the human capital of a worker is observable, the wage payments will depend upon the effective labor supply of the worker, i.e., wtht where wt=Fh(kt,(1-t)ht) is the wage rate. The economy starts at period 0 with given capital transfers and human capital endowments, b-1 and h0, respectively.
Let the bequest transfer, bt-1, the stock of human capital, ht, the effective wage rates wt, wt+1, the interest rates rt, rt+1 for dates t and t+1 be given. The tax rate at date t, t, is assumed to prevail in the next period as well. An individual chooses the levels of saving, st, capital transfer, bt, and direct investment in his offspring's education, et, so as to maximize
| ut=c1t1 c2t2 yt+13 [1-et]4, | (6) |
subject to constraints
c1t=yt-st-bt 0, | (7) |
| c2t=(1+rt+1)st, | (8) |
ht+1=(1et+2t)ht, et 0, | (9) |
where the income yt is defined by (3). Given the initial capital stock k0, transfers b-1, initial human capital h0 at the outset of period 0, and a sequence of tax rates (t)t=0, a competitive equilibrium is a sequence [c1t,c2t,st,bt,et]t=0, and a sequence of prices (wt,rt)t=0 such that for t=0,1,2,... .
(a) Given the above prices, [c1t,c2t,st,bt,et]t=0 is the optimum for ((6), (7), (8) and (9)).
(b) For the human capital levels (ht), given by (9), the market clearing conditions hold:
| wt=Fh(kt,(1-t)ht), | (10) |
| 1+rt=Fk(kt,(1-t)ht), | (11) |
| kt+1=st+bt. | (12) |
Condition (12) is a market clearing condition for the physical capital at the end of period t, equating the aggregate capital stock at date t+1 to the aggregate savings and transfers of physical capital at date t. After substituting the constraints, the first-order conditions that lead to the necessary and sufficient conditions for optimum are
| (13) |
| (14) |
| (15) |
| if et=0. | (16) |
It is clear from (15) that the optimal amount of time invested in the off-spring's education takes into account the gain to his income, due to the choice of the parent's objective function. An increase in either the parents' human capital ht or the wage at the future date wt+1 increases, ceteris paribus, the time spent on education by the parents at the expense of their leisure. Equation (15) establishes also a negative relationship between private and public education: an increase in t+1 decreases the time spent on private education et and hence, raises leisure. This substitution among types of provision of education will have a number of implications throughout this paper.
From ((8), (13) and (14)) we also obtain that
| (17) |
Using ((3), (7), (8), (13) and (14)), we obtain for et>0:
| (18) |
To simplify the subsequent analysis we assume that the aggregate production function in our economy has the Cobb–Douglas form.6 For some 0<
<1:
| F(kt,(1-t)ht)=Akt[(1-t)ht]1-. |
In equilibrium the following expressions are obtained:
(1+rt)=A(kt/(1-t)ht)-1 and wt=(1-)A(kt/(1-t)ht).
Using ((12) and (18)) we derive
| (19) |
| (20) |
The following assumption regarding the utility function and the aggregate production function guarantees that the intergenerational transfers of capital are non-negative:
Assumption 1. (2+3)>2.
This assumption holds when the capital share in aggregate output and the altruism parameter 3 are `not too small'. Substituting ((18) and (19)) in (7), while making use of ((8) and (13)), we obtain an expression for the income at date t:
| (21) |
However, using ((17) and (19)) we can also express aggregate income at any date t as a proportion of aggregate output at the same date:
| (22) |
This indicates the part of aggregate output which `young' members allocate between current consumption, saving and bequest.
From ((15) and (17)) we derive with et
0 that
Using ((1), (4) and (19)), the growth factor of human capital is given by
| (23) |
The growth factor 
t can be smaller than 1 for i sufficiently close to 1, i.e., for low returns to education, and a significant weight 4 for leisure in the utility. When et=0 and leisure equals 1, the growth factor of human capital is readily obtained from (9):
| (24) |
The time dependence of t in either (23) or (24) hinges only on the time dependence of the tax rate. From ((21) and (22)) and the production function we obtain, for et>0, that
Dividing by (23):
| (25) |
This describes the equilibrium dynamic path of the capital–labor ratio in autarky.
Given the above framework let us consider the `optimal' level of public provision of education in autarky. In each date the government is elected by the working generation Gt and the `retired' generation which does not participate in financing public education. Since education is financed by taxing wage incomes it is reasonable to assume (this is compatible with recent political economy theories) that the government chooses the level of public education etg, given the current economic conditions, in a way that maximizes the welfare of members in Gt. Due to `altruism' between parents and their children the choice of `optimal' etg takes into account, to some extent, the welfare of Gt+1 as well.7 Given the level etg that maximizes welfare of Gt we prove:
Proposition 1. Regardless of initial conditions, the optimal provision rate of public education is time independent and depends on the parameters of the utility and production functions only.
The result in this proposition depends on the choice of preferences and production function. Eliminating, for example, the assumption that the production function is Cobb–Douglas will make the initial conditions matter.
Proof. Consider the utility function (2). Let us substitute for yt+1 in (2). Assuming first that et>0 we make use of (23) to obtain an expression for leisure:
| (26) |
The substitution of ((23) and (26)) in (2) leads to an expression for the lifetime utility of the individuals in the tth generation:
ut= m(1+2t)4+(1-)(2+3)(1-t)(1-)[1+2+3+(2+3)], | (27) |
where m groups parameters and variables that are predetermined at the outset of period t. It is assumed that any chosen t will stay in place in the next period. Maximizing (27) with respect to t, we derive the optimal level of public education under the `mixed' regime, denoted by m:
| (28) |
which establishes our claim. We shall assume that the parameters on the RHS of (28) guarantee that m>0. We are now left with the determination of the optimal t when et=0. To that end let us consider the following optimization problem of the tth generation:
| Maxst,bt ut=c1t1 c2t2 yt+13, |
subject to constraints ((7) and (8)) and ht+1=2tht. Using the first order conditions, and repeating the same steps as above, we obtain
| ut=pt(1-)(2+3)(1-t)(1-)[1+2+3+(2+3)]. | (29) |
The maximization of (29) with respect to t leads to the optimal provision of public education
| (30) |
where p is independent of 2. 
It is important to note that m in (28) is decreasing in 1 and increasing in 2. If we assume 1=2 then m is independent of 2 and p>m. Also m is independent of time and of the size of private provision of education et. m is increasing in 4 (for 21), that is, the level of public education increases in the weight to leisure in the utility function. Our preferences could have been chosen with home-produced good rather than leisure without affecting our results significantly (except for `reinterpretation' of 4). This follows from our assumption that labor supply is inelastic, hence, each individual has one unit of time to allocate to non-labor activities.
The following exercise shows how optimal provision of education varies with the representative consumers' preferences. We do that by varying the parameter 4 while fixing the other parameters of the model.8
Proposition 2. The weight for leisure in the utility function determines the optimal regime of education. There exists an interval (4m,4p) such that for 4 in this interval the optimal regime sustains both public and private education. When 44p, it is optimal to provide only public education. When 4
4m, it is optimal to provide only private education.
Proof. There exists a value 4m for which m in (28) is zero:
For 44m, only private education exists. For 4>4m, it is optimal to provide public education. Substituting (23) in (26), making use of (28), one obtains
which gives the extent of leisure (and of private education) in a mixed regime of education. As 
(1-et)/4>0, et=0 at
| 4p=(1-)[1+(2+3)(2+)]1/2, |
where, under our assumptions we have 4p>4m. For 44p, private education ceases.9
The above proposition is illustrated in Fig. 1. It depicts the optimal provision of public education for combinations of 3 and 4 while assuming 1=2= and fixed values for the parameters of the model. The emphasis is on 3 and 4 because of their offsetting effects on leisure (see (15)) and hence, on utility. Fig. 1 is divided into three planes, each representing a regime of education. For any given value of 3, there is a first range of values of 4 starting from 0 which gives rise to private education only (=0); for a second range of intermediate values of 4, it is optimal to have both public and private education; finally, a third range of values of 4 justifies public education only ( taking values around 0.35).
Fig. 1. Optimal public education in autarky. Parameter values:
=0.5,
The above propositions demonstrate that the optimal rate of financing of public education is independent of a country's initial levels of physical and human capital. An important implication is that any heterogeneity observed across education systems is reflected by a nation's preferences rather than the nation's wealth.10
Let us compare now the equilibrium paths of a single economy in autarky under the three regimes of education discussed in Proposition 2. Consider the competitive equilibria for given initial conditions and compare the long-run properties of this economy under each regime.
Note first that the time independence of the tax rate in the previous section implies time independence of in (23) and (24) as well. Let us use ((22) and (21)) to obtain the expression for output and income growth (for any )
| (31) |
In the long-run kt+1/ht+1=kt/ht=k/h. From ((23) and (25)), we obtain the long-run capital–labor ratio
| (32) |
Substituting this in (31) gives
This relationship holds whatever regime of education, that is for any . Hence, the long-run economic growth in autarky coincides with the human capital growth factor , regardless of initial conditions and the education regime. Our model in the stationary state is an AK-type endogenous growth model where all variables grow at the rate (-1). This is a consequence of having incorporated externalities that yield constant returns to scale to parents' human capital in the process describing the evolution of human capital.
Now let us introduce some notations for the different education regimes allowing 4 to vary accordingly. Let i (i=a,m,p) denote the optimal provision rate of education under, respectively, private education (namely, a=0), mixed provision (m is given by Eq. (28)) and public provision (p given in (30)). Likewise, using ((23) and (24)), let i (i=a,m,p) denote the corresponding growth factors. We prove the following relationship regarding the long-run rates of growth of each regime.
Proposition 3. Regardless of initial conditions, growth factors across the various education regimes rank as follows: amp.
The proof is relegated to the appendix. An implication of Proposition 3 is that for any two economies which differ in education regime, say, due to difference in preferences, their long-run endogenous growth rates differ. If we move from private education into `mixed' education, namely, from =0 to positive, growth rate will decline. This results from the stronger impact that private education has on growth as the desirability for leisure (or home produced goods) in the utility function declines.11
Consider two economies in autarky: the domestic economy, described above, and the foreign economy, whose variables are marked with `*'. These economies are assumed to differ only in the initial capital transfers and human capital. At date t=0, the following variables are given: b-1, h0 for the domestic economy; b-1*, h0* for the foreign economy. Denote the domestic equilibrium by {(c1t,c2t,st,bt,et),(wt,rt)}. The equilibrium of the foreign economy is denoted by {(c1t*,c2t*,st*,bt*,et*),(wt*,rt*)}. As the utility and production functions are similar, both countries provide public education at the same rate, i.e., =*. In this case, since (25) holds for both economies, if (k0/h0)>(k0*/h0*) then (kt/ht)>(kt*/ht*) for all t. Thus, (1+rt)<(1+rt*) and wt>wt* for all t.
Assume that at date t=0 the domestic and foreign economies are integrated to form a single commodity market and a single capital market; namely, there is full capital mobility while labor remains internationally immobile. Upon integration of capital markets, physical capital will flow from the low return to the high return country until interest rates are equalized in the integrated economy. The type of international capital movement involves a change in the location but not the ownership of physical capital. Even though we consider `full' capital mobility between the two markets one can easily extend our analysis to the case of `restricted' capital flows between the two markets. In such an equilibrium (not defined here) we clearly do not obtain equalization of capital labor ratios; however, this `constrained' competitive equilibrium achieves (for some `joint' social welfare function) higher level of welfare compared to the autarky case.12
In the sequel, we use capital letters to distinguish post-integration variables from their autarky counterparts. Hence, (1+Rt) stand for the post-integration interest rate, Wt and Wt* are the wage rates, 
t and t* the provision rates of public education, etc.
The capital stock used in the production in the home country is Kt, while the stock of physical capital, located at home and abroad, owned by domestic residents is denoted Tt. Hence (12) becomes
| Tt+1=St+Bt. | (33) |
Similarly we define Tt* for the foreign country. Any difference (Tt-Kt) corresponds to a net outflow of domestic capital abroad. At any date t, the following international identity must hold:
| Tt+Tt*=Kt+Kt*. | (34) |
Hence, the above difference corresponds also to a foreign inflow of capital, (Kt*-Tt*). Using ((33) and (34)), the first-order conditions for both countries under integration become
| (35) |
| (36) |
These two equations are the analogs of ((19) and (20)) for the integrated economy. As for the autarky equilibrium, we obtain:
| (37) |
| (38) |
| (39) |
| (40) |
It is worth noting that (35), (36), (37), (38), (39) and (40) that describe the dynamic path of the integrated economy are similar to those obtained for the autarky case.
Following capital market integration, equal returns to physical capital implies
| (41) |
From the properties of the production functions
Combining these two expressions
| (42) |
Remarks. Eq. (42) can be generalized to the case where technologies differ in scale (i.e. AA*). In such a case the stocks of human capital should be normalized. Moreover, though (42) has been demonstrated for the incorporation of physical capital and effective labor in a Cobb–Douglas production function, it holds for a broader class of production functions including the CES with homogeneity of degree one. Thus each country's share in total output and share in the stock of physical capital of the integrated economy is equal to its share in the stock of human capital. In each date government will therefore desire to vary its human capital in the aggregate level in order to maximize the welfare of its citizens. Obviously, this situation may result in some sort of competition between governments via the provision of public education. Given this observation, a central issue that can be studied within this framework is the extent in which countries attempt to modify their public education policy as capital markets become integrated. More precisely, three main questions can be raised: How does the integration of capital markets affect the provision of public education? What can be said about the division of the gains between the capital-exporting and the capital-importing countries? How does the strategic provision of public education affect the allocation of production and capital between the two countries? These questions, which have been largely untouched in the literature, will be taken up in this section. Since the decision regarding public education is taken by the government, which attempts to maximize welfare, we face a two-stage game. In the first stage, governments select the provision of public education (by chosing or *). In the second stage, the economies achieve competitive equilibria given the levels of human capital resulting from the above selection of (,*). We shall concentrate on three types of solutions: the cooperative solution, the Nash bargaining solution and the Nash equilibrium. Let us add that the notions `cooperative' or `non-cooperative' relate only to stage 1 of this two-stage game. As it is demonstrated later, in the Nash equilibrium solution over-provision of public education may occur and attain inefficient equilibrium. Thus we shall start our analysis with the case where governments coordinate their policies taking into account the welfare of individuals in both countries.
Public education policy is called `optimal' when at each date t both countries (governments) decide jointly upon the public education provision rate in a way that maximizes certain weighted sum of the tth generations' utilities. As mentioned earlier, due to the way `altruism' is assumed here such maximization takes into account the incomes of generation t+1 as well. Now we show:
Proposition 4. Following capital market integration, under coordination of education policy, the optimal provision rate of public education is the same as that under autarky. It is independent of time, initial levels of transfers and human capital across countries.
See the appendix for the proof. Proposition 4 implies that t=t+1=t*=t+1*=. Given this result, Propositions 2 and 3Propositions 2 and 3 hold for the integrated economy as a whole. A feature of this cooperative solution is that capital market integration affects neither the optimal provision rate of formal education nor the long-run growth rate when compared to autarky. There are, however, dynamic benefits that the integrated economy achieves from international capital movements.
Proposition 5. Regardless of the education regime, total output and total capital stock of the integrated economy increase at all dates compared to the autarkic case.
The proof is relegated to the appendix. This proposition confirms the robustness of the traditional static gains from international capital mobility (see Ruffin, 1985) and extends the result to a dynamic framework with human capital and public education.
Our main purpose now is to numerically compute and compare the dynamic paths of the domestic and foreign economies for the cases of autarky and capital market integration. Initial values were taken to be: 1=2=1, 3=2, =0.5, A=4.0, k0=T0=2, k0*=T0*=1, h0=H0= h0*=H0*=1. The weight to leisure 4 takes three values: 4(a)=0.5, 4(m)=1.5 and 4(p)=5.0. Each value of 4 is chosen such that, according to Proposition 2, each education regime is attained as an optimal one (`a' stands for private, `m' for mixed and `p' for public). This implies that m=0.044 and p=0.353. We assume that 1=2= throughout these examples and the efficiency of educational regimes is chosen to be (a)=1.44, (m)=2.07 and (p)=3.06 such that the growth rate of output is the same, that is =1.08. Given this information, a representative agent in each economy, endowed with perfect foresight, attains his/her optimal consumption-bequest in two steps. First, the paths of wt, rt, ht, kt and yt are simultaneously solved for. Second, given these parameters, the paths of c1t, c2t, st, bt and ut can be computed. Simulations under capital market integration must satisfy (33) and the adding-up constraint for capital balances (34).
The simulated dynamic paths enable us to deal with three important issues. First, they illustrate the dynamic gains from trade. To that end, Fig. 2 plots the change in income following capital market integration as a percentage of their autarky values. The three panels of Fig. 2 represent the three education regimes namely, private education in panel (a), mixed in panel (b) and public in panel (c). The broken line gives the time pattern of the gains for the integrated economy resulting from capital market integration. Gains in income of the order of 1.5–2 percent are observed in the short-run but fade away with time whatever the education regime. Note that the gains in utility turn out to be much larger (not shown). Second, there is an implementation paradox. Using (42) Fig. 2 displays the individual gains of participating countries. We observe that the first generation of both countries G0 and G0* are better off after CMI whatever education regime is considered. In the case of public education foreign individuals are worse off in all generations t, t>0, following the CMI. The same applies to the domestic individuals in the private provision case in all generations t, t>2. In the latter case (and in the mixed regime), the domestic wages decrease after integration which lowers the return to private education and, hence, the stock of domestic human capital compared to its autarky level (and vice versa for the foreign country). Despite of this and although individuals in our model are altruistic, the first generation will decide in favor of capital market integration even if later generations lose. Third, they illustrate the fallacious meaning of numbers on education spending. Consider the regime of pure public education in panel (c). The path of human capital is the same in both countries and is unaffected by integration. However, as the foreign country is capital poor at the outset, foreign wages increase while domestic wages decrease after integration. CMI makes it therefore more costly to the country poor in physical capital to maintain a similar level of human capital.13
Fig. 2. Gains from capital market integration. Change in income of individual countries (as a percentage of autarky values).
Let us simplify matters by considering in the sequel the case of public education only. Since both countries have similar representative agent's utility, the same aggregate production function and the same human capital evolution process, t=t*=p. Using (24):
| Ht=2pHt-1,Ht*=2pHt-1*. |
Substitution in (42) gives
This last expression indicates that the partition of output in the integrated economy at any date t is determined by the initial levels of human capital. Thus, with a cooperative symmetric solution for the provision rates, none of the two countries can improve its relative position.
In considering non-cooperative solutions we shall take the following approach. Public education policy cannot be varied too often by governments, hence only stationary behavior will be considered now. Since the share in production of each country will depend upon its share in the stock of human capital in the integrated economy, education policy in the foreign country will affect the welfare level in the domestic country and vice versa. Thus the two governments in negotiating the CMI will have to deal also with the public education provision. In our framework it is unreasonable to expect an agreement where the public education levels differ, since it implies that in the long run only one country will produce. We assume now that the resolution of this conflict is implemented via the Nash bargaining solution. The `disagreement point' will be given by `no CMI', i.e., by the autarkic utility levels when =p and no capital flows between the two countries.
We shall consider the Nash bargaining solution for this conflict assuming that it takes place at the outset of each period to determine the provision of public education in each country. Given the initial capital holdings of residents and the human capital levels at the beginning of date t, to define the set of feasible utilities for the Nash bargaining problem let us derive the set of all utilities `attainable' at period t (under the given parameters at the outset of this period). In this case we must consider also partial capital transfers between the two markets (so far we looked at competitive equilibrium with unrestricted flow of capital which results in a `Pareto optimal' competitive equilibrium).
Let 
be a parameter in [0, 1]. We say that the two economies, or capital markets, are -integrated, 0
1, if only a proportion of the unrestricted competitive equilibrium (CE) flow of capital is allowed to move between the domestic and the foreign markets. Let us elaborate on this type of arrangement. Given the ownership pattern of capital by each country's residents at the outset of date t and the initial distribution of human capital, if the flow of capital between the two countries for this period under unrestricted CE is 
t, then we impose the restriction: only t is allowed to move from one country to the other, and under this restricted transfer of capital we consider the CE in each economy. For =1 it is the CMI case we have considered before while =0 is the autarkic case. Clearly, such equilibria will also depend upon the identical , 01, chosen by the two governments. For each given values of (t,t) at the outset of date t the (restricted) competitive equilibrium, after capital transfers at level t take place, is called: (t,t)-CE. Denote the corresponding utilities of the representative consumers in each country by Ut(t,t) and Ut*(t,t) where (t,t)
[0,1]2. To formulate our Nash bargaining problem at the outset of date t, <St,dt>, given the initial conditions of the two economies, we define
| St={(u,u*)R2 | Forsome (,)[0,1]20uUt(,) and 0u*Ut*(,)}. |
The `disagreement point' dt is given by the autarkic utility levels, i.e., when capital is not allowed to move between the two countries,
| dt=<Ut(0,p),Ut*(0,p)>. |
Proposition 6. The Nash bargaining solution yields provision levels of public education as in the cooperative solution case; namely
See the appendix for the proof. Thus, the Nash bargaining solution chooses the same provision level of public education as the cooperative solution, and hence the same allocation of production. To understand this result let us refer to Fig. 3 which displays the feasible set together with a map of iso-indifference curves for the integrated economy. It is important to note that lies on a straight line via (dt,dt*) with a slope Bt*/Bt (see the appendix). Thus the Nash bargaining solution
must coincide with
p which maximizes the weighted sum of utilities since both Ut(1,) and Ut*(1,) are either increasing in or decreasing in simultaneously. Moreover, this argument also demonstrates that the Kalai–Smorodinsky solution to this bargaining problem is also obtained at =p, since (Ut(1,p), Ut*(1,p)) is the `ideal point' and it is in St.
Fig. 3. Education game: the Nash bargaining solution. (dtdt*) are disagreement points (autarkic utility levels); N(
Consider now the case where both countries are identical at t=0 in all parameters. Starting from autarky, introducing capital market integration will result, in a Nash equilibrium, in variation of , even though we have a completely symmetric case. We can obtain explicitly the reaction curves in terms of and * using the usual Nash behavior: the domestic government chooses in a way that maximizes the utility of its representative consumer, while * is assumed to be given, and vice versa. We demonstrate now that for Nash equilibria the provision of public education differs from the cooperative or autarkic level even in the symmetric case:
Proposition 7. When we consider Nash equilibrium for this symmetric game the optimal provision of public education is either higher or lower than p in each date.
The proof is to be found in the appendix. The extent by which the Nash equilibrium (NE) differs from the cooperative one is also shown in the proof. Numerically it can be added that, when countries are fully symmetric, the cooperative solution p is 0.353 and the NE solution at date t=0 is 0.468 for the usual parameter values. Overinvestment in public education is a result of the altruism in preferences. By increasing t, each government overinvests in next period's human capital in the hope of attracting more physical capital, which will result in higher utility via yt+1. This is achieved, however, at the expense of the current return to physical capital.
Thus if governments choose at date 0 the NE 0 and 0* , assuming that this can be repeated in each date, we find deviations from the level p, which basically mean that we are not in a Pareto optimal situation.
The objective of this paper is to examine in a dynamic framework the optimal provision of education in a competitive equilibrium with and without capital market integration. Endogenous growth in our economy is attained via investment in education. Due to strong evidence that parents' human capital and parents' investment in the upbringing of their child play an important role (see, e.g., Becker and Tomes, 1986), we assume that the evolution of human capital depends on: (1) public education, (2) the parents' human capital and (3) `private education', which is, in our case, the time spent by parents to enhance their children's human capital.
When each economy is considered in isolation, the optimal provision of public education is shown to depend on the parameters of household's utility function and of the economy's aggregate production function. Whatever the regime of education, the result that international capital mobility increases production, obtained for static models, has been extended to a dynamic framework with human capital and intergenerational links. Moreover, the distribution of total output and total physical capital between the two regions depends on the region's share in the total human capital of the integrated economy. Thus, it opens the possibility for governmental manipulation, via the provision of public education, to improve the well-being of its voters. We apply one policy tool for competition: the investment in public education. We derive: (a) the optimal provision of public education in the autarkic case is the same as that of the cooperative solution and of the Nash bargaining solution; (b) the Nash equilibrium between governments results in efficiency losses: (c) international coordination might be necessary to avoid these inefficiencies and should coordinate the provision (via time) of public education, in contrast to cases where monetary spending in education is considered.
This paper uses an OLG framework with some specific assumptions, hence the robustness issue should be discussed. First, the functional forms used in our model are widespread in this branch of the literature and are needed for a closed form solution. The selection of these functional forms was, however, motivated very strongly by empirics, since the relevance of the model hinges on the relevance of empirical results. For example, the significance of the various types of intergenerational transfers has strong support in the empirical literature (see, for example, the extensive literature surveyed by Laitner, 1997). Also, the incorporation of physical capital and human capital has repeatedly been shown to have empirical relevance in production. In particular, the Cobb–Douglas production function provides a good fit on data for the USA and other industrial countries. Second, the human capital evolution process incorporates externalities that yield constant returns to scale to parents' human capital (assuming homogeneous population). In this case, growth rates of output and effective labor are independent of the distribution of human capital. Third, some outcomes are obviously sensitive to the assumptions made here. A striking feature of our model is the irrelevance of the specific levels of physical and human capital in determining the optimal share in total output allocated to public education. We believe that the specific choices of preferences and production function are the main reason for this outcome. The results may also be sensitive to the choice of the decision variable of the government, the tax rate instead of education expenditure. In our model the tax rate represents, basically, the level of public education hence it is also the strategic variable in the Nash equilibrium. However, it has been established that Nash equilibria in expenditures are more rivalrous than those in tax rates ( Wildasin, 1988).
We are grateful to G. Glomm, D.S. Hamermesh, K. Reffett and seminar participants at Florida International, Indiana, Geneva, Michigan, the Tinbergen Institute and the Midwest International Economics Meeting at Ann Arbor for their beneficial remarks. Research assistance by L.W. Punt is gratefully acknowledged. The current version has considerably benefited from the critical remarks by B. Rustem, our Editor, and by five anonymous referees.
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Proof of Proposition 3. Let us use (23) twice to obtain the growth factors a and m. Setting t=0 in (23):
| (A.1) |
where a is continuously decreasing in 4, for 44m. Substituting (28) in (23):
| (A.2) |
where m is continuously decreasing in 4, for 4(4m,4p). Now, substituting (30) in (24):
| (A.3) |
which is independent of 4, for 44p. Hence, mp if
which, by Proposition 2, is always satisfied. Likewise, am if
which, by Proposition 2, is always satisfied.
Proof of Proposition 4. We will demonstrate the result for public education only. The proof for the mixed regime can be obtained by analogy. Authorities of both countries that participate to capital market integration choose a single t(=t*) such as to maximize the following weighted sum of domestic and foreign utilities:
| max{atC1t1 C2t2 Yt+13+at*C1t*1 C2t*2 Yt+1*3}, | (A.4) |
where the weights at and at* are independent of t. Authorities also expect futures rates to be equal to the current one, that is t=t+1=t*=t+1*. Making use of the first-order conditions, the optimization problem (A.4) becomes
Making use of ((3), (7), (9) and (17)), one obtains
| (A.5) |
Likewise, one obtains Yt+1*. Under capital integration
| (A.6) |
The interest rate (1+Rt+1) and wage Wt+1 can be expressed in terms of (A.6) at period t+1. (37) and (38) give the dynamic path of the human capital and of the stock of the physical capital of the integrated economy respectively. Hence, all period t+1 variables can be expressed in terms of period t variables which cannot be influenced by the choice of t. Hence,
| (1+Rt+1)=t1-(1-t)(1-) [cst1], |
| Yt+1=t1-(1-t)(1-2) [cst2], |
where cst1 and cst2 stand for groups of parameters and of predetermined variables. Yt+1* can be obtained by analogy. Hence,
| atUt+at*Ut*=t(1-)(2+3)(1-t)(1-)[1+2+3+(2+3)] [cst3], | (A.7) |
where cst3 is another constant. Maximization of (A.7) with respect to t leads to
| (A.8) |
which is similar to p in (30).
Proof of Proposition 5. At date t=0, we have k0+k0*=K0+K0*. With capital market integration, we have
Denote
Since, at date t=0
| (1-)h0=(1-)H0 and (1-*)h0*=(1-*)H0* |
are given, we can write
Therefore, by the concavity of the production function
Thus
However, since k0+k0*=K0+K0* then
| q0+q0*<Q0+Q0*. |
This implies that
| y0+y0*<Y0+Y0*. |
Therefore
| k1+k1*<K1+K1*. |
As =* in autarky and with integration, aggregate labor supply is unaffected by integration and, hence
Rewriting this expression
| q1+q1*<F(k1+k1*,h1(1-)+h1(1-*))<F(K1+K1*,H1(1-)+H1*(1-*)), |
since h1+h1*=H1+H1*. Dividing both sides by (K1+K1*):
Hence, q1+q1*<Q1+Q1*, which implies that k2+k2*<K2+K2*. This process continues for all t=2,3,4,... proving our claim that qt+qt*<Qt+Qt*.
Before we look for the Nash bargaining solution let us note the following property for the `full CMI' case (i.e., when =1):
Claim. For any , 01,Ut(1,)/Ut*(1,)=
t where t is a constant which depends only on the initial conditions of the two economies at date t.
Proof of the Claim. Let be in [0,1], the common tax rate for both countries, i.e. t=t*=. Making use of the first-order conditions, the expression for domestic utility is
Repeating the same steps as in the proof of Proposition 4 (which makes use of (37) and (38)), all period t+1 variables can be expressed in terms of period t variables which cannot be influenced by the choice of t. Denote by Bt the constant which depends upon parameters fixed at date t such that
| Ut(1,t)=Btt(1-)(2+3)(1-t)(1-)[1+(2+3)(1+)]. |
Similarly, we obtain that
| Ut*(1,t)=Bt*t(1-)(2+3)(1-t)(1-)[1+(2+3)(1+)]. |
The ratio of the last two expressions
is a constant which depends upon parameters which cannot be influenced by the choice of t.
From the proof we see that we can write
| Ut(1,)=Bt(1-)(2+3)(1-)(1-)[1+(2+3)(1+)], |
and
| Ut*(1,)=Bt*(1-)(2+3)(1-)(1-)[1+(2+3)(1+)], |
where Bt and Bt* are given by the initial parameters at the outset of period t.
Proof of Proposition 6. To solve for the Nash bargaining solution we should note first that any <Ut(,),Ut*(,)> is not Pareto optimal if <1. This follows from our earlier analysis which shows that for =1 we have positive gains to aggregate outputs, compared to =0, but this can be generalized to any <1. We shall allow asymmetry in the bargaining power of the two countries (see Zhou, 1996); thus for some positive constants 
and 
the Nash solution will be attained by maximizing (u-dt)(u*-dt*) over the set
rather than over the whole feasible set St. Using the above expressions for the utility levels under full CMI and noting that
cannot be obtained for =0 or 1, we derive from the optimum condition:
| (1-)(2+3)(1-)(2+3)-1(1-)[1+(1+)(2+3)]- (1-)(2+3)(1-)(1-)[1+(1+)(2+3)]-1(1-)[1+(1+)(2+3)]=0, |
which yields that
Proof of Proposition 7. The proof is for the regime of public education only. The maximization problem of the domestic country's authority is to choose t given t* such as to maximize Ut:
| Maxt Ut=C1t1 C2t2 Yt+13 given t*. | (A.9) |
By analogy, the foreign country's maximization problem is
| Maxt* Ut*=C1t*1 C2t*2 Yt+1*3 given t. | (A.10) |
Making use of the first-order conditions, the maximization problem of the domestic authority is
| (A.11) |
given t*. The expression for Yt+1 is given by (5), that of (1+Rt+1) depends on (A.6). Like in the proof of Proposition 6, ((37) and (38)) can be used to express t+1 variables in terms of period t variables. After substitution, the final expression for Ut is
| Ut=3[bt+21t(1-t)t]1+2+31t(-1)(2+3), | (A.12) |
where 2 and 3 are constant grouping parameters of the model and
| 1t=[(1-t)+(1-t*)]1-/[(1-t)t+(1-t*)t*]. | (A.13) |
Maximization of Ut in (A.12) with respect to t for a given t* leads to the following FOC:
| (A.14) |
where 4=A(1-)3KtHt1-/[1+(2+3)]. By imposing symmetry, i.e. t=t*, (A.14) simplifies to
| (1+2+3)(1-2)=-(1-(1-)), | (A.15) |
where =-(1-)(2+3)(2-2)-1bt/4+(1+(2+3))/2. From (A.15), it is clear that the symmetric Nash equilibrium (NE) is t=t*=0.5 when =0. If one assumes <0 instead, then assuming [1-t(1+)]>0 leads to a symmetric NE solution Taking [1-
t(1+)]<0 the NE solution is then Hence, the NE solution at date t=0 is different from the cooperative solution
except by a fluke.
Corresponding author. Tel.: +972-3-640-9913; fax: +972-3-640-9908; email: izil@post.tau.ac.il
1 The analysis we present here suggests that the time spent learning and not education spending should be the decision variable of governments. In our model, the former is equal to the labor income tax.
2 The European Credit and Transfer System of the European Union is one of the steps in that direction. Similar conclusions were derived by Gradstein and Justman (1995) in a different model.
3 Aspects of income distribution are therefore omitted. They are discussed in models of heterogenous agents economies like in Glomm and Ravikumar (1992), Viaene and Zilcha (2001) and others.
4 See, e.g., Lillard and Willis (1996) and OECD (1997) for empirical evidence.
5 There is an analogy between our model and home-production economies analyzed in Benhabib et al. (1991), Greenwood and Hercowitz (1991) and others. These models include a homework production function and a production function for market actitivities. Likewise, our model has an aggregate production function for tradeable goods and one process describing education or production of human capital. Unlike these models, the household sector is described here by a utility function defined over consumption at two dates and leisure, where the additional argument of altruism is represented by the desirability for higher income of the offspring. The analogy with these models would have been closer if we had assumed 4=0 and replaced yt+1 by ht+1 in the utility function, since with home produced goods, no utility is usually derived from either leisure or future generation's income. In our framework inserting some home-produced good explicitly in the utility function will not affect our analysis since labor supply is inelastic.
6 The incorporation of physical capital and effective labor in this form is supported by the empirical work of Mankiw et al. (1992).
7 We do not see justification in such an OLG framework, where provision of public education in each date is determined by the current government, to assume that {etg} should maximize the `sum of discounted utilities', hence, taking into account all future generations.
8 The next result can be generalized to the case where both 3 and 4 vary. Intervals will be substituted by planes in this case.
9 Observed time allocation to non-labor activities can be calibrated using the reduced form expression for leisure (1-et) found in this proof. Characteristics of time-budget surveys have been analyzed in Lee and Lapkoff (1988), Gronau and Hamermesh (2000) and others. For example, Lee and Lapkoff report that, like in our model, the lowest amount of leisure is recorded during the active life (about 16 h), the highest during the retirement period. Consumption of home production follows a similar pattern and the publicly funded education expenditures take place exclusively during the active period.
10 Proposition 2 stresses also the importance of obtaining empirical estimates for the weight for leisure in the utility function. Though no empirical studies are available for direct comparison, useful estimates can be derived from empirical determinants of health, as there is a large weight on current health to current utility. For example, Sickles and Yazbeck (1998) suggest a health elasticity of leisure ranging from 0.23 to 0.30. These are normalized estimates of 4. In our model where the normalization of 's is not imposed, these estimates imply values for 4 in the range (1.19, 1.71). To obtain this range we assume 1=2 and 3=22, the latter satisfying Assumption 1. From Fig. 1, for 3=2 and the computed range for 4, one obtains values for in the range (0.09, 0.15).
11 Proposition 3 is basically comparing growth rates of autarkic economies that differ in education regimes. Though few empirical studies have undertaken hypothesis testing on this topic, they offer some insight into the schooling-GDP growth connection associated with the mixed regime and public provision alone. While the former gives a role to parental tutoring there is none in the latter. Using a decompositional technique, Glaeser (1994) divides the education's positive effects on economic growth into parts and concludes on higher gains to education for children in a society with educated parents as opposed to those who are alone in their learning. Also, Burnhill et al. (1990) find that parental education influences entry to higher education in Scotland over and above the influence of parental social class. A reason which is put forward is that parental education elicits more parental involvement at home.
12 We assume here that capital flows are sufficiently large to obtain factor price equalization. However, such a situation might not be optimal for a large capital-exporting country since it may strategically restrict outflows and thereby secure monopoly rents. See Wong (1995, Chapter 11) for a ranking of different policies for a country in the presence of international trade in goods and of factor mobility.
13 It is important to stress that the simulation results in Fig. 2 can be generalized to a very broad range of parameter values. Whereas the pattern of country responses to CMI is very robust with respect to changes in A, , and 's, it is sensitive to the choice of capital share . For example, for =0.35 the changes in foreign national income in Fig. 2(c) are all negative. However, independent of parameter values gains from CMI are always attained.
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Volume 26, Issue 12, October 2002, Pages 2005-2036 |
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