The Role of Education in Development

Transcription

1 MPRA Munich Personal RePEc Archive The Role of Education in Develoment Rioll Marla and Cordoba Juan Rice University Setember 2006 Online at htt://mra.ub.uni-muenchen.de/1864/ MPRA Paer No. 1864, osted 21. February 2007

2 TheRoleofEducationinDeveloment Juan Carlos Córdoba and Marla Rioll First Version: Setember 2006 This Version: February 2007 Abstract Most education around the globe is ublic. Moreover, investment rates in education as well as schooling attainment differ substantially across countries. We contruct a general equilibrium life-cycle model that is consistent with these facts. We rovide simle analytical solutions for the otimal educational choices, which may entail ure ublic rovision of education, and their general equilibrium effects. We calibrate the model to fit cross-country differences in demograhic and educational variables. The model is able to relicate a number of key regularities in the data beyond the matching targets. We use the model to identify and quantify sources of world income differences, and find that demograhic factors, in articular mortality rates, exlain most of the differences. We also use the model to study the role of ublic education, and the effects of the HIV/AIDS andemic in develoment. Keywords: schooling, life exectancy, mortality rates, cross-country income differences. JEL Classification: I22, J24, O11. We thank Mark Bils, Daniele Coen-Pirani, David N. DeJong, Pohan Fong, Paul Gomme, Tatiana Koreshkova and articiants in seminars at the University of Rochester, Concordia University, and University of Delaware for their valuable comments. Deartment of Economics, Rice University, 6100 Main Street, Houston TX 77005, jcordoba@rice.edu. Deartment of Economics, University of Pittsburgh, 4907 W.W. Posvar Hall, Pittsburgh PA 15260, rioll@itt.edu.

3 1 Introduction The rovision of education around the world has three salient features. First, ublic education is the redominant form of education. It accounts on average for 85% and 78% of rimary and secondary enrollment resectively, a fact illustrated in Figure 1 for a set of 91 countries. Second, richer countries invest significantly more resources in education er uil than oor countries. For examle, the US invests 200 times more resources er uil a year than the average of the 5 oorest countries in our samle. Figure 2 illustrates an asect of this fact. It ortrays the investment rate in education, defined as the ratio of ublic exenditures in education er uil to er-caita GDP, against er-caita GDP. This investment rate differs substantially across countries and increases with income. The third feature is well-known: schooling attainment also increases with income and differs substantially across countries (Barro and Lee, 2000). The urose of this aer is to construct a general equilibrium model of education and human caital accumulation that is consistent with the set of facts mentioned above. In articular, we study the role ublic education and investments in education, as well as years of schooling, in the formation of human caital and in the wealth of nations. Our analysis has two main distinctive features. First, in our model young individuals face credit market frictions and the government alleviates them by roviding ublic education. This is a dearture from the frictionless Ben-Porath model that has been the focus of recent research on human caital formation, including Manuelli and Seshadri (MS, 2005), Hendricks (2005), and Hugget, Ventura and Yaron (2006), among others. Second, we calibrate the key arameters of the human caital roduction function by targeting moments of the cross-country distribution of schooling and returns to schooling. Cometing aers instead only emloy U.S. evidence to calibrate key arameters. Thus, our calibration strategy avoids issues that arise due to the secial nature of the U.S. educational system. We devise a simle model that incororates key features of the data regarding education and develoment, and yet has simle closed-form solutions. This allows us to identify the mechanisms at work and roerly quantify the contribution of different fundamentals, such as TFP, ublic education exenditures, mortality rates and taxes in exlaining income differences across countries. 1

4 We show that the model erforms well in other dimensions beyond the matching targets. For examle, although we only target the average return to schooling in the calibration, the model relicates the overall distribution of returns better than alternative well-known models. Also, our model relicates the revalence of ublic education around the world even though rivate education is also available in the model. Overall, we conclude that our model rovides a lausible quantitative theory of schooling, human caital, and incomes. We then emloy the model for several uroses. First, we comute the long-run elasticity of outut to TFP. As argued by MS, this elasticity increases substantially in models with endogenous human caital and imlies that given TFP differences result in larger income differences. We find an elasticity of 2.4 in our model which is substantially lower than the one found by MS of 9. On the other hand, our estimate is close to the 2.8 value found by Erosa, Koreshkova and Restuccia (EKR, 2006) although using a different model. Our estimate imlies that smaller but still imortant TFP differences are required to exlain income differences across countries comared to models of exogenous human caital. In contrast, MS large elasticity imlies that no TFP differences are required to exlain income differences. 1 Second, we construct human caital stocks for a set of 91 countries and comare the results to existing alternatives. We find that our estimates are more diserse and significantly lower for oor countries than the ones estimated using Mincer style equations. The reason is that Mincer equations abstract from exenditures in education while our human caital roduction function includes both years of schooling and exenditures as inuts. Since ublic exenditures er uil are ositively correlated with er caita income, including differences in education exenditures across countries makes human caital much lower in oorer countries. These findings are qualitatively similar to those of MS and EKR, and quantitatively close to those of Córdoba and Rioll (CR, 2004). Our human caital estimates are significantly different from Hendricks (2002) who finds relative minor differences in human caital across countries using earnings of immigrants in the US. 1 This comarisson is between the comlete models that include, among others things, variation in demograhics andinthericeofcaital. 2

5 However, we show that both estimates can be conciliated if immigrants are ositively self-selected to some minor extent. For examle, we find that a tyical Mexican immigrant would need close to 50% more human caital than a non-immigrant with same schooling for both estimates to be identical. Our estimates are roughly consistent with those of CR who use a comletely different methodology based on country secific rural-urban wage gas to estimate human caitals. Third, we assess the sources of cross-country income differences according to the model. A simle variance decomosition exercise shows that the role of TFP in exlaining steady state income differences decreases significantly comared to models with exogenous human caital, from 60% to 34%. In addition, counterfactual exercises show that eliminating TFP differences would reduce the world variance of log incomes by 48%, while eliminating differences in mortality rates would reduce this variance by 51%. Finally, eliminating differences in the rovision of ublic education would do so by 18%. We thus find that TFP still lays a major role in exlaining income differences, but that mortality differences are at least as imortant. Finally, we use the model to assess the effects of the HIV/AIDS andemic on long run develoment. This andemic has increased mortality rates dramatically articularly in Sub-Saharan Africa. Contrary to some claims in the literature, we find that the long run consequences of the andemic are devastating for develoment. For examle, the model redicts that if rates of infection ersist at the current levels, the log variance of income would increase in 28%. Our aer is related to the recent quantitative work by MS and EKR regarding human caital differences across countries using microfounded models of human caital formation. The first aer derives the imlication of a frictionless Ben-Porath model for human caital differences, while the second builds uon a heterogenous agent model in the Bewley tradition. MS consider ure rivate education, a feature inconsistent with Figure 1. Our model is similar to MS in its use of a reresentative agent model and its treatment of demograhic factors, but we focus on the role of ublic education and credit market frictions in the formation of human caital, and our calibration strategy is very different. While MS calibrate the key arameters of the human caital technology to match life-cycle earning in the US, we calibrate these arameters to match features 3

6 of the cross-country distribution of schooling and returns to schooling. We arrive to substantially different quantitative results. On the other hand, our model is similar to EKR in the treatment of ublic education and in some aggregate results. The models, however, are very different. In their model, TFP is the only source of variation exlaining income and schooling differences, while mortality rates, as well as TFP, lay the key role in our model. EKR do not consider the role of mortality rates in generating differences in human caital accumulation across countries. In contrast, MS, Bils and Klenow (2000) and Ferreira and Pessoa (2003) find, as we do, that these differences in mortality are imortant. Finally, EKR calibrate their model to match moments of the cross sectional distribution of earnings andschoolingintheus,whilewetargetthecross-country distribution of schooling and returns. Finally, in our model ermanent TFP differences do not affect years of schooling while they do in MS and EKR. This feature of their models rovides an additional channel for TFP to affect income. However, it also imlies that their models do not ossess a balanced growth ath, while ours does. To the extent that a balanced growth ath is regarded as a desirable roerty of a growth model, our model rovides a lausible theory of human caital and income differences. A further advantage of our modeling aroach is that we are able to obtain closed-form solutions for most of our results. The remaining of the aer is organized as follows. Section 2 lays down the model. Section 3 calibrates it and rovides overidentifying tests. Section 4 reorts human caital estimates. Section 5 resents the imlications of the model for cross-country income differences as well as some counterfactual exercises. Section 6 reorts the results for HIV/AIDS andemic. Section 7 concludes. 2 The Model Consider an economy comosed of I generations of individuals, cometitive firms hiring labor and renting caital to roduce finals goods, and cometitive mutual funds managing individuals savings and the caital stock of the economy. Denote N i,fori = {1,..,I}, the oulation size of generation 4

7 i, andn the total oulation. Schooling-age individuals (i =1), or "children" for short 2,canwork or go to school while adults (i = {2,...,I}) onlywork.aeriodinthemodellastfort years, and individuals live for a maximum of I T years. Demograhics Of any given cohort only a fraction π i survive to age i. We define π 1 =1and π I+1 =0. Life exectancy at birth is given by T PI i=1 π i. The oulation size of children satisfy N 1 = fn 2,wheref is the fertility rate of young adults (i =2). Only young adults have children. We focus on a steady state demograhic situation in which total oulation is growing at a constant rate. Since N 0 1 = fn0 2 = fπ 2N 1, where ( 0 ) denotes next eriod, then the common growth rate for all grous along a steady state is: N 0 N = fπ 2. (1) Moreover, in steady state the age comosition of the oulation is stable and described by (see details in the Aendix): n i N i N = π i (fπ 2 ) i P I j=1 π. (2) j j (fπ 2 ) The Production of Human Caital Children s human caital, h 1, is assumed to be a fraction θ 1 of the average arental human caital, h. Young adults human caital, h 2, deends on human caital investments undertaken during childhood: h 2 = z (χ + s) γ 1 x γ 2 (3) where [z,γ 1, γ 2 ] > 0 are technological arameters, χ are re-school years, s are years of formal schooling, x are educational exenditures in the form of final goods. Human caital for later eriods of life are assumed to be a roortion θ i 0, i {3,..,I}, ofh 2. Thus, human caital investments during childhood determine the whole ath of human caitals during the individual s life. The arameters θ i allow to cature the life-cycle rofile of earnings while h 2 determines its level. For examle, θ i =1reresents a flat life-cycle earning rofile at the level h 2. For comleteness, 2 In the calibration below, "children" are individuals between the ages of 6 to 26 years. 5

8 define θ 2 1. Individuals Problem Individuals maximize their lifetime welfare as described by: IX π i ρ i 1 ln c i i=1 where c i is consumtion at age i, andρ is a discount factor. Moreover, individuals face the following resource and technological restrictions during their life time: (1 s/t )=c 1 + e, e 0, s [0,T]. (4) wh i +(1+r)a i 1 = c i + a i,a 1 = a I =0,i=2,..,I. (5) h 1 = θ 1 h, (6) h 2 = z (χ + s) γ 1 ( + e) γ 2, (7) h i = θ i h 2 for i =3,...,I. (8) where w =(1 τ) w is the after-tax wage, τ is a roortional tax on income, a i are savings at age i, r is the net return on savings, e are rivate exenses in education, and are ublic exenditures o in education er child. Individuals maximize over n{c i,a i,h i } I i=1,s,e,. The first restriction states that the after-tax labor income during the first eriod of life can be used to consume or to invest in education. In articular, children cannot borrow. Equations in (5) are the set of budget constraints during adulthood. The remaining equation describe the accumulation of human caital. Notice that children receive ublic subsidies for education,, which enter directly in the constraint (7). Although arents in our life-cycle formulation are not altruistic, they do hel their children in two ways. First, they rovide them with an earning otential during their schooling-age years that is tied to their own earning otential. This is a form of unintended bequest that allows children 6

9 to consume and invest in education. Moreover, it imlies that individuals in richer economies canconsumeandinvestmoreineducationthanindividuals in oorer economies. Second, arents also hel children by financing the ublic education system through roortional income taxes. This form of social bequest reinforces the ability of children in richer economies to afford more consumtion and investment. Firms Outut is roduced by a reresentative cometitive firm oerating a Cobb-Douglas technology: Y = AK α H 1 α, 0 < α < 1, where Y denotes outut, K reresent hysical caital services, and H reresents human caital services. The firm hires labor and caital in cometitive markets at the rates w er unit of human caital and r er unit of caital. Profit maximization entails the following conditions: w =(1 α) Y H,r= α Y K. (9) Mutual Funds Individuals deosit their savings in mutual funds (MFs). MFs own the caital stock of the economy, and rent it to firms at the rate r. MFs oerate a constant returns to scale technology that transform q units of outut into 1 unit of caital. Thus, q is the rice of caital in terms of final goods, the numeraire. MFs are cometitive, ay roortional taxes τ on earned income, and returns r to the surviving deositors. Without loss of generality, consider a single MF that holds all the caital stock in the economy, K 0 = P I i=1 N ia i /q. Free entry guarantees zero rofits so that: ((1 δ) q + r (1 τ)) K 0 =(1+r) IX i=1 π i+1 π i N i a i, 7

10 where δ is the dereciation rate of caital and π i+1 /π i is the fraction of the oulation of age i that reaches age i +1.Thus: 1+r =(r (1 τ) /q +1 δ) P I i=1 N ia i π i+1. π i N i a i P I i=1 Note that for the case of constant survival robability, π i = π i 1, this equation becomes the more standard exression 1+r =(r (1 τ) /q +1 δ) /π. Government The government collects roortional taxes on caital and labor income to finance government exenditures, G, thatis,g = τ(wh + rk). Using (9) this equation can be written as G = τy. Moreover, a fraction 0 < ε < 1 of G is allocated to ublic education: N 1 = εg. Thus: = ετy/n 1. (10) Resource Constraints Gross outut, which includes undereciated caital, can be used for consumtion by all individuals (C), rivate education, government services, or to accumulate caital: Y +(1 δ) qk = C + N 1 e + G + qk 0. Next eriod caital stock, K 0, is the sum of individuals savings divided by q. Finally, aggregate human caital, H, is the sum of individuals human caital available for roduction. Definition of Equilibrium Let y = Y/N, k = K/N, and h = H/N. The following is our definition of equilibrium: Definition A stationary cometitive equilibrium for given taxes τ are rices w and r, quantities o n{c i,a i,h i }I i=1,s,e, h, k, y, andh, and subsidies such that (i) given h, rices, taxes, o and subsidies, n{c i,a i,h i }I i=1,s,e solves the individual roblem; (ii) given y, k and h, w and r satisfy w =(1 τ)(1 α) y/h, 1+r = α y k (1 τ) /q +1 δ ³ P I i=1 N ia i / P I i=1 (iii) h = h 2,h=(1 s /T ) n 1 h 1 + P 2 n ih i,qk= P I i=1 n ia i,y= Akα h 1 α,= ετy/n 1. 8 π i+1 π i N i a i ;

11 Note that in a stationary equilibrium: Ã h = (1 s /T ) n 1 θ 1 + X 2 θ i n i! h 2, and (11) h 2 = z (χ + s ) γ 1 ( + e ) γ 2. (12) 2.1 Characterization of the Equilibrium Individuals Problem We now characterize the solution to the individuals roblem given rices and olicies. Otimal choices may imly corner solutions. We denote ublic education the case in which e =0and s > 0, no formal education the case in which s =0and e > 0, mixed education the case e > 0 and s > 0, and no education the case e =0and s =0. The following roosition describes otimal educational choices. All roofs are in the Aendix. First define: eρ IX i=2 ρ i 1 π i ; m eργ 2 (1 + χ/t ) ; m (1 + eργ 2) χ/t 1 1+eργ 1 eργ 1 The following assumtion facilitates the resentation of the results (reduces the number of cases to resent), but lays no essential role in the derivations or in the quantitative work. Assumtion 1 eργψ χ/t. This assumtion guarantees that re-school is not too imortant as to make formal schooling unattractive. It is easy to check that under Assumtion 1 m>m> 0. Proosition 1 Given rices and olicies, the otimal solution for schooling years, s,andrivate exenditures in education, e,are: s = h eργ 1 T 1+eρ(γ 1 +γ 2 ) m 0 for m > wh i 1 for m eργ 1 T χ 1+eργ 1 for > m m, 9

12 e wh = 1+eργ 1 1+eρ(γ 1 +γ 2 ) eργ 2 / 1+eργ for m > h 2 m for m 0 for i > m m. Proosition 1 is illustrated in Figure 3. 3 It tyifies educational choices in three categories deending on the ratio of ubic subsidies to otential earnings during childhood, /( ).Informally, the roosition shows that lack of ublic funding may revent any formal schooling to take lace, and too much ublic funding may crowd-out comletely rivate investments in education. More formally, according to Proosition 1 the ublic education case arises when >m. In this situation, individuals do not invest resources but only time in formal education. Since the threshold level deends on labor income during childhood, the model redicts that oorer individuals would be more likely to choose only ublic education than richer individuals. The mixed education case occurs when the amount of ublic funding falls within an intermediate range (m /( ) m). Finally, the case of no formal education occurs when the ublic rovision of education is below certain threshold (/( ) <m). In this case, insufficient funding discourages formal schooling. Under Assumtion 1, the case no education cannot arise. Proosition 1 also has the following imlications for the role of ublic education on educational outcomes. First, additional ublic funding increases years of schooling only if m /( ) m. Otherwise, years of schooling do not resond to changes in ublic funding. An interretation of this result is that increasing ublic funding of education does not affect years of schooling of individuals already in ublic schools, although it enhances their human caital. Second, rivate funding resonds less than one to one to changes in ublic funding. Secifically, e/ < 1 whenever e>0. Thus, additional ublic funding crows out rivate funding but not comletely, desite ublic and rivate funding being erfect substitutes in the roduction function. The reason is that ublic funding relaxes borrowing constraints which hinder rivate funding in the first lace. 3 A useful result to check the continuity of the choices at the threshold levels is to note that: m m = 1+eρψ (γ 1 + γ 2 ) (1 + eρψγ 2 ) eρψγ 1 (eρψγ 1 χ/t ). 10

13 Proosition 1 also highlights the critical role of demograhic factors in educational choices. It states that individuals in economies with lower mortality rate, larger eρ, would choose more years of schooling and larger rivate funding of education for any given size of the ublic sector. As we see below, this mechanism turns out to exlain most of the differential in schooling across countries. Moreover, Proosition 1 also states that z, the learning ability arameter, does not affect the choice of schooling years or educational exenditures. Our model redicts that children do not self-select into a ure ublic, rivate, or mixed educational regime based on their learning ability. In articular, high ability children do not invest more years or resources in education than low ability children. Finally, Proosition 1 states that rivate funding of education deends on otential earning levels ( ) but years of schooling deends on the ratio of ublic education sending to earnings (/( )). An imortant imlication of this result, to be described below in detail, is that TFP differences would induce differences in educational exenses but not in years of schooling. In this regard, our model has different imlications from the models analyzed by MS and EKR, models in which years of schooling deend on TFP levels. Education in General Equilibrium The following roosition characterizes the educational choices in general equilibrium. For this urose, the ratio /( ) canbesolvedfromequations (10), (9) and (11) as: = P ετ (1 s/t + 2 θ in i / (θ 1 n 1 )). (13) (1 τ)(1 α) Note that ετ is the share of ublic education in outut (see equation 10). Substituting this result into Proosition 1 yields Proosition 2. Proofs are in the Aendix. First define: s H (1 τ)(1 α) m (1 + χ/t + m) / (1 + eρ)+ P 2 θ in i / (θ 1 n 1 ) ; s H (1 τ)(1 α) m 1+ P 2 θ in i / (θ 1 n 1 ) Proosition 2 The otimal solution for schooling years, s, and the share of educational exen- 11

14 ditures in outut, s H,aregivenby: s = 0 for s H > ετ (ετ s H)eργ 1 Tm/s H 1+eρ(γ 1 +γ 2 +ετγ 1 /((1 τ)(1 α))) for s H ετ s H eργ 1 T χ 1+eργ 1 for ετ > s H, s H = ετ + ετ + (1 τ)(1 α) 1 s /T + P 2 θ in i /(θ 1 n 1 ) (1 τ)(1 α) 1+ P eργ 2 ετm/s H 2 θ in i /(θ 1 n 1 ) 1+eργ 2 (s H ετ)(1+eργ 1 )m/s H for s H > ετ 1+eρ(γ 1 +γ 2 +ετγ 1 /((1 τ)(1 α))) for s H ετ s H ετ for ετ > s H. Proosition 2 is analogous to Proosition 1. It characterizes educational choices in terms of the size of ublic education. If the ublic rovision of education is too large no rivate funds are invested in education, while if it is too small then no schooling takes lace. Proosition 2 also states that years of schooling deend on demograhic arameters and on ublic olicy arameters but it is indeendent of the TFP level A. Asaresult,differences in TFP in our model are not amlified through years of schooling, as is the case in MS and EKR. Due to this feature our model ossesses a balanced growth ath, while MS s and EKR s do not. On the other hand, in all three models TFP differences roduce differences in the quality of schooling : larger TFP increases human caital through its effect on educational exenditures. Similarly, the share of educational exenditures in outut deends on demograhic and olicy arameters, but it is indeendent of TFP. For the cases in which s H > ετ, s H decreases with ετ but less than one-to-one. This is because lower taxes increase rivate wealth and therefore savings in the form of rivate educational investments, which artly offsets the fall in ublic education. Production in General Equilibrium Given the general equilibrium results for educational variables, s and s H,rovidedinProosition2,wecannowderive some imlications of the model for aggregate outut, and role of different arameters in exlaining outut differences across countries. 12

15 The following Proosition rovides the solution for the equilibrium level of outut. First, define Θ j (r) π jρ j 1 (1 + r) j Φ(r) Φ a (r) IX eρ n i i=2 j=2 IX i=2 IX i=2 θ i for j 2; i (1 + r) ix (1 + r) i j (θ j Θ j (r)) ; and π i+1 n i π i ix (1 + r) i j (θ j Θ j (r)) j=2 Proosition 3 The equilibrium level of outut is given by: y = A 1 (1 α)(1 γ 2 ) µ α µ 1 k (1 α)(1 γ 2 ) h 1 γ 2 y y γ (14) 2 where h y γ 2 Ã = (1 s /T ) n 1 θ 1 + X 2 θ i n i! z (χ + s ) γ 1 (s H/n 1 ) γ 2, (15) k y = (1 α)(1 τ) /q (1 s /T ) n 1 θ 1 + P 2 θ Φ(r), (16) in i r = ³ 1 δ + α (1 τ) y k /q Φ(r) Φ a (r) 1 (17) Equation (14) in Proosition 3 rovides the determination of aggregate and er caita outut in three comonents. The first comonent, A 1 (1 α)(1 γ 2 ) = A 1+ α 1 α + γ 2 (1 α)(1 γ 2), collects the direct and indirect effects of TFP on steady state outut. The elasticity of outut with resect to TFP, 1+α/ (1 α) +γ 2 / [(1 α)(1 γ 2 )], can be itself decomosed in three arts. The first art catures the direct effect of TFP on outut; the second art catures the indirect effect of TFP on outut through its effect on caital accumulation; and the third art catures the indirect effect of TFP on outut through its effect on human caital accumulation. This last effect has being recently stressed in the works of MS and EKR and its quantitative significance deends only the size of γ 2.Ifγ 2 =0, then equation (14) adots the more standard form used in the works of Hall and Jones (HJ, 1999) and Klenow and Rodriguez-Clare (KRC, 1997), among others. 13

16 The second comonent collects the direct and indirect effects of the caital-outut ratio in outut. According to equation (14), the elasticity of outut with resect to the caital-outut ratio, α/ [(1 α)(1 γ 2 )], isalsoaffected by γ 2. This elasticity can be written as α/ (1 α) + γ 2 / [(1 α)(1 γ 2 )]. The second term catures the effect that a higher caital-outut has on the accumulation of human caital and on outut. The third comonent, (h/y γ 2 ) 1 1 γ 2, is related to the accumulation of human caital. In a standard formulation with exogenous human caital (γ 2 =0), this comonent is just the stock of human caital. If human caital is endogenous, this comonent catures the effect of educational decisions, which themselves are function of demograhic and olicy arameters, but indeendent of TFP, as shown in equation (15). Equations (16) and (17) jointly determine the caital outut ratio, k/y, andthenetreturns r. Both variables deend on demograhic factors (π i,n i ), years of schooling s, and taxes τ. Imortantly, they are indeendent of TFP, articularly because s is indeendent of TFP as stated in Proosition 2. The ositive deendence of k/y years schooling is due to the fact that a larger fraction of students reduces labor suly, increases wages for all, in articular for adults, and therefore their savings. Thus, the effect of schooling on the term k/y catures an extensive margin, as more schooling reduces labor suly, while its effect on h/y γ 2 catures an intensive margin, as schooling increases adult s human caital. Mincerian Returns Psacharooulos and Patrinos (2002) have collected Mincerian returns for a fairly large set of countries. Matching these returns rovides a natural test for any quantitative model of schooling. Mincer returns, or simly returns to schooling, are obtained from a cross-section of individuals in each country using a OLS regressions of the tye: ln(d i )=a + bs i + cx i + v i (18) where d i are earnings of individual i, b is known as the Mincerian return, and X i is a vector of controls. In order to use these estimates, we need to derive the theoretical counterart of equation 14

17 (18) in our model. This requires to reformulate the model so that individuals in a given cohort are heterogenous in terms of earnings and schooling. Suose only for this section that each cohort is comosed of a cross-section of individuals that differ both in their learning ability, z i,andin their discount arameter eρ i. Yearly earnings during young adulthood, d i,aregiveninourmodel by d i = w (z i (χ + s i ) γ 1 ( + e i ) γ 2 ) /T. Alternatively, using Proosition 1: d i = ³ ³ γ2 w z i χ γ γ2 1 γ 1 T /T for m wh ³ ³ γ2 1 w z i (χ + s i ) γ 1 +γ 2 γ2 wh γ 1 T /T for m w (z i (χ + s i ) γ 1 γ 2 ) /T for m. m Collecting terms one can write this equation more comactly as: ln d i = a 1 + γ 1 ln (χ +0)+v 1i for m a 2 +(γ 1 + γ 2 )ln(χ + s i )+v 2i for m a 3 + γ 1 ln (χ + s i )+v 3i for m. m where the terms [a 1,a 2,a 3 ] collect all constant terms, and the variables v ji, j = {1, 2, 3}, collect the terms associated to z i. A linear aroximation of the schooling term on the right hand side of this equation around the steady state roduces: ln d i ' a 1 + v 1i for m a 2 + γ 1 +γ 2 χ+s s i + v 2i for m a 3 + γ 1 χ+s s i + v 3i for m. m (19) Since according to Proosition 1, cov(s, v) =0, then this exression rovides a theoretical counterart of equation (18) in our model. We use this exression below to assess the model s imlication for Mincer returns. 15

18 3 Data and Emirical Imlementation 3.1 Calibration We now calibrate the model to match certain features of the cross-country data. For this urose we assume that the arameters {A j,q j, π 2j,..,π Ij,f j, τ j, ε j } vary across countries while the arameters [α, δ, θ 1,.., θ I,T,γ, χ,z,ρ] are identical across countries. Individuals in the model are born at age 6 which roughly corresonds to the beginning of the schooling age in the data. Consistently, we set χ =6. Moreover, we consider 5 generations (I =5), and a time eriod length T =20. In this secification, formal schooling takes lace between the ages of 6 to 26, and individuals live u a maximum of 106 years. Parameter α is set to 1/3 as in Mankiw, Romer and Weil (MRW, 1992), HJ, KRC, Bils and Klenow (BK, 2000), and others. Parameter δ is set such that the annual dereciation rate is 0.06, which is the value assumed by HJ to comute the caital stocks that we use. This requires to set δ =1 (1 0.06) 20. Calibration of π j and f j We estimate π ij as follows. We take the US life table of total oulation from National Vital Statistics Reort by the US Census Bureau and, for each country j, we adjust the US age-secific mortality rates to reroduce the life exectancy of country j as reorted by the World Bank. Mortality rates for ages 0 to 5 are adjusted using the World Bank data on mortality rates under age 5. Mortality rates over ages 5 are multilied by a factor which allows to exactly relicate country s j life exectancy. {π ij } I i=2 is comuted, using the results of the table, as the number of survivors at ages 36, 56, 76, and96 resectively divided by the number of survivors at age 6. Fertility rates, f j, are comuted using equation (2). According to this equation, the share of ³ j oulation under 26 is given by n 1 =(fπ 2 ) 1 PI / j=1 π j (fπ 2 ). We use data from the World Poulation Prosects (2004) to comute n 1 for each country. 4 An imortant consideration is the year of the data used for these comutations. Since the late 70 s mortality rates have increased significantly, articularly in Sub-Saharan Africa, due to the 4 Given data restriction, our data on n 1 actually corresonds to oulation under

19 HIV/AIDS andemic. For this reason we choose 1980 demograhic variables, which are still not affected by the andemic and are likely more relevant to exlain economic erformance u to the mid 90 s when the andemic became more ronounced. We also conduct exeriments below to assess the long run consequences of the HIV/AIDS andemic. Figure 4 shows the demograhic data and the calibrated values of fertility and survival robabilities. A salient feature of the data is that survival rates are very different across countries and articularly low for very oor countries. The Figure also shows the estimated fertility rates relative to the US. They also decrease significantly with income levels, exhibit substantial difference across countries, and are articularly large for very oor countries. Calibration of τ and ε j In the model, G = τy. Accordingly, we set τ equal to the share of government exenditures in GDP in 1995 from the Penn World Tables. Furthermore, we comute ε j for each country as ε j = s Hj /τ j,wheres Hj is the share of ublic exenditures in education to GDP for 1995 according to UNESCO. Figure 5 shows the resulting tax rate, τ j, and Figure 2 shows subsidy er uil as a fraction of er caita GDP /y = s Hj /n 1. While taxes decrease slightly with income, subsidies clearly increase with income secially because the fraction of schooling age children is systematically larger in oor countries than in rich countries. Calibration of θ We calibrate θ =[θ 1,..θ 5 ] to reroduce the life-cycle attern of earnings. For this urose, we emloy the estimates of BK. Using cross-country evidence on earnings, they estimate that earnings at age i satisfy: w(i) =a(s) e m(ex), where ex stands for years of exerience and m(ex) = ex (ex) 2.Assuming0years of exerience for children, 20 for young adults, 40 for adults, and 60 for older adults, and no earnings for elders, we estimate θ as θ =[θ 1 = e m(0) m(20), θ 2 =1, θ 3 = e m(40) m(20), θ 4 = e m(60) m(20), θ 5 = 0] which roduces θ =[0.48, 1, 1.19, 0.80, 0]. 17

20 Calibration of γ 1, γ 2 and ρ To calibrate the key arameters γ 1, γ 2, and ρ we target moments of the cross-country distributions of returns to schooling and the average years of schooling. Returns to schooling are obtained from Psacharooulos and Patrinos (2002), while average years of schooling in each country are obtained from Barro and Lee for the year Following the derivations in Section 3, we comute returns to schooling as r s =(γ 1 + γ 2 ) / (χ + s) for a mixed educational system, r s = γ 1 / (χ + s) for a ure ublic system, r s =(γ 1 + γ 2 ) /χ for a re-school only system, and r s = γ 1 /χ for a no-schooling regime. The equilibrium educational system whether mixed, ublic, or other is obtained using the model (more recisely, Proosition 2) for different otential values of γ 1, γ 2,andρ. In these calculations, the values of other arameters are set at their calibrated values already described. Imortantly, the calibration of γ 1, γ 2,andρ is indeendent of the arameter values described below (A j and q j ) and therefore robust to alternative secification of those arameters. Table 1: Calibration of γ 1, γ 2 and ρ Data (1) (2) (3) (4) (5) (6) (7) (8) γ γ ρ s r s s std r sstd Table 1 reorts statistics related to schooling and returns to schooling from the model and the data for different ossible choices of γ 1, γ 2 and ρ. In these exeriments, γ 2 is allowed to vary (between 0 to 0.5) andγ 1 and ρ are chosen to match the average years of schooling (s) andthe average returns to schooling (r s ). The table illustrates different trade-offs in the choice of γ 2.For examle, γ 2 =0.30 roduces the best results in terms of the standard deviation of returns to schooling (0.042 in the data vs in the model), while γ 2 =0.45 roduces the best results in 18

21 terms of matching the standard deviation of schooling (2.89 vs 1.658). In both cases, however, the model falls short of exlaining all variation in returns and schooling. We choose the following arameters: γ 2 =0.40, γ 1 =0.722 and ρ =0.99. Calibration of q j Given arameters (α, δ,t,ρ, θ, τ j, π ij,f j ) already obtained, and years of schooling, s j, equations (16) and (17) can be used to solve for q j k j /y j and r j for each country. On the other hand, HJ rovide caital-outut ratios for a grou of countries for They comute caital stocks using annual investments valued at common international rices rather than at domestic rices. Therefore, their caital-outut ratio corresonds to that of k/y instead of qk/y, which allows domestic rices of caital, q j, to vary across countries. Given q j k j /y j redicted from the model and k j /y j from the data, q j can be obtained. 3.2 Assessment We now assess the ability of the model to fit evidence beyond the matching targets. Schooling Panel A of Figure 6 comares schooling attainments in the data and in the model. Although we only target the mean of schooling, the model is able to relicate the entire distribution of schooling fairly well. The correlation between schooling in the data and in the model is 86%. Years of schooling in the data range from a minimum of 0.69 to a maximum of 12.18, andinthe model they range between 1.73 and Finally, the correlation between schooling and relative incomes is 85.3% in the data, versus 82.8% in the model. Returns to Schooling Panel B of Figure 6 and Table 2 comare returns to schooling in the data and the model, as well as the redictions of two alternative models. The first model is that of HJ who estimate returns to schooling as 13.4% for first four years of schooling, 10.1% for the next four years, and 6.8% for the remaining years. The second model is one of the versions in BK their intermediate version with moderate decreasing returns to schooling. In this version, returns to schooling satisfy the exression r s =0.18 s

22 Table 2: Returns to Schooling Data Model HJ Model BK Model mean(r s ) stdev(r s ) correl(r s,r smodel ) correl(y,r s ) min(r s ) max(r s ) Our model is able to reroduce the overall attern of decreasing returns as income level increases, and 71% of the disersion of returns. Similarly to the other two models, our model overredicts the correlation between returns and relative incomes and underredicts the disersion of returns. However, it is encouraging that our model is comarable to alternative models desite the fact that our calibration only targets the mean of returns, while the cometing models target the overall attern of returns. School Regime and School Exenditures Panel C of Figure 6 shows the redicted equilibrium educational system mixed, ublic, etc. for countries according to their relative income levels, and Panel D shows observed ublic, and both redicted rivate and total shares of educational exenditures in GDP. The model redicts that most countries (82 out of 91) adot a mixed system, and 9 countries adot a ure ublic system. For a subsamle of 53 countries for which informationonrivateeducationexendituresisavailable from the World Bank, the model redicts that a significant fraction of total education exenditures, 65.4% on average, corresonds to ublic education. This share is 77.5% in the data. For OECD countries, ublic education is on average 85.7% of total education exenditures, and the model redicts it is 83%. For the US, ublic education is 74.8% of total sending in education, and it is 87% in the model. Finally, for the same subsamle of 53 countries, the correlation between total education sending er uil as a fraction of GDP er caita and GDP er caita relative to the US is 62.7% in the data, and 75.6% in the model. For the case of rivate education sending er uil as a fraction 20

23 of GDP this correlation is 22.7% in the data and 38.4% in the model. 4 Human Caital Stocks 4.1 A Cross-Section In this section we use the model to estimate human caital stocks for a set of 91 countries and comare the results to some existing alternatives. According to equations (11) and (12), human caital stocks are given by: Ã h j = (1 s j /T ) n 1j θ 1 + X 2 θ ij n ij! z (χ + s j ) γ j + e j 1 γ. We use this formula to comute human caital stocks relative to the US, which eliminates the need to estimate z, a arameter assumed to be identical across countries. Notice that once the technological arameters are obtained, this formula could ma observables n ij, s j and j + e j into human caitals. However, due to the lack of reliable data on rivate education exenditures we utilize the model s redictions on total exenditures to comute human caital stocks. We also reort the results when only available information on ublic exenditures is emloyed. Figure 7 and Table 3 characterize six different estimates of human caital stocks: (1) those from our model, denoted CR; (2) our model but using only ublic exenditures of education (CR-P); (3) Hendricks (2002) estimates based on immigrants earnings in the US and no self-selection 5 ; (4) a version of BK that uses the formula h = e f(s) (age s 6) (age s 6)2,wheref(s) = (0.18/0.72) s These estimates are similar to those obtained by HJ; (5) MRW s estimates based on schooling (here we use rimary lus secondary schooling as argued by KRC); 5 Using Hendricks notation, we comute human caital in country c as w η c /w k c using the information in his Table 1. This measure of human caital includes both measured and unmeasured skills. See discussion in Section III.A in Hendricks (2002). 21

24 (6) Cordoba and Rioll s (CR, 2004) estimates based on rural-urban wage gas. Panel A of Figure 7 and the first two columns of Table 3 characterize the human caital estimates obtained with our model using either total educational exenditures or ublic exenditures only. Both estimates are highly correlated between them (correlation of 0.98) and with relative incomes. Moreover, relative human caitals are systematically higher than relative incomes: while the average relative income is 0.31 of the US, the average relative human caital is 0.42 or 0.39 of the US, deending on what exenditures are used. Although the overall roerties of both estimates are similar, there are imortant country secific differences articularly for high income countries. For examle, Hong Kong s and Singaore s relative human caital fall substantially when only ublic exenses are considered. Given that according to UNESCO, 93% of rimary and 89% of secondary enrollment in Hong Kong is in rivate institutions, it seems clear that some significant imutation for rivate exenditures is required. 6 For this reason, we favor our human caital estimates that include imuted values for rivate exenditures in education. Table 3: Estimates of Average Human Caital Relative to the US CR CR-P Hendricks BK MRW CR2004 (1) (2) (3) (4) (5) (6) mean stdev min max corre(h, y) Panels B and C of Figure 7 and columns 3 to 5 of Table 3 comare our human caital estimates to those of BK, Hendricks (2002) and MRW. They are similar for high income countries but differ substantially for low income countries. Thus, while BK and Hendricks estimates imly relative minor differences in human caital between rich and oor countries, our estimates, as well as MRW s estimates, imly substantial differences. The means of the estimates are 0.75 for Hendricks, Unfortunately, UNESCO does not reort enrollment for Singaore or Taiwan. 22

25 for BK, 0.48 for MRW, and 0.42 for our model. Moreover, our estimates exhibit significant more disersion than the alternatives. A further comarison is to the estimates of Cordoba and Rioll (2004). They construct human caital stocks for a set of countries weighting rural and urban estimates. They estimate urban stocks using a standard Mincer aroach (as in HJ) while rural stocks are Mincer tye estimates adjusted to account for the observed rural-urban wage ga in each country. Panel D of Figure 7 and the last column of Table 3 comares our estimates to those of CR (2004). The 2004 estimates are midway between our current estimates and those of BK, but both estimates rovide a substantial revision of human caital stocks relative to revious estimates, and they are at some extent similar to those obtained by just taking average years of schooling relative to the US. Finally, we can comare our human caital stocks with those of MS. Since they reort average human caital relative to the US by income deciles, we construct equivalent measures for our estimates in Table 4. As shown in the table, our human caital estimates are consistently below MS s, roortionally more so for those countries below the 50th ercentile. For instance, the average country in the 50th 60th ercentile has 60% of US s human caital according to MS, while it has only 31% of US s human caital according to our estimates. The only excetion are those countries in the lowest decile: they have 10% of US s human caital according to our estimates, and 8% according to MS s. Table 4 also reorts BK s human caital relative to the US. We use BK s estimates to comute the imlied quality differentials in human caital across deciles for both our model and MS s (see last two columns). In articular, the ratio between our human caital and BK s can be interreted as the relative human caital quality. As shown in Table 4, quality differences exists at all income decilesinourestimates,whileinmstheyareonlyrelevantforcountriesbelowthe40th ercentile. Secifically, while the average country in the 40th 50th ercentile range has about the same human caital quality of the US according to MS, it has around 1/2 of the quality according to our estimates. As noted above, the excetion is in the lowest decile. Our human caital estimates imly that a country in the lowest decile of the income distribution has a human caital quality 23

26 around 1/3 of the US, while it is around 1/4 according to MS. In conclusion, our human caital estimates imly quantitatively imortant adjustments, roortionally more so for countries below the 50th ercentile of the world income distribution. 7 Table 4: Estimates of Average Human Caital Relative to the US by Decile CR MS BK Quality index Decile (1) (2) (3) CR MS Self-Selection of Immigrants To assess how lausible our estimates are, remember that Hendricks estimates reorted in Table 3 are not adjusted for ossible self-selection of immigrants, while BK s estimates are not adjusted for quality differences. Can the differences between Hendricks s estimates and ours be exlained by self-selection? Figure 8 Panel A lots the ratio of Hendricks estimates to our estimates for the subset of 62 common countries (under the name CR). This ratio sans from 0.93 to 4.25 and has an average of This is a much lower degree of self-selection of what Hendricks (2002) argued was imlausible. In articular, Hendricks argued against the degree of self-selection imlied in his Figure 2, which sans from 1 to 12. Such large degree of self-selection would exlain all 7 At this oint we are unable to comare our human caital estimates to those of EKR, since they do not reort their estimates. 24

27 cross-country income differences without TFP differences. In contrast, our series require Mexican immigrants, the main source of immigration to the US, to have only around 50% more human caital than non-immigrants which is just a small fraction of the income ga between the two countries (of 4.36 times). Our series imly that on average immigrants have 90% more human caital than non-immigrants, while the income ga on average is times. Moreover, our 2004 estimates imly even much lower degrees of self-selection than our current estimates, as shown in Figure 8 Panel B (33% on average and 19% for Mexican immigrants). Table 5: Selection of Immigrants Uer Bound (1) (2) (3) (4) (5) (6) (7) (8) γ γ ρ ss 3.070* ss max 13.06* ss min 0.690* ss mex ss 0.810* ss max 0.999* ss min 0.320* ss mex 0.760* ss = self-selection (Hendricks human caital/our human caital), ss = average self-selection, ss = ercentile osition of immigrant, ss = average ercentile. Following Hendricks (2002, g 208-9), we can also estimate the osition of immigrants in the earnings distribution of the source country imlied by our model by assuming that earnings are lognormal distributed, and using the Klaus Deininger and Lyn Squire s (1996) Gini coefficients 25

28 for income. 8 Figure 8B describes these results. The series denoted Hendricks is the uer bound he rovided. Our series imly much lower degree of self-selection than the uer bound for most countries. Table 5 reorts additional exercises for the different arametrization described in Table 1. Thus, for examle, our benchmark arametrization (γ 2 =0.4) imlies that the tyical Mexican immigrant is drawn from the 65 ercentile of the earnings distribution. This is consistent with the findings of Chiquiar and Hanson (2005), who find intermediate ositive self-selection of Mexican immigrants. They find that the tyical immigrant is drawn from the 72 ercentile. Table 5 also illustrates that γ 2 =0.35 or γ 3 =0.4roduce the best results in terms of intermediate self-selection, which lends further (and strong) suort to our benchmark calibration. 5 The Sources of Income Differences Develoment Accounting and the TFP Elasticity of Income A key arameter in our model is γ 2. Our assessment of the revious section lends suort to our estimate of γ 2 as it gives rise to lausible redictions in terms of schooling, returns to schooling, and human caital estimates. This arameter controls, among other things, the extent u to which TFP differences exlain cross country income differences. As stated in Equation (14), the elasticity of outut to TFP in our model is given by α 1 γ 2 =(1.5) (1.6) = 2.4. Thisisasignificantly larger elasticity than what is obtained in model with exogenous human caital (of only 1.5). This elasticity is remarkably similar to the EKR estimate of 2.8 desite the substantially different modeling aroaches. A crucial difference is that our model exlains schooling differences mostly in terms of demograhic differences, while EKR abstract from demograhic differences and rely on TFP differences only. On the other hand, our elasticity is much lower than the MS estimate of around 9. Theirlarge elasticity imlies that almost no differences in TFP are required to exlain income differences, while imortant TFP differences are still required in our model. There are two differences in our modelling aroaches that hel exlain the different results. First, TFP differences affect the steady ³ 8 σ The Gini coefficient of a log-normal distribution satisfies G =2Φ 2 1 where Φ is the standard normal distribution. Thus, σ = 2Φ 1 1+G 2. 26