The welfare implications of resource allocation policies under uncertainty: The case of public education spending

Journal of Macroeconomics 33 (2011) 176–192

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The welfare implications of resource allocation policies under uncertainty: The case of public education spending Konstantinos Angelopoulos a,1, Jim Malley a,⇑, Apostolis Philippopoulos b,c,2 a

Department of Economics, Adam Smith Building, University of Glasgow, Glasgow G128RT, United Kingdom Department of Economics, Athens University of Economics and Business, 76 Patission Street, 10434 Athens, Greece c CESifo, Poschingerstr. 5, 81679 Munich, Germany b

a r t i c l e

i n f o

Article history: Received 14 December 2009 Accepted 30 November 2010 Available online 13 December 2010 JEL classification: E62 H30 H52 Keywords: Fiscal policy rules Economic fluctuations Public education spending

a b s t r a c t In this paper, we examine whether policy interventions, aimed at improving resource allocation, also have important stabilization effects over the business cycle. To this end, we employ a dynamic stochastic general equilibrium model in which public education expenditures, financed by distorting taxes, enhance the productivity of private education choices. We then calculate the welfare implications of competing operating targets using a statecontingent instrument rule for public education spending. Our main findings are: (i) there can be important cyclical effects of different resource allocation policies depending on the operating target used and the degree of macroeconomic uncertainty; (ii) it is important to use an operating target which is as close as possible to the heart of the market imperfection that justifies policy action; (iii) policy action should not be monotonic in the degree of macroeconomic uncertainty. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction It is widely recognized that education matters for growth (see e.g. Barro and Sala-i-Martin, 2004, chapter 12, and Aghion and Howitt, 2009, chapter 13). Irrespective of the economic motivation, e.g. externalities from economy-wide human capital, provision of equal opportunities, considerations relating to the distribution of income, etc., public education spending is a long-established political and economic reality in all countries. In 2005, the World Bank reported public education spending shares of GDP and of total public spending respectively of (5.9, 15.2) for the US; (4.7, 9.7) for Germany; (3.7, 10.7) for Japan; and (5.5, 11.9) for the UK. The recent 2008–2009 crisis has added new calls for even higher public spending on education. Most governments adopted big fiscal stimulus packages to mitigate the recession, and education has been a priority area in these packages. Among other arguments, a main one is that investing in education is necessary to offset the significant loss in potential output caused by the recent crisis. Indeed, in the US, public education spending is one of the strategic priorities of the new American administration, which has stated that it will focus spending on infrastructure, energy, education, health and support for the poor (see e.g. Feldstein, 2009). In particular, the 2009 American Recovery and Reinvestment Act dedicated more than $100 billion for pre-school, K-12, and higher education. In the EU, there is also ‘‘a strong case for immediately developing an ambitious strategy targeting the key drivers of potential growth’’ and this ‘‘requires considerable investments in ⇑ Corresponding author. Tel.: +44 141 330 4692; fax: +44 141 330 4940. E-mail addresses: [email protected] (K. Angelopoulos), [email protected] (J. Malley), [email protected], offi[email protected] (A. Philippopoulos). 1 Tel.: +44 141 330 5273; fax: +44 141 330 4940. 2 Tel.: +30 210 8203357, +49(0)89 9224 1410; fax: +30 210 8203301, +49(0)89 9224 1409. 0164-0704/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2010.11.007

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infrastructure, human capital, R&D and innovation’’ (see ECFIN, European Economic Brief, issue 3, June 2009). Pissarides (2010) has also recently argued for the suitability of post-compulsory education as a ‘‘counter-cyclical policy tool’’. When governments intervene in the economy, they either explicitly or implicitly follow state-contingent policies. Thus, they use their policy instruments to respond to operating targets or indicators, where the latter reflect the actual or expected state of the economy.3 For instance, in the stabilization policy literature, policy instruments react to the cyclical position of the economy as measured by the gap between current and long-term values of operating targets. Or, in the resource allocation policy literature, the government may react to indices of resource misallocation. For example, in the context of our research, in the presence of positive externalities from economy-wide human capital, policymakers can use public education spending to react to the gap between social and market returns to education implying that public education spending would be a positive function of this gap. When implementing such state-contingent policy rules, a problem is that there are many possible operating targets to choose from. Indeed, in the context of a particular model of the economy, any endogenous variable might be considered. Hence, to compare different operating targets, or to compare different instrument rules in general, a welfare ranking should generally be employed, where welfare is typically measured as the present discounted value of households’ expected lifetime utility. In the macroeconomic stabilization policy literature, the welfare implications of alternative instruments and/or operating targets is generally handled in a stochastic framework (see e.g. Andrés and Doménech, 2006; Schmitt-Grohé and Uribe, 2007 and Malley et al., 2009). In contrast, the public economics literature has typically evaluated the welfare implications of policies aimed at improving resource allocation using a deterministic setup (see e.g. Atkinson and Stiglitz, 1980, for the early literature and Turnovsky, 2000, for optimal resource allocation policy in a growth setup). To bridge these literatures, we propose to examine whether public education policy, aimed at improving resource allocation, can also have important cyclical effects over the business cycle or off steady-state. Consistent with the macroeconomic policy stabilization literature, we propose to examine the welfare (allocative and stabilizing) implications of competing operating targets, used as indicators of the state of the economy, in a state-contingent instrument rule for public education spending. To achieve our objectives, we employ a standard dynamic stochastic general equilibrium model, in which public education spending can stimulate the engine of long-term or endogenous growth, human capital accumulation. The model is essentially a variant of the Lucas (1988, 1990) and the Tamura (1991) deterministic models, augmented to allow for public education spending in a stochastic environment. Hence, we examine the welfare effects of public education spending in an economy that fluctuates around its balanced growth path, as is hit by stochastic shocks to productivity. To be in a position to realistically assess the effects of public education expenditure, we assume that it is financed by a distorting tax on income. Our empirical base of departure for our model calibration is the post-war US economy. To illustrate the importance of uncertainty and hence the welfare effects of allocative policies off steady-state, we examine competing state-contingent policy rules. These rules require that the government reacts actively in each time period to the gap between the private allocation and the social planner’s allocation, in an effort to bring market outcomes closer to their socially optimal values. In a non-stochastic world, when the degree of activism is optimally chosen, all active rules imply the same welfare gain over the benchmark case in which public spending on education is exogenously set to its historical average. In contrast, when we evaluate welfare in a stochastic setup, the results can change quite markedly. In such a setup, different allocative rules, in which the share of public education spending reacts to different operating targets, result in optimal policies that are no longer welfare equivalent. The best rule, in our experiments, is the one in which the government reacts to the gap between the market value of human capital growth and its social planner value. Nevertheless, in general, fiscal action comes at a cost: changes in public education spending can destabilize the economy in periods of increased uncertainty. Therefore, the policy messages arising from our analysis is that: (i) there can be important cyclical effects of different resource allocation policies depending on the operating target used and the degree of macroeconomic uncertainty; (ii) it is important to use an operating target which is as close as possible to the heart of the market imperfection that justifies policy action; (iii) policy action should not be monotonic in the degree of macroeconomic uncertainty. The rest of the paper is organized as follows. Section 2 presents the theoretical model. Section 3 discusses the data, calibration and long-run solution. Section 4 contains the results and Section 5 the conclusions. Finally, Appendix A presents the derivation of welfare, the social planner’s solution. 2. Theoretical model In what follows, we first set out the structure of the model by describing the behavior of households and firms and public education spending policies.4 The latter are justified by positive externalities generated by the economy-wide human capital stock. We then solve the second-order approximation of our model’s equilibrium conditions around the deterministic steadystate (see e.g. Schmitt-Grohé and Uribe, 2004) to aid in the evaluation of welfare. In contrast to solutions which impose certainty 3 The arguments for state-contingent instrument rules rely on their simplicity and transparency (for reviews, see e.g. McCallum (1999) and Walsh (2003a, chapters 9 and 11)). The most well-known rule is the Taylor rule for interest rate policy. 4 For a similar model setup aimed at identifying the optimal share of public education spending rather than welfare-evaluating alternative state-contingent policy rules, see Angelopoulos et al. (2008).

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equivalence, the solution of the second-order system allows us to take account of the uncertainty agents face when making decisions. 2.1. Households The economy is populated by a large number of identical households indexed by the superscript h and an equal number of identical firms indexed by the superscript f, where h, f = 1, 2, . . . , Nt. The population size, Nt, evolves at a constant rate n P 1, so that Nt+1 = nNt where N0 is given. Each household’s preferences are given by:

E0

1 X

h

bt UðC ht ; lt Þ

ð1Þ

t¼0 h

where E0 is the conditional expectations operator, C ht is consumption of household h at time t; lt is h’s leisure at t, and h 0 < b < 1 is the subjective rate of time preference. The instantaneous utility function, U ht ¼ UðC ht ; lt Þ, is increasing in all its arguments, concave and satisfies the Inada conditions. Specifically, we use a Cobb–Douglas form:

h U ht ¼

ðC ht Þl ðlt Þ1l h

i1r

1r

ð2Þ

where, 1/r(r > 1) is the intertemporal elasticity of substitution between consumption in adjacent periods and 0 < l < 1 is the weight given to consumption relative to leisure. At each date, household h consumes C ht , invests X ht in the production of its own human capital and saves Iht in physical capital. It receives interest income, rt K ht , where rt is the return to capital and K ht is the beginning-of-period physical capital h h stock. The household allocates one unit of time to leisure, lt , work, uht and education, eht , so that lt þ uht þ eht ¼ 1. A household h h h with a stock of human capital, Ht , receives labor income, wt ut Ht , where wt is the wage rate and uht Hht represents effective labor. Finally, each h receives dividends paid by firms, Pht , and an average lump-sum transfer/tax, Got , from the government. Thus, the budget constraint of household h is:

C ht þ X ht þ Iht ¼ ð1  st Þðr t K ht þ wt uht Hht þ Pht Þ þ Got

ð3Þ

where 0 < st < 1 is the distortionary income tax rate. Each household’s physical and human capital evolve according to:

K htþ1 ¼ ð1  dp ÞK ht þ Iht

ð4Þ

et Hhtþ1 ¼ ð1  dh ÞHht þ ðeht Hht Þh1 ðX ht Þh2 ðHt Þ1h1 h2 B

ð5Þ

and

where, 0 6 dp, dh 6 1 represent depreciation rates on physical and human capital respectively. The second term on the r.h.s. 0 0 of (5) is the quantity of ‘‘new’’ human capital created at time t. In particular, eht Hht is h s effective time at school, X ht is h s prie h3 e vate expenditure on education, Ht is the average human capital stock in the economy and B t  Bðg t Þ , where B > 0 is a constant scale parameter and g et is average public education expenditure expressed in efficiency units (see below for further detail). Finally, 0 < h1 < 1, 0 < 0 < h2 < 1 and 0 6 h3 < 1 are constant parameters.5 The assumption that individual human capital accumulation is an increasing function of both private and public expenditure on education nests several models in the literature and reflects the idea that public spending applies more to primary and secondary education, while private spending applies more to college education and on-the-job training (see e.g. the discussions in Blankenau and Simpson, 2004, p. 586, and Arcalean and Schiopu, 2010, p. 606). Jones et al. (1997) and Jones et al. (2005) use private education spending only, while the inclusion of public education spending, g et , is consistent with policy practice, as well as with previous theoretical work (see e.g. Glomm and Ravikumar (1992), Blankenau and Simpson (2004), Su (2004), Blankenau (2005) and Arcalean and Schiopu, 2010, who also provide a recent review of the literature on private versus public education spending). The assumption that individual human capital accumulation is an increasing function of the per capita level of economywide human capital, Ht , captures the popular idea that the existing know-how of the economy provides an external positive effect (see also e.g. Lucas, 1988; Azariadis and Drazen, 1990; Tamura, 1991; Glomm and Ravikumar, 1992 and Angelopoulos et al., 2008). In this environment, one agent’s return to human capital is positively affected by the human capital of other agents in the society. Hence, decentralized decision-making leads to a growth rate of human capital that is inefficiently low. These externalities can in turn justify public expenditure on education (see also e.g. Atkinson and Stiglitz, 1980 and Hanushek, 2002). 5 Note that the parameter restrictions employed in (5) imply increasing returns to scale (IRS) at social level. Lucas (1988) and Benhabib and Perli (1994) are examples of other studies which employ the IRS assumption in either or both the physical and human capital production functions. In general, some restrictions on returns to scale are unavoidable to obtain a stationary equilibrium (see below).

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Given the above setup, households act competitively taking prices, policy and aggregate outcomes as given. Thus, each h h h h h chooses fC ht ; X ht ; Iht ; lt ; uht ; eht ; K htþ1 ; Hhtþ1 g1 t¼0 to maximize (1) subject to (3)–(5), the time constraint lt þ ut þ et ¼ 1 and inih h tial conditions for K 0 and H0 (see Appendix A for details). 2.2. Firms Each firm f produces Y ft using physical capital, K ft , and effective labor, uft Hft . The production function is:

Y ft ¼ At ðK ft Þa ðuft Hft Þ1a

ð6Þ

where At is the level of Hick neutral technology and 0 < a < 1 is a parameter. Given this setup, firms act competitively taking prices as given. Thus, each f chooses K ft and uft Hft to maximize (see Appendix A for details):

Pft ¼ Y ft  r t K ft  wt uft Hft

ð7Þ

2.3. Government budget constraint Total income tax revenues finance total expenditure on public education, Get , and lump-sum transfers/taxes, Got . Thus,6

Get þ Got ¼ st

Nt X ðr kt K ht þ wt uht Hht þ Pht Þ

ð8Þ

h¼1

where, among the three policy instruments ðGet ; Got ; st Þ, we choose the income tax rate, st, to be the residually determined one. Note that, when we calibrate the model, the inclusion of Got will make the residually determined value of the income tax rate correspond to the rate which exists in the data. This will allow for a realistic assessment of the trade-offs between increased spending on public goods versus increased distortions due to higher tax rates. 2.4. Stationary decentralized competitive equilibrium A decentralized competitive equilibrium (DCE) is defined to be a sequence of allocations fC t ; X t ; It ; lt ; ut ; et ; K tþ1 ; 1 1 Htþ1 g1 t¼0 , prices fr t ; wt gt¼0 and the tax rate fst gt¼0 such that households maximize utility, firms maximize profits, markets clear and the government budget constraint is satisfied in each period. This is given the paths of two policy instruments, h h fGet ; Got g1 t¼0 , and initial conditions for the state variables, ðK 0 ; H0 Þ. Market-clearing values will be denoted without the superscripts h, f. Since the model allows for endogenous growth, we transform quantities to make them stationary. We first define per capita quantities for any variable Z as Z t  Z t =N t , where Z t  ðY t ; C t ; X t ; It ; K t ; Ht ; Get ; Got Þ, and then express these quantities as shares of per capita human capital, e.g. zt  Z t =Ht , while the gross human capital growth rate is defined as ct  Htþ1 =Ht . We thus have:

yt ¼ At ðkt Þa ðut Þ1a

ð9Þ

p

nct ktþ1  ð1  d Þkt þ ct þ xt þ h

h1

h2

nct ¼ 1  d þ Bðet Þ ðxt Þ

g et

¼ yt

ðg et Þh3

ð1  lÞct lð1  aÞð1  st Þyt ¼ ð1  et  ut Þ ut

ð10Þ ð11Þ ð12Þ

ð1  lÞet lh1 xt ¼ ð1  et  ut Þ h2 c t   ð1  stþ1 Þaytþ1 kt ¼ bðct Þlð1rÞ1 ktþ1 1  dp þ ktþ1 n h io lð1rÞ1 wt ¼ bðct Þ ktþ1 ð1  aÞð1  stþ1 Þytþ1 þ wtþ1 1  dh þ h1 Bðetþ1 Þh1 ðxtþ1 Þh2 ðg etþ1 Þh3

ð14Þ

g et

ð16Þ

þ

g ot

¼ st y t

ð13Þ

ð15Þ

where, kt and wt are the transformed shadow prices associated with (3) and (5) respectively in the household’s problem and are given by7: 6 To focus on public education, we abstract from other common types of government spending like public investment in infrastructure, utility-enhancing public consumption or redistributive transfers. Also note that Eq. (8) is as in e.g. Baxter and King (1993), in the sense that we use a balanced budget. Ignoring public debt is not important here because lump-sum taxes/transfers are equivalent to debt financing in this class of models. See also Angelopoulos et al. (2008). lð1rÞ1 lð1rÞ1 7 In particular, kt  Kt =Ht and wt  Wt =Ht where h-superscripts are omitted. See Appendix A for details.

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K. Angelopoulos et al. / Journal of Macroeconomics 33 (2011) 176–192

kt ¼ lðct Þlð1rÞ1 ð1  et  ut Þð1lÞð1rÞ lð1rÞ

wt ¼

ð1  lÞðct Þ

ð1  et  ut Þ

ð17Þ

ð1lÞð1rÞ1

ð18Þ

Bt h1 ðet Þh1 1 ðxt Þh2 ðg et Þh3

The stationary DCE is summarized by the above system of ten equations in fct ; yt ; ct ; xt ; ut ; et ; ktþ1 ; kt ; wt ; 1 the exogenous policy instruments, fg et ; g ot g1 t¼0 , and productivity, fAt gt¼0 .

st g1 t¼0 given

2.5. Fiscal instruments and technology We next specify the processes governing the evolution of the exogenous fiscal policy instruments and technology. We assume that public education spending as share of output follows a feedback policy rule. This rule consists of an exogenous component, as well as an endogenous or state-contingent part designed to react to resource misallocation:

g et ¼ g e0 ðz  zt Þf yt

ð19Þ

where, 0 6 g e0 6 1 is an exogenous constant public education to output share, zt is an indicator of the current state of the economy, z⁄ is the value of this indicator in the associated social planner’s long-run solution,8 and 0 6 f is a feedback policy coefficient. Obviously, there are many possible candidates for zt. For instance, we could choose zt to be any endogenous variable in the DCE. We discuss below (see Section 4) the rationale for considering a subset of the model’s endogenous variables. Consistent with the public economics literature, the motivation for the state-contingent part of the fiscal rule is that governmental intervention in education is primarily justified by the presence of positive externalities from the economy’s aggregate human capital. In the presence of such externalities, human capital accumulation is inefficiently low and so the government steps in to reallocate scarce public resources to education and improve efficiency. If the market and social plange ner solutions coincide, i.e. z⁄ = zt, then no public education spending can be justified and yt ¼ 0. As f increases, the governt ment reacts more to the gap between private and social planner’s outcomes, by spending more on education. To obtain a common base for comparisons using the different policy rules, we normalize public education spending to be the same in all regimes under no feedback policy, i.e. when f = 0 (see Angelopoulos et al., 2008, for this special case). In particular, we set the constant term in the rule, g e0 , to be the average of public spending on education as a share in GDP in the data. This implies that under no reaction, all the policy rules that we consider imply the same welfare since the benchmark underlying economy is the same in all cases. Hence, given the calibration of the constant term and the choice of the target zt, the optimal choice of f will determine the welfare maximizing public education spending as a share of GDP for each policy rule that we consider. By relating the policy instrument to the observed value of some endogenous variable, we implicitly exploit the information contained in observations of the endogenous variable in question. If the objective is to design the best policy, there should be no reason to restrict policy to respond only to one endogenous variable, and not to a combination of two, or more, or all, endogenous variables, unless observations of that one variable contain all of the available information relevant to achieving the policy objective (see the discussions in Friedman, 1990, pp. 1210–1212, and Walsh, 2003a, chapter 9). To make the results clearer, we focus on policy rules that involve reactions to one endogenous variable at a time and compare results across different endogenous variables. The question of what further role additional endogenous variables can play is an empirical issue. Concerning lump-sum taxes/transfers, we simply assume:

g ot ¼ g o0 yt

ð20Þ

where, 0 6 g o0 6 1 is fixed at a constant share. Finally, following the RBC literature we assume that technology, At, follows an AR(1) process: a

a

At ¼ Að1q Þ Aqt1 eet

a

ð21Þ a

where A > 0 is a constant, 0 < q < 1 is the autoregressive parameter and e  iidð0; r a t

2 aÞ

are the random shocks to productivity.

3. Data, calibration and solution The model’s structural parameters relating to preferences, production and physical and human capital accumulation are next calibrated using annual post-war data for the US. As our aim is to use the model to evaluate welfare as approximated 8 The social planner’s or policy benchmark solution (hereafter SP) is defined to be the case in which a planner solves the problem by internalizing externalities. This implies that in the steady-state, z⁄ > z. Note that our quantitative results reported below change very little (i.e. at the fourth decimal place) when the target value is not the long-run social planner solution, z⁄, but the current one, zt . Hence we use z⁄ for simplicity. Finally note that we experimented with one period lags of z rather than the current one but found that our results are basically unaffected. As above, the differences show up at the fourth decimal place.

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around the steady-state, it is important that the calibrated parameters imply a sensible long-run solution. This provides the criterion for choosing those parameters we cannot retrieve from the data or previous empirical studies, especially the exponents in the production function for human capital. Our base calibration is reported below and we then follow this, in the next section, with an exercise which searches for the values of the reaction coefficients in (19) which maximize welfare across different rules. 3.1. Data To calibrate the model, we need data for the endogenous variables as shares of human capital. Thus it is important to obtain a measure of human capital that is comparable to monetary valued quantities such as consumption, income, physical capital and government spending. To obtain this, and working as in Angelopoulos et al. (2008), we use data from Jorgenson and Fraumeni (1989, 1992a,b) on human and physical capital.9 The basic idea used in the construction of this dataset is that the output of the education sector is considered as investment in human capital. Jorgenson and Fraumeni (1989) note that ‘‘in order to construct comparable measures of investment in human and nonhuman capital, we define human capital in terms of lifetime labor incomes for all individuals in the US population. Lifetime labor incomes correspond to asset values for investment goods used in accounting for physical or nonhuman capital’’ . Jorgenson and Fraumeni’s calculations reveal that the monetary value of the stock of human capital is very large. In particular, it is about 15 times bigger than the (similarly calculated) stock of physical capital, and about 63 times larger than output. The additional (annual) data required for calibration include output (GDP), consumption, government spending on education, private spending on education, compensation of employees, GDP deflator, long-term nominal interest rates, labor force, effective average tax rates on labor and capital income, hours worked and years of education in the labor force. These are obtained from the following sources: (i) Bureau of Economic Analysis (NIPA accounts); (ii) OECD (Economic Outlook database); (iii) US Department of Labor, Bureau of Labor Statistics (BLS); (iv) ECFIN Effective Average Tax Base (see Martinez-Mongay, 2000) and (v) Ho and Jorgenson (2000). 3.2. Calibration The numeric values for the model’s parameters are reported in Table 1 and the long-run solution they imply is presented in Table 2. To calibrate the model, we work as follows. 3.2.1. Labor’s share, discount rate, leisure and population growth We set the value of (1  a) equal to labor’s share in income (i.e. 0.578) using compensation of employees data from the OECD Economic Outlook. This figure is similar to others used in the literature, see e.g. Lansing (1998). Given labor’s share, capital’s share, a, is then determined residually. The discount rate, 1/b is equal to 1 plus the ex-post real interest rate, where the interest rate data is from the OECD Economic Outlook. This implies a value 0.964 for b. Again this figure is similar to other US studies, see e.g. King and Rebelo (1999), Lansing (1998) and Perli and Sakellaris (1998). Following Kydland (1995, chapter 5, p. 134), we set (1  l), the weight given to leisure relative to consumption in the utility function, equal to the average value of leisure versus work time for the working population, which is obtained using data on hours worked from Ho and Jorgenson (2000).10 This implies l = 0.36. The population gross growth rate n is set equal to the post war labor force growth rate, 1.016, obtained by using data from Bureau of Labor Statistics. 3.2.2. Depreciation rates, technology and public spending The depreciation rates for physical, dp, and human capital, dh, are calculated by Jorgenson and Fraumeni to be, on average, 0.049 and 0.0178, respectively. We also use a value for the intertemporal elasticity of consumption (1/r) that is common in the DSGE literature (i.e. r = 2). Using a production function and time period similar to ours, Lansing (1998) provides estimates for Total Factor Productivity (TFP). Hence we use his parameters for the stationary TFP process in (20), e.g. q = 0.933 and ra = 0.01. We also require constants for government education and other government spending as shares of output. The constant part of education spending ratio is set at the data average using NIPA data, i.e. g e0 ¼ 0:053. We set other government spending, g o0 , in the government budget Eq. (8) so that the long-run solution for the tax rate gives 0.21. This value corresponds to the effective income tax reported in the ECFIN dataset.11 This implies g o0 ¼ 0:157 for the output share of other government spending.

9 Generally, empirical studies use measures of school enrolment ratios or years of schooling as general proxies of labor quality or human capital. However, in our setup, these proxies are measures of the input to the production function of human capital (time spent on education) and not of the output of this activity, new human capital. 10 To obtain this we divide total hours worked by total hours available for work or leisure, following e.g. Ho and Jorgenson (2000). For example, they assume that there are 14 hours available for work or leisure on a daily basis with the remaining 10 hours accounted for by physiological needs. 11 We calculate this as the weighted average of the effective tax rates on (gross) capital income and labor income, where the weights are capital’s and labor’s shares in income.

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K. Angelopoulos et al. / Journal of Macroeconomics 33 (2011) 176–192 Table 1 Parameters. Parameter

Value

Definition

A>0 B>0 01 0 0

0.124 0.507 0.422 0.578 0.964 1.016 0.049 0.018 0.053 0.157 2.000 0.360 0.550 0.050 0.400 0.100 0.210 0.933 0.010

Productivity in goods production Productivity in human capital production Productivity of private capital Productivity of effective labor Rate of time preference Population growth Depreciation rate of physical capital Depreciation rate of human capital Public education spending as share of output Other public spending as share of output 1/r is the intertemporal elasticity of consumption Consumption weight in utility Productivity of household human capital Productivity of private education spending Productivity of aggregate human capital Productivity of public education spending Effective direct tax rate AR(1) parameter technology Std. dev. of technology innovations

3.2.3. Human capital We next move on to the parameters h1, h2, h3 and B in the production function of human capital and A in the production function of goods. Note that the expression ðeht Hht Þh1 ðX ht Þh2 ðHt Þ1h1 h2 Bðg et Þh3 in Eq. (5) is essentially the production function for the creation of new human capital, or what Jorgenson and Fraumeni (1992a,b) call investment in human capital. Model consistent values for the scale parameters A and B are obtained by solving Eqs. (5) and (6) using data averages and long-run values for the variables y, k, x, u, e, c and ge, as well as the calibrated parameters a, h1, h2, h3, n, and dh. For this exercise, we obtain model consistent y, from Eq. (3), using NIPA data. Data on y, k, x and c are obtained using the NIPA accounts and the human capital data by Jorgenson and Fraumeni discussed above. We also acquire a proxy for e and u to calibrate A and B by employing data on the allocation of time in the labor force in education, work and leisure. Using data on hours worked from Ho and Jorgenson (2000) (see above), we calculate the share of leisure time to be 0.64. We assume that, on average, agents have the same leisure versus non-leisure time allocation independent of whether non-leisure time is used to create human capital or to work. This implies that in the steady state, nonleisure time (i.e. e + u) should be equal to 0.36. Agents spend time to educate both in formal education and in on-the job learning. Private agents allocate approximately 30% of their non-leisure time to formal education as opposed to work (assuming an average of 14 years spent on education and 35 years in work). In addition, they spend time to improve their human capital while at work, in the form of on-the-job training. According to the 1976 SRC Time Use Data (see e.g. Kim and Lee, 2007), this amounts to 20% of their work time. Taken together, these imply that on average, 35% of the labor force spends its non-leisure time in human capital creation. Given the clear upward trend in the education data, we might expect this figure to increase in the future and treat this as a lower bound for our steady-state results. In any case, for the purpose of calibrating A and B we set e and u to 0.126 and 0.234 respectively. Given the functions for the calibration of A and B, we calibrate h1, h2 and h3 so that we obtain an economically meaningful and data-consistent long-run solution. In particular, we choose the three parameters so that the long-run solution for education time, e, growth, c and private spending on education as a share of GDP, x/y , are close to the data. This can be obtained by setting values of h1 = 0.55, h2 = 0.05 and h3 = 0.1. It is also important to note the following regarding the calibrated values of h1, h2 and h3. For higher (lower) values of h2, the steady-state x/y solution becomes higher (lower). The value h2 = 0.05 implies that private education spending as a share of GDP is close to the data average (about 2%). Moreover, for higher externalities, the growth rate becomes too low, irrespective of the size of h3. This happens because, with very high externalities, there are free riding problems in the creation of human capital. On the other hand, for low externalities, the implied share of time allocated to education (e) in the long run becomes unrealistically large. By contrast, our calibrated values h1 = 0.55 and h3 = 0.1 guarantee a growth rate consistent with the data average (2%)12 and, at the same time, imply that agents in the long run will spend about 50% of their non-leisure time in acquiring human capital.

3.2.4. Long-run solution and transition The steady-state solution implied by this calibration reported in Table 2 is close to the data average for the US, (see e.g. King and Rebelo, 1999). Concerning the allocation of time, the model’s long-run solution implies that total leisure time in the 12 Note that a value of the gross growth rate of 1.02 is the US per labor input growth rate for 1949–1984 using GDP data from the NIPA accounts and labor data from Bureau of Labor Statistics.

K. Angelopoulos et al. / Journal of Macroeconomics 33 (2011) 176–192

183

Table 2 Steady-state solution. Variable

Model solution

c

1.020 0.629 0.132 0.239 0.676 0.023 2.903

l e u c/y x/y kp/y

labor force is 62.9% (see also above). In addition, the model solution indicates that education effort in the long run is about 50% of the labor force non-leisure time. As discussed above, this is consistent with increased education and training activities in the US labor force. To sum up, this model economy for the post-war US is consistent with externalities in private human accumulation and productive public education expenditure. Lucas (1988) suggests a value of human capital externality of 0.4, but (since his externality is modeled as a direct argument in the goods production function) its effect on output produced is much higher relative to our calibrated externality. The associated value of the productivity of public education expenditure, h3 = 0.1, is also well within the range assumed in the related literature (see e.g. Blankenau (2005) p. 501). Working as in Schmitt-Grohé and Uribe (2004), the model is saddlepath stable under the parameter values reported in Table 1. 4. Results Recall that the raison d’etre for government in our simple setup is that externalities in human capital accumulation imply higher social over private returns to education and hence they result in suboptimally low human capital accumulation in the DCE. Hence, it is natural to start by choosing the growth rate of human capital (ct) and the shadow price of human capital (wt), as the indicator variables zt in our fiscal policy reaction function (19).13 In the first case, the authorities react to the gap between the market value of ct and its social planner value c⁄ (we call this the c gap rule). In the second case, the authorities react to the gap between the shadow price of human capital, which measures the implicit returns to education, and its social planner value. An alternative indicator of the returns to education as perceived by the household in DCE – and thus a natural candidate for zt – is the wage rate (wt). Finally, for reasons of comparability with the macroeconomic stabilization literature that looks at output, we will also consider reaction to the output growth (cy = Yt/Yt1) gap (see e.g. Walsh, 2003b).14 We compute welfare under each of the alternative policy rules, that is, under alternative indicator variables (zt) in (19), as well as under different magnitudes of the feedback policy coefficient (f) on each of those indicator variables (see Section 4.1.1). We then compare the welfare-maximizing value of f, and the associated welfare, under each rule. Welfare is defined as the conditional expectation of the discounted sum of household’s lifetime utility. We will approximate both the equilibrium solution and welfare to second-order around the non-stochastic steady-state (see e.g. Schmitt-Grohé and Uribe, 2004, 2007). In contrast to solutions which impose certainty equivalence, the solution of the second-order system allows us to take account of the uncertainty agents face when making decisions. In addition, as pointed out by e.g. Rotemberg and Woodford (1997), Woodford (2003) and Schmitt-Grohé and Uribe (2004), the second-order approximation to the model’s policy function helps to avoid potential spurious welfare rankings of different regimes that may arise under certainty equivalence.15 Finally, the welfare gains/losses, associated with alternative policy rules, denoted as n, will be obtained by computing the flat over time percentage compensation in private consumption that the individual would require to be equally well off between two policy rules/regimes, see e.g. Lucas (1990), Schmitt-Grohé and Uribe (2007) and Malley et al. (2009). 4.1. Optimal resource allocation and net welfare In this subsection, we start our analysis by computing welfare, and the associated welfare-maximizing value of the feedback policy coefficient, f, for alternative indicators used as zt in the policy rule (19). We then study the differences between the competing policy rules as TFP uncertainty rises. Finally, we present further results regarding, according to our analysis, the best available policy rule.

13

For comparability with the other rules where the arguments in the policy functions are in ratios or rates, we also express the argument in the w gap rule as a    w 1 . w

ratio, i.e. 14

t1

Note that in the steady-state, output growth is equal to the human capital growth, so this target is fully compatible with the above reasoning for human capital growth. Obviously, along the transition path the two targets can have different welfare implications. 15 See Appendix A for further details on the derivation of welfare. Also note that we evaluate the conditional expectation in the welfare function using MonteCarlo integration. To this end we conduct 1000 simulations and approximate an agent’s infinite lifetime by 300 years since b300 ’ 0.

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Fig. 1. Optimal feedback policy.

4.1.1. Welfare under different indicators Using the baseline parameters in Table 1 above, we compute welfare for a wide range of feedback policy coefficients, f, under each feedback rule. This allows us to find the welfare-maximizing value of f under each feedback rule.16 The four diagrams in Fig. 1 (see Appendix A) plot the welfare associated with different values of f when the government reacts to the gaps of ct, wt, wt and Yt/Yt1 from their respective social planner values. The figure shows that the optimal f values are 0.16, 4.5, 0.1 and 0.13 respectively. It also shows that, at the optimum, the level of welfare associated with each active rule is approximately the same. It finally shows that this value is clearly higher than the corresponding value under the common base case of no feedback policy reaction, i.e. when f = 0 in each policy rule. 4.1.2. Policy rules under uncertainty We now compute the welfare gains/losses associated with each policy rule relative to the common base case of no feedback re-allocation policy, for each of the four indicators (ct, wt, wt and Yt/Yt1) discussed above, across a varying degree of macroeconomic uncertainty. Welfare under each feedback re-allocation policy refers to the maximum welfare resulting from the optimally chosen f in (19). The common base case of inaction is when the government’s public education share in output is held constant at its data average (i.e. when we set f = 0 in Eq. (19)). Table 3 summarizes the welfare gains/losses (i.e. the value of n) of feedback re-allocation policy relative to the common base case of inaction across different values of the standard deviation of TFP, ra. In particular, we present results for the deterministic case (i.e. ra = 0), the base calibration case where ra = 0.01 (which was also the case in Fig. 1), and for two more scenarios representing higher levels of uncertainty, i.e. ra = {0.03, 0.06}. In a deterministic environment, ra = 0, all feedback re-allocation policies imply the same welfare (and thus the same welfare gains of 1.1% over the common base). Since all endogenous variables, used as policy indicators in (19)), are obtained from the solution of the same problem (i.e. the DCE for z and the SP for z⁄), the particular choice of z makes no difference to the extent that active policies (namely, the feedback policy coefficients, f) are chosen optimally. Therefore, when fiscal reaction is optimally chosen, and there is no uncertainty, the choice of the endogenous variable serving as a policy indicator does not matter to welfare. In other words, in a deterministic environment, all endogenous variables are equally good as indicators of the market inefficiency. By contrast, in a stochastic environment, ra > 0, feedback re-allocation policies also yield benefits/costs off steady-state. A higher value of n in rows 3–5 in Table 3, relative to the value reported in row 1, indicates the extra benefits associated with a particular optimal feedback re-allocation policy’s ability to smooth the economy along the transition path. A smaller value of n suggests that active policy is destabilizing the economy, so that the welfare gains over the common base obtained in a 16

This is as in e.g. Schmitt-Grohé and Uribe (2007). On the other hand, this differs from solving an explicit second-best optimal Ramsey policy problem.

185

K. Angelopoulos et al. / Journal of Macroeconomics 33 (2011) 176–192 Table 3 Optimal rules and welfare gains/losses from re-allocation policy.

ra = 0 ra = 0.01 ra = 0.03 ra = 0.06

c gap

w gap

w gap

cy gap

0.011 0.012 0.012 0.013

0.011 0.011 0.006 0.013

0.011 0.011 0.010 0.007

0.011 0.010 0.001 0.079

Fig. 2. Uncertainty and the relative welfare gains of the growth gap.

non-stochastic world are getting smaller (and, when uncertainty is high, even negative). This happens because action always comes at a cost (here, changes in public education spending trigger changes in distorting income taxes). Hence, excessive action destabilizes the economy.17 Only when the government reacts to the gap between the market value of the growth rate of human capital and its social planner’s value, the attractiveness of a re-allocation policy is increasing in the degree of overall uncertainty (at least in the empirically plausible range of standard deviations reported in Table 3). Therefore, reaction to the c gap is superior to reaction to all other gaps in terms of welfare. This likely happens because the growth rate of human capital is the engine of perpetual growth and hence the key determinant of welfare. To put it differently, the growth rate of human capital is closer to the heart of the market imperfection than any other indicator. Therefore, in a stochastic world, the choice of the indicator variable, or operating target, does matter to the macroeconomy. 4.1.3. Gains from the c gap rule To obtain a fuller picture of these relationships, we next plot in Fig. 2 the welfare gains associated with reacting to the c gap rule relative to the common base case of no feedback re-allocation policy as well as the other policy rules. Again this is carried out for different levels of uncertainty. The results suggest that the relative welfare benefits of following the human capital growth rate rule are highest when the alternative is the output growth rate rule. In this case, the relative gains can be nearly 10% at high levels of uncertainty. The rank-ordering from highest to lowest for the remaining rules is that reacting to the wage gap is second and this is followed by reacting to the shadow price of human capital gap respectively. Thus, it appears that the c gap rule outperforms the competitors considered when the focus is varying degrees of uncertainty. To further explore what is driving the above results, we next examine the impulse response (IR) functions of the three first-moment arguments in the welfare function, i.e. the growth rate, ct, leisure, lt, and consumption, ct, for all policy rules, 17 An additional shortcoming of the w and cy rules is that for high values of volatility, zt can be greater that z⁄ for some realizations of TFP. This result is counterintuitive in a model where positive externalities from economy-wide human capital are expected to push market outcomes (zt) below their social planner values (z⁄).

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Fig. 3. Responses of g^; ^l and ^c to a temporary TFP shock relative to the base case.

in response to a 1% standard deviation shock to TFP. The IRs are presented in Figs. 3 and 4. As can be seen in these Figures, the advantage of the c gap rule off steady-state is that the path of human capital growth is relatively smoother than in the competing rules.18 The other rules involve trade-offs, as they result in making some series smoother, while increasing the volatility in others. Hence, their overall second-order result on welfare is not clear and, as we saw above, depends on the level of uncertainty. Regarding the output growth rule, in addition to making consumption more volatile, notice that the effect of this rule is that human capital growth becomes countercyclical for the first periods after the shock. This happens because, after a positive TFP shock, the current output growth rate increases, so that the gap in the feedback policy rule gets smaller. This, in turn, reduces the share of public education spending which tends to reduce ct. Although this counter-cyclical policy reaction is also present in the other rules, it is weaker in these. Therefore, the c gap rule implies additional benefits over the business cycle, when compared to the other active rules, as it makes the growth rate in the economy more stable. Although the other policy rules improve the resource allocation in the deterministic long run, they effectively destabilize the economy along the transition path. The general message is that the human capital growth rule is preferred. 4.2. Welfare gains over time We next examine the welfare gains over time, when following different policy rules, by evaluating welfare in varying time periods after a temporary TFP shock for a low level of uncertainty, i.e. ra = 0.01.19 In Table 4, we employ time horizons of 5, 10 and 20 years after the temporary shock hits the economy and record welfare in each case. In addition, we report the welfare gains from using all types of policy rules and all time horizons. Finally, we report the welfare gains from using the c gap rule as opposed to the other rules for all these time horizons. There are a number of interesting results to be observed in Table 4. The first pattern observed is that policy rules imply welfare gains over the base case for all time horizons. Hence, for low levels of uncertainty, the policy rules can improve outcomes, not only by a lifetime utility criterion, but also when the interest is in shorter term benefits. Notice that, for all rules, the welfare gains versus the base case settle down to 1–1.2% in the long run, as expected given our previous evaluations in Table 3. Another useful observation, consistent with the findings in Table 3, is that the human capital growth rule outper18 Note from the second-order approximation to the welfare function (see Appendix A) that, in addition to the steady-state values of c, l, c and the deviations of ct, lt, ct from their steady-state values, what also matters for welfare is the squared deviations and cross products of ct, lt, ct from their steady-state values. Hence the variance of the series and their covariances are also important in determining the level of welfare. 19 This is in effect the average in the US data and is consistent with the value used in most studies in the literature.

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Fig. 4. Responses of g^; ^l and ^c to a temporary TFP shock relative to the growth gap rule.

Table 4 Effects of a temporary TFP shock. Years

Base case

c gap

w gap

w gap

cy gap

Welfare 5 10 20 1

26.92 54.54 94.55 167.34

26.30 53.40 92.90 166.66

26.36 53.52 93.06 166.62

26.31 53.42 92.93 166.67

26.35 53.55 93.12 166.63

Gains relative to the base case 5 10 20 1

0 0 0 0

0.065 0.059 0.049 0.011

Relative to the c gap 5 10 20 1

0.065 0.059 0.049 0.011

0 0 0 0

0.058 0.052 0.044 0.012

0.064 0.058 0.048 0.011

0.059 0.051 0.042 0.012

0.006 0.006 0.005 0.001

0.001 0.001 0.001 0.000

0.005 0.008 0.007 0.001

forms the other rules, with implied welfare gains in the range 0.1–0.8%. Notice that, in the long run, when the effects of the shock have faded away, all policy rules give rise to the same level of welfare. Again, given the evaluations in Table 3, this is as expected since all policy rules imply the same resource allocation in the steady state.

5. Conclusions In this paper, we quantitatively assessed the welfare implications of alternative public education spending rules along both the transition path and at steady-state. The steady-state effects result from reallocative policies designed to bring

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market outcomes closer to the socially optimal ones. In contrast, the off steady-state effects are a by-product of the introduction of uncertainty through a stochastic TFP shock. We found that the overall welfare effects of resource allocation rules can be significant in the case of a rule based on the difference between the market value of human capital growth and its social planner’s value; we also found that spending changes initiated by the remaining rules considered disrupt private decisions and destabilize the economy in periods of increased uncertainty. Thus, while it appears possible to design allocationimproving education policy rules, it is also important to recognize and account for their cyclical effects. Despite establishing that it is possible to design a simple welfare improving education spending rule, much more research is needed to make these implementable in practice. This will require continual improvement in the measurement of human capital but also most probably the econometric estimation of models and policy rules along the lines of the ones developed in this paper. We leave these extensions for future research. Acknowledgements We would like to thank an anonymous referee and the editor, Theodore Palivos, for helpful comments and suggestions. The usual disclaimer applies. Appendix A A.1. Non-stationary optimality conditions On the households’ side, the first-order conditions include the constraints and the optimality conditions for C ht ; X ht ; uht ; eht ; K htþ1 ; Hhtþ1 which are respectively:

Kht ¼ lðC ht Þlð1rÞ1 ð1  uht  eht Þð1lÞð1rÞ

Wht ¼

ð22Þ

Kht

ð23Þ

e t h2 ðeh Hh Þh1 ðX h Þh2 1 ðHt Þ1h1 h2 B t t t

Kht ð1  st Þwt Hht ¼ ð1  lÞðC ht Þlð1rÞ ð1  uht  eht Þð1lÞð1rÞ1

Wht ¼

ð24Þ

ð1  lÞðC ht Þlð1rÞ ð1  uht  eht Þð1lÞð1rÞ1 e t h1 ðeh Þh1 1 ðHh Þh1 ðX h Þh2 ðHt Þ1h1 h2 B t

t

ð25Þ

t

Kht ¼ bEt Khtþ1 ½1  dp þ ð1  stþ1 Þrtþ1 

ð26Þ h

e tþ1 h1 ðeh Þh1 ðHh Þh1 1 ðX h Þh2 ðHtþ1 Þ1h1 h2 Wht ¼ bEt Khtþ1 ð1  stþ1 Þwtþ1 uhtþ1 þ bEt Whtþ1 1  dh þ B tþ1 tþ1 tþ1

i

ð27Þ

where Kht and Wht are the multipliers associated with (3) and (5) respectively. Eq. (22) equates the shadow price associated with household’s budget constraint to the marginal utility of consumption; (23) equates the marginal cost (in terms of foregone consumption) of spending on private education to the marginal benefit from augmenting human capital; (24) equates the marginal utility of leisure to the marginal utility of consumption; (25) equates the marginal cost (in terms of foregone leisure) of time allocated to education to the marginal benefit from augmenting human capital; (26) is a standard Euler equation for physical capital stating that the marginal cost of foregone consumption today equals the expected discounted net return to capital in the next period; and (27) is the Euler equation for human capital stating that the marginal cost of investing in new human capital today equals the expected discounted net labor income return plus the expected discounted marginal benefit from starting with higher human capital in the next time period. On the firms’ side, the first-order conditions for uft and K ft are:

ð1  aÞY ft uft Hft

aY ft K ft

¼ wt

¼ rt

ð28Þ ð29Þ

A.2. Welfare analysis A.2.1. Instantaneous utility Using the notation set out in the paper, first consider the per capita representation of the instantaneous utility function given by (2):

K. Angelopoulos et al. / Journal of Macroeconomics 33 (2011) 176–192

h Ut ¼

ðC t Þl ðlt Þ1l

189

i1r ð30Þ

1r

or using our notation for stationary variables:

h Ut ¼

ðct Ht Þl ðlt Þ1l

i1r ð31Þ

1r

where Ht is the beginning-of-period human capital stock. Since ct  Htþ1 =Ht , we have for t P 1: t1 Y

Ht ¼ H 0

!

cs

ð32Þ

s¼0

and H0 is given from initial conditions. Substituting (32) into (31) gives

( H0 ct

t1 Q s¼0

Ut ¼ h U0 ¼

cs

)1r

l ðlt Þ

1r i1r ðH0 c0 Þl ðl0 Þ1l 1r

1l

for t P 1 for t ¼ 0

ð33Þ ð34Þ

A.2.2. Steady-state utility We define the long run as the state without stochastic shocks and constant stationary variables. Using (33) and (34), steady state utility is written as

h U t

¼

ðH0 cct Þl ðlÞ1l

i1r ð35Þ

1r

where the ⁄ superscript denotes steady-state per capita utility. In the steady-state, non-stationary C t grows at the constant rate c, which in turn implies for r, c > 1 that the growth of U t is constant and less than unity. A.2.3. Second-order approximation of within-period Q  utility t1 Define for simplicity a variable zt  ct s¼0 cs , so that the second-order approximation of the within-period utility function in (33) around its long run is:

1 1 2 U st ’ U t þ ½U z z^zt þ ½U l l^lt þ ½U z z þ U zz z2 ^z2t þ ½U l l þ U ll l ^l2t þ ½U zl zl^zt^lt 2 2 where

^zt ¼ ^ct þ

t1 X

c^s ;

s¼0

     2  u e 1 u u ^ t Þ2 ^t  ^et  ðu u þ 1ue 1ue 2 1ue 1ue !   2  1 e e ue ^t ^et ; ð^zt Þ2 ð^et Þ2  þ  u 2 1ue 1ue ð1  u  eÞ2

 ^lt ’ 

2

¼ ð^ct Þ þ

t1 X

!2

c^s

þ 2^ct

s¼0

’

t1 X

c^s ; ð^lt Þ2

s¼0

! 2  2 u e ue ^ t Þ2 þ ^ t ^et ðu ð^et Þ2 þ 2 u 1ue 1ue ð1  u  eÞ2



and the partial derivatives in (36), evaluated at the steady-state, are:

h

Uz ¼

l

l H0 z ðlÞ1l z

i1r

ð36Þ

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K. Angelopoulos et al. / Journal of Macroeconomics 33 (2011) 176–192

Ul ¼

h i1r l ð1  lÞ H0 z ðlÞ1l l h

U zz ¼

U ll ¼

l

l½lð1  rÞ  1 H0 z ðlÞ1l

i1r

z2 h i1r l ð1  lÞ½ð1  lÞð1  rÞ  1 H0 z ðlÞ1l 2

l h

U zl ¼

l

lð1  lÞð1  rÞ H0 z ðlÞ1l

i1r

zl

The expression in (36) gives the second-order approximation, U st , for any t P 1. Therefore at t = 0, the expression for U st is the same except that z0  c0. A.2.4. Second-order approximation of lifetime utility Finally, expected lifetime utility, Vt, is given by the expected discounted sum of U s0 and U st , i.e.:

V t ’ U s0 þ E0

1 X

bt U st

ð37Þ

t¼1

In the simulations, lifetime is approximated by T = 300 years and the sample average for V is calculated using 1000 simulations. A.2.5. Welfare comparison of two regimes Say there are two regimes denoted by the superscripts A and B. Then, following e.g. Lucas (1990), we define n as the constant fraction of regime B’s consumption supplement that the household would be willing to give up to be as well off under A as under B. Hence, we write:

V At ¼ ð1  nÞlð1rÞ V Bt

ð38Þ

Solving for n, we obtain:

lnð1  nÞ ¼

1 V At  ln lð1  rÞ V Bt

! )n’

1 V At  ln lð1  rÞ V Bt

! ð39Þ

where, V Bt and V At are calculated by using the second-order approximation of welfare as defined in (37) above and averaged over 1000 simulations. A.3. Social planner’s solution This is defined to be the case in which human capital externalities are internalized. Since, in this model, taxes are used to finance public education spending, where the latter is not required in the absence of human capital externalities, we drop public education spending and set taxes at zero. Hence the planner maximizes (we omit h and f superscripts denoting households and firms and express all quantities in per capita terms):

E0

1 X t¼0

bt

8h i1r 9 > > < ðC t Þl ðlt Þ1l = 1r

> :

ð40Þ

> ;

subject to:

C t þ X t þ It ¼ At ðK t Þa ðut Ht Þ1a

ð41Þ

K tþ1 ¼ ð1  dp ÞK t þ It h

ð42Þ h1

h2

1h2

Htþ1 ¼ ð1  d ÞHt þ ðet Þ ðX t Þ ðHt Þ

B

ð43Þ

as well as, the time constraint lt + ut + et = 1 and initial conditions for K0 and H0. Working as in the DCE, we obtain a new per capita stationary equilibrium in nine equations and nine variables fct ; yt ; ct ; xt ; ut ; et ; ktþ1 ; kt ; wt g1 t¼0 .

K. Angelopoulos et al. / Journal of Macroeconomics 33 (2011) 176–192

yt ¼ At ðkt Þa ðut Þ1a

ð44Þ

p

nct ktþ1  ð1  d Þkt þ ct þ xt ¼ yt h

h1

ð45Þ

h2

nct ¼ 1  d þ Bðet Þ ðxt Þ

ð46Þ

kt ¼ lðct Þlð1rÞ1 ð1  et  ut Þð1lÞð1rÞ lð1rÞ

wt ¼

ð1  lÞðct Þ

191

ð47Þ

ð1lÞð1rÞ1

ð1  et  ut Þ

Bh1 ðet Þh1 1 ðxt Þh2

ð1  lÞct lð1  aÞyt ¼ ð1  et  ut Þ ut ð1  lÞet lh1 xt ¼ ð1  et  ut Þ h2 c t   ay kt ¼ bðct Þlð1rÞ1 ktþ1 1  dp þ tþ1 ktþ1 n h io lð1rÞ1 wt ¼ bðct Þ ktþ1 ð1  aÞytþ1 þ wtþ1 1  dh þ ð1  h2 ÞBðetþ1 Þh1 ðxtþ1 Þh2

ð48Þ ð49Þ ð50Þ ð51Þ ð52Þ

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