Income taxes, subsidies to education, and investments in human capital

Mendolicchio, Concetta; Paolini, Dimitri; Pietra, Tito Working Paper Income taxes, subsidies to education, and investments in human capital Quaderni - Working Paper DSE, No. 701 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Mendolicchio, Concetta; Paolini, Dimitri; Pietra, Tito (2010) : Income taxes, subsidies to education, and investments in human capital, Quaderni - Working Paper DSE, No. 701, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4540 This Version is available at: https://hdl.handle.net/10419/159542 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Income taxes, subsidies to education, and investments in human capital Concetta Mendolicchio IAB and Dimitri Paolini DEIR and CRENoS, Università di Sassari and Tito Pietra1 DSE, Università di Bologna Version: January 24, 2010 We study a two-sector economy with investments in human and physical capital and imperfect labor markets. Human and physical capital are heterogeneous. Workers and firms endogenously select the sector they are active in, and choose the amount of their sector-specific investments in human and physical capital. To enter the high-skill sector, workers must pay a fixed cost that we interpret as direct cost of education. Given the distribution of the agents across sectors, at equilibrium, in each sector there is underinvestment in both human and physical capital, due to non-contractibility of investments. A second source of inefficiency is related to the self-selection of the agents into the two sectors. It typically induces too many workers to invest in education. Under suitable restrictions on the parameters, the joint effect of the two distortions is that equilibria are characterized by too many people investing too little effort in the high skill sector. We also analyze the welfare properties of equilibria and study the effects of several tax-subsidy policies on the total expected surplus. JEL classification: J24; H2 Keywords:Humancapital;Efficiency; Labour income tax Corresponding author: T. Pietra, Tel.: +39 051 2098026; fax +39 051 2098040. E-mail address: [email protected] 1A previous version of the paper has been circulated as "Human capital policies in a static, two-sector economy with imperfect markets", CRENoS 08-07. The paper is part of the Ph.D. dissertation at IRES, Universitè Catholique de Louvain, of the first author. We thank for helpful comments the members of her dissertations committee (R. Boucchekine, V. Vandenberghe, M. Belot, B. Decreuse, B. Van der Linden), P. Pestieau, and participants of a seminars at the University of Luxembourg, at the Università di Salerno and at ASSET 2009. The usual disclaimers apply. We acknowledge the financial support of MIUR-PRIN 2006 and the Fondazione Banco di Sardegna. The first author acknowledges the support of the "Programma Visiting Professor" of the Università di Sassari (Italy). 1 1. INTRODUCTION Causes and consequences of investments in human capital have been a central field of research in the last few decades for several motivations. Among them, the relevance of human capital externalities in growth theory, and the issues related to the dynamics of the wage premium and, more generally, to the evolution of income distribution. Still, the analysis of human capital externalities is far from settled from both the empirical and the theoretical viewpoints. Empirically, it is not obvious that there are significant, positive differences between social and private returns, at least at the level of subsidies prevailing in most Western countries.2 From a theoretical viewpoint, the exact microeconomic mechanism generating the externality is not fully understood. A better understanding of its nature has policy relevance. This is true even if one is willing to take for granted that there are no significant, unexploited, positive externalities, because this is typically obtained with high subsidies to education.3 In this paper, we extend the microeconomic analysis of the externalities related to investments in human capital and derive some results on the welfare effects of several policies: fixed tax/subsidies on the direct cost of the acquisition of high skill human capital, and tax/subsidies on labor income, or - equivalently in our set-up - on the investment in human capital. We consider economies with three key features: 1. Workers are ex-ante heterogeneous, while firms are not, 2. Investments in human and physical capital are non-contractible, 3. There are two separate sectors employing different kinds of human and physical capital, so that an agent must choose both the level of his/her investment and its type. The economy is basically a two-sector generalization, with sector specific inputs, of the model considered in Acemoglu (1996). In his framework, firms and workers choose the amount of their investments. Then, they are matched randomly, and income distribution is determined by a bargaining process. After agents have chosen the sector they are going to be active in (hence, the type of their investment), the model considered here reduces to a pair of separated Acemoglu’s economies. In our set-up, income distribution takes place through bargaining, too. However, bear in mind that, when workers are heterogeneous, the crucial feature is noncontractibility of investments: the bargaining set-up affects several details, but not some key aspects of the welfare results. Our main departure from Acemoglu (1996) is that we adopt the notion of human capital put forth in Roy (1951): there are distinct markets for high skill and low skill labor, that we assume to be perfectly non substitutable (hence, the twosector structure). However, contrary to what is often assumed in Roy models, once a worker has selected the type of human capital she wants to acquire, she still has 2For the U.S.A., a negative conclusion is reached, for instance, by Heckman, Layne-Farrar, and Todd (1996) and by Acemoglu and Angrist (2001). For E.U. countries, the results in De la Fuente (2003) are also negative. See also Krueger and Lindhal (2000). 3In 2005, in the OECD average, 85.5% of the direct cost of education (all levels included) is financed by public sources (see OECD (2008, Table B3.1, p. 251)). The EU19 average is 90.5%. At the tertiary level, these percentages are, respectively, 73.1% and 82.5% (Table B3.2b, p. 253). 2 to decide how much effort to invest. As common in the literature, human capital translates one-to-one into efficiency units of high skill (low skill, respectively) labor.4Most of the recent literature takes a different point of view, adopting the efficiency units approach with homogeneous human capital, therefore ruling out, by assumption, all the consequences of self-selection, which are, instead, relevant from both the theoretical and the empirical viewpoints.5 With imperfect markets and self-selection of the agents into different markets, two distinct distortions are at work. In the main text, lack of contractibility of investments and the bargaining set-up generate an hold-up problem, inducing an inefficiently low level of investments, in human and physical capital of both types (hence, in each sector). Secondly, due to lack of contractibility, agents choose type and level of their investments looking at the distribution of their expected future returns, which, in turn, depends upon the distribution of the investments of the potential partners in each sector. Given that workers are heterogeneous, a switch of a subset of them from one sector to the other affects the distribution of returns of the firms, hence their optimal investments. This second potential source of distortion is independent of the random matching-bargaining set-up, and is at work even when spot labour markets are perfectly competitive.6Consequently, public policies have two distinct effects on total surplus, the index of welfare adopted here. The first is their direct impact on the optimal investments of the agents acquiring a sectorspecificasset: wewillrefertoitasincentive effect. The second is the one on the agents’ distribution across markets: we call it composition effect, following Charlot and Decreuse (2005) (see also Decreuse (2008)). In "pure" Roy models (with selfselection, but no choice of the investment effort) only the composition effect is at play. In "pure" efficiency-units models (without self-selection) only the incentive effect is at work. Our model allows us to study the interaction between the two phenomena. We consider two separate sectors, using sector specific inputs (high/low skill capital). The crucial property is that human and physical capital are heterogeneous. To identify one type of capital with one sector somewhat simplifies the set-up and sharpens the welfare results. However, the two distinct distortions would be at work even with just one productive sector employing both high and low skill labor. Bear in mind that, whenever in the sequel we mention the two-sector structure of the economy, we implicitly mean that the two sectors use different kinds of human and physical capital. Some of the results in Acemoglu (1996) survive in our set-up. For instance, in both cases, the human capital externality is related to its (sector-specific) average 4As usual, we can also interpret effort as elastic supply of labor of a given skill. 5A survey supporting this claim is in Sattinger (1993). For more recent discussions of the different empirical implications of efficiency units vs. Roy models see, for instance, Carneiro, Heckman, and Vytlacil (2001). Investments in human capital in a two-sector economy with frictions due to random matching (but with perfectly inelastic supply of human and physical capital) have been studied in Sattinger (2003), Charlot and Decreuse (2005), and Mendolicchio, Paolini, and Pietra (2008). 6This case is briefly analyzed in Appendix 2, where, we consider economies with perfectly competitive spot labor markets, non-contractibility of investments, and asymmetric information. Due to perfect competition, the hold-up problem disappears, and (taking as given the distribution of the agents across sectors) investments are at their constrained efficient level. However, due to asymmetric information and lack of contractibility, the composition effect still induces constrained inefficiency of equilibria, which are always characterized by overinvestment in education. 3 level, not to its aggregate level (as postulated in Lucas (1988)). There are, instead, sharp differences with respect to the policy prescriptions: in the one-sector model, subsidies to investments in human capital (or to labor supply) are unambiguously beneficial. This is because only the incentive effectisatplay: asubsidytothe investments in human capital of any subset of agents increases them and, therefore, their expected value as a first order effect. This has a positive impact on the firms’ investment decisions and, in turn, further increases the optimal investment of all the workers. This chain of positive feedbacks guarantees that these subsidies are (in a neighborhood of zero) welfare improving. To reformulate the point differently: in one-sector economies, there is a unique distortion induced by the hold-up problem which induces underinvestment for both firms and workers. Any policy increasing the investments of any subset of agents is welfare improving. With two sectors, the incentive effect of a policy can be strengthened, weakened, or overturned, by its composition effect. In the final section of the paper, we study the welfare effects of several kinds of balanced budget policies based on tax/subsidies to the direct costs of education, and on skill-contingent subsidies to labor supply. Consider, for instance, subsidies to low skilled labor income (or a lower marginal tax rate for low incomes) financed with lump-sum taxes. If total factor productivities are sufficiently diverse across sectors, they always increase total surplus, because their positive effect on individual effort in this sector is strengthened by the composition effect, i.e., by the improvement of the expected quality of the pool of workers in both markets. An increase in taxes on the direct costs of education (again, balanced with lump-sum taxes) also increases total surplus, because of its composition effect. On the other hand, subsidies to high skill labor incomes have a (first order) positive incentive effect on the investments of these workers, but a negative composition effect. Subsidies to the investments in the high-skill sector have always a negative impact on the equilibrium utility of low-skilled workers (and on the equilibrium profits of the firms active in that sector). The total effect for agents active in the high-skill sector may be positive or negative, according to the magnitudes of the (positive) incentive effect and the (negative) composition effect. We provide a robust example where the total effect of these subsidies on surplus is negative. There is a large literature on the effects of subsidies to education and of labor income taxes on accumulation of human capital. The usual arguments favoring subsidies hinge either on their positive externality effects, or on the existence of liquidity constraints. Additionally, subsidies to education have been analyzed as one of the components of the optimal mix of redistributive policies (see Bovenberg and Jacobs (2005), Jacobs (2005, 2007), Jacobs and Bovenberg (2008), Jacobs, Schindler and Yang (2009), Schindler and Weigert (2008, 2009)). The last two aspects may be both empirically and theoretically important, but we abstract from them, focussing the analysis on the pure efficiency issue related to the presence of an hold-up problem and of self-selection. The classical analysis of the effects of labor income tax on investments in human capital started with the seminal papers by Ben-Porath (1970), Boskin (1975) and Heckman (1976).7Aflat labor income tax has a negative impact on human capital accumulation just because of non-deductibility of the direct costs of education. On 7As mentioned above, in our set-up, one obtains substantially identical results considering direct (non-linear) subsidies to effort and subsidies to the direct costs of education. Previous, related work in this area includes Blankenau (2005), Blankenau and Camera (2006, 2009), Caucutt and Kumar (2003), Lloyd-Ellis (2000), Sahin (2004), and Su (2004). 4 the other hand, by depressing the net interest rate, in fully specified life-cycle models of consumer behavior, a tax on total income may actually have a positive effect. Eaton and Rosen (1980) extend the analysis to (uninsurable) multiplicative wage uncertainty, pointing out that a flat earning tax affects investments in human capital through its effects on their riskiness and (via an income effect) on the attitude toward risk (see, also, Anderberg and Andersson (2003), and Anderberg (2008)). Consider now a progressive income tax (compared with a revenue-neutral flat one). The canonical conclusion is that it discourages investments at the high skill level, while it may encourage them for the less skilled. While the literature provides us with many insights, it mostly deals with economies where there is no self-selection into different skills, so that one of the key mechanism at work in our economy is absent. Also, bear in mind that, in our set-up, at the equilibrium, workers face no uncertainty, so that the mechanism pointed out in Eaton and Rosen (1980) is absent. The structure of the paper is the following. Section 2 discusses the general features of the model. Section 3 and 4 analyze the benchmark, Walrasian, economy, and the one with imperfect labor markets. Section 5 studies the welfare properties of the equilibria of the economy with frictions. Most of the details are in Appendix 1. In Appendix 2, we sketch the analysis of a perfectly competitive economy with asymmetric information and self-selection into the two sectors. 2. THE MODEL The economy is composed by two separate production sectors, denoted by s∈ {ne, e}.Workers (denoted by a subscript iwhen we refer to individuals, Iwhen we refer to their set) and firms (denoted by jand J, respectively) can choose to enter one of the two sectors, paying a fixed cost. Workers’ costs, (cne I,c e I),are exogenous, and can be interpreted as private, direct, fixed costs of education (tuitions and the like). We denote firms’ costs (dne J,d e J).They are endogenously determined, and will be discussed later on. There are two intervals of equal length of workers and firms, ΩI=ΩJ≡(0,1) , both endowed with the Lebesgue measure. Each interval is partitioned into two sets, {Ωne I,Ωe I}≡ΩP Iand {Ωne J,Ωe J}≡ΩP J,determined endogenously. Let ν(Ωs I) (ν(Ωs J)) denote the measure of the set Ωs I(Ωs J,respectively). In sector s, production requires a firm j(with physical capital ks j)andaworkeri(with stock of human capital hs i).Once the partitions ΩP Iand ΩP Iare given,each sector of the economy reduces to the set-up studied in Acemoglu (1996). Firms are identical, and choose their investments in physical capital to maximize their expected profits. Workers choose their investments in human capital to maximize their expected utilities. The economy lasts one period, divided into several subperiods. We consider two versions of the basic model. We adopt as a benchmark a frictionless (or Walrasian) economy, where, in subperiod 0, firms and workers enter, paying a fixed cost, one of the two sectors. Then, at 1, each firm active in sector sismatchedwithaworker active in the same sector (we will be more precise on the matching issue later on). Matched firm and worker sign a binding contract on the amount of human and physical capital that they will supply in subperiod 2. In the final subperiod, investments are carried out, exchanges and production take place, and agents are paid on the basis of their marginal product. In the second version of the model, the one with frictions, the total output of each match is split according to the Nash bargaining solution with exogenous 5 weights βand (1 −β).8Moreover, and most important, agents cannot contract with their partners a given level of investment, because these are carried out before matches take place. To summarize: in subperiod 0, agents choose to enter one of the two sectors, paying a fixed entry cost. In subperiod 1, they choose their levels of investment. In subperiod 2, they are randomly matched and then, finally, production and exchanges take place. Technologies are described by a pair of Cobb-Douglas production functions with constant returns to scale. Therefore, in the Walrasian set-up, equilibrium profits are zero, entry costs djmust be zero for each s,andeachfirm is indifferent among sectors. Thus, the equilibrium partition is essentially determined by the labor supply side of the model. On the contrary, in the economy with frictions, expected producers’ surpluses are positive in both sectors and, as we will show later on, they are, at the equilibrium, always larger in sector e. To avoid additional complications not really germane to our main issue, we want to consider an economy with full employment at the equilibrium. This requires that, at the equilibrium, each agent is actually matched with a partner. We assume, as implicit in Acemoglu (1996), that the matching function guarantees with probability one a match to each agent, provided that ν(Ωs I)=ν(Ωs J).9Given the focus of the paper, the partition ΩP I must be determined endogenously. Hence, to guarantee full employment, we need that, at each equilibrium, ν(Ωs I)=ν(Ωs J). The easiest way to obtain this property is to introduce a feature of the economy such that equilibrium expected profits are always equal in the two sectors.10 One way to do it is to assume that the technology exploited in sector ne is free, while the one adopted in sector eis protected by a patent, owned by some outside agent.11 Rights to use the patent are auctioned offto firms before the match firm-worker obtains. Given that, at an equilibrium, expected profits in both sectors must be identical, the equilibrium royalties must be equal to the (positive) difference between the expected producer’s surpluses in the two sectors. Then, at each equilibrium, each firm is indifferent among sectors, so that we can choose ΩP Jwith ν(Ωs I)=ν(Ωs J). The property we are looking for. Without any loss of generality, the prices of both kinds of output are set equal to 1 and, therefore, we omit them. Finally, notice that there are always three additional, trivial, equilibria: the ones where all the workers and the firms are in one of the two sectors, and the one where none is active in any sector. As usual, we ignore them. 8For a rationalization of this allocation rule in this context, see the Appendix in Acemoglu (1996). We assume that βis sector-invariant. Given that it is exogenous, to let it vary across sectors would just introduce more notation without any real additional insight. 9A commonly used function which delivers this property is πs j=min{ν(Ωs I),ν(Ωs J)} ν(Ωs J),where πs j is the probability of a match for a firm active in sector s. 10An alternative solution is to assume that firms cannot move across sectors. A non-null measure of firms is exogenously assigned to each sector. We then pick a matching function which always guarantees that each firm is matched with a worker (and conversely) for each non-trivial partition of the workers. As long as there is a continuum of agents in each sector, this can be done. Of course, this approach would break down if we had a finite number of agents and, anyhow, is based on a very ad hoc trick. 11Clearly, nothing would change if each technology was subject to a distinct patent. Also: any inputusedonlyinsectoreand with perfectly inelastic supply would do. We consider the case of a patent to simplify as much as possible the model. 6 3. THE FRICTIONLESS ECONOMY When active in sector s, and matched with worker iwith human capital hs i, firm jhas production function ys ij =Ashsα iks(1−α) j, with Ae>A ne.Let µbe the unit price of physical capital, that we assume to be equal in the two sectors. This implies some loss of generality, but it simplifies notation and computations. Most important, similar results hold for µe6=µne. If active in sector s, and given a match with worker i, firm jsolves optimization problem choose ks j∈arg max Ashsα iks(1−α) j−µks j−ws ijhs i. For each worker active in sector s, the utility function is Us i(Cs i,h s i)=Cs i−1 δi hs(1+Γ) i 1+Γ, where Cs idenotes consumption, hs iis the amount of human capital (or the labor supply). Let cs Ibe the (fixed) cost of the investment in sector shuman capital. Then, in the absence of taxes and subsidies, if worker iis active in sector sand matched with firm j,Cs i=¡ws ijhs i−cs I¢.Workers are heterogeneous because of the parameter δi,indexing their marginal disutility of effort.Without any essential loss of generality, we assume that δi=i, and that δiis uniformly distributed on (0,1) . More general assumptions on the distribution of δi, or its support, would not change any essential result. Given that, in the sequel, we will introduce uniform lumpsum taxes, we must either allow for negative consumption, or assume that workers have a strictly positive, and sufficiently large, initial endowment of consumption good. Given the properties of the utility functions, purely notation-wise, the most convenient solution is the first one. It is straightforward to check that the amount of agent i’s investment in human capital in sector sis given by HWs (δi)≡∙δiαAs1 α³1−α µ´1−α α¸1 Γ ,where the superscript Wdenotes the frictionless economy. Assuming (with no loss of generality because, at the equilibrium, profits are always zero) that firm jis always matched with worker i=j, at the equilibrium, we can write the demand for physical capital of firm j=ias KWs(δi)≡∙δiαAs1+Γ α³1−α µ´1+Γ−α α¸1 Γ . Let’s now consider the equilibrium partition ΩP I.For convenience (here and in the sequel), set cne I=0and ce I>0.LetVWs(δi)≡Us i¡HWs (δi),KWs (δi)¢,be the level of utility of agent iactive in sector s, evaluated at the equilibrium. Worker ichooses to enter sector eif and only if VWe (δi)−VWne (δi)≥0.A straightforward computation shows that this inequality is satisfied if and only if δi≥δW≡∙1+Γ Γce I¸Γ⎡ ⎣Ãαµ1−α µ¶1−α α!1+Γ Γ³Ae1+Γ αΓ−Ane 1+Γ αΓ´⎤ ⎦ −Γ .(1) Hence, for ce Ipositive and sufficiently small, there is a unique threshold value δW,strictly increasing in ce I.All the agents with δi<δ Wdo not invest in education, 7 while the ones with δi≥δWdo.12 4. THE ECONOMY WITH FRICTIONS: EQUILIBRIA AND THEIR COMPARATIVE STATICS PROPERTIES Later on, we will show that, at the equilibrium, it is always Ωe I=[δF,1),where δFdenote the equilibrium value of the threshold in the economy with frictions. Hence, we can restrict the analysis to partitions ΩP Iand ΩP Jdefined by an arbitrary level of the threshold, denoted bδ. To emphasize this, we use the notation Ωs J(bδ)and Ωs I(bδ). For future reference, let’s determine the optimal amount of investments assuming that there is a public intervention defined by a pair of vectors ξs≡(τs,ζs,∆cs I,T), ξ≡(ξe,ξne),describing (possibly) sector specific subsidies and taxes. We assume that there are linear subsidies on labor income (with rates τs,s=ne, e),and on the cost of the investments in physical capital (with rates ζs,s=ne, e),and fixed taxes on the direct costs of education, ∆cs I(we will always set ∆cne I=0).T denotes a (uniform) lump-sum tax, such that the public budget is balanced. We write the subsidy rates as sector specific just to simplify the notation. At equilibrium, this system of subsidies is isomorphic to a system of step-linear subsidies to labor income and to investments in physical capital.13 Also,bearinmindthatourmain qualitative results hold if we start with an arbitrary (linear) tax system, and interpret (τe,τne)as small changes in the tax rates applied to the two classes of workers. Then, τe≤0and τne ≥0would define a move from a flat labor income tax to a progressive one. Pick an arbitrary threshold bδ. If active in sector s, firm jselects the value of ks j solving the expected profits maximization problem choose ks j∈arg max ks j EΩs I(e δ)³(1 −β)Ashsα iks(1−α) j−µ(1 −ζs)ks j´−ds J =(1−β)AsEΩs I(e δ)(hsα i)ks(1−α) j−µ(1 −ζs)ks j−ds J,(Πs) where, given any random variable xs,with xs:Ωs I→R,(or ys,with ys:Ωs J→R), EΩs I(e δ)(xs j)≡UΩs I(e δ)xs idi ν(Ωs I(e δ)) (or EΩs J(e δ)(ys i)) denotes the conditional expectation of xs i over the set Ωs I(bδ)(or of ys jover Ωs J(bδ)). The pair of maps Ks(bδ,EΩs I(e δ)(hsα i),ξ),s=ne, e, defines the optimal investment in physical capital for the firms active in the two sectors. They are j−invariant because firms in each sector are identical, and depend upon the exogenous vector ξ, the arbitrary threshold bδ, and the conditional expectations EΩs I(e δ)(hsα i).Let Πs(δi,bδ,EΩs I(e δ)(hsα i),ξ)be the surplus (because inclusive of ds J)of the firm matched with worker iin sector s. 12Evidently, the agent with δi=δWis indifferent between the two choices. We assume that any indifferent agent will actually choose to invest. 13Exactly the same closed form of the equilibrim is obtained considering a direct subsidy to effort in education of the form τshsα i,which, however, would require direct observability of effort. 8 considers taxes and subsidies balanced with lump-sum taxes. The Corollary taxes on labor income balanced by subsidies to the direct costs of education. Bear in mind that, in the sequel, we always consider changes in total surplus. We are not concerned with actual Pareto improvements. However, given that utility functions are quasi-linear, an increase in total surplus immediately translates (modulo an appropriate - and i−contingent - system of lump-sum taxes and transfers) intoaParetoimprovement. 5.1. Constrained optimal allocations The objective function of the planner is defined as the sum of the expected utilities and producers’ surpluses of the agents, i.e., P¡hs i,ks j,Ωs I,Ωs J¢≡X sZΩs I(e δ)"βEΩs J(e δ)(Ashsα iks(1−α) j)−1 δi hs(1+Γ) i 1+Γ−cs I#di +X sZΩs J(e δ)h(1 −β)EΩs I(e δ)(Ashsα iks(1−α) j)−µks jidj. The planner’s policy instruments are the partitions ΩP Iand ΩP Jand a pair of maps (HCOs(δi,bδ),K COs(bδ)).We restrict the partitions to have the structure Ωe I(bδ)=Ωe J(bδ)=hbδ,1´.Given that firms are ex-ante identical, the informational constraints embedded into the definition of P(.),and the properties of the (implicit) matching function, to impose this structure on ΩP Iand ΩP Jdoes not entail any loss of generality. Also, observe that, given that firms are identical, expected total surplus and realized total surplus coincide. We define an allocation Constrained Optimal (or CO) if and only if it solves the planner’s optimization problem above. Let δCO be the level of the threshold associated with the CO allocation. Proposition 3. Under the maintained assumptions, each economy with frictions has a CO allocation. Equilibrium allocations are never CO, and they are characterized by underinvestment in the amount of physical capital and in educational effort.Moreover, for each economy, there is a lower bound ce I>0such that, for each ce I<c e I,δ F<δ CO,i.e., overinvestment in educational level holds. Proof. See Appendix 1. Thesourceofinefficiency considered by Acemoglu (1996) reappears in our setup, because, given any threshold level bδ, HCOs(δi,bδ)>e Hs(δi,bδ),for each δi,and KCOs(bδ)>e Ks(bδ). On the other hand, the relation between δCO and δFis not univocal. When the direct costs of education are sufficiently low, we always obtain δF<δ CO.Forsufficiently high values of ce I,however, it can be δF>δ CO,as established in Example A2 in Appendix.18 Finally, one can show that, once the optimal subsidies (τ,ζ)are introduced, to implement the CO allocation we always need ∆ce I>0.Thus, CO always requires us to shrink the set of agents investing 18Bear in mind that, in computing δFand δCO,we use different investment functions: (h Hs(.),h Ks(.)) and HCOs(.),KCOs(.),respectively. Hence, there is no contradiction between this property and the fact (established below) that the surplus associated with the market equilibrium is always increasing in the threshold value, even when 1>δ F>δ CO. 15 in the high skill sector (compared to its equilibrium value in the market economy with optimal subsidies (τ,ζ)). It is easy to see that the CO distribution of investments in human and physical capital can be attained with an appropriate system of subsidies to the investments, and of fixed taxes or subsidies on the direct costs of education. Moreover, given that preferences are quasi-linear, the system of tax and subsidies can be balanced using uniform lump-sum taxes (T)on workers (notice that, in the absence of positive endowments of consumption goods, this could entail negative consumption for some subset of agents). Corollary 1. There is a (balanced budget) system of taxes and subsidies ξ such that the associated equilibrium allocation is CO. Proof. See Appendix 1. Remark 6. In our set-up (as well as in Acemoglu (1996)), equilibria of the economy with frictions are constrained inefficient for each value of β, because, at ξ=0, even if δCO =δF, e Hs(δi,δCO) HCOs(δi,δCO)=(1−β)1−α αΓβ1 Γ6=1,for each sand i, and e Ks(δCO) KCOs(δCO)=(1−β)1+Γ−α αΓβ1 Γ6=1,for each s. In the usual random matching model, efficiency obtains when the Hosios’ condition is satisfied, i.e., when βis equal to the absolute value of the elasticity of the matching function. In our economy there is always full employment, so that no congestion externality is at work. Therefore, the Hosios’ condition has no connection with Pareto efficiency.19 5.2. The effect of income taxes and subsidies to education on total surplus We conclude considering the welfare effects of alternative, balanced budget, tax schemes. In particular, we study the effect on total surplus of local changes in the vector ξ, in a neighborhood of ξ=0. We just consider the effects of (τ,∆ce I). Assume that ∂h F ∂e δ|e δ=δF>0.Our first result is that an increase in the direct cost of education (redistributing the revenues as lump-sum transfer), or an increase of the subsidies to labor income in the "low skill" sector ne (financed with lump-sum taxes) always has a positive effect on total surplus. On the contrary, an increase in the subsidy to labor income in the high skill sector (again, financed with lump-sum taxes) may decrease it. The intuition behind the result is fairly simple, also given the discussion of the comparative statics of equilibria in Prop. 2 above. A subsidy τne >0has a direct, positive incentive effect on effort in this sector,and a positive composition effect on effort in both sectors. This is because it induces an increase in the equilibrium value of δF(ξ),which, by itself, increases investments in both sectors.Due to the composition effect, a tax on higher education ∆ce I>0has an indirect, positive impact on effort in both sectors. Therefore, these two policies 19Given any threshold e δ, as observed in Acemoglu (1996, p. 789), the externalities are related to "the value of the future matches and are always positive". 16 always lead to an increase in total surplus. The third one (τe>0) makes sector emore attractive to workers. Therefore, it induces some workers with δi<δ F(0) to switch to sector e. Hence, it has an unambiguous, negative composition effect on the welfare of the workers remaining in sector ne, and on the expected profits in this sector. The negative effect on the welfare of the workers in sector e, due to the composition effect, may actually overcome the positive incentive effect in this sector, too. More generally, the net effect on total surplus is ambiguous, and there are economies where subsidies in the high skill sector induce a lower total surplus. This is established in Prop. 4 and by a final example. In showing these results, the main difficulty is that a change in the threshold induces a discontinuous jump in the expected producer’s surplus for the firms changing sectors. We provide one sufficient condition which guarantees that, at the equilibrium, the total surplus is increasing in the value of the threshold. This condition is far from necessary for our results. The condition is that, given (α, β, Γ),the threshold value δFmust be below some upper limit δ. Given Prop. 1, this essentially implies a lower bound on the ratio Ae Ane .The implicit restriction on the equilibrium threshold is not unreasonable. For instance, for α=2 3,the total expected surplus is increasing in δFif δF<0.6 and Γ=0.2,if δF<0.35 and Γ=0.5and so on. The critical value δis decreasing in αand Γ.20 To conclude, let’s make formal the heuristic argument above. Given ξ, workers and firms choose their individually optimal behavior. Let S³δF(ξ),ξ´be the expected total surplus corresponding to the equilibrium associated with the vector ξ of policy instruments.Then, S³δF(ξ),ξ´≡X sZΩs J(δF(ξ)) EΩs I(δF(ξ))(e Πs(δi,δF(ξ),ξ))dj (8) +X sZΩs I(δF(ξ)) e Vs(δi,δF(ξ),ξ)di, with total lump-sum taxes given by T(ξ)=X s τsZΩs I(δF(ξ))ws(δi,δF(ξ),ξ)di −∆ce Iν(Ωe I(δF(ξ))), so that the budget is balanced. Proposition 4. Consider an equilibrium associated with ξ=0andsuchthat ∂f(.) ∂e δ|e δ=δF(0) >0and EΩe I(δF(0))(δ α 1+Γ−α i)≥EΩne I(δF(0))(δ α 1+Γ−α i) 1−α. Then, i. ∆ce I>0,and sufficiently small, increases total surplus, ii. τne >0,and sufficiently small, increases total surplus, iii. τe>0,and sufficiently small, may decrease total surplus. 20An alternative sufficient condition is that βis "large enough". Notice that, for the class of economies considered in Appendix 2, total expected surplus is increasing in δF.Here, we need additional restrictions because of the discontinuity of the expected producer’s surplus of the firms moving across sectors. 17 The proofs of (i, ii)are in Appendix 1, where we also establish that the welfare effect of a subsidy τeis, in general, indeterminate. The third statement is shown in Example A3, in the same Appendix, where we provide a strategy to construct economies where an increase in τedecreases total surplus. Changes in expected surplus are our measure of welfare gains and losses. However, the different policy instruments have different implications also in terms of individual welfare. Abstract from the lump-sum taxes. An increase in the value of τne (or of ∆ce I)has a positive impact on the utility level of all the workers and on the expected surplus of each firm. On the contrary, an increase in τehasalwaysa negative impact on the utility of all the workers in sector ne (and on the expected surplus of all the firms active in this sector). It may have a positive or negative impact on utility and surplus of agents active in sector e. To conclude, let’s consider the policies where subsidies to effort are financed through taxes on the direct costs of education, instead of lump-sum taxes. The proof of the Corollary is a straightforward computation and, therefore, it is omitted. Corollary 2. Consider the balanced budget policies (τe,∆ce I),(τne,∆ce I).Under the assumptions of Prop. 1, and if Γis sufficiently small, (τs,∆ce I)>> (0,0) and sufficiently small increases total expected surplus. The result requires the labor/effort supply to be sufficiently elastic to the wage rate. Under this restrictions, balanced budget policies with progressive income taxation (τe<0) and subsidies to the direct costs of education (∆ce I<0) are welfare reducing. The impact of the composition effect of such a policy on total surplus is (under the maintained assumptions) always negative. Evidently, the impact of the incentive effect is also negative. The restriction on the value of Γ guarantees that its impact on individual utilities is (in absolute value) larger that the positive one due to the transfer |∆ce I|to agents investing in education.21 Finally, we have been considering a sector-contingent vector of subsidy rates (τe,τne).This is certainly an unusual feature of the policy. However, let ws(δi,δF) be agent i’s labor income in sector s. It is easy to check that max Ωne I(δF)wne(δi,δF)≤wne(δF,δF)<w e(δF,δF)≤min Ωe I(δF)we(δi,δF). Hence, given the properties of the utility functions, the same results can be obtained with a standard system of step-linear taxes or subsidies. Also, we are taking as a reference point an economy where ξ=0.Evidently, introducing a flat tax rate ton labor incomes, we would obtain exactly the same results by changing the marginal tax rates. 6. CONCLUSIONS The paper considers a class of economies where we model both extensive and intensive margins of investment choices. The main conclusion is that the results typically obtained in an efficiency unit set-up (which considers only the intensive margin) can fail to be robust to their natural extension to a Roy’s model with optimal choice of investments in human and physical capital, which accounts for 21It is worthwhile to mention that these results are at variance with the very high subsidies to education and high marginal income tax rates prevailing in most European countries, see, for instance, Figure 1 in Bovemberg and Jacobs (2005). 18 both intensive and extensive margins. By assumption, the efficiency unit framework rules out all the phenomena induced by the self-selection of the agents into different labor markets and, therefore, all the welfare consequences related to the composition effect. While an assessment of the empirical relevance of this effect is beyond the scope of this paper, from a qualitative viewpoint this is a potentially relevant phenomenon, with possible significant policy implications. Our analysis is carried out for a simple, parametric class of economies. This allows us to compute explicitly the equilibria and the welfare effects of different policies, and to compare directly our results with the canonical results of Acemoglu (1996). Evidently, to consider quasi-linear utility function is restrictive, in particular in the analysis of the welfare impact of various policies. However, first, an extension of the analysis to a richer environment is possible, but at an high cost in terms of analytical tractability. Secondly, all the results are "open", so that they certainly survive in environments where income effects are sufficiently small. What matters most, the basic intuition behind the welfare results is strong, and they should be robust to many possible extensions of the basic set-up. 7. APPENDIX 1 Existence and comparative statics of the equilibria in the economy with frictions We start with an arbitrary threshold bδ. Remember that firms are, ex-ante, identical. Then, for each firm active in s,thefirst order conditions (FOCs in the sequel) of problem (Πs)imply Ks(bδ,EΩs I(e δ)(hsα i),ξ)="(1 −β)(1−α)AsEΩs I(e δ)(hsα i) µ(1 −ζs)#1 α ,(9) and EΩs J(e δ)(Ks(.)1−α)=Ks(.)1−α. The FOCs of optimization problem (Us)imply that Hs(δi,bδ,EΩs J(e δ)(ks1−α j),ξ)=hδiαβ (1 + τs)AsEΩs J(e δ)(ks1−α j)i1 1+Γ−α.(A2) Let γ≡1+Γ 1+Γ−α,so that (γ−1) ≡α 1+Γ−α. Solving (A1)and(A2), by imposing that expectations are fulfilled, we obtain e Ks(bδ,ξ)=∙(1 −α)(1−β) µ(1 −ζs)EΩs I(e δ)(δγ−1 i)¸1+Γ−α αΓ (A3) ×((1 + τs)αβ)1 ΓAs1+Γ αΓ, and e Hs(δi,bδ,ξ)=∙(1 −α)(1−β) µ(1 −ζs)EΩs I(e δ)(δγ−1 i)¸1−α αΓ (A4) ×δ 1 1+Γ−α i((1 + τs)αβ)1 ΓAs1 αΓ. 19 Using these functions, agent i’s utility, at the bδ−conditional equilibrium and if active in sector s,is e Vs(δi,bδ,ξ)≡Us i(e Hs(δi,bδ,ξ),e Ks(bδ,ξ)) = −(cs I+∆cs I+T)(A5) +∙(1 −α)(1−β) µ(1 −ζs)EΩs I(e δ)(δγ−1 i)¸(1+Γ)(1−α) αΓ ×δγ−1 iβ1+Γ ΓAs1+Γ αΓ[(1 + τs)α]1 Γ1+Γ−(1 + τs)α 1+Γ. Similarly, given an arbitrary bδ, firm j(ex-post) surplus, if active in sector sand matched with worker i,is e Πs(δi,bδ,ξ)=(1−β)As1+Γ αΓ((1 + τs)αβ)1 Γ³δγ−1 i−(1 −α)EΩs I(e δ)(δγ−1 i)´ ×µ(1 −α)(1−β) µ(1 −ζs)EΩs I(e δ)(δγ−1 i)¶(1+Γ)(1−α) αΓ .(A6) Its expected value is EΩs I(e δ)(e Πs(δi,bδ,ξ)) = ∙(1 −α)(1−β) µ(1 −ζs)EΩs I(e δ)(δγ−1 i)¸1+Γ−α αΓ (A7) ×µ(1 −ζs)α((1 + τs)αβ)1 ΓAs1+Γ αΓ (1 −α). Proof of Prop. 1. Set ξ=0,and omit it from the notation. Pick the partition ΩP I(bδ)induced by any arbitrary bδ. Assume that there is an agent i0such that δi0=bδat bδsolving (f(bδ)−ace I)=0.It is easy to check that e F(δi,bδ)≥0if and only if δi≥bδ. Hence, each equilibrium partition ΩP Isuch that Ωs I6=∅,eachs, satisfies Ωe I(δF)=hδF,1´,as claimed in the text. By direct computation, for each threshold bδ, EΩe I(e δ)(δγ−1 i)= 1 γ 1−bδγ 1−bδand EΩne I(e δ)(δγ−1 i)=bδγ−1 γ. Evidently, both functions are continuous at each bδ∈(0,1). Given that they are conditional expectations of a strictly increasing function,both are strictly increasing in bδ. Clearly, f(bδ)is continuous at each bδ∈(0,1).GiventhatEΩs I(e δ)(δγ−1 i),each s, is bounded, lim e δ→0 f(bδ)=0.Given that lim e δ→1 1−e δγ 1−e δ=∂(e δ) ∂e δ|e δ=1 =γ, lim e δ→1 f(bδ)=(γAe1+Γ αΓ−Ane 1+Γ αΓ)(1+Γ−α)≡ _ C>0. Hence, by the intermediate value theorem, for each ce Isuch that ace I∈(0, _ C),there is an interior solution to e F(δF,δF)=0. Using (A7), and given that EΩe I(e δ)(δγ−1 i)>E Ωne I(e δ)(δγ−1 i),and Ae>A ne, deF J=hEΩe I(δF)(e Πe(δi,δF)) −EΩne I(δF)(e Πne(δi,δF))i>0. 20 Hence, all the equilibrium conditions are satisfied at δF. This establishes the firstpartofProp.1. We now proceed to study uniqueness of equilibrium and its comparative statics properties. Observe that ∂h F(.) ∂∆ce I=−a<0,and that, by direct computation, ∂f(.) ∂τs=δFγ−1(AsEΩs I(δF)(δγ−1 i)(1−α))1+Γ αΓ(1 + Γ)(1−α) Γ(−1)ϕ(s)>0, with ϕ(e)=2and ϕ(ne)=1,so that ∂f(.) ∂τe>0and ∂f(.) ∂τne <0.Itisalsoeasy to see that ∂f(.) ∂Ae>0,while ∂f(.) ∂Ane <0.Hence, uniqueness of equilibrium, and the signs of the comparative statics properties, could be immediately established if the sign of ∂f(.) ∂e δ|e δ=δFwas uniquely defined. Unfortunately, this is not the case. As established in Example A1 below, there are economies with multiple equilibria and where, obviously, sign|∂f(.) ∂e δ|e δ=δF|varies across equilibria. Hence, to establish the second part of Prop. 1, we need to impose additional restrictions on the parameter space. By direct computation, at ξ=0, ∂f(.) ∂bδ=(γ−1) 1 bδf(bδ)+(1+Γ−α)(1 −α)(1+Γ) αΓbδγ−1 bδ ×[Ae1+Γ αΓEΩe I(e δ)(δγ−1 i)(1−α)(1+Γ) αΓηe α(bδ) −Ane 1+Γ αΓEΩne I(e δ)(δγ−1 i)(1−α)(1+Γ) αΓηne α(bδ)], where ηs α(bδ)is the elasticity of EΩs I(e δ)(δγ−1 i)with respect to bδ. By direct computation, ηne α(bδ)=(γ−1) ,while ηe α(bδ)=−γe δγ(1−e δ)+e δ(1−e δγ) (1−e δ)(1−e δγ).With a straightforward manipulation, we obtain Γbδ Γ−1 Γ 1+Γ−α∙γ(1−α) Ane ¸1+Γ αΓ∂f(.) ∂bδ =⎛ ⎝Ae Ane Ã1−bδγ (1 −bδ)bδγ−1!(1−α)⎞ ⎠ 1+Γ αΓµ(1 −α)(1+Γ) αηe α(bδ)+Γ(γ−1)¶−1. If ηe α(bδ)≥0at each bδ∈(0,1) ,and 1−e δγ (1−e δ)e δγ−1is bounded away from zero, the right hand side of the eq. above is always positive, for Ae Ane sufficiently large. Therefore, for Aelarge enough, ∂f(.) ∂e δ>0at each bδand, in particular, at each equilibrium threshold. Evidently, if ∂h F(.) ∂e δ³=∂f(.) ∂e δ´>0at each solution to e F(bδ,bδ)=0,the solution must be unique. Moreover, by the implicit function theorem, ∂f(.) ∂e δ|e δ=δF>0 at each equilibrium implies that δF(.)satisfies ∂δF(.) ∂τe<0,∂δF(.) ∂τne >0,∂δF(.) ∂∆ce I>0, ∂δF(.) ∂Ae|ξ=0 <0and ∂δF(.) ∂Ane >0,as claimed. Hence, to conclude, we need the two additional results mentioned above (we omit the index "b" to simplify notation): 21 Fact 1. ηe α(δ)≥0,at each δ∈(0,1) . By direct computation, ηe α(0) = 0 and ηe α(1) = γ−1 2>0.Hence, either there is δ∈(0,1) such that ηe α(δ)=0or ηe α(δ)>0for each δ∈(0,1) ,as claimed. Consider the numerator of ηe α(δ),callitg(δ), g(δ)=−γδγ(1 −δ)+δ(1 −δγ). Given that the denominator is strictly positive for each δ∈(0,1) ,η e(δ)≤0if and only if g(δ)≤0.Clearly, g(0) = g(1) = 0.Given that ∂g(.) ∂δ =¡1−γ2δγ−1+¡γ2−1¢δγ¢, ∂g(.) ∂δ |δ=0 >0and ∂g(.) ∂δ |δ=1=0.Moreover, ∂2g(.) ∂δ2|δ=1=γ¡γ2−1¢δγ−1−γ2(γ−1) δγ−2=γ(γ−1) >0, so that δ=1is a local minimum of g(δ).Hence, if there is a eδ∈(0,1) such that g(eδ)=0,theremustalsobeaδ∈(0,1) such that g¡δ¢=0and ∂g(.) ∂δ |δ=δ> 0.Given that, by assumption, δ∈(0,1) , δ 6=0,and, therefore, g(δ) δ=0,and µ∂g(.) ∂δ |δ=δ−g(δ) δ¶>0.However, 0<∂g(.) ∂δ |δ=δ−g(δ) δ=−γ2δγ−1+¡γ2−1¢δγ+γδγ−1¡1−δ¢+δγ =¡γ−γ2¢¡1−δ¢δγ−1<0, because γ>1.A contradiction. Hence, g(δ)>0and, therefore, ηe α(δ)>0,at each δ∈(0,1). Fact 2. Let G(δ)≡³1−δγ 1−δ δ δγ´.Then, G(δ)>γ>1,for each δ∈(0,1) . The result is quite obvious from the geometrical viewpoint. Alternatively, observe that lim δ→0G(δ)=+∞and lim δ→1G(δ)=γ. Hence, to establish the Fact, it suffices to show that ∂G(δ) ∂δ <0at each δ∈(0,1) .By direct computation, ∂G(δ) ∂δ |δ=e δ=γ (1 −bδ)bδγÃ1 γ 1−bδγ 1−bδ−1!=γ (1 −bδ)bδγ³EΩe I(e δ)(δγ−1 i)−1´<0. EXAMPLE A1. We show that there are economies such that ∂f(.) ∂e δ|e δ=δF<0. Fix ξ=0.Let α=1 2,Γ=10,A ne =1,and Ae=11/10.By direct computation, f(bδ)=10.5µ105 110¶11 10 ⎡ ⎢ ⎢ ⎣bδ 1 21 ⎛ ⎜ ⎝11 10 ⎛ ⎝1−bδ 110 105 1−bδ⎞ ⎠ 1 2⎞ ⎟ ⎠ 11 5 −bδ 1 10 ⎤ ⎥ ⎥ ⎦. 22 ∂f(e δ) ∂e δis strictly positive for bδsufficiently small, and negative for all bδlarger than some critical value δ. For instance, one can check that ∂f(e δ) ∂e δ|e δ=1 2<0,while n∂f(e δ) ∂e δ|e δo∞ v=0 with bδv→0is unbounded above. Clearly, choosing appropriately ce I,we can construct an economy with δF=1 2,i.e., such that ∂f(e δ) ∂e δ|e δ=δF<0. AsarguedinRemark 1, this implies that, for some values of ce I,this economy has multiple equilibria. Inefficiency properties of the economy with frictions The optimal choice ks jis clearly j−invariant and, by assumption, ν(Ωs I(bδ)) = ν(Ωs J(bδ)).Hence, the planner’s objective function can be rewritten as P(hs i,ks,bδ)≡X sZΩs I(e δ)ÃβAshsα iks(1−α)−1 δi hs(1+Γ) i 1+Γ!di −cs Iν(Ωs I(bδ)) +X sÃ(1 −β)AsRΩs I(e δ)hsα idi ν(Ωs I(bδ)) ks(1−α)−µks!ν(Ωs J(bδ)) =X sZΩs I(e δ)ÃAshsα iks(1−α)−1 δi hs(1+Γ) i 1+Γ!di −(cs I+µks)ν(Ωs I(bδ)). Its optimization problem is max (hs i,ks,e δ) P(hs i,ks,bδ).(P) It is convenient to decompose (P)into three problems. First, given an arbitrary value bδ, we determine the maps (HCOs(δi,bδ),KCOs(bδ)) solving, for each s,the optimization problem max (hs i,ks)Ps e δ(hs i,ks)≡ZΩs I(e δ)"Ashsα iks(1−α)−1 δi hs(1+Γ) i 1+Γ#di (Ps δ∗) −(cs I+µks)ν(Ωs I(bδ)). Next, given the value functions Ps(bδ)of the two problems (Ps e δ),s=ne, e, we can recast problem (P)as max e δ P(bδ)≡Pe(bδ)+Pne(bδ),(P) finding the optimal value of bδ, δCO. ProofofProp.3. Given that optimization problem (Ps e δ)is concave, each s, its solution is completely characterized by the FOCs: i. ∂Ps e δ(hs i,ks) ∂hi=αAsks(1−α)hs(α−1) i−1 δihsΓ i=0, ii. ∂Ps e δ(hs i,ks) ∂k =(1−α)Asks(−α)RΩs I(e δ)hsα idi −µRΩs I(e δ)di =0, which imply 23 a. KCOs(bδ)=As1+Γ αΓα1 Γ³1−α µEΩs I(δγ−1 i)´1+Γ−α αΓ, b. HCOs(δi,bδ)=δ 1 1+Γ−α iα1 ΓAs1 αΓ³1−α µEΩs I(δγ−1 i)´1−α αΓ. Comparing a−bto (A3)−(A4),KCOs(bδ)>e Ks(bδ)and HCOs(δi,bδ)>e Hs(δi,bδ),for each bδ, δiand s. Therefore, equilibria are always characterized by underinvestment in physical capital and in the effort in education. Demand and supply functions are clearly well-defined and continuous at each bδ∈(0,1).By substituting in the objective function the optimal values (KCOs(bδ),HCOs(δi,bδ)),weobtain bP(bδ)≡αΓ 1+ΓX s v(Ωe I(bδ))As1+Γ αΓEΩs I(e δ)(δγ−1 i)1+Γ−α αΓ−v(Ωe I(bδ))bce I where 1 b≡α1 Γ³1−α µ´(1−α)(1+Γ) αΓ.Given that P(bδ)is a continuous function, problem ¡P¢has a solution, either internal or at one of the boundary points, and, therefore, CO allocations exist. Compare a market allocation and any CO allocation. If δCO =δF=bδ, KCOs(bδ)6=e Ks(bδ)and the market allocation is not CO. Otherwise, δCO 6=δF and constrained inefficiency follows immediately. To establish the second part of Prop. 3, observe that, by direct computation and rearranging terms, the (necessary) FOC of problem ¡P¢can be written as 0=γb∂P(bδ) ∂bδ=γαAne 1+Γ αΓEΩne I(e δ)(δγ−1 i)1+Γ−α αΓ+Ae1+Γ αΓEΩe I(e δ)(δγ−1 i)(1+Γ)(1−α) αΓ ×Ã(1 −α)1−bδγ 1−bδ−bδγ−1!+γbce I. Hence, δCO is either the solution to γbce I=MCO(bδ)≡Ae1+Γ αΓEΩe I(e δ)(δγ−1 i)(1+Γ)(1−α) αΓ×Ãbδγ−1−(1 −α)1−bδγ 1−bδ! −γαAne 1+Γ αΓEΩne I(δγ−1 i)1+Γ−α αΓ or δCO ∈{0,1}.The condition defining the equilibrium value δF(i.e., e F(bδ,bδ)=0) canberecastedas γbce I=MF(bδ)≡(β(1 −β)1−α α)1+Γ Γbδγ−1Ae1+Γ αΓEΩe I(e δ)(δγ−1 i)(1−α)(1+Γ) αΓ −(β(1 −β)1−α α)1+Γ Γbδγ−1Ane 1+Γ αΓEΩne I(e δ)(δγ−1 i)(1−α)(1+Γ) αΓ. Evidently, MF(0) = 0,while it is easy to verify that MCO(0) <0.Therefore, existence of the lower bound ce Iwith the stated properties follows by continuity. EXAMPLE A2. Let Γ=µ=Ane =1,while α=β=1 2,and Ae=2.Using the expressions above, MCO(bδ)≡24⎛ ⎝3 4 1−bδ 4 3 1−bδ⎞ ⎠ 2⎛ ⎝bδ 1 3−1 2 1−bδ 4 3 1−bδ⎞ ⎠−2 3µ3 4bδ 4 3¶3 =32 3ce I 24 The last two terms are always strictly positive. Given that, by definition of δF,the term in square brackets is zero,at δF,∂P(e δ) ∂e δ>0.Hence, δCO >δ F,so that we have always overinvestment in education level, at the equilibrium. 9. 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