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Choosing the Subscribers

by F. Gaspart (ECRU/UCL) and B. Verheyden (CEPS-INSTEAD)‏. Choosing the Subscribers. Motivations (1). How do young institutions start ? No State => local provision of public goods No enforcement => non-cooperative framework

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Choosing the Subscribers

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  1. by F. Gaspart (ECRU/UCL) and B. Verheyden (CEPS-INSTEAD)‏ Choosing the Subscribers

  2. Motivations (1) • How do young institutions start ? • No State => local provision of public goods • No enforcement => non-cooperative framework • Voluntary subscription minimal authority : a Principal choosing the group of subscribers

  3. Motivations (2) • Lack of social security => investment motive for children if they exhibit ascending altruism • Endogenous fertility = choosing the number of subscribers • Education expenses = choosing the endowments of the subscribers

  4. Literature on fertility and education • Becker & Lewis (1973) : consumption motive • Caldwell, Boldrin & Jones (2002) : investment motive with ascending altruism • Baland & Robinson (2000) : short-term income

  5. Our contribution • A theory of young institutions • A unified framework for all three motives of endogenous fertility and education • Individualized returns to education, revealed after birth => unequal education expenses • The role of savings in demographic transitions

  6. Timing • (1) The Principal P chooses n, the size of the group N of subscribers. • (2) Nature reveals the vector k of individual returns to education. • (3) P chooses the vector e of education expenses and the level of savings s. • (4) Each agent i in N chooses ti, his contribution to the public good.

  7. Utilities P's utility is V(a;b;B(n,f))‏Short-term consumption : a = I(n) - s - eiOld-age consumption (public good) : b = rs + T Transfers : T =  ti*Consumption motive : B(n,f) Agent i's program is ti* = argmax Ui ( ci ; b )‏Private consumption : ci = fi-ti (C =  ci)Returns to education : fi = f(ei,ki) (F =  fi*)

  8. Relevant assumptions ? (1) Strict binormality of agent's goodssay U( c,b ) = c + u(b) => T = u'-1(1) – r s => s* = e* = 0 and n* = argmax I(n)‏ The vehicle of causality : income effects on transfers to parents Eviction of savings Collective action between children

  9. Relevant assumptions ? (2) I(n)‏ is concave and has an interior maximum. P is risk neutral. The component B(n,f) : For the sake of clarity, say B is a constant for now. The discussion will state whenever other assumptions mitigates our results.

  10. Voluntary subscription : a reminder For a clear explanation of the proofs, see Cornes & Hartley (2007). Existence and unicity Comparative statics :A transfer among positive subscribers leaves equilibrium consumptions unaffected. leaves total contribution unaffected Agents with the same utility function contribute the same amount.

  11. The Principal's problem (1) Partly a profit maximization problem :for a given total expenditure E on education, e* = argmax T = argmax F is necessary. Convex returns to education : Empirically relevant Theoretically trivial : invest in one agent only, s*=0, « stopping rule » on n* (as Ejrnaes & Pörtner, 2004). Concave returns to education in the sequel.

  12. The Principal's problem (2) Basic idea : the marginal rate of substitution between components a and b is equal to the marginal return on savings, eviction included. to the marginal impact of education on transfers, dilution included. Take the sum of all agents' budgets in equilibrium :ci*(b) = fi-ti* => T = -C(T+rs) + F(E,k)

  13. The Principal's problem (2b) Call m = 1 / ( 1+C'(b) ) ; from the implicit function theorem, we have : Eviction : dT/ds = -r(1-m) < 0 Dilution : dT/dE = dT/dei = f' m < f' Proposition 1: for all agent with a positive education, r = f'(ei,ki) Proof : dV/ds = -Va + Vb (r+dT/ds) = 0dV/dE = -Va + Vb dT/dE = 0

  14. Choosing the number of subscribers Risk-neutrality : simply EΣ f(ei,ki) = F*(E,n) . Proposition 2 : if s* is interior, n* ≥ argmax I(n)‏ Proof :Having a value T for a given E, V is maximized w.r.t. E and s. Take expectations : T* = F*-C(T*+rs*) .n* = argmax V ( I(n)-E*-s*, T*+rs*, B) By the envelope theorem, we have :dV/dn = Va I'(n*) + Vb dT*/dn For E* fixed, dT*/dn ≥ 0 ; therefore if I'(n)>0, dV/dn >0 .The only case where dT*/dn=0 is when the marginal agent doesn't transfer anything.

  15. Predictions Proposition 3 : if s* is interior, dn*/d‏r < 0 Proof : T* = F*(E*,n)-C*(T*+rs*) => d(dT*/dn*)/dr <0 . Empirically relevant case : convexity between school types, strong local concavity at completion years. Becker's consumption motive mitigates inequalities among children, directly affects n* and diminishes | dE*/dr | .

  16. Borrowing or saving constraints Proposition 4 : if s* is at a corner in 0, n*>argmax I(n), but n* is smaller than if r were as high as F'(E). Proof of the first part :Again, dT/dn* ≥ 0 and I'(n)>0 => dV/dn >0 If s*>0 but a constraint is limiting savings, the Principal chooses more subscribers because of the constraint.

  17. Conclusions No concavity of returns to education means no investment motive for fertility. The eviction of savings and dilution of altruistic incentives cancel each other. Education is driven by return rates. A demographic transition can occur early in the modernization process if good savings opportunities arise. Borrowing constraints reduce fertility, saving constraints stimulate fertility.

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