Choosing the Subscribers - PowerPoint Presentation

Slide1

Choosing the Subscribers

by

F. Gaspart (ECRU/UCL)

and B. Verheyden (CEPS-INSTEAD)‏Slide2

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 subscribersSlide3

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 subscribersSlide4

Literature on fertility and education

Becker & Lewis (1973) : consumption motive

Caldwell, Boldrin & Jones (2002) : investment motive with ascending altruism

Baland & Robinson (2000) :

short-term incomeSlide5

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 transitionsSlide6

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 t

i

, his contribution to the public good.Slide7

Utilities

P's utility is V(a;b;B(n,

f

))‏

Short-term consumption : a = I(n) - s -



e

i

Old-age consumption (public good) : b = rs + T

Transfers : T =



t

i

*

Consumption motive : B(n,

f

)

Agent i's program is t

i

* = argmax U

i

( c

i

; b )

‏

Private consumption : c

i

= f

i

-t

i

(C =



c

i

)

Returns to education : f

i

= f(e

i

,k

i

) (F =



f

i

*)Slide8

Relevant assumptions ? (1)

Strict binormality of agent's goods

say 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 childrenSlide9

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.Slide10

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.Slide11

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.Slide12

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 :

c

i

*(b) = f

i

-t

i

* => T = -C(T+rs) + F(E,

k

)Slide13

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/de

i

= f' m < f'

Proposition 1:

for all agent with a positive education, r = f'(e

i

,k

i

)

Proof :

dV/ds = -V

a

+ V

b

(r+dT/ds) = 0

dV/dE = -V

a

+ V

b

dT/dE = 0Slide14

Choosing the number of subscribers

Risk-neutrality : simply

E

Σ

f(e

i

,k

i

) = 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 = V

a

I'(n*) + V

b

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.Slide15

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 | .Slide16

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.Slide17

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.