Estimating the Technology of Cognitive and Noncognitive Skill Formation

θ2 = γ1θ1 + γ2I1 + (1 - γ1 - γ2) θP .

For period two it is:

θ3 = min{θ2,I2,θP} .

These patterns of complementarity are polar cases that represent, in extreme form, the
empirical pattern found for cognitive skill accumulation: that substitution possibilities are
greater early in life compared to later in life.

The problem of society is to choose how much to invest in child A and child B in periods
1 and 2 to maximize total aggregate skills, θ₃^A + θ₃^B , subject to the resource constraint
I₁^A + I₂^A + I₁^B + I₂^B ≤ M , where M is total resources available for investment. Formally, the
problem is

min γ1θ₁^A + γ2I₁^A + (1 - γ1 - γ2) θ_P^A, I₂^A, θ_P^A +
min {γi^θB + Y2I1B ⁺ (1 ^- Y1 ^- Y2) ^θB^, IB^, θp }

subject to: I₁^A + I₂^A + I₁^B + I₂^B ≤ M

When the resource constraint in (4.6) does not bind, which it does not if M is above a
certain threshold (determined by θP), optimal investments are

I2^B = θP^B

Notice that if child A is disadvantaged compared to B on both measures of disadvantage,
(θ₁^A < θ₁^B and θ_P^A < θ_P^B), it can happen that

I₁^A > I₁^B , but I₂^A < I₂^B

if

θA - θB >   ^γ1   (θA - θΒ

P P   γ₁ + γ₂   1    1

Thus, if parental endowment differences are less negative than child endowment differences
(scaled by _γ^γ _γ ), it is optimal to invest more in the early years for the disadvantaged and
less in the later years. Notice that since (1 - γ1 - γ2) = γP is the productivity parameter on
θP in the first period technology, we can rewrite this condition as (θA — θB) > ɪ (θA — θB).
The higher the self-productivity (γ₁) and the higher the parental environment productivity,
γ_P , the more likely will this inequality be satisfied for any fixed level of disparity.

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