Estimating the Technology of Cognitive and Noncognitive Skill Formation

skills matter for performance in most tasks in life and have different effects in different tasks
in the labor market and in other areas of social performance. Outcomes include test scores,
schooling, wages, occupational attainment, hours worked, criminal activity, and teenage
pregnancy.

In this paper, we identify and estimate a CES version of technology (2.1) where we assume
that θ_C,t, θ_N,t, I_C,t, I_N,t, θ_C,P , θ_N,P are scalars. Outputs of skills at stage s are governed by

and

1

φ        __ ^φs      φΦN,N  I _       Λ^φs,N  I _       T-^φs,N  I _       Λ^φs,N  I _       Λ^φs,N  ^φs^,N

θN,t+1 = γs,N,1θC,t + γs,N,2θN,t + γs,N,3IN,t + γs,N,4θC,P + γs,N,5θN,P
where γs,k,l ∈ [0, 1], _l γs,k,l = 1 for k ∈ {C, N}, l ∈ {1, . . . , 5}, t ∈ {1, . . . , T } and s ∈
{1,...,S}. _1-φ _fe is the elasticity of substitution in the inputs producing θ_k,_t+₁, where
φs,k ∈ (-∞, 1] for k ∈ {C, N}. It is a measure of how easy it is to compensate for low levels
of stocks θC,t and θN,t inherited from the previous period with current levels of investment
I_C,t and I_N,t. For the moment, we ignore the shocks η_k,t in (2.1), although they play an
important role in our empirical analysis.

A CES specification of adult outcomes is:

Qj = nρj (Θc,t +ι)^φQ^j + (1 - Pj) (Θn,t +ι)^φQ^j O^φQ^j ,                  (2.5)

where ρj ∈ [0,1], and φQ_j∙ ∈ (-∞, 1] for j = 1,..., J. j—1— is the elasticity of substitution
1-φQ,j

between different skills in the production of outcome j . The ability of noncognitive skills
to compensate for cognitive deficits in producing adult outcomes is governed by φQ,j . The
importance of cognition in producing output in task j is governed by the share parameter
ρj.

To gain some insight into this model, consider a special case investigated in Cunha and
Heckman (2007) where childhood lasts two periods (T = 2), there is one adult outcome (“hu-
man capital”) so J = 1, and the elasticities of substitution are the same across technologies
(2.3) and (2.4) and in the outcome (2.5), so φ_s,C = φ_s,N = φQ = φ for all s ∈ {1, . . . , S}.
Assume that there is one investment good in each period that increases both cognitive and
noncognitive skills, though not necessarily by the same amount, (Ik,t ≡ It, k ∈ {C, N}). In
this case, the adult outcome is a function of investments, initial endowments, and parental

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