Estimating the Technology of Cognitive and Noncognitive Skill Formation

Skills evolve in the following way. Each agent is born with initial conditions θ1 =
(θC,1 , θN,1). Family environments and genetic factors may influence these initial conditions
(see Olds, 2002, and Levitt, 2003). We denote by θ_P = (θ_C,P , θ_N,P ) parental cognitive and
noncognitive skills, respectively. θt = (θC,t, θN,t) denotes the vector of skill stocks in period
t. Let ηt = (ηC,t, ηN,t) denote shocks and/or unobserved inputs that affect the accumulation
of cognitive and noncognitive skills, respectively. The technology of production of skill k in
period t and developmental stage s depends on the stock of skills in period t, investment at
t, Ik,t, parental skills, θP , shocks in period t, ηk,t, and the production function at stage s :

θ_k,t+1 = f_k,s (θ_t, I_k,t, θ_P , η_k,t) ,                                      (2.1)

for k ∈ {C, N}, t ∈ {1, 2, . . . , T }, and s ∈ {1, . . . , S}. We assume that fk,s is monotone
increasing in its arguments, twice continuously differentiable, and concave in I_k,t . In this
model, stocks of current period skills produce next period skills and affect the current pe-
riod productivity of investments. Stocks of cognitive skills can promote the formation of
noncognitive skills and vice versa because θt is an argument of (2.1).

Direct complementarity between the stock of skill l and the productivity of investment
Ik,t in producing skill k in period t arises if

∂2^fk,s(∙) > 0, t ∈{1,...,T}, l,k ∈{C,N}.

k,t l,t

Period t stocks of abilities and skills promote the acquisition of skills by making investment
more productive. Students with greater early cognitive and noncognitive abilities are more
efficient in later learning of both cognitive and noncognitive skills. The evidence from the
early intervention literature suggests that the enriched early environments of the Abecedar-
ian, Perry and Chicago Child-Parent Center (CPC) programs promoted greater efficiency in
learning in schools and reduced problem behaviors.⁹

Adult outcome j , Qj , is produced by a combination of different skills at the beginning of
period T + 1:

Qj = gj (θC,T +1, θN,T+1) , j ∈ {1, . . . , J}.¹⁰                        (2.2)

These outcome equations capture the twin concepts that both cognitive and noncognitive

⁹See, e.g., Cunha, Heckman, Lochner, and Masterov (2006), Heckman, Malofeeva, Pinto, and Savelyev
(2010), Heckman, Moon, Pinto, Savelyev, and Yavitz (2010b), Heckman, Moon, Pinto, Savelyev, and Yavitz
(2010a), and Reynolds and Temple (2009).

¹⁰ To focus on the main contribution of this paper, we focus on investment in children. Thus we assume that
θT +1 is the adult stock of skills for the rest of life contrary to the evidence reported in Borghans, Duckworth,
Heckman, and ter Weel (2008). The technology could be extended to accommodate adult investment as in
Ben-Porath (1967) or its generalization Heckman, Lochner, and Taber (1998)

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