I'm not sure how convincing I find this piece, by Alessandro Lizzeri and Marciano Siniscalchi, but it's worth a read if you've been following the debates over twin adoption studies. In the model, parents both expose their children to learning and protect them. Parents also judge a successful child by how much that child has their abilities, so in equilibrium the more a child differs from a parent, the more the parent intervenes to direct the path of the child's development.
Again, in the model children of "better" parents have better outcomes on average, above and beyond genetic transmission as a mechanism. When it comes to adopted children, parents intervene more and parents also bring more similar interventions to bear on identical twins than on fraternal twins.
In twin adoption studies it will appear that parenting does not matter when in fact it does. Here is a neat passage from the paper:
Thus, differential sheltering by biological and adoptive parents provides a countervailing force to common rearing…We can also reinterpret this countervailing force…Differential sheltering by biological and adoptive parents implies that, despite being reared apart, adopted twins are in fact subject to a shared environmental influence–namely, adoption itself. This intuitively leads to greater phenotypic correlation by compensating for the lack of the direct commonrearing effect.
The piece ended up being published in the QJE 2008.
For the pointer I thank a loyal MR writer.