Table 1.

Prove for a formal model of inference of parental signals

Panel A

A normal person expects a gain from social interaction. Formally, this means their assessment of s, the gain from social interaction, is positive, as the chance p of a positive interaction G exceeds their chance of a negative interaction L: Individuals are socially anxious if they have subjective beliefs p < p* = −L/(G−L), so that they expect a loss from social interaction (in the simple case when G = −L, an individual is socially anxious if his or her subjective assessment of p is below ½). Such individuals will rationally seek to minimize social interaction

Panel B

The child interprets the father’s action aF and mother’s action aM in an assessment of paternal and maternal social confidence. The child’s own belief is a weighted average of the two signals, each weighted by its perceived precision. Let the perceived precision (statistically, the reciprocal of the perceived variance of the estimate error) of the paternal signal be θF, while it is θM for the mother. The child’s resulting belief on p will be: Our assumption is that the child attributes to the father greater expertise regarding the external social environment relative to the mother. Proposition 1 is immediately proven under this simple condition: which is true if the father is very anxious (i.e., pF is low) or if paternal behavior is very influential (i.e., θF is high). Proof: Suppose the father is anxious, but the mother is not (if both are anxious, the proof is trivial). Because θF > θM (the child assumes that the father’s information about the external world is more precise than the mother’s), the child inference E(p) will be dominated by the weight of paternal anxiety. Under Condition I, their posterior belief is below the threshold for social anxiety p*

Panel C

The mother faces increasing costs of providing more comfort, as described by a function f(c), increasing and convex in c. This implies that more care is increasingly costly as it reduces income, sleep, or personal freedom. The optimal choice of maternal care c will maximize her preference, which equals child comfort minus her cost of providing maternal care: The optimal level of care c* satisfies the first order condition for maximization: which states that the mother will rationally provide care till the point where there is no net gain, namely, when the marginal increase in child comfort equals her marginal cost in providing it. Note that social confidence E(s) is not directly affected by care, and vice versa. Proposition 2 states that maternal care increases when a child has an anxious father and thus a lower E(s). The proof follows directly from the optimal mother’s choice of care c*, which says that she will supply care until its marginal cost equals the child’s marginal utility. Since lower social confidence E(s) reduces the child’s overall well being, it increases the comfort value of maternal care. The mother responds by increasing the amount of care she supplies, till the child’s marginal utility equals her increased cost of care