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Field
Primary Outcomes (Explanation)
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Before
Outcome 1: choice probability of the larger-family option, estimated via a linear probability model
with respondent fixed effects (and via a scaled/heteroscedastic logit as the structural
counterpart), on the reduced-form contrasts in money price (log income ratio) and time price
(non-leisure hours).
Outcome 2 -- the normal-good test: the registered output is the confidence interval on the
income-interaction parameter (mu) from the reduced-form specification, not a reject/fail-to-reject
verdict. Because the reduced-form model is a linear probability model, its coefficients measure
changes in choice probability, not utility; we report mu, its confidence interval, and the implied
change in the predicted probability of choosing the larger-family scenario across the displayed
income range (p25 to p90), with a confidence interval obtained by propagating coefficient
uncertainty. A tight interval bounding mu near zero is the substantive finding that the value of
children is flat as a share of income (normal in dollars but not in share). The structural
interaction is reported as a corroborating cross-check.
Outcome 3 -- willingness to pay: reported in dollars and in leisure/sleep hours for one and for two
additional children, by education stratum, from the structural model (which supplies the utility
metric; WTP is not derived from the reduced-form probability coefficients). WTP for each child
contrast is defined as the compensating change in income that equates utility between child levels
under the full structural utility function, including the income interaction, holding time fixed;
reported at each displayed income level with the p50 value as the headline within-stratum quantity.
We also report the parity-zero, one-versus-none willingness to pay as the first-birth extensive-
margin value.
Confirmatory tests (hidden until unlock): H1, demand slopes down in the money price; H3, demand
slopes down in the time price (both one-sided at 5%); H2, the income-interaction (normal-good)
test, reported as the confidence interval on mu. H1-H3 are treated as three distinct hypotheses
with no family-wise multiplicity correction.
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After
Outcome 1: choice probability of the larger-family option, estimated via a linear probability model
with respondent fixed effects (and via a scaled/heteroscedastic logit as the structural
counterpart), on the reduced-form contrasts in money price (log income ratio) and time price
(non-leisure hours).
Outcome 2 -- the normal-good test: the registered output is the confidence interval on the
income-interaction parameter (mu) from the reduced-form specification, not a reject/fail-to-reject
verdict. Because the reduced-form model is a linear probability model, its coefficients measure
changes in choice probability, not utility; we report mu, its confidence interval, and the implied
change in the predicted probability of choosing the larger-family scenario across the displayed
income range (p25 to p90), with a confidence interval obtained by propagating coefficient
uncertainty. A tight interval bounding mu near zero is the substantive finding that the value of
children is flat as a share of income (normal in dollars but not in share). The structural
interaction is reported as a corroborating cross-check.
Outcome 3 -- willingness to pay: reported in dollars and in leisure/sleep hours for one and for two
additional children, by education stratum, from the structural model (which supplies the utility
metric; WTP is not derived from the reduced-form probability coefficients). WTP for each child
contrast is defined as the compensating change in income that equates utility between child levels
under the full structural utility function, including the income interaction, holding time fixed;
reported at each displayed income level with the p50 value as the headline within-stratum quantity.
We also report the parity-zero, one-versus-none willingness to pay as the first-birth extensive-
margin value.
Confirmatory tests: H1, demand slopes down in the money price; H3, demand
slopes down in the time price (both one-sided at 5%); H2, the income-interaction (normal-good)
test, reported as the confidence interval on mu. H1-H3 are treated as three distinct hypotheses
with no family-wise multiplicity correction.
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