I encourage you to precisely spell out the consequences of assuming exogenous fertility, and the precise assumptions you are making. In the data, people choose location and fertility jointly.
What is a Location?
a tuple (wage, price of childcare type j, relative price of C)
all generated outside of the model - there is no mechanism in the model to set prices and wages.
In the real world some places have very limited supply of market childcare - rural areas. Why not incorporate those constraints explicitly?
Locations differ starkly by housing costs for people with and without children - absent from the model. (My bias from above.)
Calibration of opportunity cost to (observed) mothers’ wages: multi-way Selection problem?!
Using observed wage for non-participants yields biased estimates (Heckman)
high wage mothers move to expensive city (Urban literature)
might have lower fertility. (?)
Effect of tenure: cost of childcare after 7 years of tenure might drop because…the kid is now 7+ years old!
Counterfactuals and the Lucas Critique?
The plan is to use the estimated model to perform different policies.
The model is in partial equilibrium: all prices are estimated from data, there is no mechanism to set them inside the model.
Policy will change allocation of people across space, presumably changing prices, how will we interpret the experiments?
Suggestions
Suggestions
Focus more on the childcare story (starting in title?)
The choice of different childcare mixes adds a lot of complexity - try to make more out of this! Argue that we need this level of complexity (figure 6).
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