During the last month I found myself entrapped in a series of discussions concerning the best time to have a child. While not my favourite topic in the world, deciding when to have a child is one of the biggest decisions a household has to make. This initiated my interest to examine this decision by using simple economic modelling. (Note: I omit the mathematical characterisation of the model in this post, but hopefully I will post it in the near future.)
Of course my simple analysis omits several factors that come into play in the decision making process, but I find that key features of such a decision are still illustrated. Before moving on to the results, I should identify my simplifying assumptions:
A1: Each household can only have one child.
A2: Each household has two "parents".
A3: Before a child is born, both parents earn wages and receive the same wage in each period.
A4: After the child is born, only the highest paid parent continues to work.
A5: Parents live for T years.
A6: Parents derive a constant pleasure each period from having a child and exert a constant effort each period by raising a child. But total effort increases the longer it takes to have a child (i.e., younger parents are more “efficient”).
The key assumptions are A2 & A4, which collectively boil down to the requirement that parents decrease their time at work in order to raise their child. So, A2 & A4 can be replaced by requiring parent(s) to shift some of their "working time" towards "raising time". Assumption A1 is taken so as to simplify the analysis, while A3 is made in order to generate a difference in income after a child is born. Finally, A5 can be considered as the life expectancy.
In essence, the problem of deciding when to have a child is an optimal stopping problem, where the household stops to be childless and switches to parenthood. So, given that households have finite lives A5 is our constraint.
In the simplest model, households cannot accumulate savings and yet the results seem intuitive. On the one hand, I find that the higher the wages the household has to forfeit from when it has a child, the longer it takes to have a child. Similarly, the longer the household expects to live, the longer it will postpone parenthood. On the other hand, the higher the pleasure the household derives from having a child, the sooner it will decide it have a child. Effort has exactly the same effect as pleasure. That is, the higher the per period effort that needs to be exerted, the sooner the child will come. It's also interesting to note that pleasure and effort have the most profound effect, followed by forfeited wages and life expectancy.
Then I proceed to extend the model in order to allow for savings, child and general costs (note: costs are the same each period). Moreover, I impose the requirement that in equilibrium the household must either spend or leave as bequest all of its earned income. This is a neat condition as it gives us the optimal time to have a child (since it must hold in equilibrium).
In the extended model, the opportunity cost of having a child comprises of lost wages and child costs. The effect of the latter is ambiguous, while the effect of lost wages is clear-cut. In particular, the higher the wage (that has to be given up) the sooner the household will choose to have a child. This might seem counterintuitive at first as it contradicts the results of the simple model above. However, recall that this wage is earned prior to the child birth, so the higher it is, the faster the household will accumulate sufficient savings in order to have a child.
This channel allows the wage earned after the child is born to also affect the optimal time to have a child. As in the case of forfeited wages, I derive a negative association. That is, the higher the wages after the child is born are, the sooner the child will come. In this case, the higher wage will enable the household to avoid depleting their savings too early, and thus will be able to support its child. Finally, as in the simple model, I find that effort and pleasure decrease the optimal time to have a child.
In conclusion, despite the model's simplicity it's interesting to find the varying effects wages can have. My initial hunch was that of a positive relationship, which was confirmed in the simplest model. However, it was intriguing to unravel the effects of savings and how they can reverse the effects of wages.
