Very difficult to speculate on a global basis. We see effects of less growth locally, e.g. when in country regions more and more people move to the cities (because of job opportunities and night life/culture), starting a cycle where only the old people stay behind - schools are closed when the last children are gone.
But these regional effects are balanced out by movements from other areas. Or in India, where the increase of abortion of girls (although it’s now officially forbidden to determine gender before birth, it’s apparently still easy to do; and the still high dowry makes girls too expensive for poor families) has lead to villages full of 20-something males, with nary a girl to marry in sight, leading to stealing girls from other villages, or buying them from the Philippines.
Similarly, when the reproduction rates in Western Europe go down partly because many couples can’t receive, some turn to fertility clinics, others look at Eastern European countries (in the US China and Asian countries or Africa) for easy-bought adoption.
But a global decrease? Do the people know that your device is causing this, or will they try desperately to receive? Is your set rate the standard replacement of one girl, one boy to exactly replace the parent pair, or (at least at first) less?
How will people (not only the economy) react if something - an accident, an illness, a famine/war, natural disaster - kills their children, and they can’t replace it?
What will the device do to couples who already can’t receive? If women stop ovulating once they’ve reached their limit, then all the selling of eggs for fertility research (and sperm for sperm banks), will stop. Also, all the research into stem cells will stop.
And the designer babies - babies specifically selected pre-implantantion to posess a cure for a sick sibling - will those be possible?
It could lead to much more overprotection of children, if everybody knows they can’t be replaced. Or to even more restrictions of their rights, because there are less children around than before.
Less children means not only less payers for all forms of generation-based retirement and social security and health insurance. It also means less workers - that can be a bonus, apart from the thorny tax/ social security net problem, because a lot of work is already done by machines, and employment places are becoming scarcer in the industrialized nations.
But it also means less consumers. If the machines make all the cars, but nobody buys them, then the current economy model won’t work anymore. We could start with a different model.
Depending on how your device works, individually every pair gets two children or statistically, every nation gets so and so many children each year, you could still have couples with no children and couples with five or six; or you could have lots of couples not ready for children suddenly having one or two. Maybe they will sell them off/ give them away to other couples, or there will be big state homes. Maybe all the third world countries will give their children to the first world countries.
Most likely, once people figured out that your device was causing this, they would send an army against you to destroy the device and kill you.
Because any changes a drastic globabl change like this would cause would require so many other adaptions to continue living that you could simply change the system without a magical device.