Revisiting the Marriage Supermarket

In comments to yesterday's post on the effects on dating style of a declining number of university men a number of people asked why a relatively small change in the sex ratio (m:w) from 50:50 to say 40:60 should make such a big difference.  In the Logic of Life, Tim Harford gave a characteristically excellent explanation.

Imagine, says Tim, a marriage supermarket.  In this supermarket any man and woman who pair up get $100 to split between them.  Suppose 20 men and 20 women show up at the supermarket, it's pretty clear that all the men and women will pair up and split the $100 gain about equally, $50,$50.  Now imagine that the sex ratio changes to 19 men and 20 women.  Surprisingly, a tiny change in the ratio has a big effect on the outcome.

Imagine that 19 men and women have paired up splitting the gains $50:$50 but leaving one woman with neither a spouse nor any gain.  Being rational this unmatched woman is unlikely to accede to being left with nothing and will instead muscle in on an existing pairing offering the man say a $60:$40 split.  The man being rational will accept but this still leaves one women unpaired and she will now counter-offer $70:$30.  And so it goes.

If you follow through on the logic it becomes clear that in the final equilibrium no married (paired) woman can be significantly better off than the unmarried woman (otherwise the unmarried woman would have an incentive to muscle in with a better deal) and so because the unmarried woman gets nothing the married women can't get much more nothing.  Thus when the sex ratio is 20:20 the split is $50:$50 and when the sex ratio is 19:20 the split is more like to $99:$1 in favor of the men.

The key simplification of the marriage supermarket is that the next best option to marriage (pairing) is worth $0–thus there is a long way to fall from the equal sex ratio equilibrium of $50.  If the outside option is worth more then changes in the sex ratio will have smaller effects.  Nevertheless, the logic of the marriage supermarket explains why a relatively small change in the sex ratio can lead to a large change in sexual and other mores affecting the marriage equilibrium.


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