A simple model of unemployment, wage stickiness and ZMP

Following up on yesterday’s discussion of wage stickiness for the unemployed and the employed (Tyler, Alex).

Imagine a farmer whose farm produces 100 bushels of wheat. He hires 10 workers to bring in the wheat, paying each of them 9 bushels. Thus, each worker carries 10 bushels, the wage is 9, the wage bill is 90, and the farmer earns 10.

Now suppose that due to climate change or a swarm of locusts the farm only produces 90 bushels of wheat. If wages were fully flexible then an equilibrium exists in which each worker is paid a wage of 8 leaving the farmer with 10 bushels as before. The farmer doesn’t want to reduce everyone’s wages, however, because that will reduce morale so he fires one worker leaving nine. Each worker now brings in 10 bushels, as before, and is paid a wage of 9, for a total wage bill of 81 leaving the farmer with 9 bushels. The unemployment rate is 10%.

The unemployed worker doesn’t want to be unemployed and offers to work for less, a lot less, say 5 bushels. Even at the lower wage, however, the farmer doesn’t want to hire the worker because the worker doesn’t generate enough additional output to justify even a low wage. In fact, in this scenario the worker has ZMP.

The best the farmer can do in response to the lower wage offer by the unemployed worker is to fire an employed worker and hire the unemployed worker at the lower wage. Eventually this will restore equilibrium but it takes time to cycle through enough firings and hirings to reach full employment. Note also that in this model the farmer only has a weak incentive to do this since in the equilibrium with 10% unemployment he earns 9, almost as much as before.  As an aside, also note that in my model the unemployed workers are simply unlucky (as I argued earlier). If they were to switch places with the employed, productivity would be just as high. The unemployed worker has ZMP but is not a ZMP worker.

Since the driving shock that lowers productivity in this model is a real factor (weather, locusts), this is a real business cycle model . That raises a very interesting point. The most that wage flexibility can do in this model is to restore full employment; wage flexibility cannot restore full output. Thus, the workers in this model have a very good reason to dislike wage flexibility. In the equilibrium in which wages fall the unemployed worker is better off by a lot but the 9 employed workers are all worse off than in the unemployment equilibrium.  In contrast, in a Keynesian model wage flexibility can restore full output not just full employment. Thus, and somewhat surprisingly, it’s easier to justify wage stickiness in an RBC model than in a Keynesian model since the gains from wage flexibility are so much higher in the latter!

Even in a Keynesian model along the lines of the Sweeney/Krugman babysitter model it will still be the case that lower wages by the unemployed don’t get you far enough to restore equilibrium–although as noted, we will need a coordination failure story in the Keynesian model since in principle everyone would be better off in that model with wage flexibility.


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