First published: February 2016
This post is a short formalized and English version of this previous post written in French.
Most economists consider that the lump of labor – the idea that the amount of work available to workers is fixed – is a fallacy and that, contrary to the naive belief, increasing productivity, immigration, women labor, and increasing the legal number of hours worked won’t mechanically increase unemployment. In this post, I clarify this issue and show that the lump of labor is not always a fallacy. In particular, if the economy is in a state of classical unemployment – which is likely to characterize the state of the French labor market-, the theory suggests that reducing the number of hours unambiguously reduces unemployment.
The WS-PS Model
The economy is populated by many workers/consumers and firms. The firms behave competitively in the sense that they choose their amount of labour and capital according to the marginal productivity principle: taking the real wage and real interest rate as exogenous, they increase their demand for labour and capital until the point where their marginal productivity is equal to their cost. This gives in particular the labor demand schedule, also called the PS curve (for price-setting): the lower the unemployed population, the lower the wage – because of decreasing marginal returns.
where L is labor, w hourly wage, K capital (held fixed), u the unemployment rate, A the hourly productivity of labor, H the number of hours and N the size of the working age population.
The second important block of the model describes the bargaining that determines the wage. The wage is the solution to a bargaining game between Labor Unions and Companies and the resulting schedule – labelled the WS curve for wage-setting – is, for common parameters, assumed to be a decreasing schedule of unemployment because high unemployment means low outside option and low bargaining power on the part of workers.
Where n is the rate of working-age population growth, q the probability of losing one’s job, Ve is the value of having a job and Vr is the value of being unemployed.
The consumers are intertemporal optimizers and are characterized by a Euler equation. However since we consider only steady-state equilibrium, the Euler simply related the psychological time discount factor of the consumers to the real interest rate. We abstract from this part of the model to focus on the labor market – as far as our main results are concerned this assumption is innocuous.
Steady-state Equilibrium in WS-PS
At steady-state, the two equations write:
WS is decreasing in (u,w) while PS is increasing. They cross only once. The equilibrium unemployment-wage is (u**,w**).
Two extreme cases have to be considered.
- The perfect market equilibrium in which wages are not determined by a bargaining process but chosen by the auctioneer or naturally reached through the Invisible Hand’s actions to clear the labor market is a situation in which WS becomes mixed-up with the w-axis. The equilibrium unemployment-wage is (0,w*).
- The other extreme case is the case of wage rigidities – such as the minimum wage regulations. In such case, the WS schedule is horizontal and the equilibrium unemployment-wage is (u***,w***).
Comparative Static : decrease in number of hours worked
What happen if the number of hours worked get decrease? From the two equations, it is clear that WS remains unaffected and PS moves upward. Since there is less supply of hours in the economy, for any wage, the number of hours available decreases which says exactly that PS moves upward. Besides, if one considers that the labor productivity is decreasing in the number of hours – because a worker is less tired and becomes more productive – then a second effect kicks in that makes labor more productive and therefore WS moves upward even more.
The upward shift in PS decreases unemployment (from black to red) unless the true WS is vertical – which is the free-market economy in which case the decrease in the supply of labor is entirely absorbed by a wage increase. In the more realistic theoretical cases of non vertical WS, the decline in unemployment is all the larger as the WS is flat, i.e. as the wages don’t increase with the proportion of unemployed people in the working age population, which occurs when firms have a large bargaining power or unions care a lot about unemployment risk. The decrease in unemployment is largest in the fixed wage economy – which occurs when the wages are set by the State which is the case for workers paid at the minimum wage in many developed economies such as France and the US.
References
Johnson and Kayard (1986), “The Natural Rate of Unemployment: Explanation and Policy”, Handbook of Labor Economics
Cahuc and Zylberberg (1999), “Le Modèle WS-PS”, Revue d’Economie et de Statistique