Good point about workers preferring negative L when they have a high passive income!
I think you’re right about using a CES utility function. My guess would be that for an elasticity of substitution (between consumption and leisure) less than 1, the equilibrium would go down as goes up (and vice versa for an elasticity greater than 1). I actually can’t decide whether I’d predict most people’s elasticity to be more or less than 1. A complicating factor, which you mentioned in your footnote 3, is that people’s preferences for wealth are highly influenced by the people around them (both by the wealth of those around them, and by the culture). I’ve always wanted to explore the economic consequences of this—my intuition is that it messes up all the classic “welfare theorems” in economics.
Casey Barkan
You’re right that the result could change if there were heterogeneous agents with an uneven distribution of capital. However, the model above can be reformulated to include an uneven distribution of capital, and the result remains mostly the same. Specifically, suppose that instead of the representative agent, there were agents with identical preferences (the same utility function as above) but different capital allocations . Then, total labor supply (the sum of all agents’ labor supplies) would be
This leads to an equivalent result for total labor supplied: . So the model still predicts that total labor supplied is independent of the capital stock, regardless of how capital is distributed. But, how much each agent contributes to that total labor supply does depend on the distribution of capital. If the distribution of capital changes, agents who gain a larger share of total capital will work less than before, and those whose share of total capital falls will work more than before.
Interesting question! As @harsimony alluded to, there are opposing effects here: increasing returns to capital may reduce labor supplied as people shift toward leisure, but increasingly scarce labor pushes wages up and pushes interest rates down, causing people to shift away from leisure toward labor. Let’s see how these competing effects play out in a simple macroeconomic model. The TLDR is that, in the simple model below, the two effects exactly cancel.
Suppose there is a competitive economy with production function , and a representative agent with utility function where is consumption, is labor, is capital, is the wage, and is the rental rate of capital (a.k.a. the interest rate). Note that the representative agent values both consumption and leisure (where leisure is ). Labor supplied is the function that maximizes under the constraint (this just says that the agent consumes all income, and we’ll neglect savings for simplicity). The labor supply that solves this maximization problem is
Exactly as you suggested in your question, Oly, is decreasing in returns to capital . In other words, if wage is held fixed, people work less when they have more passive income . However, is increasing in (as long as people have at least some passive income, otherwise is independent of ). Now let’s ask, as increases through capital accumulation, will workers work more or less?
In equilibrium, and . Now we have a system of three equations (these two equations and the labor supply above) and, for fixed , three unknowns (). The equilibrium labor, , that solves this system is
The equilibrium labor is independent of capital stock ! Hence, in this simple model, the two opposing effects exactly cancel: workers work a constant amount as their passive income from capital increases.
It’s certainly also true that one could write down a different model where these two effects don’t exactly cancel, and it’s possible you could cook up a model where something really pathological happens (perhaps where workers work so much less as their passive income increases that total output goes down?). But I think the simple model above is a good baseline for what to expect.
Agreed that successful sandbagging will likely require Schelling coordination, and my guess is that this will be extremely difficult for models to pull off! Great to see that you’re investigating this topic.
Definitely agree! The “don’t worry about other people’s stuff” argument gets thrown around a lot, and is often assumed to be equivalent to “don’t be envious”, but I think that argument actually contains a logical mistake. Suppose person A doesn’t care directly about their relative wealth (i.e. their utility function is independent of their relative wealth), but their utility function depends on their interactions with other people (such as friendships, job interviews, etc) and those other people interact with person A in a way that depends on person A’s relative wealth. Then, it can be instrumentally useful for person A to increase their relative wealth despite their utility function being independent of relative wealth! So person A has no envy, but their utility is (indirectly) affected by their relative wealth.