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 F(K,L)=AKαL1−α, and a representative agent with utility function U(C,L)=Cβ(Lmax−L)1−β where C is consumption, L is labor, K is capital, w is the wage, and r 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 Lmax−L). Labor supplied is the function L(w,r) that maximizes U under the constraint C=wL+rK (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
L(w,r)=βLmax−(1−β)rKw
Exactly as you suggested in your question, Oly, L is decreasing in returns to capital rK. In other words, if wage is held fixed, people work less when they have more passive income rK. However, L is increasing in w (as long as people have at least some passive income, otherwise L is independent of w). Now let's ask, as K increases through capital accumulation, will workers work more or less?
In equilibrium, w=∂LF(K,L)=(1−α)A(K/L)α and r=∂KF(K,L)=αA(L/K)1−α. Now we have a system of three equations (these two equations and the labor supply above) and, for fixed K, three unknowns (w,r,L). The equilibrium labor, L∗, that solves this system is
L∗=(1−α)β1−αβLmax
The equilibrium labor is independent of capital stock K! 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.
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 F(K,L)=AKαL1−α, and a representative agent with utility function U(C,L)=Cβ(Lmax−L)1−β where C is consumption, L is labor, K is capital, w is the wage, and r 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 Lmax−L). Labor supplied is the function L(w,r) that maximizes U under the constraint C=wL+rK (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
L(w,r)=βLmax−(1−β)rKw
Exactly as you suggested in your question, Oly, L is decreasing in returns to capital rK. In other words, if wage is held fixed, people work less when they have more passive income rK. However, L is increasing in w (as long as people have at least some passive income, otherwise L is independent of w). Now let's ask, as K increases through capital accumulation, will workers work more or less?
In equilibrium, w=∂LF(K,L)=(1−α)A(K/L)α and r=∂KF(K,L)=αA(L/K)1−α. Now we have a system of three equations (these two equations and the labor supply above) and, for fixed K, three unknowns (w,r,L). The equilibrium labor, L∗, that solves this system is
L∗=(1−α)β1−αβLmax
The equilibrium labor is independent of capital stock K! 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.