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.
This is great! A good concrete operationalisation.
I think this is tacitly assuming that capital ownership/income is already uniformly distributed across all workers, so the only exogenous variable is the capital accumulation. That’s addressing part of the question quite nicely—though I was also wondering about the changing distribution (and prevalence) of capital ownership. I don’t think this single representative agent is what cashes out if we imagine that distribution changing...? But it’s a nice illustration of some kind of ‘universal ownership’ endpoint.
I’m also not sure of the Cobb-Douglasness of utility on consumption and leisure inputs! This seems to imply that, for a given leisure allowance, utility grows faster in consumption than I’m given to believe it does in practice. It’s super interesting that things cancel out in this case, though. That said, it’s an equivalent maximisation to treating utility as logarithmic in consumption plus logarithmic in leisure, which is not crazy.
This gives me some great inspiration if I want to come back to this and look at more modelling variations. Thanks,
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 N agents with identical preferences (the same utility function as above) but different capital allocations Ki. Then, total labor supply (the sum of all agents’ labor supplies) would be
L(w,r)=βNLmax−(1−β)r∑iKiw This leads to an equivalent result for total labor supplied: L∗=(1−α)β1−αβNLmax. 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.
You’re totally right, the representative agent actually was representative! So (unsurprisingly) this directs attention to how capital ownership might be distributed between (workers over different) sectors.
I noticed something funny about the optimal L: it’s negative if wages are sufficiently dwarfed by capital interest. But this makes sense: if they could, these workers would buy more time in the day for leisure by sacrificing some income.
I’d guess (haven’t mathed it through though) that general CES utility and production functions don’t produce this independence of total labour on total capital, and that this is a special Cobb-Douglass case. But it’s also not totally implausible. (After all, indeed humans have tended to work a lot on the whole throughout the last century or two of unprecedented capital accumulation, despite Keynes’ predictions.)
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 L∗ would go down as rK 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.
Right, I think positional goods and the like are among several distortions of the basic premises of the welfare theorems (and indeed empirically many people are sad, lonely, etc. in our modern world of abundance) - I sometimes think those theorems imply a sort of normative ‘well, just don’t worry about other people’s stuff!’ (i.e. non-envy, which is, after all, a deadly sin). cf Paretotopia, which makes exactly this normative case in the AI futurism frame.
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.
Well, why do we envy? Evopsych just-so story says that, of course, others having much impinges on me whether I want it to or not (my security/precarity, liberty, relative success in interactive situations, …).
I think you can express that in a ‘goods’ framing by glossing it as the consumption of ‘status and interactive goods’ or something like that (but noting that these are positional and contingent on the wider playing field, which the bog standard welfare theorem utility functions aren’t).
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.
This is great! A good concrete operationalisation.
I think this is tacitly assuming that capital ownership/income is already uniformly distributed across all workers, so the only exogenous variable is the capital accumulation. That’s addressing part of the question quite nicely—though I was also wondering about the changing distribution (and prevalence) of capital ownership. I don’t think this single representative agent is what cashes out if we imagine that distribution changing...? But it’s a nice illustration of some kind of ‘universal ownership’ endpoint.
I’m also not sure of the Cobb-Douglasness of utility on consumption and leisure inputs! This seems to imply that, for a given leisure allowance, utility grows faster in consumption than I’m given to believe it does in practice. It’s super interesting that things cancel out in this case, though. That said, it’s an equivalent maximisation to treating utility as logarithmic in consumption plus logarithmic in leisure, which is not crazy.
This gives me some great inspiration if I want to come back to this and look at more modelling variations. Thanks,
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 N agents with identical preferences (the same utility function as above) but different capital allocations Ki. Then, total labor supply (the sum of all agents’ labor supplies) would be
L(w,r)=βNLmax−(1−β)r∑iKiw
This leads to an equivalent result for total labor supplied: L∗=(1−α)β1−αβNLmax. 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.
You’re totally right, the representative agent actually was representative! So (unsurprisingly) this directs attention to how capital ownership might be distributed between (workers over different) sectors.
I noticed something funny about the optimal L: it’s negative if wages are sufficiently dwarfed by capital interest. But this makes sense: if they could, these workers would buy more time in the day for leisure by sacrificing some income.
I’d guess (haven’t mathed it through though) that general CES utility and production functions don’t produce this independence of total labour on total capital, and that this is a special Cobb-Douglass case. But it’s also not totally implausible. (After all, indeed humans have tended to work a lot on the whole throughout the last century or two of unprecedented capital accumulation, despite Keynes’ predictions.)
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 L∗ would go down as rK 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.
Right, I think positional goods and the like are among several distortions of the basic premises of the welfare theorems (and indeed empirically many people are sad, lonely, etc. in our modern world of abundance) - I sometimes think those theorems imply a sort of normative ‘well, just don’t worry about other people’s stuff!’ (i.e. non-envy, which is, after all, a deadly sin). cf Paretotopia, which makes exactly this normative case in the AI futurism frame.
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.
Well, why do we envy? Evopsych just-so story says that, of course, others having much impinges on me whether I want it to or not (my security/precarity, liberty, relative success in interactive situations, …).
I think you can express that in a ‘goods’ framing by glossing it as the consumption of ‘status and interactive goods’ or something like that (but noting that these are positional and contingent on the wider playing field, which the bog standard welfare theorem utility functions aren’t).