Thanks for this. If I understand correctly, the result is primarily driven by the elastic labor supply, which is a function of W and not of R, and the constant supply of capital. This seems most relevant for very fast takeoff scenarios.
My intuition is that as people realize that their jobs are being automated away, they will want to work more to bolster their savings before we move into the new regime where their labor is worthless and capital is all that matters. This would require fully modeling the household’s intertemporal utility and endogenizing the capital supply. This might be tricky, however, with your production function, because if A_{auto} is increasing at a constant rate and households are allowed to save, you will get superexponential growth.
Hi Caleb, thanks for your comment! That’s a great point that workers’ desire for savings prior to their jobs being automated may drive people to work more (similarly, people may work more as wages drop in order to maintain a subsistence level of consumption). On the other hand, if wages drop so low that workers can’t subsist, then labor supplied will drop regardless.
When you say it would be tricky to model savings with my production function, is the reason that superexponential growth with exponential time discounting in the utility function will lead to diverging utilities? That’s an interesting point. Do you know if there has been work on more realistic utility functions to address this issue?
Diverging utilities can be an issue. You can also get infinite output in finite time. The larger issue is that the economy has no steady state. In economic growth models, a steady state (or balanced growth path (BGP)) represents a long-run equilibrium where key economic variables per capita (like capital per worker, output per worker, consumption per worker) grow at constant rates. This greatly simplifies the analysis.
For example, I have a paper in which I analyze how households would behave if they expected TAI to transform the economy. To do this, I calculated the steady state the economy was in prior to households learning about the potential of TAI as well as the post-TAI steady state. I could then calculate the transition path between these two steady states. I thought of using the same production function you used here, but then there wouldn’t have been a post-TAI steady state, which is necessary to be able to find the transition path.
A production function that I have mused about is one like yours, but with land. This should solve the issue, as the post-TAI economy will no longer be AK. It also addresses another issue I have with growth models: it’s really tricky to get wages to decrease in the long run. For the production function you use, if A_{old} was also growing at a constant rate, then wages would eventually rise. This is because the new production technology doesn’t harm the old technology other than by taking capital away. Each production technology could work next to the other without interfering. In reality, there is a limited amount of space on earth and once the new technology is more efficient, it isn’t profit maximizing to ‘waste space’ on the old style of production. I don’t know if it is worth modeling for what you are doing, might be too many bells and whistles, but it is something I’ve been thinking about.
Ah I didn’t realize that a balanced growth path is important for analytical tractability, thanks for that insight. And yes I’m reading your paper (haven’t gone thoroughly through the model yet though), really interesting!
That’s a great idea to add land to the production function to bring back diminishing returns to capital. I am hoping to extend the model to include capital accumulation, and I’ll think about including land when I do that. I suppose this will lead to a balanced growth path if A_auto is constant, but if A_auto is growing then growth could still be super exponential. (Regarding whether wages will eventually rise, I actually think wages would remain at zero in this model with capital accumulation and without land, because the marginal product of labor with the old tech will remain below the reservation wage).
I’m imagining something that is Cobb-Douglas between capital and land. Growth should be exponential (not super exponential) when A_auto is growing at a constant rate, same as a regular Cobb-Douglas production function between capital and labor. Specifically, I was thinking something like this:
Thanks for this. If I understand correctly, the result is primarily driven by the elastic labor supply, which is a function of W and not of R, and the constant supply of capital. This seems most relevant for very fast takeoff scenarios.
My intuition is that as people realize that their jobs are being automated away, they will want to work more to bolster their savings before we move into the new regime where their labor is worthless and capital is all that matters. This would require fully modeling the household’s intertemporal utility and endogenizing the capital supply. This might be tricky, however, with your production function, because if A_{auto} is increasing at a constant rate and households are allowed to save, you will get superexponential growth.
Hi Caleb, thanks for your comment! That’s a great point that workers’ desire for savings prior to their jobs being automated may drive people to work more (similarly, people may work more as wages drop in order to maintain a subsistence level of consumption). On the other hand, if wages drop so low that workers can’t subsist, then labor supplied will drop regardless.
When you say it would be tricky to model savings with my production function, is the reason that superexponential growth with exponential time discounting in the utility function will lead to diverging utilities? That’s an interesting point. Do you know if there has been work on more realistic utility functions to address this issue?
Diverging utilities can be an issue. You can also get infinite output in finite time. The larger issue is that the economy has no steady state. In economic growth models, a steady state (or balanced growth path (BGP)) represents a long-run equilibrium where key economic variables per capita (like capital per worker, output per worker, consumption per worker) grow at constant rates. This greatly simplifies the analysis.
For example, I have a paper in which I analyze how households would behave if they expected TAI to transform the economy. To do this, I calculated the steady state the economy was in prior to households learning about the potential of TAI as well as the post-TAI steady state. I could then calculate the transition path between these two steady states. I thought of using the same production function you used here, but then there wouldn’t have been a post-TAI steady state, which is necessary to be able to find the transition path.
A production function that I have mused about is one like yours, but with land. This should solve the issue, as the post-TAI economy will no longer be AK. It also addresses another issue I have with growth models: it’s really tricky to get wages to decrease in the long run. For the production function you use, if A_{old} was also growing at a constant rate, then wages would eventually rise. This is because the new production technology doesn’t harm the old technology other than by taking capital away. Each production technology could work next to the other without interfering. In reality, there is a limited amount of space on earth and once the new technology is more efficient, it isn’t profit maximizing to ‘waste space’ on the old style of production. I don’t know if it is worth modeling for what you are doing, might be too many bells and whistles, but it is something I’ve been thinking about.
Ah I didn’t realize that a balanced growth path is important for analytical tractability, thanks for that insight. And yes I’m reading your paper (haven’t gone thoroughly through the model yet though), really interesting!
That’s a great idea to add land to the production function to bring back diminishing returns to capital. I am hoping to extend the model to include capital accumulation, and I’ll think about including land when I do that. I suppose this will lead to a balanced growth path if A_auto is constant, but if A_auto is growing then growth could still be super exponential. (Regarding whether wages will eventually rise, I actually think wages would remain at zero in this model with capital accumulation and without land, because the marginal product of labor with the old tech will remain below the reservation wage).
I’m imagining something that is Cobb-Douglas between capital and land. Growth should be exponential (not super exponential) when A_auto is growing at a constant rate, same as a regular Cobb-Douglas production function between capital and labor. Specifically, I was thinking something like this:
X_old^beta(A_old K_old^alpha L^{1-alpha})^(1-beta) + X_auto^beta(A_auto K_auto)^(1-beta)
st X_old + X_auto = X_total (allocating land between the two production technologies)
As to your second point, yes, you are correct, as long as A_old is constant wages would not increase.
Ah yes that makes sense that growth will be exponential if A_auto has a fixed growth rate. Thanks!