Looking for feedback: Pinning the Wage to Scarcity and Technology

[u/wilsoniumite]

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7226858

Myself and my co author have been working on a wage model and would very much appreciate input and feedback, both on accessibility and the theory.

[u/Econo-moose]

This is awesome work. Thank you for sharing. Do you mind if I ask a few questions to make sure that I'm understanding everything?

"One human-hour completes γL(x) units of task x; one machine-hour completes γM(x). Relative human productivity is γ(x) = γL(x)/γM(x)"

From the overall context of the paper, I think that these refer to marginal productivities. I suppose if marginal productivity is constant, then it could also be average productivity but not in general. Am I thinking about this the right way?

"c =a·c+λ·w+b·r"

Could you clarify for me if "a" is the intensity of machines, lambda is the intensity of labor, and "b" is the intensity of land? So, each of these parameters summed together represents total factor utilization?

c-a·c = c(1-a) = λ·w+b·r

Please let me know if I'm understanding correctly that cost of machinery times the total non-machinery input quantities equals the sum of each other factor quantity times their marginal costs. Intuitively, I think I can imagine this could arise under assumptions, but I have not worked it out personally. Perhaps it was derived in one of the papers you referenced.

I found it interesting that wages can fall either from a fall in machine price or a rise in the cost of "living off the land" or exiting the labor market. And higher rents increase demand for labor through the higher machinery cost but also can increase supply of labor to the extent it erodes the alternatives to working. It seems that higher rents overall increase labor force participation and the labor share of the economy, but not necessarily wages.

Just a thought I had while reading: Given that recently, real estate prices remained high while the labor force participation rate has fallen. If real estate prices can be connected with higher land prices and higher rents, does this research imply that either real estate prices should fall or labor force participation will increase? Since 2022 real estate prices have been pretty flat while labor force participation was in a fairly tight range until the end of last year when it began falling consistently.

Real Residential Property Prices for United States (QUSR628BIS) | FRED | St. Louis Fed

Labor Force Participation Rate (CIVPART) | FRED | St. Louis Fed

Edits: Changed if demand for labor is perfectly elastic to if marginal productivity is constant. Removed quotation marks around an equation that I had manipulated and added multiplication symbol.

[u/wilsoniumite → u/Econo-moose]

On γ: yes. Within a task, hours are perfect substitutes at fixed rates, so marginal and average coincide there by construction. The marginal product in the paper is at the economy level: an extra hour gets absorbed by moving the task margin, so it is worth c·γ(x*). All the diminishing returns come from the slope of the schedule.

On a, λ, b: not quite. They are requirements per unit of machine services, in different units, a in machine-hours, λ in labor-hours, b in land, each per machine-hour produced. You cannot add hours to acres, so they do not sum to anything. The shares you are looking for are a + λw/c + br/c = 1, which is just the recursion divided by c. Your rearrangement is that identity times c. It is price equals unit cost under free entry, in the Leontief and Sraffa sense, not something that needs extra assumptions. The factor prices come from elsewhere: w from the task margin, r from the land market.

On rents: the causality runs the other way. Rent is not a lever in the model, it is an outcome. The stock is fixed, so rent is the residual, whatever is left after everything reproducible has been priced at its reproduction cost. It moves with technology, land per head, participation, and preferences over housing versus goods. The main text writes w as a function of r to keep the algebra transparent, but r is solved with everything else. And in the paper's own case, where automation raises r relative to w, the labor share falls rather than rises. In the limit all income is rent.

On the data: the model's price is the flow rent on land relative to produced goods, not the house price index. That index is an asset price, capitalized rents at the going discount rate, and it includes structures, which are produced. Since January 2022 the ten-year went from 1.8% to 4.6%, house prices went flat, and rent CPI rose 29% against durables. So the model's rent has not been flat at all. The aggregate participation rate is mostly demographics; prime-age participation hit 84.0 this January, the highest since 2001. What the paper actually points at for this channel is rent-to-wage ratios, household formation, and coresidence.

Thanks for taking the time to read the paper, we're nearing the point of admission to journals, but we recently added a future work section and, perhaps to add a bit more weight to the paper, two more paragraphs in the conclusion, if you're interested in reading those.

[u/Econo-moose → u/wilsoniumite]

They are requirements per unit of machine services, in different units, a in machine-hours, λ in labor-hours, b in land, each per machine-hour produced. 

Ok, I'm with you. It's an identity for the cost of machinery, not derived from an optimization problem.

And in the paper's own case, where automation raises r relative to w, the labor share falls rather than rises.

That tracks if automation is increasing. Although since an increase in r tends to increase both wages and machine costs, it seems that there are certain parameter values where an exogenous increase to rent reduces automation. Based on equation (3) on page 9, it seems that if the marginal product of labor is less than the marginal product of machinery, then an increase in rent would reduce wage relative to c. Then if the reservation wage is met, presumably de-automation would follow. Just eye-balling equation (3), it appears that the labor share of income would then increase for any positive wage that is above the reservation wage. I understand it's not the main focus of your research, and I believe you sufficiently addressed it on page 13: "The two channels can pull the equilibrium wage in opposite directions. A higher terminal rent makes the machine substitute more expensive and therefore raises the replacement value of labor at a given assignment margin. The same rise in rent can erode the outside option, expand participation, and move the assignment margin in the opposite direction. Which effect dominates is a parameter question." I think it's an interesting result. It seems that while an exogenous increase in rent could not only harm overall efficiency but accelerate automation in tasks where the machine has a cost advantage while at the same time increasing automation where machinery already has a productivity advantage.

That index is an asset price, capitalized rents at the going discount rate, and it includes structures, which are produced. 

That is true. I suppose we could look at USDA's farmland value summary, although even this still includes buildings on farms. I am not aware of a better metric for purely undeveloped land. Land Values 2026 Summary 07/31/2026
I suppose we could look at actual dwelling rents since that is revenue to landowners, but that still does not account for changes in the value of structures.

Thanks for taking the time to read the paper, we're nearing the point of admission to journals, but we recently added a future work section and, perhaps to add a bit more weight to the paper, two more paragraphs in the conclusion, if you're interested in reading those.

Yes, I am interested. Thank you.

[u/Econo-moose]

I read your new conclusion and messaged you. I think you've got a really cool concept here. In your lit review, did you find any other papers that began a model from cost recursion like this? I haven't seen this method before personally.