I'm writing a function to simulate the Ricardian
Law of Rent for part of an EU4 mod (something for fun), and think I've reached an acceptable solution but am not positive that all the steps I took are sound. The units of the final rent function seem to check out and it exhibits the properties that I would expect, but I still have doubts if it's conceptually correct.
In the mod, there is a function labor(x) which represents the amount of labor necessary to produce a unit of goods in some industry as a function of x, land. Less productive land requires more labor to produce the same unit of goods, and it's assumed that the most productive land will be used first. So for example the excellently-situated 0.1th unit of land may require 15 people to produce a bushel of turnips in a month, but to make that same bushel of turnips on the crummy 7.8th unit of land may require 50 people, and that's what this function represents. I'm using simple polynomials to model the labor function composed of one x variable and two constants 'a' and 'c', e.g. labor(x) = a*x + c, labor(x) = a*x^2 + c, or more generally as labor(x) = a*x^n + c.
The function I want to arrive at is the total amount of rent (in money) that would be appropriated under these conditions given the total amount of land in production and the labor function. The Law of Rent suggests that the amount a landlord can charge in rent for producing on their land is limited by the difference between the amount a laborer can produce on that land compared to the amount the same laborer could produce if they were working on the best rent-free land available (or take everything beyond subsistence if no land is available or they are forcibly prevented from moving). I took the following steps to try and reach a function of that rent value from the labor function:
1) The difference in the amount of labor to produce one unit of goods on a unit of land 'x' compared to the final unit of land in production would be given by labor(x) - labor(T), where 'T' is the total amount of land in production. The units of this would be in labor.
2) The difference in the amount of money a laborer could produce on a unit of land compared to the last unit of land would be P*s/labor(x) - P*s/labor(T), where 'P' is price (money per goods) and 's' is general productivity of all units of land (goods per land). The units of this function would be in money per land per labor. This function could be factored to P*s(1/labor(x) - 1/labor(T))
3) Since the above is the difference per laborer, to get the difference per unit of land I
think I need to multiply the function again by labor(x). I'm not positive if this step actually makes sense conceptually, and this is where most of my doubts come from, but the result after some simplification is r(x) = P*s(1 - labor(x)/labor(T)), in units of money per land.
4) From there by taking the integral from 0 to T with respect to x of r(x), I should get a function R(T) that I think represents the total rent appropriated in units of money. Substituting back in the labor function -- labor(x) = a*x^n + c -- the integral results in a nice and clean function R(T) = P*s (n*a*T^(n+1) / ( (n+1)(a * T^n + c) ) in units of money. This function is attractive because it exhibits all the behaviors I would expect, can be computed with simple arithmetic, and appears to be in the correct units, but I just don't know if it's logically consistent with the Law of Rent and actually represents the amount of rent that would be appropriated under those conditions.
So what I'd like to ask is, if anyone actually made it to the bottom of this post:
1) Did I goof the math
2) Does the logic seem consistent with what I'm trying to model?
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