What are the bounds on the arbitrary decimals and the physical meaning of K and Y in this model? It seems AD_1 + AD_2 = 1 and K = development? If so, I think I see what you mean. I used the same model for originally determining EP in Blackened Skies. I decided (with help from players) that I didn't really like a production-based model and I wanted to measure competitiveness, resource availability, and real vs. nominal growth. So I kept the Y value as a reference and started trying to measure supply and demand (specifically demand which I started to see as the most relevant indicator of an economy's size). Eventually I realized I could derive an EP stat based on models of market interactions and I decided to go with it.
Usually I make the exponents of both equal to 1. So if K's is .3, L's is .7. Any lower and you start getting diminishing returns to scale, and any higher you get increasing returns to scale, both of which might be an issue for balance.
K is capital. It represents physical property used in production including factories.
Y is national income.
Anyway, my assumption earlier was that Y is a linear system of GDP/capita and total capita and by superposition a 50% reduction of total capita produces exactly a 50% reduction in Y. Obviously this is a derived quantity, and GDP/capita (tentatively "productivity") would be attached in no small degree to population also, making the entire thing nonlinear -- however, allow us to assume the basic unit of demand is the population. Hence we can say there is a "base demand" which is the nominal demand produced by any population and is linearly proportional to that. The true demand will be equal to this base demand multiplied by some non-dimensional factor. By this transformation a 50% reduction in population results in a 50% reduction in base demand.
If Y = Consumption + Investment + Government Spending + Net Exports, a 50% reduction of the population wouldn't lead outright to a 50% reduction in the size of the economy still. Consumption would decrease, yes, but investment and government spending don't necessarily fall by 50% as well.
Going back to the example above of K = 100, L = 100 (with L also being the population), the same .3/.7 split, we get this.
Y = 100
Wages = 100 * .7 (assuming no rents or anything of the sort)/100 = .7
Savings Rate = 5% (using America as the example)
Savings = .7 * .05 = .035
Consumption = .7 - .035 = .665
We're going to be generous and say that owners of capital consume at the same rate, and therefore consume 95% of their income and save 5%, meaning 95 out of 100 is consumed. Consumption per person = .95
Reducing L by 50 leads to this.
Y = 61
Wages = .854
Savings overall = 3.05
Consumption = 57.95
Consumption per person = 1.159
The economy is smaller overall, and savings are smaller overall. The 50% reduction in the economy reduced output by roughly 40%. This takes into account "demand" through consumption, which in this scenario is 95% of output.
From a logical point of view, after 50% of the population disappeared, their wages increased because the infrastructure, factories, capital, etc., was left behind. Demand per person, likewise, increased with wages. If the 50% reduction in population includes a 50% reduction in capital, then you would get 50% reduction in GDP in the model because .7 + .3 = 1, so the model assumes constant returns to scale.
Very rarely does it work out that neatly though.