The model cannot approximate consumption as a function only of total earnings, because savings equate with spending only so far as capital accumulation occurs; further savings have no stimulative value. Capital accumulation stops at a certain threshold determined by existing available technology and the cost of labor inputs, ie wages. Further savings has no real effect on the economy.
Another possible factor in the relationship between wages and consumption is total imports and exports. Imported goods replace goods that would be produced domestically and slightly reduce price levels. Exported goods are additional produced goods that are not then consumed locally, and so represent a slight increase in the price level. The issue of currency exchange appears here, and since the value of accumulated currency depends on goods available in that currency, exports also tend to decrease the cost of imports, while imports decrease the value of exports.
Taking these elements together, we have:
Consumption = Wages * (1 - wage saving rate - wage deduction rate) + Profits * (1 - profit saving rate - profit deduction rate) + min(Capital replacement + New capital accumulation, Wages * wage saving rate + Profits * profit saving rate)
Total economic payments are as follows:
Payments = wages + profits = Consumption + Domestic Government Spending - Imports + Exports
Assume that all employed people receive wages. For this purpose, and elsewhere in the model, count self-employment profits from small enterprises as wages. Then people employed can be calculated as follows:
Employment = wages / average wage
We can now start plugging the model around and we get:
wages = Consumption + Domestic Government Spending - Imports + Exports - profits <=>
wages = Wages * (1 - wage saving rate - wage deduction rate) + Profits * (1 - profit saving rate - profit deduction rate) + min(Capital replacement + New capital accumulation, Wages * wage saving rate + Profits * profit saving rate) + Domestic Government Spending - Imports + Exports - profits
Which has two cases based on the minimum condition:
A. Excess savings exist:
Wages = [Capital replacement + New capital accumulation + Domestic Government Spending - Imports + Exports - Profits * (Profit saving rate + Profit deduction rate)] / (wage saving rate + wage deduction rate)
B. Savings shortage:
Wages = [Domestic Government Spending - Imports + Exports - Profits * Profit deduction rate] / wage deduction rate
Since Employment * Average wage = wages, we can stop here and use as our data the statistics that comprise the relationship with wages, then transform them into the final employment estimates. This can then be modeled econometrically.
We will analyze the range 2001-2009 in the United States, and assume that all years fall into category A. First, a two variable model that compares just the most significant 2 variables, Government Spending and Private saved or deducted profits, which I call "profit holdings".
This is a decent result. The R-squared value is 0.946, but the degrees of freedom is only 6. Government Spending is very strongly positively correlated with wages, and profit holdings are strongly discorrelated: as corporate profits increase, ceteris paribus, one can expect wages to decrease.
One can note the oscillation of the estimator above and below the trend lines for employment and wages. That is because employment and wages actually endogenously affect each other in a mutually moderating fashion: worker layoffs compete with wage cuts and worker hirings compete with wage increases in the marketplace. The estimators do not incorporate this trade off and so tend to over-estimate economic changes. That is, however, a good thing. It shows that even as growth occurs, an underlying dynamic can develop that is destructive. Autocorrelation is not controlled for here.
An estimate was also run with imports and exports included, to examine the claim that trade policies have had a detrimental effect on employment levels. This model is unable to establish a significant effect from either of those factors. It can be argued, though, that a larger range of dates is necessary, and different variable transformations utilized in order to allow perception of the worker replacement dynamic that is hypothesized. This regression did not make those transformations and is based on only 9 data points. Since the alternative, 4 variable model leaves only 4 degrees of freedom while increasing the R-squared value to 0.97, the 2 variable model is preferable from the perspective of statistical robustness.