Model 0’s adjustments of a household sector’s utility function are analogous to a financial intermediary’s adjustment of capital placements to the shape of an economy’s industrial production functions: both operations can be portrayed as responses to the financial signal of money’s marginal value, as indicated by a separation of a sector’s marginal value of money c_IK from unity:

Defining optimality as a labor/leisure sector LK’s c_LK equaling unity, this equation can be used to compute LK’s optimal current dividend e_LK':

Recurring to the definition of b_LK, and substituting optimal current dividend -e_LK' for the ideal dividend f, we have a second equation in b_LK and e_LK':

Eliminating b_LK from these equations yields ...

... where the variable w_LK has been introduced for our notational convenience:

Any separation between optimal passive income e_LK' and the current passive income e_LK is to be resolved by re-shaping household utility.

It must be noted that economics proper has long regarded household utility as having no unit of measure, hence no price, and therefore no possibility of quantification. In the alternative, we advance the variable Z_LK of the household utility function as a cardinal measure of utility. Changing Z_LK alters the ‘marginal cost’ of producing labor; but its reformation of LK’s utility function occurs without altering the shape of the utility isoquants governing tradeoffs among a sector LK’s inputs (i.e.: items of household consumption). And Z_LK does not enter the market’s valuation of labor.

Model 0 uses the difference between e_LK' and e_LK to create its primitive adjustment regime for Z_LK. This difference must first be scaled in respect to the price of LK’s product, the wage rate P_LK. The term (e_LK' -e_LK)/P_LK has the units of labor, t_LK +Y_LK. A relationship between Z_LK and leisure -Y_LK  is then derived from Equation 3-1’s generic hyperbolic production function:

And our further notational convenience will abbreviate this expression’s last term as simply _LK. As shown in below, these specifications determine the driving function for a state variable representing a labor/leisure sector LK’s Z_LK:

If this figure’s household Z is operative as a quantification of household utility, then utility’s unit of measure is the labor hour; its price is (minus) the wage; and its manifestation is respite from work in the form of leisure.