My mock-up didn't capture this. Instead, I used as eligibility the total clients placed divided by the total walking in the door. Since everyone reflected in the quarterly data was employed for at least thirty days, I don't think that should be a problem. It also avoids getting into an unnecessary level of detail that fails to change the outcome even though it more closely tracks what's happening.
My data were generated randomly, so they are not realistic. But the same calculations should work for your actual data if they are aggregated the same way. If, however, payments are calculated individually for each client there could be some adjustment needed, because x would not be as assumed.
With the data available, which is limited to only the four quarters, this is more of a descriptive than a modeling exercise because payout is completely determined according to the quarterly input, number of clients, and the payout levels. It is possible to do scenario planning by varying assumptions made for future periods as to the quarterly data expected as some proportion of the historical (perhaps on an agency-by-agency basis), client intake levels (perhaps based on general economic conditions expected), assessing the budgetary impact of changing payout unit amounts, etc.
No. At this point you should have enough of a start to modify the x and y examples to conform to the specific business problem, understand how the calculations performed by f work and then make the appropriate adjustments.
Come back with a reprex. See the FAQ in a new thread if you have difficulties with specific steps.