Minimizing total cost (TC) of a fund of cash (or anything else) Mason Gaffney Notes, 17 May 87, rev. 3/92 1 Define: W = monthly cash receipt (W is evidently from "wage") M = mean amount of money held during month x = number of transfers into cash holding during month (this is not explicit in H&T, but making it explicit makes things clearer) k = cost per transfer. (Note, this is independent of the amount transferred. Hence, it represents economies of scale. 2 ) R = opportunity interest rate, monthly basis T = amount of money per deposit T = W/x M = T/2 = W/2x (1) (2)
Note that Mx equals a constant, W/2. M and x are inversely related. In general, M symbolizes a fund of capital, x symbolizes turnover, and W/2 symbolizes flow (volume) of funds: remember Volume = Capital x Turnover. (Here, "volume" is another word for "flow.") Monthly Deposit Cost (DC) = kx Monthly Carrying Cost (CC) = RM Monthly Total Cost (TC) = DC + CC = kx + RM Objective: Choose a value of M to minimize TC. This is a typical optimizing problem, because it is a trade-off between DC and CC. As M rises, DC falls, and CC rises. To do this, we must express kx as a function of M. From (2), x=W/2M. Substituting: TC = kW/2M + RM Method: Use differential calculus 1) to find the value of M that minimizes TC, for given values of W, k, and R. (We will denote this value of M as M*); 2) having found M*, to find the optimal value of x that it entails (note, the value of x is not among the three that are given). Most of you understand something, at least, about calculus. I use little calculus in this course, but here it makes things easier. Differentiate (6), and set its derivative equal to zero, to find (6) (3) (4) (5)
1Notation taken from Hall and Taylor, pp. 296-98 2It is like one-stop shopping, or one-pass, large-unit construction. You gain by doing things in one shot; but you pay the price in higher interest cost.
2 where TC is a max or a min (in this case...