Cash management as an inventory problem, the square-root rule for money holdings.
Warming up the engine…
The economics of cash management: holding money sacrifices interest, but going to the bank costs time and fees, so optimal money holdings balance shoe-leather costs against forgone interest.
M* = √(c·Y / 2i), c = cost per withdrawal, Y = spending, i = interest rate
Withdraw rarely and you hold big idle balances losing interest; withdraw constantly and you waste time and fees. The square-root rule says optimal cash rises with spending but LESS than proportionally, and falls when interest rates rise, giving money demand its interest elasticity from pure optimization rather than assumption.
The square root is the answer to remember: doubling spending raises optimal money holdings by √2, not 2. Money demand falls in i, rises in transaction costs.
Now prove you have it
Move the curve to where you think it lands, and get told exactly which part you got right.
Real-world scenarios on this model
Equilibrium at M 63, C 3.2.
Equilibrium: M* = 63.2, C* = 3.2
Current equations
How much cash should you carry? Big withdrawals mean fewer trips to the bank (falling blue curve) but more interest forgone (rising amber line). The cheapest point is where the two costs cross.