You decide to make a new monetary system from scratch. You give everyone a chequing account on your computer, with an initial balance of 0 units. If Andy buys bananas from Betty and pays her 100 units, Betty now has a positive balance and Andy now has a negative balance. The Net money supply remains at 0 units, but the Gross money supply is now 200 units.
If every individual's payments and receipts of money (which is not the same as income and expenditure, because we buy and sell non-money assets for money too) were perfectly synchronised (so that Andy sells 100 units worth of apples to Carol, who sells 100 units worth of carrots to Betty, who sells 100 units worth of bananas to Andy, all at exactly the same time) then the Gross money supply would always be zero. In a representative agent model (like the simplest version of the New Keynesian model) that perfect synchronisation is exactly what happens (which is why money seems to disappear from the simplest New Keynesian model).
But if agents are heterogenous, perfect synchronisation of payments and receipts of money won't happen automatically. And it is costly for individuals to synchronise their payments and receipts of money. So the Gross money supply will be strictly positive, and will be an increasing function of the extent of non-synchronisation of payments and receipts of money.
One way for individuals to increase the degree to synchronisation is for individuals with positive balances in their chequing accounts to lend to those with negative balances. Their incentive to synchronise depends on the spread between the interest rate paid on positive balances (the "deposit rate") and charged on negative balances (the "bank rate"). The larger the spread, the bigger the incentive for individuals to borrow and lend to each other directly. But if the spread is zero, and if the central bank sets no limits on how large a negative balance an individual can hold, there is no incentive for individual borrowing and lending.
If the central banker who creates the monetary system sets no limits on how large a negative balance an individual can hold (overdraft limits), the system is open to abuse. Transversality conditions won't just enforce themselves. Each individual has an incentive to buy an unlimited amount of consumption goods and then die with an infinitely large negative balance and negative net worth. And if accounts were numbered anonymous accounts, that could not be traced to any specific individual, individuals would have an incentive to open an unlimited number of numbered accounts, with a negative balance in each, unless the central bank restricted numbered accounts to having a non-negative balance. (The numbered accounts would work like paper currency, which is equally anonymous, and cannot be held in negative quantities.)
It is not easy to understand why individual borrowing and lending would exist. If the central bank is unwilling to allow an individual to have a larger negative balance, why would some other individual be willing to lend to him? The central bank has an additional penalty that it can impose on those who default: it can threaten to deny future access to the payments system. Perhaps the individual lender has private information on the borrower's credit-worthiness? And why should the central bank charge a spread between deposit rate and bank rate leaving gains from trade to individual borrowing and lending? Wouldn't the central bank prefer to capture those gains for itself? Perhaps because the central bank has monopoly power, faces a downward-sloping demand curve, and cannot or will not price-discriminate by offering smaller spreads to those who have better ability to borrow or lend?
If the limits on negative balances are non-binding, the gross money supply responds automatically to accommodate fluctuations in the degree of synchronisation of payments and receipts, even if the central bank keeps the net money supply fixed exogenously. But in the real world, where limits on negative balances are sometimes binding, the central bank must adjust the net money supply to accommodate fluctuations in the degree of synchronisation.