Two headlines appear on the same morning. The government has reduced its deficit; the public debt has reached a new high. The first is offered as evidence of restraint, the second as evidence that the restraint cannot be real. Both can describe the same account. The deficit measures a shortfall over a period, while the debt records obligations still outstanding at a date. Reducing a positive shortfall slows the addition to those obligations without necessarily reversing it. The distinction is the bathtub's in its most consequential public form: an inflow can become smaller while the level continues to rise. But the debt account also requires something the image alone cannot show. Borrowing can acquire assets, exchange rates can change the value of liabilities, and obligations can enter or leave the recorded boundary. A serious reckoning must explain the whole passage from the opening stock to the closing stock, then ask what that stock requires of the future.
Begin with the simple case in which every overall deficit is financed by additional debt and nothing else changes it. An opening debt of one thousand, followed by a deficit of fifty, becomes one thousand and fifty. A deficit of thirty in the next year brings it to one thousand and eighty. The annual shortfall has fallen by forty percent; the debt has risen in both years. A balanced budget would stop the additions on these assumptions, while a surplus applied to repayment would reduce the stock. The distinction reaches further than terminology. An announced reduction relative to a projected deficit is different from a reduction relative to last year's deficit, and each is different from paying down debt. The comparison needs a date for the stock, a period for the flow, and a stated baseline for the claimed improvement. Without them, “less borrowing” can slide into “less owed,” though the first describes a change in the tap and the second a change in the water.1
The complete account makes those exclusions visible. A government can borrow to purchase a financial asset, increasing gross debt without an equal increase in its overall budget deficit. It can finance part of a deficit by drawing down a cash balance, so debt rises by less than the shortfall. A liability denominated in foreign currency can become larger in domestic-currency terms when the exchange rate changes, even though the foreign-currency amount owed has not increased. And the market price of a bond can move while its contractual payments remain fixed. These changes belong in a reconciliation of transactions, revaluations, and other changes in the stock. Suppose a deficit of thirty is financed with twenty of new debt and ten of existing cash. Gross debt rises by twenty; the cash asset falls by ten; net debt rises by thirty if that cash is the only changing asset deducted from gross debt. The books fit, but they fit only after both sides of the balance sheet have been followed. Calling every unexplained increase in a debt number fresh overspending would lose the distinction the account was built to preserve.2
Interest then joins yesterday's stock to today's flow. The primary balance excludes interest; the overall balance includes it. With debt of one thousand and an effective interest rate of five percent, the interest bill is fifty. A primary surplus of twenty still leaves an overall deficit of thirty and, in the simple financing case, closing debt of one thousand and thirty. Yet the debt's size relative to the economy may fall. If annual nominal output grows from one thousand to one thousand and fifty, the debt ratio declines from one hundred percent to about ninety-eight point one percent. The numerator has risen, but the denominator has risen faster. More generally, with a consistent debt definition and the other adjustments excluded, the new ratio equals the old ratio multiplied by one plus the effective nominal interest rate, divided by one plus nominal output growth, and then reduced by the primary surplus as a share of current output. This relation makes the interest-growth comparison exact. When the two rates are equal, a zero primary balance preserves the ratio even though interest-financed borrowing enlarges the nominal stock.3
The ratio still describes only the liabilities selected for its numerator. Gross debt counts the covered debt obligations; net debt deducts the corresponding classes of financial assets. A broader net-worth account also considers other assets and liabilities, including the public capital that borrowing may have financed. These measures answer different questions. A loan held by the government may produce receipts, yet be difficult to sell in a crisis; a road may provide valuable services without producing cash that can be used on a bond's payment date. The borrowing has to be understood alongside the asset acquired and the obligation undertaken. The boundary matters just as much. Central-government debt, general-government debt, and the liabilities of the whole public sector cover different institutions. A bond held by a domestic household is an asset to that household and a liability to the government; its domestic ownership does not relieve the government of the need to make the payment. Consolidating accounts can remove an internal claim from a wider total while leaving its distributional and financing consequences for the separate parties intact.4
To judge the burden, the payments must be placed on a calendar. A large stock of long-term, fixed-rate domestic-currency debt presents a different exposure from an equal stock that must be refinanced next month, resets its interest rate frequently, or is payable in a foreign currency. A government may be able to service its obligations over time yet face an immediate shortage of acceptable funding; an asset-rich account can have the same difficulty. The simple debt-ratio equation clarifies the sources of a change, but its interest rate, growth rate, and primary balance are themselves outcomes that policy and circumstances affect. A favorable relation between interest and growth in one year does not promise that it will endure. Nor does a falling ratio by itself establish that borrowing financed worthwhile activity, that the distribution of its costs is acceptable, or that the next maturity can be met. These are questions about resources, institutions, creditors, and the adjustment that can actually be carried out. Accounting puts them on the right base; it cannot choose their answers.5
The discipline of the debt is therefore a discipline of joined accounts. The standing obligation belongs to a date; the deficit belongs to a period; their connection must include the financing transactions and valuation changes that actually occurred. A falling deficit can accompany rising debt, rising debt can accompany a falling debt ratio, and each observation can coexist with a different conclusion about the wisdom of the policy that produced it. The next step requires the use of the borrowed resources and the terms of their repayment to be examined, rather than a verdict borrowed from the direction of one number. For the creditor, moreover, the same obligation is a holding expected to furnish future payments. What the debtor must provide is what the owner hopes to receive. The next plate turns that connection around, asking how a stock held today can be converted into a scheduled stream: the annuity.