Stock and flow

• 1. Introduction

  ›

  The Polish economist Michał Kalecki called economics "the science of confusing stocks with flows." He meant it as a sharp jab at his own profession. The target was a pair of ideas that look almost too simple to go wrong. What exactly are stocks and flows, and why would a distinction this basic cause so much trouble? What happens to an economist's arithmetic when one is mistaken for the other? And how did a habit from the account books end up describing bathtubs, hotels and the atmosphere of the planet? The answers begin with a single date on the calendar.

  2

• 2. Moments and Intervals

  ›

  Picture the 31st of December 2004. A quantity measured on that day, and only on that day, is a stock. It may have piled up over many earlier years, but the measurement itself is a snapshot. A flow works differently. It can only be measured across an interval, such as a year, so time is built into its units. In that sense a flow is roughly analogous to a rate or a speed.

  United States nominal gross domestic product shows the contrast. It counts the dollars spent over a period such as a year, so its units are dollars per year. The nominal capital stock of the country is a different kind of number. It adds up the value of equipment, buildings and other real productive assets, so its units are simply dollars.

  A diagram of the two together shows a pool being filled and drained. The stock of capital available today is increased by the flow of new investment. It is depleted by the flow of depreciation. That difference in units is not a technicality, because it decides what arithmetic is allowed.

  2

• 3. Ratios Without Comparison

  ›

  The debt to GDP ratio is measured in years. Debt is counted in dollars and GDP in dollars per year, so dividing one by the other leaves time. The ratio can be read as the number of years needed to pay off all debt, if every dollar of GDP went to repayment.

  That result rests on a firm rule. Stocks and flows carry different units, so they cannot be meaningfully compared, equated, added or subtracted. Multiplying and dividing them is another matter, and ratios are perfectly meaningful. Some economics students confuse the valid act of taking a ratio with the invalid act of comparing.

  Turn the fraction upside down and the units turn with it. The velocity of money is nominal GDP divided by the nominal money supply. Its units work out to dollars per year over dollars, which is 1 per year. A stock over a flow gives time, and a flow over a stock gives 1 over time. Accountants use the same trick on their own books, with a vocabulary of their own.

  2

• 4. The Balance Sheet Habit

  ›

  On a balance date, an accountant records what an asset is worth at that instant. That figure is a stock. Over the whole accounting period, the totals of sales and purchases, or of incomes and expenditures, are flows. Divide the flow of an economic activity by the average stock during the period and the result counts the turnovers of that stock, which are also called rotations.

  Some entries are always flows. Profit and income are the usual examples. Others can be shown either way, and capital is the standard case. Capital as a stock yields a periodic income, and that income is a flow.

  A person or a country can hold many kinds of stock. The list includes money, financial assets, liabilities, wealth, real means of production, capital, inventories, and human capital, which is also called labor power. The matching flows are income, spending, saving, debt repayment, fixed investment, inventory investment and labor utilization. Tracking how any of these stocks moves from one date to the next takes only subtraction, as a tally of machines makes plain.

  2

• 5. Counting Machines

  ›

  On the 1st of January 2010, an imagined country owns 20 machines. A year later it owns 23. The stock of physical capital rose by 3, so the flow of net investment during 2010 was 3 machines per year. On the 1st of January 2012 the count reaches 27. Across 2010 and 2011 together, that is 7 more machines in two years, an average flow of three and a half machines per year.

  This is the discrete-time version of the idea. The change in a stock between two dates one time unit apart, its first difference, equals the flow per unit of time. Continuous time gives the same answer in calculus form. The time derivative of a stock is a flow, and the stock at any moment is the integral of the flow from a starting moment set at zero.

  Capital shows how this plays out. Gross investment adds to it gradually and depreciation subtracts from it gradually. The instantaneous rate of change of the capital stock is net investment, which is gross investment minus depreciation. A worked diagram of two stocks changed by a flow goes further. It lists its equations in order of execution and steps through time from 1 to 36.

  2

• 6. Bathtubs and Levels

  ›

  A bank balance in yen is a stock. Deposits and interest pour into it, withdrawals drain out of it, and the flows are counted in yen per month. A lumber inventory, measured in board feet, rises with incoming lumber and falls with outgoing lumber, at so many board feet per week. A housing stock can be tallied in dollars or in housing units. New homes built feed it and housing lost to depreciation drains it, each counted per year. Shareholdings grow with purchases and shrink with sales, in shares per month.

  Everyday cases fit the same mold. Guests in a hotel rise with arrivals and fall with departures, in persons per day. A population changes through births and immigration on one side and deaths and emigration on the other, in persons per year. Water in a bathtub, in liters, grows as water pours in and shrinks as it drains out, in liters per second. Waste at a disposal site grows by dumping and shrinks by decay, in tons per week. A fuel tank, in gallons, is refilled by refueling and emptied by fuel consumption, in gallons per month.

  Behind all of them stands one definition. A stock, which can also be called a level variable, is anything accumulated over time by inflows or depleted by outflows. Stocks can only be changed by flows. Mathematically, a stock is the accumulation of its flows, with the outflows subtracting. A flow, or rate, changes a stock over time and is measured across an interval, such as the births over a day or a month.

  Jay Forrester originally used the words levels and rates of flow. Together the pair form the basic building blocks of system dynamics models. The same pair also travels under other names, including integral, derivative and state variable. The same arithmetic governs the atmosphere, where the units are tons rather than liters.

  1

• 7. Earth and Body

  ›

  Durable greenhouse gases in the atmosphere form a stock, and climate change mitigation is a fairly straightforward problem of managing it. The goal is to reduce the concentration of those gases. The tools are flows. Emissions into the atmosphere are an inflow to be reduced, and outflows such as carbon dioxide removal are to be increased.

  The concepts reach well beyond money. They apply to conserved quantities such as energy, and to materials in stoichiometry and in water reservoir management. They also apply to durable pollutants that accumulate in the environment or inside organisms.

  In the human body, energy homeostasis describes a linear relationship between flows and a stock. The flows are the food we eat, the energy we expend and the wastes we excrete. The stock shows up as the gain or loss of body weight over time.

  Earth system science is full of such problems. The carbon cycle, the nitrogen cycle, the water cycle and Earth's energy budget can all be framed as stocks moved by flows. Long before any of these applications, the idea lived in accounting practice.

  1

• 8. Old Ledgers, Sharp Tongues

  ›

  Accounting practice has drawn this line for centuries. An asset, which is a stock, was always a different thing from income, which is a flow. The distinction is elementary, and nobody needed an economist to invent it.

  Economics took longer to formalize the terms. In that effort, Irving Fisher formalized capital as a stock, giving the discipline a precise anchor for the idea.

  Kalecki's famous remark came around 1936. It appeared in his critique of the quantity theory of money, where he judged the confusion between the two kinds of quantity to be at the heart of the problem. Joan Robinson frequently quoted the line.

  23