Stocks and ﬂows In economics (as in the physical sciences), empirical observations and measurements are made either at a point in time or over a period of time. As a result, the basic variables can be either stocks or ﬂows. A stock measure refers to the value of a variable at a given point in time. A ﬂow measure refers to the value of a variable over a given period of time. On top of these basic measures, a host of derivative measures (e.g. ratios of ﬂows to ﬂows, ﬂows to stocks, stocks to ﬂows, and stocks to stocks) can be determined. This distinction is extremely useful in accounting, economics, and ﬁnance.
Stocks and ﬂows Consider this ﬁgure: It shows a container with a certain liquid (e.g. water), one pipe delivering it into the container, and a pipe leaking it out. Physicists and engineers call it a simple “hydrodynamic system.”
Stocks and ﬂows Basically, we can measure the performance of this system by measuring the amount of liquid in the container at given points in time (e.g. on Sundays at noon, on the last day of each month) and the amount of water that ﬂows through the container over given periods of time (e.g. a week, a month). Then, by combining both forms of measurement, we get a richer view of the functioning of the system. The stock measures may help us correct errors in the measurement of ﬂows or, vice versa. (The term stock has several meanings in economics. It also refers to the “equity” or legal ownership of a company traded in the market. The context should make it clear when we use the term in one or the other sense.)
The algebra of stocks and ﬂows Let the upper-case letters denote stocks and the lower-case ones ﬂows. Let Xt be the amount of water in the vessel at point in time t, xit the amount of water ﬂowing into the vessel during the month, and xot the amount of water ﬂowing out of the vessel during the month. Then: Xt+1 = Xt + xit − xot The amount of water in the container at t + 1 is equal to the amount of water in the container at t plus the water that got in it during period (t, t + 1) minus the water that leaked out during the same period.1 1 Assume no evaporation or, alternatively, that the water leaked out in the period includes evaporated water.
The algebra of stocks and ﬂows Example: On November 1 (t = 0), the amount of water in the container is 10 gallons (X0 = 10). The water ﬂowing into the vessel during November is 40 gallons (xit = 40) and the water ﬂowing out of the vessel is 39 gallons (xot = 39). Calculate the amount of water in the container on December 1: X1 = X0 + xi1 − xo1 = 10 + 40 − 39 = 11. If the stock of water on December 1 is any diﬀerent from (greater than or less than) the result above, we have made errors in our measurements. The new measure of the water level can help us correct them.
Some accounting The two basic ﬁnancial statements that accountants produce are the balance sheet and the income statement. These ﬁnancial statements provide a detailed picture of the ongoing ﬁnancial performance of a business.
Balance sheet The balance sheet of any organization (e.g. a business) has two sides. The left-hand side reports the value of all the organization’s assets at a given point in time, typically at the end of a year. The assets are the resources the organization manages measured at a point in time. The right-hand side reports the source of the assets listed on the left-hand side. They are either owed to others (liabilities) or “owned” by the legal “owners” (equity or net worth). The fundamental balance-sheet equation is: At = Lt + Et where t indicates a point in time (e.g. the last day of the year), A is total assets, L is total liabilities, and E is total equity.2 2 To separate an organization from its individual owners, it may be convenient to state the equation as A = L, i.e. assets equal liabilities. They are liabilities to either others or to the individual “owners” of the organization. In this interpretation, the equity of the legal owners of, e.g., a business is considered a special type of liability.
A typical balance sheet ABC Inc. Balance Sheet 12/31/06 Assets Liabilities and Equity Cash and liquid securities $ 10 Payables $ 20 Inventories 50 Other short-term debt 40 Receivables 60 Mortgages 80 Trucks (net) 25 Other long-term debt 250 Oﬃce equipment (net) 10 Liabilities 390 Machinery (net) 45 Equity 120 Buildings (net) 220 Other ﬁxed assets (net) 90 Assets $ 510 Liabilities plus Equity $ 510
Income statement The income statement reports on its top line the total ﬂow of gross income (e.g. sales revenues) received by the organization during a period of time (e.g. a year). The next lines report the various expenses that the activity of the organization incurred during the period to sustain its gross income. These expenses – sorted out as production costs, operating expenses, ﬁnancial expenses, and taxes – are deducted or subtracted from the top line. The bottom line of the income statement indicates the ﬂow of net income or net proﬁt during the period. The fundamental equation is: NIt = GIt − PCt − OEt − FEt − Tt where t is the period of time from point in time t − 1 to point in time t, NI is the residual or net income (proﬁt if positive, loss if negative), GI is the gross income (typically, sale revenues), PC is the total cost of goods sold (e.g. the cost of raw materials, storage costs, wages and beneﬁts of factory-ﬂoor workers), OE are the operating expenses (sales and administrative expenses, including salaries and commissions of administrative and sales personnel,
Income statement A typical income statement: ABC Inc. Income Statement for 1/1/07 through 12/31/07 Sales revenues $ 200 (-) Cost of goods sold 90 Gross proﬁt 110 (-) Operating expenses (includes depreciation) 40 Operating proﬁt 70 (-) Interest paid 10 Taxable proﬁt 60 (-) Taxes 10 Net proﬁt $ 50
Summary Note that: all items in the balance sheet are stock measures (“water in a container” at a point in time), all items in the income statement are ﬂow measures (“water that ﬂows in/our” over a period of time), the balance sheet and the income statement are related in multiple ways, assets are “positive water stocks” while liabilities and equity items are “negative water stocks” (if that makes sense), and sale revenues are “positive water ﬂows” and costs and expenses are “negative water ﬂows’.’
Summary Every time an organization conducts an operation or transaction, every time a business takes raw materials from its inventories and have its workers process them on the factory ﬂoor, every time its sales people sell a batch of goods or its administrative personnel orders a shipment from its suppliers, every time a payment is made or received, there is “water” ﬂowing from one balance-sheet container into another one. At the end of the given period (and beginning of the next period), the balance sheet reports the adjusted levels of “water” in each container at that point in time. Also at the end of the given period (beginning of the next), each spurt of “water” that ﬂowed from container to container during the period is added up (aggregated) into its respective category and recorded in the income statement. The legal owners of the organization (if a corporation, the legal owners are called stockholders) pay most attention to the level of “water” in their equity “container.”