Sterman,J.D.2000.Businessdynamics systemsthinkingandmodelingfora complexworld.boston:irwinmcgrawhill Chapter7:Dynamicsofstocksandflows(p.231241)
7 Dynamics of Stocks and Flows Nature laughs at the of integration. -Pierre-Simonde Laplace (1749-1827) The successes of the differential equation paradigm were impressive and extensive. Many problems, including basic and important ones, led to equations that could be solved. A process of set in, equations that could not be solved were automatically of less interest than those that could. Stewart (1989, p. 39). Chapter 6 introduced the stock and flow concept and techniques for mapping the stock and flow networks of systems. This chapter explores the behavior of stocks and flows. Given the dynamics of the flows, what is the behavior of the stock? From the dynamics of the stock, can you infer the behavior of the flows? These tasks are equivalent to integrating the flows to yield the stock and differentiating the stock to yield its net rate of change. For people who have never studied calculus, these concepts can seem daunting. In fact, relating the dynamics of stocks and flows is actually quite intuitive; it is the use of unfamiliar notation and a focus on analytic solutions that deters many people from study of calculus. What if you have a strong background in calculus and differential equations? It is generally not possible to solve even small models analytically due to their high order and nonlinearities, so the mathematical tools many people have studied are of little direct use. If you have more mathematical background you will find this chapter straightforward but should still do the graphical integration examples and challenges to be sure your intuitive understanding is as solid as your technical 231
232 Part Tools for Systems knowledge. Modelers, no matter how great or small their training in mathematics, need to be able to relate the behavior of stocks and flows intuitively, using graphical and other nonmathematical techniques. The chapter also illustrates how stock and flow dynamics give insight into two important policy issues: global warming and the war on drugs. 7.1 RELATIONSHIP BETWEEN STOCKS AND FLOWS Recall the basic definitions of stocks and flows: the net rate of change of a stock is the sum of all its inflows less the sum of all its outflows. Stocks accumulate the net rate of change. Mathematically, stocks integrate their net flows; the net flow is the derivative of the stock. 7.1 Static and Dynamic Equilibrium A stock is in equilibrium when it is unchanging (a system is in equilibrium when all its stocks are unchanging). For a stock to be in equilibrium the net rate of change must be zero, implying the total inflow is just balanced by the total outflow. If water drains out of your tub at exactly the rate it flows in, the quantity of water in the tub will remain constant and the tub is in equilibrium. Such a state is termed a dynamic equilibrium since the water in the tub is always changing. Static equilibrium arises when all flows into and out of a stock are zero. Here not only is the total volume of water in the tub constant, but the tub contains the same water, hour after hour. The number of members of the US senate has been in dynamic equilibrium since 1959 when Hawaii joined the union: the total number of senators remains constant at 100 even as the membership turns over (albeit slowly). The stock of known Bach cantatas is in static equilibrium since we are unlikely to lose the ones we know of, the odds of discovering previously unknown cantatas are remote, and Bach can t write any new ones. 7.1.2 Calculus without Mathematics To understand dynamics, you must be able to relate the behavior of the stocks and flows in a system. Given the flows into a stock, what must the behavior of the stock be? Given the behavior of the stock, what must the net rate of change have been? These questions are the domain of the calculus. Calculus provides rules to answer these questions mathematically provided you can characterize the behavior of the stocks or flows as mathematical functions. Calculus is one of the most beautiful and useful branches of mathematics but one far too few have studied. Happily, the intuition behind the relationship between stocks and flows is straightforward and does not require any mathematics. If you are shown a graph of the behavior of the flows over time, you can always infer the behavior of the stock. This process is known as graphical integration. Likewise, from the trajectory of the stock you can always infer its net rate of change, a process known as graphical Integration and differentiation are the two fundamental operations in the calculus. Table 7-1 provides the definitions graphically and in plain language. The amount added to a stock during any time interval is the area bounded by the curve defining its net rate of change. Why? Consider the bathtub metaphor
Chapter7 Dynamics of Stocks and Flows 233 TABLE 7-1 Integration and differentiation: definitions and examples Integration Stocks or integrate their net flow. The quantity added to a stock over any interval is the area bounded by the graph of the net rate between the start and end of the interval. The final value of Ihe stock is the initial value plus the area under the net rate curve between the initial and final times. In the example below, the value of the stock at time = the area under the net rate curve between times and increases the Differentiation The slope of a line tangent to any point of the trajectory of the stock equals the net rate of change for the stock at that point. The slope of the stock trajectory is the derivative of the stock. In the example below, the slope of the stock trajectory at time is so the net rate at = At time the slope of the stock is larger, so the net rate at = is greater than The stock rises at an increasing rate, so the net rate is positive and increasing. Change in Stock 0 u) Y s, in Stock... 8 again. How much water is added to the tub in any time interval, such as between time and in Table Divide the entire interval into a number of smaller segments, each small enough that the net flow of water is not changing significantly during the segment (Figure 7-1). The length of each segment is called dt for delta How much water flows in during each small interval of duration dt? The quantity added is the net flow during the interval, say R, multiplied by the length of the interval, that is, the area of the rectangle dt periods wide and R high: Quantity added during interval of length dt = R dt (Units) = (Time) Note the units of measure: The flow in units per time, accumulated for a period of time yields the quantity added to the stock. To use a concrete example, suppose = 1 minute and = 2 minutes. The question is how much water flows into the tub during that minute. Divide the
234 Part Tools for Systems FIGURE 7-1 Graphical integration Divide time into small intervals of length dt. Each rectangle represents the amount added during the interval dt, assuming the net rate at that time remains constant during the interval. The area of each rectangle is The total added to the stock between and is then the sum of the areas of the rectangles. Dividing time into smaller increments increases the accuracy of the I minute up into six 10-second intervals and assume the flow is constant throughout each of these intervals. If at the start of the first interval the flow was 6 liters per minute (that is, 0.1 then the amount added would be (0.1 seconds) = 1 liter. At the start of the second 10-second interval, the flow has increased, perhaps to 7 or about 0.117 The area of the second rectangle is then 1.17 liters. Calculating the area of all six rectangles and adding them together gives an approximation of the total volume of water added during the minute. The approximation isn't perfect because the net flow is actually changing during each 6-second interval. In Figure 7-1, the flow is actually rising, so the calculated value of the stock will be too small. To increase the accuracy of the approximation, simply divide time into even finer intervals, increasing the number of rectangles. Computer simulations integrate the stocks in the model in precisely this fashion; the modeler must choose the time step dt so that the approximation is acceptable for the In the limit, as the time interval becomes infinitesimal, the sum of the areas of all the rectangles becomes equal to the total area under the net rate curve. Calculus provides formulas that give the exact areaunder the net rate-provided the net rate can be expressed as a certain type of mathematical function. But whether the net rate can be integrated analytically or not, the amount added to a stock is always the area under the net rate. Graphical integration is the process of estimating that area from a graph of the net rate. 7.1.3 Graphical Integration To illustrate graphical integration, consider the most basic stock and flow system: a single stock with one inflow and one outflow. Assume the flows are there are no feedbacks from the stock to either flow. Suppose the outflow from the 'The procedure described above is known as Euler integration and is the most commonly used method for numerical simulation. Other methods such as Runge-Kutta integration use more sophisticated methods to estimate the area and select the step. See Appendix A.
Chapter7 Dynamics of Stocks and Flows 235 FIGURE 7-2 Graphical integration: example While the rate steps up and steps down, the stock and remains at a higher level. Note the different units of rneasure for the rate and stock. 400 0' I 0 10 20 30 Time (seconds) stock is zero. Suppose also that the inflow to the stock follows the pattern shown in Figure 7-2. The inflow begins at zero. At time 10 the inflow suddenly increases to 20 remains at that level for 10 seconds, then steps back down to zero. If the initial level of the stock is 100 units, how much is in the stock at time 30, and what is the behavior of the stock over time? Table 7-2 shows the steps involved in graphical integration. Applying these steps to Figure 7-2, first make a set of axes for the stock, lined up under the graph for the flows. Next calculate the net rate. Since there is only one inflow and one outflow, and since the outflow is zero at all times, the net rate of change of the stock (Total Inflow - Total Outflow) simply equals the inflow. Initially, the stock has a value of 100 units. Between time 0 and time 10, the net flow is zero units/ second, so the stock remains constant at its initial value. At time 10, the net rate jumps to 20 and remains there for 10 seconds. The amount added is the area under the net rate curve (between the net rate curve and the zero line). Since the rate is constant, the area is a rectangle 20 high and 10 seconds long, so the stock rises by 200 units, giving a total level of 300 units by time 20. Because the net rate is positive and constant during this interval, the stock rises linearly at a rate 20 (the slope of the stock is 20 At time 20, the inflow suddenly ceases. The net rate of change is now zero and remains constant, and the stock is again unchanging, though now at the level of 300 units. Note how the process of accumulation creates inertia: though the rate rises and falls back to its original level, the stock does not return to its original level. Instead, it remains at its maximum when the net rate falls back to zero. In this fashion, stocks provide a memory of all the past events in a system. The only way for the stock to fall is for the net rate to become negative (for the outflow to exceed the inflow). Note also how the process of accumulation changed the shape of the input. The input is a rectangular pulse with two discontinuous jumps; the output is a smooth, continuous curve.
236 Part Tools for Systems TABLE 7-2 Steps in graphical integration 1. Calculate and graph the total rate of inflow to the stock (the sum of all inflows). Calculate and graph the total rate of outflow from the stock (the sum of all outflows). 2. Calculate and graph the net rate of change of the stock (the total inflow less the total outflow). 3. Make a set of axes to graph the stock. Stocks and their flows have different units of measure (if a stock is measured in units its flows are measured in units per time period). Therefore stocks and their flows must be graphed on separate scales. Make a separate graph for the stock under the graph for the flows, with the time axes lined up. 4. Plot the initial value of the stock on the stock graph. The initial value must be specified; it cannot be inferred from the net rate. 5. Break the net flow into intervals with the same behavior and calculate the amount added to the stock during the interval. Segments might be intervals in which the net rate is constant, changing linearly, or following some other pattern. The amount added to or subtracted from the stock during a segment is the area under the net rate curve during that segment. For example, does the net flow remain constant from time to time t,? If so, the rate of change of the stock during that segment is constant, and the quantity added to the stock is the area of the rectangle defined by the net rate between and If the net rate rises linearly in a segment, then the amount added is the area of the triangle. Estimate the area under the net rate curve for the segment and add it to the value of the stock at the start of the segment. The total is the value of the stock at the end of the segment. Plot this point on the graph of the stock. 6. Sketch the trajectory of the stock between the start and end of each segment. Find the value of the net rate at the beginning of the segment. Is it positive or negative? If the net flow is positive, the stock will be increasing at that time. If the net flow is negative, the stock will be decreasing. Then ask whether it is rising or falling at an increasing or decreasing rate, and sketch the pattern you infer on the graph. If the net rate is positive and increasing, the stock increases at an increasing rate (the stock accelerates upward). If the net rate is positive and decreasing, the stock increases at a decreasing rate (the stock is decelerating but still moving upward). If the net rate is negative and its magnitude is increasing (the net rate is becoming more negative), the stock decreases at an increasing rate. If the net rate is negative and its magnitude is decreasing (becoming less negative), the stock decreases at a decreasing rate. 7. Whenever the net rate is zero, the stock is unchanging. Make sure that your graph of the stock shows no change in the stock everywhere the net rate is zero. If the net rate remains zero for some interval, the stock remains constant at whatever value it had when the net rate became zero. At points where the net rate changes from positive to negative, the stock reaches a maximum as it ceases to rise and starts to fall. At points where the net rate changes from negative to positive, the stock reaches a minimum as it ceases to fall and starts to rise. 8. Repeat steps 5 through 7 until done.
Chapter 7 Dynamics of Stocks and Flows 237 FIGURE 7-3 The accumulation process creates delays. Note the me-quarter cycle lag between the peaks of the net flow and the peaks of the stock. 200 150.100 u) 50-50 600 400 0 3 6 912 24 36 48 Time (months) Now consider the flows specified in the top panel of Figure 7-3. The outflow is constant at 100 but the inflow fluctuates around an average of 100 with a period of 12months and an amplitude of At the start, the inflow is at its maximum. Assume the initial value of the stock is 500 units. Since the outflow is constant, the net inflow is a fluctuation with amplitude and a mean of zero. The stock begins at its initial value of 500 units, but since the inflow is at its maximum, the stock initially rises with a slope of 50 However, the net flow falls over the first 3 months, so the stock increases at a decreasing rate. At month 3 the net flow reaches zero, then goes negative. The stock must therefore reach a maximum at month 3.The amount added to the stock in the first 3 months is the area under the net rate curve. It is not easy
238 Part Tools for Systems to estimate the area from the graph because the net rate curve is constantly changing. You could estimate it by approximating the area as a set of rectangles, as described above, though this would take time. Using simulation to carry out the accumulation shows that a little less than 100 units are added to the stock by the time the net rate falls to zero at month 3. From month 3 to month 6, the net rate is negative. The stock is therefore falling. Just after month 3, the net rate is just barely negative, so the rate of decline of the stock is slight. But the magnitude of the net rate increases, so the stock falls at an increasing rate. At 6 months, the net rate has reached its minimum (most negative) value of -50 The stock is declining at its maximum rate; there is an inflection point in the trajectory of the stock at month 6. How much did the stock lose between month 3 and month Assuming the fluctuation in the net rate is symmetrical, the lossjust balanced what was gained in the first 3 months, reducing the stock back to its initial level of 500 units. From month 6 to month 9, the net flow remains negative, so the stock continues to fall, but now at a decreasing rate. By month 9 the net flow again reaches zero, so the stock ceases to fall and reaches its minimum. Again using the assumption of symmetry, the quantity lost from months 6 to 9 is equal to the quantity lost from months 3 to 6, so the stock falls to a level just above 400 units. From months 9 to 12 the net flow is positive, so the stock is rising. During this time the net rate rises, so the stock increases at an increasing rate, ending with a slope of 50 as the net rate reaches its maximum. Again, the stock gains the same amount, recovering its initial level of 500 units exactly at month 12. Beyond month 12 the cycle repeats. The example illustrates the way in which the process of accumulation creates delays. The input to the system is a fluctuation with a 12-monthperiod reaching its peak at time = 0, 12, 24,... months. The stock, or output of the system, also fluctuates with a 12-month period but lags behind the net inflow rate, reaching its peaks at t = 3, 15, 27,...months. The lag is precisely one-quarter cycle. The lag arises because the stock can only decrease when the net flow is negative. If the net flow is positive and falls to zero, the stock increases and reaches its maximum. Analytical Integration of a Fluctuation The example in Figure 7-3 can be made precise using a little basic calculus. The stock S is the integral of the net rate R. Assuming the net flow is a cosine with period 12 months and amplitude 50 R = then J J S = Rdt = = + The stock follows a sine wave with the same period and amplitude times that of the net flow. The delay caused by the accumulation process is easily seen since = - S - The stock follows the same trajectory as the net flow but with a phase lag of (one-quarter cycle). Equation (7-2) also shows that the amplitude of the stock is (50 (12 96 units, so the stock fluctuates between about 404 and 596, as seen in the figure.
Chapter7 Dynamics of Stocks and Flows 239 7.1.4 Graphical Differentiation The inverse of integration is differentiation, the calculation of the net rate of change of a stock from its trajectory. Given a graph of a stock, it is always possible to infer the net rate of change and plot it. As in the case of integration, there are analytic methods to calculate the net rate of a stock if the function describing the stock s path is known. However, in most dynamic models no analytic function for the stocks is known, so you must develop the of graphical differentiation. Graphical differentiation is straightforward. Simply estimate the slope of the stock at each point in time and plot it on a graph of the net rate. Figure 7-5 provides an example.
240 Part Tools for Systems FIGURE 7-5 Graphical differentiation 2000 1750 1500 1250 1000 5 weeks 5 weeks Time (weeks) The initial stock is 2000 units. For the first 10 weeks the stock declines linearly, so the net rate during this interval is negative and constant. The stock falls from 2000 to 1000 units in 10 weeks, so the net rate (the slope of the stock) is -100 At week 10the stock suddenly starts increasing. Drawing a line tangent to the stock curve at time 10 gives an estimate of the slope of 200 week. The net rate therefore steps up from -100 the instant before the start of week 10 to just after it starts. From weeks 10 to 20 the stock increases at a decreasing rate, so the net rate is positive but At time 20 the stock reaches a maximum so the net rate is zero. There are no kinks or bumps in the stock trajectory, implying a steady, linear decline in the net rate from 200 in week 10 to zero in week 20. From week 20 to week 30 the stock is falling. By week 30 it is falling rapidly; the slope of a line tangent to the stock trajectory at week 30 has a slope of -200 Again, there are no in the trajectory, so the net rate declines linearly from zero in week 20 to -200 week in week 30. At week 30 the stock suddenly stops changing and remains constant afterwards. The net rate suddenly steps up from-200to zero and remains at zero thereafter. Graphical differentiation of a stock reveals only its net rate of change. If the stock has multiple inflows and outflows it is not possible to determine their
Chapter 7 Dynamics of Stocks and Flows 241 individual values from the net rate alone: a firm s stock of cash remains constant whether revenues and expenditures both equal $1 million per year or $1 billion per year. 7.2 SYSTEM DYNAMICS IN ACTION: GLOBAL WARMING Much of the power of the system dynamics perspective comes from understanding how the process of accumulation creates dynamics, even before considering the feedbacks coupling the stocks and their flows. To illustrate, consider global warming. Is the earth warming? Is the warming caused by emissions of greenhouse gases caused by human activity? How much warming is likely over the next century? What changes in climate patterns, rainfall, growing season, storm incidence and severity, and sea level might ensue, and how much damage would these changes cause to humanity and to other species? These questions are difficult to answer, and legitimate scientific debates about the impact of anthropogenic GHG emissions continue. Despite the scientific uncertainty, several facts are not in dispute. The temperature at the earth s surface-the land, lower atmosphere, and surface layer of the ocean (the so-called mixed layer, the top 50 to 100meters, where most sea life exists)-is primarily determined by the balance of the incoming solar radiation and the outgoing reradiated energy. The earth is a warm mass surrounded by the cold of space and like all such masses emits so-called black body radiation whose frequency distribution and intensity depends on its surface temperature. The warmer the mass, the more energy it radiates. Incoming solar energy warms the earth.as it warms, more energy is radiated back into space. The temperature rises until the earth is just warm enough for the energy radiated back to space to balance the incoming solar energy.