3.5. Experiments with the Model of Drug-related Crime
3.5.1. A Tour of the Model
Open the file called 'Crime and Drugs' in the CD folder for Chapter 3. An overview diagram like Figure 3.16 appears showing the main sectors, stock accumulations and links.[] First let's tour the model and note the starting conditions in this imaginary world. Move the cursor into the world of the drug users, to the phrase 'drug-related crime'. This is a good place to begin. A little box pops up containing the number 200, meaning that, when the simulation begins, drug addicts are already committing 200 crimes per month. Move backwards in the causal chain. You will find that addicts collectively need £200 000 per month to fund their addiction (from demand of 500 kilograms per month at a street price of £400 per kilogram) and that each crime yields £1000 on average. Now visit the community and check the 'sensitivity to crime' and 'call for police action'. Typically a crime leads to five complaints, so the call for police action takes a value of 1 000 complaints per month (200 * 5) received from the community.[]
[] The reinforcing feedback loop that drives growth in drug-related crime is not visible in the sectorised diagram in Figure 3.16. The reason is that concepts which are the link pins between sectors are duplicated in each of the joined sectors. So, for example, drug seizures appear in the police department and again in the street market. Similarly, the street price appears in the street market and again in the drug users sector. Drug-related crime appears in the drug users sector and again in the community. Finally, the call for police action appears in the community and again in the police department. This duplication is achieved in the software by 'ghosting' a variable name. A special ghost icon is placed on top of a variable and when clicked a copy of the variable's icon appears which can be moved and placed at any point on the model page. When deposited, a ghost-like manifestation of the variable appears with a dotted outline. This replica carries with it the variable's underlying algebraic formulation for use in the new location. Other system dynamics software packages offer similar functionality under different names, such as clone. It is not essential to use ghosting or cloning in a sectorised model. Active links can be drawn between variables in different sectors. However, in large models the use of ghosting or cloning is to be recommended as it greatly reduces visual complexity and the daunting sight of a spaghetti-like tangle of connections. Even in small models like drugs and crime there is merit in such clarity. Moreover (and this is important) the absence of overt visual links between sectors is a reminder of the invisibility of feedback loops that weave their way across organisational boundaries.
[] Note of warning: the parameter values in this small pedagogical model have not been carefully calibrated and would likely benefit from more field research and logical thought. For example, the street price of drugs is crudely initialised at £400 per kilogram (kg) and the total supply of drugs on the street is set at 500 kg per month. In a real-world modelling project both these numbers should be set as accurately as possible based on experts' judgements and/or published data in government reports or the press; and should then be double-checked with a dose of logical reasoning. Some numbers are easier to pin down than others and once set can help clarify the values of other related parameters in a kind of logical bootstrapping. For example, in the drug–crime model the average yield per crime incident can be estimated on common-sense grounds. Knowledge of that number may then, through a process of logical reasoning, shed more light on the street price of drugs in £ per kg.
Figure 3.16. Overview of the drug-related crime model showing sectors, stocks and links
Next, move to the police department to probe what's happening there. Initially there are 10 police officers allocated to drug busting but, due to the volume of complaints, the indicated allocation (the number of officers really needed) is 13.[] Police effectiveness in drug busting, a productivity measure, is 10 kilograms per police officer per month. Hence, drug seizures start at 100 kilograms per month (10 officers each seizing 10 kilograms monthly on average).
[] The numerical value of the change in allocation of police officers is displayed as 0.833333. Obviously some judgement is needed in interpreting this numbingly precise value. It means that, on average, at the start of the simulation, the re-allocation of police officers to drug busting increases by the equivalent of about 80 per cent of a single full-time equivalent police officer per month. Modellers should beware of misleading precision in a simulator's calculations and round the numbers to a precision consistent with the problem situation.
Now step out of the police department and into the street market. The total supply of drugs brought to the neighbourhood is assumed to be 500 kilograms per month, but the supply that actually reaches the street is 400 kilograms per month due to the effect of seizures. Demand for drugs is 500 kilograms per month, so the drug supply gap is 100 kilograms per month. This initial shortage leads to pressure for price increase of one per cent per month. That completes our tour of initial conditions.
3.5.2. Escalating Crime – The Base Case
Now we are ready to simulate. Press the 'Run' button and watch the stock accumulations for street price and the number of police allocated to drug busting. They start small and, as time goes by, they quickly escalate. Press the run button again and now watch the little dials. What you see, over 48 simulated months, are the changes and knock-on consequences that unfold when drug addicts commit crimes, disturb a community, and invoke a response from police. There is a lot going on so it is a good idea to make several runs and study the dials sector by sector. After 48 months (4 years) of drug busting and escalating crime, conditions are much worse.
To investigate how much conditions have changed by the end of the simulation, repeat the same tour of variables you began with. In the world of the drug user, crime has risen to 446 incidents per month and is still growing fast. In the police department, 30 police officers are allocated to drug busting and seizures have reached 296 kilograms per month. Meanwhile, the supply of drugs on the street has dwindled to 204 kilograms per month and the street price has reached to £893 per kilogram.
To explore the dynamics in more detail, open the graphs located in the lower right of each sector. The time charts shown in Figure 3.17 will appear showing how conditions unfold from the perspective of each stakeholder. Begin with the police department (Graph 1). The call for police action (line 1) starts at 1000 complaints per month and grows steadily to more than 2 000 after 48 months. More complaints lead to growth in drug seizures (line 2) from 100 to almost 300 kilograms per month. The underlying re-allocation of police officers to drug busting is shown on a separate page of the graph, which can be viewed by clicking the page tab in the bottom left of the chart. A new page appears (not included in Figure 3.17) that shows the indicated allocation of police (line 1) growing in response to complaints. Following the indicated allocation with a slight time lag, there is a corresponding increase in the number of police allocated to drug busting (line 2).
Figure 3.17. Dynamics as seen in each sector
Meanwhile, in the street market (Graph 2), demand for drugs (line 1) remains steady at 500 kilograms per month while drug seizures (line 4) grow. Therefore, the supply of drugs on the street falls (line 2) and the drug supply gap (line 3) rises. This shortage forces up the street price of drugs (line 5) from £400 to almost £900 per kilogram over four years. The result, in the world of the drug user (Graph 3), is a steady escalation of funds required (line 2) that induces a corresponding increase in drug-related crime on a parallel trajectory (line 3). Meanwhile, demand for drugs (line 1) remains steady, at the same value already seen in the street market. Finally. from the community's viewpoint (Graph 4), drug-related crime is rising (line 1), provoking more complaints and an ever greater call for police action (line 2).
3.5.3. Drilling Down to the Equations
Feedback structure gives rise to dynamic behaviour, and the most detailed expression of feedback structure is to be found at the level of algebraic equations. The equations of the drug-related crime model can be found by 'drilling down' below the diagram. Click the 'equation' tab on the left of the screen and a whole page of equations is revealed, which are reproduced in Figures 3.18 and 3.19. The formulations are identical to the ones presented earlier in the chapter and you are invited to browse them. Notice they are conveniently organised by sector, starting with stock accumulations, then algebraic converters and finally graphical converters. Moreover, there is a one-to-one correspondence between the equations and the diagram, making it easy to move back and forth between the visual representation and its algebraic equivalent. The equations are readable like sentences and replicate the full variable names used in the diagram. The intention is to create 'friendly algebra', easy to communicate and interpret. For example, in the police department, the change in the allocation of police is equal to the difference between the 'indicated allocation of police' and the 'number of police allocated to drug busting', divided by the 'time to move staff'. This expression, though algebraic, is pretty much plain English, and normally modellers strive to achieve such transparency, both to clarify their own thinking and to communicate the algebra to others.
The same transparency applies to stock accumulations and even to non-linear relations in graphical converters. For example, in the street market, the street price of drugs at time t is equal to the street price at the previous time (t-1) plus the change in street price over the period dt (the interval of time between t-1 and t). The change in street price is then expressed as the product of 'street price' and 'pressure for price change'. In turn, the pressure for price change is formulated as a non-linear graph function of the drug supply gap. The shape of this graph is specified by the paired numbers shown at the end of the equation, where the first number is the size of the drug supply gap (in kilograms per month) and the second number is the corresponding value of the pressure for price change (the fractional change per month). The actual shape can be seen by double clicking on the equation. A window appears. On the right are two columns containing the paired numbers in two columns and on the left is a graph constructed from these numbers. The drug supply gap is on the horizontal axis and the pressure for price change is on the vertical axis.
Figure 3.18. Equations for the Community and Police Department
Figure 3.19. Equations for the street market and world of the drug users
3.5.4. Anomalous Behaviour Over Time and Model Boundary
A simulator is a relentless and unforgiving inference engine – it shows the logical implications of all the assumptions it contains; the behaviour over time that will unfold given the structure. This enforced consistency between assumptions and outcomes has the power to surprise people and is a major benefit of simulators for strategy development. Surprise behaviour often reveals flaws and blind spots in people's thinking. Modellers themselves are the first beneficiaries of such ruthless consistency during the early diagnostic testing of a new model. The drug-related crime model provides a simple example.
Recall the dynamic hypothesis behind the model. Escalating crime is attributable to police drug busting that removes drugs (and drug dealers) from the streets. A side effect is to push up the street price of drugs and this price inflation inadvertently forces addicts to commit more crime, leading to more drug busting and so on. The structure is a reinforcing loop and the simulator shows that crime escalation is possible given reasonable operating assumptions about the police department, street market, the community and addicts themselves.
An extension of the simulation from 48 to 60 months pushes this logic beyond the limits of common sense and reveals a world in which the price of drugs is sky high, crime has increased six fold and the supply of drugs on the street is negative! Figure 3.20 shows this anomalous scenario in more detail. To create these charts open the model called 'Crime and Drugs – 60 months'[] and press the 'Run' button. Open Graph 1 in the police department. The call for police action (line 1) begins at 1 000 complaints per month and escalates to more than 6000 after 60 months (five years). Notice the steep upward trajectory typical of exponential growth from a reinforcing feedback loop. Meanwhile, drug seizures rise in a similar exponential pattern until month 55 when the rate of increase begins to moderate. The reason for this slowdown is that police are being overstretched and it is increasingly difficult for department leaders to re-allocate officers to drug busting, despite the rising call for police action. Turn to page 2 of the police department chart to see the overstretch effect in more detail. The indicated allocation of police (line 1) rises exponentially to 70 police officers in month 54, as the police department recognises the escalating call from the community for police action. However, in the final six months of the simulation the department encounters its own manpower limit to drug busting, since (by assumption) no more than 95 officers can be allocated to the task, no matter how great the call from the community. The indicated allocation gradually levels out at 95 officers and the growth in the actual number of police allocated to drug busting (line 2) begins to slow down.
[] This model is identical to the previous one but the length of the simulation has been increased to 60 months by changing 'run specs' in the pull-down menu headed 'Run'. In addition, the vertical scale of chart variables has been adjusted so the simulated trajectories do not go out of bounds.
So far so good, in the sense that the trajectories are plausible from an operational viewpoint. But the full consequences are quite surprising. The cap on police effort does not stop crime escalation. To understand why step into the 'street market' and open Graph 2. The demand for drugs (line 1) is steady at 500 kilograms per month throughout the entire 60 months, reflecting the assumption that addicts will seek their daily fix regardless of supply. Meanwhile drug seizures (line 4) are rising. Between months 30 and 45 they increase from 160 to 244 kilograms per month. By month 54 seizures are 500 kilograms per month – equal to the entire supply. So according to this scenario by month 54 there are no more drugs available on the street – they have all been seized. Moreover, by month 60, police officers are seizing more drugs than the dealers themselves can obtain and the supply of drugs on the street (line 2) is negative. Clearly this extreme situation is illogical - a fallacy in the model revealed by simulation. Common sense says police cannot seize more drugs than are supplied. There is one more subtle twist in this diagnostic simulation. The drug supply gap (line 3) fuels an exponential rise in street price (line 5), which shows no sign of slackening, despite the levelling-off of police effort at the end of the simulation.
Figure 3.20. Anomalous dynamics in a 60-month simulation
A modeller faced with these contradictions returns to the model's assumptions to find the fallacy. A few possibilities come to mind. The simplest, and least disruptive to the integrity of the model, is that police effectiveness in drug busting is not constant (as assumed) but depends on the supply of drugs on the street. As supply is reduced through drug seizures it becomes more and more difficult for police to trace the few drugs that remain – an example of the 'law of diminishing returns'. This formulation requires a new causal link and a graphical converter that shows police effectiveness as a non-linear function of the supply of drugs on the street.[] Such a re-formulation would solve the absurdity of negative supply, but would not necessarily deal with price escalation (line 5). The problem here is that any sustained supply shortage, even a 20 per cent shortfall or less, will invoke an upward drift in price, because there is no cost anchor for the price of drugs. Recall that the rate of change of price is proportional to the current street price and the drug supply gap. Such price behaviour is plausible in a market where demand is totally inelastic, yet endless price escalation is clearly unrealistic. This inconsistency suggests the need for a more radical change to the model – a rethink not only of individual formulations but also an expansion of the model boundary. One idea is to include the dynamics of supply. The current model assumes the total supply of drugs is fixed, so drug seizures create a permanent shortage on the street. However, if the street price is high then, sooner or later, the supply of drugs will increase to compensate for drug busting, thereby re-establishing an equilibrium of supply and demand.
[] If you try toadd this connection to the diagraman error message isgenerated saying 'sorry, but that would create a circular connection'. The software is pointing out a new fallacy. In a dynamical system every feedback loop must contain at least one stock accumulation that stores the changes generated around the loop. In this case, the supply of drugs on the street could be reformulated as a stock accumulation:
Supply of drugs on the street (t) = supply of drugs on the street (t−dt) + (change in street supply) * dt
INIT supply of drugs on the street = 400 {kilograms per month}
INFLOWS:
Change in street supply = (indicated supply of drugs on the street − supply of drugs on the street) / time to affect street supply
Indicated supply of drugs on the street = total supply of drugs - drug seizures Time to affect street supply = 1 (month)
To complete the new formulation simply add diminishing returns to police effectiveness:
Drug seizures = number of police allocated to drug busting * police effectiveness in drug busting Police effectiveness in drug busting = 10 * GRAPH (supply of drugs on the street) {kilograms per officer per month}
Sketch your own graph, which should be non-linear on a scale between zero and one. When the supply of drugs on the street is high then police effectiveness should be at a maximum value of one. When the supply of drugs on the street is significantly reduced (say less than half its initial value) then police effectiveness begins to decline, and should reach zero when the supply on the street is zero.
Not surprisingly, the small pedagogical model of drugs and crime in this chapter has its limitations. It illustrates principles of model building but would not be an adequate model to address drugs policy. However, the same modelling principles can and do lead to useful policy models in context of serious applications projects, exemplified in Homer's (1993) study of national cocaine prevalence in the USA.