Job tenure dynamics

This is a simple model of the dynamics of employment in a sector. I built it for a LinkedIn article that describes the situation and the data.

The model is interesting and reasonably robust, but it has (at least) three issues you should know about:

  • The initialization in equilibrium isn’t quite perfect.
  • The sector-entry decision (Net Entering) is not robust to low unemployment. In some situations, a negative net entering flow could cause negative Job Seekers.
  • The sector-entry decision also formulates attractiveness exclusively as a function of salaries; in fact, it should also account for job availability (perceived vacancy and unemployment rates).

Correcting these shortcomings shouldn’t be too hard, and it should make the model’s oscillatory tendencies more realistic. I leave this as an exercise for you. Drop me a note if you have an improved version.

The model requires Vensim (any version, including free PLE).

Download employees1.mdl

Social network valuation with logistic models

This is a logistic growth model for Facebook’s user base, with a very simple financial projection attached. It’s inspired by:

Quis pendit ipsa pretia: facebook valuation and diagnostic of a bubble based on nonlinear demographic dynamics

Peter Cauwels, Didier Sornette

We present a novel methodology to determine the fundamental value of firms in the social-networking sector based on two ingredients: (i) revenues and profits are inherently linked to its user basis through a direct channel that has no equivalent in other sectors; (ii) the growth of the number of users can be calibrated with standard logistic growth models and allows for reliable extrapolations of the size of the business at long time horizons. We illustrate the methodology with a detailed analysis of facebook, one of the biggest of the social-media giants. There is a clear signature of a change of regime that occurred in 2010 on the growth of the number of users, from a pure exponential behavior (a paradigm for unlimited growth) to a logistic function with asymptotic plateau (a paradigm for growth in competition). We consider three different scenarios, a base case, a high growth and an extreme growth scenario. Using a discount factor of 5%, a profit margin of 29% and 3.5 USD of revenues per user per year yields a value of facebook of 15.3 billion USD in the base case scenario, 20.2 billion USD in the high growth scenario and 32.9 billion USD in the extreme growth scenario. According to our methodology, this would imply that facebook would need to increase its profit per user before the IPO by a factor of 3 to 6 in the base case scenario, 2.5 to 5 in the high growth scenario and 1.5 to 3 in the extreme growth scenario in order to meet the current, widespread, high expectations. …

(via the arXiv blog)

This is not an exact replication of the model (though you can plug in the parameters from C&S’ paper to replicate their results). I used slightly different estimation methods, a generalization of the logistic (for saturation exponent <> 1), and variable revenues and interest rates in the projections (also optional).

This is a good illustration of how calibration payoffs work. The payoff in this model is actually a policy payoff, because the weighted sum-squared-error is calculated explicitly in the model. That makes it possible to generate Monte Carlo samples and filter them by SSE, and also makes it easier to estimate the scale and variation in the standard error of user base reports.

The model is connected to input data in a spreadsheet. Most is drawn from the paper, but I updated users and revenues with the latest estimates I could find.

A command script replicates optimization runs that fit the model to data for various values of the user carrying capacity K.

Note that there are two views, one for users, and one for financial projections.

See my accompanying blog post for some reflections on the outcome.

This model requires Vensim DSS, Pro, or the Model Reader. facebook 3.vpm or (The .zip is probably easier if you have DSS or Pro and want to work with the supplementary control files.)

Update: I’ve added another set of models for Groupon: groupon 1.vpmgroupon 2.vpm and

See my latest blog post for details.


Economic Cycles: Underlying Causes

Nathaniel Mass’ model of economic cycles, replicated from his 1975 book, Economic Cycles: An Analysis of Underlying Causes, which unfortunately seems to have disappeared from the Productivity Press site (though you can still find used copies).

I haven’t checked, but I’m guessing that the model is quite similar to that in his PhD thesis, which you can get from MIT libraries here. Here’s the abstract:

The models: mass2.mdl mass2.vpm

These don’t have units defined, unfortunately – I’d love to have a copy with units if you’re so inclined.

The Dynamics of Commodity Production Cycles

These classic models are from Dennis Meadows’ dissertation, the Dynamics of Commodity Production Cycles:

While times have changed, the dynamics described by these models are still widespread.

These versions should work in all recent Vensim versions:

DLMhogs2.vpm DLMhogs2.mdl

DLMgeneric2.vpm DLMgeneric2.mdl


A System Zoo

I just picked up a copy of Hartmut Bossel’s excellent System Zoo 1, which I’d seen years ago in German, but only recently discovered in English. This is the first of a series of books on modeling – it covers simple systems (integration, exponential growth and decay), logistic growth and variants, oscillations and chaos, and some interesting engineering systems (heat flow, gliders searching for thermals). These are high quality models, with units that balance, well-documented by the book. Every one I’ve tried runs in Vensim PLE so they’re great for teaching.

I haven’t had a chance to work my way through the System Zoo 2 (natural systems – climate, ecosystems, resources) and System Zoo 3 (economy, society, development), but I’m pretty confident that they’re equally interesting.

You can get the models for all three books, in English, from the Uni Kassel Center for Environmental Systems Research, Follow the Archiv(e) link on the home page and enter the downloads Archiv(e). This will put you in a file browser. Choose the Software folder, then the Zoo folder to obtain a .zip archive of the zoo models for the whole series, in Vensim .mdl format.

To tantalize you, here are some images of model output from Zoo 1. First, a phase map of a bistable oscillator, which was so interesting that I built one with my kids, using legos and neodymium magnets:

Continue reading “A System Zoo”

The Rise and Fall of the Saturday Evening Post

Replicated by David Sirkin and Julio Gomez from Hall, R. I. 1976. A system pathology of an organization: The rise and fall of the old Saturday Evening Post. Administrative Science Quarterly 21(2): 185-211. (JSTOR link). Just updated for newer Vensim versions.

This is one of the classic models on the Desert Island Dynamics list.

There are some units issues, preserved from the original by David and Julio. As I update it, I also wonder if there are some inconsistencies in the accounting for the subscription pipeline. Please report back here if you find anything interesting.



Market Growth

John Morecroft’s implementation of Jay Forrester’s Market Growth model, replicated by an MIT colleague whose name is lost to the mists of time, from:

Morecroft, J. D. W. (1983). System Dynamics: Portraying Bounded Rationality. Omega, 11(2), 131-142.

This paper examines the linkages between system dynamics and the Carnegie school in their treatment of human decision making. It is argued that the structure of system dynamics models implicitly assumes bounded rationality in decision making and that recognition of this assumption would aid system dynamicists in model construction and in communication to other social science disciplines. The paper begins by examining Simon’s “Principle of Bounded Rationality” which draws attention to the cognitive limitations on the information gathering and processing powers of human decision makers. Forrester’s “Market Growth Model” is used to illustrate the central theme that system dynamics models are portrayals of bounded rationality. Close examination of the model formulation reveals decision functions involving simple rules of thumb and limited information content. …

Continue reading “Market Growth”

Rental car stochastic dynamics

This is a little experimental model that I developed to investigate stochastic allocation of rental cars, in response to a Vensim forum question.

There’s a single fleet of rental cars distributed around 50 cities, connected by a random distance matrix (probably not physically realizable on a 2D manifold, but good enough for test purposes). In each city, customers arrive at random, rent a car if available, and return it locally or in another city. Along the way, the dawdle a bit, so returns are essentially a 2nd order delay of rentals: a combination of transit time and idle time.

The two interesting features here are:

  • Proper use of Poisson arrivals within each time step, so that car flows are dimensionally consistent and preserve the integer constraint (no fractional cars)
  • Use of Vensim’s ALLOC_P/MARKETP functions to constrain rentals when car availability is low. The usual approach, setting actual = MIN(desired, available/TIME STEP), doesn’t work because available is subscripted by 50 cities, while desired has 50 x 50 origin-destination pairs. Therefore the constrained allocation could result in fractional cars. The alternative approach is to set up a randomized first-come, first-served queue, so that any shortfall preserves the integer constraint.

The interesting experiment with this model is to lower the fleet until it becomes a constraint (at around 10,000 cars).

Documentation is sparse, but units balance.

Requires an advanced Vensim version (for arrays).

carRental.vpm carRental.vmf

A Behavioral Analysis of Learning Curve Strategy

Model Name: A Behavioral Analysis of Learning Curve Strategy

Citation: A Behavioral Analysis of Learning Curve Strategy, John D. Sterman and Rebecca Henderson, Sloan School of Management, MIT and Eric D. Beinhocker and Lee I. Newman, McKinsey and Company.

Neoclassical models of strategic behavior have yielded many insights into competitive behavior, despite the fact that they often rely on a number of assumptions-including instantaneous market clearing and perfect foresight-that have been called into question by a broad range of research. Researchers generally argue that these assumptions are “good enough” to predict an industry’s probable equilibria, and that disequilibrium adjustments and bounded rationality have limited competitive implications.  Here we focus on the case of strategy in the presence of increasing returns to highlight how relaxing these two assumptions can lead to outcomes quite different from those predicted by standard neoclassical models. Prior research suggests that in the presence of increasing returns, tight appropriability and accommodating rivals, in some circumstances early entrants can achieve sustained competitive advantage by pursuing Get Big Fast (GBF) strategies: rapidly expanding capacity and cutting prices to gain market share advantage and exploit positive feedbacks faster than their rivals. Using a simulation of the duopoly case we show that when the industry moves slowly compared to capacity adjustment delays, boundedly rational firms find their way to the equilibria predicted by conventional models.  However, when market dynamics are rapid relative to capacity adjustment, forecasting errors lead to excess capacity, overwhelming the advantage conferred by increasing returns. Our results highlight the risks of ignoring the role of disequilibrium dynamics and bounded rationality in shaping competitive outcomes, and demonstrate how both can be incorporated into strategic analysis to form a dynamic, behavioral game theory amenable to rigorous analysis.

The original paper is on ; it was eventually published in Management Science. You can get the MS version from John Sterman’s page here.

Source: Replicated by Tom Fiddaman

Units balance: Yes

Format: Vensim (the model uses subscripts, so it requires Pro, DSS, or Model Reader)

Behavioral Analysis of Learning Curve Strategy (Vensim .vmf)

New update: