Veblen Model Explanation, Envy Rule

WHAT IS IT?

Veblen Rev 5.3, September 23, 2004, by Ralph Abraham This is a NetLogo model for conspicuous consumption, with the envy rule.

HOW IT WORKS

The initial setting

The strategy space, or action set, A, is the unit interval, [0, 1], in this model. The agents (called turtles in NetLogo) are consumers. Each consumer is shown as a triangle on the strategy space. They have different colors just for the visual effect. When several consumers are on the same patch (discretized interval of the strategy space) only the top one can be seen in entirety, but the x position is a floating point number, so parts of lower turtles may be seen.

The strategy space is shown as five horizontal rows in the upper half of the display. These are to be regarded as superimposed layers on a single interval. It represents a unit interval corresponding to the choice of strategy, x. All consumers have the same income, 1, but choose variously how much to spend on ordinary consumption, x, and how much to spend on conspicuous consumption, 1-x. Thus x = 0 represents 100% conspicuous consumption, such as diamond rings, and x = 1 represents 100% discrete consumption, such as savings.

A chosen number of consumers begin at initial positions in the strategy space. This initial density is important to the outcome of a run. This model is arranged so that the initial density is the sum of an arbitrary number of square waves. Thus the operator may approximate an arbitary initial density. Interesting choices include a single square wave or herd, two herds, a tent shape or heap, two heaps, and so on. In any case, the operator begins by adding square waves, or sub-herds, until a desired initial distribution is obtained. Each addition of a sub-herd is called a "puff".

The distribution, F(x)

The instantaneous state of the system is represented by the density of consumers in the strategy space, f ("rho" in Friedman 2001), a probability measure, or equivalently by its cumulative distribution, F (D in Friedman 2001), the integral of f, a monotone function increasing from zero to one.

The density, f, is also shown as a graph in the upper plot window, labelled "Density of Consumers", showing the average density of turtles on each patch.

The payoff, phi(x, F)

The most important function in the model is the payoff function, phi. It is a real-valued function, depending on both x and F. The function phi is the landscape in this example of landscape dynamics. The definition of phi used here is the sum of two nonpositive functions:

+ amp * ln(x): negative as x is in the interval (0, 1], and
- integral of F(y)dy from 0 to x, negative as F(y) is nonnegative.
Note that we avoid the troublesome value x = 0. The graph of phi is shown as below the graph of f, and on the same horizontal scale, x in (0, 1]. The two plots are updated after every 10th step.

The slope

The slope of the landscape, or gradient of phi, phisubx, aka the fitness gradient, is given by the formula:

phisubx (x, F) = amp/x - F(x).
This is called the envy rule in (Friedman 2001). Another formula, the "pride rule", will be realized in another model. The first term in the envy rule is the direct utility a consumer receives from ordinary consumption. It is monotone decreasing to the value "amp" at x=1. The constant "amp" ("c" in Friedman 2001) may be set with a slider. It represents the importance of ordinary consumption relative to conspicuous consumption. The second term is also monotone decreasing. We will be especially interested in the zero-crossing of this function.

Here the slope is shown with colors on the color bar below the gray row in the black screen. Its color code is:

+0.1 or higher [ red ], positive, step to the right
-0.1 to 0.1 [ yellow ], small step to the left or right
-0.1 or lower [ green ], negative, step to the left
This is chosen to emphasize the zero-crossing of the slope. As the slope depends on F, which is time-dependent, the yellow segment of the slope color bar will be expected to move about.

The step

Consumers step uphill on the landscape. With each step of discrete time, each consumer adjusts her strategy, x, by an increment proportional to the slope. The proportion (stepsize) may be set with a slider. Each turtle moves uphill by an increment: stepsize * slope. This is the Euler method for integrating the partial differential equation representing the envy rule.

HOW TO USE IT

STEP #1: THE INITIAL HERD

  • (1a) Set the "population" slider to determine the number of consumers in the initial subherd.
  • (1b) Set the 'center" slider to the center of the initial subherd of turtles (that is, the first puff into the distribution of consumers).
  • (1c) Set the "width" slider to the width of the initial subherd, as a percentage of the total width of the window.
  • (1d) Press the "setup" button to create the initial distribution of consumers on the choice interval. The initial plot shows a narrow herd of turtles around your chosen center, on row 1.
  • (1e) Choose a layer with the "puff-row" drop-down choice list. Then repeat (1a), (1b), and (1c), and press the "puff" button. A second square-wave subherd joins the herd on the chosen row. The total population is now indicated by the "totalpop" counter. Repeat as required to create the initial distribution desired.

STEP #2: ACTION

  • (2a) Press the "step" button to activate a single step in the consumption game. Every turtle will take one step, proportional to the slope of the landscape at its current position (the value of phisubx at its current x). You will see the turtles move, and then (after ten steps) the plots will be redrawn.
  • (2b) Adjust the "stepsize" slider so that the herd moves very slowly, to avoid numerical artifacts, such as gaps.
  • (2c) Press "step" several times to judge the stepsize choice.
  • (2d) Press "go" to trigger a rapid sequence of steps. Press "go" again to halt the action.

THINGS TO TRY

Try different numbers initial distributions, and change the value of the payoff constant with the amp slider. In case of suspicioous behavior, try decreasing the stepsize.

THINGS TO NOTICE

Try to predict what will occur in the long run. Clumping? Herd spreading? Bifurcation? In all case, check the color bar and the plots to see if the behavior follows the rules.

CREDITS AND REFERENCES

Many thanks to Uri Wilensky for his cleverness and industry in creating and evolving NetLogo, and to the NSF for support.

Revised 23 Sept 2004 by Ralph Abraham