Notes toward a macro version of the Nee–DellaPosta–Opper model of institutional change

One of the virtues of the Visiting Scholars program of the Russell Sage Foundation is that someone like me, a card-carrying macroeconomist by trade, came to read this paper by Victor Nee (co-authored with Daniel DellaPosta and Sonja Opper). The S-curves in Figure 3 triggered the natural reaction of the proverbial child with a hammer (or a model of a hammer): there is a purely aggregative way to get to this outcome, and it tells an interesting story on its own. So here are some preliminary notes toward a macromodel corresponding to the nearest-neighbor micromodel of the DellaPosta, Nee, and Opper’s paper. In their paper, chance innovations give rise to utility gains that through positive externalities diffuse through networks and communities. “The greater the utility gain and larger the network externalities, the more likely political actors will accommodate endogenous institutional change.” This “bottom-up” approach highlights the importance of understanding the forces that give rise to innovation and deviance, as well as the motives.

Let N(t) be the number of deviants at time t. I will treat time as continuous, though everything could easily be translated to discrete time, which would be more convenient for computer simulation and for adding random fluctuations.

In the absence of any suppression of deviance by the state (or anyone else), N(t) evolves according to

So any small N (<m) will decay back to zero, but if N should happen to exceed m, it will trace out a standard logistic curve with upper asymptote M. In the paper by Nee et al., M is the population size, but it could be determined in some other way; it is the largest sustainable number of deviants.

The state can choose to suppress an arbitrary number S of deviants per unit time in which case N evolves according to

If S is constant in time, the phase diagram is simple and familiar. Draw the parabola given by the right-hand side (RHS) of equation (1) and superimpose a horizontal line at height S. It will normally intersect the parabola at two points, N′ and N″, say N′ < N″. If N takes a value to the left of N′ or to the right of N″, N will decrease with time (because, as Figure 1 shows, dN/dt < 0 there). At any value between N′ and N″, N will increase. So if N ever gets above N′; it will follow a logistic-like path to a stable equilibrium at N″ < M. At N″ the natural endogenous rate of increase in N is S and is just offset by the suppressive activities of the state. (Notice that if the horizontal at S lies above the peak of the parabola, the state snuffs out all deviance as soon as it appears, and N is always zero except for occasional random bursts, immediately suppressed.)

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Figure 1.

Here, the number of deviants suppressed per unit time is a constant. N′ is an unstable equilibrium. Any displacement to the left leads to the disappearance of deviants. Any displacement to the right leads to an S-curve ending at N″.

Of course, S(t) = constant is not an intelligent policy. It makes more sense that the state has a policy S = s(N): the amount of suppression is tuned to the amount of deviance. (More complicated policies are possible.) What should be assumed about the function s(N)? How might a policy be determined in this abstract world?

Suppose the state has an implicit valuation function defined on N: V(N). The state can assign a numerical value to the (lack of) desirability of a situation with N deviants. So V is a decreasing function of N. Also suppose that the cost to the state of suppressing S deviants per unit time is c(S). (The cost of suppression might also depend on N. This is only a minor complication and I avoid it.) Then, a fairly rational (though not hyper-rational) policy for the state might be to choose S at every time t to

If the state did nothing, there would be N deviants, and the state wouldn’t like that. It would feel better if there were fewer deviants, but bringing that about is costly. (Note the tacit assumption that V and c are commensurable. The Office of Management and Budget [OMB] will have to figure it out.) If that is the policy, the appropriate S will satisfy

and

The conventional assumption is that c″ > 0 (increasing marginal cost) so equation (2b) does not fix the sign of V″(N − S). Now equation (2a) implicitly determines the function S = s(N); it gives the maximizing S for any value of N. It does this by finding the value of S such that the marginal gain from suppressing one more deviant is just equal to the marginal cost of suppressing that deviant. I warned you that I am an economist.

Should S be larger for larger N? One might think so, but there are those increasing marginal costs of suppression to worry about. Harsh suppression may be very costly in terms of resources and/or discontent. We can calculate dS/dN by implicit differentiation of equation (2a)

The denominator of the RHS of equation (3) is negative, by equation (2b). So the sign of dS/dN is the opposite of the sign of V″(N − S): the “optimal” S increases with N if V is concave at N − S and decreases if V is convex at that point. This makes sense: S increases with N if the incremental “harm” done by an additional deviant gets bigger as there are more deviants to begin with. Alternatively, it may be that once there are a lot of deviants, another deviant doesn’t make much difference (V is convex); in that case, once N gets large, the state may suppress less. The shape of V(N) describes how and how much the authorities care about deviance from approved social norms. That will depend on the nature of “the authorities” and probably on the nature of the deviation. In any case, specification of V(N) and therefore s(N) has to be an important part of a theory of institutional change.

Whatever one thinks about that, once S = s(N) is determined from equation (2a), it can be plotted along with the underlying parabola and the path of N (and S) read off. There are many possibilities. If suppression is valuable and cheap, the state may always choose to drive N to zero. Or the situation may be like the case of constant S described earlier: Figure 2 illustrates a stable equilibrium at N″, kept there because s(N″) = dN/dt at N″. Or, if s′(N) < 0, the state may sort of “give up” if N gets large, and let N go all the way to M. This last case is illustrated in Figure 3.

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Figure 2.

Here, the suppression of deviance is more intense the more deviants there are. Qualitatively, the model behaves much as in Figure 1.

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Figure 3.

Here, the presence of more deviants discourages suppression. Once N exceeds N′, deviance increases, suppression is eventually abandoned as too costly, and the number of deviants grows to include all M eligibles.

One could complicate the model by supposing that the state cannot perform even this fairly myopic but instantaneous optimization, and instead merely gropes its way toward an optimum. Then, one could add another differential equation

This says that if at any time S is less than the optimum for the current N, S will increase; if S exceeds the current optimum, it will decrease. The parameter b is a speed of adjustment. Nothing basic changes. The main effect is that the more complicated dynamics—two variables—may cause the path to the ultimate equilibrium to overshoot and then reverse course. This phenomenon can easily be illustrated on (considerably busier) phase diagrams, but is not worth the trouble at this stage.

It should be kept in mind that the optimization described here is very short-sighted. The state reacts only to the current state of the system, making no attempt at a long-run policy. Even at this simpler level, we (or at least I) have little grasp of what V(N) should look like. The key issue, so far as this simple model is concerned, is whether dislike of and resistance to deviance increase as the number of deviants increase or whether this effect eventually attenuates as deviance becomes commonplace. (And in a model with foresight, would this consequence be anticipated? And if so, what then?) This (dubious?) higher level of rationality would require the state to have a long-run goal and to formulate a plan to achieve it in the most useful satisfying way. To model that situation would require a much more elaborate formulation.

Sticking with the myopic model, it’s easy to work out special cases, for example, when V(N) is quadratic (and, say, c(S) = cS²). Then, s(N) is linear. If V(N) = p − qN², s(N) slopes upward, various pictures are possible. On the other hand, if V(N) = p − qN + rN² (downward-sloping part only), s(N) can have negative slope, allowing still further possibilities.