What is the optimal degree of centralization

What is the optimal degree of centralization?
- And how should the question be discussed?

by Hans O. Melberg

Introduction
Recently the Minister of Justice in Norway announced that the number of police districts in Norway should be reduced from 19 to 5. The argument was that this would result in “more police officers on the streets” since fewer police officers would have to sit behind desks to do “paper work.” This, of course, sounds very nice to those who believe in “more police on the streets.” The reason this caught my interest, however, was not the prospect of having more police on the street, but the fact that the argument given for the reform in no way justified the reduction to five. In fact, if the underlying logic is that “fewer districts leads to more police on the street” then it seems that the optimal number of police districts is 1, not 5. Obviously this would be dismissed as “too much centralization.” But on what grounds? Why is 1 too much, 19 not enough and 5 optimal?

Clearly the answer must involve some kind of trade-off. Centralization has some good effects and some bad effects. At some point the bad effects outweigh the good effects and we have “too much” centralization. All this sounds obvious (and it is), but I still want to examine it a bit closer in order to see whether it is possible to find more “surprising” lessons. Since I do not know much about the organization of the police, I will discuss the topic by focusing on another sector: The organization of treatment for drug abuse.

The question
Admission to the treatment system in Norway is relatively decentralized in the sense that the drug user and the treatment institutions themselves to a large degree jointly decide who should go where. An extreme alternative to this would be a centralized system in which a single institution – the “Central” – decides who should go to which institution. The question is then if and to what extent centralization produces better and cheaper allocation of clients to the different institutions.

Let me first of all note that I have no illusions about being able to give the answer. I simply want to explore the question in a half-rigorous way in order to see whether something interesting shows up. To do so I shall start with the two “classic” effects of centralization. One often noted problem of centralization is that when the decision-makers are far away they – perhaps due to poor information – make decisions that seems silly at a local level. For instance, a “Central” might send “wrong” clients to a treatment institution because they are not fully aware of local conditions. Sending somebody who is allergic to animals to a institution located on a farm or something like that. Hence, the “quality” of the decisions might decrease in average because of the information problems a Central faces. This is a bad effect of centralization. On the good side, one might argue that centralization allows for specialization which, in turn, means that fewer people are required to do the job. These two effects together would mean that centralization produces worse but cheaper decisions.

To be slightly more rigorous, say that there are n decision to be made (e.g. every year n people have to be allocated to a treatment). The cost of taking a decision (c) is simply a function of the number of workers (l) you have to use; and – due to specialization – as centralization (s; 0<s<1) increases the number of workers needed decreases. However, centralization also worsens the information on which the decision is made (since the “Central” knows less about local conditions compared to a local decision-maker). Because of this, the quality (q) of the decision is reduced (in average). The overall net benefit of the decisions (O) is:

(1) O = n (q-c)

Just for the sake of experimenting, I now imagine some explicit functions that relate quality and cost to the degree of centralization in order to explore the consequences of centralization. Assume, for instance, that the mechanism that reduces the quality of information and hence the quality of the decision is as follows:

(2) q = (1-s)²

Moreover, as centralization increases, costs goes down because the need for labour goes down. This produces, say, the following relationship:

(3) c = 1-s

Putting all this back into (1) we have:

(4) O = n [(1-s)² – (1-s)]

You want to decide upon the appropriate degree of centralization, which can easily be found (in this case) by differentiating O with respect to s and setting equal to zero.

O’ = n (-2s + s²)

n (-2s + s²) = 0

s* = ½

Hence, in my highly stylised example the optimal level of centralization is 0.5.

Conclusion
Can we learn anything from the example? I am, of course, aware of many potential problems that I have ignored. For instance, I have ignored many other possible effects of centralization, focusing on only two possible effects. Moreover, I have not justified the functional forms I have chosen. Why, for instance, does centralization decrease the quality of decisions non-linearly while it decreases costs linearly? I have not even defined my terms too closely (What exactly does centralization means; Is it not more a discrete choice in which we choose between different levels than different degrees? And, how do you measure quality and costs in the same units? )

These are all issues I will discuss in another paper, but even at this stage I believe I have learned something. When faced with the question of “optimal centralization” I will immediately start to think about which micro-mechanisms that are relevant and what kind of functional form they will have in aggregate (linear or non-linear). This may not be much, but it a slightly more specific than just saying that centralization has some good and some bad consequences.

What is the optimal degree of centralization? - And how should the question be discussed?

What is the optimal degree of centralization?
- And how should the question be discussed?