Money Should Follow Mission, Not Just Functions

Who should get how much ?

One Man One Shilling. That has been, for a long time, a phrase commonplace in the minds of many Kenyans and especially in the senate. It alludes that money should follow population; loosely put, one man one shilling would ignore the fact that serving 10 people isn’t ten times as expensive as serving one person. This doesn’t have to a be a binomial problem where it is either land or population because both of them matter and should be well taken into consideration without hurting the other.

When designing a criteria for revenue distribution, the first important step is identifying the correct parameters and that’s where the CRA formulas (old and proposed new) have failed. CRA formula fails because of what I will call overgeneralization and composite parameterization. Overgeneralization means that lots of parameters are grouped together into one making it not only hard to measure but also lose its significance in the formula. Composite parameterization is where a parameter appears more than once in the formula, usually hidden as a sub-parameter of the other more general parameters. Here is an example: Health Index = f(population, area,…) — Health index is a function of both population and area among others. If you look at number of hospitals, as an example, the aim is to either achieve a certain number of hospitals per square kilometer or number of hospitals per thousand people. The same would apply to the number of doctors, nurses et al. Agriculture = f(population, area,….) — Agriculture is also a function of both population and area besides others like suitability of the land for the specific agricultural practice. The same can be said of roads, and urban services. In this case, population and area are entrenched within all the other parameters and having them again as independent parameters corrupts the entire formula leading to unfair imbalances. The first task would then be to remove both the land area and population parameters.

Look at the poverty parameter, as an example. This is not only a composite parameter but also an extremely generalized parameter. What are the factors that influence poverty levels ? The list is long but just to mention a few: education, road networks, agriculture, electricity, climatic conditions, health, urbanity among many others. Poverty in itself, is already covered by nearly all the other parameters combined; why we would have it as a separate parameter beats logic.

In machine learning, one of the most important steps is usually to get the correct parameters otherwise the predictions will be misleading. One of the methods of parameter tuning is correlation analysis and in which case, when you get a set of parameters with high correlation score, the general rule of thumb is to pick only one of them. Principal Component Analysis (PCA), for instance, allows one to only get the principal components and ignore the rest. Where generalized parameters exist, the general rule of thumb is to break them down and I see no other place where that practice is more applicable than in the parameterization of the revenue sharing formula. In this case, we would eliminate population, land area and break down poverty, health et al into granular sub-components.

The parameterization of the revenue sharing model should follow county functions which are well listed in our Constitution:

  • Agriculture,
  • County health services,
  • Control of air pollution, noise pollution, other public nuisances and outdoor advertising,
  • Cultural activities, public entertainment and public amenities,
  • County transport,
  • Animal control and welfare,
  • Trade development and regulations,
  • County planning and development,
  • Pre-primary education, village polytechnics, home craft centres and childcare facilities,
  • Implementation of specific national government policies on natural resources and environmental conservation,
  • County public works and services, Fire station services and disaster management,
  • Control of drugs and pornography and
  • Ensuring and coordinating the participation of communities and locations in governance at the local level and assisting communities and locations to develop the administrative capacity for the effective exercise of the functions and powers and participation in governance at the local level.

At the very least, we should have those 13 parameters to reduce composite parameterization. However, we will still succumb to the problem of generalized parameters and so we need to go more granular. Look at Agriculture, as an example, where the specific functions of the county government can be listed: Crop husbandry, Animal husbandry, Livestock sale yards, County abattoirs, Plant disease control, animal disease control, Fisheries. Now instead of using agriculture as a parameter, we should replace it with these 7 parameters. Instead of using health, we could pick the specific functions under health: County health facilities and pharmacies, Ambulance services, Promotion of primary health care, Licensing and control of undertakings that sell food to the public, Veterinary services (excluding regulation of the profession), Cemeteries, funeral parlours and crematoria, Refuse removal, refuse dumps and solid waste disposal.

The parameters of the revenue sharing model should follow county functions and be as granular as possible to avoid generalization and composite parameterization that create imbalances and inequity

The problem with our parameterization is that it looks at the results and not the causes and that needs to be reversed. Instead of looking at poverty levels, look at the status of the factors contributing to poverty, instead of looking at how healthy people are, look at the quality of services contributing to the health quality. To solve a problem, we must look at the contributing factors and fix them at the root and money should be allocated to target these root causes.

When we go granular, we get higher accuracy. As an example, when we look at the specific parameter of control of drugs, we know that Mombasa should get the biggest chunk because it is where the pandemic is most prevalent. Similarly, when we look at control of HIV, then my home county Homa Bay should get the highest chunk as it is where the disease is most prevalent. But when we use a general parameter like “health index”, we lose this specificity as important things are covered/masked by other trivial issues on the larger scale. Another advantage of going granular is that we make it easy to measure the parameters. It’s extremely hard to measure “health index” but is is very easy and uncontroversial to measure “opioid pandemic” or “HIV prevalence” in a region.

The problem with our parameterization is that it looks at the results and not the causes

We have removed the polarizing parameters like population and land area and also broken down the list of parameters to their most granular form. We have also said that parameterization should follow functions of the county governments but how exactly should we weight these parameters?

The parametric revenue sharing ratio should follow mission not functions. Mission in this case defines where we want to be in the next ‘x’ years. One of the things our constitution mandates the CRA to do is that it should ensure equitable revenue sharing so as to resolve historical marginalizations. We can look at these parameters based on a few set of features:

  • Initial Condition (the current value of the parameter)
  • Limit/target (Where we want the parameter to be in the next ‘t’ years)
  • Capacity (The maximum value the parameter can ever achieve while maximizing returns and minimizing expenditure)
  • Growth Rate (The rate at which the parameter needs to increase for it to move from the initial condition to the limit within the set ‘t’ years)

These set of 4 features can be used to ensure equity and that no county is “losing” — I’m not talking about losing money as compared to what they used to get before but it means not losing in as far as functional requirements and economic growth are concerned. Let me take the simplest example I can think of. The question is: how many hospitals(just a general word, could be dispenaries et al) should each county have ? I will look at a situation of 4 hypothetical counties making up the Republic of Utopia.

Here, we are saying that number of hospitals is a function of both area and population ie H = f(p,a)

As can be seen in the figure, using either area or population indepdently produces different results. With land as the only parameter, county C will get the second highest share but with population, it gets the third highest share. So let’s not use either of them indepdently. As a matter of fact, in a situation like this, the general ideal formula is to have the minimum limited by area parameter and the maximum limited by the population parameter. However, we are not looking into that in this article so let’s look at what we do next with the mission-oriented formula

Achieving equity means that counties that are historically marginalized should get more money so that they can build more hospitals but in so doing, we should not “punish” those that are already ahead. Again, for simplicity of explanation, we will be using only the area parameter in the figure above. The best county has 0.67 hospitals per square kilometer while the worst has only 0.1 hospitals per square kilometer. How do we bridge this gap ? My approach suggests setting the optimum target that we want to achieve ie the mission. In this case, we say that the target is 0.7 and is the same for all counties as shown in the figure. In a practical situation, however, the target is limited by the capacity (earlier explained) and as such, some counties may have a higher target than others based on the capacity of the parameter, wherever applicable. With the target, we can get the deficit ie how much more the county needs to achieve in order for them to achieve the target but this isn’t an important value as we won’t be using it.

Now we have values for each of the factors earlier mentioned: initial condition is the current value of hospitals/area in each county, the limit is the target (capped by capacity) and our aim is to get the growth rate. A simple approach would be to look at the growth rate as the gradient of the straight line joining the initial value and the target

Straight line growth. Gradient needed to achieve this is 0.05

However, straight lines rarely exist in real world. In most cases, the growth rate is lower at the beginning and increase with time which gives us kind of an exponential curve. Again, exponential curves do not exist in the practical world due to the carrying capacity which eventually slows down the rate of growth (like the spread of a rumor or corona virus) — this is a scenario best described by a logistic function

Logistic curve is limited by the capacity

Now we know how our formula should look like; it should be a logistic function with the parameters as shown below:

The logistic formula applied as a single-parameter revenue sharing formula

This formula describes how a parameter can be moved from its initial value to the target value. The manner of growth is controlled by the logistic growth rate which is akin to the revenue share the county should get to achieve this growth. With this approach, we use the growth rate as the basis for revenue share but within it we have equity taken into consideration as well as ensuring we do not “punish” any county.

So how do we get required growth rate? To achieve this, I considered it an optimization problem. My initial approach was to use Bayes Inference — specifically Markov Chain Monte Carlo methods to estimate the parameter values but I am not a huge fan of implementing MCMC so I decided to use Limited-Memory BFGS minimization algorithm (any other type should just work fine). There are infinite logistic growth curves from a given minimum (y-intercept) and maximum limit (the asymptote) and so the task is to get the best curve ie the curve that gives you closest to the target at the end of the target period. For our sample counties shown earlier, the below curves were achieved:

Curve of Best Fit: Ensures that each county gets as close as possible to the target within the set time period

The curves show that all the counties will move towards achieving the optimal target of how many hospitals per unit area they should have over a period of time. As can be seen, no county is stagnating and all of them are growing and at the same time, we ensure that the historically underserved counties like county B also get an equitable share; in the end, we have less disparities as compared to when we started. The attained growth rates for each of the counties are: A(0.41), B(1), C(0.72) and D(0.95). As can be noticed for county A, the linear gradient was 0.05 but the logistic growth rate is 0.41. We can then get their equitable share as shown below:

Equitable share for each county.

This example is done for one very simplistic parameter (number of hospitals per square kilometer) and would be done for each of the hundreds of parameters that could be found. With automated processes using statistical tools like the one I have used here, the process is pretty fast and takes just a few seconds to complete.

The parametric revenue sharing ratio should follow mission not functions. We first need to establish where we want to be in the future and then determine how we should get there.

The aim is to ensure equity while not punishing anyone. Thus the task of the CRA is to ensure that each county is, first and foremost, able to run its current operations eg pay salaries and then all the other factors eg reducing poverty levels, improving health, ensuring better rural access index, promoting agriculture, mining, tourism et al are done based on a formula such as this. As an example, when we look at tourism as a parameter, a county that has zero tourism sites and zero potential for the same would have its capacity set to zero meaning it receives no money for tourism. So the limiting capacity is an important factor in this formula. We need to know the potential of every county and the possible returns on investment when determining the target and capacity so that we don’t pump a lot of money for maize farming (as an example) to a county that has no capacity for the same.

The overral revenue share would thus be:

Revenue Sharing Formula

The formula defines a minimum share for the county (every county must be able to run without disruption of services) and then the amount corresponding to each of the other parameters. The decision on the value of ‘x’ which defines how much each parameter gets is an economic question (in which I am no expert) but the ration ‘Ri’ is achieved as described in the previous formula.

Money should follow functions and mission in the manner described. Functions help us to get the parameters to use for sharing and mission helps us get the share ratio for each of those parameters. Should money follow population and land area? Yes but not as direct parameters; as we have already seen, population and land area inherently control nearly all of the other functional parameters and including them as independent parameters only serve to skew the share ratio and create a lot more imbalance and future historical injustices. Obviously money alone cannot help our counties, the policies of how to spend the money and fighting graft among many others will be the major differentiating factors but those are for another day.

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