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DCLG Problem – Business rates and pooling
DCLG Problem – Business rates and pooling Introduction
Business rates and pooling Cameron Hall, Ian Hewitt, Mark Holland, - - PowerPoint PPT Presentation
Business rates and pooling Cameron Hall, Ian Hewitt, Mark Holland, - - PowerPoint PPT Presentation
DCLG Problem Business rates and pooling Business rates and pooling Cameron Hall, Ian Hewitt, Mark Holland, Owen Jones, Zoe Lawson, Neeraj Oak, Eddie Wilson ESGI 91, Bristol 19 April, 2013 DCLG Problem Business rates and pooling
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DCLG Problem – Business rates and pooling Introduction
The business rates retention scheme
Terminology
The new business rates retention scheme calculates the funding received by each LA on the basis of three inputs: f : the baseline funding level (i.e. what the LA is expected to need), r: the business rates baseline (i.e. the business rates the LA is expected to collect), and x: the business rates income (i.e. the business rates collected and, potentially, available for the LA). The variables f and r are known from the outset (and have been set for the coming years), but x is unknown and must be forecast from historic data.
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DCLG Problem – Business rates and pooling Introduction
The business rates retention scheme
Tariff rate
Another important variable is the tariff rate, v, which is a monotone decreasing function of f
r :
v f r
- =
1 2,
0 < f
r ≤ 1 2,
1 − f
r , 1 2 ≤ f r ≤ 1,
0,
f r ≥ 1.
If an LA has a high funding requirement (f ) but a low expected income (r), then it receives a favourably low tariff rate (v).
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DCLG Problem – Business rates and pooling Introduction
The business rates retention scheme
Payoff function
The payoff function, P, describes how much money the LA receives from the business rates retention scheme: P(f , r, x) = (1 − s) f ,
x−r f
≤ −s, f + x − r, −s ≤ x−r
f
≤ 0, f +
- 1 − v
f
r
- (x − r),
x−r f
≥ 0, where s = 0.075 is the safety net threshold. P is an increasing function of x (the funds collected), but this is discounted by the tariff rate if x > r.
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DCLG Problem – Business rates and pooling Introduction
Pooling
LAs are being encouraged to form geographically contiguous pools in the hope of increasing the funds that they receive and reducing risk. In a pool, the f , x, and r values for the LAs are summed before the expressions for v and P are applied. For a pair of LAs (or, equivalently, for a pair of pools that might consider amalgamating into a larger pool), pool formation will theoretically be favourable if P(fA + fB, rA + rB, xA + xB) > P(fA, rA, xA) + P(fB, rB, xB).
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DCLG Problem – Business rates and pooling Introduction
Questions
Given the known values for f and r, and projecting from past data
- n x, where is it favourable to pool?
How big will ‘optimal’ pools be? Are there simple tools that can enable LAs to determine whether pool formation is favourable in their case? How might changing the details of the tariff rate affect optimal pool formation?
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DCLG Problem – Business rates and pooling Introduction
Our approaches
Analysis of the two LA/pool case We used analytic methods to explore the simple problem of determining when it’s favourable for two LAs (or, equivalently, two pools) to combine. Data processing We converted the available data into easily-usable forms, and we explored methods for predicting future values of x from past data. Simulations and pooling algorithms We used r, f and generated x data to develop possible pooling strategies across the entire country.
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DCLG Problem – Business rates and pooling Introduction
An interesting point
If no LA is going to need the safety net, the maximum amount received from central government can be achieved (albeit non-uniquely) by minimising the total tariff. Moreover, we find that
- f >
- r,
and hence the tariff for a pool made of every LA is zero! Hence, from a purely mathematical point of view, maximum
- verall return can be achieved if every LA combines to form one
big pool. (!)
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DCLG Problem – Business rates and pooling When should two individuals pool?
Two LAs considering a pool
Imagine two LAs (A and B) considering whether they’d receive a greater income if pooled together or if apart: We already know f and r (and hence v) for both LAs. How does the change in return associated with pooling depend on x? Because P is piecewise linear with its domains of definition determined by x−r
f , we introduce q = x−r f
as a dimensionless ‘x’ variable, and plod through all of the different cases...
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DCLG Problem – Business rates and pooling When should two individuals pool?
Seventeen different cases...
Dashed lines showing critical values of qA, qB and qAB −s −s qA = xA−rA
fA
qB = xB −rB
fB
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DCLG Problem – Business rates and pooling When should two individuals pool?
Is there an advantage to pooling?
If vA < vB, what is the sign of PAB − PA − PB? −s −s zero zero −ve −ve +ve +ve −ve qA = xA−rA
fA
qB = xB −rB
fB
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DCLG Problem – Business rates and pooling When should two individuals pool?
Is there an advantage to pooling?
If vA = vB > 0, what is the sign of PAB − PA − PB? −s −s zero zero −ve zero +ve +ve −ve qA = xA−rA
fA
qB = xB −rB
fB
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DCLG Problem – Business rates and pooling When should two individuals pool?
Is there an advantage to pooling?
If vA = vB = 0, what is the sign of PAB − PA − PB? −s −s zero −ve zero −ve qA = xA−rA
fA
qB = xB −rB
fB
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DCLG Problem – Business rates and pooling When should two individuals pool?
Some observations from the two individual analysis
Pooling can only be advantageous if at least one of the potential participants expects x > r (i.e. business rates income exceeds baseline). Pooling has no advantages if both of the potential participants have a zero levy rate.
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DCLG Problem – Business rates and pooling When should two individuals pool?
Some observations from the two individual analysis
Pooling can only be advantageous if at least one of the potential participants expects x > r (i.e. business rates income exceeds baseline). Pooling has no advantages if both of the potential participants have a zero levy rate. If one participant has income over the baseline and the other participant has income between the safety net and the baseline, and v = 0 for at least one participant, pooling is always advantageous if the business rates income for the pool is less than the baseline. (!)
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DCLG Problem – Business rates and pooling Data processing
Challenges in data preparation
The heterogeneity of English local government
The big challenge with data processing in this project was to bring everything into a consistent form. We needed to:
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DCLG Problem – Business rates and pooling Data processing
Challenges in data preparation
The heterogeneity of English local government
The big challenge with data processing in this project was to bring everything into a consistent form. We needed to: Build tables for ri, fi and (for the billing authorities only) the historic x values.
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DCLG Problem – Business rates and pooling Data processing
Challenges in data preparation
The heterogeneity of English local government
The big challenge with data processing in this project was to bring everything into a consistent form. We needed to: Build tables for ri, fi and (for the billing authorities only) the historic x values. Deal with inconsistent spelling in the data: e.g. & or ‘and’; the apostrophe (or lack thereof) in ‘King’s Lynn’; etc.
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DCLG Problem – Business rates and pooling Data processing
Challenges in data preparation
The heterogeneity of English local government
The big challenge with data processing in this project was to bring everything into a consistent form. We needed to: Build tables for ri, fi and (for the billing authorities only) the historic x values. Deal with inconsistent spelling in the data: e.g. & or ‘and’; the apostrophe (or lack thereof) in ‘King’s Lynn’; etc. Understand and implement the funding differences between unitary authorities, shire counties, shire districts, London boroughs, fire authorities etc.:
xi for billing authorities are the business rates collected multiplied by a conversion factor depending on type, xi for non-billing authorities (e.g. counties) are the sum of the income of the constituent billing authorities multiplied by (different) factors.
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DCLG Problem – Business rates and pooling Data processing
Challenges in data preparation
Which LAs are allowed to pool?
In addition to cleaning up the financial data, it’s essential to know which LAs are allowed to pool together. To join a pool, LAs can be logically adjacent (e.g. Stroud and Gloucestershire), or geographically adjacent (e.g. Bristol and North Somerset). Working out logical adjacency required yet more data preparation... Moreover, no geographical adjacency matrix was easily available, so we constructed our own on the basis of publicly available geographical data (postcode centroids).
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DCLG Problem – Business rates and pooling Data processing
Constructing an adjacency matrix
Figure: The postcodes and local authorities of England (and more...)
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DCLG Problem – Business rates and pooling Data processing
Constructing an adjacency matrix
2 3 4 5 6 7 x 10
5
1 2 3 4 5 6 x 10
5
Figure: The postcodes and local authorities of England (and more...)
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DCLG Problem – Business rates and pooling Data processing
Approaches to modelling xt
Since x can be expected to increase exponentially with inflation, an appropriate stochastic model for x might be xt = A exp{(µ − 0.5 σ2) t + σ Wt}, where t is time, Wt is a Wiener process, and A, µ and σ are all constants that need to be determined. However, there are only nine historic data points (2003/04 to 2011/12) and the data is (sometimes) incredibly noisy. As a result, linear regression seems to provide satisfactory fits (and is easy to implement).
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DCLG Problem – Business rates and pooling Data processing
Some contrasting historic x data
2003 2004 2005 2006 2007 2008 2009 2010 2011 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 x 10
7
Uttlesford
2003 2004 2005 2006 2007 2008 2009 2010 2011 4 4.5 5 5.5 6 6.5 7 7.5 x 10
8
City of London non police
Figure: The contrasting business rates income trends of Uttlesford and the City of London.
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DCLG Problem – Business rates and pooling Pooling simulations
Games for pooling
Our approach to pooling was to consider the growth of pools from pairwise interactions. Consider two adjacent ‘pools’ and apply some decision process to determine whether they would do better together than apart. While easier to implement than an approach that allows pools to rearrange, this means that different runs lead to different results (path-dependency), we are not guaranteed to achieve a globally optimum solution.
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DCLG Problem – Business rates and pooling Pooling simulations
An example algorithm
We performed simulations of pooling using the following heuristic algorithm:
1 Initialise each ‘pool’ to contain a single LA. 2 Pick a ‘pool’ at random, and find all of its neighbouring
‘pools’.
3 Choose the neighbour that leads to the largest expected value
- f PAB − PA − PB.
4 If this is above a certain (nonnegative) threshold, θ, then
combine the two pools.
5 Return to Step 2, and repeat until convergence is reached.
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DCLG Problem – Business rates and pooling Pooling simulations
A caveat
Expected values versus risk aversion?
These calculations are all based on expected values of PAB − PA − PB, but these are all drawn from distributions.
−4 −2 2 4 6 8 x 10
−3
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Figure: Distribution of PAB−PA−PB
PA+PB
for Croydon to join with the GLA.
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DCLG Problem – Business rates and pooling Pooling simulations
Development of pools with θ = 0
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DCLG Problem – Business rates and pooling Pooling simulations
Development of pools with θ = 0.001
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DCLG Problem – Business rates and pooling Pooling simulations
Development of pools with θ = 0.002
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DCLG Problem – Business rates and pooling Conclusions
Conclusions
For fixed f and r, there are (relatively) simple conditions on x that indicate when two LAs would benefit from combining.
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DCLG Problem – Business rates and pooling Conclusions
Conclusions
For fixed f and r, there are (relatively) simple conditions on x that indicate when two LAs would benefit from combining. LAs should have simple, universally-used alphanumeric labels.
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DCLG Problem – Business rates and pooling Conclusions
Conclusions
For fixed f and r, there are (relatively) simple conditions on x that indicate when two LAs would benefit from combining. LAs should have simple, universally-used alphanumeric labels. A range of different models for business rates income (x) are possible. What if the new scheme leads to behavioural changes?
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DCLG Problem – Business rates and pooling Conclusions
Conclusions
For fixed f and r, there are (relatively) simple conditions on x that indicate when two LAs would benefit from combining. LAs should have simple, universally-used alphanumeric labels. A range of different models for business rates income (x) are possible. What if the new scheme leads to behavioural changes? Pairwise algorithms can be used as a heuristic to obtain ‘good’ pools. Without a threshold on PAB−PA−PB
PA+PB
, our pairwise algorithm produces very large pools!
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DCLG Problem – Business rates and pooling Conclusions
Other problems to consider
Games for divvying up P in a pool: What bargaining power does a top-up authority (where v = 0) have when dealing with tariff authorities (where v > 0)?
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DCLG Problem – Business rates and pooling Conclusions
Other problems to consider
Games for divvying up P in a pool: What bargaining power does a top-up authority (where v = 0) have when dealing with tariff authorities (where v > 0)? Risk reduction: Everything that we’ve done has concentrated on expected values. How does the distribution of P change when LAs pool?
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DCLG Problem – Business rates and pooling Conclusions