Devolution and grant-in-aid design for the provision of impure public goods

In the previous section we showed that centralised provision does not allow to reach
the FB allocation; here we consider the alternative solution of devolving production
to lower tiers that know local preferences and can provide a good that fits best user
needs. In this section we consider and compare two possible alternatives:

“pure devolution” where LGs are solely responsible for the provision of the impure
public good (the case will be indicated by the letter F);

devolution where LGs provide the good, but CG influences their decisions using a
matching grant.⁹

Pure devolution

If the goods produced at local level were local public goods, devolution would always
allow to reach the welfare level of FB (Oates 1972]). However, the presence of spillovers means that also devolution is a second best
option. In this case each Local Government (LG) maximises its utility function, but
it does not take into account the accrued utility experienced by users in other local
authorities. As for centralisation, each LG has to find the optimal user charge that maximises the aggregate utility of jurisdiction j. The subsidy will be financed using a (local) linear income tax at rate ; the budget constraint in this case is

LG has to find the value of that maximises the following objective function:

Table 3 shows the results derived in Appendix 2 in terms of the optimal subsidy , quantities , total quantity and total welfare .

Table 3. Results for devolution with no matching grant

Comparing Table 1 with Table 3 it is straightforward to see that the two results coincide for , i.e. when there are no spillovers. In all the other cases, the subsidy is too low,
total quantity falls short of the optimal level, and the welfare level attained is
lower than in FB.

This is the first result of our model: the findings of the traditional literature
on fiscal federalism are valid also for the provision of impure public goods, provided
that there are no spillovers among regions. In all the other cases, “pure” devolution
produces a welfare loss which is equal to .

Devolution with a matching grant

CG may try to influence the choice of the local subsidy using a matching grant.¹⁰
This solution has some drawbacks: the asymmetry of information that prevents Central
Government from providing the optimal quantity of y may also influence the optimal matching grant setting decision, for two main reasons:

coordination problems, fiscal illusion and spillovers: due to the specific characteristics
of y, any change in affects the level of utility in authority j and its decision on . The matching grant introduces another interdependence in decisions, because the
level of local expenditure has an impact on the national tax rate, hence on the welfare
of each jurisdiction. If local decision makers misperceive the effects of their actions,
CG may be unable to attain FB, even when it can observe the reaction function of LGs;

asymmetry of information: CG cannot observe local preferences parameters and the
reaction function of each local government.

The environment is characterised by the following assumptions: y is subsidised by LG, which does not take into account the spillovers created by its
production. CG influences the behaviour of LGs using a matching grant.

The timing of the game is as follows: (a) CG sets the grant to maximise total welfare
using its beliefs on LGs’ behaviour and users’ preferences; (b) LGs set their reaction
function and their local tax rate.

Although CG cannot observe some relevant parameters, LGs are followers, i.e. we rule
out the possibility that they may act strategically in setting their reaction function.¹¹
The problem can be solved through backward induction: in the first stage LG decisions
and reactions to a grant setting are considered; in the second stage CG finds the
optimal grant, given its information set.

The analytical model is presented in Appendix 3. In what follows we discuss the intuition
behind the findings.

LG reaction function

LG receives a matching grant at rate from CG. If it thinks that the good should be further subsidised, it will introduce
a supplementary subsidy at rate , to be financed by a proportional tax on local income at rate . The user charge for the service will be equal to . LG decision has a twofold impact on total welfare: on the revenue side it will change
the national tax rate; on the expenditure side it will alter the quantity of the impure
public good. This is one of the novel elements of our model: given the nature of y, each LG has to foresee the behaviour of the other local authority and should take
into account the impact of increasing its expenditure. These effects may not be correctly
perceived by LGs. In our model we consider three alternative behaviours for the LGs.

(FC) Each LG thinks that the other local authority will replicate the same strategy
(full coordination—FC), i.e. it will subsidise local production by the same amount.
On the revenue side the overall change in the national tax rate is taken into account
and on the expenditure side for each j the quantity is updated accordingly.

(PC) Each LG thinks that the other local government does not further subsidise the
good (partial coordination—PC). In this case the quantity will not change, while a one-sided correction of the national tax rate is considered.

(FR) Each LG thinks that the effects of its expenditure on the tax rate are marginal,
so that its decision influences neither the rate, nor Q (free rider, FR). The latter hypothesis may not be reasonable in a model with two
local authorities, but in a more general context where the number of jurisdictions
is fairly large this behaviour may be quite plausible. It is interesting to note that
this is the hypothesis that has been used by the traditional literature in defining
grant in the presence of spillovers.

The detailed derivation of the formulas for the local subsidy is presented in Appendix 3 and reported in Table 4. For each LG behaviour two values are reported, because a second source of asymmetry
of information has to be taken into account. The actual subsidy set at local level
also depends on the local preferences , which cannot be observed; as in section “Centralisation” we assume that an estimate
z, equal for both local authorities is used by CG to guess the subsidy level that will
be set by LGs.

Table 4. Local subsidy in case of devolution with a matching grant

From Table 4 we note that the subsidy set by a LG behaving as a FR is higher than that in the
PC case, as one might expect. A “free rider” behaviour implies that the LG does not
take into account the increase in the national tax rate that is action is causing,
i.e. LG underestimate the tax price for good y and will be prepared to subsidise it at a higher rate. A more general conclusion
cannot be drawn: the relative effects of spillovers and national matching grant will
determine the result. The use of a matching grant may allow CG to improve on pure
devolution only if the grant is set correctly, but CG can observe only a subset of
the parameters. For this reason, it will have to devise a strategy to minimise the
negative effects due to the lack of information.

Grant setting

In the second stage CG has to set the matching grant, based on his beliefs about the
behaviour of the LGs and the estimation of the unobservable preferences. We assume
that CG expects LGs to have the same behaviour (i.e. both are either FC, PC or FR),
thus the matching grant will be equal for the two regions (i.e. ). The estimated reactions of the LGs (the subsidy in the last column in Table 4) are then used to determine the welfare function for B=FC,PC,FR using (3). CG has to decide which of the LG reaction is more plausible and assigns a probability
to each of the three possible reactions; the matching grant will be then found by maximising the expected welfare . The analytic derivation of the optimal grant in the general case is presented in
Appendix 3. The solution depends on the probabilities ; in Table 5 the optimal subsidies for the following relevant cases are shown: CG believes that
LGs will behave as either FC, PC or FR (i.e. for each possible value of B) and the case where CG assigns equal probability to
each of the behaviours, i.e. for all B (the abbreviation “Equi” is used for this case). Note that if LGs act as
FC, CG cannot influence the expected welfare and no matching grant will be used. For
PC the grant is twice the size than for FR, as one might expect. Finally, if all the
reactions functions are taken in consideration with equal weight (Equi) the optimal
grant is slightly higher than for FR, but quite close. The traditional literature
and most actual grant formulae use FR assumption to model the behaviour of LGs. FR
behaviour has a boosting effect on expenditure; by assuming the worst scenario in
terms of effects on expenditure, CG tries to reduce the negative impacts on its expenditure
of LG choices.

Table 5. Matching grant

Ex-post welfare analysis

Once the grant has been set by CG, the LGs will further subsidise the good following
the scheme presented in the second column in Table 4. In general, for each case of the grant setting, three different reactions of the
LGs are possible. The state contingent solution is presented in Table 8 in Appendix 3. In what follows we examine the results from a more qualitative point
of view; the discussion will be supported by the graphical visualisation of the main
findings.

Let us start by examining the case where CG assumes that LGs reaction is of the “fully
coordinated” type (FC). It turns out that the difference of the user charge with respect
to the FB case does not depend on CG grant, which will therefore be set to zero. If
the action of the LGs is either PC or FR, the outcome of the “pure” devolution case
is replicated. The difference in quantities with respect to the FB case is:

and the welfare difference is:

Note also that, if the reaction of LGs is FC, the outcome does not depend upon the
action of CG: the total quantity is equal to the one in FB, but if the preferences
in the two regions differ, it is not correctly distributed among them. In fact the
difference in each region is and this causes a welfare loss with respect to the FB case equal to .

If instead LGs reaction is either PC or FR, CG can reduce the difference in the subsidy
and total quantity using a matching grant, by an amount that depends on z. If CG beliefs are fulfilled, the result is the same in both cases (PC and FR) and,
if the total quantity produced is optimal, while the welfare loss is half the one obtained
when LGs react as FC. Again, the total quantity is optimal, but its distribution across
local authorities is different from FB and the welfare level is lower.

Fig. 1. Comparison of the mean welfare loss produced by the matching grant for various levels
of z and under the different assumptions on the behaviour of LGs

When the reaction of the local authorities is uncertain, the analysis has to be carried
out by comparing the mean value of the welfare functions. In all cases the term can be factored out, and the comparison only depends on z, i.e. on the quality of the information that CG has on the preferences of the two
regions. The analytical comparison can be made by standard algebraic calculations;
Fig. 1 illustrates the results by showing the relative position of the average welfare losses
(wrt FB) under the different assumptions of CG about the reaction function of LGs.
With the exception of the FC case, welfare losses are convex in z. The assumption that LG reacts as PC minimises the welfare loss only if CG considerably
underestimates the average of local preferences . Let us call and the ex-post average welfare differences (the last column in Table 8) under the two assumptions “FR” and “Equi” of CG on the behaviour of LG. If z ranges in then is the lowest and has a minimum for . The welfare loss that can be expected using FR performs better for higher values
of z and has its minimum value for .

increases at a higher rate than as the distance of z from increases because the grant under FR is lower than with the other assumptions. At
the left of the grant may be too low to make local authorities react optimally and local preferences
are underestimated. At the right of the two effects may offset each other. The two welfare losses are equal for ; this implies that for values of z lower than or very close to the mean, using a grant that minimises the expected welfare
loss is preferred to assuming that LGs are free riders. In setting the matching grant,
most traditional literature on fiscal federalism implicitly or explicitly¹²
assumes that CG may observe local preference on average and sets the grant as if local
authorities were free riders. If CG can observe the mean of true preferences (i.e.
), it should use the grant that minimises the welfare loss, but the mistake made using
FR is small.

The comparison with the “pure devolution” case is less clearcut: the welfare loss
does not depend on z and in a graphical comparison similar to the one in Fig. 1 it is represented as a horizontal line. Its relative position in the picture depends
on the ratio of the preferences in the two regions. If the ratio is not too high,
and obviously any solution with a matching grant is preferable. As the ratio gets
larger, “pure devolution” could anyhow be a viable option only in extreme cases, where
either z greatly underestimates , or if the magnitude of the two parameters and is extremely different.¹³
Thus, ruling out unrepresentative cases, we can then conclude that a matching grant
generally improves welfare, i.e. in the presence of spillovers it is optimal for CG
to induce LGs to change their expenditure patterns using a matching grant.