Insuring against Public Sector Cost Shocks :Transfers or Uniformity?

Sam BucovetskyEconomics, LAPS, York University

October 31, 2012

1 the problem

Risk averse jurisdictions are subject to uncertainty about the cost of thelocal public output. If they are sufficiently risk averse, they would like toinsure against these shocks, by having transfers from low–cost jurisdictionsto high–cost jurisdictions.

But if the realizations of the costs are private information, then a (verystandard) selection problem arises. Getting jurisdictions to reveal cost infor-mation requires distortion of their public output decisions.

On the other hand, a central government could eliminate all uncertainty1byrequiring jurisdictions to contract with a private firm to provide a given quan-tity of the local public good, before anyone observes any cost realizations.Competition among prospective providers ensures that these providers willprovide the local good at its expected cost. But now there’s uniformity ofoutput levels : no jurisdiction gets its most–preferred level of public output,given the cost parameter realization it has observed.

So which method provides higher expected utility?

1if the number of jurisdictions is large, and the shocks are independent, so that thereis no aggregate risk

1

2 the model

There is a large number of regions. Each region has (certain) income y,which can be spent on private good consumption x or on local public outputprovision g. Preferences are represented by a utility function u(x, g) whichis strictly concave. (Risk aversion is the whole issue here.)

The unit cost of the public output can be either cH or cL, with cH > cL.The probability of a high cost realization is π. Initially, it will be assumedthat all regions are identical ex ante : each will have a high–cost public sectorwith probability π, and regions’ draws are independent.

The number of regions is large enough so that there is no aggregate un-certainty.

The income elasticity of demand for the local public good is non–negative,so that its uncompensated demand curve slopes down.

3 which way to transfer?

With income uncertainty, concavity of the utility function ensures that peo-ple want to insure (at fair odds) against the “bad” state, in which income islow.

With price uncertainty, there are two offsetting effects. Concavity of theutility function implies that the marginal utility of income is higher whenutility is low, holding prices constant. But one additional euro can buy moregoods when prices are low (the indirect utility function is convex in prices).

Let the private good be the numeraire. Suppose that a jurisdiction wereto choose the optimal level of public good provision, given the realizationof the unit cost : the Marshallian demand for the public good. This levelof public output provision will be denoted G(y, c) and is the level of publicoutput g for which ug

ux(y − cg, g) = c.

Roy’s identity implies that the marginal utility of income, which here isa function of the person’s income y and the cost of the public good c, λ(y, c)will be an increasing function of c under the following condition :

Lemma 1 ∂λ∂c> 0 if and only if |ελy| > εGy, where εab is the elasticity of a

with respect to b.

The elasticity |ελy| measures the extent of aversion to income uncertainty.In fact, it is exactly the Arrow–Pratt measure of relative risk aversion.

2

In all the examples and analysis that follows, attention is restricted tothe case in which the marginal utility of income is higher in the high–coststate, so that an expected–utility–maximizing central government will wantto transfer income from jurisdictions with low costs to those with high costs.This is the case which seems best to fit actual practice, if intergovernmentaltransfers are to be explained as insurance against cost shocks.

ASSUMPTION : |ελy| > εGy

4 decentralization under asymmetric informa-

tion

The term “decentralization” here is not meant to describe a devolution ofpower to regions. Instead it refers to a centrally–run federation, in which pub-lic output is allowed to vary among regions. It is the solution to a principal–agent problem2 in which the central government is the principal, and theregions are the agents.

Each region observes its own cost realization, but no–one else observesit. The central government can observe (and verify) individual regions’ pub-lic output levels g, but it cannot observe private consumption. So it cancondition transfers on the output levels chosen by jurisdictions.

The central government wants to transfer income from low–cost to high–cost regions because of the declining marginal utility of income. (This is thesame story as Varian’s model of progressive income taxation as insuranceagainst income uncertainty.)

So its problem is to choose grants αL and αH to regions which choosepublic output levels gL and gH , so as to maximize expected utility subjectto the budget constraint, and the constraint that a region with cost ci wantsto choose local public output level gi and receive a transfer αi.

Its problem is thus to maximize

πu(y − cHgH + αH , gH) + (1− π)u(y − cLgL + αL, gL)

subject to the budget constraint

παH + (1− π)αL ≤ 0 (1)

2with no participation constraint

3

and the selection constraint

u(y − cLgL + αL, gL)− u(y − cLgH + αH , gH) ≥ 0 (2)

The first–order conditions for this maximization are

π(uHg − cHuHx )− µ(umg − cLumx ) = 0 (3)

(1− π + µ)(uLg − cLuLx ) = 0 (4)

πuHx − πν − µumx = 0 (5)

(1− π + µ)uLx − (1− π)ν = 0 (6)

where subscripts refer to partial derivatives, and the superscripts “H”,“L” and “m” refer to the high–cost types consuming (y−cHgH+αH , gH), low–cost types consuming (y−cLgL+αL, gL) and would–be mimickers consuming(y− cLgH + αH , gH) respectively. The multipliers µ and ν correspond to theselection constraint (2) and the budget constraint (1) respectively.

Equation (4) is the usual “no distortion at the top” result, that low–costregions choose an efficient local public sector, at which the marginal rate ofsubstitution equals the marginal cost of the local public output. So equa-tion (4) says that gL = G(y+αL, cL). As long as gH < gL, then equation (4)and the fact that the selection constraint binds, so that (y − cLgL + αL, gL)and (y − cLgH + αH , gH) are on the same indifference curve, imply thatumg > cLu

mx . Then the fact that µ > 0 if the selection constraint binds, and

equation (3) imply that uHg < cHuHx , so that gH < G(y+αH , cH). The usual

distortion occurs : the high–cost region provides too little of the local publicoutput, to make mimickry less attractive.

The inability to redistribute without distortion also means that insuranceis incomplete : equations (5) and (6), and the fact that µ > 0 imply thatuHx > ν > uLx , so that marginal utility of income is higher in high–costregions.

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5 uniformity

Uniform provision here means that every region consumes the same level gof the local public output, and that everyone pays the same amount cg intaxes, where c is the unit cost of uniform provision.

c ≡ πcH + (1− π)cL

Note that a central government would choose to have all regions pay anequal share of the cost, if cost shares were subject to choice, because of theconcavity of the utility function.

The information requirements for uniformity are discussed a little morein the following section (6). The decision on how much to provide is made exante, before costs are observed. And all regions are uniform ex ante. Sinceoutput levels are observable, governments can enforce their contract with aprivate provider to provide g units in every jurisdiction. A private providerknows, since there is no aggregate uncertainty, that its costs will be cg if itcommits to providing g in every region.

The optimal level of g to provide is G(y, c), and all regions agree that thisis the best level of public output to provide — given that all regions get thesame level, and that all regions pay the same share of the costs.

6 information : elaboration and explanation

Several features of the information structure here are worth noting :

• The (uninformed) central government can observe, and make transferscontingent upon, the level of public output g in a region ; the levelof expenditure cig is unobservable to the central government.

• The option of central provision implies that it is possible to providepublic output region, at a unit cost of ci, without knowing what ci is.That is, not only does the central government know that a proportionπ of regions have high costs, but it can provide public output there atthat cost — provided that it provides the same level of output in allregions.

The first assumption is similar to those in Boadway et al (1999). InCornes and Silva (2000, 2002) both the unit cost ci and the level of public

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output g in a region are observable by the central government, but this unitcost is determined by two unobservable components, an inherent cost and anendogenous effort level.

In contrast, the public output level g is not observable to the centralgovernment in Cornes and Silva (2003), and in Huber and Runkel (2006),but the total expenditure level cig is.3

Lockwood (1999) considers both cases : cig observable but not g, andvice versa.4

Bordignon et al (2001) assume that it is incomes, not unit costs of thepublic sector, which differ among regions, and which are not observable bythe central government. In Bucovetsky et al (1998) and in Dhillon et al(1999) the unobservable variable is the taste for the public output.

For many categories of government expenditure, it may seen more rea-sonable to assume that expenditure levels are more easily and accuratelyobserved than quantity levels of public output. It may be fairly easy toget data on the total school, or police, budget for a jurisdiction. But thelevel of educational attainment, or of public safety, achieved by that budgetmay depend on the characteristics of the local population. It seems naturalthat local politicians would be better–informed about these characterisitcsthan outsiders. It also seems natural that a low–cost jurisidction could hidethat fact by padding its budget. In such a setting, the notion of central-ized provision, by an uninformed central government, would have to implyuniformity of expenditure, rather than uniformity of public output. Cen-tral provision would add nothing to the options attainable by the centralgovernment through standard incentive–compatible menus.

On the other hand, student test scores and local crime rates can be ob-served, and these data are often collected by a central government. Thatis the framework used here, In particular, uniform central provision entailsguaranteeing that a given standard (an average test score, or an overall crimerate) to each region. The second point above makes the central governmentcapable of such a commitment, even if cannot observe the cost of attain-ing that standard in each region. The notion here is that any cost–paddingthat a region could do, in order to disguise its true nature as a low–cost re-

3There are two public outputs in each of these papers ; neither quantity can be observeddirectly by the central government, but aggregate expenditure on the two outputs can beobserved.

4He also allows for unobservable variation in public output costs, tastes, or regionalincome.

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gion, requires control by the local government. Central government officals,or private intermediaries with whom they contract, are assumed to be ableto prevent malfeasance within regions, even if they do not know the actualproductivity of workers there.

An alternative framework would bring similar results. A considerable pro-portion of public expenditure is devoted to provision of insurance of varioustypes. Suppose then that g is the level of some type of insurance provided ina region, and that ci is the probability of any individual there requiring thatinsurance. Insurance here is not being provided (solely) against monetaryloss. If it were, full insurance would be optimal in each region, so that thequantity of insurance provided could not serve as a signal of the probabilityof the event.

Instead, the insurance helps mitigate some catastrophic event, such as amajor illness. The utility of any resident of a region will be U(x) if she ishealthy, and U(x) + S(g) if she is sick, where x is her (net–of–tax) income,and where S(g) < 0 and S ′(g) > 0. So the government provision of g helpsmitigate the damage of illness, but cannot completely eliminate it. If eachresident of region i faces the same probability of illness ci, and if the regionalgovernment provides g to every sick person, financing this expenditure byan equal tax on all residents, then the expected utility of a resident of theregion is

U(y − cig) + ciS(g)

where y is the (exogenous, identical) income. If the functions S and U areconcave, it follows that the ideal quantity of public output, g∗(ci, y) must bea strictly decreasing function of ci. The marginal utility of income, U ′(x)must be increasing in ci when the region chooses its optimal policy g∗(ci, y).

So qualitatively, this case of public insurance with asymmetric informa-tion about the regional claim rate, is identical to the model considered in thepaper. The central government wants to redistribute to high–probability re-gions. Mimickry cannot be prevented directly if the central government can-not make transfers contingent on the claim rate to be realized subsequently.And here it is certainly possible for the central government to provide uniforminsurance directly.

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7 can decentralization be better?

The first–best solution, if cost realizations were observable, would be to pro-vide different levels of public output in low– and high–cost regions (so thatuig = ciu

ix for i = L,H), and to transfer income from low–cost to high–cost

regions, so as to equalize the marginal utility of income. In general, neitheruniformity nor decentralization under asymmetric information can achievethis optimum.

But if the two goods x and g are very good substitutes, then decentral-ization will achieve a higher level of expected utility. In the extreme, supposethat

u(x, g) = Ψ(x+ g)

for some strictly concave, increasing function Ψ(·)5, and suppose as well thatcL = 0.5, cH = 1.5 and π = 0.5. Since the goods are perfect substitutes, thebest plan is to consume none of the public good when ci is high, and noneof the private good when ci is low. The first–best solution is to set gL = 4

3y,

gH = 0, and to transfer y/3 from each low–cost region to each high–costregion, giving a utility level of Ψ(4

3y) in each region.

Under uniformity, the choice of gU is immaterial, since c = 1, but everyonegets a utility of Ψ(y).

But the first–best solution can still be achieved under asymmetric infor-mation, since mimicking requires a low–cost region not to provide any of thelocal public output. The first–best solution gL = 4

3y, gH = 0, αL = −1

3y,

αH = 13y satisfies the selection constraint (2).

8 can uniformity be better?

The whole rationale for transfers (or uniformity) here is the declining marginalutility of income. Given that regions are identical ex ante, transfers from thecentral government, or uniform provision administered by a central govern-ment serve to redistribute from low–cost regions to high–cost regions, whichis only justified if the marginal utility of income is higher when costs arehigh.

5with −uΨ′′(u)Ψ′(u) > 1 ; otherwise expected utility would be maximized by transferring

income to the low–cost jurisdiction

8

To see the role of risk aversion here, consider a concave transformationΨ(u(x, g)) of the original utility function, where Ψ′(·) > 0 and Ψ′′(·) < 0.This transformation has no impact on the choice of policy under uniformity: gU is chosen to maximize u(y − cg, g). This transformation does affect thepolicy chosen under decentralization : a more concave utility function leadsto bigger transfers.

Under decentralization, there is an ex post utility frontier : the set of(uH , uL) combinations which solve the expected utility maximization problem(subject to the constraints (1) and (2)) for some transformation Ψ(·) of thegiven initial utility function u(x, g). Equivalently, consider directly the expost Pareto optimality problem : policies (gH , gL, αH , αL) which satisfy thebudget constraint (1), the selection constraint (2) and the reverse selectionconstraint u(y− cHgH +αH , gH) ≥ u(y− cHgL +αL, gL), and for which thereis no other feasible policy which is Pareto preferred.

If u(y − cgU , gU) > uH for every (uH , uL) combination on this Paretofrontier, then there is some transformation Ψ(·) so that expected utility ishigher under uniformity than under decentralization with asymmetric infor-mation, if jurisdictions’ utility functions are Ψ(u(x, g)). The assumption thatu(y − cgU , gU) > uH implies that uniformity leads to higher expected utilityif Ψ(u) ≡ 1

1−βu1−β for β sufficiently large and positive.

That is, if the solution to the problem of maximization of u(y − cHgH +αH , gH) subject to (1) and (2) leads to a solution uH for which u(y−cgU , gU) >uH , then uniformity yields higher expected utility for sufficiently high β inthe special case of Ψ(·) defined in the previous paragraph.6

To show that uniformity can be better, suppose that7

u(x, g) = x+ ln g

that y = 5, cH = 1.5, cL = 0.5, π = 0.5.In this case gU = 1, and u(y− cgU , gU) = 4. But maximization of uH(y−

cHgH + αH , gH) subject to (1) and (2) yields uH = 3.72. Therefore, there issome Ψ(·) which makes uniform provision better. For example, if

Ψ(u) =√u

6And if this solution yields uH ≥ u(y− cgU , gU ), then uniformity yields lower expectedutility for any concave utility function which is ordinally equivalent to the given u(x, g).

7The quasi–linear form for u(x, g) is chosen for ease of computation. With quasi–linearity, gL will not vary along the Pareto frontier when the selection constraint (2)binds.

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then uniformity gives an expected utility level of 4, whereas decentralizationunder asymmetric information results in an expected utility of 3.99574.

9 increasing risk aversion

The previous examples suggest that increasing risk aversion on the part of thecentral planner, implying here a greater taste for redistribution towards high–cost regions, may lead to a preference for uniformity over decentralizationwith transfers.

This suggestion is true in general. That is,

Proposition 1 For given (π, cL, cH), suppose that uniformity yields higherexpected utility than decentralization for some utility function u(x, g). IfΨ(·) is any concave, increasing transformation, then uniformity will leadhigher expected utility than decentralization for the utility function U(x, g) ≡Ψ[u(x, g)].

Proof The Proposition can be proved graphically. The solution underuniformity, the (x, g) combination such that ug/ux = c and x + cg = y, willnot change when the concave transformation Ψ(·) is applied.

Consider next the set of outcomes which can be achieved under decen-tralization, that is the set of outcomes which satisfy conditions (1), (2) andthe analogous “reverse selection” constraint that u(y + αH − cHgH , gH) ≥u(y + αL − cHgL, gL). This set does not change when u(x, g) is replaced bythe new, more concave, utility function U(x, g).

Now consider the set S of utility combinations (uH , uL) which can beachieved under decentralization. If uH is graphed along the horizontal, thenthis set must lie everywhere on or above the 45–degree line. The fact thatu(y − α− cLg, g) > u(y − α− cHg, g) if g > 0, and the selection constraints,imply that residents of low–cost regions must do strictly better under decen-tralization if both regions provide some of the public output.

The outcome under uniformity is a point on the 45–degree line when uHis graphed on the horizontal, and uL on the vertical. Expected utility will behigher under uniformity if and only if the central planner’s indifference curvethrough the uniform outcome, the line πuH + (1 − π)uL = u, lies strictlyabove the set S, as in figure 1.

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Now transform the utility measure. The outcome under uniformity hasnot changed. Neither has the set of outcomes under decentralization. Theset S has not changed either, except now the expected utility from a pointin (uH , uL) space is πΨ(uH) + (1− π)Ψ(uL). Concavity of Ψ(·) implies thatthe central planner’s indifference curve through the uniform outcome (u, u)is convex to the origin, and has the same slope −π/(1 − π) at the uniformoutcome. Therefore, if the uniform outcome (u, u) were preferred originallyto decentralization, so that the line πuH + (1− π)uL = u lies strictly abovethe set S, then the new indifference curve through (u, u), corresponding tothe utility function U(x, g), must lie strictly above the set S. •

If the decision maker is sufficiently risk averse, uniformity must be prefer-able. For given (π, cL, cH) and given “initial” utility measure u(x, g), letΨ(u) = u1−γ

1−γ . If γ is large enough, the outcome under uniformity must

be preferable, since it always must be the case that uL > u > uH , where(uH , uL) are utilities attained in the decentralized optimum, and u utilityunder uniformity.

But if the decision maker’s risk aversion is low enough, will decentral-ization be preferred? The answer is “yes”, at least if the decision maker issufficiently risk averse so as not to want “reverse” transfers.8.

Proposition 2 Let v(y, c) be the indirect utility corresponding to u9. Sup-pose that vx(y, cL) = vx(y, cH). Then decentralization yields higher expectedutility than uniformity.

Proof As discussed in section 3, marginal utility of income is assumedhere to decrease with income. Since it also decreases with the price of thepublic output, the hypothesis of the proposition will hold if utility is concave,but not “too concave”.

Consider the central planner’s first–best utility frontier, the set of utilitycombinations possible if lump–sum transfers between regions were possible.This set contains the set S of utility combinations possible when selectionconstraints must be satisfied. The set must also contain the utility combi-nation (u, u) achieved under uniformity ; in fact this point must be strictlyinside this first–best set.

8as discussed in section 3 above9so that v(y, c) ≡ u(y − cG(y, c), G(y, c))

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Now there is one point common to the first–best utility frontier, and theboundary of S. That is the “laissez–faire” point, where no transfers occur,and where jurisdiction i gets consumption bundle (x∗i , g

∗i ). The first–best

frontier and the boundary of the second–best set S must be tangent at thispoint L, as in figure 2. (The two frontiers coincide there, and the first–bestfrontier cannot cross the boundary of S, so they must be tangent.)

If ux(x∗L, g

∗L) = ux(x

∗H , g

∗H), then this common point L must be the out-

come under decentralization. Concavity of the utility function implies thateach v(y, ci) is concave in y alone, where v(y, c) denotes the indirect utilityfunction, maxg u(y − cg, g).

Thus the central planner’s indifference curve through L must be every-where on or above the first–best utility frontier, which is above the outcomeof uniformity (u, u), proving the proposition. •

The implication of the two propositions together is that there is a thresh-old degree of risk aversion, for any given ordinal representation of prefer-ences. There must be some positive, but low, degree of risk aversion, forwhich decentralization, with transfers from low–cost to high–cost regions, ispreferable. And above the threshold, uniformity will be preferred.

10 increasing substitutability

The example in section 7 of the superiority of decentralization rested on theassumption that the public and private outputs were perfect substitutes.

But perfect complementarity also seems condusive to decentralization,since the costs of distorting public output provision in the high–cost regions(in order to satisfy the selection constraint) are lower. In fact, decentraliza-tion must always be at least as good as uniformity when the 2 goods areperfect complements.

Proposition 3 If u(x, g) = min (x, g), then the outcome uL = uH = u isfeasible under decentralization.

Proof Under uniformity, with perfect complementarity, g = x = yc+1

=u.

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Under decentralization, suppose that low–cost regions are assessed a taxof

−αL =y

c+ 1(π(cH − cL))

so that high–cost regions get a transfer of

αH =y

c+ 1((1− π)(cH − cL))

With these levels of income, the first–best public output levels are

g∗L = g∗H =y

c+ 1

Low–cost jurisdictions have no incentive to mimic high–cost regions, sincethe extra income they would receive could only be spent on the private good,which is useless when x > g. •

Proposition 3 implies that the utility possibility frontier is a curve goingup and to the left above the point (uL, uH) = (u, u) : the selection constraintwill only bind if αH > y

c+1((1−π)(cH−cL)), so that the central planner wishes

to give higher utility to high–cost regions (which can never happen under theself–selection constraint). So for moderate levels of risk aversion, the centralplanner will choose a transfer αH strictly less than y

c+1((1−π)(cH−cL)), and

get expected utility strictly higher than under uniformity.With perfect complementarity, if the planner’s welfare measure is some

W (u) which is continuously differentiable, where u is the cardinal measureof utility u(x, g) = min (x, g), then the planner will always strictly preferdecentralization. At the uniform outcome (uL, uH) = (u, u) the slope of theplanner’s iso–welfare measure is

duLduH W=W

= − π

1− π

The utility possibility frontier under decentralization is a straight line, goingup and to the left from the point (u, u) (when uL is depicted on the verticalaxis, and uH on the horizontal), with slope

duLduH decent

= − π

1− π1 + cH1 + cL

which is steeper than the iso–welfare curve through (u, u).

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So, unless the planner has extreme “max–min” preferences (in which caseW (·) is not continuously differentiable), decentralization will be stricly pre-ferred to uniformity.

Consider then holding the planner’s degree of risk aversion fixed in somefashion, and changing the degree of substitutability between the two goods.For example, consider the family of utility functions

U(x, g) =1

1− β(xρ + gρ)(1−β)/ρ

for some β > 2. If β is held constant, and ρ decreases from 1 to −∞, theelasticity of substitution between goods falls from infinity to zero.

If ρ is sufficiently negative (that is, if the elasticity of substitution 1/1−ρ issmall enough), then the planner must prefer decentralization. The argumentabove showed decentralization is preferred strictly to uniformity in the limitas ρ→ −∞, so that this must be true when −ρ is large but finite.

More generally, for an economy defined by some (y, cH , cL, π), consider a1–parameter family of utility measures U(x, g;σ), where σ measures substi-tutability, with the properties that

• U(x, g;σ) is quasi–concave, increasing in both arguments, and well de-fined for all σ ≥ 0

• the consumption bundle (x, g) which maximizes U(x, g;σ), subject tothe uniformity constraint x+ cg ≤ y, does not vary with σ

• U(x, g;σ) is continuously differentiable at this (x, g) for all σ ≥ 0

• σ equals the elasticity of substitution between goods, at (x, g)

Then the above argument shows that, if the planner’s preferences arerepresented by the utility measure U(x, g;σ), then there is some strictlypositive bound σ such that the planner will strictly prefer decentralizationwhenever 0 < σ < σ.

However, for the family of preferences just described, it will not alwaysbe the case that the planner’s relative preference for decentralization overuniformity is “monotonic”. Consider again the CES example. Here if ρ isvery negative, decentralization is preferred. If cL < 1 < cH , the example ofsection 7 applies ; decentralization must Pareto dominate uniformity if thegoods are perfect substitutes. And a calculated example confirms this. If

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y = 2, cH = 1.5, cL = 0.5 and π = 0.5, then, for CES preferences, if ρ = 0.9(so that the elasticity of substitution is 10), a transfer of 0.2 from the low–cost jurisdiction to the high–cost jurisdiction achieves the first–best outcome(xH , gH) = (0.0035, 3.59) ; (xL, gL) = (2.1442, 0.0372) Pareto–dominatesthe outcome under uniformity, (x, g) = (1, 1).

So in this CES example, decentralization Pareto dominates uniformitywhen the elasticity of substitution is high. It weakly Pareto dominates whenthe elasticity of substitution is 0. For a given welfare measure U(x, g) =

11−β (xρ+gρ)(1−β)/ρ, holding β constant, utility is higher under decentralizationwhen the elasticity of substitution is small but positive, or when the elasticityis very large.

But for intermediate values of the elasticity of substitution, uniformitymay be preferred. If β = 10, and ρ = −1, uniformity gives a higher value forwelfare than the maximum which can be achieved under decentralization.

11 mixing the two

Implicit in the analysis so far is that the central planner faces an all–or–nothing choice : either uniform central provision (and no regional provision),or regional provision with no central provision.

This would be the case if there were set–up costs which were very impor-tant.

But if there were no such costs, then the planner could choose, simulta-neously, three levels of public output provision : a level gL to be provided(and financed) by every region reporting a low cost, a level gH to be provided(and financed) by every region reporting a high cost, and a level gU to beprovided for everyone, financed equally by residents of all regions. In such acase, a resident of a type–i region would have the consumption bundle

(y + αi − cgU − cigi, gU + gi)

As might be expected, in this situtation, one of the planner’s instrumentsis redundant. If the selection constraint binds, then the no public outputshould be provided by the high–cost region.

Proposition 4 If the selection constraint binds at the planner’s optimum,then gH = 0.

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Proof Suppose, to the contrary, that gH > 0 at the planner’s optimum,and that the selection constraint was binding.

Consider a policy adjustment : lower gH by ε, lower gL by ε, raise gU byε, raise αL by (c− cL)ε and lower αL by p

1−p(c− cL)ε.This adjustment is feasible : at a planner’s optimum, it must be the case

that gU + gL = G(y + αL, cL) and gU + gH ≤ G(y + αH , cH). The fact thatthe low–cost residents are better off than the high–cost residents and theassumption that the public output is normal imply that G(y + αL, cL) >G(y + αH , cH), so that gL > gH .

By construction, the adjustment leaves public output the same in thelow–cost region. It also leaves unchanged private consumption there, sincethe increased transfer exactly compensates for the higher cost of the publicsector for residents of low–cost regions.

But the fact thatc = pcH + (1− p)cL

implies that private income in the high–cost region is also unchanged :

∆xH = ∆αH + (cH − c)ε = −1− pp

(c− cL)ε+ (cH − c)ε = 0

Since gH + gU is unchanged, residents of the high–cost regions have the sameconsumption bundle as before.

But the attractiveness of mimickry has been reduced. The private incomea low–cost region would get from claiming to be a high–cost region has fallenby 1−p

p(c− cL)ε+ (c− cL)ε > 0.

Hence the change has left all residents’ consumption unchanged, but hasrelaxed the selection constraint.

Therefore, the planner could lower αL slightly (raising αH) and raiseoverall social welfare.

The adjustment, which increased social welfare strictly (when the selec-tion constraint was binding) was possible whenever gH > 0. Therefore it cannever be optimal to have gH > 0 and the selection constraint binding at thesame time. •

The adjustment in the proof shows as well that there is no harm in re-stricting gH to equal zero : if the selection constraint were not binding, thenno harm would be done by the adjustment constructed in the proof.

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12 the best mix

Proposition 4 says that, if mixing is possible, the planner’s problem can bereduced to the choice of gU , gL and transfers αL and αH so as to maximize

(1− π)u(y + αL − cgU − cL(gL − gF ), gL) + πu(y + αH − cgU , gU)

subject to the budget constraint (1 − π)αL + παH = 0, and the selectionconstraints.

Since gH = 0, the selection constraint for the low–cost regions is simplythat they get as least as high utility as the high–cost regions.

But the perfect–information planner’s solution must give at least as highutility to the low–cost regions : if transfers from low–cost regions to high–cost regions resulted in an outcome at which v(yH +αH , cH) ≥ v(yL+αL, cL),then the assumption that the public good is a normal good would imply themarginal utility of income is higher in the low–cost region.

Lemma 2 If cL < cH , and if v(yL, cL) = v(yH , cH) then λ(yL, cL) > λ(yH , cH)

if and only if the public output is a normal good (where λ(y, c) ≡ ∂v(y,c)∂y

).

Proof Define the income level y(c) so that v(y(c), c) = v for somereference level of utility v. Then

dλ(y(c), c)

dc=∂λ

∂c+G(y(c), c)

∂λ

∂y(7)

From Roy’s identity∂v(y, c

∂c= −λ(y, c)G(y, c) (8)

so that

vyc = −G(y, c)∂λ

∂y− λ(y, c)

∂g

∂y= vcy =

∂λ

∂c(9)

where vab indicates a second derivative.Plugging (9) into (7),

dλ(y(c), c)

dc= −λ(y(c), c)

∂G(y, c)

∂y(10)

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proving that when the public output is a normal good, then the marginalutility of income must be lower in the higher–cost region, if transfers haveresulted in the same utility level in both regions. •

So implementing the planner’s first–best optimum will must satisfy (strictly)the selection constraint for low–cost regions : the planner would want toredistribute income so as to equalize the marginal utility of income ; thisredistribution must result in a higher utility level for the lower–cost regions; therefore the selection constraint is satisfied when gH = 0.

But the planner will want to do some redistribution from low–cost tohigh–cost regions : the assumption that −ελy > εGy > 0 ensures that themarginal utility of income would be higher in the high–cost regions int heabsence of transfers.

And if αL > 0 > αH , then the selection constraint must be satisfied forthe high–cost regions : mimicking a low–cost region would lose them transferincome, and require them to provide more public output than they want.

Therefore

Proposition 5 If mixing of centralized and decentralized provision is possi-ble, then the central planner can achieve the full–information optimum evenwhen public sector costs are private information.

13 Ex Ante Differences among Regions

So far, it has been assumed that there is a large number of regions, andthat each region’s cost parameter ci is an independent draw from the samedistribution : for each region the probability of a high cost realization is thesame π. If the central government can commit to a policy before regionsobserve the realization of their own cost parameter, then there will be una-nimity as to which central government policy is best : that which maximizesthe expected utility of each region. Moreover, each region will prefer strictlyto subject itself to the rules of this federation : the central government’soptimal policy offers strictly higher expected utility to each identical regionthan the expected utility (1 − π)v(y, cL) + πv(y, cH) it would receive on itsown (when it is able to choose its public expenditure after observing its owncost parameter).10

10Proof : The “autarky” solution gi = G(y, ci), i = L,H, with no transfers, is feasi-ble for a decentralized federation with an uninformed central government. The central

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Now suppose instead that regions differ in their probability of havinga high cost cH for their public sector. That is, suppose that there is acontinuum of regions ; each region has a probability ρ of a high cost shock; the proportion of regions with probability ρ or less is Φ(ρ), where Φ(·) isa differentiable distribution function. [Φ : [0, 1] → [0, 1] with Φ(0) = 0 andΦ(1) = 1.] If φ(ρ) ≡ Φ′(ρ) is the density function, and if ρ is the lowest valueof ρ for regions in the federation, then the expected value of ρ is

π ≡ 1

1− Φ(ρ)

∫ 1

ρ

ρφ(ρ)dρ (11)

The formulation in (11) allows for the possibility that some types of regionwill choose not to join a federation. Implicit in the definition is that it is thelowest–cost regions, ex ante, who would choose not to join.

******

—more to come

References

[1] R. Boadway, I. Horiba, and R. Jha. The Provision of Public Services byGovernment Funded Decentralized Agencies. Public Choice, 100:157–184, September 1999.

[2] M. Bordignon, P. Manasse, and G. Tabellini. Optimal Regional Redis-tribution under Asymmetric Information. American Economic Review,91:709–723, 2001.

government could make no transfers, and order regions reporting a cost parameter of cito provide a public output level G(y, ci). With such a policy, a region with a low costwould do strictly better telling the truth, rather than trying to mimic. So it is also feasi-ble for the central government to make a small transfer. It could pay some small αH toeach region reporting a high cost, tax each region −αL = − π

1−παH if it reported a lowcost, and require a region reporting ci to choose a public output level G(y + αi, ci). Withsmall enough αH , the self–selection constraint will still hold as an inequality. And theassumption that |ελy| > εgy means that this actuarially fair insurance scheme will raiseeach region’s expected utility.

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[3] S. Bucovetsky, M. Marchand, and P. Pestieau. Tax Competition andthe Revelation of Preferences for Public Expenditure. Journal of UrbanEconomics, 44:367–390, 1998.

[4] R Cornes and E. Silva. Local public goods, risk sharing and privateinformation n federal systems. Journal of Urban Economics, 47:39–60,2000.

[5] R Cornes and E. Silva. Local public goods, inter-regional transfers, andlocal public goods. European Economic Review, 46:329–356, 2002.

[6] R Cornes and E. Silva. Public good mix in a federation with incompleteinformation. Journal of Public Economic Theory, 5(2):381–397, 2003.

[7] A. Dhillon, C. Perroni, and K. Scharf. Implementing Tax Coordination.Journal of Public Economics, 72(2):243–268, May 1999.

[8] B. Huber and M. Runkel. Optimal Design of Intergovernmental Grantsunder Asymmetric Information. International Tax and Public Finance,13:25–41, 2006.

[9] B. Lockwood. Inter-regional Insurance. Journal of Public Economics,72:1–39, 1999.

[10] H. Varian. Redistributive Taxation as Social Insurance. Journal ofPublic Economics, 14:49–68, 1980.

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14 Figures

Figure 1

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Figure 2

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