Division of the Humanitiesand Social Sciences

A public goods example:Quasilinearity with nonnegativity

KC BorderApril 2012

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In a public goods economy with quasilinear utilities it is sometimes asserted that there isa unique Pareto efficient level of the public good, and it is found by maximizing the sum ofutilities.1 It is true that maximizing the sum of utilities will find an efficient outcome, buttypically there are others, provided we impose the reasonable restriction that consumption ofthe private good be nonnegative.2

1 The simplest economy with public goodsThere are two consumers and two goods, a public good z and a private good x. Consumptionsets are R2

+. Note that this rules out negative consumption of goods. Each consumer has aquasilinear utility function, of the form

ui(z, xi) = xi + vi(z),

where each vi is strictly increasing and strictly concave. In order to allow the case v(z) = ln(z)we allow v to assume the value −∞ at zero.

The public good must be produced from the private good according to the linear produc-tion function

z = f(x) = x.

The total endowment of the private good is denoted ω.

2 AllocationsAn allocation a for this economy is a triple a = (z, x1, x2) ≧ 0 satisfying

x1 + x2 + z = ω.

Let A denote the set of allocations, and observe that it is a convex subset of R3+. It is clear that

(z, x1) determines the allocation completely, as x2 = ω − x1 − z, so that A is 2-dimensional.3

1I need to find some published assertions of this, but for now take my word for it that one of my colleagueswho has worked on public goods for decades believed this to be true.

2Some authors explicitly allow arbitrarily large negative levels of consumption of the linear utility good, e.g.,MWG [2], Definition 3.B.7, but this is silly.

3I am taking the hard line position that disposal of goods is a “production” technology that is not available.Assuming it is available would not change anything important.

1

KC Border Public goods with nonnegativity 2

We can represent an allocation by a single point in the “Samuelson triangle:”4 See Figure 1.Here the level of the public good is displayed on the horizontal axis, and x is measured on thevertical axis. The upper boundary can be interpreted as the production possibility frontier ofthe economy.

x1

z

ω

ω

z

x1

x2

Figure 1. Samuelson diagram for the allocation (z, x1, x2).

We can also use this triangular diagram to represent indifference curves over allocations.See Figure 2. The indifference curves extend outside the Samuelson triangle just as indifferencecurves can extend outside an Edgeworth box.

3 Avoiding trivialitiesAssumption 1 Assume that for each consumer i = 1, 2, there is a (unique) z∗

i > 0 that satisfies

v′i(z∗

i ) = 1.

Since each vi is assumed to be strictly concave, v′i is strictly decreasing so such a z∗

i is unique.One way to guarantee the existence is to assume that each v satisfies the Inada conditions,

limz→0

v′(z) = ∞ and limz→∞

v′(z) = 0.

The significance of z∗i is this. Imagine that consumer i is the only consumer. Then he or she will

choose to maximize ui(z, x) = x + vi(z) subject to z + x ⩽ ω and the nonnegativity constraintsx ⩾ 0 and z ⩾ 0. The solution to this problem is

(z, x) =

(ω, 0) if ω ⩽ z∗i

(z∗i , ω − z∗

i ) if ω ⩾ z∗i .

4This sort of diagram was employed by Paul Samuelson [3] in 1955. It is much easier to work with than theKolm triangle diagram introduced by Serge-Christophe Kolm [1]. Here is an on-line chapter on Kolm triangles.

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1 2 3 4 5 6z

-1

1

2

3

4

5

6

x1

Figure 2. Indifference curves. Consumer 1 is in red, and consumer 2 is in blue. (The coloredline segments indicate Pareto efficient points.)

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That is, the consumer will use the private good up to the amount z∗i to produce the public

good and consume what is left over if anything. We shall call z∗i consumer i’s individually

optimal level of the public good.The next assumption is more of an ordering convention, but it rules out the case of equality.

Assumption 2z∗

2 > z∗1 .

The next assumption would be unnecessary if we had assumed the Inada conditions.

Assumption 3 There is a (unique) z∗ that satisfies

v′1(z∗) + v′

2(z∗) = 1.

We shall see presently that any strictly positive Pareto efficient allocation will satisfy z = z∗

if ω > z∗, and of course feasibility implies z ⩽ z∗ whenever ω ⩽ z∗. (Note that this relies onxis being nonnegative.)

Note also that since each v′i is strictly positive and strictly decreasing,

z∗ > z∗1 > 0 and z∗ > z∗

2 > 0. (1)

4 Pareto efficiencySince the set A of allocations is convex, and each utility ui is concave, the utility possibility setU ⊂ R2 defined by

U ={(

u1(z, x1), u2(z, x2))

: (z, x1, x2) ∈ A}

is a convex. Thus we can find Pareto efficient allocations by maximizing a semipositive weightedsum of utilities, subject to being an allocation. (We refer to the weights α on the utilities aswelfare weights.) Here is the Lagrangean:

α1(x1 + v1(z)

)+ α2

(x2 + v2(z)

)+ µ(ω − x1 − x2 − z).

The first-order conditions are usually written as:

∂

∂x1: α1 − µ = 0

∂

∂x2: α2 − µ = 0

∂

∂z: α1v′

1(z) + α2v′2(z) = 0.

You can tell right away that something is wrong. The weights α1 and α2 are chosen arbitrarily,and varying them gives different efficient points, but the first-order conditions imply α1 =µ = α2. What went wrong is this: I ignored the nonnegativity constraints on my variables.Observe that no matter what the level z of the public good is, then x1 and x2 must maximizeα1x1 + α2x2. So if α1 > α2 ⩾ 0, then x2 = 0 and x1 = ω − z. The first-order conditions I wrote

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down above are necessary only if z > 0, x1 > 0, and x2 > 0, which can occur only if α1 = α2.So let’s write down the Kuhn–Tucker conditions:

∂

∂x1: α1 − µ ⩽ 0, x1 > 0 =⇒ = (2a)

∂

∂x2: α2 − µ ⩽ 0, x2 > 0 =⇒ = (2b)

∂

∂z: α1v′

1(z) + α2v′2(z) − µ ⩽ 0, z > 0 =⇒ = . (2c)

4.1 Case I: ω > z∗

This is the usual assumption in the literature. If we assume the Inada conditions and drop thenonnegativity of the xis we would not need it. We now characterize Pareto efficiency in thiscase.

There are three subcases to consider.

Subcase 1: α1 = α2

This is the only case in which we might get both x1 > 0 and x2 > 0. Since α1 = α2, we may aswell set both to unity. Then the Lagrangean is

x1 + v1(z) + x2 + v2(z) + µ(ω − x1 − x2 − z).

The first-order Kuhn–Tucker conditions are thus

1 − µ ⩽ 0, x1 > 0 =⇒ = (3a)1 − µ ⩽ 0, x2 > 0 =⇒ = (3b)

v′1(z) + v′

2(z) − µ ⩽ 0, z > 0 =⇒ = . (3c)

By (3a) and (3b) we have µ ⩾ 1, with µ = 1 if either x1 > 0 or x2 > 0 at the optimum. In factwe must have x1 + x2 > 0. To see this, suppose x1 = x2 = 0 at the optimum. Then z = ω > 0by feasibility, and (3c) becomes v′

1(ω) + v′2(ω) = µ ⩾ 1. But each v′

i is strictly decreasing andby assumption ω > z∗. Thus v′

1(ω) + v′2(ω) < v′

1(z∗) + v′2(z∗) = 1, a contradiction. Thus x1 > 0

or x2 > 0 (or both), so x1 + x2 > 0 and µ = 1.Since µ = 1, (3c) implies v′

1(z) + v′2(z) ⩽ 1 = v′

1(z∗) + v′2(z∗). But (again as v′

1 and v′2 are

strictly decreasing) this implies that z ⩾ z∗ > 0. But then (3c) implies v′1(z) + v′

2(z) = 1, so

z = z∗.

Thus the following are efficient allocations:

z = z∗, x1 + x2 = ω − z∗ > 0.

Let us look at two special cases of these allocations. Consumer 1’s most preferred is

a∗12 = (z∗, ω − z∗, 0),

and consumer 2’s favorite isa∗21 = (z∗, 0, ω − z∗).

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Subcase 2: α1 > α2

Since α1 > α2, we may as well set α1 = 1, so α2 < 1. Then the Lagrangean is

x1 + v1(z) + α2(x2 + v2(z)

)+ µ(ω − x1 − x2 − z).

The Kuhn–Tucker conditions are thus

1 − µ ⩽ 0, x1 > 0 =⇒ = (4a)α2 − µ ⩽ 0, x2 > 0 =⇒ = (4b)

v′1(z) + α2v′

2(z) − µ ⩽ 0, z > 0 =⇒ = . (4c)

Again, by (4a) we have µ ⩾ 1, but 1 > α2, so by (4b) we must have

x2 = 0.

An argument similar to the one above shows that x1 > 0. To see this, suppose x1 = 0. Thensince x2 = 0 too, we have z = ω > 0 by feasibility. So (4c) implies v′

1(ω) + α2v′2(ω) = µ ⩾ 1.

Thereforev′

1(ω) + v′2(ω) > v′

1(ω) + α2v′2(ω) ⩾ 1,

so ω < z∗, a contradiction. Therefore, x1 > 0 and µ = 1. This in turn implies

v′1(z) + α2v′

2(z) ⩽ 1.

When α2 = 0, this reduces to v′1(z) = 1, so z = z∗

1 , and as α2 → 1, we see that z → z∗.That is, by increasing the welfare weight on consumer 2, we still do not give him any of theprivate good, but raise the level of the public good toward the joint optimum.

Thus the efficient allocations (z, x1, x2) are of the following form:

z∗1 ⩽ z < z∗, x1 = ω − z > 0, x2 = 0.

Subcase 3: α1 < α2

I leave it as an exercise to use the same logic as in subcase 2 the efficient allocations (z, x1, x2)are of the following form:

z∗2 < z < z∗, x1 = 0, x2 = ω − z > 0.

SummaryWe summarize the efficient allocations in the following proposition.

Proposition 4 Let ω > z∗. Then an allocation (z, x1, x2) is Pareto efficient if and only ifx1 + x2 > 0 and:

• If x1 > 0 and x2 > 0, then z = z∗.

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KC Border Public goods with nonnegativity 7

• If x1 > 0 and x2 = 0, then z∗1 ⩽ z ⩽ z∗.

• If x1 = 0 and x2 > 0, then z∗2 ⩽ z ⩽ z∗.

Figure 2 also shows the Pareto efficient set for the case where

v1(z) = ln(z), v2(z) = 2 ln(z), and ω = 6.

In this casez∗

1 = 1, z∗2 = 2, and z∗ = 3.

The magenta vertical line corresponds to the case of equal welfare weights, α1 = α2; it connectsthe allocation a∗12 = (z∗, ω − z∗, 0) and a∗21 = (z∗, 0, ω − z∗). The red line segment onthe upper boundary corresponds to the case α1 > α2, so x2 = 0; it connects the allocationa∗1 = (z∗

1 , ω − z∗1 , 0) and a∗12 = (z∗, ω − z∗, 0). The blue segment corresponds to the case

α1 < α2, so x1 = 0; it connects the allocations a∗2 = (z∗2 , 0, ω − z∗

2) and a∗21 = (z∗, 0, ω − z∗).Figure 3 shows the corresponding utility possibility set. The Pareto set is made of three

sections. The magenta line segment corresponds to the case of equal welfare weights, α1 = α2; itsendpoints correspond to a∗21 (upper left) and a∗12 (lower left). The solid red curve correspondsto the case α1 > α2; its endpoints correspond to a∗12 (upper left) and a∗1 (lower left). Thesolid blue curve corresponds to the case α1 < α2; its endpoints correspond to a∗2 (upper left)and a∗21 (lower left).

5 Small endowmentsCase II: z∗

2 < ω ⩽ z∗

If the endowment satisfies z∗2 < ω ⩽ z∗, we get two kinds of efficient allocations: For the first

kind we set the welfare weights α1 = 1 and α2 ⩽ 1. Then efficient allocations take the form:

{(z, x1, x2) ≧ 0 : z∗1 ⩽ z ⩽ ω, x1 = ω − z, x2 = 0}.

The second class uses weights α2 = 1 and α1 ⩽ 1. Efficient allocations take the form:

{(z, x1, x2) ≧ 0 : z∗2 ⩽ z ⩽ ω, x1 = 0, x2 = ω − z}.

See Figures 4 and 5.

Case III: z∗1 < ω ⩽ z∗

2

Assumption 2 says that z∗1 < z∗

2 . For endowments with z∗1 < ω ⩽ z∗

2 , any welfare weights willyield x2 = 0, but positive weight on consumer 2 requires a level of public good above z∗

1 . Theset of efficient allocations is:

{(z, x1, x2) ≧ 0 : z∗1 ⩽ z ⩽ z∗, x1 = ω − z, x2 = 0.}

See Figures 6 and 7.

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-1 1 2 3 4 5u1

-1

1

2

3

4

5

u2

Figure 3. The upper portion of the utility possibility set for v1(z) = ln z, v2(z) = 2 ln z, andω = 6. (The set is unbounded below.)

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0.5 1.0 1.5 2.0 2.5z

-1

1

2

3

x1

Figure 4. Efficient allocations for v1(z) = ln z, v2(z) = 2 ln z, and z∗2 < ω < z∗ with ω = 2.5.

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-1.0 -0.5 0.5 1.0 1.5 2.0u1

-1.0

-0.5

0.5

1.0

1.5

2.0

u2

Figure 5. Utility possibility set for v1(z) = ln z, v2(z) = 2 ln z, and z∗2 < ω ⩽ z∗ with ω = 2.5.

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KC Border Public goods with nonnegativity 11

0.2 0.4 0.6 0.8 1.0 1.2 1.4z

-0.5

0.5

1.0

1.5

x1

Figure 6. Efficient allocations for v1(z) = ln z, v2(z) = 2 ln z, z∗1 < ω ⩽ z∗

2 with ω = 1.5.

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-1.0 -0.5 0.5 1.0u1

-1.0

-0.5

0.5

1.0

u2

Figure 7. Utility possibility set for v1(z) = ln z, v2(z) = 2 ln z, z∗1 < ω ⩽ z∗

2 with ω = 1.5.

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Case IV: ω ⩽ z∗1

Finally, if ω < z∗1 , the only efficient allocation is to use the entire endowment to produce the

public good:

z = ω, x1 = x2 = 0.

See Figures 8 and 9.

SummaryThe following proposition summarizes the small endowment cases.

Proposition 5 Recall that for any allocation, z + x1 + x2 = ω.

• Assume z∗2 < ω ⩽ z∗. Then an allocation (z, x1, x2) is Pareto efficient if and only if either

x2 = 0 and z∗1 ⩽ z ⩽ ω or x1 = 0 and z∗

2 ⩽ z ⩽ ω.

• Assume z∗1 < ω ⩽ z∗

2. Then an allocation (z, x1, x2) is Pareto efficient if and only if

x2 = 0 and z∗1 ⩽ z ⩽ ω.

• Assume 0 < ω ⩽ z∗1. Then an allocation (z, x1, x2) is Pareto efficient if and only if z = ω.

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0.1 0.2 0.3 0.4 0.5z

-0.2

0.2

0.4

0.6

x1

Figure 8. Efficient allocations for v1(z) = ln z, v2(z) = 2 ln z, ω ⩽ z∗1 with ω = 0.5.

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-1.0 -0.8 -0.6 -0.4 -0.2u1

-2.0

-1.5

-1.0

u2

Figure 9. Utility possibility set for v1(z) = ln z, v2(z) = 2 ln z, ω ⩽ z∗1 with ω = 0.5.

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6 Voluntary contributionFor this section we have private ownership of the private good. For convenience we assumethat each individual endowment ωi satisfies

ωi > z∗i .

We model voluntary cooperation as a noncooperative game. Each player chooses to con-tribute a quantity 0 ⩽ ti ⩽ ωi of their endowment to be used to produce the public good. Whatis the Nash equilibrium of this game?

The payoffs to the consumers are given by

π1(t1, t2) = ω1 − t1 + v1(t1 + t2), π2(t1, t2) = ω2 − t2 + v2(t1 + t2).

A Nash equilibrium (t̄1, t̄2) satisfies (by definition)

π1(t̄1, t̄2) ⩾ π1(t1, t̄2) for all 0 ⩽ t1 ⩽ ω1,

andπ2(t̄1, t̄2) ⩾ π2(t̄1, t2) for all 0 ⩽ t2 ⩽ ω2.

For the moment ignore the constraint that ti ⩽ ωi, as I doubt it will bind, but if we take thefree rider problem seriously, we shouldn’t ignore the 0 ⩽ ti constraints.

The first-order Kuhn–Tucker conditions for t̄1 are

−1 + v′1(t̄1 + t̄2) ⩽ 0 with = 0 if t1 > 0.

and for t̄2 are−1 + v′

2(t̄1 + t̄2) ⩽ 0 with = 0 if t2 > 0.

which imply (once more by the strict decreasingness of v′i and the definition of z∗

i ) that

t̄1 + t̄2 ⩾ z∗1 and t̄1 + t̄2 ⩾ z∗

2 .

Clearly the first inequality must be strict as z∗2 > z∗

2 by Assumption 2. Then the Kuhn–Tuckercondition imply t̄1 = 0. Thus t̄2 ⩾ z∗

2 > 0, so the second condition must hold with equality,namely t̄2 = z∗

2 . Thus the Nash equilibrium is

t̄1 = 0, t̄2 = z∗2 .

Note that this is exactly what consumer 2 would choose if he were on his own. It results in theallocation x1 = ω1, x2 = ω2 − z∗

2 , z = z∗2 . Since both x1 > 0 and x2 > 0, efficiency requires

z = z∗ (Proposition 4), which is greater than z∗2 . Thus the Nash equilibrium allocation is

inefficient—not enough public good is produced.

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7 Lindahl equilibriumFor this section we shall assume that there are private endowments ωi and profit shares θi, andthat

ωi > z∗i , i = 1, 2.

ω1 + ω2 > z∗.

Definition 6 A Lindahl equilibrium is a list

(z̄, x̄1, x̄2; p, q1, q2) ≧ 0

satisfying

1. (z̄, x̄1, x̄2) is an allocation, that is, x̄1 + x̄2 + z̄ = ω1 + ω2.

2. z̄ maximizes profit (q1 + q2)z − pz (subject to z ⩾ 0). Let π = (q1 + q2)z̄ − pz̄.

3. For each consumer i = 1, 2, (z̄, x̄i) maximizes xi + vi(z) subject to pxi + qiz ⩽ pωi + θiπ(and nonnegativity).

We cannot have a maximum for a consumer if p = 0 or qi = 0, so without loss of generalitywe may set

p = 1.

Then order for a profit maximum to exist we must have

π = 0 and q1 + q2 ⩽ 1 with equality if z̄ > 0.

Since π = 0, consumer i’s Lagrangean is

xi + vi(z) + λi(ωi − xi − qiz).

The Kuhn–Tucker conditions are

1 − λi ⩽ 0 = 0 if x > 0 (5a)v′

i(z) − λiqi ⩽ 0 = 0 if z > 0. (5b)xi + qiz ⩽ ωi = ωi if λi > 0. (5c)

The first order condition (5a) implies that the multiplier λi ⩾ 1 > 0, so (5c) implies thateach budget holds with equality.

Lemma 7 z̄ > 0:

Proof : Suppose for the sake of contradiction that z̄ is zero. Then x̄i = ωi > 0 (5c), so λi =1 (5a). Thus qi ⩾ v′

i(z̄) (5b). Summing gives 1 ⩾ q1 + q2 ⩾ v′1(z̄) + v′

2(z̄), which impliesz̄ ⩾ z∗ > 0, a contradiction. Therefore z̄ > 0.

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Since z̄ > 0, (5b) impliesv′

i(z̄) = λiqi. (6)

Moreover since z̄ > 0, profit maximization implies

q1 + q2 = 1. (7)

Lemma 8 z̄ ⩽ z∗.

Proof : Nowv′

1(z̄) + v′2(z̄) = λ1q1 + λ2q2 ⩾ q1 + q2 = 1 = v′

1(z∗) + v′2(z∗).

By the decreasingness of v1 + v′2 we have z̄ ⩽ z∗.

Lemma 9 In a Lindahl equilibrium, at least one xi > 0.

Proof : If x̄1 = x̄2 = 0, thenz̄ = ω1 + ω2 > z∗,

which contradicts Lemma 8.

Proposition 10 Every Lindahl equilibrium is efficient.

Proof : From Proposition 4 there are three cases:Case 1: x̄1 > 0 and x̄2 > 0.In this case, λ1 = λ2 = 1 by (5a), so v1(z̄) + v2(z̄) = q1 + q2 = 1 (7), so z̄ = z∗. Efficiency

follows from Proposition 4.Case 2: x̄1 > 0 and x̄2 = 0.Then λ1 = 1 by (5a). So by (6) we have v′

1(z̄) = q1 ⩽ 1v′1(z∗

1), so z̄ ⩾ z∗1 . But z̄ ⩽ z∗ by

Lemma 8. Again efficiency follows from Proposition 4.Case 3: x̄1 = 0 and x̄2 > 0.This proof is similar to case 2.

References[1] Kolm, S.-C. 1963. Les fondements de l’economie publique - introduction à la théorie du rôle

economique de l’etat. Paris: IFP.

[2] Mas-Colell, A., M. D. Whinston, and J. R. Green. 1995. Microeconomic theory. Oxford:Oxford University Press.

[3] Samuelson, P. A. 1955. Diagrammatic exposition of a theory of public expenditure. Reviewof Economics and Statistics 37(4):350–356.

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