Discussion Papers In Economics
And Business

The State Finance and Public Goods from The
General Equilibrium Viewpoint: Fundamental
Welfare Theorems for Lindahlian General
Equilibrium with Money
KEN URAIHIROMI MURAKAMIDAISUKE KOBAYASHI

Discussion Paper 23-08

June 2023

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe State Finance and Public Goods from The
General Equilibrium Viewpoint: Fundamental
Welfare Theorems for Lindahlian General
Equilibrium with Money∗
Ken Urai†
Hiromi Murakami‡
Daisuke Kobayashi§

Abstract
This paper deals with the concept of Lindahl general equilibrium with money and its relation
to the ﬁrst and second fundamental theorems of welfare economics and other problems like corearguments. This paper is concerned with the question of how the optimal supply of public goods
can be achieved in an ideal situation based on market transactions. The paper also points out thatunless the (Lindahlian) ﬁnancing problem is treated together with the ﬁscal problem of the state (asa monetary equilibrium), the fundamental theorems of economics may lead to the wrong message
in terms of policy. The ideal state described in this paper is one in which public and private ﬁrms
and their proﬁts are clearly distinguished, and in which the optimal allocation is feasible only undera constant government deﬁcit. This is not addressed in the traditional Lindahl general equilibrium
and cost-sharing equilibrium. Also, under such a general situation, the activity criterion for publicﬁrms in the ideal state is organized as a special type of marginal cost pricing. The framework ofthis paper, including government activities (optimal money issuance and taxation), is of extremely
urgent signiﬁcance, especially as seen in the recent problems of health care costs and state ﬁnances.

Keywords: Lindahl Equilibrium, General Equilibrium, Fundamental Theorems of Welfare Economics,
Public Goods, Satiation, Money
JEL Classiﬁcation: D51, D62, H53∗ We thank Prof.Yamamoto (Kyoto Bunkyo University) for helpful comments in the presentation of our research on an
early draft of the core concept in an economy with public goods, which was the basis for this paper.
† Graduate School of Economics, Osaka University, E-mail: urai@econ.osaka-u.ac.jp‡ Faculty of Fine Arts, Kyoto City University of Arts, E-mail: h murakami@kcua.ac.jp§ Department of Community Medical Support, Toyama University Hospital, E-mail: kobadai@med.u-toyama.ac.jp

1

Introduction
In this paper, we consider the problem of the state ﬁnance (including the issuance of ﬁat money) andpublic goods from the general equilibrium perspective (the Lindahl general equilibrium with income
transfers). Although it is usually diﬃcult to present a realistic implication of Lindahl equilibria, withrespect to merit goods like medical cares, it would be available to bring about arguments on an idealisticﬁscal state of government. Here, the government is treated as the supplier of ﬁat money as well as aprovider of public goods. In such a setting, several important, previously unknown results can beobtained regarding the government’s ﬁscal balance, the roles and norms of public and private ﬁrms, andespecially the most desirable way to share the costs of public goods.
The general equilibrium argument including public goods begins with Foley (1970) and extended
by Milleron (1972) to the case with multiple ﬁrms. These arguments, however, do not allow ﬁrms
to have non-convex technologies. Mas-Colell and Silvestre (1989) treats the case with the non-convex
technology though their arguments depend on a special cost-function structure and do not incorporate
multiple ﬁrm settings. In our model, we treat public and private multiple ﬁrms as well as deﬁcits ofthe government because of the non-convexity of the public-ﬁrm technology in order to describe an ideal
resource allocation (to establish the ﬁrst and second welfare results) for public-goods economies.
One of the most important purposes of our model is to describe a situation such that a certain activity
arrangement that is necessary for Pareto optimality can never be realized as an equilibrium until thegovernment prepares for a budgetary deﬁcit. The problem is related, for one thing, to the saturation of
consumers’ preferences (or to the necessity of savings and capital accumulations in view of dynamics).
The equilibrium concept in this paper is generally called a dividend equilibrium (competitive equilibriumwith slack in Mas-Colell 1992), in which government ﬁat money often plays an essential role. At thesame time, since the public-ﬁrm technology is not necessarily convex, we have to present a criterion
for the public-ﬁrm production decision alternative to the proﬁt maximization. Let us ﬁrst review thesepoints with the following simple example:(Example 1) Suppose that there are two commodities, a public good (commodity 1) anda private good (commodity 2). Here the public good is assumed to have both the nonexcludability and non-rivalness. Consider a public ﬁrm and a private ﬁrm, and two consumers(a worker and a capitalist). The worker (iw ) has one unit of private good (identiﬁed with one
unit of labour) as the initial endowment. The capitalist (ic ) owns the private ﬁrm (technology
Y1 ⊂ R2 ) and having no private good. Government possesses the public technology Y0 ⊂2
R2 . Each consumer has consumption set R+
with a utility function such that u(x1 , x2 ) =max{x2 , 8}. (That is, each consumer has a satiated preference and does not obtain positive
utility from the public good.) Furthermore, suppose that Y0 = {(0, 0), (−1, 1)} and Y1 =√ √{(y1 , y2 )| y2 = 1010 10 − y1 − 1, 0 ≦ y1 ≦ 10}. In other words, Y0 is a technology thattransforms one unit of private good into one unit of public good (or does nothing), and Y1
is a well behaved technology that transforms one unit of public good into 10 units of private
good (see Figure 1).

1y2(−1, 1)u(−1, 1) + Y1
u

−1 0

9

y1

Figure 1: Y0 = {(0, 0), (−1, 1)} and (−1, 1) + Y1
In the setting of Example 1 above, it is clear that the production of the public good must take placefor Pareto optimality. Furthermore, for the competitive private ﬁrm, only p = (p1 , p2 ) = (0, 1) is theprice (with norm 1) to support production behavior that produces 10 units of private good with one
unit of public good input, while satisfying the proﬁt-maximization condition. It follows that if we do not
use any income-transfer policy, the unique competitive (Lindahl) equilibrium, where dividends dw = 0
for worker and dc = 10 for capitalist (the proﬁt of private ﬁrm), is the allocation, xw = (1, 1) for
worker and xc = (1, 9) for capitalist.1 The allocation is not Pareto optimal involving a government
budget deﬁcit −1 = p · (−1, 1), even though the result is better than a situation where no production
occurs. If we allow the government to further increase its budget deﬁcit and allow a positive dividend(ﬁat money issuance) dw = 1 to worker, we obtain a Pareto optimal (dividend) competitive equilibrium
allocation, xw = (1, 2) and xc = (1, 8), with a government budget deﬁcit −2. For these Lindahl generalequilibrium with dividends, the satiation of preference of capitalist or, from a dynamic perspective,
savings (especially under the overlapping-generations framework) is essential. 2 The savings of capitalist
is equal to the government budget deﬁcit for both cases. Such a government deﬁcit can be viewed as theloan from the private sector, i.e., credit given to the state from the private sector. In fact, the private
sector’s faith in the value of its savings is precisely the same as its faith in the state, money, and themarket system. Can we really say that such a government deﬁcit is a bad thing and must be eliminated?
Moreover, it is only with such a government deﬁcit (unless we assume a lump-sum tax from the privateﬁrm) that a Pareto-optimal allocation of resources can be realized.
This paper deals with the ﬁrst and second fundamental theorems of welfare economics for (Lindahl)market equilibrium with public goods, including the distinction between public and private ﬁrms, as wellas the government’s ability to issue money (credit). Such a treatment is important in today’s society,where public goods (as merit goods) such like medical cares, account for a large share of state ﬁnances
through public and private institutions. In particular, our model provides a description of an ideal stateof cost-sharing for public goods, while taking into account what the state’s contribution is and what thealternative norm is for proﬁt maximization in a public ﬁrm.3
1 Note that the price of public good is 0 and the preferences are satiated, so u(1, 9) = u(1, 8) maximizes the utility ofcapitalist.
2 See Appendix A.
3 As emphasized in the above example, the essential role played by the satiation in a static model can be replaced by2

Denote by R the set of real numbers. For ﬁnite set A, ♯A denotes the number of elements of A. We
write RA instead of R♯A to represent the ♯A-dimensional topological vector space. Order relations on RA ,≧ and >, are deﬁned respectively as (xa )a∈A ≧(ya )a∈A iﬀ xa ≧ ya for all a ∈ A, and (xa )a∈A > (ya )a∈A
iﬀ (xa )a∈A ≧(ya )a∈A and (xa )a∈A 6= (ya )a∈A . We also deﬁne relation  as (xa )a∈A  (ya )a∈A iﬀ
AA
xa > ya for all a ∈ A. By R+
and R++, we represent the sets {x ∈ RA | x ≧ 0} and {x ∈ RA | x  0},respectively.

2

The Model

2.1

Basic Settings

Denote by K ∪ L, the ﬁnite set of commodities, where K 6= ∅ is the index set of public goods and
L 6= ∅ is the set of private goods (K ∩ L = ∅). As in Foley (1970), a vector of public and private
goods is written as ((xk )k∈K ; (zℓ )ℓ∈L ) = (x; z) ∈ RK∪L . Notation I is used to denote the non-empty
index set of ﬁnite agents. Each agent i ∈ I is represented by (≿i , ωi ), where ≿i is the preferenceK∪Land ωi ∈ RL is the initial endowment. The preferences are
relation on consumption set Xi = R+assumed to satisfy reﬂexivity, transitivity, completeness and continuity. The preferences, therefore, canbe represented by utility functions. Note, however, that the preferences are allowed to be satiated.
An economy, E, is identiﬁed with a ﬁnite list of consumers, (≿i , ωi )i∈I , a ﬁnite list of private ﬁrm(convex) technologies, (Yj )j∈J and a public ﬁrm (possibly non-convex) technology Y0 . As Milleron
K
K
K
K(1972), technologies are identiﬁed with subsets of R+× R+
× RL , where (x− , x+ ; y) ∈ R+
× R+
× RL

means that x− is an input vector of public goods, x+ is an output vector of public goods, and y is a
net production vector of private goods. (We consider the situation that output and input prices are
diﬀerent for public goods.) We assume the following:KK(1-1) For each j ∈ J, technology Yj is a closed subset of R+× R+
× RL having 0 as its element.
KK(1-2) Technology Y0 is a non-empty closed (possibly non-convex) subset of R+× R+
× RL .

Note that we use the most general framework including multi-ﬁrms like Milleron (1972) and positive
proﬁts like Mas-Colell and Silvestre (1989), while not using a special condition for public good inputs
like Foley (1970)’s condition (B.5) “No public good is necessary as a production input”.
By the meaning of public goods, we may view the set of all technically possible production plans in
the economy, Y , as the following set.
Y = {(x; y) | x =∑j∈J∪{0}x+
j ,y =∑

yj , (x, x+
j ; yj ) ∈ Yj for all j ∈ J ∪ {0}}.

(1)j∈J∪{0}such things as savings under a dynamic overlapping-generations structure (see, Appendix A). In other words, the mostimportant question is how much savings (the value left by someone) contribute to society, taking into account production(including the future), and the appropriate government budget deﬁcit should be determined on this basis.

3

For ﬁrms, we also assume that there is a list of shareholding rates, ((θij )j∈J )i∈I .
Technology Y is used to deﬁne a feasible allocation of economy E. A consumption allocation for a listK∪Lof agents S ⊂ I is a sequence of elements of consumption sets, ((xi ; zi ) ∈ R+
)i∈S . For an economyK∪L
E = ((≿i , ωi )i∈I , Y0 , (Yj )j∈J , ((θij )j∈J )i∈I ), if consumption allocation ((x; zi ) ∈ R+
)i∈I satisﬁes

∑
i∈I

zi =

∑

ωi + y for some (x; y) ∈ Y ,

(2)

i∈I

we say that (x; zi )i∈I is feasible under (x; y) ∈ Y . The list of production actions associated with (x; y),()∑+
i.e., (x, x+
0 ; y0 ) ∈ Y0 , ((x, xj ; yj ) ∈ Yj )j∈J , where y = y0 +j∈J yj , is called a production allocation.
+
We call the (m + n + 1)-tuple ((x; zi )i∈I , (x, x+
0 ; y0 ), (x, xj ; yj )j∈J ) a state of E.

2.2

Lindahl Equilibrium with Income Transfers

The list of price vectors, (p∗K , p∗L ) ∈ RK∪L , input price vectors for consumers, (piI ∗ ∈ RK )i∈I , pro∑∑
ducers, (pjJ ∗ ∈ RK )j∈J , and public ﬁrm, p0∗ ∈ RK , such that p∗K = p0∗ + i∈I piI ∗ + j∈J pjJ ∗ ,income transfers to consumers, (d∗i )i∈I ∈ RI , and feasible consumption allocation, (x∗ ; zi∗ )i∈I under
∑
∗+ ∗
∗
∗
(x∗ ; y ∗ ) ∈ Y with y ∗ = j∈J∪{0} yj∗ , where (x∗ , x∗+
0 ; y0 ) ∈ Y0 and (x , xj ; yj ) ∈ Yj for each j ∈ J,
is called a Lindahl quasi-equilibrium with income transfers, (p∗K , p∗L , (piI ∗ )i∈I , (pjJ ∗ )j∈J , p0∗ , (d∗i )i∈I ), for
E = ((≿i , ωi )i∈I , Y0 , (Yj )j∈J , ((θij )j∈J )i∈I ), if the following conditions are satisﬁed.
j∗
j∗
−
+
∗
∗
∗
∗
∗
(i) Proﬁt maximization for private ﬁrms: p∗K · x∗+
j − p J · x + p L · y j ≧ p K · x j − p J · x j + pL · y j+∗
for all (x−j , xj ; yj ) ∈ Yj for each j ∈ J. We denote by πj the proﬁt of ﬁrm j (the value of the left

hand side of this inequality) for each j ∈ J.
+
−
0∗
∗
∗
∗
∗
0∗
∗
(ii) Proﬁt Maximality for the public ﬁrm: p∗K · x∗+
0 − p · x + pL · y0 ≧ pK · x0 − p · x0 + pL · y0 for+−+
all (x−
0 , x0 ; y0 ) ∈ Y0 such that (x0 , x0 ; y0 ) is associated with an allocation that is Pareto superior

to the status quo allocation.4 We denote by π0∗ the proﬁt of the public ﬁrm (the value of the left
hand side of this inequality).
(iii) Expenditure minimization for consumers: for each i ∈ I, (x∗ ; zi∗ ) satisﬁes piI ∗ ·x∗ +p∗L ·zi∗ ≦ p∗L ·∑ωi + j∈J θij πj∗ + d∗i and for all (x; zi ) such that (x; zi ) ≿i (x∗ ; zi∗ ), we have piI ∗ · x + p∗L · zi ≧ piI ∗ ·x∗ + p∗L · zi∗ .
The expenditure minimization implies the following utility maximization if we can redeﬁne d∗i so as to∑satisfy piI ∗ · x∗ + p∗L · zi∗ = p∗L · ωi + j∈J θij πj∗ + d∗i and the minimum wealth condition, inf{piI ∗ · x +∑p∗L · zi | (x; z) ∈ Xi } < p∗L · ωi + j∈J θij πj∗ + d∗i , is satisﬁed (Debreu 1959[p.69,(1)]).
(iv) Utility maximization for consumers: for each i ∈ I, (x∗ ; zi∗ ) is the ≿i -greatest element of∑{(x; zi ) ∈ RK∪L | piI ∗ · x + p∗L · zi ≦ p∗L · ωi + j∈J θij πj∗ + d∗i }.
4 The condition is automatically satisﬁed under the ordinary proﬁt maximization condition.

4

If conditions (i), (ii), and (iv) instead of (iii) are satisﬁed, the same list of prices and feasible allocations
for E is called a Lindahl equilibrium with income transfers. It follows that if there exists a Lindahl
quasi-equilibrium, the allocation and the Lindahl price system can also be identiﬁed with a Lindahl
equilibrium by redeﬁning each di ∈ R to satisfy the equation,∑piI ∗ · x∗ + p∗L · zi∗ = p∗L · ωi +
θij πj∗ + d∗i ,(3)j∈J

as long as the minimum wealth condition, inf{piI ∗ · x + p∗L · zi | (x; z) ∈ Xi } < p∗L · ωi +∑∗∗j∈J θij πj + di ,for every i ∈ I is satisﬁed.
Note that in this paper, we do not use assumptions like local non-satiation that necessarily makeequation (3) hold for each i in conditions (1) or (2). This is because several important examples, asseen in the Introduction, are provided along with the existence of satiation points of preferences in an
equilibrium state.5
If equation (3) holds for all i ∈ I, by incorporating the feasibility equation (2), we have the next
lemma.
Lemma 1: A feasible allocation (x∗ ; zi∗ )i∈I under (x∗ , y ∗ ) ∈ Y such that equation (3) for all i ∈ I holds
j∗0∗∗
relative to a certain Lindahl price system with income transfers, (p∗K , p∗L , (pi∗I )i∈I , (pJ )j∈J , p , (di )i∈I ),
∑
∑
∑
∑
∗
∗
∗
∗
∗
satisﬁes i∈I pi∗
I ·x +
j∈J∪{0} pL · yj =j∈J πj +i∈I di . (Without equation (3), ≦ holds.)
∑
∑
∗
∗
∗
∗
Proof: Summing up equation (3) of each i ∈ I, we have i∈I (pi∗
I · x + pL · z i ) =i∈I (pL · ωi +∑∑∗∗j∈J θij πj + di ). By considering the feasibility equation (2) together withi∈I θij = 1 for each j ∈ J,■

we obtain the result.
In Lemma 1, the left hand side amount,∑i∗∗i∈I pI · x +∑∗∗j∈J∪{0} pL · yj , can be identiﬁed with thesum of all proﬁts of private and public ﬁrms.
Lemma 2: For feasible allocation (x∗ ; zi∗ )i∈I under (x∗ , y ∗ ) ∈ Y , amount∑j∈J∪{0} pL · yj +∑ii∈I pI · x

relative to a certain Lindahl price system, (pK , pL , (piI )i∈I , (pjJ )j∈J , p0 ), is equal to the sum of all proﬁts∑
j+
0
of private and public ﬁrms, j∈J (pK · x+
j − pJ · x + pL · yj ) + (pK · x0 − p · x + pL · y0 ).
Proof: We may calculate the sum of proﬁts of private and public ﬁrms as follows.
∑j+0(pK · x+j − pJ · x + pL · yj ) + (pK · x0 − p · x + pL · y0 )
j∈J

=

∑

j
(pK · x+j − pJ · x + pL · y j ) + (

j∈J

=

∑

(pL · yj − pjJ · x) + (

j∈J

=

∑

j∈J∪{0}pL · yj +

∑

∑
i∈I

∑
i∈I

piI +

piI +

∑ j
∑
p J ) · x − pK ·x+
j + pL · y 0j∈J∑ jp J ) · x + pL · y 0

j∈J

j∈J

piI · xi∈I

5 Usually, assumptions such as local non-satiation are also used to show that condition (iv) means condition (iii). For
example, (iv) implies (iii) if preferences are convex and point (x∗ ; zi∗ ) is not a satiation point (Debreu 1959 [p.71,(2)]).
We also note that if preferences are strongly convex, every satiation point, if such an exists, is unique, so condition (iii) is
automatically satisﬁed.

5■In the above, note that the feasibility condition is not necessary.
Hence, by Lemma 2, Lemma 1 also means the following.

Lemma 3: A feasible allocation (x∗ ; zi∗ )i∈I under (x∗ , y ∗ ) ∈ Y such that equation (3) for all i ∈ I holds
j∗0∗∗
relative to a certain Lindahl price system with income transfers, (p∗K , p∗L , (pi∗I )i∈I , (pJ )j∈J , p , (di )i∈I ),∑satisﬁes π0∗ = i∈I d∗i . (Without equation (3), ≦ holds.)

Remark 1 (Wealth Transfer): We may identify p∗L · ωi +∑∗j∈J θij πj + di with the amountthat Samuelson (1969) proposed to call a lump-sum redistribution because he seems to assume
that ωi = 0 for all i. Foley (1970) also uses the same amount minus p∗L · ωi as a lump-sumtransfer, which he assumes to be 0 under the situation like Lemma 1 since he treats only
cone-shaped technologies with 0-proﬁts. A positive proﬁt case is treated by Mas-Colell andSilvestre (1989) under a (possibly) non-convex single (aggregate) technology.
Remark 2 (Dividend Equilibrium): Because of the existence of satiation and income
transfers together with the rigorous feasibility equation (2), we base our equilibrium concept
on a dividend equilibrium that is ﬁrstly deﬁned by Aumann and Drèze (1986) for pureexchange economies and extended by Mas-Colell (1992) to production economies. As Kajii(1996) discussed, such dividends or non-negative wealth transfers can also be reinterpreted
as money (see Murakami and Urai 2017). Urai et al. (2022) further extended the concept tothe problem of monetary equilibrium in overlapping generations economies.
Remark 3 (Non-convex Second Welfare Theorem): In this paper, we treat second
welfare theorem for economies including public goods, monetary transfers and non-convex
technologies. For second welfare problems in economies with non-convex technologies, see,e.g., Kamiya (1995).

Discussion 1 (Lindahl Equilibrium with Transfers and Production Core)
Example 1 in the Introduction also provides an important setting for a production core problem foreconomies including public goods. In the example, the government’s cost of producing the public goodis 1. Let such a cost be shared equally (half and half) between the worker and the capitalist. The
method is simple: d0 = −0.5 and d1 = 10 − 0.5 = 9.5. Furthermore, suppose that there is no satiation in
the consumers’ utility. Although the Lindahl equilibrium (worker consumes 0.5 and capitalist consumes
9.5 of the private good) is Pareto optimal, it is not in the core (obviously, blocked by the worker evenin the weakest condition that production technologies cannot be used for any deviation). However, it“apparently” satisﬁes Mas-Colell and Silvestre’s suﬃcient condition for core allocations, that is, “thecost-burden of each agent is non-negative.” Here, we need to rethink “each agent’s cost-burden” inlight of the diﬀerence between the settings in Mas-Colell and Silvestre (1989) and ours. In our settings,
public and private ﬁrms are distinguished, and the public good produced by the public ﬁrm only makes
6

sense as an input for the private ﬁrm. The proﬁt of the private ﬁrm, 10, is calculated, therefore, without
deducting the cost necessary for producing the public good. It can be said that the “true” proﬁt thatthe capitalist should have received is 9, so the fact that he receives a dividend of 9.5 means that “thecost-burden is (essentially) negative for the capitalist.” It follows that under an income transfer policy(tax system) that attributes the burden of public-good production to the wrong place, the Lindahl
equilibrium is no longer a core allocation.

Discussion 2 (Optimal Tax Policy)
Example 1 and the discussion in Remark 4 shows that in the general situation, where multiple ﬁrms
exist and public goods are used in production, the “true” cost burden of public-goods production isdiﬃcult to determine because of the existence of private proﬁts that free-ride on the action of the publicﬁrm. This also means that as long as such free-riding exists through corporate proﬁts, the goal ofbalanced budgets alone will lead to a non-core Lindahl income transfer equilibrium through a wrong
income distribution (wrong tax system and cost-bearing requirements). Conversely, however, we canconsider what the optimal tax system (cost-sharing for public goods) is by asking about the income
transfer conditions for the Lindahl equilibrium to be a core allocation. Of course, in this case, theoptimality of the tax system would depend on the deﬁnition of the production core, taking into accounta certain “fairness”. For example, if all production technologies can be used at the time of deviation,then a fair tax system would be based on the idea that “the knowledge of production technology itself
can be used by everyone. If, on the other hand, a portion of the production technology can be usedat the time of deviation in proportion to the shareholding ratio, then the fairness of the tax system
will take into account the meaning of “intellectual property and shareholding” in the private ownership
economy as much as possible. Furthermore, if we add the requirement to “ﬁx the behavior of the publicﬁrm” at the time of deviation, then the fair tax system is to be considered with the greatest emphasis
on “without free-riding on the current behavior of the public ﬁrm.” It follows that the problem of
optimal taxation is possible to be generalized as a question like the existence of a Lindahl equilibriumwith income transfers leading to a core allocation.

3

Fundamental Welfare Theorems for Lindahl Equilibria
Now we provide the ﬁrst fundamental theorem of welfare economics for Linhahl equilibria. (Condition(iii) must be added here in addition to condition (iv). See footnote 5.)
Theorem 1 (The First Fundamental Theorem): A Lindahl equilibrium allocation, (x∗ ; zi∗ )i∈I , under asystem of equilibrium Lindahl price vectors and income transfers, (p∗K , p∗L , (piI ∗ )i∈I , (pjJ ∗ )j∈J , p0∗ , (d∗i )i∈I ),is Pareto optimal if condition (iii) is satisﬁed.
Proof: Assume the contrary. Then we have a feasible allocation, (x′ ; zi′ )i∈I , under (x′ ; y ′ ) ∈ Y suchthat (x′ ; zi′ ) i (x∗ ; zi∗ ) at least one i ∈ I and (x′ ; zi′ ) ≿i (x∗ ; zi∗ ) for all i ∈ I. By considering the utility
7maximization condition (iv) and the expenditure minimization condition (iii), we have the following
inequality.

∑(piI ∗ · x′ + p∗L · zi′ ) >i∈I∑(piI ∗ · x∗ + p∗L · zi∗ ).

(4)

i∈I

∑
∑
∑
Feasibility condition (2) means that the left hand side is equal to i∈I piI ∗ · x′ + p∗L · i∈I zi′ = i∈I piI ∗ ·∑x′ + p∗L · (y ′ + i∈I ωi ). On the other hand, by the same equation, the right hand side is equal to
∑
∑
∑
∑
i∗
∗
∗
∗
i∗
∗
∗
∗
i∈I pI · x + pL ·i∈I zi =i∈I pI · x + pL · (y +i∈I ωi ). It follows that we have,
∑
∑
∑
∑
(5)
piI ∗ · x′ + p∗L · y ′ + p∗L ·ωi >piI ∗ · x∗ + p∗L · y ∗ + p∗L ·
ωi ,
i∈I

and

i∈I

∑

i∈I

piI ∗ · x′ + p∗L · y ′ >

i∈I

∑

i∈I

piI ∗ · x∗ + p∗L · y ∗ .

(6)

i∈I

By Lemma 2, the both sides of the above inequality are equal to the sum of all proﬁts of private andpublic ﬁrms, which contradicts to the proﬁt maximization and maximality conditions (i) and (ii) of■

Lindahl equilibria.

Next we show the second welfare theorem for Lindahl equilibria. The second theorem is given in theform of quasi-equilibrium as in Debreu (1959; p.95,6.4). As stated before, a quasi-equilibrium is anequilibrium as long as the minimum wealth condition is satisﬁed.
Theorem 2 (The Second Fundamental Theorem): Every Pareto optimal allocation, (x∗ ; zi∗ )i∈I , feasible()∑
under (x∗ ; y ∗ ) ∈ Y with y ∗ = y0∗ + j∈J yj∗ for E = (≿i , ωi )i∈I , Y0 , (Yj )j∈J , ((θij )j∈J )i∈I , where each
Yj , j ∈ J, is convex and there exists at least one i′ such that (x∗ ; zi′ ) is not satiated, is a Lindahl
quasi-equilibirum under price vectors and income transfers, (p∗K , p∗L , (piI ∗ )i∈I , (pjJ ∗ )j∈J , p0∗ , (d∗i )i∈I ).
Proof: As Milleron (1972), we identify in this proof the subspace, RK , of commodity space RK∪L with(♯I + ♯J + 1)-times replication of RK in order to describe input vectors of public goods for consumersi ∈ I, private ﬁrms (Yj )j∈J , and public ﬁrm Y0 . The identiﬁcation enables us to treat public goods
as if they were ordinary private goods at least from the viewpoint of all decision makers. Let (Ki )i∈I ,(Kj )j∈J , and K0 be disjoint family of (♯I + ♯J + 1)-times replication of index set K for public goods,
and consider the following extended commodity space E of RK∪L .
∏∏E=
R Ki ×R Kj × R K0 × R L .
i∈I

(7)

j∈J

Let us identify Y0 with subset Ŷ0 of E through the next canonical identiﬁcation mapping:
∏
∏
+
+
+
+
−
R Kj × R K0 × R L .
h0 : Y0 3 (x−R Ki ×0 , x0 , y0 ) 7→ (x0 , ..., x0 , x0 − x0 ; y0 ) ∈ Ŷ0 ⊂

(8)

j∈J

i∈I

In the same way, identify Yj for each j ∈ J with Ŷj by
+
+
+
+
−
+
+
hj : Yj 3 (x−j , xj , yj ) 7→ (xj , ..., xj , xj − xj , xj , ..., xj ; y0 ) ∈ Ŷj ⊂

∏
i∈I

8

R Ki ×∏j∈J

RKj × RK0 × RL , (9)−Kjwhere the entry for x+. Note that for each j ∈ J, thej − xj is the coordinate corresponding to R

convexity of Yj clearly implies the convexity of Ŷj . Moreover, identify Xi for each i ∈ I with its image
through the next mapping,hi : Xi 3 (xi ; zi ) 7→ (0, ..., 0, xi , 0, ..., 0; zi ) ∈ hi (Xi ) ⊂ E + =

∏

Ki
R+
×

i∈I

∏

K

K0
L
R + j × R+
× R+
,

(10)

j∈J

Ki
. We deﬁne X̂i as the following canonicalwhere the entry for xi is the coordinate corresponding to R+

extension of the range, hi (Xi ), to the open subset of E + , such that
K
K
K
K
K
X̂i = hi (Xi ) + R+
× · · · × R+
× {0} × R+
× · · · × R+
× R+
× {0},(11)

Ki
L
where the two entries of {0} are representing for those of R+
and R+. Each preference, ≿i , can naturally
ˆ on X̂i by deﬁning it as the relation depending only on the range of hi . Then, thebe extended to ≿i

ˆ , X̂ hi (xi ;zi ) = {w ∈ X̂i | w≿
ˆ hi (xi ; zi )}, is nothing but the above
upper contour set at hi (xi ; zi ) for ≿iii(x ;z )

canonical extension of image hi (Xi i i ) of the upper contour set at (xi ; zi ) for ≿i .
∑∑Note that for each point (x∗ ; y ∗ ) ∈ Y such that x∗ = j∈J∪{0} x∗+
and y ∗ = j∈J∪{0} yj∗ , wherej∗K∪L(x∗ , x∗+with its image in E under the mapping,j ; yj ) ∈ Yj for each j ∈ J ∪ {0}, if we identify Y ⊂ R∏∏
h : Y 3 (x∗ ; y ∗ ) 7→ (x∗ , ..., x∗ , 0, ..., 0; y ∗ ) ∈ h(Y ) ⊂(12)
R Ki ×R Kj × R K0 × R L ,
i∈I

j∈J

∏

where the entries for x∗ are coordinates corresponding to i∈I RKi , the point in h(Y ) can be represented
as the sum of the canonically identiﬁed points of Ŷj , j ∈ J ∪ {0}, like
∑
∗+
∗+
∗−
∗+
∗+ ∗
(x∗ , ..., x∗ , 0, ..., 0; y ∗ ) =
(x∗+
(13)
j , ..., xj , xj − xj , xj , ..., xj ; yj ),j∈J∪{0}where the entry for x∗+− x∗−
for each j ∈ J ∪ {0} is the coordinate corresponding to RKj . Then, byj∑jdeﬁning Ŷ as Ŷ = j∈J∪{0} Ŷj , we may identify Y with a subset of Ŷ canonically through h as∑∑∗+ ∗Ŷj = Ŷ .
h(Y ) 3 h(x∗ ; y ∗ ) =hj (x∗−(14)
j , xj , yj ) ∈j∈J∪{0}j∈J∪{0}Moreover, note that a list of Lindahl prices, ((piI ∗ )i∈I , (pjJ ∗ )j∈J , p0∗ , p∗L ), can be identiﬁed with a+price system over E. For each j ∈ J ∪ {0}, the value of action (x−j , xj , yj ) ∈ Yj of ﬁrm j under thej∗−∗Lindahl price system, p∗K · x+
j − pJ · xj + pL · yj , can be represented as
j∗
−
+
−
+
0∗ ∗
((pi∗
(15)
I )i∈I , (pJ )j∈J , p , pL ) · hj (xj , xj , yj ), for hj (xj , xj , yj ) ∈ Ŷj ,
∑
∑
j∗
0∗
because Lindahl price system always satisﬁes p∗K = i∈I pi∗
I +j∈J pJ + p . For each i ∈ I, the value∗of action (x; zi ) ∈ Xi , pi∗
I · x − pL · zi , can also be represented as
j∗
0∗ ∗
((pi∗
I )i∈I , (pJ )j∈J , p , pL ) · hi (x, zi ) for hi (x, zi ) ∈ X̂i .

(16)

Hence, conditions (i),(ii) and (iii) of Lindahl equilibrium can be assured through the extended actionsj∗0∗∗
in Ŷj and X̂i for each j ∈ J ∪ {0} and i ∈ I with the price ((pi∗I )i∈I , (pJ )j∈J , p , pL ) over the extended

commodity space, E.
9Now we can prove the second fundamental theorem. With respect to the allocation, (x∗ ; zi∗ )i∈I ,()∑
feasible under (x∗ ; y ∗ ) ∈ Y with y ∗ = y0∗ + j∈J yj∗ for E = (≿i , ωi )i∈I , Y0 , (Yj )j∈J , ((θij )j∈J )i∈I , we∗+ ∗suppose, at ﬁrst, that the action of public ﬁrm, (x∗−0 , x0 ; y0 ), is the only element of Y0 and transform∗∗+ ∗∗economy E on RK∪L into economy Ê on E. Deﬁne Y0∗ as Y0∗ = {(x∗−0 , x0 ; y0 )} and Yj as Yj for each()∗∗
j ∈ J, and consider economy E = (≿i , ωi )i∈I , Y0∗ , (Yj∗ )j∈J , ((θij )j∈J )i∈I and its transformation, Ê()ˆ , hi (ωi ))i∈I , Ŷ ∗ , (Ŷ ∗ )j∈J , ((θij )j∈J )i∈I . Even though Y ∗ is convex, it is not unclear that the
= (≿
i

0

0

j

set,
Y ∗ = {(x; y) | x =∑j∈J∪{0}is convex. However, set Ŷ ∗ =x+
j ,y =∑∗
yj , (x, x+
j ; yj ) ∈ Yj for all j ∈ J ∪ {0}},(17)j∈J∪{0}∑∗∗j∈J∪{0} Ŷj ⊃ h(Y ) is obviously convex as the sum of convex sets in E.
h (x∗ ;zi∗ )Hence, as in Debreu (1959)[Ch.6, 6.4], by considering the class of upper contour sets, X̂i i, i ∈ I,j∗i∗
0∗
by incorporating feasibility equation (2), we obtain a supporting price, ((pI )i∈I , (pJ )j∈J , p , p∗L ), at∑∑h (x∗ ;zi∗ )0 of the convex set, −Ŷ ∗ − i∈I hi (ωi ) + i∈I X̂i i
in E.
j∗
∗
It is clear under price represented by (pj )j∈J and pL , condition (i) If Lindahl equilibrium is satisﬁed
for all j ∈ J since every Ŷj∗ is convex. For condition (ii), note that the status quo allocation is Pareto

optimal, so there is no allocation that is Pareto superior to it, hence the condition is automatically∗∗∗∗satisﬁed. For condition (iii), take d∗i ∈ R appropriately so as to satisfy pi∗
I · x + p L · zi = p L · ω i +∑∗∗■j∈J θij πj + di for every i ∈ I.

4

Conclusion
In this paper, fundamental theorems of welfare economics for economies with public goods are given,

including cases in which a Pareto-optimal Lindahl equilibrium allocation is possible only through aninevitable government budget deﬁcit. In particular, (1) medical cares as merit goods give concrete
signiﬁcance to the fact that the fundamental theorems of welfare economics may be asserted on thebasis of Lindahl equilibrium. (2) The possibility of deﬁcit ﬁnancing is a general equilibrium theoretical
conﬁrmation that the private sector does not necessarily share all the costs of public goods, and thus
what is unnecessary tax collection. (3) With regard to tax collection, our results show the necessity ofﬁnancing that cannot be realized by the private price in Lindahl equilibria (the undistributed proﬁts
means the necessity of lump-sum taxes for private ﬁrms). Furthermore, (4) in addition, the activity
criteria of the public ﬁrm is given as a kind of maximality of proﬁt, i.e., further Pareto improvement
actions inevitably lead to lower proﬁts. (5) Compared to the cost-sharing equilibrium in Mas-Colell
and Silvestre (1989), here the question of “what is true cost?” is re-examined through the separationof public and private ﬁrms. Savings in the private sector are the purchase of tomorrow s ﬁat money,
and it is appropriate from the standpoint of general equilibrium theory to treat the amount of suchsavings as value creation by the state. Thus, in an economy with the concept of saturation or savings(i.e., capital accumulation), this amount must be subtracted from the cost of public good production aslong as we follow the spirit of the Second Fundamental Theorem of Welfare Economics.

10

REFERENCES
Aumann, R. J. and Drèze, J. H. (1986): “Values of Markets with Satiation or Fixed Prices,” Econometrica 53, 1271–1318.
Debreu, G. (1959): Theory of Value. Yale University Press, New Haven, CT.
Foley, D. K. (1970): “Lindahl’s Solution and the Core of an Economy with Public Goods,” Econometrica
38(1), 66–72.
Kajii, A. (1996): “How to discard non-satiation and free-disposal with paper money,” Journal of Mathematical Economics 25, 75–84.
Kamiya, K. (1995): “Optimal Public Utility Pricing: A General Equilibrium Analysis,” Journal of
Economic Theory 66(2), 548–572.
Mas-Colell, A. (1992): “Equilibrium theory with possibly satiated preferences,” in Equilibrium and dynamics: Essays in honor of David Gale., (Majumdar, M. ed) , Chapter 9, pp. 201–213, Macmillan,
London.
Mas-Colell, A. and Silvestre, J. (1989): “Cost Share Equilibria: A Lindahlian Approach,” Journal of
Economic Theory 47, 239–256.
Milleron, J.-C. (1972): “Theory of Value with Public Goods: A Survey Article,” Journal of Economic
Theory 5, 419–477.
Murakami, H. and Urai, K. (2017): “Replica core limit theorem for economies with satiation,” Economic
Theory Bulletin 5(2), 259–270.
Urai, K., Murakami, H., and Moriya, T. (2022): “Existence of Monetary Equilibrium for Overlapping
Generations Economies with Satiated Agents,” Discussion Paper No. 22-03, Graduate School of
Economics, Osaka University.

Appendix A
Toward what we described in Example 1 in a static framework, relying for simplicity on ‘preference
saturation’, we would like to add here a supplementary discussion on some dynamic considerations.
As mentioned in the explanation of the basic example, the budgetary surplus of a preference-saturated
agent in a static dividend equilibrium can be considered as savings, i.e., a loan to the government, but
at the same time it must be the ﬁnal or eternal savings (undistributed proﬁt) under Walras’ law. It is
the purpose of this appendix to show that such savings can exist even without the preference saturation.
If we allow for ‘undistributed proﬁts’, directly to the ﬁrms, it is not diﬃcult to consider such eternal savings (linked to government budget deﬁcits) in a dynamic framework (think, for example, ﬁrms’
eternally undistributed internal deposits).6 In Example 2, we present a dynamic framework in which
6 Here, we note that the coexistence of undistributed proﬁts with optimality is an interesting feature of the problem concerned when contrasted with the transversality condition in optimal programming theorems for production with discountfactors.

11

similar matters arise without having to consider things like ’undistributed proﬁts’. Under the overlapping generations (consumption loan) framework, the problem can be represented by the seigniorage inSamuelson-type savings economies. In the following, we will restate it as a model including production,
closer to the basic example in the introduction.
(Example 2): Let us consider a two-period overlapping generations economy such that foreach period t = 1, 2, ..., there are two commodities (a public good and a private good) andan agent t who lives in periods t and t + 1 and has a well-behaved technology Yt that can
produce 2 units of private good from one unit of public good in period t under supporting
price (1, 0) at action (private, public) = (2, −1). Suppose that public-ﬁrm technology Y0 issuch that by using one unit of private good in period 1, it can provide one unit of public
good for each period t = 1, 2, . . . (or does nothing). Moreover, suppose that each agent t has
no initial endowment and has a utility function depending only on the amounts of private
2
goods in R+
as u(xy , xo ) = xy + xo , where superscripts denote young = y or old = o period

for private goods.
Public Good(−1, 1)u(−1, 1) + Ytu

−1 0

Private Good

1

Figure 2: Y0 = {(0, 0), (−1, 1)} and (−1, 1) + Yt
In this case, we also have a trivial (unique) public ﬁrm action that is Pareto optimal, “using one unitof private good in period 1 and provide 1 unit of public good for every periods.” If otherwise, we have(0, 0), (0, 0), ..., as the only feasible allocation. As long as the public ﬁrm chooses the action, we havea (unique) Pareto optimal (Lindahl) general equilibrium allocation, (1, 1), (1, 1), ..., under price system,(0, 0, ...) for public goods and (1, 1, ...) for private goods with dividend dt = 2 for each t = 1, 2, ... thatis equal to the proﬁt for technology Yt under price (private, public) = (1, 0). For this Lindahl generalequilibrium, the proﬁt of public ﬁrm is −1 = (−1) · 1 + 1 · 0 + 1 · 0 + · · ·, so that the unique Pareto
optimal allocation can only be achieved with an inevitable government budget deﬁcits.

12