The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No.2012-11(accepted in March 22, 2013)

Welfare Analysis of Fiscal Policies
in a Fixed Price Overlapping Generations Model

Jumpei Tanaka1
The University of Kitakyusyu

Abstract
The purpose of this paper is to study the effects of fiscal policies such as a wastefulpublic spending and both inter-generational and intra-generational income transfers in
a Keynesian under-employment economy. Using a non-Walrasian fixed priceoverlapping generations model with different groups of households concerning theconsumption propensity, we present detailed analyses of the welfare effects of the above
policies under both balanced and loan budgets. Our analyses can be regarded as a
general equilibrium extension of Ono(2011) which reconsidered the effectiveness of suchpolicies in the simple textbook multiplier model.
JEL Classification: D50, E12, E13, E60
Key word: fiscal policy, income transfer, public debt, overlapping generations,
non-Walrasian, disequilibrium, fixed price, Keynesian economics
1

E-mail: j-tanaka@kitakyu-u.ac.jp
11. Introduction
The textbook Keynesian multiplier analysis states that a fiscal expansion policy canraise GDP and therefore is effective in an under-employment economy, even if it iswasteful in the sense that it contributes to neither utilities of households nor
productivities of firms. However, since an increase of GDP does not necessarily mean
improving utilities of households, the correctness of the above statement needs to bechecked carefully. Recently, Ono(2011) reconsidered the effectiveness of fiscal policies
such as a wasteful public spending and an income transfer under the Keynesian
multiplier framework, and showed the followings.
(Ⅰ) An increase in a wasteful public spending under a balanced budget raises GDP, but
it is not effective because the household’s consumption does not change. Namely, it isequivalent to an income transfer between households with the same propensity toconsume.
(Ⅱ) An increase in a wasteful public spending under a loan budget raises both GDP and
the household’s consumption in the presence of fiscal illusion. However, in the absenceof fiscal illusion the policy effect of it is equivalent to that under a balanced budget.
(Ⅲ) An income transfer from households with higher propensity to consume to thosewith lower one reduces GDP. Furthermore, the balanced-budget multiplier can be lessthan 1 if the government increases a public spending by imposing tax on householdswith higher propensity to consume.
Ono’s analysis is based on the simple textbook multiplier model and therefore it isvery brief. Since the non-Walrasian fixed price model, which was pioneered by
Clower(1967) and Barro-Grossman(1971, 1976) and was developed by Malinvaud(1977),
Benassy(1986, 2002) and so on, provides a proper general equilibrium basis of the2multiplier analysis, we can expect that the welfare effect of Keynesian fiscal policies canbeexamined more precisely by using it. In fact, Benassy(1986, ch3) has alreadydiscussed such a question and has shown (Ⅰ) and the first half of (Ⅱ) of Ono’s
conclusion above. In his analysis, however, the effect of income transfers between
different types of households is not considered. Furthermore, since the discussions ofboth Ono and Benassy are based on a static framework, it is not necessarily clear thatwhat effect such fiscal policies have on the welfare of the next generation. So, in thispaper we investigate these issues in detail by using a non-Walrasian overlapping
generations model with two different groups of households concerning the propensity toconsume2.
We first consider “Case 1” where at period t the government imposes a lump-sum
tax on each group in generation t (young generation at period t ), and conducts thefollowing three kinds of policies under a balanced budget: (a) a wasteful public spending,(b) an inter-generational income transfer from generation t to generation t  1(namely, from the young to the old), (c) an intra-generational income transfer between
different groups of households in generation t , and demonstrate the followings.
(ⅰ) A spending policy (policy (a)) raises GDP. The (balanced-budget) multiplier is 1, but
it becomes less than 1 when the government imposes a heavier tax on the group withhigher propensity to consume. The consumption of each group in each generation does
not change (namely, a policy (A) is neutral from the viewpoint of welfare). Furthermore,
the same result holds even when the government implements a spending by borrowing
from generation t and repays at the next period by imposing a lump-sum tax to

To my knowledge, Rankin(1986) is the first to present a non-Walrasian OG model. He
examined in the model the effects of a permanent increase in public debt stock on
capital accumulation and the welfare of future generations in the new steady state.
2

3

generation t again (namely, the Ricardian neutrality holds true).
(ⅱ) An inter-generational income transfer from the young to the old (policy (b)) raises
GDP by the same size as a policy (a). The consumption of each group in generation t
does not change, but that in generation t  1 increases, which means that it is a
Pareto-improving policy.
(ⅲ) Concerning an intra-generational income transfer (policy (c)), if it is implemented
from households with lower (resp. higher) propensity to consume to that with higher(resp. lower) one, then GDP rises (resp. falls). The consumption of the group who
receives (resp. does not receive) a transfer necessarily increases (resp. decreases). The
consumption of each group in generation t  1 remains unchanged.
Among these results, the result (ⅰ) has already been shown by Ono and Benassy, but
the results (ⅱ) and (ⅲ) have not yet and they can be interpreted as an extended versionof Ono’s result (Ⅲ).
We next investigate “Case 2” where the government conducts each policy at period t
by issuing public debt (namely, under a loan budget) and repays it at period t  1 by
imposing a lump-sum tax on each group in generation t  1 , and show the followings.
(ⅳ) At period t both GDP and the consumption of each group in each generation inCase 2 become larger than in Case 1 in all three policy cases. (This implies that a public
spending under a loan budget can raise the consumption at period t even in theabsence of fiscal illusion.)(ⅴ) At period t  1 GDP becomes larger in Case 2 than in Case 1 in all three policy
cases. The consumption of each group in generation t  1 does not change between twocases, despite of the fact that the government shifts a tax burden to each group in
generation t  1 in Case 2 while it does not in Case 1.

4

Accordingly, in all three policy cases the government can improve the welfare in thesense of Pareto by shifting a tax burden to the next generation. This is because such ashift is essentially equivalent to implementing an income transfer from the young to theold at period t  1 and such an inter-generational transfer can improve theconsumption of the young without harming that of the old, as shown in the result (ⅱ) ofCase 1. This result means that under a non-Walrasian fixed price model there is noburden of public debt, which is investigated in more general setups by Tanaka(2008)
and Ogawa and Ono(2010)3.
The organization of the rest of this paper is as follows. In section 2, before proceeding
to the policy analysis, we present the basic structure of the “benchmark case” where thegovernment activity is omitted. In section 3 and section 4 we examine “Case 1” and“Case 2” mentioned above, and prove the results (ⅰ)～(ⅴ) rigorously. Finally, in section
5 we conclude the paper.

2.

The Benchmark CaseIn this section we present the basic structure of the benchmark case without the
government sector before proceeding to the policy analysis. We consider a discrete-timeclosed-economy overlapping-generations model where each generation lives for two
periods (young and old). Each generation consists of two different groups (the group A
and the group B ) of households who are identical except for the form of utility function(in other words, the propensity to consume). Households of the group A (resp. theEspecially, Ogawa and Ono (2010) shows the result under a fairly general frameworkof Rankin(1986)’s type where the household has a money in the utility (MIU) function
and the firm makes an investment decision.
3

5

group B ) exist in a continuum at the interval [0, h] (resp. the interval [h,1] ) where 0＜ h ＜1, so the total population size of each generation is 1 and constant over time. It is
assumed that there is no capital good in this economy and the only way of saving is tohold a fiat money, so young households spend a part of their income to purchasing
money for the purpose of saving and then they sell it at their old period.
Since the purpose of this paper is to investigate the welfare effects of fiscal policies in
a Keynesian under-employment economy, we adopt a non-Walrasian fixed price model
where both price and wage are exogenously fixed and accordingly the “equilibrium” ofthe (output) market is achieved by quantity adjustment. We especially focus on theregime of so called “Keynesian unemployment”, where both the output and labormarkets face excess supplies and therefore young households (resp. firms) supply thelabor (resp. the output) which quantity is equal to the firm’s labor demand (resp. thehousehold’s aggregate output demand).
In what follows we formulate the behavior of each agent in detail, and then derive the
non-Walrasian equilibrium.

Households of generation t
At the beginning of period t each household of generation t is endowed with L
units of labor. It is assumed, however, that because the fixed real wage is high andaccordingly the firm’s labor demand ( Lt ) is smaller than L at the wage level, eachddhousehold is forced to supply Lt units of labor (in other words, each household faces“demand constraint” in the labor market).
The utility function of the groupj ( A, B) ’s household is denoted by U t , j ＝

6

U j (Cty, j , Cto1, j ) , where C ty, j (resp. Cto1, j ) is the young (resp. old) period consumption of
the group j at period t (resp. t  1 ). For simplicity of analysis the disutility of labor
supply is not considered4. We assume that U t , j is homothetic and satisfies the usual
list of conditions. Each household of the group j maximizes U t , j under the following
budget constraint:(1)

Pt Cty, j ＋ M td, j ＝ Wt Ldt ＋  t , Pt 1Cto1, j ＝ M td, j ,where M t , j is the nominal money demand, Pt is the nominal price, Wt is thednominal wage, and  t is the nominal profit. For simplicity, we assume throughout the
paper that both nominal price and wage are fixed at a constant level.
(2)

Pt ＝ P , Wt ＝ W

So the budget constraint (1) can be rewritten as

C ty, j ＋ mtd, j ＝ wLdt ＋  t , Cto1, j ＝ mtd, j ,ddwhere mt , j  M t , j / P , w  W / P and  t   t / P . From the assumption of thehomothetic utility function the optimal consumption and saving plans of the groupj ( A, B) ’s household can be derived as(3)

C ty, j ＝  j [ wLdt ＋  t ]mtd, j ＝ (1   j ) [ wLdt ＋  t ],where we assume 0＜  j ＜1.

Households of generation t  1
At the beginning of period t households of the group j ( A, B) in generation t  1
Since we focus on an under-employment situation where households works less thanthey want to do, such an assumption is not problematic.
4

7

have mt 1, j units of money which was stored in their young period, and they spend all
dof them to their old period consumptions. So their behaviors at period t can be
expressed by(4)

C to, j ＝ mtd1, j .

Note that the following relationship holds:(5)hmtd1, A ＋ (1  h)mtd1, B  m ,where m denotes the exogenous real money stock, which is assumed to be fixed at aconstant level.

Firms
There are many identical firms and the number of them is normalized to 1. Each firm
has the usual concave production function Yt ＝ F ( Lt ) , where Yt is the output and Lt
is the labor input. Since the real profit of each firm is defined as  t  F ( Lt ) － wLt , thefirm’s optimal labor demand and output supply would be(6)F ( Lt ) ＝ w →Lt ＝ F  1 (w) and Yt ＝ F [ F  1 ( w)] ,if the optimal output supply Yt was smaller than the aggregate output demand Yt :d

(7)

Yt d ＝[ hCty, A ＋ (1  h)Cty, B ]＋[ hCto, A ＋ (1  h)Cto, B ].

We assume, however, that the firm faces output demand constraint (namely, Yt ＜ Yt )ddue to a high fixed real wage and therefore each firm is forced to supply Yt
output. So, the actual labor demand and real profit are(8)

Yt d ＝ F ( Ldt )→Ldt ＝ Ldt (Yt d ) ,  t ＝ F ( Ldt ) － wLdt .

8dunits ofNon-Walrasian market equilibrium
Substituting the third equation in (8) into the first equation of (3), the consumption of
the group j ( A, B) of generation t can be rewritten as(9)

C ty, j ＝  j Yt d .

From (4) and (5) the aggregate consumption demand of generation t  1 is(10)

hCto, A ＋ (1  h)Cto, B ＝ m

Thus, by substituting (9) and (10) into (8), we can derive the “equilibrium” output in thebenchmark case as(11)

Yt bench ＝ m /  .

（   h(1   A ) ＋ (1  h)(1   B ) , 0＜  ＜1）

Here,  can be interpreted as the saving rate at a macro-level. From (11) we canconfirm the followings.
・ An increase in the exogenous real money stock m raises the equilibrium output, andthe multiplier ( dYtbench/ dm ) is 1 /  (＞1)

・ An increase in the saving rate  reduces the equilibrium output (namely, “theparadox of thrift” holds).
The young and old period equilibrium consumptions of the group j ( A, B) ofgeneration t are(12)(Cty, j ) bench ＝  j m /  ,(Cto1, j ) bench （＝ (mtd, j ) bench ）＝ (1   j )m /  .

The old period consumption of the group j ( A, B) in generation t  1 , on the other
hand, is given by(13)(Cto, j ) bench ＝ (1   j )m /  ,9because the same equilibrium state is reproduced at every period in the benchmark
case5 and therefore (Ct , j )
o

bench

o

＝ (Ct 1, j )benchholds true.

(Remark)
The regime of “Keynesian unemployment” discussed above holds under the conditionsdd
1of Lt ＜ L in the labor market and Yt ＜ Yt ＝ F [ F  ( w)] in the output market. Since

Ldt ＜ L can be rewritten as F ( Ldt ) ＝ Yt d ＜ F (L) and Yt d is equal to m /  inequilibrium, the range of exogenous price ( P ) and wage ( W ) under which such ascheme is realized is2.

 1  W MM
＜ F (L ) and＜ F  F    .
PP
 P 
Case 1In this section we introduce the government sector into the benchmark case of theprevious section and investigate the welfare effects of various fiscal policies. Moreconcretely, in this section we consider “Case 1” where at period t the government
imposes a lump-sum tax on each group in generation t and conducts the following
three kinds of policies under a balanced budget: (a) a wasteful public spending, (b) aninter-generational income transfer from generation t to generation t  1 (namely,
from the young to the old), (c) an intra-generational income transfer between different
groups in generation t . The main conclusions of this section can be summarized asfollows.

This is because the real money stock (the only stock variable in our model) is constantover time.
5

10

(ⅰ) A spending policy (policy (a)) raises GDP, but the consumption of each group in eachgeneration does not change, which means that it is neutral from the viewpoint ofwelfare.
(ⅱ) An inter-generational income transfer (policy (b)) raises GDP by the same size as apolicy (a). The consumption of each group in generation t does not change, but that ingeneration t  1 rises, which means that it is a Pareto-improving policy.
(ⅲ) Concerning an intra-generational income transfer (policy (c)), if it is implemented
from the group with lower (resp. higher) propensity to consume to that with higher (resp.
lower) one, then GDP rises (resp. falls). The consumption of the group who receives(resp. does not receive) a transfer necessarily increases (resp. decreases). The
consumption of each group in generation t  1 remains unchanged.

The government
At period t the government imposes a lump-sum tax  in real terms on each groupin generation t , so the total revenue is h  ＋ (1  h) ＝  . This revenue is used for (a) awasteful

publicspending,

or

(b)

an

inter-generationaltransfer,

or

(c)

an

intra-generational transfer.

Households of generation t
In Case 1 the problem of the group A ’s household in generation t is formulated as

max

Cty, A ,Cto1, A

U t,A

s.t.

Cty, A ＋ mtd, A ＝ wLdt ＋  t － ＋  1

h,Cto1, A ＝ mtd, A ,where  1 is a dummy variable which is equal to 1 when an intra-generational transfer(policy (c)) is implemented and 0 otherwise. (Notice that an intra-generational transfer
is implemented from the group B to A .) The optimal consumption and saving plans

11

are
(14)

Cty, A ＝  A [ wLdt ＋  t －  ＋  1


h

],

mtd, A ＝ (1   A ) [ wLdt ＋  t －  ＋  1


h

].

The problem of the group B , on the other hand, is given bymax

Cty, B ,Cto1, B

U t ,Bs.t.

Cty, B ＋ mtd, B ＝ wLdt ＋  t － , Cto1, B ＝ mtd, B ,so the optimal consumption and saving plans are(15)

Cty, B ＝  B [ wLdt ＋  t －  ],mtd, B ＝ (1   B ) [ wLdt ＋  t －  ].

Households of generation t  1
Each household in generation t  1 receives a transfer which size is  in real terms
if the government implements an inter-generational transfer (a policy (b)). She spends
all her incomes (the sum of an initial money holding and a transfer) to the old period
consumption, so the behavior of her can be expressed by(16)

C to, j ＝ mtd1, j ＋  2 ,( j  A, B )where  2 is a dummy variable which is equal to 1 when an inter-generational transfer
is implemented and 0 otherwise.

Firms
The behavior of the firm is the same as the benchmark case, so the actual labordemand and real profit are(17)

Yt d ＝ F ( Ldt )→Ldt ＝ Ldt (Yt d ) ,  t ＝ F ( Ldt ) － wLdt .

Non-Walrasian market equilibrium

12

In what follows we derive the non-Walrasian equilibrium in each case of three policies
in turn.

(a) The case of a wasteful public spending
In this case  1 ＝  2 ＝0 holds and the size of a public spending is  , so the aggregate
demand for output is given by(18)

Yt d ＝[ hCty, A ＋ (1  h)Cty, B ]＋[ hCto, A ＋ (1  h)Cto, B ]＋  .

Here, notice that a public spending  is newly added in the component of aggregate
output demand. From (14)～(17), the consumption of each group j  A, B in eachgeneration can be rewritten as(19)

C ty, j ＝  j [ Yt d －  ],C to, j ＝ mtd1, j . （where hmtd1, A ＋ (1  h)mtd1, B  m ）.

So the equilibrium output in Case 1 (a) is（20）

Yt1,a ＝m
＋  . （   h(1   A ) ＋ (1  h)(1   B ) ）

From (11) and (20), the equilibrium output in Case 1 (a) is larger just by the size of apublic spending than in the benchmark case, which means that the balanced budget
multiplier is equal to 1. The reason why Yt

1, a

＞ Yt

bench

holds is as follows. In Case 1 (a)the consumption demand of each group in generation t lowers due to an imposition oftax, but its extent is smaller than the size of tax (or public spending) itself because of
the inter-temporal consumption smoothing behavior (see (19)). As a result, theaggregate demand for output in Case 1 (a) is more stimulated than the benchmark caseand the equilibrium output becomes higher.
However, this result does not necessarily means that a wasteful public spending
policy is welfare-improving, because the welfare of each household depends not on GDP

13

but on the equilibrium consumption level. Substituting (20) into (19), we have(21)

(generation t )(Cty, j )1,a ＝  j m /  , (Cto1, j )1,a ＝ (1   j )m /  ,(generation t  1 )(Cto, j )1,a ＝ (1   j )m /  .

( j  A, B )From (12), (13) and (21) the consumption level of each group in each generation is thesame between Case 1 (a) and the benchmark case. It is especially notable that theconsumptions of generation t are the same despite the fact that generation t bears a
tax in Case 1 (a) while it does not in the benchmark case. The reason why it holds is thatin Case 1 (a) a tax burden is exactly offset by an increase of income caused by a rise in
the equilibrium output.
After period t  1 the equilibrium states are the same between two cases, because
the real money stock m (the only stock variable in our model) is constant and thegovernment plays no role after the period. We can therefore conclude that a wastefulpublic spending (policy (a)) under a balanced budget is neutral from the viewpoint ofwelfare. This result is plausible in a sense, because the increased output caused by apublic spending is used for a wasteful purpose and the amount of output utilized for
consumption remains unchanged between two cases.

(Remark 1)
We can easily show that the same result holds even if we consider the case where thegovernment implements a public spending by borrowing from each group in generationt and repays at the next period by imposing a lump-sum tax to generation t again.
That is to say, the timing of taxation (namely, whether it is imposed at the young or old
period) has no influence on the consequence as far as the tax burden is attributed to an

14

identical household, which means that the Ricardian Neutrality holds in our model.

(Remark 2)
We can obtain the different result from Case 1 (a) in the case where the government
imposes the different size of a lump-sum tax on each group in generation t . Consider
the situation where the government imposes  A (resp.  B ) units of tax on the groupA (resp. B ) in generation t while the total amount of tax revenue is fixed at thesame level as Case 1 (a) (namely, h A ＋ (1  h) B =  ). In this situation the equilibrium
output can be calculated as

Yt ＝m   [h A A  (1  h) B B ]＋. (   h(1   A ) ＋ (1  h)(1   B ) )

Notice that this is reduced to (20) if  A ＝  B ＝  holds. The balanced budget multiplierwhen the additional spending is financed from the group A only is

1 A
dYt
＝
.

d d B 0This multiplier becomes less than 1 if the propensity to consume of the group A islarger than the group B (namely,  A ＞  B ), as was pointed out by Ono(2011). This
result shows that the balanced budget multiplier becomes lower when the additional
spending is financed from the group with higher propensity to consume.
The equilibrium consumption of each group in generation t is1 A y m (1  h)(1   B )( B   A ) o, Ct 1, A ＝Cty, A ＝  A  
Ct , AA
1B y m h(1   A )( A   B ) o, Ct 1, B ＝Cty, B ＝  B  Ct , BB
From (21), both the young and old period consumptions of the group A (resp. B )decrease (resp. increases) by a public spending policy if the tax burden of the group A

15

is larger than the group B (namely,  A ＞  B ). Accordingly, the result in Case 1 (a) thata wasteful public spending is neutral from the viewpoint of welfare does not hold when
the different size of tax is imposed on each group in generation t .

(b)

The case of an inter-generational transfer

In this case  1 ＝0 and  2 ＝1 hold and the size of a public spending is zero, so theaggregate demand for output is

Yt d ＝[ hCty, A ＋ (1  h)Cty, B ]＋[ hCto, A ＋ (1  h)Cto, B ].

(22)

From (14)～(17), the consumption of each group j  A, B in each generation is(23)

C ty, j ＝  j [ Yt d －  ],C to, j ＝ mtd1, j ＋  . （where hmtd1, A ＋ (1  h)mtd1, B  m ）Accordingly, the equilibrium output in Case 1 (b) is(24)

Yt1,b ＝m
＋  . （   h(1   A ) ＋ (1  h)(1   B ) ）

From (20) and (24), the equilibrium output in Case 1 (b) is equal to that in Case 1 (a).
This is because in the case of an inter-generational transfer all of the transferredincomes to old generation are spent to consumption expenditure and therefore theaggregate demand level of this case is equivalent to Case 1 (a) where all of the tax
revenue is spent for government purchase.
Concerning the equilibrium consumption of each group j  A, B in each generation,
we have(25) (generation t )(generation t  1 )(Cty, j )1,b ＝  j m /  ,(Cto1, j )1,b ＝ (1   j )m / (Cto, j )1,b ＝ (1   j )

m
＋ .


From (21) and (25), both the young and old period consumptions of each group in16

generation t are the same between Case 1 (b) and Case 1 (a), but the consumption ofeach group in generation t  1 becomes higher in Case 1 (b) due to the transfer. Since
after period t  1 the equilibrium states are identical between two cases, we canconclude that Case 1 (b) is Pareto-superior to Case 1 (a). This result stems from the factthat in Case 1 (b) the tax revenue from generation t is utilized to raise theconsumption of generation t  1 while in Case 1 (a) it is used for a wasteful public
spending.

(Remark)
We can easily confirm that as far as the total amount of a transfer is fixed theequilibrium output (24) does not change even when the size of a transfer to each groupin generation t  1 is different. (For example, each household of the group A receives

 / h units of transfers while each household of the group B does not receive anytransfers). This is because the consumption propensity of both groups in generationt  1 is 1 and therefore the total size of consumption demand of generation t  1 does
not change.

(c)

The case of an intra-generational transfer

In this case  1 ＝1 and  2 ＝0 hold and the size of a public spending is zero, so theaggregate demand is(26)

Yt d ＝[ hCty, A ＋ (1  h)Cty, B ]＋[ hCto, A ＋ (1  h)Cto, B ].

From (14)～(17), the consumption of each group j  A, B in each generation is(27)

Cty, A ＝  A [ Yt d －  ＋ ], Cty, B ＝  B [ Yt d －  ], C to, j ＝ mtd1, j .
h17So, the equilibrium output in Case 1 (c) is(28)

Yt1,c ＝m (1  h)( A   B )＋.
（   h(1   A ) ＋ (1  h)(1   B ) ）From (11) and (28), in the case of  A ＞  B (namely, the group A ’s propensity toconsume is larger than the group B ’s), the equilibrium output in Case 1 (c) becomes
higher than in the benchmark case. This is because an income transfer from the groupwith lower propensity to consume to the group with higher one stimulates the aggregate
consumption demand. However, even in such a case the equilibrium output in Case 1 (c)is lower than Case 1 (a) or Case 1 (b), because in Case 1 (a) or Case 1 (b) all of the tax
revenue is spent for government purchase or old period consumption while in Case 1 (c)a part of the transferred tax revenue to the group A in generation t is saved.
Concerning the equilibrium consumptions of each group in each generation, we have(29)

(generation t  1 )(Cto, j )1,c ＝ (1   j )m

（ j  A, B ）(generation t )(group A )(group B )1 A m (1  h)(1   B ) o1,c, (Ct 1, A ) ＝(Cty, A )1,c ＝  A  (Cty, A )1,c

h
A


1B
 m (1   A ) o1,c, (Ct 1, B ) ＝(Cty, B )1,c ＝  B  (Cty, B )1,c .


B



From (12), (13) and (29), the consumption of each group in generation t  1 is equal
between Case 1 (c) and the benchmark case, and the young and old consumptions of thegroup A who receives a transfer (resp. the group B who does not receive it) is higher(resp. lower) in Case 1 (c) than in the benchmark case. Note that this result does notdepend on magnitude relation of each group’s consumption propensity. In the case of A ＜  B (resp.  A ＞  B ) the equilibrium output decreases (resp. increases) relative
to the benchmark case, but nevertheless the consumptions of the group A who
receives a transfer becomes higher (resp. the group B who does not receive it becomes18

lower) than the benchmark case. In other words, the size of a transfer is necessarily
larger than the variation of the equilibrium output (income) caused by a policy (c).

４．Case 2
In this section we consider “Case 2” where the government implements three kinds offiscal policies (a)～(c) at period t by borrowing from each group in generation t andrepays at the next period by imposing a lump-sum tax to each group in generation t  1(the next generation), and re-examine the welfare effect of each policy. The main
conclusions of this section can be summarized as follows.
(ⅳ) At period t both GDP and the consumption of each group in generation t becomelarger in Case 2 than in Case 1 in all three policy cases.
(ⅴ) At period t  1 GDP becomes larger in Case 2 than in Case 1 in all three policy
cases. The consumption of each group in generation t  1 is equivalent between twocases, despite of the fact that a tax burden is shifted to each group of generation t  1
in Case 2 while it is not in Case 1. This means that a shift of a tax burden to the next
generation is a Pareto-improving policy.

4.1

Equilibrium at period t

The government
The government issues  units of public debt in real terms (which is equal to the tax
revenue in Case 1) and sells them to each group in generation t . Denoting the amountof public debt purchased by the group A and B as bt , A and bt , B respectively, the19following holds:  ＝ hbt , A ＋ (1  h)bt , B . This revenue is utilized for (a) a wasteful public
spending, or (b) an inter-generational transfer, or (c) an intra-generational transfer.

Households in generation t
In Case 2 the problem of the group A ’s household in generation t ismax

Cty, A ,Cto1, A

U t,A

Cty, A ＋ mtd, A ＋ bt , A ＝ wLdt ＋  t ＋  1

s.t.


h

,

Cto1, A ＝ mtd, A ＋ bt , A .

Note that in Case 2 there are two ways of saving: money and public debt. Since the priceof output is assumed to be constant over time (see (2)) and therefore the return rate ofholding money is zero, then the return rate of public debt must be also zero by thearbitrage condition. The optimal consumption and saving plans of the group A are(30)

Cty, A ＝  A [ wLdt ＋  t ＋  1


h

],

mtd, A ＋ bt , A ＝ (1   A ) [ wLdt ＋  t ＋  1


h

].

The problem of the group B is given bymax

Cty, B ,Cto1, B

U t ,B

Cty, B ＋ mtd, B ＋ bt , B ＝ wLdt ＋  t , Cto1, B ＝ mtd, B ＋ bt , B ,s.t.

so the optimal consumption and saving plans are(31)

Cty, B ＝  B [ wLdt ＋  t ],mtd, B ＋ btd, B ＝ (1   B ) [ wLdt ＋  t ].

Households in generation t  1 and Firms
The behaviors of each group j  A, B in generation t  1 and firms are the same as
Case 1, so we have(32)

C to, j ＝ mtd1, j ＋  2 ,(33)

Yt d ＝ F ( Ldt )→Ldt ＝ Ldt (Yt d ) ,  t ＝ F ( Ldt ) － wLdt .

20

Non-Walrasian market equilibrium
In what follows we derive the non-Walrasian equilibrium in each case of three policies
in turn.

(a) The case of a wasteful public spending
In this case  1 ＝  2 ＝0 holds and the size of a public spending is  , so the aggregate
demand for output is given by (18), and from (30)～(33) the consumption of each groupj  A, B in each generation is(34)

C ty, j ＝  j Yt d , C to, j ＝ mtd1, j . （where hmtd1, A ＋ (1  h)mtd1, B  m ）Thus, the equilibrium output in Case 2 (a) can be calculated as(35)

Yt 2 , a ＝

m 
＋ .
 

（   h(1   A ) ＋ (1  h)(1   B ) ）From (20) and (35), the equilibrium output at period t in Case 2 (a) is larger than inCase 1 (a), which means that shifting a tax burden to the next generation stimulates theoutput. The reason is that in Case 2 (a) each group in generation t does not bear a tax
burden and therefore the consumption demand is larger than in Case 1 (a) (see (19) and(34)).
Concerning the equilibrium consumption of each group j  A, B in each generation,
we have(36)(generation t )1 jm  
(Cty, j ) 2,a ＝  j    , (Cto1, j ) 2,a ＝(Cty, j ) 2,aj  

(generation t  1 )(Cto, j ) 2,a ＝ (1   j )m.
From (21) and (36) the young and old period consumptions of each group in generationt are higher in Case 2 (a) than in Case 1 (a), while the consumption of each group in21

generation t  1 is the same between two cases.

(b)

The case of an inter-generational transfer

In this case  1 ＝0 and  2 ＝1 hold and the size of a public spending is zero, so theaggregate demand for output is given by (22), and from (30)～(33) the consumption ofeach group j  A, B in each generation is(37)

C ty, j ＝  j Yt d ,C to, j ＝ mtd1, j ＋  .

Hence, the equilibrium output in Case 2 (b) is(38)

Yt 2 ,b ＝m ＋ . （   h(1   A ) ＋ (1  h)(1   B ) ） The equilibrium output at period t in Case 2 (b) becomes larger than in Case 1 (b) (see(24) and (38)) by the same reason as in Case 2 (a) discussed above. The equilibrium
consumption of each group j  A, B in each generation is(39)(generation t )1 jm  
(Cty, j ) 2,b ＝  j    , (Cto1, j ) 2,b ＝(Cty, j ) 2,b ,  
j(generation t  1 )(Cto, j ) 2,b ＝ (1   j )m.
From (25) and (39) the consumptions in generation t are higher in Case 2 (b) than inCase 1 (b) while those in generation t  1 are the same between two cases, as in Case 2
(a).

(c)

The case of an intra-generational transfer

In this case  1 ＝1 and  2 ＝0 hold and the size of a public spending is zero, so theaggregate demand for output is given by (26) and the consumption of each group in eachgeneration is22
Cty, A ＝  A [ Yt d ＋ ], Cty, B ＝  B Yt d , C to, j ＝ mtd1, j . ( j  A, B )h(40)

Accordingly, the equilibrium output in Case 2 (c) is(41)

Yt 2,c ＝m  A＋.
（   h(1   A ) ＋ (1  h)(1   B ) ）The equilibrium output at period t in Case 2 (c) becomes larger than in Case 1 (c) (see(28) and (41)) by the same reason as in Case 2 (a). The equilibrium consumption of eachgroup in each generation is(42)(generation t )m 1  B
1 A
 
  B   , (Cto1, A ) 2,c ＝(Cty, A ) 2,c ＝  A   (Cty, A ) 2,cA   h1Bm   (Cty, B ) 2,c ＝  B   A  , (Cto1, B ) 2,c ＝(Cty, B ) 2,c ,
 
B

m
o 2,c
(generation t  1 ) (Ct , j ) ＝ (1   j ). （ j  A, B ）The consumptions in generation t are higher in Case 2 (c) than in Case 1 (c) while
those in generation t  1 are the same between two cases, as in Case 2 (a).

(Remark)
When the government transfers the revenue financed by debt issue not to the group

A only but to each group equally, the young period consumption of each groupj  A, B is C ty, j ＝  j [ Yt d ＋  ], and the equilibrium output in such a situation isYt ＝m [h A  (1  h) B ]＋.
（   h(1   A ) ＋ (1  h)(1   B ) ）By comparing this with (42), we haveA  B






Yt

2 ,c

 Yt .
23

This shows that if the consumption propensity of the group A is larger (resp. smaller)
than the group B , the equilibrium output when the government transfers the revenue
to the group A only is larger (resp. smaller) than that when it does to each groupequally. The equilibrium consumption of each group j  A, B in each generation is(generation t )1 j ym  
C ty, j ＝  j    , Cto1, j ＝Ct , j ,j  

(generation t  1 )C to, j ＝ (1   j )m.
The young and old period consumptions of the group A (resp. the group B ) ingeneration t become lower (resp. higher) than Case 2 (c), while the consumption ofeach group in generation t  1 remains unchanged. Notice that this equilibrium
consumption allocation is the same as that in Case 2 (a).

4.2

Equilibrium at period t  1

In the previous subsection 4.1 we showed that both the equilibrium output at periodt and consumption of each group in generation t become larger in Case 2 than in Case
1 in all three policy cases. However, in Case 2 the welfare of each group in generationt  1 may be harmed since a tax burden is shifted to her. In this subsection we examine
the effect of such a tax shift on the equilibrium output at period t  1 and the welfare
of generation t  1 .

The government
At period t  1 the government imposes a lump-sum tax 

on each group in

generation t  1 (so the total tax revenue is also  ), and repays a debt to each group in
generation t .

24

Households in generation t  1
The problem of each group j  A, B in generation t  1 ismax

Cty1, j ,Cto 2 , js.t.

U t 1, j ＝ U j (Cty1, j , Cto 2, j )Cty1, j ＋ mtd1, j ＝ wLdt1 ＋  t 1 －  , Cto 2, j ＝ mtd1, j .

So, the optimal consumption and saving plans are(43)

Cty1, j ＝  j [ wLdt1 ＋  t 1 －  ],mtd1, j ＝ (1   j ) [ wLdt1 ＋  t 1 －  ].

Households in generation t
Each group in generation t spends all her incomes (the sum of a money holding anda debt repayment) to the old period consumption. Although the levels of old period
consumptions are different among three policy cases (see (36), (39) and (42)), but theaggregate level becomes equal among them:(44)m  
hCto1, A ＋ (1  h)Cto1, B ＝[ h(1   A ) ＋ (1  h)(1   B ) ]    .
  

Namely, the aggregate consumption of generation t at period t  1 does not depend
on which policy is implemented at period t .

Firms
The behavior of the firm is the same as before, so we have(45)

Yt d1 ＝ F ( Ldt1 )→Ldt1 ＝ Ldt1 (Yt d1 ) ,  t 1 ＝ F ( Ldt1 ) － wLdt1

Non-Walrasian market equilibrium

25

The aggregate demand for output at period t  1 is given byYt d1 ＝[ hCty1, A ＋ (1  h)Cty1, B ]＋[ hCto1, A ＋ (1  h)Cto1, B ].
Here, from (43) and (45) the consumption of each group j  A, B in generation t is

Cty1, j ＝  j [ Yt d1 －  ],and the aggregate consumption of generation t is given by (44). So the equilibrium
output at period t  1 in Case 2 can be derived asYt 21 ＝m＋ .
（   h(1   A ) ＋ (1  h)(1   B ) ）This is larger than the equilibrium output at period t  1 in Case 1 (which is equal tom /  in all three policy cases). The equilibrium young and old period consumptions ofeach group j  A, B in generation t  1 are(Cty1, j ) 2 ＝  jm,(Cto 2, j ) 2 ＝ (1   j )m.
This result shows that in spite of the fact that a tax burden is shifted to each group in
generation t  1 in Case 2 while it is not in Case 1, the consumption of each group in
generation t  1 is equal between two cases. Why does such a result hold? This isbecause a shift of the tax burden to generation t  1 is equivalent to an income transferfrom (the young) generation t  1 with lower consumption propensity to (the old)
generation t with higher one, and that such an inter-generational transfer brings
about an increase of the equilibrium output which just offsets the tax burden ofgeneration t  1 as is closely discussed in Case 1 (b).
After period t  2 the equilibrium states are identical between Case 2 and Case 1,
because the real money stock m is constant and the government plays no role after theperiod. We can therefore conclude that in all three policy cases Case 2 is Pareto-superior
to Case 1. That is to say, in a fixed price model the government can improve the welfare26

of generation t without harming the welfare of future generations by shifting a tax
burden to generation t  1 , which is shown also in Tanaka (2008) and Ogawa andOno(2010) in more general setups.

5.

Final Remarks

We examined the effects of three kinds of fiscal policies (a wasteful public spending
and both inter-generational and intra-generational income transfers) on output andwelfare by using a non-Walrasian fixed price overlapping generations model withdifferent groups of households concerning the consumption propensity. Among theresults derived in this paper, the result of Case 1 (b) that an inter-generational transferfrom the young to the old can improve the consumption of the old without harming thatof the young seems to be most interesting and important, because it shows that demand
stimulation policies can be effective in a Keynesian under-employment economy. We caneasily confirm that this result holds true even when we introduce a certain kind of
bequest motive or the money in the utility function concerning the preference ofhouseholds, which means that the result is robust to some extent.
In this paper we consider a “pure” fixed price model where both price and wage are
fixed, which corresponds to the simple textbook multiplier model. By introducing theendogenous nominal price determination, the model can be extended to the so called
Keynesian AD-AS model. To investigate the welfare effects of fiscal policies in such amodel is one of the important questions for future researches.

27

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