can pay-as-you-go pensions raise the capital stock?

CAN PAY-AS-YOU-GO PENSIONS RAISE THE CAPITAL STOCK?

byMARK A. ROBERTS*

University of Nottingham

We reconsider pay-as-you-go (PAYG) pension policy in a version of theDiamond overlapping generations model with an imperfectly competi-tive financial sector and with a low rate of tax on its profits. PAYG thenhas two effects on capital: the well-known negative crowding-out effectin displacing savings and a positive effect in reducing a second form ofcrowding-out caused by the displacement of ‘productive savings’ incapital by ‘non-productive savings’ in financial sector equity. These twocountervailing effects may generate a hump-shaped steady-state rela-tionship between the PAYG contribution rate and the capital stock.

1 I

The reform of unfunded, pay-as-you-go (PAYG), pensions systems is oftopical concern, largely because of the problem of fiscal sustainability causedby the ageing of the population, which places an increasing burden onyounger tax payers. In addition, there are the known long-run financial ben-efits of reform where the market rate of return on savings exceeds the implicitrate of return on PAYG, which Samuelson (1958) first pointed out to be thegrowth rate of the tax base. Breyer (1989), however, shows that Pareto lossesare intrinsic to any reform of unfunded pensions, because the current gen-eration has contributed into a fund pre-reform and will be unable to draw outfrom another fund post-reform.1 These losses are also likely to raise politicalbarriers to reform, especially in democratic systems where median voters bearthe burden of transition.

Aside from these sustainability and relative rate of return issues, a pre-diction of standard life-cycle models is that PAYG pensions crowd out private

© Blackwell Publishing Ltd and The Victoria University of Manchester, 2003.Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK, and 350 Main Street, Malden, MA 02148, USA.

1

The Manchester School Supplement 20031463–6786 1–20

*This paper is part of the research project ‘Social Security Initiatives and Economic Growth’under the Evolving Macroeconomy Programme initiated by the ESRC, to which I am grate-ful for finance. I would also like to thank Friedrich Breyer in particular and the partici-pants of seminars presented at the University of Manchester and the University ofKonstanz and of the MMF-UEM Conference at the University of Warwick. I take soleresponsibility for any errors and shortcomings.

1Reform may still be Pareto-improving if there are various accompanying side benefits, as inHomburg (1990) who considers the accompanying labour supply effect of lowering dis-tortionary taxation. In a very strict theoretical sense, it is then not pensions reform per sewhich is Pareto-improving, since the side benefits could be introduced independently of thereform, e.g. by switching from distortionary to non-distortionary taxation. In a less strictsense, pension reform may still open the door for a Pareto improvement if it is practicallyinfeasible to obtain the side benefits independently of the reform, e.g. where taxes areinevitably distortionary.

household savings and capital through consumption-smoothing effects.2 Thecrowding-out effect of PAYG, as an elemental feature of the analysis, hassome bearing upon the issues of sustainability and relative rate of return. Ithas also been stated in its own right as an argument in favour of reform,namely by Feldstein (1995), which is valid whenever the level of economicactivity is a policy objective in its own right.

The present paper is not concerned with the questions of whether reformis either necessary or desirable, although its conclusions will inevitably haveimplications for both of these concerns. Instead, it develops a theoreticalmodel to test the general validity of the contention that unfunded pensionsnecessarily reduce the capital stock.3 We can also provide a theoretical ration-ale for the empirical research results, which are that the extent of crowding-out is not nearly as great as the standard life-cycle models predict, and theremay even be a crowding-in.

The time series evidence in Feldstein (1974) suggests that social securityreduces household saving only by 30–50 per cent, and with more data, inFeldstein (1996), by 60 per cent. Leimer and Lesnoy (1982) find that socialsecurity can lead to either a small decrease or even an increase in privatesavings.4 The theoretical results of this paper show both that the scale ofcrowding-out may not be so large and that crowding-in may not be an aberration but a real possibility for certain economies.

We explicitly analyse the effects of the optimizing behaviour of financialfirms which act as intermediaries between saving households and borrowingproductive firms. A version of the Diamond (1965) overlapping generationsmodel is considered with Nash–Cournot competition between N financialfirms where the integer N can take any value. The model has some general-ity, since it embraces the extreme monopoly–monopsony or monempory5 caseof a single firm or of a cartel where N = 1 and the standard, implicitly com-petitive case where N Æ •.

Recent literature has also investigated the economic effects of monopo-listic finance on economic activity. Partial equilibrium models have focusedon considering how monopoly banking may help to deal with the moralhazard and adverse selection problems that arise under imperfect informa-

2 The Manchester School

© Blackwell Publishing Ltd and The Victoria University of Manchester, 2003.

2If the market rate of interest equals the rate of time preference, so that an individual’s con-sumption is constant over time, the temporary equilibrium effect of a unit increase in PAYGwill be a reduction of household savings by (2 + n)/(2 + r) per unit where r is the marketrate of interest and n is the growth rate of the tax base.

3There are other reasons why PAYG may have either a small or a negative effect on the capitalstock. Generally, much of savings may be unrelated to life-cycle factors and may be madefrom precautionary and bequest motives. Concerning life-cycle factors, households facingborrowing constraints would be unable to offset a pension increase by dissaving (Hubbard,1986). Pension increases may even increase savings by inducing early retirement (Feldstein,1974) and by making workers more aware of their retirement needs (Cagan, 1965).

4In their survey of the literature, Engen and Gale (2000) conclude that it is difficult to assess theeffects from time series analysis.

5Nichol (1943) first used the terms ‘monempory’ and ‘oligempory’.

tion. In particular, Petersen and Rajan (1995) argue that a higher concentra-tion of firms supplying credit makes it easier for them to internalize exter-nalities, leading to an easing of credit constraints.

More recent papers by Cetorelli (1997), Smith (1998) and Guzman(2000a) have looked at the effects of monopoly finance on macroeconomicperformance in the general equilibrium of overlapping generations models.A general conclusion in the survey by Guzman (2000b) is that the effects ofbanking monopoly on performance are more harmful in dynamic generalequilibrium models, because of the additional and adverse effect on capitalaccumulation.

Although models of imperfectly competitive financial markets haveestablished a welcome beachhead within dynamic general equilibriummodels, the macroeconomic analysis of pensions has generally been relegatedto life-cycle models. These models implicitly assume perfect competition inthe finance sector through an imposition of equality a priori between thesavings rate of interest and the marginal product of capital, which is theinvestment rate of interest where the production sector also is perfectly com-petitive. In the present paper we investigate the effects of unfunded pensionson capital accumulation in a setting of perfect information and imperfectcompetition with endogenously determined savings and investment rates ofinterest.

We find that imperfect competition and PAYG pension policy eachpotentially have two effects on capital accumulation. The first effect of imper-fect competition is the wedge effect, where the savings rate of interest is belowthe borrowing rate of interest. The wedge effect causes savings and invest-ment to be lower than their levels under competitive finance. The lowersavings rate of interest reduces the likelihood that the economy is dynami-cally efficient, which is usually6 defined where the borrowing rate of interestexceeds the economic growth rate.

A second effect of imperfect competition is an equity price effect, whichexists because of the assumption that financial profits are endogenouslyreturned to households as the dividend returns on their holdings of financialsector equity.7 Households may hold either deposits or financial sector equity.These two assets are assumed to be perfect substitutes from the point of viewof individual households, but are widely different in their effects on theeconomy. Only deposit savings generate the funds that are transferred from

Can Pay-as-you-go Pensions Raise the Capital Stock? 3


6This definition no longer applies under imperfect competition.7The alternative is where profits are returned exogenously by a fiscal distribution rule where the

authority seizes the totality of financial sector profits and then reallocates the proceeds tothe household sector. The effect of the rule depends on to whom and how profits arereturned and these choices inevitably have powerful effects in life-cycle models where aggre-gate savings are determined by the intergenerational distribution of income. Consumptionsmoothing implies higher savings where profits are distributed to the young rather than theold, unless the old receive them in the form of a pro-rata savings subsidy (Roberts, 1998).

the household sector to the production sector for the purpose of investment.The intermediation process of the financial sector engineers the transfer.

We assume that financial equity serves only as a store of value, becausethere are no new issues since there are no investment needs in the financialsector.8 Financial firms have already acquired their existing stock of non-depreciable operating capital by issuing shares at some starting point in timefor perpetuity. Hence, financial equity plays a role equivalent to money, non-productive government debt, a fixed factor like land or a bubble. Conse-quently, savings in financial equity merely constitute asset trading within thehousehold sector, as the old sell this particular store of value on to the young.So while deposit saving raises economic activity, saving in financial equityleads to the crowding-out of capital.9 Consequently, we may use the termi-nology of Tirole (1985) to refer to the holding of these respective assets asproductive and non-productive savings.

The extent of crowding-out positively depends on the market price offinancial equity, because, in equilibrium, the higher the price the young payfor this store of value, the less total savings are left over to go into depositsavings. The equity price is naturally the present value of financial sectorprofits. It then follows that the abnormal profits arising under imperfect com-petition generate a second form of crowding-out by inflating financial equityprices to reduce deposit savings.10

To recap, imperfect competition reduces capital through two routes. Thewedge effect of imperfect competition in the financial sector reduces totalsavings, while the equity price effect reduces that part of total savings whichgoes into deposit holding and capital accumulation. In what follows, we showthat PAYG pension policy also has two related effects on capital accumula-tion. The first is the well-known consumption smoothing effect in reducingtotal savings and the second is another wedge effect which, this time, increasesboth the amount of savings and the proportion of productive savings.

First, an increase in the level of PAYG, by definition, raises later-periodretirement income allocations and lowers initial-period working incomes,because of the requirement of higher payroll taxes to finance the pension payments. A consumption smoothing response then reduces total savings.This first effect does not depend on imperfect competition and is, in fact,



8This is a reasonable approximation, if the aggregate economic gains from investing in a newbanking technology are insignificant compared with those from investing in productivecapital. It is also possible to consider financial profits net of some operating or deprecia-tion cost without losing the main point of the analysis.

9In a broader setting, financial intermediation may raise the activity level or growth rate bychanging the composition of savings. In a key paper, Bencivenga and Smith (1991) arguethat financial intermediaries provide liquidity, which enables a switch from savings held inliquid assets to productive assets. As usual, this effect is already implicitly incorporated,since liquid assets are not considered.

10This second effect has not, to our knowledge, been considered in models with imperfectly com-petitive finance.

even stronger under perfect competition, since the scale of crowding-out ismagnified by an already larger capital stock.

The second effect is more interesting, because it can seriously undermineor even reverse the standard comparative statics predictions of PAYG. Itworks through the positive effect of PAYG pensions on the interest elasticityof savings.11 An increase in PAYG will cause a greater wealth effect of therate of interest on savings by raising the later-period allocation. A rise in the interest elasticity of the savings function then reduces the scope for themonopsonistic exploitation of households by profit-maximizing financialfirms. The effect will cause financial sector profits and equity prices to fall,causing a crowding-in of productive savings as households put more of theirsavings into deposits. So, while the consumption-smoothing effect of PAYGreduces total savings, the importance of the wedge effect is in increasing theproportion of this total which is channelled into investment.12

We should also point out that the effect depends on imperfect competi-tion in the financial sector and works through changing the interest elastic-ity of the savings function.13 It does not depend on the PAYG nature of thepension per se, and the second effect would also arise under state pensionswhich are funded, provided they alter the relevant elasticity in the prescribedmanner.14

These two countervailing effects of PAYG on the capital stock create thepossibility of a non-monotonic relationship. We show that with standardfunctional form specifications—logarithmic for the utility function andCobb–Douglas for the production function—the capital stock is a hump-shaped function of the PAYG contribution rate. Thus, there is an interiorlevel of the PAYG contribution rate which maximizes the steady-state valueof the capital stock. The most basic policy implications, where the level ofeconomic activity is an objective, are that there is some intermediate level ofPAYG provision that would benefit a developing economy without any pro-vision and that only partial reform may be desirable for a mature welfarestate.

The rest of the paper is set up as follows. In Section 2 the basic modelis laid out, presenting the behaviour of the household, productive and finan-cial sectors. Section 3 derives the main results, which are contained in twopropositions. Section 4 discusses a range of other related issues and Section5 provides a general summary.



11The analysis is not symmetric regarding the interest elasticity of investment, since PAYG policydirectly affects only the savings schedule. There would be a second-order strengthening ofthe result with a rise in the interest elasticity, but this is fixed by the assumption of aCobb–Douglas technology.

12The wedge effect may also provide some offsetting rise in total savings.13A referee has correctly indicated that, even under competition, a reduction in hypothetical inter-

mediation costs could also raise the capital stock.14In fact under funded pensions the first effect can even lead to a crowding-in (for example, see

Fisher and Roberts, 2003), so that both effects raise the capital stock.

2 T M

2.1 The Household Sector

There is a representative household, j, which has a two-period utility function:

(1)

where ctY, j and cO, j

t+1 are the consumption levels of a household, j, which isyoung at time t and old at time t + 1. The budget constraint is

(2)

where wt is the common first-period wage income, bt is the amount of payrolltaxation raised to finance current PAYG payments and bt+1 is second-periodretirement income, consisting of PAYG receipts.

Savings by an individual j at time t may be held in two assets, fixed-pricedeposits at

j and a portfolio of financial equity etj, which has the variable price

vt. Thus j’s savings are given by

(3)

Individuals are distributed on the interval [0, 1] and aggregate savings aregiven by

(4)

where

Savings in fixed-price deposits at time t earn an interest rate of return rSt+1 at

time t + 1. The other asset, financial equity, is purchased by the young fromthe old. To simplify the analysis, it is assumed that there are no new issues offinancial equity. At some initial point in time, the financial sector made aonce-and-for-all equity issue to fund its own perpetual and non-depreciatingstock of operating capital. Thus young households acquire equity only fromold households. At time t, after they have received the financial profit or divi-dend income dt

j, the old sell the financial equity on to the next generation ofyoung at the price vt. Apart from the dividend payment, there is also the pos-sibility of a capital gain or loss outside the steady state, for the old who sellthe equity at the price vt would have purchased it when they were young atthe price vt-1, and generally vt π vt-1.

First- and second-period consumption levels are, respectively,

(5)c w b a v e w b stj

t t tj

t tj

t t tjY, = - - - = - -

s s j a a j e e jt jj

t jj

t jj= = =Ú Ú Ú

0

1

0

1

0

1

d d d

s a e vt t t t= +

s a e vtj

tj

tj

t= +

cr

c w br

btj

tt

jt t

tt

Y O, ,++

= - +++

++

+1

11

111

11

U U c ctj

tj

tj= ( )+

Y O, ,, 1



(6)

We assume certainty equivalence, as if the interest rate, the dividendincome and the equity price at time t + 1 are all known at time t. The maxi-mization of utility in equation (1), subject to the budget constraint in theform of equations (5) and (6) with respect to the choices of deposit holdingsand equity holdings by the young, gives the first-order condition for the individual’s time profile of consumption

(7)

and an asset price equilibrium

(8)

Equation (7) is the Euler equation, which gives the representative individual’stime profile of consumption or asset accumulation, and equation (8) is therisk-neutral no-arbitrage condition between deposits and equity. Certaintyequivalence first implies that the composition of the final portfolio has nobearing on asset accumulation, which is given by equation (7). It may besolved to give the savings equation

(9)

We assume that the savings function is a monotonic function of the rate ofinterest.

Second, as there is no incentive to hold a balanced portfolio, individ-uals will only hold one of the two assets in the presence even of infinitesimaltransaction costs. Thus the household sector will be split between the depositholders and equity holders, respectively, the customers and the owners of thefinancial sector. This feature has no implications for the solution, but doesendorse the underlying assumption of value maximization, which is discussedfurther in Section 4.

The predetermined stock of equity is normalized to unity, et = 1, "t ≥ 0, so that equations (4) and (9) determine aggregate deposit savings as

(10)

2.2 The Production Sector

There is a representative firm operating under constant returns to scale, whichenables the production function to be presented in per capita terms

(11)y f k f ft t= ( ) ¢ > ¢¢ <0 0,

a s w b b r vt t t t t t= -( ) -+ +, ,1 1S

s s w b b rt t t t t= -( )+ +, ,1 1S

vd v

rtt t

t

=+

++ +

+

1 1

11 S

∂∂

∂∂

U

cr

U

ct

tt

t

tY

SO= +( )++

1 11

c b r a d v e

b r s d v r v et

jt t t

jt t t

j

t t tj

t t t t tj

+ + + + +

+ + + + +

= + +( ) + +( )= + +( ) + + - +( )[ ]

1 1 1 1 1

1 1 1 1 1

1

1 1

O S

S S

,



where yt and kt are output and the capital stock per head at time t. The pro-duction function has the usual properties.

Firms hire capital at the separate borrowing rate of interest rtL; the vari-

able wt is both the wage they pay their workers and the per capita wage bill.There is perfect competition in the production sector. Per capita profits are

The first-order condition for a maximum with full depreciation over theperiod of production determines the investment function as

(12)

and, with the constant returns and free entry assumptions, the wage is deter-mined as

(13)

where

2.3 The Financial Sector

There are N finance firms, indexed z, z = 1, . . . , N, which engage inNash–Cournot quantity competition. Each finance firm z creates deposits attime t, aZ,t, by borrowing from households and then lends these funds to pro-duction firms, which then acquire capital at time t + 1, kZ,t+1.15 At time t + 1,the bank pays householders the market rate of interest on savings, rS

t+1, andcharges firms the market rate of interest on loans, rL

t+1, where rLt+1 ≥ rS

t+1. Eachproduction firm uses its loan for investment, and with 100 per cent depre-ciation the aggregate capital stock is determined by

(14)

where

Profits accruing at time t + 1 for each financial firm z are

(15)

To confirm, there are no issues of equity, because the outstanding stockof non-depreciable working capital for each financial firm was introduced at

P Z t t t Z tr r k, ,+ + + += -( )1 1 1 1L S

k k a kt Z tZ

N

t Z tZ

N

+ += =

= =Â Â1 11 1

, ,

k at t+ =1

∂∂

wk

f k kt

tt t= - ¢¢( ) > 0

w f k f k kt t t t= ( ) - ¢( )

¢( ) =f k rt tL

Yt t t t tf k r k w= ( ) - -L



15To the extent that deposits are created for life-cycle savings, these financial firms will also consistof pension companies as well as investment banks.

some initial point in time for perpetuity. The assumption of symmetry acrossfinancial firms allows us to work with the assumption that households arealso indifferent to the composition of the equity portfolio.16

Equations (10) and (14) imply that the aggregate capital stock is givenby

(16)

The acquisition of financial market equity by the young is captured by thevaluation term vt, which crowds out the capital stock.

As all financial profits are paid out as dividends, PZ,t+1 = dZ,t+1, equations(8) and (15) together give

(17)

2.3.1 Some Properties of the Steady State. At this point, it is possible toconsider some of the steady-state properties of the model before proceedingwith the financial sector optimization. Substituting the aggregate form ofequation (17) with symmetric equilibrium

into (16) and then considering the steady-state levels, defined by x = xt, "t,where xt = kt, vt, bt, wt, rt

S, rtL, the capital stock and the financial equity

valuation are

(18)

(19)

First, equation (18) shows that the proportion of crowding out in the steadystate is 1 - rS/rL. In the competitive finance case where rS = rL, this is, of course,zero. Second, equation (19) shows that the equity price in the steady state isa multiple of this same crowding-out ratio. Third, returning to equation (18),we see that PAYG pensions policy has potentially two effects on the capitalstock: one in affecting total savings s(.), and the other in affecting the pro-portion of total savings rS/rL which goes into deposits and capital. To com-plete the model, it only remains to determine the savings rate of interest fromoptimization in the financial sector.

vr

rs w b b r= -Ê

ËÁˆ¯̃ -( )1

S

LS, ,

kr

rs w b b r= -( )

S

LS, ,

vr r k v

rtr t t t

t

=-( ) +

++ + + +

+

1 1 1 1

11

L S

S

vr r k v

rZ tt t Z t Z t

t,

, ,=-( ) +

++ + + +

+

1 1 1 1

11

L S

S

k s w b b r vt t t t t t+ + += -( ) -1 1 1, , S



16The normalization e = 1 implies that eZ = 1/N "z.

2.3.2 Financial Sector Optimization. Each firm z acts to maximize thevalue of its financial equity vZ,t at time t, as given by equation (17). We con-sider the dynamic programming solution where the future value vZ,t+1 is takenas given at time t, so that the current maximizer takes as given the decisionsof the future maximizer and, by recursions, all future maximizers. Although,by assumption, no game is played over time, Nash–Cournot behaviour isplayed within any period of time, because the interest rates are commonprices determined by aggregate quantities to which each firm makes a non-negligible contribution where N < •.

It is convenient to invert equation (16) to express the household savingsfunction in terms of the savings rate of interest17

and then to use equation (12) for the borrowing rate of interest. Equation(17) becomes

(20)

Note that the aggregate financial value or the aggregate equity portfolio pricevt enters the right-hand side of equation (20) (twice), because the savings rateof interest is determined by aggregate savings and because some part ofaggregate savings consists of aggregate equity. Consequently, we use theimplicit function theorem to obtain the derivative.

Each financial firm z maximizes the value of its equity vZ,t in equation(20) by choosing a level of deposits aZ,t, which is equivalent to choosing alevel of lending kZ,t+1 (from equation (14)). The firm takes into account theeffect both of its individual choice kZ,t+1 on the aggregate outcome kt+1 andof its own equity valuation vZ,t on the aggregate equity portfolio vt. The firstderivative is

(21)

We consider (i) an interior solution where ∂vZ,t/∂kZ,t+1 = 0, "z, and (ii) a symmetric equilibrium where kZ,t+1 = (1/N)kt+1, vZ,t = (1/N)vt, "z. Using st = vt + kt+1 gives the first-order condition

¢( ) - ( ) + ¢¢( ) - =+ + + ++f k r k f k

kN

rsNt t t t

t t1 1 1 1

11 0S S, . . .

∂∂

∂∂

v

k

v

a

f k r k f k k v k r k

r k v k

Z t

Z t

Z t

Z t

t t t t Z t Z t Z t t

t t Z t Z

,

,

,

,

, , ,

, ,

, . . . , . . .

, . . .

+

+ + + + + + +

+ +

=

=¢( ) - ( ) + ¢¢( ) - +( ) ( )

+ ( ) + +

1

1 1 1 1 1 1 1 1

1 11

S S

Stt r+( )1 1

S

vf k r k v w b b k v

r k v w b bZ tt t t t t t Z t Z t

t t t t t,

, ,, ,

, ,=

¢( ) - + -( )[ ] ++ + -( )

+ + + + +

+ +

1 1 1 1 1

1 11

S

S

r r k v w b bt t t t t t+ + += + -( )1 1 1S S , ,



17The inversion can be made because the monotonicity of the capital accumulation equation (16)depends on the monotonicity of the savings equation (9), which is assumed.

We also define the interest elasticities of savings and investment as

(22)

so that the equilibrium may be presented as a relationship between the ratioof two interest rates, the two elasticities and the number of firms:

(23)

With imperfect competition, N < •, and with less than infinite interest elasticities, e < • and h < •, the savings rate of interest will be less than the borrowing rate of interest, so that the rate of profit is positive. The crowding-out ratio in the steady state is calculated as (1 + e /h)/(1 + Ne).

3 A

Equation (23) shows that the ratio of the two interest rates is constant if thetwo interest elasticities are constant (for a given number of firms). Equation(18) then implies that, in the steady state, the rate at which non-productivesavings crowd out productive savings is also constant. In this constant elas-ticity case, PAYG policy will always reduce the capital stock through crowd-ing out private savings. The extent to which PAYG crowds out capital is lessunder imperfect financial market competition, because the capital–savingsratio is already lower by the factor rS/rL.

A more plausible and interesting case is where the interest elasticitiesvary with the level of PAYG. There are two broad possibilities. If the ratiorS/rL fell with PAYG, it would cause even greater damage to capital accumu-lation. However, the existence of wealth or discounting effects of the rate ofinterest suggests the opposite. PAYG is a future income allocation, which isdiscounted by the savings rate of interest when current savings decisions aremade. As PAYG rises, any change in the savings rate of interest will have alarger negative effect on wealth and current consumption, so that theresponse of savings will become more positive.18

The rise in the interest elasticity of savings e implies that there will beless scope for the monopsonistic exploitation of households by value-maximizing financial firms. This effect reduces the rate of profit, which is thegap between the borrowing and savings rates of interest. It is then clear, fromequation (23), that for rS/rL to rise it is necessary that there should be no coun-tervailing decrease in the interest elasticity of investment h—if there shouldbe one—of a magnitude which swamps the effect of the higher e. This is guar-

r

r NNS

L =+

+ee

hh1

1

e h∫ ∫¢( )

¢¢( ) >r

r s

f kf k k

S

S1

1



18There could be a sign reversal, so that the PAYG would generate a backward-bending savingsfunction instead of a negatively sloped function where the intertemporal elasticity of sub-stitution is inelastic.

anteed within the approximation of a Cobb–Douglas production technologywhere h is constant.19 We consider also the following specification for thehousehold utility and savings functions.

3.1 A Logarithmic Household Utility Function

We specify a logarithmic household utility function:

(24)

This is the most tractable case and one commonly used, since the income andsubstitution effects of the rate of interest exactly cancel out to leave thewealth effect of the rate of interest the only one of concern. It also allows usto obtain an analytical solution to the model.

The weighting q relates to the rate of time preference d, where d ∫(2q - 1)/(1 - q), and conventionally q ≥ 1/2 so that d ∫ (2q - 1)/(1 - q) ≥ 0.Household optimization generates a savings function of the form20

(25)

which has the steady-state interest elasticity

(26)

A constant elasticity Cobb–Douglas production function is used

which gives the following solutions for the borrowing rate of interest, thewage and the (constant) interest elasticity of investment:

(27)

We also consider the case of an exogenous contribution rate b, where b ∫ b/w, so that the ratio of pensions to current wages rather than the levelof the pension is exogenous. The pension level is then also endogenous to the capital stock from equation (27):

(28)b w Ak= = -( )b b a a1

r Ak w AkL = = -( ) = - -( )-a a h aa a1 1 1 1

y Akt t= a

eq

q q=

-( ) +( ) -( ) - +( )br

r w b r b

S

S S1 1 12

s w br

bt t tt

t= -( ) -( ) -+

£ £+

+11

0 1 21

1qq

q

V w b s b r st t t t t t t= - -( ) + -( ) + +( )[ ] < <+ +q q qln ln1 1 0 11 1S



19If there is a general constant elasticity of substitution form for the production function,y = A[a-1(1 - a) + k(s -1)/s ]s/(1-s), then h = s [1 + a(1 - a)-1k(s -1)/s ]. The effect of a rising ewill be dampened or reversed with an elastic degree of factor substitution, where s > 1,and enhanced with an inelastic degree where (0 <) s < 1. This follows from the reasoningthat the temporary equilibrium effect of PAYG will reduce the capital stock and from thesign of the derivative ∂h/∂k = (s - 1)a(1 - a)-1k-1/s.

20The property of monotonicity in the interest rate allows for inversion.

This reflects the real-world feature that the level of welfare provision tendsto increase as economies become richer.

Equations (18), (25), (27) and (28) together give a solution for the savingsrate of interest:

which has the quadratic form

with the solution

(29)

where

We use the positive-valued solution to ensure the general non-negativity ofthe interest rate.21 Note that the savings rate of interest is increasing in b. Asb Æ 0, rS Æ a/(1 - a)(1 - q), and as b Æ 1, rS Æ •.

Returning to (26), we see that the required property of the savings elas-ticity, ∂e/∂b > 0, obtains at least locally, since as b Æ 0, e Æ 0, while b > 0implies e > 0.

Equations (23) and (26)–(28) give

Eliminating the interest rate term by using both forms of equation (29),removing superfluous terms and applying the definitions g and h also used inequation (29) gives

(30)

Equation (30) is the ultimate statement of the model which shows that thelevel of the capital stock in the steady state depends on the parameters of thehousehold savings function q and b and of the firms’ production functions

g h∫-( ) + - + -( )

-( ) -( )∫

-( ) -( )a a q q b

q bqb

q b1 1 2 1

2 1 1 1 1

kN A

g h h g N h g N

1

2 2 1 2

1

2 1 1 1 1 2- =

- -( )[ ]( ) + -( )[ ] - -( )[ ]{ } -

a a a

kN A

r r Nr

12

1

1 1 1 1- =

- -( )[ ]-( ) -( ) +( ) - +( ) +

a a a qb

q b qb qbS S S

hr

∫-( ) -( )

>qb

q b∂∂ b1 1

0S

g ∫-( ) + - + -( )

-( ) -( )a a q q b

q b1 1 2 1

2 1 1

1 2 1 2+ = + -( )r g g hS

1 1 02

+( ) - +( ) + =r g r hS S

aa

q bqb

11 1

1-= -( ) -( ) -

+ÈÎÍ

˘˚̇

rr

SS



21For the negative root, as b Æ 0, g - (g2 - h)1/2 Æ 0, so rS Æ -1.

A and a, the number of financial firms N and the government policy param-eter b. We now consider the effects of policy on the capital stock and savingsin the steady state.

Proposition 1: There is a non-monotonic steady-state relationship betweenthe PAYG contribution rate b and the capital stock (where N < •).

Proof: (i) Generally, where 0 < b < 1, (0 <) g < •, 0 < h < •, so that k > 0. (ii)In the limit b Æ 0, we find that h Æ 0 and that g Æ [a/(1 - a) + (1 - q)]/2(1 - q), so that 0 < g < •. This implies k Æ 0 where N < •.22 (iii) In the other limit where b Æ 1, we find that g2/h Æ • and that h/g Æ 2q/[a/(1 - a)+ q], so that 0 < h/g < •. This also implies that k Æ 0 where N < •.

Therefore, there is a low range of values for b where ∂k/∂b > 0 and ahigh range of values where ∂k/∂b < 0.

The monopoly or cartel case, where N = 1, is the most accessible to analy-sis, since the ratio term h/g then drops out of the denominator of equation(30) to give

The remaining ratio term g2/h, which enters the denominator twice, has anunambiguously negative effect on k. Thus, the value of b which minimizesg2/h is the same value which maximizes k and is determined at b* = [1 -q (1 - a)]/(1 + a). If b < b*, ∂k/∂b > 0, and if b ≥ b*, ∂k/∂b £ 0.

We consider the effects of varying the interest elasticity of investment.In the limiting case of an infinitely elastic investment function where the pro-duction function exhibits constant returns to scale, a Æ 1, b* = 1/2. We findthe case of a unitary elasticity of investment where the production functionis logarithmic, where a Æ 0, so that b* Æ 1 - q. Here b < 1/2 with a posi-tive rate of time preference, where q > 1/2. So generally, with a single bankor a cartel and positive rates of time preference, PAYG pensions will gener-ally raise the steady-state capital stock until a benefit rate at some point lessthan 50 per cent. The exact point depends positively on the interest elasti-city of investment.

For more than a single cartel firm and non-cartel behaviour, where N >1, the ratio h/g remains in the denominator of equation (30). Any positiveeffect of PAYG on the capital stock is then dampened, because ∂(h/g)/∂b >0. As N increases, the dampening increases, so that the level of the pension

kA

g h h g

12

2 2 1 22 1 1 1

- =( ) + -( )[ ] -

a a



22The result that the capital stock vanishes in the steady state where the pension tends to zero issurprising, but is itself based on a very strong implicit assumption which we now makeexplicit. Besides a pension at zero, there is absolutely no other source of expected second-period income (e.g. from receiving a gift or from scavenging or begging). Alternatively, ifthere is some positive amount e (e ª 0) of expected second-period income from anothersource, the capital stock will not vanish in the steady state.

ratio which maximizes the steady-state capital stock is decreasing in N, themeasure of financial market competition. As N Æ •, the capital-stock-maximizing pensions ratio tends to zero.

Proposition 2: There is a non-monotonic steady-state relationship betweenthe PAYG contribution rate b and savings (where N < •).

Proof: Equations (18) and (27) imply that steady-state savings are given by s = aAka/rS. The value of b which maximizes steady-state savings will be lowerthan the corresponding value which maximizes steady-state capital k because,while the numerator of s = aAka/rS is a hump-shaped function of b fromProposition 1, the denominator rS is an increasing function of b from equa-tion (29).

4 F D

4.1 Welfare Considerations

The model has been considered only in its steady state, because of the ana-lytical difficulties of the dynamics that arise when forward-looking assetprices are combined with backward-looking capital stock dynamics in a non-linear model. Consequently, it is not possible to say a great deal about thewelfare implications of pension changes, although there are some basicstatements which can be made.

In all models, PAYG generally has three effects on household utility: (i)a relative rate of return effect, which is favourable only where the growth rate(especially of the tax base) exceeds the market rate of interest on savings, (ii)a favourable effect from an increase in the savings rate of interest and (iii) anunfavourable effect from the decrease in capital and, hence, wages.

In the standard model, where both the goods and finance markets arecompetitive, if the implicit rate of return on pensions exceeds the marginalproduct of capital, the first effect becomes favourable and the secondfavourable effect dominates the third unfavourable effect. Thus, dynamic effi-ciency requires that the implicit rate of return on pensions is no greater thanthe savings rate of interest, which is equal to the marginal productivity ofcapital. In the present model, this equality no longer holds, so that thedynamic efficiency condition becomes more complex. Furthermore, there isa sign change where PAYG pension increases raise the capital stock.

In all models, the generation alive at the time of a policy change will beunaffected by the third, wage, effect because the capital stock is predeter-mined. In this particular model if PAYG eventually raises the capital stock,the utility gains (losses) of later generations are likely to exceed (be less than)those of earlier generations. This suggests that a utility gain for the currentgeneration, following an increase in PAYG, may be a sufficient condition forthe economy to be dynamically inefficient.



It is known in the standard competitive model with overlapping genera-tions that the economy is dynamically inefficient if the capital stock is toohigh. We can demonstrate that in this present model there will be dynamicinefficiency also if the capital stock is too low. The proof of Proposition 1demonstrates that where the pension rate tends to zero, so does the capitalstock, so that the economy vanishes in the absence of any level of PAYG pen-sions (or of any other allocation for retirement). An obvious corollary is thatthere must be some low level of pension provision which cannot make anygeneration worse off, since no one can do worse than zero. Thus, the economymust be dynamically inefficient for some very low range of values for thepension rate.

4.2 Taxing Financial Sector Profits or Dividends

PAYG can increase the capital stock only because high net-of-tax dividendson financial sector equity already draw savings away from deposits, whichwould get channelled into investment. If the overriding objective of policy isto raise economic activity and hence the capital stock, the first recourseshould be to tax either financial profits or dividend payments at 100 percent.23 With regard to financial equity price determination and capital accu-mulation, taxation of financial sector profits, reducing gross and net divi-dends, is equivalent to direct taxation of gross dividends.

The capital stock may generally be maximized at an intermediate levelof the PAYG contribution rate in the absence of an adequate dividend tax.If dividends are completely taxed away, the capital stock will be highest undera zero contribution rate, because there will be no second equity price effectas equity prices will be constant at zero. If the objective is to raise the level of economic activity, a PAYG pensions reform may even be counter-productive in the absence of an accompanying policy to raise sufficiently the rate of taxation on either financial sector profits or dividends. It will bebeneficial if accompanied by this separate policy.

A policy to raise either of these taxes may not be pursued for the samereasons that there may be resistance to reforming an excessively burdensomepension system. Raising these taxes will lead to losses for the constituencylocked into holding financial sector equity, which has intragenerational aswell as intergenerational welfare implications. Furthermore, it has also beenargued that raising the level of pensions from a very low level can be Pareto-improving. In this state, the same welfare and political economy considera-tions might point to a limited measure of pension provision along with ahands-off attitude towards financial profits or dividend taxation.



23In an alternative environment of uncertainty, there may also be benefits in allowing financialfirms a minimum level of expected net-of-tax profits, making dividend taxation the betteroption.

4.3 A Resolution of the Customer–Owner Dilemma

There is a well-known problem of general equilibrium with imperfect com-petition, because households are also the ultimate owners of profit-makingfirms as well as their customers. A conflict may arise which may be describedas the customer–owner dilemma. Demand functions may become less wellbehaved but, more importantly, the underlying objective of profit or valuemaximization itself may be in question (Hart, 1985). It even becomes validto question the assumption of value maximization and to ask whether insteadthe customer–owners would prefer a better deal on prices (interest rates) thanprofits.24

This problem can be resolved by two intrinsic features of the model andby an additional, but very weak, assumption. The first feature is that younghouseholds can undertake savings in two ways, by acquiring either financialsector equity or deposits. The second feature is that with certainty, or cer-tainty equivalence, the two assets, equity and deposits, are perfect substitutes.The additional assumption is that acquisition of financial equity and finan-cial deposits involves separate transactions, each requiring a negligible trans-action cost. The negligibility of costs implies that households will alwaysmake asset transactions, but the mere presence of costs along with the perfectsubstitutability of the assets implies that each household will transact in onlyone of the two assets.

Thus, the economy will generally be divided into two groups of house-holds, those who hold deposits and those who hold financial equity. This divi-sion avoids the customer–owner dilemma, since the customers, the holders offinancial deposits, will not be the same people as the owners, the holders offinancial equity. The owners will be quite happy with the objective of valuemaximization, as they are not the customers, while the customers, as non-shareholders, will have no voting rights.

4.4 Fully Funded Pensions

The model has demonstrated that the sign of the effect of changes in the levelof unfunded PAYG pensions on the capital stock and economic activitycannot be taken for granted. An immediate question is whether the sameanalysis applies where state pensions are funded, with the fund invested likeany other form of savings. We note that the ‘financial sector’ has been looselydefined in this model and will also consist of pensions companies in so faras household savings are made for the life-cycle. A large part of householddeposit savings in this model will in fact be the acquisition of private fundedpensions.



24This may be resolved by the existence of mutual societies in the UK sense of the term, which,in principle, have intended to offer better deals on interest rates.

An established result is that the provision of fully funded state pensionswill not crowd out total savings, because voluntary private savings by indi-viduals will then be replaced one-for-one by forced savings through taxation,which go into the acquisition of a state savings fund. There may even be thecrowding-in of total savings with other forms of financial market imperfec-tion, such as lending constraints, especially the discriminatory lending con-straints considered in the model with unemployment by Fisher and Roberts(2003).

There will be a further source of crowding-in comparable to the interestelasticity effect, which has been considered with respect to PAYG pensions.The managers of state pension funds will appear as big players to the finan-cial institutions, so there will be little scope for the same monopsonisticexploitation which individual savers, as small players, would be subjected tounder imperfect competition. This effectively makes the aggregate supply ofsavings more interest elastic. The total provision of funded pensions and totalsavings will increase because state funded pensions will then crowd out privatefunded pensions by a factor less than one—even in the absence of lendingconstraints. The fall in pension company profits, as before, will cause privatesavings to switch from non-productive savings to productive savings. Fur-thermore, if the government is concerned with activity levels, the managersof the public pensions fund, as agents of the government, would ensure that the fund was invested in capital-increasing activities rather than in theprofitability of other pension funds.

4.5 Functional Form Specifications

The case of the logarithmic utility function has been considered. This has theproperty that income and substitution effects of the interest rate directlycancel out, which leaves only the discounting effect, which is itself zero in theabsence of retirement income in the two-period model. This ensures thatsavings are infinitely inelastic in the absence of a pension and that the elas-ticity is increasing in the level of the pension. Generalizing the specificationcan either weaken or strengthen the results, depending on whether substitu-tion or income effects dominate. If substitution effects dominate incomeeffects as PAYG provision increases, the results are strengthened, but weak-ened vice versa.

4.6 The Form of Imperfect Competition

Finally, another generalization would be to consider other forms of financialsector competition. In terms of quantity competition, one extension mightbe a Stackelberg game between a single large institution and the rest of themarket. Another consideration would be to see how the central bank deter-mination of base interest rates and market intervention would affect the



behaviour of the financial system. Price competition is also likely to be rele-vant because of the heterogeneity of financial services.

5 S C

We have looked at the effects of PAYG pensions on the steady state of thecapital stock where the financial sector is imperfectly competitive. A key effectis the absorption of household savings into financial sector equity, drawn bythe abnormal profits that are made under imperfect competition. This effectis greatest where the level of pension provision is lowest. Consequently, a non-monotonic, specifically hump-shaped, steady-state relationship mayarise—with standard specifications of functional forms—between the PAYGcontribution rate and the capital stock on one hand and total savings on theother.

The ambiguity of the sign of the response, which depends on the levelof pensions, may help to explain some of the variability in the empiricalresults to the extent that the steady-state properties of the economy will bereflected in the time series data. There may also be structural changes becauseof the non-linear effects of changes in the level of PAYG over long time spans.There may also be other factors, not least in the state of competition in thefinancial sector, as well as changes in public policy and the institutionalframework regarding the regulation of this particular sector.

One of the main reasons for advocating pension reform is the belief thatPAYG pensions inevitably reduce the capital stock. We have presented amodel which shows that this can no longer be a foregone conclusion, butdepends on the environment of financial market competition. If an overrid-ing concern is capital accumulation, then a radical overhaul of PAYG pen-sions may be ineffective or even counter-productive in the absence of a fullertaxation of financial sector profits. If this is infeasible, then the scale of anyreform of PAYG should be measured against the weight of competition inthe financial sector and the level of PAYG provision. If a mature welfare stateis so overburdened with a PAYG pension fund that the supply side is sub-stantially constricted, a partial reform may be the best outcome between theoptions of full reform and no reform, while a modest provision of PAYG ina developing economy without a welfare state may be a prescription forraising the capital stock.

R

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can pay-as-you-go pensions raise the capital stock?

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