Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

(17.10)

rs receive their respective ro):

The crucial thing to note d by the currently old, Kt, interest upon which the

• Tt+i in (17.3) and (17.6))

(17.13)

e ultimately wish to study uilibrium, it is useful to

Chapter 14 for details):

be written as follows:

(17.17)

7uation (17.17) says that ft-hand side) can be either f capital (right-hand side).

AKt+i +(Mt representing

!--qa1 does not depreciate at all 4 is quite long (in historical period (B = 1). Blanchard (17.10) is a net production :s from the capital demand

Chapter 17: Intergenerational Economics, II

Aggregate consumption is the sum of consumption by the young and the old agents in period t:

(17.18)

Since the old, as a group, own the capital stock, their total consumption in period t is the sum of the undepreciated part of the capital stock plus the rental payments received from the firms, i.e. Lt_i = (rt + 8)1<t- + (1 — 8)Kt. For each young agent consumption satisfies (17.2) so that total consumption by the young amounts to: Lt cr = WtLt — StLt . By substituting these two results into (17.18), we obtain:

where we have used the fact that Yt = (rt + 8)Kt + WtLt in going from the first to the second line. Output is fully exhausted by factor payments and pure profits are zero.

Finally, by combining (17.17) and (17.19) we obtain the expression linking this period's savings decisions by the young to next period's capital stock:

so that (17.20), in combination with (17.6), can be rewritten in per capita form as:

The capital market is represented by the demand for capital by entrepreneurs (equation (17.16)) and the supply of capital by households (equation (17.22)).

17.1.4 Dynamics and stability

The dynamical behaviour of the economy can be studied by substituting the expressions for Wt and rt+i (given in, respectively, (17.15) and (17.16)) into the capital supply equation (17.22):

This expression relates the future to the present capital stock per worker and is thus suitable to study the stability of the model. By totally differentiating (17.23) we obtain:

593

The Foundation of Modern Macroeconomics

where Sw and Sr are given, respectively, in (17.7) and (17.8). We recall from Chapter 2 that local stability requires that the deviations from a steady state must be dampened (and not amplified) over time. Mathematically this means that a steady state is locally stable if Idkr±i jar I < 1. It is clear from (17.24) that we are not going to obtain clearcut results on the basis of the most general version of our model. Although we know that the numerator of (17.24) is positive (because Sw > 0 and f" < 0), the sign of the denominator is indeterminate (because S r is ambiguous).

Referring the interested reader to Galor and Ryder (1989) for a rigorous analysis of the general case, we take the practical way out by illustrating the existence and stability issues with the unit-elastic model. Specifically, we assume that technology is Cobb-Douglas, so that yr = kti ', and that the felicity function is logarithmic, so that U(x) = log x and 0-(x) = 1/0(x) = 1. With these simplifications imposed the savings function collapses to Sr = Wt/(2 + p), the wage rate is 147r- = ELkti-EL , and (17.23) becomes:

Equation (17.25) has been drawn in Figure 17.1. Since limk_,0 g'(k) = oc and limk_,,„, gi(k) = 0, the steady state, satisfying is unique and stable. The diagram illustrates one stable trajectory from /co. The tangent of g(.) passing through the steady-state equilibrium point E0 is the dashed line AB. It follows from the diagram (and indeed from (17.25)) that the unit-elastic Diamond-Samuelson model satisfies the stability condition with a positive slope for g(.), i.e. 0 <

kt+1

kt+1=g(kt)

Figure 17.1.The unit-elastic Diamond-Samuelson model

17.1.5 Efficiency

It is clear from the di , sonable setting in whit unique steady-state eqt general model also has 1 keep things simple, and we restrict attention to we compare the market A golden-age path is kt+1 = kt = k. Such a highest possible utility, 1965, p. 1128). Formal of a "representative" inc

subject to the economy-

f (k) - (n + 3)k = C y

Note that we have drop the fact that we are look note about this formul. to consumption during (17.27) C1' and C° refer t at a

apples and oranges—fot can ignore these differ, The first-order condi:,

state resource constraint 1

U' (C°) 1 + p INCY) - 1+ n' I

r(k) n + S.

Samuelson (1968a) calls consumption golden ru (17.29) with their rev:

3 The steady-state resource

(17.17) and the resulting exp (kt+1 = kt = k), and all time

d (17.8). We recall from - m a steady state must be this means that a steady

.24) that we are not going version of our model. tive (because SW > 0 and

- I use Sr is ambiguous). for a rigorous analysis

•-ating the existence and e assume that technology function is logarithmic, nplifications imposed the rate is Wt = ELkti ', and

(17.25)

e limk_o g'(k) = oo and s unique and stable. The of g(.) passing through

3.It follows from the dia- mond-Samuelson model i.e. 0 < g/(k*) < 1.

1 = kt

B

kt+i g (k)

icr

I model

Chapter 17: Intergenerational Economics, II

17.1.5 Efficiency

It is clear from the discussion surrounding Figure 17.1 that there is a perfectly reasonable setting in which the Diamond-Samuelson model possesses a stable and unique steady-state equilibrium. We now assume for convenience that our most general model also has this property and proceed to study'its welfare properties. To keep things simple, and to prepare for the discussion of social security issues below, we restrict attention to a steady-state analysis. Indeed, following Diamond (1965) we compare the market solution to the so-called optimal golden-age path.

A golden-age path is such that the capital-labour ratio is constant over time, i.e. = kt = k. Such a path is called optimal if (i) each individual agent has the highest possible utility, and (ii) all agents have the same utility level (Diamond, 1965, p. 1128). Formally, the optimal golden-age path maximizes the lifetime utility

of a "representative" individual,

subject to the economy-wide steady-state resource constraint:

Note that we have dropped the time subscripts in (17.26)-(17.27) in order to stress the fact that we are looking at a steady-state situation only. 3 An important thing to note about this formulation is the following. In (17.26) CY and C° refer, respectively, to consumption during youth and retirement of a particular individual. In contrast, in (17.27) CY and C° refer to consumption levels of young and old agents, respectively, at a particular moment in time. This does, of course, not mean that we are comparing apples and oranges—for the purposes of selecting an optimal golden-age path we can ignore these differences because all individuals are treated symmetrically.

The first-order conditions for the optimal golden-age path consist of the steadystate resource constraint and:

Samuelson (1968a) calls these conditions, respectively, the biological-interest-rate consumption golden rule and the production golden rule. Comparing (17.28)- (17.29) with their respective market counterparts (17.5) and (17.16) reveals that

3 The steady-state resource constraint (17.27) is obtained as follows. First, (17.18) is substituted in (17.17) and the resulting expression is divided by Lt . Then (17.14) is inserted, the steady state is imposed (kt± i = kr = k), and all time indexes are dropped.

595

The Foundation of Modern Macroeconomics

they coincide if the market rate of interest equals the rate of population growth:

As is stressed by Samuelson (1968a, p. 87) the two conditions (17.28)-(17.29) are analytically independent: even if k is held constant at some suboptimal level, so that production is inefficient as f'(k) n + 8, the optimum consumption pattern must still satisfy (17.28). Similarly, if the division of output among generations is suboptimal (e.g. due to a badly designed pension system), condition (17.28) no longer holds but the optimal k still follows from the production golden rule (17.29).

If the steady-state interest rate is less than the rate of population growth (r < then there is overaccumulation of capital, k is too high, and the economy is dynamically inefficient. A quick inspection of our unit-elastic model reveals that such a situation is quite possible for reasonable parameter values. Indeed, by computing the steady-state capital-labour ratio from (17.25) and using the result in (17.16) we find that the steady-state interest rate for the unit-elastic model is:

Blanchard and Fischer (1989, p. 147) suggest the following numbers. Each period of life is 30 years and the labour share is EL = 3/4. Population grows at 1% per annum so n = 1.01 3° -1 = 0.348. Capital depreciates at 5% per annum so 8 =1- (0.95)3° = 0.785. With relatively impatient agents, the pure discount rate is 3% per annum, so p = (1.03) 3° -1 = 1.427 and (17.30) shows that r = 0.754 which exceeds n by quite a margin. With more patient agents, whose pure discount rate is 1% per annum, p = (1.01) 3° - 1 = 0.348 and r = 0.269 which is less than n.

17.2 Applications of the Basic Model

In this section we show how the standard Diamond-Samuelson model can be used to study the macroeconomic and welfare effects of old-age pensions. A system of social security was introduced in Germany during the 1880s by Otto von Bismarck, purportedly to stop the increasingly radical working class from overthrowing his conservative regime. It did not help poor Otto—he was forced to resign from office in 1890—but the system he helped create stayed. Especially following the Second World War, most developed countries have similarly adopted social security systems. Typically such a system provides benefit payments to the elderly which continue until the recipient dies.

In the first subsection we show how the method of financing old-age pensions critically determines the effects of such pensions on resource allocation and welfare. In the second subsection we study the effects of a demographic shock, such as an ageing population, on the macroeconomy.

17.2.1 Pensions

In order to study the into the Diamond-Sam vides lump-sum transk young. It follows that changed from (17.2) an

Cr + St = vv, - 7-, 1

Ct°,1 = (1+ rt _

so that the consolida

Zt--1

Wr - Tr + 1 + rt,i

The left-hand side of 11 during youth plus the p Depending on the I\ can distinguish two prof government invests th in the next period in the

we have:

Zt+1 ( 1 + rt-Fi)Tt-

In contrast, in an unfu.. are covered by the taxe! Lt_1 old agents (each Tt in taxes) a PAYG syst (17.21) to:

Zt = (1 + n)Tt .

Fully funded pensions

A striking property of a this we mean that an ecc aspects to an economy- be demonstrated as folk First, we note that, by Zt+ i, disappear from th, these variables also do

plan, i.e. Cr and

e of population growth:

(golden rule)

•:fIns (17.28)-(17.29) are me suboptimal level, so consumption pattern

.it among generations ml), condition (17.28) no 'n golden rule (17.29). population growth (r < n) the economy is dynammodel reveals that such a es. Indeed, by computing _: the result in (17.16) we

model is:

(17.30)

numbers. Each period of n grows at 1% per annum num so 8 = 1- (0.95)3° = rate is 3% per annum, so which exceeds n by quite

• rate is 1% per annum, n.

icon model can be used - pensions. A system of Os by Otto von Bismarck, s from overthrowing his - yd to resign from office Ily following the Second 42I social security systems. elderly which continue

-icing old-age pensions :e allocation and welfare. - is shock, such as an

Chapter 17: Intergenerational Economics, II

17.2.1 Pensions

In order to study the effects of public pensions we must introduce the government into the Diamond-Samuelson model. Assume that, at time t, the government provides lump-sum transfers, Zt , to old agents and levies lump-sum taxes, Tt , on the young. It follows that the budget identities of a young household at time t are changed from (17.2) and (17.3) to:

so that the consolidated lifetime budget constraint of such a household is now:

The left-hand side of (17.33) shows that lifetime wealth consists of after-tax wages during youth plus the present value of pension receipts during old age.

Depending on the way in which the government finances its transfer scheme, we can distinguish two prototypical social security schemes. In a fully funded system the government invests the contributions of the young and returns them with interest in the next period in the form of transfers to the then old agents. In such a system we have:

In contrast, in an unfunded or pay-as-you-go (PAYG) system, the transfers to the old are covered by the taxes of the young in the same period. Since, at time t, there are 14_1 old agents (each receiving Zt in transfers) and Lt young agents (each paying Tt in taxes) a PAYG system satisfies Lt_iZt = Lt Tt which can be rewritten by noting

(17.21) to:

Fully funded pensions

A striking property of a fully funded social security system is its neutrality. With this we mean that an economy with a fully funded system is identical in all relevant aspects to an economy without such a system. This important neutrality result can be demonstrated as follows.

First, we note that, by substituting (17.34) into (17.33), the fiscal variables, Tt and Zt+ i, disappear from the lifetime budget constraint of the household. Consequently, these variables also do not affect the household's optimal life-cycle consumption plan, i.e. ci and C°±1 are exactly as in the pension-less economy described in

597

The Foundation of Modern Macroeconomics

section 17.1.1 above. It follows, by a comparison of (17.2) and (17.31), that with a fully funded pension system saving plus tax payments are set according to:

where S(Wt , rt+i ) is the same function as the one appearing in (17.6).

As a second preliminary step we must derive an expression linking savings of the young to next period's stock of productive capital. The key aspect of a fully funded system is that the government puts the tax receipts from the young to productive use by renting them out in the form of capital goods to firms. Hence, the economy-wide capital stock, Kt , is:

where Kr and Lt_1Tt_i denote capital owned by households and the government, respectively. The economy-wide resource constraint is still as given in (17.17) but the expression for total consumption is changed from (17.19) to: 4

Finally, by using (17.17), (17.38), and (17.36) we find that the capital market equilibrium condition is identical to (17.22). Since the factor prices, (17.15)—(17.16), are also unaffected by the existence of the social security system, economies with and without such a system are essentially the same. Intuitively, with a fully funded system the household knows that its contributions, Tt , attract the same rate of return as its own private savings, St . As a result, the household only worries about its total saving, St + Tt , and does not care that some of this saving is actually carried out on its behalf by the government. 5

= Yt + (1 - cS)Kt - Lt(St Tt) [Lt-iZt - (1 + rt)K il •

This final expression collapses to (17.38) because the term in square brackets on the right-hand side vanishes:

14_1 Zt - (1 + rt)K = Lt_i [Zt - (1 + rt)Tt_i] = 0.

5 An important proviso for the neutrality result to hold is that the social security system should not be too severe, i.e. it should not force the household to save more than it would in the absence of social security. In terms of the model we must have that Tt < (1 + n)kt+ i (see Blanchard and Fischer, 1989, p. 111).

Pay-as-you-go pensions

Under a PAYG system ti to (17.35). Assuming t time (so that Tt± i = Tr dation of (17.31)—(17.3 household:

(rt .“ - Wt — ,1+ ri

This expression is us, existence of a PAYG for young agents if th( population. Put daft.. lump-sum tax (subsidy

The household ma N. straint (17.39). Since th Euler equation is still g restrict attention to t.. (and technology is Co satisfies Cr = (1 + p function is: 1

St = Wt — T — Cl.

1-

=W t — T — (F.

=( 1 \

+p

It is easy to verify .

SW < 1, Sr > 0, —1 < S]

Since the PAYG pt erations it does not only private saving au; By combining (17.2L

6 Consumption by the

have Ltdr = Lt [Wt - St -

I

Ct = (rt (5)Kt + (1 - = Yr + (1 - 3)K, -

This final expression coital)! vanishes under the PAYG

1

and (17.31), that with a e set according to:

(17.36)

^sz in (17.6).

, n linking savings of the ry aspect of a fully funded

• young to productive use Hence, the economy-wide

(17.37)

►seholds and the govern- t is still as given in (17.17) ri (17.19) to: 4

(17.38)

that the capital market prices, (17.15)—(17.16), r system, economies with ely, with a fully funded attract the same rate of hold only worries about

s saving is actually carried

►.zents is Lt _i q = (rt + 6)K' that aggregate consumption

kets on the right-hand side

al security system should not ould in the absence of social e Blanchard and Fischer, 1989,

Chapter 17: Intergenerational Economics, II

Pay-as-you-go pensions

Under a PAYG system there is a transfer from young to old in each period according to (17.35). Assuming that the contribution rate per person is held constant over time (so that Tr± i = Tr = (17.35) implies that Z t+ 1 = (1 + n)T so that consolidation of (17.31)—(17.32) yields the following lifetime budget constraint of a young household:

This expression is useful because it shows that, ceteris paribus the factor prices, the existence of a PAYG system contracts (expands) the consumption possibility frontier for young agents if the interest rate exceeds (falls short of) the growth rate of the population. Put differently, if rt+i > n (rt±i < n) the contribution rate is seen as a lump-sum tax (subsidy) by the young household.

The household maximizes lifetime utility (17.1) subject to its lifetime budget constraint (17.39). Since the rate of return on household saving is rr+i , the consumption Euler equation is still given by (17.5). To keep matters as simple as possible we now restrict attention to the simple unit-elastic model for which utility is logarithmic (and technology is Cobb—Douglas). In that case, the optimal consumption plan satisfies Cr = (1 + p)Wt /(2 + p) and C'')+1 = (1 + rt+i )T/Vt /(2 + p) and the savings function is:

Sr = — T —

It is easy to verify that the partial derivatives of the savings function satisfy 0 < SW < 1, Sr > 0, —1 < ST < 0 (if rt+i > n), and 5T < —1 (if rt+i < n).

Since the PAYG pension is a pure transfer from co-existing young to old generations it does not itself lead to the formation of capital in the economy. Since only private saving augments the capital stock, equation (17.20) is still relevant. 6 By combining (17.20) with (17.40) we obtain the expression linking the future

6 Consumption by the old agents is L t, q.) = (rt + + (1 - 8)Kt + Lt _iZt . For young agents we have LtCr = Lt [Wt — St — Tt ] so that aggregate consumption is:

Ct = (rt + 8)Kt + (1 - 8)Kt + Lt-iZt + Lt [Wt - St - Tr]

= Yt + (1 - 8)Kt + [Lt _iZt - Lt Tt ] - Lt St .

This final expression collapses to (17.19) because the term in square brackets on the right-hand side vanishes under the PAYG scheme. Combining (17.17) and (17.19) yields (17.20).

The Foundation of Modern Macroeconomics

kMN k"(7) k*

Figure 17.2. PAYG pensions in the unit-elastic model

capital stock to current saving plans:

With Cobb-Douglas technology (yt 14') equations (17.15) and (17.16) reduce

to, respectively, Wt W (kt) = ELkti-EL and rt+i r(kt+i) = ( 1 - EL)kr13 - 8. By using these expressions in (17.41) we obtain the fundamental difference equation (in

implicit form) characterizing the economy under a PAYG system, k r±i = The partial derivatives of this function are:

where Sw and Sr are obtained from (17.40). We illustrate the fundamental difference equation in Figure 17.2. 7

7 The fundamental difference equation can be written as:

The second term on the right-hand side vanishes as k t±i 0 (sincer(kt+ i) +00 in that case). Hence, W(kMIN) = T. For kt < kmiN the wage rate is too low (W(kt ) < T) and the PAYG scheme is not feasible. By differentiating the fundamental difference equation we obtain:

In Figure 17.2, the (I,: unit-elastic Diamond-& Figure 17.1 and point B absence of social secur,,, t = 0 when the economy Members of the old gc_ not contributed anythin

(see equati spent entirely on addiul time t = 0 is now:

Co = (1 + n)[(1 +

and, since k0 is predeten In contrast, members introduction of the PAY( they must pay T in the o (1 + n)T in the next per

mined, the net effect resources (W0 defined ,a.

where the sign is ambig,, growth rate n. Furtherr depends on the capital s the savings behaviour of (17.43), however, that t to reduce saving by the dki /dT = g7- < 0. This ad by the vertical differ.,,,,

As a result of the poli■ from C to the ultimate it would have been with new steady state (i.e. since W'(x) > 0 and r'tx higher than it would ha,

1

It is straightforward to show for kr 00, and W'(kmIN) > 0. larger values of k,, and be, rm two intersections with the k,

= kt

-------- +1 = g(kt,0)

k,1 --=g(kt,T)

kr

del

(17.41)

'7.15) and (17.16) reduce

=( 1 - EL)kt13 - 8. By using al difference equation (in

system, kt+1 = g(kr, T).

(17.42)

(17.43)

he fundamental difference

.) +oo in that case). Hence, ue PAYG scheme is not feasible.

Chapter 17: Intergenerational Economics, II

In Figure 17.2, the dashed line, labelled "kt+ 1 = g(kt , 0)" characterizes the standard unit-elastic Diamond-Samuelson model without social security, i.e. it reproduces Figure 17.1 and point B is the steady state to which the economy converges in the absence of social security. Suppose now that the PAYG system is introduced at time t = 0 when the economy has an initial (non-steady-state) capital-labour ratio of ko. Members of the old generation at time t = 0 cannot believe their luck. They have not contributed anything to the PAYG system but nevertheless receive a pension of Z = (1 + n)T (see equation (17.35)). Since the old do not save this windfall gain is spent entirely on additional consumption. Consumption by each old household at

and, since ko is predetermined, so is the interest rate and dC0() / dT = (1 + n).

In contrast, members of the young generation at time t = 0 are affected by the introduction of the PAYG system in a number of different ways. On the one hand, they must pay T in the current period in exchange for which they receive a pension (1 + n)T in the next period. Since the wage rate at time t = 0, W(ko), is predetermined, the net effect of these two transactions is to change the value of lifetime resources (Wo defined in (17.39)) according to:

where the sign is ambiguous because r(ki ) may exceed or fall short of the population growth rate n. Furthermore, (17.45) is only a partial effect because the interest rate depends on the capital stock in the next period (k 1 ), which is itself determined by the savings behaviour of the young in period t = 0. It follows from (17.41) and (17.43), however, that the total effect of the introduction of the PAYG system is to reduce saving by the young and thus to reduce next period's capital stock, i.e. dki / dT = g7- < 0. This adverse effect on the capital stock is represented in Figure 17.2 by the vertical difference between points A and C.

As a result of the policy shock, the economy now follows the convergent path from C to the ultimate steady state E0 . It follows from Figure 17.2 that kt is less than it would have been without the PAYG pension, both during transition and in the new steady state (i.e. the path from C to E0 lies below the path from A to B). Hence, since W'(x) > 0 and r'(x) < 0, the steady-state wage is lower and the interest rate is higher than it would have been. The long-run effect on the capital-labour ratio is

It is straightforward to show that ,k(kt+i ) +oo for kt+ , -± 0, Vi(kt,i ) 0 for kt+1 oo, W' (kt ) 0 for kt oo, and 14P(kmiN) > 0. It follows that g(kt , T) is horizontal in kt = kMIN, is upward sloping for

larger values of kt , and becomes horizontal as kt gets very large. Provided T is not too large there exist two intersections with the kr + 1 = kt line.

601

The Foundation of Modern Macroeconomics

obtained by using (17.41) and imposing the steady state (kt + 1 = kr):

where 0 < gk < 1 follows from the stability condition.

The upshot of the discussion so far is that, unlike a fully funded pension system, a PAYG system is not neutral but leads to crowding out of capital, a lower wage rate, and a higher interest rate in the long run. Is that good or bad for households? To answer that question we now study the welfare effect on a

of a change in the contribution rate, T. As in our discussion of dynamic efficiency above we thus continue to ignore transitional dynamics for the time being by only looking at the steady state.

To conduct the welfare analysis we need to utilize two helpful tools, i.e. the indirect utility function and the factor price frontier. The indirect utility function is defined in formal terms by:

where A Y (CY , C°) is the direct utility function (i.e. (17.1)). The lack of subscripts indicates steady-state values and VAV represents lifetime household resources under the PAYG system:

For example, for the logarithmic felicity function (employed regularly in this chapter) the indirect utility function takes the following form:

where wo is a constant. 8

8 The explicit functional form of the indirect utility is obtained by plugging the optimal consumption levels, as chosen by the household, back into the direct utility function (17.1). The reader should verify the properties stated in (17.50)-(17.52).

The indirect utility t to be very useful below:

0AY = aw acY aAY s ar

r — n) a

aT

According to (17.50)—(1 wage rate and the into

W' (k)dk I dT < 0) but tf run, the effects of fact absence of a PAYG syst,_ But both W and r de

classical model) and dependency we obtain t property:

The slope of the factor tiating (17.15) and (17.1 that dWt ldrt = —kt . Fror

9 These properties are d AY [CY(W,r, T), C°( W, r, T

periods of life. By using this ic

aAY anY [acY aw 801 aw

It follows from the constra. same steps we obtain for a.k

aAY _anY pc)"

ar acY L ar

Using C° - (1 + r)T = (1

aAY anY racy aT = 80' aT

where the final result folio% 10 The factor price frontier I

( 1 - EL W = EL+ )

where the reader should ver.