Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

= kr) :

(17.46)

p

Ily funded pension system, f c - nital, a lower wage rate, 7- bad for households? To

on a steady-state generation In of dynamic efficiency for the time being by only

!pful tools, i.e. the indirect ity function is defined in

1)). The lack of subscripts usehold resources under

(17.48)

;)loyed regularly in this form:

(17.49)

I

_:ing the optimal consumption 17.1). The reader should verify

Chapter 17: Intergenerational Economics, II

The indirect utility function (17.47) has a number of properties which will prove to be very useful below:9

According to (17.50)—(17.51), steady-state welfare depends positively on both the wage rate and the interest rate. Since we saw above that the wage falls (dW I dT = W' (k)dk dT < 0) but the interest rate rises (dr I dT = r' (k)dk I dT > 0) in the long run, the effects of factor prices on welfare work in opposite directions even in the absence of a PAYG system (if T = 0).

But both W and r depend on the capital-labour ratio (as in the standard neoclassical model) and are thus not independent of each other. By exploiting this dependency we obtain the factor price frontier, Wt = 0(0, which has a very useful property:

The slope of the factor price frontier is obtained as follows. In general, by differentiating (17.15) and (17.16) (for rt ) we get drt = f"(kt )dkt and dWt = —kt f"(kt )dkt so that dWt ldrt = —kr . From this it follows that d2 Wt Idil = —dkt ldrt = — (ko.io

9 These properties are derived as follows. We start with the identity AY (W, r, T)

A Y [CY(W, r, T), C° (W , r, T)1, where OW , r, T) are the optimal consumption levels during the two periods of life. By using this identity, partially differentiating (17.1), and using (17.5) we obtain:

It follows from the constraint in (17.47) that the term in square brackets is equal to unity. Using the same steps we obtain for 8A Y/ar:

Using C° - (1 + n)T = (1 + r)S we obtain (17.51). Finally, we obtain for a/1- Y/8 T:

where the final result follows from the constraint in (17.47).

10 The factor price frontier for the Cobb-Douglas technology is given by:

0--,) /EL

W = r + EL)

where the reader should verify the property stated in (17.53).

603

The Foundation of Modern Macroeconomics

We now have all the necessary ingredients to perform our welfare analysis. By differentiating the indirect utility function with respect to T we obtain in a few steps:

where we have used (17.48) and (17.50)—(17.52) in going from the first to the second line and (17.53) as well as S = (1 + n)k in going from the second to the third line. The term in square brackets on the right-hand side of (17.54) shows the two channels by which the PAYG pension affects welfare. The first term is the partial equilibrium effect of T on lifetime resources and the second term captures the general equilibrium effects that operate via factor prices.

The expression in (17.54) is important because it illustrates in a transparent fashion the intimate link that exists between, on the one hand, the steady-state welfare effect of a PAYG pension and, on the other hand, the dynamic (in)efficiency of the initial steady-state equilibrium. If the economy happens to be in the golden-rule equilibrium (so that r = n) then it follows from (17.54) that a marginal change in the PAYG contribution rate has no effect on steady-state welfare (i.e. dA Y /dT = 0 in that case). Since the yield on private saving and the PAYG pension are the same in that case, a small change in T does not produce a first-order welfare effect on steady-state generations despite the fact that it causes crowding out of capital (see (17.46)) and thus an increase in the interest rate (since r'(k) < 0).

Matters are different if the economy is initially not in the golden-rule equilibrium (so that r n) because the capital crowding out does produce a first-order welfare effect in that case. For example, if the economy is initially dynamically inefficient (r < n), then an increase in the PAYG contribution rate actually raises steady-state welfare! The intuition behind this result, which was first demonstrated in the pensions context and with a partial equilibrium model by Aaron (1966), is as follows. In a dynamically inefficient economy there is oversaving by the young generations as a result of which the market rate of interest is low. By raising T the young partially substitute private saving for saving via the PAYG pension. The latter has a higher yield than the former because the biological interest rate, n, exceeds the market interest rate, r. The reduction in the capital stock lowers the wage but this adverse effect on welfare is offset by the increase in the interest rate in a dynamically inefficient economy. To put it bluntly, capital crowding out is good in such an economy.

Equivalence PAYG and deficit financing government debt

As was shown by Auerbach and Kotlikoff, a PAYG social security scheme can also be reinterpreted as a particular kind of government debt policy (1987, pp. 149-150).

In order to demonstrate debt into the model. Ili. between the pension insi debt as set out by Diam , Assume that the gore:: the old generations, and it of interest as capital. Ig; identity is now: I

Bt+1— Bt = rtBt +

where Bt is the stock of ill on existing debt (rtBt ) plu tax on the young and/or , Because government dt household is indifferent Consequently, the you1/2, in order to maximize Wei and (17.32). The savings I

St = S(I/Vt, rt+i),

where Wt is given by convenience:

1717t = Wr — Tt + , z

It remains to derive th capital formation. There saving is StLt . Saving can capital market equilibriur

LtSt = Bt+i +

We are now in the pa was proved by 1 (1988). Buiter and Klett-, librium with governmen

I

11 Consumption by old ag• we have LtC1 = Lr [Wt — Tr — $

Ct = (rt + 8)Kt +(1 — 81K. -

=Yt + (1 — 3)Kt + [(1 I

=Yr + (1 — s)K, + Br- . -

By combining the final expressi

Jr welfare analysis. By --= obtain in a few ste-s:

(17.54)

nn the first to the sec- ,.e second to the third - .54) shows the two first term is the par- )nd term captures the

in a transparent fash- steady-state welfare is (in)efficiency of the be in the golden-rule a marginal change in (i.e. dAY /dT = 0 pension are the same rder welfare effect on g out of capital (see

0).

lien-rule equilibrium

ta first-order welfare am ically inefficient v raises steady-state

nstrated in the pen- 1966), is as follows. In oung generations as a e young partially sub- ' r has a higher yield Is the market interest ut this adverse effect

.imically inefficient Ich an economy.

- scheme can also be 11987, pp. 149-150).

Chapter 17: Intergenerational Economics, II

In order to demonstrate this equivalency result, we now introduce government debt into the model. This model extension also allows us to further clarify the link between the pension insights of Aaron (1966) and the macroeconomic effects of debt as set out by Diamond (1965).

Assume that the government taxes the young generations, provides transfers to the old generations, and issues one-period (indexed) debt which yields the same rate of interest as capital. Ignoring government consumption, the government budget identity is now:

where Bt is the stock of public debt at the beginning of period t. Interest payments on existing debt (rtBt) plus transfers to the old are covered by the revenues from the tax on the young and/or additional debt issues (Bt-± i - Be).

Because government debt and private capital attract the same rate of return, the household is indifferent about the composition of its savings over these two assets. Consequently, the young choose consumption in the two periods and total saving in order to maximize lifetime utility (17.1) subject to the budget identities (17.31) and (17.32). The savings function that results takes the following form:

where ii7t is given by the left-hand side of (17.33) which is reproduced here for convenience:

It remains to derive the expression linking private savings plans and aggregate capital formation. There are Lt young agents who each save Sr so that aggregate saving is StLt . Saving can be in the form of private capital or public debt. Hence the capital market equilibrium condition is now: 11

We are now in the position to present an important equivalence result which was proved inter alia by Wallace (1981), Sargent (1987a), and Calvo and Obstfeld (1988). Buiter and Kletzer state the equivalence result as follows: "... any equilibrium with government debt and deficits can be replicated by an economy in

11 Consumption by old agents is Lt _i = (rt +8)Kt + (1 — 8)Kt + (1+ rt )Bt + L t _iZt . For young agents we have Ltd = Lt [Wt — Tt — St ] so that aggregate consumption is:

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

=Yr + (1 — 8)Kt + [(1 + rt)Bt + Lt-iZt —ItTt] — LtSt

=Yt + (1 - 3)Kt + Bt-Fi — LtSt •

By combining the final expression with the resource constraint (17.17) we obtain (17.58).

605

The Foundation of Modern Macroeconomics

which the government budget is balanced period-by-period (and the stock of debt is zero) by appropriate age-specific lump-sum taxes and transfers" (1992, pp. 2728). A corollary of the result is that if the policy maker has access to unrestricted age-specific taxes and transfers then public debt is redundant in the sense that it does not permit additional equilibria to be supported (1992, p. 28).

The model developed in this subsection is fully characterized (for t > 0) by the following equations:

where bt Bt /Lt is per capita government debt and where k0 and A) are both given. Equation (17.59) is consumption of an old household, (17.60) is the consumption Euler equation for a young household (see also (17.5)), (17.61) is (17.31) combined with (17.58), and (17.62) is the government budget identity (17.55) expressed in per capita form. Finally, we have substituted the rental expressions W t = W(kt) and rt = r(kt ) in the various equations (see (17.15) and (17.16) above).

The first thing we note is that the fiscal variables only show up in two places in the dynamical system. In (17.59) there is a resource transfer from the government to each old household (Fr) consisting of debt service and transfers:

(1 + r(kt))(1 + n)bt + Zt. (government to old)

Similarly, in (17.61) there is a resource transfer from each young household to the government (FIG) in the form of purchases of government debt plus taxes:

rt (1 + n)bt+i + Tt. (young to government)

Since there are Lt _i old and Lt young households, the net resource transfer to the government is Lt = 0, where the equality follows from the government budget constraint (17.62). Hence, in the absence of government consumption, what the government takes from the young it must give to the old. Once you know FIG you also know rtG° (1 + n)rtYG and the individual components appearing in the government budget identity (such as b t+i , bt , Zt , and Tt ) are irrelevant for the determination of the paths of consumption and the capital stock (Buiter and Kletzer, 1992, p. 17).

The equivalence result is demonstrated by considering two paths of the economy which, though associated with different paths for bonds, taxes, and transfers, nevertheless give rise to the same paths for the real variables, namely the capital stock and consumption by the young and the old. For the reference path, the

sequence tbt,20iVit-0 gi )r ict yt- c, given k(

{bt rt we can always fin‘ resulting sequences for ti

= trYtto,

The key ingredient of t resource transfers from ti ment to the old (F °) are to the following expre-

-= (1 + ?it

bt+1— bt+1 =

(1-i. n,

By using (17.63) in (17.3; for the same real variab! paths. Obviously the gu path satisfies the govemt As a special case of the PAYG system (studied abl One (of many) alternativ the young generations, i.

From PAYG to a funded s,

In the previous subsectl deficit financing and a there we showed how this section we continue without and then with b Up to this point we ha steady-state generations facing the economy, i.e. with the transition froiDiamond (1965, pp. 11: transitional welfare eft, As we argued above, ti pre-existing one) affects ations at the time of the the shock confers on t' gain utility to the tune effect on generations bo

A (and the stock of debt transfers" (1992, pp. 27— as access to unrestricted ndant in the sense that it

2, p. 28).

cterized (for t > 0) by the

(17.59)

(17.60)

(17.61)

(17.62)

e ko and bo are both given. 17.60) is the consumption - .61) is (17.31) combined ntity (17.55) expressed in 7essions Wt = W(kt) and

0above).

how up in two places in

r from the government id transfers:

(government to old)

young household to the

• • debt plus taxes:

(young to government)

rf• resource transfer to the lows from the government iment consumption, what - -.e old. Once you know al components appearing

nd Tr ) are irrelevant for apital stock (Buiter and

g two paths of the econ- :ds, taxes, and transfers, - ibles, namely the capi- x the reference path, the

Chapter 17: Intergenerational Economics, II

sequence ibt,2t,ttr0 gives rise to a sequence for the real variables denoted by {4, 'at°, kt rt given ko and bo. We can then show that for any other debt sequence

{kr,. we can always find sequences for taxes and transfers {Z r , tt }tx)0 such that the resulting sequences for the real variables are the same as in the reference path, i.e.

{ 'tY }tc'to = {s t°};to = {°171o, and fkrI tcto = Iktl;to-

The key ingredient of the proof is to construct the alternative path such that the resource transfers from the young to the government (riG) and from the government to the old (Fr) are the same for the two paths. These requirements give rise to the following expressions:

Zr — Zr = (1+ n) [( 1 + r(kr))bt — (1 + r(kr))bt], (17.63) bt+i — bt+i = ( 1 +1 n )[tt — tr]. (17.64)

By using (17.63) in (17.59) and (17.64) in (17.61) we find that these equations solve for the same real variables. As a result, the Euler equation (17.60) is the same for both paths. Obviously the government budget identity still holds. Finally, if the reference path satisfies the government solvency condition then so will the alternative path.

As a special case of the equivalence result we can take as the reference path the PAYG system (studied above), which has bt = 0, Tr = T, and Zr = + n)T for all t. One (of many) alternative paths is the deficit path in which there are only taxes on the young generations, i.e. Zr = 0, br = 77(1+0, and Tr = T — (1 + n)bt±i for all t.

From PAYG to a funded system

In the previous subsettion we have established the equivalence between traditional deficit financing and a PAYG social security system. As a by-product of the analysis there we showed how public debt affects the equilibrium path of the economy. In this section we continue our analysis of the welfare effects of a PAYG system, first without and then with bond policy.

Up to this point we have only unearthed the welfare effect of a PAYG system on steady-state generations (see (17.54)) and we have ignored the initial conditions facing the economy, i.e. we have not yet taken into account the costs associated with the transition from the initial growth path to the golden-rule path. As both Diamond (1965, pp. 1128-1129) and Samuelson (1975b, p. 543) stress, ignoring transitional welfare effects is not a very good idea.

As we argued above, the introduction of a PAYG system (or the expansion of a pre-existing one) affects different generations differently. The welfare of old generations at the time of the shock unambiguously rises because of the windfall gain the shock confers on them. From the perspective of their last period of life, they gain utility to the tune of LT (C?)dq? /dT = MT) > (17.44)). The welfare effect on generations born in the new steady state is ambiguous as it depends on

607

The Foundation of Modern Macroeconomics

whether or not the economy is dynamically efficient (see (17.54)). In a dynamically inefficient economy, r < n, all generations, including those born in the new steady state, gain from the pension shock. Intuitively, the PAYG system acts like a "chain letter" system which ensures that each new generation passes resources to the generation immediately preceding it. In such a situation a PAYG system which moves the economy in the direction of the golden-rule growth path is surely "desirable" for society as a whole.

As Abel et al. (1989) suggest, however, actual economies are not likely to be dynamically inefficient. If the economy is dynamically efficient, so that r > n, then it follows from, respectively, (17.44) and (17.54) that whilst an increase in T still makes the old initial generation better off, it leaves steady-state generations worse off than they would have been in the absence of the shock. Since some generations gain and other lose out, it is no longer obvious whether a pension-induced move in the direction of the golden-rule growth path is "socially desirable" at all.

There are two ways in which the concept of social desirability, which we have deliberately kept vague up to now, can be made operational. The first approach, which was pioneered by Bergson (1938) and Samuelson (1947), makes use of a so-called social welfare function. In this approach, a functional form is typically postulated which relates an indicator for social welfare (SW) to the welfare levels experienced by the different generations. Using our notation, an example of a social welfare function would be:

Once a particular form for the social welfare function is adopted, the social desirability of different policies can be ranked. If policy A is such that it yields a higher indicator of social welfare than policy B, then it follows that policy A is socially preferred to policy B (i.e. SWA, > Note that, depending on the form of the social welfare function w(.), it may very well be the case that some generations are worse off under policy A than under policy B despite the fact that A is socially preferred to B. What the social welfare function does is establish marginal rates of substitution between lifetime utility levels of different generations (i.e. (aw/aAi 1 )/(0w/ant), etc.). 12

The second approach to putting into operation the concept of social desirability makes use of the concept of Pareto-efficiency. Recall that an allocation of resources in the economy is called Pareto-optimal (or Pareto-efficient) if there is no other feasible allocation which (i) makes no individual in the economy worse off and (ii) makes at least one individual strictly better off than he/she was. Similarly, a policy is called Pareto-improving vis-a-vis the initial situation if it improves welfare for at least one agent and leaves all other agents equally well off as in the status quo.

Recently, a number of authors have applied the Pareto-criterion to the question of pension reform. Specifically, Breyer (1989) and Verbon (1989) ask themselves

12 An application of the social welfare function approach is given in the next subsection.

the question whether of a fully funded syster economy. This is a rely

generations gain if the Fl from equation (17.54) time of the policy shock system when it was you its contribution during o Pareto-improving.

Of course bond policy losses of the different ge it breaks the link betty, receipts by the old in the key issue is thus whether in the PAYG contribution found. It is thus not posy shock without making ‘:

17.2.2 PAYG pensions

In a very influential artic affects a household's s‘: this point) but also its di the model in order to de endogenous retirement. ment by assuming that lal keep the model as simp, work at all during the stk. assume furthermore th,: of a proportional tax on fair, i.e. an agent who wo, than an agent who has b possible that the PAYG s)

Households

The lifetime utility fur, general form by:

At'i = A Y (CtY 'i

—

17.54)). In a dynamically horn in the new steady system acts like a "chain resources to the gen- ; system which moves

kith is surely "desirable"

lies are not likely to be 'ent, so that r > n, then t an increase in T still

.-state generations worse

. Since some generations pension-induced move desirable" at all. irability, which we have Ina!. The first approach, (1947), makes use of a :tional form is typically

tr) to the welfare lev- )tation, an example of a

(17.65)

dopted, the social desirthat it yields a higher that policy A is socially --riding on the form of case that some genera- iespite the fact that A is Des is establish marginal rent generations (i.e.

pt of social desirability ', location of resources in -.ere is no other feasible worse off and (ii) makes "irly, a policy is called welfare for at least one

tatus quo.

rion to the question ( 1989) ask themselves

'e next subsection.

Chapter 17: Intergenerational Economics, II

the question whether it is possible to abolish a pre-existing PAYG system (in favour of a fully funded system) in a Pareto-improving fashion in a dynamically efficient economy. This is a relevant question because in such an economy, steady-state generations gain if the PAYG system is abolished or reduced (since r > n it follows from equation (17.54) that dAY /dT < 0 in that case) but the old generation at the time of the policy shock loses out (see (17.44)). This generation paid into the PAYG system when it was young in the expectation that it would receive back 1 + n times its contribution during old age. Taken in isolation, the policy shock is clearly not

Pareto-improving.

Of course bond policy constitutes a mechanism by which the welfare gains and losses of the different generations can be redistributed. This is the case because it breaks the link between the contributions of the young (Lt Tt ) and the pension receipts by the old in the same period (1, t_iZt)—compare (17.35) and (17.55). The key issue is thus whether it is possible to find a bond path such that the reduction in the PAYG contribution is Pareto-improving. As it turns out, no such path can be found. It is thus not possible to compensate the old generation at the time of the shock without making at least one future generation worse off (Breyer, 1989, p. 655).

17.2.2 PAYG pensions and endogenous retirement

In a very influential article, Feldstein (1974) argued that a PAYG system not only affects a household's savings decisions (as is the case in the model studied up to this point) but also its decision to retire from the labour force. We now augment the model in order to demonstrate the implications for allocation and welfare of endogenous retirement. Following the literature, we capture the notion of retirement by assuming that labour supply during the first period of life is endogenous. To keep the model as simple as possible, we continue to assume that households do not work at all during the second period of life. To bring the model closer to reality, we assume furthermore that the contribution to the PAYG system is levied in the form of a proportional tax on labour income and that the pension is intragenerationally fair, i.e. an agent who works a lot during youth gets a higher pension during old age than an agent who has been lazy during youth. Within the augmented model it is possible that the PAYG system distorts the labour supply decisions by households.

Households

The lifetime utility function of a young agent i who is born at time t is given in general form by:

1 Assuming an interior op the two periods and labor

Equation (17.72) is the tional form. The optimal 1 Equation (17.73) is the ul wage rate and the mari,L during youth. Equation potential to distort the 11 which determines whet: the effective tax rate, ttr . the right to a pension. Ce1 ally be negative, i.e. it 1993, p. 82).

Since all agents of a F drop the index i. In suct constant growth rate of tl

tt wt-1

t 1—Wt

Holding constant labour s ment subsidy (and tf, of the population and v, .

In the symmetric equilil of cr, Cto+i , and Nt as a sentative agent (We, rt- ,

C°±1 = C° (WtIv ,rt+i), and equilibrium) effect of a

labour supply decision ca

Nt ah =EIN/V\ (1 .I

where EwN is the uncomp that the effect of the c( sons. First, it depends

(17.70)

's contribution to the PAYG ere the proportional tax, :ant over time. Equation it (as in Breyer and Straub,

-1) is the total tax revenue

..agent i gets a share of this uring youth (the second

In system (as formalized in constraint, upon which

(17.71)

setting the household's n the aggregate supply of problem, the household as usual, by suppressing n (17.71) that the agent is

Equation (17.72) is the familiar consumption Euler equation in general functional form. The optimal labour supply decision is characterized by (17.73)—(17.74). Equation (17.73) is the usual condition calling for an equalization of the after-tax wage rate and the marginal rate of substitution between leisure and consumption during youth. Equation (17.74) shows to what extent the PAYG system has the potential to distort the labour supply decision. It is not the statutory tax rate, tL, which determines whether or not the labour supply decison is distorted but rather the effective tax rate, q t . By paying the PAYG premium during youth one obtains the right to a pension. Ceteris paribus labour supply, the effective tax rate may actually be negative, i.e. it may in fact be an employment subsidy (Breyer and Straub, 1993, p. 82).

Since all agents of a particular generation are identical in all aspects we can now drop the index i. In such a symmetric equilibrium we have M: = Nt and with a constant growth rate of the population (L e+1 = (1 + n)Lt ) (17.74) simplifies to:

Holding constant labour supply we find that the pension system acts like an employment subsidy (and if t < 0) if the so-called Aaron condition holds, i.e. if the growth of the population and wages exceeds the rate of interest (Aaron, 1966).

In the symmetric equilibrium, equations (17.71)—(17.73) define the optimal values of cr, c, +1' , and Nt as a function of the variables that are exogenous to the repre-

sentative agent (We , rt+i, and tL We write these solutions as ci = c, (wpT,rt+i), c?+1 = co(wp, , rt+ i), and Nt = N(V; \I , rt+i ), where Vt7pT -- WO — /I t ). The (partial-

equilibrium) effect of a change in the statutory tax rate, tL , on the household's labour supply decision can thus be written in elasticity format as:

The Foundation of Modern Macroeconomics

violated (if t > 0). Second, it also depends on the sign of E wN . We recall that clvw, > 0 (<0) if the substitution effect in labour supply dominates (is dominated by) the income effect. If the labour supply is upward sloping and the Aaron condition is satisfied then, for given factor prices, an increase in the statutory tax rate decreases labour supply.

The macroeconomy

We must now complete the description of the model and derive the fundamental difference equation for the economic system. We follow the approach of Ihori (1996, pp. 36-37). With endogenous labour supply, the number of agents (Li ) no longer coincides with the amount of labour used in production (LiNi ). By redefining the capital-labour ratio as ki Kt atNt), however, the expressions for the wage and the interest rate are still as in (17.15)-(17.16) and the factor price frontier is still as given in (17.53). Current savings leads to the formation of capital in the next period, i.e. Li Si = Kt± i. In terms of the redefined capital-labour ratio we get:

To characterize this fundamental difference equation we note that the labour supply and savings equations can be written in general functional form as:

By using these expressions in (17.77) we obtain the following expression:

s[wt(i-trt),rt+i,hwt±iNt+d=u+romwt+1(1-tb+i), rt+2,1(t+1. ( 17.80)

Clearly, since Wt = W(kt) and rt = r(kt), this expression contains the terms

and kr+2 so one is tempted that it is a second-order difference equation in the capital stock. As Breyer and Straub (1993, p. 82) point out, however, the presence of future pensions introduces an infinite regress into the model, i.e. since ti t depends on Nt±i (see (17.75)), it follows that if t+1 depends on Nt+2 which itself depends on kt+2, ki+3, and ti, t+2 . As a result, (17.80) depends on the entire sequence of present and future capital stocks, so that, even though we assume perfect foresight, the model has a continuum of equilibria." Since we assume that the population growth rate is constant, however, we can skip over the indeterminacy issue by first studying the steady state.

14 Indeterminacy and multiple equilibria are quite common phenomena in overlapping-generations models of the Diamond-Samuelson type. Azariadis (1993) gives a general discussion and Reichlin (1986) deals specifically with the case of endogenous labour supply.

The steady state

We study two pertinent geneity of labour suppi : show that in the unit-eh As before, the long-run which is defined as folio

AY (w r, h)

{0,c,

subject to: WN [1 -

11

Retracing our earlier deril utility function:

anY

aw

anY ar

The effect of a marginal now easily computed:

dAY aAY dW dtL — aw dtL

anY [

= a 0' N (1 -

= -N (r - n 1 + r

where we have used (17. (17.53) and (17.77) in g . worthy conclusions that is initially in the golden not produce a first-order

labour supply decision is that case (tt = tiAr -

rule equilibrium (r n) sign of the term in squa - case with lump-sum con through lifetime resour, (second term). It turns o of dr /d1-1, (and thus the s