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Constrained Pareto-optimal taxation of labour and capital incomes
Journal
of Public
Economics
CONSTRAINED
34 (1987) 355366.
North-Holland
PARETO-OPTIMAL TAXATION CAPITAL INCOMES* Lam-Gunnar
Department
of Economics,
OF LABOUR
AND
SVENSSON**
University
of Lund, S-22005
Lund, Sweden
Jijrgen W. WEIBULL** Institute for International
Received
Economic Studies, Vnioersity
May 1986, revised version
of Stockholm,
received
S-106 91 Stockholm,
September
Sweden
1987
With the Pareto principle as the sole normative criterion, necessary conditions for optimal tax rates on labour and capital incomes are established in an overlapping-generations model. The individuals in the economy have differing earning abilities and their labour supply is endogenous. The analysis focuses on intragenerational aspects and is restricted to linear taxation in steady state.
1. Introduction
The traditional approach to optimal taxation is based on the maximization of a social welfare function, the use of which requires interpersonal utility comparisons and a specified trade-off between efficiency and equity. In contrast, the present study derives necessary conditions for optimal taxation from the Pareto principle alone. In the context of optimal commodity taxation this more general normative approach can be found, for example, in Harris (1979) [cf. also Diewert (1978) and Guesnerie (1977)]. The present study, however, focuses on income taxation in a steady-state economy, taxation here being a means to obtain intragenerational distributional objectives (all generations living under identical conditions in steady state). Within this framework we analyze the range of factor prices, tax rates on labour and capital incomes, and public debt *Earlier versions of this paper were presented at the Stockholm School of Economics in December 1984, at the Institute for International Economic Studies, Stockholm University, in April 1985, and at the Fifth World Congress of the Econometric Society in Cambridge, MA, August 1985 [see Svensson and Weibull (1985)]. The authors are grateful for comments from Agnar Sandmo and two anonymous referees. **This research was supported by the Swedish Council for Research in the Humanities and Social Sciences, and part of Weibull’s work was also supported by the Bank of Sweden Tercentenary Foundation.
0047-2727/87/%3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland)
356
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
levels that are compatible with the Pareto principle. In particular, upper bounds on efficient marginal tax rates are derived. The positive side of our model combines the approaches to labour income taxation of Mirrlees (1971) and Sheshinski (1972) with the overlappinggenerational model of Diamond (1965). The Diamond model is also used by Atkinson and Sandmo (1980) in a study of the consequences of labour and capital income taxation for the intergenerational allocation of consumption. In theirs as well as in our analysis the importance of capital income taxation is clear - distortions in the labour market caused by the labour income tax may be reduced when combined with a capital income tax. More closely related to the present study, however, are the optimaltaxation approaches to intragenerational welfare by Hamada (1972), Ordover and Phelps (1975) and Ordover (1976). As in our study, Hamada derives an upper bound on the labour-income tax rate [cf. also Sheshinski (1972) and Svensson and Weibull (1986)] but our upper bound is generally lower, in part depending on the absence of capital income taxation in Hamada. Ordover and Phelps (1975) and Ordover (1976) assume that individual preferences are identical and social preferences Rawlsian. We show that their efficiency result is a consequence merely of the Pareto principle. To the best of our knowledge, the present study is the first welfare analysis of labour and capital income taxes which does not presuppose interpersonal utility comparisons. The analysis is restricted to linear taxation of labour and capital incomes, generally at different rates, in combination with a lump-sum transfer. Government may hold claims to productive capital and issue bonds. As in Diamond’s model the consumers live for two periods, work (if at all) and save in the first period only, and leave no bequests. They hold their savings in the form of claims to capital goods or as government bonds, the two forms being perfect substitutes in the present model. In each generation there is a linite number of individuals with different preferences and, just as in Mirrlees’ and Sheshinski’s models, differing earning capabilities. The population grows at a constant, nonnegative rate and the analysis is confined to steady-state equilibria - this seems to be the smallest possible extension from taxation of labour incomes to include capital income taxation as well. In such a framework, we derive global and local conditions that are satisfied by all linear tax schedules which are optima of Paretian social orderings. The global conditions are given in terms of aggregate supply and demand functions (Proposition 4), and the local conditions in terms of (uncompensated) elasticities of aggregate labour supply and private savings Ceqs. ( 12)< 1611. Tax rates which are efficient in this sense require efficient factor prices and an efficient level of public debt (credit). As in Ordover and Phelps (1975) and Ordover (1976), but unlike Hamada (1972), the Golden Rule holds for a
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
351
closed economy, i.e. the gross (real) interest rate should equal the population growth rate (Proposition 1). Somewhat extending Theorem 1 in Samuelson (1975) we show that Golden Rule equilibrium can in fact be supported by means of public debt policy alone - without any taxes or transfers whatsoever (Proposition 2). 2. The model Only one good is produced in the economy, a good that can be used both for consumption and capital formation. Let Y,,C, and K, be gross output, aggregate consumption and capital, in period t. Assuming a constant rate 6 of capital depreciation, net output is divided between consumption and investment, Y - 6K, = C, + (K, + 1 -K,). In steady-state growth at a constant rate n over time, this equation may be written Y - (n + 6)K = C. We assume n and 6 to be nonnegative and their sum to be positive throughout the study. 2.1. The household sector
Let the working time of a young individual of type i be xie [0, l), his leisure being zi= 1 -xi. The corresponding quantity of supplied labour is mixi, where m, is a positive scalar, his ‘working ability’. Let ci be his first period consumption, and e, his consumption in the retirement period. We assume that each individual of type i (i= 1,. . . , N) chooses a triplet (ci,ei,zi) so as to maximize his utility ui(ci,ei,zi) in the ‘consumption’ set X = (0, + co)’ x (0, 11, subject to the budget constraints C~+S~=~MQX;+CJ and
eis(l+p)si.
(1)
Here o =( 1 - t,)w and p =( 1 - t,)r are the net-of-tax wage and interest rates, respectively, c a lump-sum transfer, and si the savings of individual i from his first (active) period to his second (‘retirement’) period.’ Assuming a perfect capital market, the temporary budget constraints in (1) may be consolidated to a life-time budget constraint: Ci +
eJ( 1 + p) +
OmiZi
5
wmi
+
0.
(2)
The budget set Bi of individual i is hence the subset of points in his consumption set X which satisfy (2). Each utility function ui is continuous, strictly increasing in all three ‘More generally, if g1 (u2) is the transfer given in the first (second) period of life, then the equivalent life-time transfer is CT= ol + uJ( 1+ p). F or simplicity, we assume that all transfers are given in the first period, i.e. c2 = 0. A treatment of the general case is given in an earlier version of the paper, see Svensson and Weibull (1985).
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
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arguments and strictly quasi-concave, with the closure of its indifference surfaces contained in the consumption set X. All individuals will then choose positive leisure, and positive consumption in both periods, by definition of X, while it is not excluded that some (or all) will choose not to work at all. If the population growth rate n is zero, there is a constant population consisting of N young and equally many old individuals, and part of the steady state assumption is that in every generation there is exactly one individual with preferences ui and working ability m,. Likewise, in the case of a positive growth rate, there are N utility functions and ability parameters, and each sub-population of young individuals of type i grows over time at the same rate n. Without loss of generality, the number of young individuals of each type i in the period under consideration is normalized to one. Hence, there are N young and N/( 1 + n) old individuals in the ‘representative’ period. The aggregates associated with the household sector in a period are: the consumption by the young, C, =I Ci, and old, C, =CeJ( 1 +n), and the associated aggregate Cd = C, + C,. The young generation supplies labour LS=Cmixi, and saves S=csi for the subsequent period. All these aggregates are functions of the triplet (0, p, a).’ 2.2. The production sector Production possibilities are represented by a constant returns-to-scale (aggregate) production function F(K, L). Letting k= K/L be the capitallabour ratio, production per unit of labour is f(k) = F(k, 1). As usual, f is taken to be twice continuously differentiable on (0, + co) with f’>O, f” ~0, and f’(k) ranging from plus infinity to zero as k increases from zero to infinity. Let r and w be the (real) factor prices for capital and labour services paid by the production sector. Profit maximization gives the factor-price frontier 4 defined by w=+(r)
r=_/-‘(k)-6
iff w=f(k)-kf’(k)
for ‘Orne k’“’
Let 5 be the corresponding capital demand function, i.e. k= t(r) iff f’(k) = and note that q%(r)and t(r) are defined for all r exceeding minus 6.
r+6,
2.3. The public sector The public sector selects a transfer r~20 and tax rates t,, t, S 1 such that all individual budget sets Bi are non-empty. Consequently, the public sector’s *In the more general case of one transfer in each period of life, S is a function each transfer, fll and c2, not only of their discounted sum 6, see footnote 1.
of o,p
and
L.G. Suensson and J.W Weibull, Pareto-optimal taxation
359
budget balance, with G denoting its (net) holdings of capital in the period under consideration, is (4) The first term represents tax revenues from wage earnings of the currently young, the second tax revenues from interest earnings of the currently old, who are a factor (1 +n)-’ fewer. The third term, finally, represents the interest-augmented value of the public sector’s net capital holdings G/( 1 + n) ‘inherited’ from the previous period. Subtracting current transfer payments to the N young individuals, we obtain current public capital holdings, to be ‘handed over’ to the next period. Employing the aggregate notation introduced above, this budget equation may be written: t,wL + t,rs/( 1 +n) +(I-
n)G/( 1 + n) = Na.
(5)
Note that if the (real) interest rate equals the population growth rate, then public (net) asset holdings vanish from the public budget sheet - because then their growth matches interest payments. 2.4. Equilibrium At given exogenous data, all decisions taken by the economic agents in the economy are functions of the vector q= (w, r, t,, t,, r~,G), the state of the economy. Let E be the set of (steady-state, tax) equilibria defined as follows: q E E iff there is a factor combination (K, L) > 0 which is (i) profit maximizing (with respect to w and r), and (ii) market clearing, i.e. satisfying
Remark.
F(K, J!,)-(n + 6)K = Cd (product market),
(6)
L=L”
(labour market),
(7)
(l+n)K=S+G
(capital market).
(8)
If eqs. (6)-(8) hold, then so does eq. (5), by Walras’ law.
3. Efficient factor prices Let aeXN be a consumption allocation, i.e. a = 0}, where C and L are the aggregates associated with the allocation in question. Let E** be the subset of equilibria q such that the corresponding allocation a(q) is not
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
360
Pareto dominated by another allocation in A, i.e. E** is the set of Pareto equilibria in the usual sense. 3 Let AE be the subset of technically feasible allocations (a) which correspond to equilibria in the economy [i.e. such that a=a(q) for some 4 in E] and let E* be the subset of equilibria q that are constrained Pareto efficient in the sense that the corresponding allocation a(q) is not Pareto dominated by an allocation in AE (i.e. by another equilibrium allocation). Moreover, let Eg be the set of Golden Rule equilibria, E*= {qE E; r=n}, and let E” be the set of tax-free equilibria, i.e. ejkient
E’={qEE;t,=t,=O}. Evidently E**
is a subset of E*, and E* contains the set of equilibria relevant for optimal taxation, since Paretian social preferences never select equilibria which are not constrained Pareto efficient. In fact, all tax-free Golden Rule equilibria, and for practical purposes no other, are Pareto efficient, and all constrained Pareto efficient equilibria satisfy the Golden Rule: Proposition
1.
Egn E” c E** c E* c Eg. Ifpreferences are smooth,4 then E** =
Eg n E”.
(Proofs are given in an appendix at the end of the paper.) This result is similar in spirit to the (static) production-efficiency result in Diamond and Mirrlees (1971, Theorem 4), and relies on the assumption that the government may freely levy taxes on both capital and labour incomes, and freely choose the level of public debt. This is the reason why Hamada (1972) reaches another conclusion: restricting t, to be zero he finds that generally E* is not a subset of Eg. Ordover and Phelps (1975) and Ordover (1976), on the other hand, show that E* is a subset of Eg, under the assumptions that social preferences are Rawlsian and that all individuals have identical preferences fulfilling certain normality conditions [see Ordover and Phelps (1975, p. 665 and eq. (50)) and Ordover (1976, pp. 142-143 and Proposition 12)].5 The following proposition shows the non-emptiness of E** and hence also of E*, and states that Golden Rule equilibrium can be supported without any taxes or transfers whatsoever, simply by a suitable choice of public capital holdings. In order to construct such a tax-free Golden Rule equilibrium 4 (hence an element in E**, by Proposition l), let 2, = f, =0 and i= n. Then S =0 by the 3As usual, an allocation a is said to be Pareto dominated by an allocation a’ if ui(ai) 5 ui(a;) for all i, with strict inequality for some i. 4The preferences of an individual i are said to be smooth if the indifference surfaces in X have a unique normal vector at every point. 5Recall the assumption n+6 >O. Were instead n+6 SO, then Eg would be empty - efficiency would require an infinite productive capital stock.
L.G. Svensson and J.K
Weibull, Pareto-optimal taxation
361
public budget equation (5) (since ?=n). Let 4 =4(n), ft= r,+(n),e= L’(fi, n, 0) and R =LL. Then profits are maximized and the labour market clears. By Walras’ law, also the product market clears. Moreover, any level of public capital G satisfies the public budget equation (5) since i= n. Hence, by choosing G = (1 + n)R - 9, also the capital market clears. Letting B denote the aggregate savings ratio of the young generation, i.e. G=s/(i?L), we have established: Proposition 2. 4~ E**. If preferences G>O if&<(l+n)<(n)/+(n).
are smooth, then E**=
(4). Moreover,
In other words, at a given growth rate and technology, the Pareto optimal public capital stock G is positive if the private savings ratio is small, negative if it is large. Proposition 2 extends the observation in Diamond (1965) that a laissezfaire economy generally does not achieve Golden Rule equilibrium, from his case of exogenous labour supply to the present case of elastic labour supply: generally G#O. Moreover, Proposition 2 generalizes Theorem 1 in Samuelson (1975) a result saying that Golden Rule equilibria can be supported by appropriate social security systems (p. 541). Like Diamond (1965), Samuelson refers to an economy with a representative household in each generation who supplies labour inelastically. Hence, social security systems may then be financed by lump-sum taxes and transfers. In the present model, however, earning abilities differ among individuals so social security programs generally require distortionary taxation. However, Proposition 2 states that Golden Rule equilibrium can be supported without any such program - it is sufficient to choose the appropriate level of public net asset holdings. 4. Efficient tax rates By Proposition 1 all efficient tax rates, i.e. tax rates supporting equilibria in E*, are such that the corresponding interest rate equals the growth rate. What more can be said about them? In fact, some quite explicit necessary conditions can be obtained if aggregate leisure is non-inferior and aggregate consumption normal: NL: L’(o, p, CT)is non-increasing
in e.
NC: Cd(o, p, a) is strictly increasing in g. Generally, any one of the two tax rates (but not both) may be negative. However, the subsequent analysis is restricted to the identification of inefficient combinations of nonnegative tax rates. One easily shows
J.PE.
D
362
L.G. Svensson and 1.W Weibull, Pareto-optimal taxation
Proposition 3. Suppose n >O. For every pair of tax rates (t,, t,) in [0,1)’ there exists a corresponding Golden Rule equilibrium. The associated transfer a is unique under NL and NC. This proposition allows us to define $:[O,l)‘+R+ by tj(t,,t,)=a.6
a (per-capita)
tax-revenue
function
Corollary 3.1. $ is continuous. It is positive on (0, 1)2 and vanishes where t,=t,=O, and where t,= 1. We now identify a range of tax rates that are inefficient. The idea is to exclude such equilibria (or tax rates) for which there is another Golden Rule equilibrium with lower tax rates t, and t,, but not lower life-time transfer a since then all individuals’ budget sets are larger in the other equilibrium. For this purpose, let
T= {(t,, t,) E (0, l12;$(t’,, ti) < $(t,, t,) forall K, ti) #_(t,,t,) suchthat Ok,t3 S(t,, t,)>.
(9)
Clearly all pairs of (nonnegative) efficient tax rates lie in the set T - for otherwise there wou!d exist another Golden Rule equilibrium with at least one of the tax rates lower but still not a lower transfer. In sum: Proposition 4. Suppose NL, NC and n>O. If (t,, t,) 20 tax rates, then (t,, t,) E 7:
is an efficient pair of
There is a local counterpart to this global result. Granted I++is differentiable, Proposition 4 implies that of t, and t, are positive and belong to T, then a$/&, 20 and &j/&,20. Differentiation of I+$ gives the following necessary conditions for efficiency:
c; 2 WL;,
(11)
where r =n, w is the corresponding wage rate 4(n), and subscripts signitiy partial derivatives.’ If we let et=wLi/LS, et=pL;/L”, e~=oS,/S and e; = pS,/S, then (10) is equivalent to (12) below, and (11) to (13): 6Note that aggregate tax revenues equal Na in Golden Rule equilibrium, see eq. (5). However, this equivalence between tax revenues and transfers does not hold in the more general case when some transfers are given in the second period of life as well, see Svensson and Weibull (1985). ‘Here NL and NC have been slightly strengthened to require L:sO and Cd,>@ see appendix for a derivation.
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
363
( 1 + eE)t, s 1 -ye&,
(12)
(1 +ez)t,s
(13)
1 --YP1eitw,
where y = [rS/(wL)]/( 1 + n), r = n and w = 4(n). As expected, the constraint on efficient tax rates on labour (capital) incomes is stricter the more elastic is the aggregate supply of labour (private capital). When t, =0 and/or the ‘cross elasticity’ ei is zero (or n= 0), then (12) reduces to (1 +e&,~
1.8.9
(14)
On the other hand, if capital incomes are taxed (t,>O), and the cross elasticity e”, is positive (negative), then the necessary condition (12) imposes a harder (softer) constraint on efficient tax rates t, on labour incomes, ceteris paribus. The additional term on the right-hand side of (12), which accounts for distortions due to t,, is proportional to y, hence proportional also to the ratio between the capital-income tax base and the labour-income tax base. Likewise, when t, = 0 and/or ei = 0, then (13) becomes (l+e:)t,51,
(15)
a private-savings counterpart to (14), while if labour incomes are taxed and aggregate labour supply is sensitive to the net interest rate, then the efficiency constraint (13) on t, is harder (softer) when the cross elasticity e,Lis positive (negative). Finally, if the tax rates t, and t, for some reason are constrained to be equal, then this additional constraint may render some otherwise inefficient combinations of tax rates efficient. Hence, the upper bound on a pure income tax rate t is generally less strict than the bounds given above. With the notation of eqs. (12) and (13) one obtains the following necessary condition (when n > 0):
(t,>O),
[l +eE+$+y(l
+ei+ez)]tsl
+y.
(16)
This condition is in fact implied by conditions (12)-(13) together with the condition t w = t,= t.” On the other hand, the last condition and (16) do not ‘One easily shows that (14) indeed is the inequality obtained in the zero growth case. In that case, there is no correspondence to (13), since capital incomes are then zero in Golden Rule equilibrium. The efficiency condition (14) has been derived for taxes in labour incomes when capital is fixed and production is linear in labour inputs, see section 4.1 in Svensson and Weibull (1986). ‘Assuming identical individual preferences and utilitarian social preferences, Hamada (1972) obtains Sheshinski’s (1972) upper bound l/( 1 +e,J, where emin is the lowest individual laboursupply elasticity, never exceeding et and generally falling short of it. Hence the upper bound in (14) is in general lower than Hamada’s. “‘Multiply both sides of (13) by y>O, and add (12) and (13).
364
L.C.
Soenssonand J.W Weibull, Pareto-optimal taxation
together imply conditions (12) and (13). In this sense, the necessary conditions for efficiency are less strict when the additional constraint t,= t, is imposed.
Appendix Lemma A. Suppose r=nzO, 05 t,< 1 and Oj t,< 1. Then (TE R + and a G E R such that q = (4(r), r, t,, t,, o, G) E E. Proof
Let r=n,
w=+(n),
Ost,
Ost,
and k=t(n).
there
exists
a
Define g:R++R
by
g(0) =
Cd(o,p,4 - WLYW, p, 4.
Clearly g is continuous, and by (2) and monotonicity of preferences: aci+ ae,/(l+p)zNg. Hence ~c,-++co and/or Ce,-++cc as o++co. It follows that g(o)++co as r~-++cc (Cd=~ci+~ei/(l+n)++co, and L” is bounded by C mi). Moreover Cd 5 C ci + C e,/( 1 + p), and C Ci+ 1 eJ( 1+ p) = oL” when g=O, by (2). Thus g(0) s(ow)L”50, so g(a) =0 for some aZ0. It is easily verified that such a (T gives equilibrium: eqs. (6) and (7) are met, and let G=(l+n)K-S. Proposition I. To prove E* c Eg, first define a: (0, + co)+ R by a(k) = f(k) (6 +n)k. Since f is strictly concave, so is a. Let b:( -6, + co)-+R be defined by b(r) =a(t(r)). Since (‘~0, b is differentiable and strictly quasi-concave, and by (3) b achieves its maximum at r=n: b’= [f’-(6+n)]t’. Note finally that, for any q=(w,r,t,,,,t,,a,G), we have g(a)=Cd(W,p,a)-b(r)L”(o,p,a), by (3), where g:R + -+ R now is (more generally) defined by
g(o)= Cd(o,p,4 - Cw-(I - n)lL”(wp,4. Now suppose q = (w, r, t,, t,, 6, G) E E, where r # n. We may then find some r’ between r and n, and some associated t: 5 1 such that (1 - tjr’ = p. For if p=O, let r’=n and tL=l. If pO, then t,< 1 and r>O, and we may choose r’ > 0 between r and n, and again let t;= 1 -p/r’. Thus p’ = p, and by letting tk be such that (1 - t’,)w = w, we clearly have tk 5 1 and w’ =w as well. Finally note that b(r’)> b(r) since r lies between r and n. Since q E E, we have Cd(o, p, a) = [w +(r - n)k]L”(o, p, a), by (3), (6) and (7). Hence, O=g(a) > Cd(o, p, 6) - b(r’)L”(o, p, a). By continuity of L” and Cd, and since L” is bounded while Cd is not (see proof of Lemma A), there is g’>g such that Cd(o, p, a’) = b(r’)L”(o, p, 0’). But then q’=(w’, r’, t:, t:, cr’,G’) E E if
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
365
w’= +(r’) and G’ = (1 + n)K - S. It follows that q does not belong to E*: 0’ > c, w’=o and p’=p. To prove EBn E” cE**: Suppose q* belongs to Eg n E” but not to E**. Then there is a feasible allocation, a’ = (c’, e’, x’) E A, which Pareto dominates the equilibrium allocation a* = a(q*). Hence by (2):
c:+ e;/( 1 + I*) 2 w*m,x; for all i, with strict inequality for some i. [Note that q* lEg n E” implies, by eq. (5), that u* =O.] Aggregation of this equation gives C’ = C; + C;>w*L’. Let K’ be a capital stock such that a’ can be produced. Then F(K’, I,‘)-(6+n)K’zC’> w*L’=[f(k*)-k*f’(k*)],!,‘=[f(k*)-(6+n)k*]L’by (3). Hence f(k’) -(6 + n)k’>f(k*) -(6 + n)k*, which is a contradiction since f(k) -(6 + n)k is maximal at k = k*. To prove E** c EB n E”: First note that E** c E* n E* by the first part of the proof. Moreover, differentiability of all indifference surfaces implies unique prices (w, r, o, p) in equilibrium, and hence t, = t,=O for a Pareto efficient equilibrium, by standard arguments. Proposition 3. Existence follows directly from Lemma A. Clearly the function g in the proof of that lemma is strictly decreasing by NC and NL. Thus fs is unique. Corollary 3.1. Let g be as in the proof of Lemma A. Then (~=$(t,+,, t,) is its unique zero. Since g is strictly increasing, and L” and Cd are continuous in (t,, t,), also II/ is continuous. By eq. (5), 0 >O unless t,= t,=O. For every a>O, L’(o,p, cr)+O when t,+l while Cd(m,p, o)-+o. Hence, for every a>0 there exists a t, < 1 such that g(a) > 0, while g(0) SO. Thus +(t,, t,) -+O when t,-+l. Inequalities
(1+06).
If qEE, then, by (3), (6) and (7):
Cd(o, p, a) = [w +(r-- n)k]L”(o, p, a). Under NL and NC, and the hypothesis w +(I -n)k>O, this equation defines 1(1by $(t,, t,) = 0‘. Differentiation with respect to t, and t, gives (Cj-[w+(r-n)k]L”,)d$/&,=w(C~--[w+(r-n)k]LS,)
and
Hence, if NL and NC are slightly strengthened to require L”,sO and Ct> 0, respectively (actually Cz > wL”, is sufficient), then
366
L.G. Svensson and .l.W Weibull, Pareto-optimal
&,9/&,,,~0 and, provided
iff
C;z[w+(r-n)k]LS,
iff
Cz2[w+(r--n)k]Li.
taxation
r > 0,
d$/&,20 This establishes equilibrium
inequalities
(lo)41
1): let r = n, where n > 0 by hypothesis.
In
[by aggregation of (2) and (3)]. Derivation w.r.t. o and p gives (12)413). If t, is constrained to equal t,, then the necessary condition for efficiency becomes dll/(t, t)/dt 20, where d+(t, t)/dt = a$(t, t)/dt, + at&t, t)/&,. Hence, when n=r, t,=t,=t, LzsO and Cz>O: d$/dt 2 0 the last inequality
iff
w( Cd, - wL”,) + I( Cz - wL;) 2 0,
being equivalent
to (16).
References Atkinson, A.B. and A. Sandmo, 1980, Welfare implications of the taxation of savings, Economic Journal 90, 529-549. Diamond, P.A., 1965, National debt in a neoclassical growth model, American Economic Review 55, 11261150. Diamond, P.A., 1970, Incidence of interest income tax, Journal of Economic Theory 2, 21 l-224. Diamond, P.A. and J.A. Mirrlees, 1971, Optimal taxation and public production, American Economic Review 61, 8-27 and 261-278. Diewert, W.E., 1978, Optimal tax perturbations, Journal of Public Economics 10, 139-178. Guesnerie, R., 1977, On the direction of tax reform, Journal of Public Economics 7, 179-202. Hamada, K., 1972, Lifetime equity and dynamic efficiency on balanced growth path, Journal of Public Economics 1, 379-396. Harris, R.G., 1979, Efficient commodity taxation, Journal of Public Economics 12, 27-39. Mirrlees, J.A., 1971, An exploration in the theory of optimum income taxation, Review of Economic Studies 38, 175-208. Ordover, J.A.. 1976, Distributive justice and optimal taxation of wages and interest in a growing economy, Journal of Public Economics 5, 139160. Ordover, J.A. and E.S. Phelps, 1975, Linear taxation of wealth and wages for intragenerational lifetime justice: Some steady-state cases, American Economic Review 65, 66C673. Phelps, E.S., 1961, The golden rule of accumulation: A fable for growthmen, American Economic Review 51, 638-643. Samuelson, P.A., 1975, Optimum social security in a life-cycle growth model, International Economic Review 16, 539-544. Sheshinski, E., 1972, The optimal linear income tax, Review of Economic Studies 39, 297-302. Svensson, L.-G. and J.W. Weibull, 1985, Efficient income taxation in steady state, Institute for International Economic Studies, Seminar Paper no. 329, Stockholm University. Svensson, L.-G. and J.W. Weibull, 1986, An upper bound on optimal income taxes, Journal of Public Economics 30, 165-182.
of Public
Economics
CONSTRAINED
34 (1987) 355366.
North-Holland
PARETO-OPTIMAL TAXATION CAPITAL INCOMES* Lam-Gunnar
Department
of Economics,
OF LABOUR
AND
SVENSSON**
University
of Lund, S-22005
Lund, Sweden
Jijrgen W. WEIBULL** Institute for International
Received
Economic Studies, Vnioersity
May 1986, revised version
of Stockholm,
received
S-106 91 Stockholm,
September
Sweden
1987
With the Pareto principle as the sole normative criterion, necessary conditions for optimal tax rates on labour and capital incomes are established in an overlapping-generations model. The individuals in the economy have differing earning abilities and their labour supply is endogenous. The analysis focuses on intragenerational aspects and is restricted to linear taxation in steady state.
1. Introduction
The traditional approach to optimal taxation is based on the maximization of a social welfare function, the use of which requires interpersonal utility comparisons and a specified trade-off between efficiency and equity. In contrast, the present study derives necessary conditions for optimal taxation from the Pareto principle alone. In the context of optimal commodity taxation this more general normative approach can be found, for example, in Harris (1979) [cf. also Diewert (1978) and Guesnerie (1977)]. The present study, however, focuses on income taxation in a steady-state economy, taxation here being a means to obtain intragenerational distributional objectives (all generations living under identical conditions in steady state). Within this framework we analyze the range of factor prices, tax rates on labour and capital incomes, and public debt *Earlier versions of this paper were presented at the Stockholm School of Economics in December 1984, at the Institute for International Economic Studies, Stockholm University, in April 1985, and at the Fifth World Congress of the Econometric Society in Cambridge, MA, August 1985 [see Svensson and Weibull (1985)]. The authors are grateful for comments from Agnar Sandmo and two anonymous referees. **This research was supported by the Swedish Council for Research in the Humanities and Social Sciences, and part of Weibull’s work was also supported by the Bank of Sweden Tercentenary Foundation.
0047-2727/87/%3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland)
356
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
levels that are compatible with the Pareto principle. In particular, upper bounds on efficient marginal tax rates are derived. The positive side of our model combines the approaches to labour income taxation of Mirrlees (1971) and Sheshinski (1972) with the overlappinggenerational model of Diamond (1965). The Diamond model is also used by Atkinson and Sandmo (1980) in a study of the consequences of labour and capital income taxation for the intergenerational allocation of consumption. In theirs as well as in our analysis the importance of capital income taxation is clear - distortions in the labour market caused by the labour income tax may be reduced when combined with a capital income tax. More closely related to the present study, however, are the optimaltaxation approaches to intragenerational welfare by Hamada (1972), Ordover and Phelps (1975) and Ordover (1976). As in our study, Hamada derives an upper bound on the labour-income tax rate [cf. also Sheshinski (1972) and Svensson and Weibull (1986)] but our upper bound is generally lower, in part depending on the absence of capital income taxation in Hamada. Ordover and Phelps (1975) and Ordover (1976) assume that individual preferences are identical and social preferences Rawlsian. We show that their efficiency result is a consequence merely of the Pareto principle. To the best of our knowledge, the present study is the first welfare analysis of labour and capital income taxes which does not presuppose interpersonal utility comparisons. The analysis is restricted to linear taxation of labour and capital incomes, generally at different rates, in combination with a lump-sum transfer. Government may hold claims to productive capital and issue bonds. As in Diamond’s model the consumers live for two periods, work (if at all) and save in the first period only, and leave no bequests. They hold their savings in the form of claims to capital goods or as government bonds, the two forms being perfect substitutes in the present model. In each generation there is a linite number of individuals with different preferences and, just as in Mirrlees’ and Sheshinski’s models, differing earning capabilities. The population grows at a constant, nonnegative rate and the analysis is confined to steady-state equilibria - this seems to be the smallest possible extension from taxation of labour incomes to include capital income taxation as well. In such a framework, we derive global and local conditions that are satisfied by all linear tax schedules which are optima of Paretian social orderings. The global conditions are given in terms of aggregate supply and demand functions (Proposition 4), and the local conditions in terms of (uncompensated) elasticities of aggregate labour supply and private savings Ceqs. ( 12)< 1611. Tax rates which are efficient in this sense require efficient factor prices and an efficient level of public debt (credit). As in Ordover and Phelps (1975) and Ordover (1976), but unlike Hamada (1972), the Golden Rule holds for a
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
351
closed economy, i.e. the gross (real) interest rate should equal the population growth rate (Proposition 1). Somewhat extending Theorem 1 in Samuelson (1975) we show that Golden Rule equilibrium can in fact be supported by means of public debt policy alone - without any taxes or transfers whatsoever (Proposition 2). 2. The model Only one good is produced in the economy, a good that can be used both for consumption and capital formation. Let Y,,C, and K, be gross output, aggregate consumption and capital, in period t. Assuming a constant rate 6 of capital depreciation, net output is divided between consumption and investment, Y - 6K, = C, + (K, + 1 -K,). In steady-state growth at a constant rate n over time, this equation may be written Y - (n + 6)K = C. We assume n and 6 to be nonnegative and their sum to be positive throughout the study. 2.1. The household sector
Let the working time of a young individual of type i be xie [0, l), his leisure being zi= 1 -xi. The corresponding quantity of supplied labour is mixi, where m, is a positive scalar, his ‘working ability’. Let ci be his first period consumption, and e, his consumption in the retirement period. We assume that each individual of type i (i= 1,. . . , N) chooses a triplet (ci,ei,zi) so as to maximize his utility ui(ci,ei,zi) in the ‘consumption’ set X = (0, + co)’ x (0, 11, subject to the budget constraints C~+S~=~MQX;+CJ and
eis(l+p)si.
(1)
Here o =( 1 - t,)w and p =( 1 - t,)r are the net-of-tax wage and interest rates, respectively, c a lump-sum transfer, and si the savings of individual i from his first (active) period to his second (‘retirement’) period.’ Assuming a perfect capital market, the temporary budget constraints in (1) may be consolidated to a life-time budget constraint: Ci +
eJ( 1 + p) +
OmiZi
5
wmi
+
0.
(2)
The budget set Bi of individual i is hence the subset of points in his consumption set X which satisfy (2). Each utility function ui is continuous, strictly increasing in all three ‘More generally, if g1 (u2) is the transfer given in the first (second) period of life, then the equivalent life-time transfer is CT= ol + uJ( 1+ p). F or simplicity, we assume that all transfers are given in the first period, i.e. c2 = 0. A treatment of the general case is given in an earlier version of the paper, see Svensson and Weibull (1985).
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
358
arguments and strictly quasi-concave, with the closure of its indifference surfaces contained in the consumption set X. All individuals will then choose positive leisure, and positive consumption in both periods, by definition of X, while it is not excluded that some (or all) will choose not to work at all. If the population growth rate n is zero, there is a constant population consisting of N young and equally many old individuals, and part of the steady state assumption is that in every generation there is exactly one individual with preferences ui and working ability m,. Likewise, in the case of a positive growth rate, there are N utility functions and ability parameters, and each sub-population of young individuals of type i grows over time at the same rate n. Without loss of generality, the number of young individuals of each type i in the period under consideration is normalized to one. Hence, there are N young and N/( 1 + n) old individuals in the ‘representative’ period. The aggregates associated with the household sector in a period are: the consumption by the young, C, =I Ci, and old, C, =CeJ( 1 +n), and the associated aggregate Cd = C, + C,. The young generation supplies labour LS=Cmixi, and saves S=csi for the subsequent period. All these aggregates are functions of the triplet (0, p, a).’ 2.2. The production sector Production possibilities are represented by a constant returns-to-scale (aggregate) production function F(K, L). Letting k= K/L be the capitallabour ratio, production per unit of labour is f(k) = F(k, 1). As usual, f is taken to be twice continuously differentiable on (0, + co) with f’>O, f” ~0, and f’(k) ranging from plus infinity to zero as k increases from zero to infinity. Let r and w be the (real) factor prices for capital and labour services paid by the production sector. Profit maximization gives the factor-price frontier 4 defined by w=+(r)
r=_/-‘(k)-6
iff w=f(k)-kf’(k)
for ‘Orne k’“’
Let 5 be the corresponding capital demand function, i.e. k= t(r) iff f’(k) = and note that q%(r)and t(r) are defined for all r exceeding minus 6.
r+6,
2.3. The public sector The public sector selects a transfer r~20 and tax rates t,, t, S 1 such that all individual budget sets Bi are non-empty. Consequently, the public sector’s *In the more general case of one transfer in each period of life, S is a function each transfer, fll and c2, not only of their discounted sum 6, see footnote 1.
of o,p
and
L.G. Suensson and J.W Weibull, Pareto-optimal taxation
359
budget balance, with G denoting its (net) holdings of capital in the period under consideration, is (4) The first term represents tax revenues from wage earnings of the currently young, the second tax revenues from interest earnings of the currently old, who are a factor (1 +n)-’ fewer. The third term, finally, represents the interest-augmented value of the public sector’s net capital holdings G/( 1 + n) ‘inherited’ from the previous period. Subtracting current transfer payments to the N young individuals, we obtain current public capital holdings, to be ‘handed over’ to the next period. Employing the aggregate notation introduced above, this budget equation may be written: t,wL + t,rs/( 1 +n) +(I-
n)G/( 1 + n) = Na.
(5)
Note that if the (real) interest rate equals the population growth rate, then public (net) asset holdings vanish from the public budget sheet - because then their growth matches interest payments. 2.4. Equilibrium At given exogenous data, all decisions taken by the economic agents in the economy are functions of the vector q= (w, r, t,, t,, r~,G), the state of the economy. Let E be the set of (steady-state, tax) equilibria defined as follows: q E E iff there is a factor combination (K, L) > 0 which is (i) profit maximizing (with respect to w and r), and (ii) market clearing, i.e. satisfying
Remark.
F(K, J!,)-(n + 6)K = Cd (product market),
(6)
L=L”
(labour market),
(7)
(l+n)K=S+G
(capital market).
(8)
If eqs. (6)-(8) hold, then so does eq. (5), by Walras’ law.
3. Efficient factor prices Let aeXN be a consumption allocation, i.e. a =
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
360
Pareto dominated by another allocation in A, i.e. E** is the set of Pareto equilibria in the usual sense. 3 Let AE be the subset of technically feasible allocations (a) which correspond to equilibria in the economy [i.e. such that a=a(q) for some 4 in E] and let E* be the subset of equilibria q that are constrained Pareto efficient in the sense that the corresponding allocation a(q) is not Pareto dominated by an allocation in AE (i.e. by another equilibrium allocation). Moreover, let Eg be the set of Golden Rule equilibria, E*= {qE E; r=n}, and let E” be the set of tax-free equilibria, i.e. ejkient
E’={qEE;t,=t,=O}. Evidently E**
is a subset of E*, and E* contains the set of equilibria relevant for optimal taxation, since Paretian social preferences never select equilibria which are not constrained Pareto efficient. In fact, all tax-free Golden Rule equilibria, and for practical purposes no other, are Pareto efficient, and all constrained Pareto efficient equilibria satisfy the Golden Rule: Proposition
1.
Egn E” c E** c E* c Eg. Ifpreferences are smooth,4 then E** =
Eg n E”.
(Proofs are given in an appendix at the end of the paper.) This result is similar in spirit to the (static) production-efficiency result in Diamond and Mirrlees (1971, Theorem 4), and relies on the assumption that the government may freely levy taxes on both capital and labour incomes, and freely choose the level of public debt. This is the reason why Hamada (1972) reaches another conclusion: restricting t, to be zero he finds that generally E* is not a subset of Eg. Ordover and Phelps (1975) and Ordover (1976), on the other hand, show that E* is a subset of Eg, under the assumptions that social preferences are Rawlsian and that all individuals have identical preferences fulfilling certain normality conditions [see Ordover and Phelps (1975, p. 665 and eq. (50)) and Ordover (1976, pp. 142-143 and Proposition 12)].5 The following proposition shows the non-emptiness of E** and hence also of E*, and states that Golden Rule equilibrium can be supported without any taxes or transfers whatsoever, simply by a suitable choice of public capital holdings. In order to construct such a tax-free Golden Rule equilibrium 4 (hence an element in E**, by Proposition l), let 2, = f, =0 and i= n. Then S =0 by the 3As usual, an allocation a is said to be Pareto dominated by an allocation a’ if ui(ai) 5 ui(a;) for all i, with strict inequality for some i. 4The preferences of an individual i are said to be smooth if the indifference surfaces in X have a unique normal vector at every point. 5Recall the assumption n+6 >O. Were instead n+6 SO, then Eg would be empty - efficiency would require an infinite productive capital stock.
L.G. Svensson and J.K
Weibull, Pareto-optimal taxation
361
public budget equation (5) (since ?=n). Let 4 =4(n), ft= r,+(n),e= L’(fi, n, 0) and R =LL. Then profits are maximized and the labour market clears. By Walras’ law, also the product market clears. Moreover, any level of public capital G satisfies the public budget equation (5) since i= n. Hence, by choosing G = (1 + n)R - 9, also the capital market clears. Letting B denote the aggregate savings ratio of the young generation, i.e. G=s/(i?L), we have established: Proposition 2. 4~ E**. If preferences G>O if&<(l+n)<(n)/+(n).
are smooth, then E**=
(4). Moreover,
In other words, at a given growth rate and technology, the Pareto optimal public capital stock G is positive if the private savings ratio is small, negative if it is large. Proposition 2 extends the observation in Diamond (1965) that a laissezfaire economy generally does not achieve Golden Rule equilibrium, from his case of exogenous labour supply to the present case of elastic labour supply: generally G#O. Moreover, Proposition 2 generalizes Theorem 1 in Samuelson (1975) a result saying that Golden Rule equilibria can be supported by appropriate social security systems (p. 541). Like Diamond (1965), Samuelson refers to an economy with a representative household in each generation who supplies labour inelastically. Hence, social security systems may then be financed by lump-sum taxes and transfers. In the present model, however, earning abilities differ among individuals so social security programs generally require distortionary taxation. However, Proposition 2 states that Golden Rule equilibrium can be supported without any such program - it is sufficient to choose the appropriate level of public net asset holdings. 4. Efficient tax rates By Proposition 1 all efficient tax rates, i.e. tax rates supporting equilibria in E*, are such that the corresponding interest rate equals the growth rate. What more can be said about them? In fact, some quite explicit necessary conditions can be obtained if aggregate leisure is non-inferior and aggregate consumption normal: NL: L’(o, p, CT)is non-increasing
in e.
NC: Cd(o, p, a) is strictly increasing in g. Generally, any one of the two tax rates (but not both) may be negative. However, the subsequent analysis is restricted to the identification of inefficient combinations of nonnegative tax rates. One easily shows
J.PE.
D
362
L.G. Svensson and 1.W Weibull, Pareto-optimal taxation
Proposition 3. Suppose n >O. For every pair of tax rates (t,, t,) in [0,1)’ there exists a corresponding Golden Rule equilibrium. The associated transfer a is unique under NL and NC. This proposition allows us to define $:[O,l)‘+R+ by tj(t,,t,)=a.6
a (per-capita)
tax-revenue
function
Corollary 3.1. $ is continuous. It is positive on (0, 1)2 and vanishes where t,=t,=O, and where t,= 1. We now identify a range of tax rates that are inefficient. The idea is to exclude such equilibria (or tax rates) for which there is another Golden Rule equilibrium with lower tax rates t, and t,, but not lower life-time transfer a since then all individuals’ budget sets are larger in the other equilibrium. For this purpose, let
T= {(t,, t,) E (0, l12;$(t’,, ti) < $(t,, t,) forall K, ti) #_(t,,t,) suchthat Ok,t3 S(t,, t,)>.
(9)
Clearly all pairs of (nonnegative) efficient tax rates lie in the set T - for otherwise there wou!d exist another Golden Rule equilibrium with at least one of the tax rates lower but still not a lower transfer. In sum: Proposition 4. Suppose NL, NC and n>O. If (t,, t,) 20 tax rates, then (t,, t,) E 7:
is an efficient pair of
There is a local counterpart to this global result. Granted I++is differentiable, Proposition 4 implies that of t, and t, are positive and belong to T, then a$/&, 20 and &j/&,20. Differentiation of I+$ gives the following necessary conditions for efficiency:
c; 2 WL;,
(11)
where r =n, w is the corresponding wage rate 4(n), and subscripts signitiy partial derivatives.’ If we let et=wLi/LS, et=pL;/L”, e~=oS,/S and e; = pS,/S, then (10) is equivalent to (12) below, and (11) to (13): 6Note that aggregate tax revenues equal Na in Golden Rule equilibrium, see eq. (5). However, this equivalence between tax revenues and transfers does not hold in the more general case when some transfers are given in the second period of life as well, see Svensson and Weibull (1985). ‘Here NL and NC have been slightly strengthened to require L:sO and Cd,>@ see appendix for a derivation.
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
363
( 1 + eE)t, s 1 -ye&,
(12)
(1 +ez)t,s
(13)
1 --YP1eitw,
where y = [rS/(wL)]/( 1 + n), r = n and w = 4(n). As expected, the constraint on efficient tax rates on labour (capital) incomes is stricter the more elastic is the aggregate supply of labour (private capital). When t, =0 and/or the ‘cross elasticity’ ei is zero (or n= 0), then (12) reduces to (1 +e&,~
1.8.9
(14)
On the other hand, if capital incomes are taxed (t,>O), and the cross elasticity e”, is positive (negative), then the necessary condition (12) imposes a harder (softer) constraint on efficient tax rates t, on labour incomes, ceteris paribus. The additional term on the right-hand side of (12), which accounts for distortions due to t,, is proportional to y, hence proportional also to the ratio between the capital-income tax base and the labour-income tax base. Likewise, when t, = 0 and/or ei = 0, then (13) becomes (l+e:)t,51,
(15)
a private-savings counterpart to (14), while if labour incomes are taxed and aggregate labour supply is sensitive to the net interest rate, then the efficiency constraint (13) on t, is harder (softer) when the cross elasticity e,Lis positive (negative). Finally, if the tax rates t, and t, for some reason are constrained to be equal, then this additional constraint may render some otherwise inefficient combinations of tax rates efficient. Hence, the upper bound on a pure income tax rate t is generally less strict than the bounds given above. With the notation of eqs. (12) and (13) one obtains the following necessary condition (when n > 0):
(t,>O),
[l +eE+$+y(l
+ei+ez)]tsl
+y.
(16)
This condition is in fact implied by conditions (12)-(13) together with the condition t w = t,= t.” On the other hand, the last condition and (16) do not ‘One easily shows that (14) indeed is the inequality obtained in the zero growth case. In that case, there is no correspondence to (13), since capital incomes are then zero in Golden Rule equilibrium. The efficiency condition (14) has been derived for taxes in labour incomes when capital is fixed and production is linear in labour inputs, see section 4.1 in Svensson and Weibull (1986). ‘Assuming identical individual preferences and utilitarian social preferences, Hamada (1972) obtains Sheshinski’s (1972) upper bound l/( 1 +e,J, where emin is the lowest individual laboursupply elasticity, never exceeding et and generally falling short of it. Hence the upper bound in (14) is in general lower than Hamada’s. “‘Multiply both sides of (13) by y>O, and add (12) and (13).
364
L.C.
Soenssonand J.W Weibull, Pareto-optimal taxation
together imply conditions (12) and (13). In this sense, the necessary conditions for efficiency are less strict when the additional constraint t,= t, is imposed.
Appendix Lemma A. Suppose r=nzO, 05 t,< 1 and Oj t,< 1. Then (TE R + and a G E R such that q = (4(r), r, t,, t,, o, G) E E. Proof
Let r=n,
w=+(n),
Ost,
Ost,
and k=t(n).
there
exists
a
Define g:R++R
by
g(0) =
Cd(o,p,4 - WLYW, p, 4.
Clearly g is continuous, and by (2) and monotonicity of preferences: aci+ ae,/(l+p)zNg. Hence ~c,-++co and/or Ce,-++cc as o++co. It follows that g(o)++co as r~-++cc (Cd=~ci+~ei/(l+n)++co, and L” is bounded by C mi). Moreover Cd 5 C ci + C e,/( 1 + p), and C Ci+ 1 eJ( 1+ p) = oL” when g=O, by (2). Thus g(0) s(ow)L”50, so g(a) =0 for some aZ0. It is easily verified that such a (T gives equilibrium: eqs. (6) and (7) are met, and let G=(l+n)K-S. Proposition I. To prove E* c Eg, first define a: (0, + co)+ R by a(k) = f(k) (6 +n)k. Since f is strictly concave, so is a. Let b:( -6, + co)-+R be defined by b(r) =a(t(r)). Since (‘~0, b is differentiable and strictly quasi-concave, and by (3) b achieves its maximum at r=n: b’= [f’-(6+n)]t’. Note finally that, for any q=(w,r,t,,,,t,,a,G), we have g(a)=Cd(W,p,a)-b(r)L”(o,p,a), by (3), where g:R + -+ R now is (more generally) defined by
g(o)= Cd(o,p,4 - Cw-(I - n)lL”(wp,4. Now suppose q = (w, r, t,, t,, 6, G) E E, where r # n. We may then find some r’ between r and n, and some associated t: 5 1 such that (1 - tjr’ = p. For if p=O, let r’=n and tL=l. If p
L.G. Svensson and J.W Weibull, Pareto-optimal taxation
365
w’= +(r’) and G’ = (1 + n)K - S. It follows that q does not belong to E*: 0’ > c, w’=o and p’=p. To prove EBn E” cE**: Suppose q* belongs to Eg n E” but not to E**. Then there is a feasible allocation, a’ = (c’, e’, x’) E A, which Pareto dominates the equilibrium allocation a* = a(q*). Hence by (2):
c:+ e;/( 1 + I*) 2 w*m,x; for all i, with strict inequality for some i. [Note that q* lEg n E” implies, by eq. (5), that u* =O.] Aggregation of this equation gives C’ = C; + C;>w*L’. Let K’ be a capital stock such that a’ can be produced. Then F(K’, I,‘)-(6+n)K’zC’> w*L’=[f(k*)-k*f’(k*)],!,‘=[f(k*)-(6+n)k*]L’by (3). Hence f(k’) -(6 + n)k’>f(k*) -(6 + n)k*, which is a contradiction since f(k) -(6 + n)k is maximal at k = k*. To prove E** c EB n E”: First note that E** c E* n E* by the first part of the proof. Moreover, differentiability of all indifference surfaces implies unique prices (w, r, o, p) in equilibrium, and hence t, = t,=O for a Pareto efficient equilibrium, by standard arguments. Proposition 3. Existence follows directly from Lemma A. Clearly the function g in the proof of that lemma is strictly decreasing by NC and NL. Thus fs is unique. Corollary 3.1. Let g be as in the proof of Lemma A. Then (~=$(t,+,, t,) is its unique zero. Since g is strictly increasing, and L” and Cd are continuous in (t,, t,), also II/ is continuous. By eq. (5), 0 >O unless t,= t,=O. For every a>O, L’(o,p, cr)+O when t,+l while Cd(m,p, o)-+o. Hence, for every a>0 there exists a t, < 1 such that g(a) > 0, while g(0) SO. Thus +(t,, t,) -+O when t,-+l. Inequalities
(1+06).
If qEE, then, by (3), (6) and (7):
Cd(o, p, a) = [w +(r-- n)k]L”(o, p, a). Under NL and NC, and the hypothesis w +(I -n)k>O, this equation defines 1(1by $(t,, t,) = 0‘. Differentiation with respect to t, and t, gives (Cj-[w+(r-n)k]L”,)d$/&,=w(C~--[w+(r-n)k]LS,)
and
Hence, if NL and NC are slightly strengthened to require L”,sO and Ct> 0, respectively (actually Cz > wL”, is sufficient), then
366
L.G. Svensson and .l.W Weibull, Pareto-optimal
&,9/&,,,~0 and, provided
iff
C;z[w+(r-n)k]LS,
iff
Cz2[w+(r--n)k]Li.
taxation
r > 0,
d$/&,20 This establishes equilibrium
inequalities
(lo)41
1): let r = n, where n > 0 by hypothesis.
In
[by aggregation of (2) and (3)]. Derivation w.r.t. o and p gives (12)413). If t, is constrained to equal t,, then the necessary condition for efficiency becomes dll/(t, t)/dt 20, where d+(t, t)/dt = a$(t, t)/dt, + at&t, t)/&,. Hence, when n=r, t,=t,=t, LzsO and Cz>O: d$/dt 2 0 the last inequality
iff
w( Cd, - wL”,) + I( Cz - wL;) 2 0,
being equivalent
to (16).
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