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Taxation and diverging expectations using energy policy as an illustration
Journal
of Public
Economics
17 (1982) 23349. North-Holland
Publishing
Company
TAXATION AND DIVERGING EXPECTATIONS USING ENERGY POLICY AS AN ILLUSTRATION M. MARCHAND C.O.R.E.,
I348 Louvain-la-Neuve,
Bel@un
P. PESTIEAU* Universite’ de Li?ge, 4000 LiPge, Belgium Received July 1980. revised version received
May 1981
This paper attempts to determine the optimal taxation policy a government ought to pursue when its expectations of future prices are quite different from those of the consumers. The government is assumed to possess the correct information a5 to future price increases and to maximize the consumers’ ex post welfare by the use of appropriate taxes or substdies. It appears that it 1s not always desirable to tax the good for which the consumers tend to understate the price increase. There are indeed a large number of factors. conflicting at times. which contribute to the derivation of a rather intrtcate tax formula. To help develop an intuitlye understanding of the model, some simple cases are analysed and a numerical example presented.
1. Introduction There are various reasons why public authorities would try to interfere with market prices and impose taxes. Among these, one traditionally thinks of externalities, market imperfections, revenue needs for distributive purposes, etc. These reasons can in a certain way be catalogued in one of the three Musgravian branches: stabilization, distribution and allocation.’ This paper introduces as further grounds for government intervention the possible divergence between social and private expectations of future prices. To again use Musgrave’s vocabulary, public policy is called for because of the presence of ‘merit goods’ which emerge here from incomplete consumer information. The starting point of this paper is a conjecture we made about the optimal taxation policy the government ought to pursue were it true that consumers would tend to underestimate a future increase in the price of energy and that the government would know the true future price. Our conjecture, hasty as it might appear, was that besides subsidizing investment in energy-saving equipment, the government should levy a tax on the current consumption of *Helpful comments by Roger Guesnerie, Claude Henry and acknowledged. We are particularly indebted to John Weymark manuscript and to Henri Born for his computational assistance. ‘See Musgrave (1959).
0047-2727/82/0000-0000/$02.75
0
1982 North-Holland
Roger Sherman are gratefully for his careful reading of the
energy. This tax was allegedly aimed at reinforcing the effect of that subsidy in encouraging energy-saving investments. However, we soon realized that our conjecture could be overturned by the following relative-price argument: since the expected ratio of current and future energy prices was smaller for the government than for the consumer, the discrepancy could be reduced by subsidizing the current use of energy. Of course, the income effect might call for action in the opposite direction. The purpose of the paper is to disentangle the truth of each of those conjectures. In this paper a two-period model is developed for a small price-taker economy. It enables us to show that the type of taxation policy which ought to be adopted for the ex post good of the consumers depends on a large number of factors: the extent to which the expectations of both the consumer and the government diverge; the degree of substitution between present and future consumptions; the quality of the energy-saving equipment; the way price expectations are formed, etc. These factors often act in opposite directions, so that it is not always possible to derive any firm conclusions as to the sign and the importance of the taxes. To help develop an intuitive understanding of the model, some simple cases are analysed and a numerical example presented. Throughout the paper the example of energy is used, although the model could apply to any case of diverging expectations. Energy is a good case because of its relevancy and its contemporary nature. According to some experts the energy crisis has been partially caused by inadequate expectations that economic agents held towards future energy price increases. Our paper is not without relation with earlier studies. Particular reference should first be made to the concept of ex post optimality developed by Harris and Olewiler (1979) when they consider cases such that ‘one is explicitly forced to adopt a social welfare function in which individuals’ tastes “count” but their probability beliefs do not’ (p. 137). They thus define the welfare foundations of our study. Divergence of beliefs do not occur only in the field of probabilities but also in that of preference towards the future. In that respect, Atkinson and Sandmo (1980) treat, for instance, the optimal taxation of savings, assuming at one point that private and social rates of time preference do differ. When dealing with expectations, it is hard not to evoke both the theory of rational expectations and that of temporary equilibrium. In the former, individuals are usually assumed to possess the same information as the government, thus making monetary policy ineffective. However, if the monetary authority has superior information to individuals, it can affect the economy [for instance, see Sargent and Wallace (1975)]. The temporary equilibrium literature, on the other hand, focuses only on individuals’ expectations of future price but it ties them to current prices. In that respect, we also look at the case of expectations which depend on observed prices. Dixit (1976) introduces public finance in models of temporary equilibrium,
but he does it to derive the effect of various public instruments and not to achieve any sort of social welfare. The paper is organized as follows. Section 2 introduces the model and presents the main results. Section 3 specializes to the simplest case with two goods and one instrument. It enables us to do justice to the relative-price argument. Two assumptions on the formation of expectations are dealt with: (i) future price expectations by the consumers are exogenously given or (ii) they depend on the current price of energy. In section 4, we look at the case with three goods and provide a numerical illustration of this situation. It appears that for a plausible range of the parameter values of our model, one can indeed expect a tax on energy and a subsidy for the energy-saving device. Finally, section 5 concludes the paper.
2. The basic model and its main results We consider a two-period, one-individual economy. In each period a representative individual consumes, besides a composite good, some type of energy, e.g. domestic oil. In the beginning of the first period he also makes two investments in durable goods which, jointly with the amount of energy purchased in each period, yield a certain level of service, e.g. home temperature, both in the present and in the future. The first durable is used in conjunction with energy to produce heat, e.g. a furnace, and the second is some kind of heat-saving equipment, e.g. home insulation, which can be used as a partial substitute for energy. In the beginning of the first period, when the individual chooses the level of his investments and that of his current consumption, he does not know what the price of energy will be in the second period. He thus bases his current choices on his expectation of the future price of energy (this expectation is represented by a distribution of subjective probabilities). This price uncertainty will, however, be lifted in the beginning of the second period, at which time he allocates his saving the part of his initial resources which has not been invested or spent during the first period ~ between his consumption of energy and that of the composite good. As stated above, the government has its own expectation of the future price of energy, an expectation which differs from that of the individual. This implies that they do not share the same opinion as to the amounts to be invested and saved in the first period. By both taxing (or subsidizing) the current consumption of energy and subsidizing (or taxing) the investment in the heat-saving equipment, the government will seek to influence the consumer in a direction which is consistent with its own expectations. The tax is not constrained to be equal to the subsidy, and therefore the difference ~ positive or negative ~ is transferred back to the consumer in a lump-sum way. In fact, both goods could be taxed (or subsidized).
The utility function of the consumer is assumed additively separable; it can be written as
to be quasi-concave
and
(1)
U,(Y,,c,)+U*(Y,,c,),
where yi and ci respectively denote the consumption of the energy-based service and that of the composite good in period i (i= 1,2). The service yi is the result of the joint combination of energy, xi, the complementary durable good, u, and the heat-saving device, v. This relation can be expressed by the following separable production function : Yizhikitxi,
#), v),
i= 1,2,
(2)
where both hi(.) and gi(.) are assumed to be quasi-concave. In terms of our example, g,(.) is viewed as the production function of heat where xi and u are assumed to be highly complementary. As for hi(.), it gives the home temperature as a function of heat and insulation. ‘These two arguments are assumed to be quite close substitutes. To simplify the notation, one can express (2) as Yi=fitxi,
k
2.1. The consumer’s
v)3
i= 1,2.
(3)
problem
The consumer is faced with a two-stage problem to be solved by backward induction. So we first consider the problem that he faces in the beginning of the second period, at which time he knows the true price of energy. His problem is simply to allocate his savings between the consumption of energy and that of the composite commodity: (4)
subject
to p2xZ+c2=s(1+r),
(5)
where pz is the world market price of energy, s is the amount of savings and r is the interest rate. The first-order condition for an interior maximum is
au, au, &,
dx,
1 P2’
(6)
which yields the following demand functions for the second period: x,(p,, u, u, s) and c,(p,, u, v, s). Clearly, the second-period level of utility the individual can expect to reach will depend on his first-period decisions as to his saving s and his investments in u and v. These are the three decisions the government will seek to influence by its taxation policy. In the beginning of the first period the individual is uncertain as to the future price of energy; he thus chooses his consumption and investment plan by maximizing the expected value of his intertemporal utility. That is,
subject
to Cl
+7c1x1
+cpp+q,u+s=w,
(8)
where n, denotes the current price of energy (including any government tax or subsidy), cpVis the cost of heat-saving equipment (including any subsidy or tax), q, is the price of the other durable good u (which is neither taxed nor subsidized), and w is the individual’s income (including the lump-sum transfer). In the maximand (7) Ei denotes the expected-value operator based on the subjective probability distribution of the future price of energy, p2, as expected by the individual. This distribution may depend on 7ci. Using (5) and (6) the solution to the problem of maximizing (7) subject to (8) can be written after some rearrangements as
(12)
These conditions yield the demand functions of the first period: xi (rt,, cp,, w), u( .), u( .) and cr( .), the three arguments of which are the variables which can
be controlled by the government. From (8), one can also derive the saving function, s(.). We need not introduce other prices than r~i and cpCas explicit arguments in the demand functions. It is indeed assumed that all producer prices are fixed (small price-taker economy) and that only the first-period consumption of energy and the investment in energy-saving equipment can be taxed or subsidized. From (11) and (12) one can furthermore derive the following two equations: dx,
1
E,[dU,/&]
711dv+I+-rEi[(c?U,/il,)(l/p,=l(P’
(13)
and dx,
1
” du+
1 +r Ei[(iU,/Sxz)(lip,ll=q”’
where we use the following dx, _ i3fJdv dv
E,[?U,/iu]
dfJ3x1
(14)
notation:
and
dfl/dU dx, _=du - ilf;/ax,
(15)
Eqs. (13) and (14) are the first-order conditions for optimal investment in the face of uncertain future price of energy. In particular, the left-hand side of (13) can be interpreted as the present value of the heat savings which can be obtained from equipment u.
2.2. The goaernment’s
problem
The price rci differs from the first-period price of energy on the world market p1 by the tax (or subsidy) levied by the government. Let z denote this tax (z =rri -pl), which is negative if it turns out to be a subsidy. Likewise, cp, differs from the producer price 4, by the government subsidy (or tax). Let 8 denote the negative of this subsidy (0 = (~~-4,). As mentioned earlier, the taxes and subsidies are transferred back to the individual in a lump-sum way. So his income is w=(~n,-p,)x,+(cp,-q,)v+w, where w stands for the initial endowment economy). For further reference, let us remark
(16) of the individual (and the that combining (8) and (16)
yields Cl
+q,u+q,u+s=~,
+p1x,
(17)
which is the true resource constraint of the economy. Before stating the government problem, we define the transfer-included price effects to be used shortly. To do so, let us introduce the demand functions into (16) and take the differential of the resulting expression with respect to rri and w, which leads to
(18) This ratio effect:
of differentials
dx,
3x,
is used
in the so-defined
transfer-included
i;x, dw (19)
dn, t-dn,+iwd?r, which takes into account the income effect of the lump-sum by a tax (or subsidy) change. Likewise, we obtain:
dw -=
x1 +
1
price
(~1 -~l)(W%c)+
(cpu-q,)W@,)
(7r1-pi)(c?x,/2w)-
(cp,-q,)(&/2w)
transfer
induced
(20)
(21) with identical shown that
expressions
for the price derivatives
of ci, u, u and s. It is easily
(22)
(and the same for q,) so that the transfer-included price effects are consistent with the overall resource constraint of the economy [see relation (17)]. It should, however, be clear that the transfer-included price effects are not utility-preserving as are the usual income-compensated price effects. On the basis of its own expectations of the future price of energy, the government maximizes the consumer’s expected utility with respect to two instruments: the current price of energy, rci, and the price of the heat-saving
30
M. Mavchund und P. Pestiruu,
equipment,
cp”. Formally,
its problem
max U,C.L (xi, u, c),
Tilxtrtion und diuerging expectations
is
cll (23)
subject
to
(711,cp”> WI,
Xl =x1
u = 4%
cp”, WI,
v = 4%
cp”,
Cl =c1
(q,
wh
(24)
cp”> WI,
w = (711 -P1)X, + (cp,-4,)~+~. In the maximand E, denotes the expectation operator based on the government’s subjective probability distribution of pZ. The first live constraints account for the reaction functions of the individual, i.e. his demand functions defined earlier as the solution of problems (7) and (8). The last constraint is eq. (15), in which the individual’s income is adjusted for the net-of-subsidy tax proceeds. The above optimization problem is solved in appendix A (using the lirstorder conditions for the individual found earlier). We obtain the following rather intricate first-order conditions for an interior maximum; we shall seek to interpret these formulae in the rest of the paper:
(n,-_pl)$
1
1t f+(v,--4.)$
(25)
M. Murchand
md P. Pestieuu,
Ttrvation
tmd diverging
expectutions
31
(26) where
(27)
(28) and 3
k
”
E,W,lW E,[XJ,/&]
’
(29)
In formulae (25) and (26), (1 -k,) can be interpreted as measuring the discrepancy between the expected marginal utilities of saving as viewed by the government and the individual. Indeed the numerator of k, in (27) is based on the government’s expectation of the future price of energy whereas its denominator is based on that of the individual. The same interpretation can be given to (1 - k,) and (1 - k,). Conditions (25) and (26) provide the rules for the optimal tax-subsidy policy which can be interpreted along the lines of optimal commodity taxation theory.’ The terms (rri -p,), (cp,--q,), (1 -k,), (1 -k,) and (1 -k,) represent the wedges between private and social valuations of xi, v, s, v and u (both in their second-period uses) respectively.3 These terms weighted by the tax derivatives of the corresponding demands add up to zero. Thus, the total dead-weight loss attributable to those wedges is minimized. It is quite easy to derive the tax or subsidy formulae from these conditions if only one of the two instruments were allowed. If the government relies
*For instance, see Diamond and Mirrlees (1971). ‘There are two wedges concerning u, namely (i) cp, -y,. the tax on c’ in the first period, (ii) (1 -k, ), the difference between the social and private valuation oft’ in the second period.
and
only on a change
in the present
price of energy, one obtains
from (25):
Tl
(30)
‘l-P’=dx,/djr,(,’ where T, denotes the right-hand side of (25). Alternatively, is only allowed on the heat-saving device, (26) yields
if a tax or subsidy
(31) where T, stands for the right-hand side of (26). Clearly, it is difficult to give any intuitive interpretation of the general tax formulae (25) and (26). For this reason, in the following sections we consider special cases and numerical illustrations.
3. The simplest case: A two-good economy with a single instrument subsidy on the first-period consumption of energy)
(a tax or
In this section the quantities consumed of all goods other than and second-period energy are assumed to be chosen exogenously. (30) then reduces to
the firstFormula
n,-p,~X-,-l Pl
(32)
and the first-best can be attained. To do so, only one decision variable (x1) need indeed be controlled for which one instrument (rrr) suffices. Will it be a tax or a subsidy? To answer such a question, we first assume that the government and the individual both have point expectations of the second-period price of energy. For illustrative purposes, we consider the case where the price expected by the government (P~,~) is higher than that expected by the individual (P~,~). This particular situation can be easily depicted in diagrams. We first take the case where the individual’s expectation is independent of the first-period price of energy (xi). Given his optimistic expectation, the individual’s budget constraint is higher than that of government; he is thus led to choose a present consumption of energy (x,,~) different from that of the government (xi, J. Two cases are depicted in figs. 1 and 2. In both figures, A represents the first-best choice that would be made by the individual were his expectation
n/l. Marchtrnd
und P. Pesrieau,
Taxation
md dirvging
expecttrtions
'1.g ‘l,i
x1
Fig. 1. Tax cas
Fig. 2. Subsidy
33
(independent
case (independent
expectations).
expectations).
34
M. Murchand
crnd P. Pestieau.
Tuxution trnd diuerging
expectations
the same as that of the government. But on the basis of his own expectation, the individual chooses B in the absence of tax or subsidy; he would then end up on the point C suboptimal on the basis of the government’s expectation. The government policy is to induce the individual to choose D and, thus, to consume the optimal amount of xi. In fig. 1, ~i.~x~,~, and the government subsidizes the current consumption of energy up to x~,~. It is noteworthy that in the above geometrical reasoning we do not require the additive separability of the utility function. The natural question is how to relate the sign of (~i,~--xi,~) to the characteristics of the underlying utility function. Let us assume that pZ,g is infinitesimally higher than p2.i. In this case we have
(33)
Now, a well-known
result is [see, for instance,
Atkinson
and Stiglitz
(1980, p.
76)l (34)
where u is the elasticity
of substitution
between
x2 and xi:
(35)
So, a tax (subsidy) is called for if the elasticity of substitution is smaller (larger) than the income elasticity of the first-period energy demand. It shows that the curvature of the indifference curves plays a key role in the policy choice. This appears clearly in figs. 1 and 2, where we focus on the elasticities of substitution (assuming identical income elasticities). In the case where today’s consumption of energy appears to be a close substitute for
M. ,Marchund
und P. Pestieau,
Tuxution
and diverging
expectutions
35
tomorrow’s consumption, the government encourages the individual to consume more energy today by subsidizing its current use (see fig. 1). On the contrary, in the case where the individual gets higher utility by making more equal his first- and second-period consumptions of energy, the government taxes its current use to reduce the imbalance in consumption that will result from the individual’s overly optimistic expectations (see fig. 2). Relation (34) also shows that even in the simple model (where we do not pay any attention to investment in energy-saving equipment), the relativeprice argument in favor of a subsidy presented in the introduction is wrong if the income effect outweighs the elasticity of substitution. Returning to the assumption of an additive utility that we need in the more complex cases, we prove in appendix B the following:
(36) where R is defined
R
=
_
as the elasticity
of the marginal
~,(a2u,iw
of x2:
(37)
au,lax, Furthermore,
utility
it is easily seen that
au2 x2,
g
Rglok,=
2x2
x2
*
(38)
au2 xZ,i
ax2 x2.i
since x2,g is always (here infinitesimally) smaller than x~,~. So the equivalence between our geometrical results and condition (32) is proved. The elasticity R can be expressed more explicitly as R
=
_
w2
f
;x2
fix2
(39) u;r,
f;’
where U; and U; (f‘; and f ;) denote the first and second derivatives of U, (f,) with respect to y, (x2). This expression emphasizes that R depends on both the individual’s preferences and the technology used to transform primary energy into a consumption good. It indeed involves the elasticity of the marginal utility of y, and that of the marginal productivity of x2 in units of y,. Under the standard convexity assumption, both terms of the right-
non-positive, which implies that R is always nonare inversely related to the elasticity of substitution to that between x2 and the two durable goods. section, we briefly consider the case where the the individual’s expectations through its choice of rri (dependent expectations). To see how our results are affected by this new specification, only the simple expectation function can be dealt with. It will be assumed that the expectations of the government and those of the individual are just equal to the current price multiplied by a constant and a random term. That is hand side of (39) are negative. Their values between y, and y2 and To complete this government can modify
p2.,=aplr
and
p2.i =bnitl,
(40)
where a and b are some scaling factors (a > b) and r] a random variable with unitary expected value. To provide a benchmark case, we shall assume that in the case of independence the individual’s expectations are formed according to (41)
Starting from the definition of k, given in (27), it can easily be understood that in the subsidy case (k, < 1) the dependency will further widen the difference between the expectations of the government and those of the individual and, therefore, the difference between the numerator and the denominator of the right-hand side of (27). This will call for a higher subsidy. On the contrary, in the tax case (k,> l), the spread between the numerator and the denominator will be reduced because the tax narrows the difference between the expectations. So a lower tax will suffice. The two cases are represented in figs. 3 and 4 for point expectations (i.e. y = 1 with probability one). In these figures A, B, C and D are defined in the same way as in figs. 1 and 2. Thus, D (on the dotted line) is the choice induced by the government policy in the benchmark case of independent expectations (p2,i = bp,). It should be compared to point E (on the dashed line) which the government induces the individual to choose in the case of dependent expectations (p2,i = bn,). Let us remark that in this case the budget line perceived by the individual (the equation of which is rrixi + br,x2 = w) has the same slope as that going through B (the equation of which is plxl + bp,x, ‘co). From this remark and the quasi-concavity of the utility function, it then follows that E is bound to lie between A and D in fig. 3 and above D in fig. 4. This implies that with dependent expectations, the optimal policy is still to tax (respectively subsidize) in the case of fig. 3 (respectively fig. 4). Furthermore,
A-f. Morchand
\
\
\
\
md P. Pestieuu,
Tuuation
\
'1.g
'1,i
Fig. 3. Tax case (dependent
'1.i
Fig. 4. Subsidy JPE-
c
and diverging
expectationsl.
xl,g
case (dependent
expectations)
expecttrtions
3-l
38
M. Mtrrchtrnd
trnd P. Pestieccu. Taution
trnd diwrging
expectations
the relative slopes of the budget lines going through D and E imply that the tax (subsidy) should be lowered (increased). The main conclusion to be drawn from this section is that the starting conjecture put forward in the introduction needs qualifications; in the oneinstrument case, subsidizing the current consumption of energy might well prove the optimal policy. This is more in line with the relative-price argument.
4. Introducing the energy-saving
equipment: A three-good economy
As soon as one leaves the two-good economy studied in the previous section, it proves very difficult to draw practical conclusions from the formulae obtained at the end of section 2. To illustrate the complexity of the problem, we introduce the energy-saving equipment in addition to x1 and x2 and take again the case of a single instrument, i.e. a tax or subsidy on the current consumption of energy. In this case, condition (30) reduces to
(42)
For point expectations satisfying pZ, g >P~,~ (as earlier), we shall attempt to infer the sign of z from (42). We have shown in the previous section that, depending on the curvature of the indifference curves in the (xi, x2) plan, (k, - 1) can be positive or negative. From condition (13), (with cp, =qJ, the term [q,- rcl (dx,/dtl)] is clearly positive. The sign of (k, - 1) is likely to be positive because an increase of the energy-saving equipment is expected to carry out more benefit if the future price of energy is higher [see the definition of k, given in (29)]. It is, however, difficult to determine the sign of or negative. the transfer-included price effects; they could be positive Furthermore, the signs of the various terms appearing in (42) could be related to each other. So, attempting to decide on the basis of (42) whether the current consumption of energy ought to be taxed or subsidized looks worthless. It is for this reason that we now turn to numerical examples. To illustrate the foregoing model, we use a CES specification for both the utility and the production functions:
U,+U,=[x;~+u-~]“‘~+[,;~+,-~]~‘~, where F and /?z - 1. With substitution in consumption
(43)
this specification, E and fl measure the degree of and in production, respectively. A low value of c
M. Mtrrchand
und P. Pestieuu,
Tuxctrion trnd dicerging
expecrtrrions
39
implies that present and future consumptions are close substitutes. Conversely, a high value implies a strong complementarity. Similarly, xi (or x2) and u are close substitutes if B is small. Four cases will be studied, in each of which we shall assume point expectations. In the first two there is a single instrument, a tax or a subsidy on the current consumption of energy. In case I, there is no way for the government to expectations (independent modify the consumer’s expectations): the consumer expects a future energy price pZ,i = 2, lower than the price expected by the government pZ, g =4. In case II (dependent expectations), the consumer expects a price equal to the current price of energy, i.e. pZ,i =x1. In the last two cases two instruments are allowed for: taxes or subsidies on the first-period consumption of energy and on energy-saving equipment with independent and dependent expectations respectively (cases III and IV). Through the calculations, the following values for the parameters are adopted : w=lO, The optimal interpretation
q,=l
and
pi=2.
values for z and 0 are given of which we now turn to.
4.1. A single instrument:
Cases I und
in tables
1, 3, 4 and
5, the
II
The figures on table 1 can be given an intuitive interpretation. For a given p, as E increases, the subsidy first decreases and then turns into an increasing tax. This is the result we had in the two-good case. Indeed, note that E is simply the consumption elasticity of the marginal utility for either period. For a given E, as fl increases, the tax does not change monotonically. We will focus attention on the two extreme values of E. First, let us take a= 5, that is when the tax policy ought to narrow the gap between y, and y,.. The effect of the tax is twofold: (i) it stimulates investment in energy-saving equipment and (ii) it tends to equalize the ex post consumption of energy in the two periods. For /?=O.S, namely a high substitution between v and xi, the first effect dominates and the tax can be very small to induce a change in u, the energy-saving equipment. On the contrary, for /I= 5, i.e. a low substitution between u and xi, the second effect dominates. Because of this low substitution, nothing can be gained by stimulating energy-saving investment, whose level furthermore shows almost no sensitivity to the tax. So the government will try to equalize the ex post values of x1 and x2. Because xi and x2 are very complementary to U, it will require a very high tax rate to approach equalization (in the limit, were u and xi perfect
M.
Murchund
ctnd P. Pestierru,
Trrxution
rrnd dicerging
expectrttions
complements, xi =x2 ex ante and thus x2 =$.x1 ex post, whatever the tax level might be). Let us now look at the other polar case, E= -0.9. Now y, and y, are strong substitutes; there is no longer a need to make them equal. Instead, the government policy will try using a subsidy to increase y, relatively to y,. The effect of such a policy will be (i) to discourage the investment in energysaving equipment and (ii) to raise the first-period consumption of energy at the expense of its second-period consumption. In this case the first effect, unlike the second, is unwanted. With high complementarity in production (e.g. p= 5) the subsidy rate can be high because it hardly affects the investment in energy-saving equipment so that the first effect does not play any significant role. This can be verified by comparing the value of c’ in the second-best solution (table 1) and its value in the absence of subsidy or tax (table 2). Conversely, in the case of high substitution in production (e.g. fi= -0.5) both effects have a sizeable impact and thus the subsidy is lower. Finally, for intermediate values of E, the outcome is expectedly not as clearcut as for polar values. Table 1 Con,umrr’>
choice in the abence
of taxation
1.~~.LI.
Table 3 yields the second-best values of z, x1, u and x2 in the case where the expectations are not independent, i.e. P~,~=K~. In broad terms, the trend of r is the same as in table 1. The big difference is that the tax tends to be lower and the subsidy higher, as explained in the previous section. In both tables 1 and 3, for realistic values of E, i.e. when the two consumptions cannot be easily substituted for each other, there is a tax which increases as p decreases. In other terms, were a better energy-saving technology (in terms of a good substitute for energy) introduced in the economy, a lower tax on energy would be needed. In terms of policy implications, this is an interesting point.
4.2. Two instruments:
Cases
III
und IV
Tables 4 and 5 consider the two-instrument case. Two important remarks are worth making. First, with two instruments a first-best allocation is possible so that the values for xi, x2 and v given in table 4 are the optimal
p=
\.
5
2
0.5
-0.1
-0.5
i:=
-0.2 5.03
0.1 X.21
-0.55 2.50 2.75
2.20
-0.4 2.95
-0.35 1.93 3.64
1.38
0.42
-0.9
0.56
0.66
0.63
0.55
0.24
Table 3
-0.1 5.22
0.1 8.22
1.79
1.75 0.15 2.31
0.05 2.82
-0.15 1.68 3.79
1.26
0.42
-0.5
1
0.92
0.71
0.57
0.24
1.63
1.61
1.49
1.20
0.42
0.24
0.4 2.17
0.25 2.75
0.05 3.28
1.14
1
0.79
- 0.05 5.31 0.57
0.125 x.22
-0.1
[,?-J
Case II: Tax on energy with dependent
1.55
1.49
1.38
1.07
0.39
0.55 2.10
0.45 2.6X
0.2 3.93
0.1 5.53
0.15 8.31
0.5
expectation
1.2
1.09
0.83
0.58
0.023
1.48
1.39
1.20
0.92
0.36
(p2,, =n, 1.
0.7 2.03
0.65 2.61
0.5 3.98
0.3 5.76
0.2 X.40
2
1.25
1.15
0.91
0.6
0.22
1.41
1.33
1.11
0.81
0.33
0.85 1.97
0.8 2.56
0.7 3.49
0.5 5.94
0.25 x.47
5
1.3
1.2
0.95
0.61
0.3
M.
Marchand
cmd P. Pestiruu.
III i;
/
II =L
Ttrxrrtion
tmd dicerging
rxpecrtrtions
43
44
M. Murchantl trnd P. Pestieuu,
Tuxtrtion cmd diverging exprctL!rions
ones. Secondly, one surprisingly notes that except in a few cases both energy and equipment are at the same time either taxed or subsidized. The reason for that is to be found in the presence of a lump-sum compensation which, though implicit. plays a very active role in displacing up or down the individual’s budget constraint. What really matters in the comparison is the ratio of the two taxes. In that respect, one notes that for E=P, the two tax rates are equal. When E> /$ the complementarity in consumption tends to be stronger than that in production and thus the misallocation in consumption is also stronger than that in production. A higher tax on energy is then needed. Conversely, when B>E a relatively higher tax (or lower subsidy) on equipment is called for. Another way to interpret the results of tables 4 and 5 is to compare them with those in tables 1 and 2. With a single instrument, the tax has not to be pushed so far as with two instruments because of its perverse effect on equipment. With two instruments, the level of equipment can simultaneously be controlled. Similarly, the subsidy on energy is not high enough in the one-instrument case as compared to the two-instrument case. Indeed, a subsidy on equipment can correct for the negative effect of the energy tax.
5. Conclusion The starting point of this paper was the conjecture that when consumers expect a lower price for energy than the government a tax should be imposed on the current consumption of energy and a subsidy granted to investments in energy-saving equipment. The foregoing theoretical analysis and numerical example lead us, first, to give more weight to the relative-price argument. There are indeed cases in which a subsidy on the current consumption of energy can be desirable. Secondly, even when a tax on energy is imposed, in some instances, a tax on equipment is also needed. However, those two results call for the following remarks. In what is usually considered as the most realistic cases, i.e. a strong intertemporal complementarity in consumption (large a), one can expect a tax on energy. Furthermore, simultaneous taxation of energy and equipment is mainly due to the presence of the positive lump-sum income transfer; to the extent that the tax rate on energy is relatively higher than that on equipment, one can then conclude that energy-saving equipment is effectively subsidized.
Appendix A: Derivation of the tax formulae (25)-(26) As a first step, we replace .x1, u, v, ci and s in the objective function and the last constraint of problem (23))(24) by the right-hand sides of the five
first constraints.
The Lagrangian
can then be written
as
c,(~,,u,v,s)l}+~~Cw-(n,-~,)x,-(cp,-qq,)v-ol.
64.1)
where xi, u, v, ci and s are to be taken as the demand functions (having rcl, cp, and w as arguments). The first-order condition for a maximum are obtained by making equal to zero the derivatives of the Lagrangian with respect to xi, cp, and w. It yields for 7ci:
au, ax, au, au, au au, ac, ac, an,
axK+au+-K+-1 1
1
au, ax, as ax, au ax, au +E, i ~8x2 ---++--++,3s an, au &cl GV (?7l,
1
(
au ac, au
ac, as
dc,
aS anI
aU OK1
---+--,+,-
02) anI
-
+A
[
1
-P&X1 -(q,,-q,,)E =o,
- (711
1
1
(A.21
Similar expressions are obtained for the derivatives with respect to cp, and w. Taking the derivative of (17) with respect to pi, we obtain:
as
-=
8%
Using
? ac, PI ~-Y’.fJ-(1u(:r. 3
&cl
1
(6) and differentiating
?
1
(5) with respect
(A.3)
1
to s enables
us to write
^
~!s+!p&?=(l+r)l%. 2
P2 ax2
(A.4)
M. Marchand
Substituting
&/dx,
and P. Pestieau,
72xution
and diverging
41
expecttrtions
from (A.3) and using (A.4), (A.2) can be written
as
(A.5)
where 5, is defined
by
(A.61 The last but one term on the left-hand side of (A.5) drops derivatives of (5) with respect to u and u give
ax2
ac,
pzu+,7,=0
By means
and
ax2
+i
-(q
-p,g 1
the
ac,
p2~+~=0.
of (9)-(12), (A.5) can be rewritten
out because
(A.7)
as
-xl-(q”-q”kg 1=o
(A.81
1
where Ti is defined in the same way as in (A.6)
(A.9)
By the same developments, it can be shown Lagrangian with respect to w reduces to
Eliminating
,I between
that
the derivative
of the
(A.@ and (A.lO) and using (18)-(21), we obtain
with the same expressions in brackets as in (A.8) and (A.lO). Dividing through by ci, and using (22) and (27)-(29), (A.ll) can be transformed into
+ E,{~U,/~u}
5i
+Ck,-112
[k/1,$
=o.
(A.12)
1 t
1 t
Formula (25) then follows from (A.12) by using (13) and (14). Starting from the derivative of the Lagrangian with respect (26) is obtained in the same way.
to cpU,formula
Appendix B: Proof of (36) Simplifying
subject
the notation,
the problem
reduces
to
maxU,(x,)+U,(x,)
(B.1)
PlXl
U3.2)
to +tP,x2
=w.
49
Denoting
by D the determinant
of the Hessian
matrix:
(B.3)
we have with obvious 6x,
+2
D,,i+x,D,, D '
where 3. is the marginal
(since pz/l= U,). and the positivity
notation
utility
(B.4)
of income.
The equivalence of D.
It then follows that
expressed
in (36) results
then
from
(BS)
References Atkinson, A.B. and A. Sandmo. 1980, Welfare implications of the taxation of savings, The Economic Journal 90, 529-549. Atkinson, A.B. and J. Stiglitr, 1980, Lectures on public economics (McGraw-Hill, London]. Diamond, P. and J. Mirrlees, 1971, Optimal taxation and public production. 1, American Economic Review 61, 8-27. Dixit. A., 1976, Public finance in a Keynesian temporary equilibrium. Journal of Economic Theory 12. 2422258. Harris, R. and N. Olewiler. 1979, The welfare economics of ex post optimality, Economica 46, 137-147. Musgrave. R., 1959, The theory of public finance (McGraw-Hill, New York). Sargent. T.J. and N. Wallace, 1975, Rational expectations, the optimal monetary instrument and the optimal money supply rule. Journal of Political Economy 83, 241-254.
of Public
Economics
17 (1982) 23349. North-Holland
Publishing
Company
TAXATION AND DIVERGING EXPECTATIONS USING ENERGY POLICY AS AN ILLUSTRATION M. MARCHAND C.O.R.E.,
I348 Louvain-la-Neuve,
Bel@un
P. PESTIEAU* Universite’ de Li?ge, 4000 LiPge, Belgium Received July 1980. revised version received
May 1981
This paper attempts to determine the optimal taxation policy a government ought to pursue when its expectations of future prices are quite different from those of the consumers. The government is assumed to possess the correct information a5 to future price increases and to maximize the consumers’ ex post welfare by the use of appropriate taxes or substdies. It appears that it 1s not always desirable to tax the good for which the consumers tend to understate the price increase. There are indeed a large number of factors. conflicting at times. which contribute to the derivation of a rather intrtcate tax formula. To help develop an intuitlye understanding of the model, some simple cases are analysed and a numerical example presented.
1. Introduction There are various reasons why public authorities would try to interfere with market prices and impose taxes. Among these, one traditionally thinks of externalities, market imperfections, revenue needs for distributive purposes, etc. These reasons can in a certain way be catalogued in one of the three Musgravian branches: stabilization, distribution and allocation.’ This paper introduces as further grounds for government intervention the possible divergence between social and private expectations of future prices. To again use Musgrave’s vocabulary, public policy is called for because of the presence of ‘merit goods’ which emerge here from incomplete consumer information. The starting point of this paper is a conjecture we made about the optimal taxation policy the government ought to pursue were it true that consumers would tend to underestimate a future increase in the price of energy and that the government would know the true future price. Our conjecture, hasty as it might appear, was that besides subsidizing investment in energy-saving equipment, the government should levy a tax on the current consumption of *Helpful comments by Roger Guesnerie, Claude Henry and acknowledged. We are particularly indebted to John Weymark manuscript and to Henri Born for his computational assistance. ‘See Musgrave (1959).
0047-2727/82/0000-0000/$02.75
0
1982 North-Holland
Roger Sherman are gratefully for his careful reading of the
energy. This tax was allegedly aimed at reinforcing the effect of that subsidy in encouraging energy-saving investments. However, we soon realized that our conjecture could be overturned by the following relative-price argument: since the expected ratio of current and future energy prices was smaller for the government than for the consumer, the discrepancy could be reduced by subsidizing the current use of energy. Of course, the income effect might call for action in the opposite direction. The purpose of the paper is to disentangle the truth of each of those conjectures. In this paper a two-period model is developed for a small price-taker economy. It enables us to show that the type of taxation policy which ought to be adopted for the ex post good of the consumers depends on a large number of factors: the extent to which the expectations of both the consumer and the government diverge; the degree of substitution between present and future consumptions; the quality of the energy-saving equipment; the way price expectations are formed, etc. These factors often act in opposite directions, so that it is not always possible to derive any firm conclusions as to the sign and the importance of the taxes. To help develop an intuitive understanding of the model, some simple cases are analysed and a numerical example presented. Throughout the paper the example of energy is used, although the model could apply to any case of diverging expectations. Energy is a good case because of its relevancy and its contemporary nature. According to some experts the energy crisis has been partially caused by inadequate expectations that economic agents held towards future energy price increases. Our paper is not without relation with earlier studies. Particular reference should first be made to the concept of ex post optimality developed by Harris and Olewiler (1979) when they consider cases such that ‘one is explicitly forced to adopt a social welfare function in which individuals’ tastes “count” but their probability beliefs do not’ (p. 137). They thus define the welfare foundations of our study. Divergence of beliefs do not occur only in the field of probabilities but also in that of preference towards the future. In that respect, Atkinson and Sandmo (1980) treat, for instance, the optimal taxation of savings, assuming at one point that private and social rates of time preference do differ. When dealing with expectations, it is hard not to evoke both the theory of rational expectations and that of temporary equilibrium. In the former, individuals are usually assumed to possess the same information as the government, thus making monetary policy ineffective. However, if the monetary authority has superior information to individuals, it can affect the economy [for instance, see Sargent and Wallace (1975)]. The temporary equilibrium literature, on the other hand, focuses only on individuals’ expectations of future price but it ties them to current prices. In that respect, we also look at the case of expectations which depend on observed prices. Dixit (1976) introduces public finance in models of temporary equilibrium,
but he does it to derive the effect of various public instruments and not to achieve any sort of social welfare. The paper is organized as follows. Section 2 introduces the model and presents the main results. Section 3 specializes to the simplest case with two goods and one instrument. It enables us to do justice to the relative-price argument. Two assumptions on the formation of expectations are dealt with: (i) future price expectations by the consumers are exogenously given or (ii) they depend on the current price of energy. In section 4, we look at the case with three goods and provide a numerical illustration of this situation. It appears that for a plausible range of the parameter values of our model, one can indeed expect a tax on energy and a subsidy for the energy-saving device. Finally, section 5 concludes the paper.
2. The basic model and its main results We consider a two-period, one-individual economy. In each period a representative individual consumes, besides a composite good, some type of energy, e.g. domestic oil. In the beginning of the first period he also makes two investments in durable goods which, jointly with the amount of energy purchased in each period, yield a certain level of service, e.g. home temperature, both in the present and in the future. The first durable is used in conjunction with energy to produce heat, e.g. a furnace, and the second is some kind of heat-saving equipment, e.g. home insulation, which can be used as a partial substitute for energy. In the beginning of the first period, when the individual chooses the level of his investments and that of his current consumption, he does not know what the price of energy will be in the second period. He thus bases his current choices on his expectation of the future price of energy (this expectation is represented by a distribution of subjective probabilities). This price uncertainty will, however, be lifted in the beginning of the second period, at which time he allocates his saving the part of his initial resources which has not been invested or spent during the first period ~ between his consumption of energy and that of the composite good. As stated above, the government has its own expectation of the future price of energy, an expectation which differs from that of the individual. This implies that they do not share the same opinion as to the amounts to be invested and saved in the first period. By both taxing (or subsidizing) the current consumption of energy and subsidizing (or taxing) the investment in the heat-saving equipment, the government will seek to influence the consumer in a direction which is consistent with its own expectations. The tax is not constrained to be equal to the subsidy, and therefore the difference ~ positive or negative ~ is transferred back to the consumer in a lump-sum way. In fact, both goods could be taxed (or subsidized).
The utility function of the consumer is assumed additively separable; it can be written as
to be quasi-concave
and
(1)
U,(Y,,c,)+U*(Y,,c,),
where yi and ci respectively denote the consumption of the energy-based service and that of the composite good in period i (i= 1,2). The service yi is the result of the joint combination of energy, xi, the complementary durable good, u, and the heat-saving device, v. This relation can be expressed by the following separable production function : Yizhikitxi,
#), v),
i= 1,2,
(2)
where both hi(.) and gi(.) are assumed to be quasi-concave. In terms of our example, g,(.) is viewed as the production function of heat where xi and u are assumed to be highly complementary. As for hi(.), it gives the home temperature as a function of heat and insulation. ‘These two arguments are assumed to be quite close substitutes. To simplify the notation, one can express (2) as Yi=fitxi,
k
2.1. The consumer’s
v)3
i= 1,2.
(3)
problem
The consumer is faced with a two-stage problem to be solved by backward induction. So we first consider the problem that he faces in the beginning of the second period, at which time he knows the true price of energy. His problem is simply to allocate his savings between the consumption of energy and that of the composite commodity: (4)
subject
to p2xZ+c2=s(1+r),
(5)
where pz is the world market price of energy, s is the amount of savings and r is the interest rate. The first-order condition for an interior maximum is
au, au, &,
dx,
1 P2’
(6)
which yields the following demand functions for the second period: x,(p,, u, u, s) and c,(p,, u, v, s). Clearly, the second-period level of utility the individual can expect to reach will depend on his first-period decisions as to his saving s and his investments in u and v. These are the three decisions the government will seek to influence by its taxation policy. In the beginning of the first period the individual is uncertain as to the future price of energy; he thus chooses his consumption and investment plan by maximizing the expected value of his intertemporal utility. That is,
subject
to Cl
+7c1x1
+cpp+q,u+s=w,
(8)
where n, denotes the current price of energy (including any government tax or subsidy), cpVis the cost of heat-saving equipment (including any subsidy or tax), q, is the price of the other durable good u (which is neither taxed nor subsidized), and w is the individual’s income (including the lump-sum transfer). In the maximand (7) Ei denotes the expected-value operator based on the subjective probability distribution of the future price of energy, p2, as expected by the individual. This distribution may depend on 7ci. Using (5) and (6) the solution to the problem of maximizing (7) subject to (8) can be written after some rearrangements as
(12)
These conditions yield the demand functions of the first period: xi (rt,, cp,, w), u( .), u( .) and cr( .), the three arguments of which are the variables which can
be controlled by the government. From (8), one can also derive the saving function, s(.). We need not introduce other prices than r~i and cpCas explicit arguments in the demand functions. It is indeed assumed that all producer prices are fixed (small price-taker economy) and that only the first-period consumption of energy and the investment in energy-saving equipment can be taxed or subsidized. From (11) and (12) one can furthermore derive the following two equations: dx,
1
E,[dU,/&]
711dv+I+-rEi[(c?U,/il,)(l/p,=l(P’
(13)
and dx,
1
” du+
1 +r Ei[(iU,/Sxz)(lip,ll=q”’
where we use the following dx, _ i3fJdv dv
E,[?U,/iu]
dfJ3x1
(14)
notation:
and
dfl/dU dx, _=du - ilf;/ax,
(15)
Eqs. (13) and (14) are the first-order conditions for optimal investment in the face of uncertain future price of energy. In particular, the left-hand side of (13) can be interpreted as the present value of the heat savings which can be obtained from equipment u.
2.2. The goaernment’s
problem
The price rci differs from the first-period price of energy on the world market p1 by the tax (or subsidy) levied by the government. Let z denote this tax (z =rri -pl), which is negative if it turns out to be a subsidy. Likewise, cp, differs from the producer price 4, by the government subsidy (or tax). Let 8 denote the negative of this subsidy (0 = (~~-4,). As mentioned earlier, the taxes and subsidies are transferred back to the individual in a lump-sum way. So his income is w=(~n,-p,)x,+(cp,-q,)v+w, where w stands for the initial endowment economy). For further reference, let us remark
(16) of the individual (and the that combining (8) and (16)
yields Cl
+q,u+q,u+s=~,
+p1x,
(17)
which is the true resource constraint of the economy. Before stating the government problem, we define the transfer-included price effects to be used shortly. To do so, let us introduce the demand functions into (16) and take the differential of the resulting expression with respect to rri and w, which leads to
(18) This ratio effect:
of differentials
dx,
3x,
is used
in the so-defined
transfer-included
i;x, dw (19)
dn, t-dn,+iwd?r, which takes into account the income effect of the lump-sum by a tax (or subsidy) change. Likewise, we obtain:
dw -=
x1 +
1
price
(~1 -~l)(W%c)+
(cpu-q,)W@,)
(7r1-pi)(c?x,/2w)-
(cp,-q,)(&/2w)
transfer
induced
(20)
(21) with identical shown that
expressions
for the price derivatives
of ci, u, u and s. It is easily
(22)
(and the same for q,) so that the transfer-included price effects are consistent with the overall resource constraint of the economy [see relation (17)]. It should, however, be clear that the transfer-included price effects are not utility-preserving as are the usual income-compensated price effects. On the basis of its own expectations of the future price of energy, the government maximizes the consumer’s expected utility with respect to two instruments: the current price of energy, rci, and the price of the heat-saving
30
M. Mavchund und P. Pestiruu,
equipment,
cp”. Formally,
its problem
max U,C.L (xi, u, c),
Tilxtrtion und diuerging expectations
is
cll (23)
subject
to
(711,cp”> WI,
Xl =x1
u = 4%
cp”, WI,
v = 4%
cp”,
Cl =c1
(q,
wh
(24)
cp”> WI,
w = (711 -P1)X, + (cp,-4,)~+~. In the maximand E, denotes the expectation operator based on the government’s subjective probability distribution of pZ. The first live constraints account for the reaction functions of the individual, i.e. his demand functions defined earlier as the solution of problems (7) and (8). The last constraint is eq. (15), in which the individual’s income is adjusted for the net-of-subsidy tax proceeds. The above optimization problem is solved in appendix A (using the lirstorder conditions for the individual found earlier). We obtain the following rather intricate first-order conditions for an interior maximum; we shall seek to interpret these formulae in the rest of the paper:
(n,-_pl)$
1
1t f+(v,--4.)$
(25)
M. Murchand
md P. Pestieuu,
Ttrvation
tmd diverging
expectutions
31
(26) where
(27)
(28) and 3
k
”
E,W,lW E,[XJ,/&]
’
(29)
In formulae (25) and (26), (1 -k,) can be interpreted as measuring the discrepancy between the expected marginal utilities of saving as viewed by the government and the individual. Indeed the numerator of k, in (27) is based on the government’s expectation of the future price of energy whereas its denominator is based on that of the individual. The same interpretation can be given to (1 - k,) and (1 - k,). Conditions (25) and (26) provide the rules for the optimal tax-subsidy policy which can be interpreted along the lines of optimal commodity taxation theory.’ The terms (rri -p,), (cp,--q,), (1 -k,), (1 -k,) and (1 -k,) represent the wedges between private and social valuations of xi, v, s, v and u (both in their second-period uses) respectively.3 These terms weighted by the tax derivatives of the corresponding demands add up to zero. Thus, the total dead-weight loss attributable to those wedges is minimized. It is quite easy to derive the tax or subsidy formulae from these conditions if only one of the two instruments were allowed. If the government relies
*For instance, see Diamond and Mirrlees (1971). ‘There are two wedges concerning u, namely (i) cp, -y,. the tax on c’ in the first period, (ii) (1 -k, ), the difference between the social and private valuation oft’ in the second period.
and
only on a change
in the present
price of energy, one obtains
from (25):
Tl
(30)
‘l-P’=dx,/djr,(,’ where T, denotes the right-hand side of (25). Alternatively, is only allowed on the heat-saving device, (26) yields
if a tax or subsidy
(31) where T, stands for the right-hand side of (26). Clearly, it is difficult to give any intuitive interpretation of the general tax formulae (25) and (26). For this reason, in the following sections we consider special cases and numerical illustrations.
3. The simplest case: A two-good economy with a single instrument subsidy on the first-period consumption of energy)
(a tax or
In this section the quantities consumed of all goods other than and second-period energy are assumed to be chosen exogenously. (30) then reduces to
the firstFormula
n,-p,~X-,-l Pl
(32)
and the first-best can be attained. To do so, only one decision variable (x1) need indeed be controlled for which one instrument (rrr) suffices. Will it be a tax or a subsidy? To answer such a question, we first assume that the government and the individual both have point expectations of the second-period price of energy. For illustrative purposes, we consider the case where the price expected by the government (P~,~) is higher than that expected by the individual (P~,~). This particular situation can be easily depicted in diagrams. We first take the case where the individual’s expectation is independent of the first-period price of energy (xi). Given his optimistic expectation, the individual’s budget constraint is higher than that of government; he is thus led to choose a present consumption of energy (x,,~) different from that of the government (xi, J. Two cases are depicted in figs. 1 and 2. In both figures, A represents the first-best choice that would be made by the individual were his expectation
n/l. Marchtrnd
und P. Pesrieau,
Taxation
md dirvging
expecttrtions
'1.g ‘l,i
x1
Fig. 1. Tax cas
Fig. 2. Subsidy
33
(independent
case (independent
expectations).
expectations).
34
M. Murchand
crnd P. Pestieau.
Tuxution trnd diuerging
expectations
the same as that of the government. But on the basis of his own expectation, the individual chooses B in the absence of tax or subsidy; he would then end up on the point C suboptimal on the basis of the government’s expectation. The government policy is to induce the individual to choose D and, thus, to consume the optimal amount of xi. In fig. 1, ~i.~
(33)
Now, a well-known
result is [see, for instance,
Atkinson
and Stiglitz
(1980, p.
76)l (34)
where u is the elasticity
of substitution
between
x2 and xi:
(35)
So, a tax (subsidy) is called for if the elasticity of substitution is smaller (larger) than the income elasticity of the first-period energy demand. It shows that the curvature of the indifference curves plays a key role in the policy choice. This appears clearly in figs. 1 and 2, where we focus on the elasticities of substitution (assuming identical income elasticities). In the case where today’s consumption of energy appears to be a close substitute for
M. ,Marchund
und P. Pestieau,
Tuxution
and diverging
expectutions
35
tomorrow’s consumption, the government encourages the individual to consume more energy today by subsidizing its current use (see fig. 1). On the contrary, in the case where the individual gets higher utility by making more equal his first- and second-period consumptions of energy, the government taxes its current use to reduce the imbalance in consumption that will result from the individual’s overly optimistic expectations (see fig. 2). Relation (34) also shows that even in the simple model (where we do not pay any attention to investment in energy-saving equipment), the relativeprice argument in favor of a subsidy presented in the introduction is wrong if the income effect outweighs the elasticity of substitution. Returning to the assumption of an additive utility that we need in the more complex cases, we prove in appendix B the following:
(36) where R is defined
R
=
_
as the elasticity
of the marginal
~,(a2u,iw
of x2:
(37)
au,lax, Furthermore,
utility
it is easily seen that
au2 x2,
g
Rglok,=
2x2
x2
*
(38)
au2 xZ,i
ax2 x2.i
since x2,g is always (here infinitesimally) smaller than x~,~. So the equivalence between our geometrical results and condition (32) is proved. The elasticity R can be expressed more explicitly as R
=
_
w2
f
;x2
fix2
(39) u;r,
f;’
where U; and U; (f‘; and f ;) denote the first and second derivatives of U, (f,) with respect to y, (x2). This expression emphasizes that R depends on both the individual’s preferences and the technology used to transform primary energy into a consumption good. It indeed involves the elasticity of the marginal utility of y, and that of the marginal productivity of x2 in units of y,. Under the standard convexity assumption, both terms of the right-
non-positive, which implies that R is always nonare inversely related to the elasticity of substitution to that between x2 and the two durable goods. section, we briefly consider the case where the the individual’s expectations through its choice of rri (dependent expectations). To see how our results are affected by this new specification, only the simple expectation function can be dealt with. It will be assumed that the expectations of the government and those of the individual are just equal to the current price multiplied by a constant and a random term. That is hand side of (39) are negative. Their values between y, and y2 and To complete this government can modify
p2.,=aplr
and
p2.i =bnitl,
(40)
where a and b are some scaling factors (a > b) and r] a random variable with unitary expected value. To provide a benchmark case, we shall assume that in the case of independence the individual’s expectations are formed according to (41)
Starting from the definition of k, given in (27), it can easily be understood that in the subsidy case (k, < 1) the dependency will further widen the difference between the expectations of the government and those of the individual and, therefore, the difference between the numerator and the denominator of the right-hand side of (27). This will call for a higher subsidy. On the contrary, in the tax case (k,> l), the spread between the numerator and the denominator will be reduced because the tax narrows the difference between the expectations. So a lower tax will suffice. The two cases are represented in figs. 3 and 4 for point expectations (i.e. y = 1 with probability one). In these figures A, B, C and D are defined in the same way as in figs. 1 and 2. Thus, D (on the dotted line) is the choice induced by the government policy in the benchmark case of independent expectations (p2,i = bp,). It should be compared to point E (on the dashed line) which the government induces the individual to choose in the case of dependent expectations (p2,i = bn,). Let us remark that in this case the budget line perceived by the individual (the equation of which is rrixi + br,x2 = w) has the same slope as that going through B (the equation of which is plxl + bp,x, ‘co). From this remark and the quasi-concavity of the utility function, it then follows that E is bound to lie between A and D in fig. 3 and above D in fig. 4. This implies that with dependent expectations, the optimal policy is still to tax (respectively subsidize) in the case of fig. 3 (respectively fig. 4). Furthermore,
A-f. Morchand
\
\
\
\
md P. Pestieuu,
Tuuation
\
'1.g
'1,i
Fig. 3. Tax case (dependent
'1.i
Fig. 4. Subsidy JPE-
c
and diverging
expectationsl.
xl,g
case (dependent
expectations)
expecttrtions
3-l
38
M. Mtrrchtrnd
trnd P. Pestieccu. Taution
trnd diwrging
expectations
the relative slopes of the budget lines going through D and E imply that the tax (subsidy) should be lowered (increased). The main conclusion to be drawn from this section is that the starting conjecture put forward in the introduction needs qualifications; in the oneinstrument case, subsidizing the current consumption of energy might well prove the optimal policy. This is more in line with the relative-price argument.
4. Introducing the energy-saving
equipment: A three-good economy
As soon as one leaves the two-good economy studied in the previous section, it proves very difficult to draw practical conclusions from the formulae obtained at the end of section 2. To illustrate the complexity of the problem, we introduce the energy-saving equipment in addition to x1 and x2 and take again the case of a single instrument, i.e. a tax or subsidy on the current consumption of energy. In this case, condition (30) reduces to
(42)
For point expectations satisfying pZ, g >P~,~ (as earlier), we shall attempt to infer the sign of z from (42). We have shown in the previous section that, depending on the curvature of the indifference curves in the (xi, x2) plan, (k, - 1) can be positive or negative. From condition (13), (with cp, =qJ, the term [q,- rcl (dx,/dtl)] is clearly positive. The sign of (k, - 1) is likely to be positive because an increase of the energy-saving equipment is expected to carry out more benefit if the future price of energy is higher [see the definition of k, given in (29)]. It is, however, difficult to determine the sign of or negative. the transfer-included price effects; they could be positive Furthermore, the signs of the various terms appearing in (42) could be related to each other. So, attempting to decide on the basis of (42) whether the current consumption of energy ought to be taxed or subsidized looks worthless. It is for this reason that we now turn to numerical examples. To illustrate the foregoing model, we use a CES specification for both the utility and the production functions:
U,+U,=[x;~+u-~]“‘~+[,;~+,-~]~‘~, where F and /?z - 1. With substitution in consumption
(43)
this specification, E and fl measure the degree of and in production, respectively. A low value of c
M. Mtrrchand
und P. Pestieuu,
Tuxctrion trnd dicerging
expecrtrrions
39
implies that present and future consumptions are close substitutes. Conversely, a high value implies a strong complementarity. Similarly, xi (or x2) and u are close substitutes if B is small. Four cases will be studied, in each of which we shall assume point expectations. In the first two there is a single instrument, a tax or a subsidy on the current consumption of energy. In case I, there is no way for the government to expectations (independent modify the consumer’s expectations): the consumer expects a future energy price pZ,i = 2, lower than the price expected by the government pZ, g =4. In case II (dependent expectations), the consumer expects a price equal to the current price of energy, i.e. pZ,i =x1. In the last two cases two instruments are allowed for: taxes or subsidies on the first-period consumption of energy and on energy-saving equipment with independent and dependent expectations respectively (cases III and IV). Through the calculations, the following values for the parameters are adopted : w=lO, The optimal interpretation
q,=l
and
pi=2.
values for z and 0 are given of which we now turn to.
4.1. A single instrument:
Cases I und
in tables
1, 3, 4 and
5, the
II
The figures on table 1 can be given an intuitive interpretation. For a given p, as E increases, the subsidy first decreases and then turns into an increasing tax. This is the result we had in the two-good case. Indeed, note that E is simply the consumption elasticity of the marginal utility for either period. For a given E, as fl increases, the tax does not change monotonically. We will focus attention on the two extreme values of E. First, let us take a= 5, that is when the tax policy ought to narrow the gap between y, and y,.. The effect of the tax is twofold: (i) it stimulates investment in energy-saving equipment and (ii) it tends to equalize the ex post consumption of energy in the two periods. For /?=O.S, namely a high substitution between v and xi, the first effect dominates and the tax can be very small to induce a change in u, the energy-saving equipment. On the contrary, for /I= 5, i.e. a low substitution between u and xi, the second effect dominates. Because of this low substitution, nothing can be gained by stimulating energy-saving investment, whose level furthermore shows almost no sensitivity to the tax. So the government will try to equalize the ex post values of x1 and x2. Because xi and x2 are very complementary to U, it will require a very high tax rate to approach equalization (in the limit, were u and xi perfect
M.
Murchund
ctnd P. Pestierru,
Trrxution
rrnd dicerging
expectrttions
complements, xi =x2 ex ante and thus x2 =$.x1 ex post, whatever the tax level might be). Let us now look at the other polar case, E= -0.9. Now y, and y, are strong substitutes; there is no longer a need to make them equal. Instead, the government policy will try using a subsidy to increase y, relatively to y,. The effect of such a policy will be (i) to discourage the investment in energysaving equipment and (ii) to raise the first-period consumption of energy at the expense of its second-period consumption. In this case the first effect, unlike the second, is unwanted. With high complementarity in production (e.g. p= 5) the subsidy rate can be high because it hardly affects the investment in energy-saving equipment so that the first effect does not play any significant role. This can be verified by comparing the value of c’ in the second-best solution (table 1) and its value in the absence of subsidy or tax (table 2). Conversely, in the case of high substitution in production (e.g. fi= -0.5) both effects have a sizeable impact and thus the subsidy is lower. Finally, for intermediate values of E, the outcome is expectedly not as clearcut as for polar values. Table 1 Con,umrr’>
choice in the abence
of taxation
1.~~.LI.
Table 3 yields the second-best values of z, x1, u and x2 in the case where the expectations are not independent, i.e. P~,~=K~. In broad terms, the trend of r is the same as in table 1. The big difference is that the tax tends to be lower and the subsidy higher, as explained in the previous section. In both tables 1 and 3, for realistic values of E, i.e. when the two consumptions cannot be easily substituted for each other, there is a tax which increases as p decreases. In other terms, were a better energy-saving technology (in terms of a good substitute for energy) introduced in the economy, a lower tax on energy would be needed. In terms of policy implications, this is an interesting point.
4.2. Two instruments:
Cases
III
und IV
Tables 4 and 5 consider the two-instrument case. Two important remarks are worth making. First, with two instruments a first-best allocation is possible so that the values for xi, x2 and v given in table 4 are the optimal
p=
\.
5
2
0.5
-0.1
-0.5
i:=
-0.2 5.03
0.1 X.21
-0.55 2.50 2.75
2.20
-0.4 2.95
-0.35 1.93 3.64
1.38
0.42
-0.9
0.56
0.66
0.63
0.55
0.24
Table 3
-0.1 5.22
0.1 8.22
1.79
1.75 0.15 2.31
0.05 2.82
-0.15 1.68 3.79
1.26
0.42
-0.5
1
0.92
0.71
0.57
0.24
1.63
1.61
1.49
1.20
0.42
0.24
0.4 2.17
0.25 2.75
0.05 3.28
1.14
1
0.79
- 0.05 5.31 0.57
0.125 x.22
-0.1
[,?-J
Case II: Tax on energy with dependent
1.55
1.49
1.38
1.07
0.39
0.55 2.10
0.45 2.6X
0.2 3.93
0.1 5.53
0.15 8.31
0.5
expectation
1.2
1.09
0.83
0.58
0.023
1.48
1.39
1.20
0.92
0.36
(p2,, =n, 1.
0.7 2.03
0.65 2.61
0.5 3.98
0.3 5.76
0.2 X.40
2
1.25
1.15
0.91
0.6
0.22
1.41
1.33
1.11
0.81
0.33
0.85 1.97
0.8 2.56
0.7 3.49
0.5 5.94
0.25 x.47
5
1.3
1.2
0.95
0.61
0.3
M.
Marchand
cmd P. Pestiruu.
III i;
/
II =L
Ttrxrrtion
tmd dicerging
rxpecrtrtions
43
44
M. Murchantl trnd P. Pestieuu,
Tuxtrtion cmd diverging exprctL!rions
ones. Secondly, one surprisingly notes that except in a few cases both energy and equipment are at the same time either taxed or subsidized. The reason for that is to be found in the presence of a lump-sum compensation which, though implicit. plays a very active role in displacing up or down the individual’s budget constraint. What really matters in the comparison is the ratio of the two taxes. In that respect, one notes that for E=P, the two tax rates are equal. When E> /$ the complementarity in consumption tends to be stronger than that in production and thus the misallocation in consumption is also stronger than that in production. A higher tax on energy is then needed. Conversely, when B>E a relatively higher tax (or lower subsidy) on equipment is called for. Another way to interpret the results of tables 4 and 5 is to compare them with those in tables 1 and 2. With a single instrument, the tax has not to be pushed so far as with two instruments because of its perverse effect on equipment. With two instruments, the level of equipment can simultaneously be controlled. Similarly, the subsidy on energy is not high enough in the one-instrument case as compared to the two-instrument case. Indeed, a subsidy on equipment can correct for the negative effect of the energy tax.
5. Conclusion The starting point of this paper was the conjecture that when consumers expect a lower price for energy than the government a tax should be imposed on the current consumption of energy and a subsidy granted to investments in energy-saving equipment. The foregoing theoretical analysis and numerical example lead us, first, to give more weight to the relative-price argument. There are indeed cases in which a subsidy on the current consumption of energy can be desirable. Secondly, even when a tax on energy is imposed, in some instances, a tax on equipment is also needed. However, those two results call for the following remarks. In what is usually considered as the most realistic cases, i.e. a strong intertemporal complementarity in consumption (large a), one can expect a tax on energy. Furthermore, simultaneous taxation of energy and equipment is mainly due to the presence of the positive lump-sum income transfer; to the extent that the tax rate on energy is relatively higher than that on equipment, one can then conclude that energy-saving equipment is effectively subsidized.
Appendix A: Derivation of the tax formulae (25)-(26) As a first step, we replace .x1, u, v, ci and s in the objective function and the last constraint of problem (23))(24) by the right-hand sides of the five
first constraints.
The Lagrangian
can then be written
as
c,(~,,u,v,s)l}+~~Cw-(n,-~,)x,-(cp,-qq,)v-ol.
64.1)
where xi, u, v, ci and s are to be taken as the demand functions (having rcl, cp, and w as arguments). The first-order condition for a maximum are obtained by making equal to zero the derivatives of the Lagrangian with respect to xi, cp, and w. It yields for 7ci:
au, ax, au, au, au au, ac, ac, an,
axK+au+-K+-1 1
1
au, ax, as ax, au ax, au +E, i ~8x2 ---++--++,3s an, au &cl GV (?7l,
1
(
au ac, au
ac, as
dc,
aS anI
aU OK1
---+--,+,-
02) anI
-
+A
[
1
-P&X1 -(q,,-q,,)E =o,
- (711
1
1
(A.21
Similar expressions are obtained for the derivatives with respect to cp, and w. Taking the derivative of (17) with respect to pi, we obtain:
as
-=
8%
Using
? ac, PI ~-Y’.fJ-(1u(:r. 3
&cl
1
(6) and differentiating
?
1
(5) with respect
(A.3)
1
to s enables
us to write
^
~!s+!p&?=(l+r)l%. 2
P2 ax2
(A.4)
M. Marchand
Substituting
&/dx,
and P. Pestieau,
72xution
and diverging
41
expecttrtions
from (A.3) and using (A.4), (A.2) can be written
as
(A.5)
where 5, is defined
by
(A.61 The last but one term on the left-hand side of (A.5) drops derivatives of (5) with respect to u and u give
ax2
ac,
pzu+,7,=0
By means
and
ax2
+i
-(q
-p,g 1
the
ac,
p2~+~=0.
of (9)-(12), (A.5) can be rewritten
out because
(A.7)
as
-xl-(q”-q”kg 1=o
(A.81
1
where Ti is defined in the same way as in (A.6)
(A.9)
By the same developments, it can be shown Lagrangian with respect to w reduces to
Eliminating
,I between
that
the derivative
of the
(A.@ and (A.lO) and using (18)-(21), we obtain
with the same expressions in brackets as in (A.8) and (A.lO). Dividing through by ci, and using (22) and (27)-(29), (A.ll) can be transformed into
+ E,{~U,/~u}
5i
+Ck,-112
[k/1,$
=o.
(A.12)
1 t
1 t
Formula (25) then follows from (A.12) by using (13) and (14). Starting from the derivative of the Lagrangian with respect (26) is obtained in the same way.
to cpU,formula
Appendix B: Proof of (36) Simplifying
subject
the notation,
the problem
reduces
to
maxU,(x,)+U,(x,)
(B.1)
PlXl
U3.2)
to +tP,x2
=w.
49
Denoting
by D the determinant
of the Hessian
matrix:
(B.3)
we have with obvious 6x,
+2
D,,i+x,D,, D '
where 3. is the marginal
(since pz/l= U,). and the positivity
notation
utility
(B.4)
of income.
The equivalence of D.
It then follows that
expressed
in (36) results
then
from
(BS)
References Atkinson, A.B. and A. Sandmo. 1980, Welfare implications of the taxation of savings, The Economic Journal 90, 529-549. Atkinson, A.B. and J. Stiglitr, 1980, Lectures on public economics (McGraw-Hill, London]. Diamond, P. and J. Mirrlees, 1971, Optimal taxation and public production. 1, American Economic Review 61, 8-27. Dixit. A., 1976, Public finance in a Keynesian temporary equilibrium. Journal of Economic Theory 12. 2422258. Harris, R. and N. Olewiler. 1979, The welfare economics of ex post optimality, Economica 46, 137-147. Musgrave. R., 1959, The theory of public finance (McGraw-Hill, New York). Sargent. T.J. and N. Wallace, 1975, Rational expectations, the optimal monetary instrument and the optimal money supply rule. Journal of Political Economy 83, 241-254.