Ramsey-type results dictate that an optimal pattern of taxes must tax more heavily those goods which have a more inelastic(compensated)demand. Corlett and Hague (1953) investigated the optimal revenue-neutral movements from an initial uniform tax. They obtained that the goods (relatively) more complementary to the untaxed good (leisure)should see their taxes increased-which in a revenue-neutral seeting implies that the other goods see their taxes disminished. In a three-good economy (with only two goods being subject to taxation) the Ramsey-type rule and the Corlett-Hague result can be easily related.Optimal taxation; Ramsey; Corlett-Hague; Demand elasticity;

Ley, Eduardo

English

Research Papers in Economics

Working Paper 92-14  Division de Economia 
April 1992  Universidad Carlos III de Madrid 
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28903  Getafe (Spain) 
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A NOTE ON  RAMSEY AND CORLETT-HAGUE RULES 
Eduardo Ley* 
Abstract  _ 
Ramsey-type results dictate that an optimal pattern of taxes must tax more heavily 
those  goods  which  have  a  more  inelastic  (compensated)  demand.  Corlett  and Hague 
(1953)  investigated the optimal revenue-neutral movements from an initial uniform tax. 
They obtained  that  the goods  (relatively)  more  complementary  to  the untaxed good 
(leisure) should see their taxes increased-which in a revenue-neutral setting implies that 
the other goods  see  their taxes disminished.  In  a  three-good economy  (with only  two 
goods being subject to taxation) the Ramsey-type rule and the Corlett-Hague result can 
be easily related. 
Key Words:
 
Optimal taxation, Ramsey, Corlett-Hague, demand elasticity.
 
*Departamento de Economia, Universidad Carlos III de Madrid.
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Economagica, Vol. 99, No. 9 (April 10,  1992) pp. 9-99 
A  NOTE ON
 
RAMSEY AND CORLETT-HAGUE RULES
 
By EDUARDO  lE}' 
Ramsey-type results dictate that an optimal pattern of taxes must tax 
more heavily  those goods which have a  more inelastic (compensated) de­
mand.  Corlett and Hague (1953) investigated the optimal revenue-neutral 
movements from  an initial  unifonn tax.  They obtained  that  the goods 
(relatively) more complementary to the untaxed good (leisure) should see 
their taxes increased-which in a revenue-neutral setting implies that the 
other goods see their taxes disminished.  In a  three-good economy (with 
only two goods being subject to taxation) the Ramsey-type rule and the 
Corlett-Hague result can be easily related. 
THE  INVERSE  ELASTICITY  RULE, often referred to as  the Ramsey rule, is  one of 
the most well-known results in public finance.  The most simple inverse elasticity 
rule is  of the form ti!(Pi +ti) =klf.ii,  where ti is  the tax on  good  i, Pi  is  i's 
before-tax price,  k  is  a  constant, and  f.ii  is  i's own-price elasticity of demand. 
It ignores any  cross effects between the taxed goods, and, if iii is  meant to be 
the marshallian demand price elasticity, any  income effects.  More complicated 
expressions are obtained  when  all  the general  equilibrium effects  are  properly 
taken into account.  Corlett and Hague (1953) investigated the optimal revenue­
neutral movements from an initial uniform tax.  They obtained that the goods 
(rel~tively) more complementary to the untaxed good  (leisure) should see their 
taxes increased-which in u revenue-neutral setting implies that the other goods 
see their taxes disminished. 
Relating both themes, if th/; optimal pattern is a uniform ad valorem tax then 
the optimal Corlett-Hague movement would be no movement at all.  Therefore, 
all  goods must behave equally with respect to the untaxed good (and symmet­
rically  between  them).  On  the other hand,  if the  Corlett-Hague  conceptual 
experiment implies  that good  j  must see  his  tax burden  increased  and good 
k  decreased,  it has to be the case  that the Ramsey-type rule suggests that j 
must be  relatively more heavily  taxed than k.  An  explicit relationship will  be 
developed in  this paper. 
In  the first section, I shall derive  the measure of excess burden  used  in  the 
paper. In section two I shall introduce Ramsey-type results, and in section three 
the Corlett-Hague model.  Finally, in  the last section I shall relate both results. 
Financial support from a Bank of Spain Fellowship is gratefully acknowledged. [
 
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2	  EDUARDO 1# 
1.  MEASUREMENT OF LOSS  FROM DISTORTING TAXES 
I  will  introduce,  first,  the  assumptions about the  model  used  throughout  the 
paper. These are: 
•  Three-good economy.  Good	 1,  leisure,  will be the numeraire and will  be  un­
taxed (otherwise proportional taxation would be optimal). 
•	 Agents have no exogenous income  (otherwise a lump-sum tax would  be opti­
mal) and have the same preferences. 
•	 The technology is  a linear one.  Therefore, producer's prices, p, are constant. 
Final consumer prices, q, are the sum of the producer prices and the vector of 
taxes, t. Thus, q = p +t. 
The loss  for  any  given  tax  vector  can  be  measured  as  the  negative of the 
compensating variation after the revenue collected has been returned to the 
consumers. The compensating variation will  be given  by 
cv =J.l(q;q,O) - J.l(q;p,O) =-J.l(q;p,O); 
where  J.lO  is  the money-metric  utility  function. l  Let  UO  _  v(p,O),  then  the 
burden L(t) due to the tax vector t  can be expressed as 
L(t) =- (~tiXi(q, e(q, UO))  - J.l(q; p, 0)) 
3 
= e(q,'uO) - Ltihi(q,UO) 
i=2 
3 
= L(qi - t;)hi(q, UO) 
i=2 
3 
= LPihi(q,  UO); 
i=2 
which  is  the value of the expenditure function at the after-tax price vector less 
the tax revenues  collected~  and (conceptually)  returned lump sum.  From the 
last expression, it can  be  s~en  that the loss equals the value,  at pre-tax prices, 
of the cheapest post-tax bunule that yields the initial level of utility. 
1  I will be using Varian (1984) textbook's notation throughout the paper. Thus, the marshal­
lian demands will be denoted by Xi(p, m); hicksian demands by hi(P, u); expenditure function 
by e(p, u) and indirect utility function by v(p, m). Therefore, Il(qj p, m) ==  e(q, v(p, m)). Also, 
price vectors are row  vectors while quantity vectors are column vectors. 
2  If the consumer were compensated so that she could stay at her initial utility level uO,  her 
demand would be h(q, UO)  and, in turn, the taxes collected would be th(q, UO).  Actual taxes 
collected, tx(q, 0)  are irrelevant here. 3 
[ 
RAMSEY  AND  CORLETT-HAGUE 
[	  The marginal loss from a small change in  one of the unit taxes is  given by 
(1) 
The optimal pattern of taxes to raise T  while minimizing the loss inflicted to 
the economy can be obtained by solving3 
mm	  L(t) 
3 [J	 
t 
s.t.	  L tihi(q, UO)  = T. 
i=2 
Both  demand and production  are  homogeneous of degree  zero  in  p  and q  re­
spectively.  Therefore,  two  normalizations  are  allowed.  It's  useful  to  make 
PI  ==  ql  ==  1, which implies tl :: O. 
[	  The first order conditions give 
k =2,3. 
Rearranging, we get
lJ  ,\ 
1-,\ 
(2) 
(3) 
(4) 
3  IT either the equivalent variation were used instead of the compensating one, or the indirect 
utility function  were  maximized subject to the revenue constraint, similar results  would be 
obtained.  However,  h(·)  would be evaluated at different points in the resulting formulae­
i.e.,  at  h(q,v(p,O))  with  the  CV  and at x(q,O)  with  the EV.  Thus,  maxt  v(q,O) + fJ  ,\ (Li tiXi(q,O) - T)  is  equivalent to mint  L(t) + '1 (Li  tih;(q,V(q,O»  - T)i  when L(t) 
uses  the equivalent variation measure.  This holds because we  have only  one consumer (or 
any arbitrary number of them as long they all have the same preferences).  In a  many-person 
economy, the expenditure approach and the indirect utility function one do not give the same 
answers.  Fw-thermore, in this context the concept of loss in not well defined. 
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for  k =  2,3.  To exchange the approximately-equal sign  for  a  plain-equal sign 
between equations (2)  and (3), all  we  need is  to have linear hicksian demands 
over the pertinent range. 
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Equation (2)  makes evident that the distortionary effects of any tax pattern 
come through the substitution effects.  Two alternative tax patterns that raise 
the same revenues must have the same income effects and, therefore, can only 
be  compared on the grounds of the inefficiencies  they arise by  the reallocation 
of resources.  Equation (4)  suggests that the optimal tax structure involves an 
equal proportionate change in  the hicksian demands for  all  goods. 
o	 
Equation (4)  does not provide  an explicit rule for  the pattern of taxes but 
indicates what it must satisfy.  The only way  the first order conditions can be 
simultaneously satisfied is  having goods (factors) whose compensated demands 
(supplies) are relatively inelastic subject to relatively higher taxes.4 
2.  RAMSEY-TYPE RESULTS 
In our two-good economy, it's easy to manipulate (2) further to get 
[	  (5) 
[	 
which implies that we  are going to have tdq2 > t3/q3 whenever 
(6) 
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L	  Equation (6)  confirms all the elasticity arguments discussed above and will  be 
used  to link  up  with  the  Corlett-Hague  results.  Let  (ij  denote  the hicksian 
elasticity of good i  with respect to fs price.  Then, we  can rewrite (6)  as 
in particular, if the cross-price elasticities were zero, we'd need 
n Therefore, good 2 must be relatively more inelastic than good 3.  LJ 
4  Atkinson and Stiglitz (1972) investigated under which conditions equal (uniform) propor­
tional taxation of all commodities (except labor, the numeraire) is optimal. They proved that 
uniform taxation is  optimal if either (i) labor is in absolutely fixed supply-Le., uniform tax­
ation amounts to a  lump-sum tax on labor rents as  argued before--and (ii) preferences are 
homothetic.  Also,  if preferences have an additive representation then tax rates are inversely 
proportional to each commodity's income elasticity of demand.  If preferences are additive in 
logarithms with equal coefficients, then all income elasticities equal one, and uniform taxation 
is optimal. 
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RAMSEY  AND  CORLETT-HAGUE  5 
From equation  (5)  we  can  also  get  a  related  expression  which  captures  the 
same ideas, 
t2  1 (A  ta  )
q2 =(22  1 - A - qa (23  ; 
if cross-price effects are zero, this equation collapses to 
tk  A 1 
-=----;
qk  1- A(H 
which is  the appealing inverse elasticity rule.  Note that the relevant elasticity is 
the hicksian elasticity which won't necessarily agree with the marshallian elastic­
ity unless income effects are zero.  Also,  the assumption that cross effects vanish 
is  a rather heroic one." 
We  can  also obtain expressions in  terms of the (observable)  marshallian de­
mands.  Using the identity hi(q,u) ==  Xi(q, e(q, u)) we  can derive Slutsky's equa­
tion, 
which can be used to substitute ahk/aqj in  (2), we get 
1  ~  (aXk(q, e(q, Uo))  aXk(q, e(q, Uo))  (  (0)))  A 
L,; ti  a  +  a  Xi  q, e q, u  =  1 _  \ ; 
Xk (q,e(q,uO)) i==2  qi m  A 
rearranging, we obtain 
(7) 
a a 
1  ""  axdq,e(q,uO))  A  TJk  ""  ° .
(  (o))L,;t;  a.  =1-\- (  O)L,;t;Xi(q,e(q,u)),
Xk  q,e q,u  i=2  q.  A  e q,u  i=2 
or, dropping arguments to make it look more compact, 
a a
-.!.- Lti  aXk  = _A_ _  'T/k  LtjX;; 
Xk  i=2  aqi  1- A  e(q,uO)  i=2 
where 'T/k  is  good's i income elOl!lticty, 
Assuming the aXk/aq;'s are  constant over  the relevant  range,  the left-hand
 
side of (7)  gives  the percel;tage change in  the marshallian demand (supply) of
 
the  kth  good  (factor).  Goods  with high  income  elasticities should  change  by
 
greater amount.  (Notice  that this  argument  would  suggest light  taxation on
 
necessities against the conventional wisdom based on  the inverse elasticity rule
 
that suggests heavy  taxation on  necessities.)
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6  EDUARDO  I.@' 
3.  CORLETT-HAGUE RESULTS 
In this section, I will  consid,~r  revenue-neutral substitution among taxes to derive II,  the Corlett-Hague results. Total revenue is given by 
L . .J 
3 
T = Ltihi(q,uO), 
i=2 
differentiating T  with respect to t, we get 
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In particular,
 
(8) 
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Setting T =Rand dT =0 uetermines all possible tax substitutions that keep 
revenue unchanged.  Since we only have two"taxes here, we  must have 
o  or 
(9) 
The marginal loss due to this change is given by 
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Consider (marginally)  substituting a  tax on  good  3  by  a  tax on good  2,  the 
proposed change does produce a movement towards efficiency if 
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RAMSEY  AND  CORLETT-HAGUE  7 
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Corlett and Hague (1953) examined the efficiency gains obtained by  moving 
away from equal proportional taxes on two goods in  this three-good economy. 
Assume initially t2 =TP2  and t3 =TP3,  so qi =(1 +T)Pi.  From the expression 
above, 
dL  oL dt21  oL 
dt3  = Ot2 dt3  T=R + Ot3 
= _ (t  2oh2 +t3 Oh3)  dt21  _ t2 oh2 _ t3 oh3. 
OQ2  Oq2  dt3  T=R  Oq3  OQ3 
The homogeneity conditions imply 
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(10) 
c 
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also 
so 
dL  T (Oh  l  dt21  Oh l ) LJ  dt3  =  1 + T  OQ2  dt3 T=R + OQ3  . 
Next, differentiating hI  with respect to t3  subject to the normalization and the 
revenue constraint, 
dh l  oh  l  dt21  oh  l 
dt3 = OQ2  dt3  T=R + OQ3'
1I  Therefore, substituting this last expression we  get u 
dL  T dh l
(11 )  dt3  - 1 + T dt3  . 
The sign of dhddt3  depenCls on the relative magnitude of the elasticities of the 
two goods with respect to leisure.  Using equations (9)  and (8), we  have that 
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(12) 
Assuming the denominator is positive,5 whenever (a1 > £21  we'll have dL/dta > 0
[J  so  the loss  will  decrease  if ta decreases.  If,  on  the other hand,  (a1  <  £21  then 
lowering the tax on good 2 will  produce a movement towards efficiency. 
o 
Thus, the sign  of dhddta depends on  the substitution terms.  In  particular, 
if 2  and  1 are substitutes, 3 and  1 must  be  complements and dhddta < 0  so 
o 
the loss  decreases.  Hence  the good that is  complementary to leisure should be 
taxed.  If  both were  substitutes, the tax should be raised on the good relatively 
more complementary with leisure. 
4.  RELATING RAMSEY AND CORLETT-HAGUE RUL-ES 
n	  Equation  (6)  stated the condition  that would  lead  to an  optimal  tax  pattern 
with t2/q2 > ta/qa, which  was 
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using equation (10), it can  he rewritten as
 
(13) 
Ll	  or 
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(14) 
Thus, if both goods are leisure substitutes-in a hicksian sense-the above elas­
ticities are positive and 2 would  be more complementary-i.e., less substitute­ o 
than 3.  The other way  equation (13) might hold is  with 2 being complementary 
to leisure and 3 being substItute, in which case the left hand side of the inequality 
is  negative and the right hand side is  positive.  Finally, if the hicksian  demand 
of good 2 (good  3)  does  not respond to changes in  the price of leisure we  need 
good 2 (good 3)  to be substitute (complementary) to leisure. 
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In  summary, equations (12)  and (14) convey  the same message that the good 
(relatively) more complementary to leisure should bear a (relatively) higher tax. 
Squash racquets, tennis shoes and good books are to be the innocent victims of 
Ramsey, Corlett and Hague. 
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5  Equation (12) could give perverse results if goods 1 and 2 were hicksian-substitutes and it 
turned out that the denominator were negative.. 
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RAMSEY  AND  CORLETT-HAGUE  9 
REFERENCES 
o  ATKINSON,  A.  AND  J. STIGLITZ:  "The Structure of Indirect Taxation and Eco-
nomic Efficiency,"  Journal of Public Economics 1 (1972),97-119.  o  ATKINSON,  A.  AND  J. STIC,LITZ  Lectures on Public Economics. New York: Mac-
Graw-Hill, 1981. 
o￿  CORLETT,  W.  AND  D.  HAGUE:  "Complementary  and  the  Excess  Burden of 
Taxation,"  Review of Economic Studies 21  no.  1(1953),21-30. 
RAMSEY,  F.P.: "A Contribution to the Theory of Taxation,"  Economic Journal 
37  (1927),47-61. 
o￿ 
TRESCH,  R.  Public  Finance:  A  Normative  Theory.  Georgetown,  Ontario: Ir-￿
win-Dorsey,  1981. 
o￿ 
VARIAN,  H.  Microeconomic Analysis. New  York:  Norton, 1984.￿ 
o￿ 
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A note on Ramsey and Corlett-Hague rules.

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A note on Ramsey and Corlett-Hague rules.

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