AmherstCollege
Departmentof Economics
Economics
58
Fall2009
Name
First Hour Test
SOLUTIONS
There are three questionson this 80 minute examination. Each is of equal weight in
grading.
1. Supposethat a personconsumestwo goods,x andy, in fixed proportions. He or she
always consumesI unit ofx together with 3 units ofy no matter what the relative prices
are.
a. What is the mathematicalform for this person's utility function?
Wanty : 3 x So math form is U(x, y) = Minf3x,y)
b. Calculatethe Marshallian demandfunction for both goods for this person.
Substitutey:-?x into budget constraint : I = p,x + 3p nx .
So, Demandfunctionsore x = --!p,*3p,
t:3x:
3I
p,*3pn
c. Use the results from part b to calculatethe indirect utility function and the expenditure
function in this case-
v(p,.Pr,t)=
v =#6
E( p, . Pr , v ) = U* D
d. In classwe discussed
why expenditurefunctionsareconcavein prices. Is the
expenditurefunctionyou calculatedin part c concavein p,? Explainyour result
intuitively.
Here
AE
op,
:V l3
A2F,
3
= 0. Concavityrequiresthe abilityto substitute
away
ap",
from a good whose price has risen. In this casethere is no substitution. Holding Z
constantrequiresholding x constant,so expendituresincreaselinearly with the price of x.
e. Oneway to computethe Marshallianprice elasticityof demandin this problemis to
usethe Slutskyequationin elasticityform. Write downthat equationandthenuseit in
conjunctionwith the demandfunctioncalculatedin part b to derive ,,.0, inthis case
(note:this will be a functionof the prices,not a specificnumber).Whatis unusualabout
theprice elasticityin this case?(Note- the price elasticitycanalsobe computedfrom the
Marshalldemandfunctionin part b. - you may usethis asa checkif you wish, but not as
your primaryproof)
In this example,substitutioneffectsarezero.Sothe Slutskyequationin elasticityform is:
€,,p,= 0 - s,e,,,. Usingpartb, €,.t =I,
elasticityir, ,,.n, = -
-- l';
+ 5Py
P'
definition of elasticitv.
s,^ = ry
I
= -4.^
p,*3pn
price
Sothe Marshallian
. Canalsoget this directlyfrom part b by applyingthe
2. John Hicks' reformulation of demandtheory focusedon compensateddemand
functions rather than on the more widely used Marshallian demandfunctions.
a. Explain the difference betweenthesetwo concepts:
Marshallian demandallows the analyst to hold income constant.
Hicks demandallows the analyst to hold utility constant.
b. One advantageof the Hicks approachto demandis that crossprice effects are
symmetric. Explain what "symmetric" meansin this context and prove that this
symmetry holds for any pair of goods.
Symmetry meansthat the effect of the price of x on the quantity ofy demandedis the
sameis the effect of the price ofy on the quantity of x demanded. This follows from the
envelopetheorem and Young's Theorem:
a*: : a2E = a'E =ar',
.
0pi
0p,0p,
6p,0pi
6p,
c. Explain in intuitive terms why Marshallian crossprice effects generally are not
symmetric.
Marshallian crossprice effects contain income effects and theseare generally not
symmetric. One good may have a larger income effect than anotherbecauseone is a
luxury and the other a necessity. This difference in income elasticitieswill result in
differing gross cross-priceeffects.
d. Explainwhy cross-priceeffectsareindeedsymmetricfor the Marshalliandemand
functionsderivedfrom a Cobb-Douglas
utility function. Thenprovethat this is a special
caseof the moregeneralresultthat Marshalliancross-priceeffectsaresymmetricif a
personalwaysspendsa constantshareof his or her incomeon eachgoodregardless
of
prices. (Hint: You will needto usethe cross-priceSlutskyequationto showthis)
The substitutiontermsin the Slutskyequationaresynrmetricby pafi b. If the individual
spendsa constantshareon eachgood, * = L!
Pi
= -r,?:.
represented
by - x i + =-U/
pipl
AI
AI
* =t
The incomeeffectsare
Pl
Hence,in this casethe incomeeffectsare
also symmetric. So the overall Marshallian cross-priceeffects are symmetric.
e. Hicks' "third law" statesthat "most goods are substitutes". This is proved using
Euler's theoremfor homogeneousequationswhich statesthat if a function .f (xr,...,x,) is
homogeneous
of degreefrthen xrfr *....xn.fn= kf (xrt...;xn). Showhow this
mathematicaltheorem can be used to prove Hicks result and then show how to statethe
Hicks result in elasticity terms.
Apply the theoremto the Hicks demandfunction xi(p,...p,,2)
which is homogeneous
9'iof degreezeroin the prices:pry+...
+ p,y=
< 0, the other
0. But, ,inr"
oh
0p,
oPt
terms must be predominatelypositive - the sign of substitutes.
Dividing the equationin this proof by x7 yields €,i,p,* ... r €,i.r, = 0. In words,the sum
of all of the compensatedprice elasticities for a good must be zero. Since the own price
elasticity is negative,the crossprice elasticitiesmust be predominatelypositive.
3. Much of the theory of optimal taxation is basedon principles that can be derived from
simple demandtheory. In this question you are askedto take a primarily graphical
approachto this topic.
a. Define the terms "lump sum principle" and "excessburden of a tax".
The "lump-sum principle statesthat non-distortionary taxes on generalpurchasing
power involve smaller lossesin utility than do taxes on individual goods. The "excess
burden of a tax is the extra utility lost from a given tax relative to a lump-sum tax that
raisesthe samerevenue.
b. Showgraphicallyhow a simpletwo goodmodelof utility maximizationcanbe used
to illustratethesetwo concepts.Thatis showthe excessburdenof a tax on a singleitem
graphically.(Note: This questionasksyou to usethe utility maximizationgraph- not a
demandcurve).Be sureto explainyour answerfuIly.
f.,ha\
Buotpeh
Ir.trf -9 "un ln*
raz,*(
(
evlu
5 Avt|r< nc,v
qr-u, :
e.xc<99
bvrJ,en-
u3
ta* ar. x
x
c. If x andy are consumedin fixed proportions, the lump sum principle doesnot hold
and there is no excessburden from a commodity tax. Show this result with a carefullv
drawn graph.
vr+1?-5tJ!\^ fa *
t-tt, 5-14v!|.c.flVehug
\ B" Jse t
fa>t on X
Al.o eqagg
f'urde,t,
.
d. Whatconclusiondo you draw from comparingyour answersto partsb andc? That
is, what determines
the sizeof the excessburdenfrom a tax?Illustrateyour conclusion
with a demandcurye graph. Be sureyou indicatewhetheryou are drawing a Marshallian
or Hicksiandemandcurvefor your illustration.
Excessburdenarisesfrom the substitutioneffectsinducedby a tax. If thereareno
substitutioneffectsthereis no excessburden.The graphbelowshowstwo Hicks demand
curves.Onefor a situationwith largesubstitutioneffects,the otherfor a situationwith
smallsubstitutioneffects.Noticethat the excessburdenis muchsmallerin the second
case.
?*
ftr
4
e. Another applicationof this line of theory concemsoptimal pricing of electricity. If
usersexhibit different responsesto electricity prices, how should prices be arrangedso as
to yield the minimal deadweightloss from an electricity monopoly?
The utility could chargehigher prices to consumerswith less elastic demands. This
would minimize the deadweightloss from such monopoly pricing. The graph would look
much like that in part d. This sort of pricing is sometimescalled "Ramsey pricing" after
the economistfrank Ramseywho discoveredthe idea in the 1920's.