Economics First Hour Test Solutions 2009

College 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.