Consumer Theory III:Duality1The Dual Problem1.1Expenditure Minimization1.The expenditure minimization problem:p:xminxs:t:u(x) u;x 0:The solution to this problem tells us the minimum expenditure that is necessary such that the consumer’s utility is not lower than u.2.Solving this problem tells the decision-maker(e.g.the government)the payment necessary to keep a person’s utility above a minimum level.3.De…nitions:A solution to the expenditure minimization problemis known as the Hicksian demand function and is denoted byx h(p;u):Substituting x h(p;u)into the objective function gives us the ex-penditure functione(p;u) p x h(p;u):4.Properties of Hicksian Demand:Suppose u is continuous and strictlyincreasing,then x h has the following properties:(a)Homogeneity of degree zero in p.That is,x h(tp;u)=x h(p;u)for all scalar t>0.(b)No excess utility:u x h(p;u) =u.5.These properties are similar to those of the Marshallian demand.Brie‡y,(a)is obvious from the defn.of the minimization problem;(b)follows from strict monotonicity.6.Theorem:Suppose u is strictly increasing and x h(p;u)is uniquefor all p.Then for any p0and p00;(p00 p0): x h(p00;u) x h(p0;u) 0:7.Proof.We note thatu x h(p00;u) =u x h(p0;u) :By de…nition,x h(p00;u)and x h(p0;u)are the cheapest way to achieve u under p00and p0,respectively.Hencep00 x h(p00;u) p00 x h(p0;u);p0 x h(p0;u) p0 x h(p00;u):These two inequalities imply that(p00 p0): x h(p00;u) x h(p0;u) 0: Note that this result follows entirely from the fact that x h min-imizes expenditure.It does not use any other properties of the utility functions.8.This result is called the compensated law of demand.It statesthat an expenditure-minimizing consumer should substitute low-price goods for high-price goods.Note that if p00and p0di¤er only in the price of good i,then the result becomes(p00i p0i) x h i(p00;u) x h i(p0;u) 0;which means that the Hicksian demand for good i is downward sloping.9.Properties of the Expenditure Function:If u(:)is continuous andstrictly increasing,then the expenditure function e(p;u)has the following properties:(a)Zero when u takes on the lowest level of utility(i.e u(0)whenthe utility function is increasing).(b)Continuous on its domain R n++ U.(c)For all p>>0;strictly increasing and unbounded above in u.(d)Increasing in p.(e)Homogeneous of degree1in p.(f)Concave in p:10.Proof:(a)The consumer can achieve u(0)by not spending anything.(b)Intuitively,the result follows from the continuity of u and thecontinuity of the budget set.(c)This is straightforward.Suppose e is not strictly increasing inu.Then there exist p,u0and u1,u1>u0,such that e(p;u1)e(p;u0).Since u is continuous and strictly increasing in x,itis possible to reduce consumption by some small"such thatu x h(p;u1) " >u0andp:(h(p;u1) ")<e(p;u0):Here"is a vector:This violates the de…nition of e(p;u0).Sup-pose that e is bounded by some e (meaning that e(p;u) efor all p and y).Since e is strictly increasing in u,e(p;v(p;e )+1)must be greater than e ,a contradiction.(d)If p1 p0(that is,p1i p0i for all i),then any consump-tion bundle x that is feasible under p1is also under p0.Sincee(p;u)is by de…nition the lowest expenditure that is neces-sary to achieve u,e p0;u p0x h p1;u p1x h p1;u e p1;u :(e)This immediately follows from the fact the Hicksian demandis homogeneous in degree zero in p.(f)Let x1=x h(p1;u)and x2=x h(p2;u).By de…nition,for anyx such that u(x) ue p1;u =p1x1 p1xande p2;u =p2x2 p2x:It follows that for any t2[0;1]and for any x such thatu(x) ute p1;u +(1 t)e p2;u tp1+(1 t)p2 x:The above equation holds for all x such that u(x) u,in-cluding x h(tp1+(1 t)p2;u).Hence,te p1;u +(1 t)e p2;u tp1+(1 t)p2 x h tp1+(1 t)p2;ue tp1+(1 t)p2 ;u :It is important to note that the concavity of e follows from thede…nition of e and does not require that u be quasiconcave.11.Di¤erentiable Utility.Solving a minimization problem involves thesame technique as solving a maximization problem.However,inthis problem the last equation comes from the utility constraint,whereas in the maximization problem it comes from the budgetconstraint.The Lagrangian function isL=p:x+ (u u(x)):The…rst order conditions are(assuming x>>0):@L @x i =p i @u@xi=0for all i=1;:::;n;@L@=u u(x)=0: The…rst n equations imply@u@x1 p1=::::=@u@x np n12.Shephard’s lemma:If e(p;u)is di¤erentiable in p at(p0;u0)withp0>>0,then@e(p0;u0)@p i=x h i p0;u0 ;i=1;:::;n:The results follows directly from the envelope theorem.@e @p i =@L@p i=x h i.The Shephard’s lemma is the counterpart of Roy’s Identity.Same Intuition.1.2Relationship Between Marshallian and Hick-sian Demand Functions1.The indirect utility function and expenditure function are inti-mately connected.So are the Marshallian and Hicksian demand functions.We have already seen that the two demand functions share a common tangency condition.Regardless of whether one’s objective is to maximize utility or minimize expenditure,one should consume in such a way that the marginal utility per dollar spent on each good is the same.2.Theorem1.8.Let v(p;I)and e(p;u)be the indirect utility func-tion and expenditure function for a consumer with a continuous and strictly increasing utility function.Then(a)e(p;v(p;I))=I;(b)v(p;e(p;u))=u:3.Part1says that if the maximum utility the consumer can obtaingiven p and I is v,then the minimum expenditure needed to achieve v is I.Part2says that if the minimum expenditure needed to achieve utility u given p is e,then maximum utility one can obtain given p and e is u.4.Proof:I will only prove part1.Part2can be proven in the sameway.Since e(p;v(p;I))is by de…nition the minimum expenditure needed to attain v(p;I),it must be that e(p;v(p;I)) I.Suppose e(p;v(p;I))<I.Then,there exists x such that px <I and u(x ) v(p;I).By strict monotonicity,p(x +")<Iand u(x +")>v(p;I)for some small".This contradicts the supposition that v(p;I)is the highest utility attainable given p and I.5.Note that the theorem is incorrect if the utility function is notstrictly increasing.For example:if u(x)=10for all x,thenv(p;5)=10;e(p;10)=0:6.Theorem1.9.Suppose u is continuous,strictly quasiconcave,andstrictly increasing.Then for all p>>0,I 0,and u2U,we have(a)x(p;I)=x h(p;v(p;I)).(b)x h(p;u)=x(p;e(p;u)).7.The…rst relation says that the Marshallian demand at prices pand income I is equal to the Hicksian demand at prices p and the maximum utility level that can be achieved at prices p and incomeI.The second says that the Hicksian demand at any prices pand utility level u is the the same as the Marshallian demand at those prices and an income level equal to the minimum expenditure necessary to achieve the utility level.8.Proof:Strict quasiconcavity means that both x(p;I)and x h(p;u)are unique.To prove part1,note that by Theorem1.8the mini-mum expenditure needed to attain v(p;I)is I.Since u(x(p;I))= v(p;I)(by de…nition of v)and px(p;I)=I,x(p;I)is the solu-tion of the expenditure minimization problem.The proof for the second part is similar.9.There are two ways to derive the Hicksian demand.The direct wayis to solve expenditure minimization problem.But we can also get the marshallian demand by solving the utility maximization problem.Substitute the demand back into the utility function.This gives the indirect utility function.From the indirect utility function,we know how much income need to change to keep utility constant as price change.Substitute this back into the Marshallian demand gives the Hicksian demand.10.Cobb-Douglas example:u(x1;x2)=x 1x 2, + =1x1= Ip1;x2=Ip2The indirect utility function isv(I;p1;p1)=u(x(p1;p2;I);x2(p1;p2;I))= I p1 I p2 = p1 p2 I Rearranging the termse(p x;p y;u)=up1 p2 = p1 p2 u:The Hicksian demand for Cobb-Douglas utility function is x h1= p1 1 p2 u;x h2= p1 p2 1u:1.3An Example:CES(continued)We continue the example u(x1;x2)=(x 1+x 2)1= :We want to derive the corresponding expenditure function,then compare it with the indirect utility function,and show the duality between Marshallian and Hicksian demand functions.1.3.1Expenditure functionBecause the preferences are monotonic,we can formulate the expenditure minimization problem as:minx1;x2p1x1+p2x2subject to:(x 1+x 2)1= =u;x1;x2 0And its LagrangianL(x1;x2; )=p1x1+p2x2+ [u (x 1+x 2)1= ] Assuming the interior solution in both goods,and by the…rst-order condition we have:@L@x1=p1 (x 1+x 2)(1= ) 1x 11=0@L@x2=p2 (x 1+x 2)(1= ) 1x 12=0@L@=u (x 1+x 2)1= =0By the…rst two,we can getx1=x2 p1p2 1=( 1)By the third one,we haveu=(x 1+x 2)1=1.3.2Hicksian demandsBy the above two equations,we can get the Hicksian demandsx h1(p;u)=u(p r1+p r2)(1=r) 1p r 11;x h2(p;u)=u(p r1+p r2)(1=r) 1p r 12where r= =( 1):1.3.3Expenditure functionTherefore,we can obtain the expenditure function:e(p;u)=p1x h1(p;u)+p2x h2(p;u)=u(p r1+p r2)1=r:1.3.4Relationship between indirect utility and expenditurefunctionsThey actually imply each other.When you get one,you can get the other very quickly.In previous lecture,we’ve already got the indirect utility function:v(p;I)=I(p r1+p r2) 1=r:For income level I=e(p;u);we must havev(p;e(p;u))=e(p;u)(p r1+p r2) 1=rWe’ve already know from theorem1.8,that v(p;e(p;u))=u;there-fore we haveu=e(p;u)(p r1+p r2) 1=rWe can get the expenditure functione(p;u)=u(p r1+p r2)1=rAnd you can try to get the indirect utility function given the expen-diture function is known.1.3.5Relationship between Marshallian and Hicksian demands We know the Marshallian demands in the previous lecture asx i(p;I)=p r 1iIp r1+p r2;i=1;2;x i(p;e(p;u))=e(p;u)p r 1ip r1+p r2=u(p r1+p r2)(1=r) 1p r 1i=x h i(p;u)2Income and Substitution E¤ects1.Slutsky equation:sincex h i(p;u )=x i(p;e(p;u ))@x h i(p;u )@p j =@x i(p;I)@p j+@x i(p;I)@I@e(p;u )@p j=@x i(p;I)@p j+@x i(p;I)@Ix h j(p;u )(by Sheppard’s lemma) =@x i(p;I)@p j+x j(p;I)@x i(p;I)@IRearranging the terms,we have@x i(p;I)@p j =@x h i(p;u )@p j x j(p;I)@x i(p;I)@I(1)The Slutsky equation decompose the e¤ect of the price of good j on the quantity demand of good i into a substitution e¤ect and an income e¤ect.The former captures the e¤ect of relative price,and the latter the e¤ect of purchasing power.To separate the income e¤ect from the substitution e¤ect,we can imagine that the consumers are compensated for the change in purchasing power due to a change in price so that their utility remain unchanged,and study how their demand for a good will change purely as a result of relative price changes.Hence,the Hicksian demand is also called the compensated demand,and the Marshallian Demand uncompensated demand.From the theorem,we certainly have@x i(p;I)@p i =@x h i(p;u )@p i x i(p;I)@x i(p;I)@I@x h i(p;u )i 0.The reason is because the expenditure function is a concave function of p;therefore we have@2e(p;u)@p2i =@x h i(p;u)@piThe income e¤ect can be positive or negative.Thus,(1)tells that for normal goods(@x i>0),@x i @p j <@x h i@p jmeaning that the Marshallian demand has a‡atter slope.The opposite is true for inferior goods.2.1The Slutsky Substitution Matrix1.Given a Marshallian demand function x (p;I ),let s ij@x i (p;I )@p j +x j (p;I )@x i (p;I )@I :2.The Slutsky matrix is de…ned as 266664s 11s 12::s 1n s 21:::::s n 1s nn3777753.Theorem:Suppose x (p;I )is a Marshallian demand function gener-ated by some continuous,strictly increasing utility function.Then the Slutsky matrix of x is symmetric and negative semide…nite.4.Proof:The proof is not hard,but it makes use of several results that we have just learnt.From the Slutsky equation,we haves ij =@x h i @p j:(The reason we need the utility function to be strictly increasing or at least locally non-satiated,is that without it Theorem 1.9and,hence,the Slutsky equation does not hold.)By the Sheppard’s lemma,we have for any i and j@2e @p i @p j =@x h i @p j =@x h j @p i:This implies that the Slutsky matrix is symmetric.Finally,since e is concave,its Hessian matrix:26666664@x h 1@p 1@x h 1@p 2::@x h 1@p n @x h 2@p 1:::::@x h 1@p n @x h n @p n37777775must be negative semide…nite.2.2Compensating Variation and Consumer Surplus1.When the government decides whether to cut or raise sales taxes,it has to know how will the tax a¤ect consumer welfare.In an-titrust law suit,an important factor that the court will look into is whether consumers are hurt by the monopolist.2.The expenditure function provides us a formal way to measurehow a change in price a¤ects consumer welfare.De…nition:Com-pensating variation(CV)associated with a change in the price of good i is the income compensation required to keep the utility ofa consumer constant.That is,CV e(p0i;p i;u) e(p i;p i;u):Or,alternatively,v(p0i;p i;I+CV)=v(p i;p i;I):By Shephard’s Lemma,@e@p i=x h.It follows thatCV=Z p0i p i@e@p i dp i;=Z p0i p i x h(p i;p i;u)dp i:Thus,the compensating variation is the area to the left of the Hicksian demand between p i and p0i.(Draw diagram)This is an extremely important result because it allow us to evaluate the wel-fare consequence of a policy.3.Since Hicksian demand is not directly observable,it is more con-venient to measure welfare change using Marshallian rather than Hicksian demand.The consumer surplus(at p i),CS,is the area bound by the Marshallian demand and p i.The change in CS when the price of good i changes from p i to p0i is4CS=Z p i p0i x(p i;p i;I)dp i:Recall from the Slutsky equation that@x i@p j p;I=@x h i@p j p;v(p;I) @x i@I p;I x i(p;I): 11For normal goods,x i(p0i;I)<x h i p0i;v(p i;p i;I) when p0i>p i, that is,Marshallian demand decreases more when it is a price increase.x i(p0i;I)>x h i p0i;v(p i;p i;I) when p0i<p i,that is, Marshallian demand increases more when price goes down.The converse is true for inferior goods.Hence,for normal goodsj4CS j>j CV jif price decreases,andj4CS j<j CV jwhen price increases.For normal goods,the change in consumer surplus underestimates the welfare loss due to price increase and overestimate the welfare gain due to a price decrease.4.Although,j4CS j=j CV j,in practice the di¤erence is pretty smalland is insigni…cant compared to measurement errors.12。
