Economics 352: Intermediate Microeconomics. Notes and Assignment Chapter 5: Income and Substitution Effects

Transcription

1 EC 352: ntermediate Miroeonomis, Leture 5 Eonomis 352: ntermediate Miroeonomis Notes and Assignment Chater 5: nome and Substitution Effets A Quik ntrodution To be lear about this, this hater will involve looking at rie hanges and the resonse of a utilit maimizing onsumer to these rie hanges. The resonse of a onsumer will be broken down into two arts: an inome effet and a substitution effet. Before things get unneessaril omliated, would like to la these two arts out. First, for a utilit maimizing onsumer a rie hange (a derease in the rie of good X, for eamle) atuall looks like this: A grah showing the effet of a derease in the rie of good on a onsumers utilit maimizing onsumtion deision. When the rie of X falls, the budget line rotates out and the onsumer s utilit maimizing bundle of goods hanges from oint A to oint B, taking her from utilit level U 1 u to utilit level U 2.

2 EC 352: ntermediate Miroeonomis, Leture 5 Now, this move from A to be an be thought of as ourring in two arts. Alternativel, ou might think about deomosing this move from A to B into two searate stes. The first of these two arts is the substitution effet. The substitution effet reflets the idea that when the rie of X hanges the relative ries of X and Y hange and the sloe of the budget line hanges. Put somewhat differentl, the rate at whih the onsumer trades off X for Y hanges. Eressed another wa, the sloe of the budget line is and when the rie of X,, hanges, this sloe hanges. The ke to the substitution effet is that this hange in the sloe of the budget line is made holding the level of utilit onstant. n terms of the grah, the substitution effet is shown b rotating the original budget line around the initial indifferene urve until it ahieves its new sloe: A grah showing the substitution effet assoiated with a derease in the rie of good. The substitution effet moves the onsumer from the bundle labeled A to the bundle labeled A. The utilit level remains at the original level U 1, but the hange in the relative ries of X and Y means that the ombination of X and Y hanges. So, to be lear, the hange in the bundle resulting from the substitution effet ours beause of the hange in relative ries and if, as shown here, the rie of X falls then more X will be onsumed and less Y will be onsumed. Now, having disussed the substitution effet, let us turn to the inome effet. Starting from the bundle A, the budget line shifts outward to the new budget line. This shift will be arallel and reflet the idea that when the rie of X falls, the onsumer s real inome rises. That is, she is able to afford a larger set of bundles than she ould afford reviousl. This inome effet, the arallel shift, takes the onsumer u to the new, higher utilit level:

3 EC 352: ntermediate Miroeonomis, Leture 5 A grah showing the inome effet of a derease in the rie of good on a onsumer s utilit maimizing onsumtion deision. So, the total effet of the derease in the rie of X is the move from oint A to oint B. This move an be deomosed into two arts. The move from A to A, the substitution effet, has no hange in utilit level and is onl a result of the hange in relative ries. This an be thought of as a rotation of the budget line around the original indifferene urve, U 1. The move from A to B, the inome effet, takes the onsumer to a new (higher) utilit level and is a arallel shift, reresenting the inrease in the set of affordable bundles that haens with the rie of X falls. Now, to some disussion of the hater in the tet. Demand Funtions As shown in Chater 4, it is ossible to start with a utilit funtion UU(,) and an inome onstraint + and from these alulate demand funtions that give the quantities of and what will be demanded as funtions of ries and inome: * (,, ) * (,, ) The onet of homogeneit sas that if all ries and inome are multilied b the same number, then nothing hanges. For eamle, if inome doubled and all ries doubled as well, the same quantities of both goods would still be demanded and the resulting utilit level wouldn t hange. More seifiall, we sa that demand funtions are homogeneous of degree zero in ries and inome. Tehniall, this means:

4 EC 352: ntermediate Miroeonomis, Leture 5 * (,, ) (t, t, t) t 0 (,, ) (,, ) * (,, ) (t, t, t) t 0 (,, ) (,, ) That is, saing that the demand funtions are homogeneous of degree zero means that multiling all ries and inome b t is equivalent to multiling the value of the demand funtion b t 0 1. So, in the end, nothing hanges. Changes in nome A hange in inome is reresented in an indifferene urve diagram as a arallel shift of the budget line. This is shown below for the situation where U(,), 1, 2 and inome rises from 12 to 18: A grah showing the effet of a hange in inome from 12 to 18 for the above eamle. You should onfirm that the numbers shown here are orret. When inome inreases and the budget line shifts out, onsumtion of an one good ma either inrease or derease. f onsumtion of a artiular good rises when inome rises, this good is alled a normal good. Normal goods are high qualit things that ou find ver desirable and lan to onsume more of as our inome rises. f onsumtion of a artiular good falls when inome rises, this good is alled an inferior good. nferior goods are, erhas, lower qualit things that ou eet to onsume less of as our inome rises. These might inlude low qualit food and low qualit housing.

5 EC 352: ntermediate Miroeonomis, Leture 5 t should be noted that the onets of normal and inferior goods deend on the inome level that a erson starts with. A erson with ver low inome might onsider a good normal that ou would onsider inferior. A erson with ver high inome might onsider inferior a good that ou onsider normal. n terms of diagram, this is how normal and inferior goods are reresented: Two grahs showing inome eansion aths for two normal goods and for one normal good and one inferior good. When X is normal, the quantit onsumed inreases as inome inreases. When X is inferior, the quantit onsumed falls as inome inreases. t should be noted that not all goods that a erson onsumes an be inferior. At least one good must be normal. Changes in Prie The analsis of hanges in rie resented in the book follows the disussion of inome and substitution effets shown at the beginning of these leture notes. The additional oint of interest is the deomosition of the hange in the quantit of X onsumed into substitution effets and inome effets. From the above eamle, this an be shown as:

6 EC 352: ntermediate Miroeonomis, Leture 5 Two grahs showing the substitution effet of a derease in the rie of and the inome effet of a derease in the rie of. n this ase, both the substitution and the inome effets inrease the quantit of X onsumed. However, if X were an inferior good then the inome effet would be negative. That is, the inome effet would slightl redue the quantit of X onsumed: Two grahs showing the substitution and inome effets assoiated with a derease in the rie of if is an inferior good. The net effet of the derease in the rie of X would still be an inrease in the quantit of X onsumed, but this net inrease will not be as large as if X had been a normal good.

7 EC 352: ntermediate Miroeonomis, Leture 5 n the diagram shown here, the substitution effet is larger than the inome effet, so the quantit of X onsumed rises when the rie of X falls. Put a slightl different wa, if the substitution effet is larger than the inome effet (if the substitution effet dominates the inome effet) then the net result of a derease in the rie of X will be an inrease in the quantit of X onsumed, even if the inome effet redues the quantit of X onsumed. There is a bizarre, but theoretiall ossible ase where the inome effet outweighs the substitution effet. This is alled a Giffen good, whih the tetbook desribes under the heading Giffen s arado. f the inome effet is negative and outweighs or dominates the substitution effet, then it ould be ossible that a derease in the rie of X will lead to less, rather than more, X being onsumed. The standard eamle of a Giffen good is otatoes in nineteenth entur reland. Potatoes were a stale of the eole s diet and when their rie rose eole beame muh oorer in a real sense. These eole substituted awa from other, normal goods, and bought more of the relativel inferior otatoes, with the net effet that onsumtion of otatoes rose even as their rie rose. ndividual Demand Curves Demand urves, the relationshi between the rie of a good and the quantit demanded holding other ries and inome onstant, an be derived from indifferene urve diagrams. Consider how the quantit of X that a erson demands hanges as the rie of Y and inome remain onstant and the rie of X falls from to to :

8 EC 352: ntermediate Miroeonomis, Leture 5 A grah of indifferene urves and budget lines showing the quantities of good demanded for three different ries of. This grah hels to establish the relationshi between indifferene urve analsis and demand urves The resulting demand urve will involve the ries (, and ) and quantities ( 1, 2 and 3 ) from the indifferene urve diagram: A demand urve grah relating ries and quantities demanded to the revious indifferene urve grah.

9 EC 352: ntermediate Miroeonomis, Leture 5 For a math eamle, imagine that a erson has the Cobb-Douglas utilit funtion U t is ossible to show, and ou should be able to do this, that the demand funtions for and are: 0.4 * and 0.6 *. f inome,, is $240, ou an diagram the demand urve for : A grah showing the demand urve for good based on the utilit funtion U and inome of $240. As the rie of X hanges, the quantit of X demanded hanges aording to the demand urve. The demand urve for X doesn t shift when the rie of X hanges. So, a hange in the rie of a good will move a onsumer from one oint on the demand urve for that good to another oint on the same demand urve. This is referred to as a hange in the quantit demanded. That is, the demand urve itself doesn t hange, onl the quantit seleted on that demand urve hanges.

10 EC 352: ntermediate Miroeonomis, Leture 5 Shifts in the Demand Curve or Changes in Demand (The re the same thing!) A hange in demand for a good ours when something other than the rie of that good hanges. For eamle, an inrease in inome will inrease demand for a normal good. This means that the demand urve will shift out (that is u or to the right) and the quantit demanded at an rie will inrease. An inrease in inome will derease the demand for an inferior good. This means that the demand urve will shift in (that is down or to the left) and the quantit demanded at an rie will derease. A variet of other things might shift the demand urve or hange demand. Colder weather will inrease the demand for sweaters and mittens, rain will inrease demand for umbrellas and salt snaks will inrease demand for soft drinks and beer. Comensated Demand Curves The ordinar demand urves disussed above (also known as Marshallian demand urves) are onstruted holding inome onstant and allowing the rie of the good to hange. There is an alternative aroah to demand urves. The alternative aroah onstruts demand urves holding utilit onstant to reate omensated demand urves (also known as Hiksian demand urves). Ordinar or Marshallian inome held onstant * (,, ) Comensated or Hiksian utilit held onstant * (,, U) The omensated demand urve an be thought of as reresenting a ure substitution effet with no inome effet. That is, instead of rotating about a oint on an ais, the budget line related to a omensated demand urve rolls along an indifferene urve:

11 EC 352: ntermediate Miroeonomis, Leture 5 Two grahs of indifferene urves and budget lines showing the underling assumtions of Marshallian and Hiksian demand urves. The Hiksian demand urve is alled the omensated demand urve beause the onsumer is omensated for the rie hange. That is, when the rie hanges the reeive omensation that allows them to remain on their original indifferene urve. f the rie of the good rises this omensation is ositive. Of ourse, if the rie falls this omensation is negative. The effet of this omensation is to redue the size of hanges in the quantit of the good that is onsumed. f the rie rises, the quantit demanded doesn t fall quite so muh beause the onsumer reeives etra mone to send. When the rie falls the onsumer doesn t inrease her onsumtion quite so muh beause some mone is taken awa. The result is that the omensated demand urve is steeer than the ordinar demand urve, refleting the idea that the hanges in the quantit demanded resulting from a rie hange are smaller. n a iture, this looks like:

12 EC 352: ntermediate Miroeonomis, Leture 5 A grah showing the relationshi between Marshallian and Hiksian demand urves. To state this slightl differentl, the Marshallian demand urve is more elasti and the Hiksian demand urve is less elasti. Eamle 5.3 This eamle revolves around deriving unomensated and omensated demand funtions and omaring them. t starts with the utilit funtion: U(,) The budget onstraint is: + Solving the utilit maimization roblem gives: L L λ λ and taking the ratio of these gives: or

13 EC 352: ntermediate Miroeonomis, Leture 5 and substituting this into the budget onstraint gives the demand funtions: + or * and 2 *. 2 Now, if these demand funtions are ut into the original utilit funtion, we get the indiret utilit funtion that eresses utilit as a funtion of the ries and inome: U (, ) V, (, ) Now, to get the omensated demand funtions, we will rewrite the indiret utilit funtion as: V rewritten as.5 2V 0 Plugging this into the demand funtions from above gives: * 2 * 2 2V 2 2V 2 V 0. 5 V These are the omensated demand funtions. As disussed in the tet, doesn t enter into the unomensated demand funtion for, but it does enter into the omensated demand funtion for. The question is asked in the eamle, Are the omensated demand funtions homogeneous of degree zero in and if utilit is held onstant? The answer for the funtions given here is es. That is, holding utilit (V) onstant, if both ries double, then the otimal quantities of and do not hange. n addition, this is true for all omensated demand funtions. To see this, imagine the utilit funtion: U(,) Following the stes above we get:

14 EC 352: ntermediate Miroeonomis, Leture 5 L L λ λ and the ratio of these is: or 3 3 and with the budget onstraint this is: * and 4 3 *. 4 This gives us: U (, ) V, (, ) V rewritten as V Plugging this into the demand funtions from above gives: * 4 3 * V V V V Both of these demand funtions are homogeneous of degree zero in the ries, holding utilit onstant. That is, if both ries double (for eamle) and utilit is held onstant, the quantities of and that are demanded will not hange.

15 EC 352: ntermediate Miroeonomis, Leture 5 A Mathematial Develoment of the Resonse to Prie Changes and the Slutsk Equation The resonse to a hange in the rie of will be a hange in the quantit of demanded. This movement along the demand urve (and not a shift in the demand urve) is reresented b the artial derivative:. The diret aroah to alulating this in the most general situation is to set u the Lagrangian: ( 1 2 n ) ( n n L U,,... + λ... ) and take the n+1 artial derivatives (the artial derivative with reset to eah good, i lus the derivative with reset to λ) to get n+1 equations, whih an then be solved simultaneousl. However, this is a lot of work and, in general, these n+1 equations ma not be solvable. 1 Let us tr an indiret aroah and see what we get. We ll start with two goods, leverl names and, and the omensated demand funtion: (,, U) whih eresses the quantit of that will be demanded for given ries of and and a given utilit level. The relationshi between the ordinar demand funtion, (,, ) and the omensated demand funtion ma be derived b thinking of a minimum eenditure funtion that desribed the minimum eenditure neessar to ahieve a given utilit level: E(,, U) then we an define the omensated demand funtion to be: (,, U) (,, E(,, U ) 1 A nast fat that the don t usuall tell ou in math lasses is that most mathematial roblems don t atuall have an analtial solution.

16 EC 352: ntermediate Miroeonomis, Leture 5 where the inome level is relaed b the minimum neessar eenditure in the regular, unomensated, demand funtion. Now, to ome u for air for just a minute, these are all different was of eressing the basi idea of the standard utilit maimization grah: A grah showing an otimal onsumtion bundle from a onsumer s utilit maimization roblem. The otimal bundle, (*,*) an be thought of as maimizing utilit given the budget onstraint, ielding (,, ) and the indiret utilit funtion V(,, ). t ma also be thought of as minimizing the eenditure needed to ahieve the given utilit level, ielding (,, U) and the eenditure funtion E(,, U). To jum ahead a bit, these an be set u as two was of looking at the same roblem, either utilit maimization given inome or as eenditure minimization given utilit. Primal maimize U(,) subjet to + ndiret utilit funtion V(,, ) Marshallian demand funtions (,, ) Dual minimize E(,) subjet to UU(,) Eenditure funtion E(,, U) Comensated demand funtions (,, U) t should also be noted that the indiret utilit funtion and eenditure funtion are inverses of eah other. That is, if ou solve the indiret utilit funtion for ou get the

17 EC 352: ntermediate Miroeonomis, Leture 5 eenditure funtion. f ou solve the eenditure funtion for U, ou get the indiret utilit funtion. OK, bak to the show. We had the following relationshi between the omensated and unomensated demand funtions: (,, U) (,, E(,, U) Now, if we take the artial derivative with reset to on eah side of the equation, we get: E + E whih an be rewritten as: E E This reresents the hange in ordinar, unomensated or Marshallian quantit demanded as the ombination of two effets, a substitution effet E inome effet. E and an The substitution effet is the hange in quantit demanded of along the original indifferene urve. This is eatl. Let s look at the inome effet a bit more. First, beause eenditures (E) are equivalent to inome () at an otimal solution (and the re two sides of the same thing) we an relae with. E E Seond, the urious term is atuall ver familiar. This is the artial derivative of eenditures (holding utilit onstant) with reset to the rie of. Now, imagine that ou are onsuming five units of and the rie of rises b $1. B how muh will eenditures on rise? At the margin, the answer is $5. This artial derivative is

18 EC 352: ntermediate Miroeonomis, Leture 5 atuall the quantit of that ou are onsuming, known as. So, we an relae with to get the inome effet: E Now, the entire funtion from above an be rewritten as the Slutsk equation 2, whih relates the hange in the Marshallian quantit demanded as a ombination of the substitution and inome effets: or U U Now, to onsider whether eah effet is ositive or negative: The substitution effet is alwas negative. The sign of the inome effet deends on the sign of, or whether the good is a normal good > 0 or is an inferior good < 0. Usuall, the net effet will be that < 0. However, if the good is an inferior good and if the magnitude of the inome effet is greater than the magnitude of the substitution effet, we get the odd result that quantit demanded rises as the rie rises, the ase of the so-alled Giffen good. So, to ut this together a bit: Te of good Normal good Slutsk nterretation < 0 > 0 2 am morall omelled to oint out that Eugenio Slutsk was a great statistiian as well as a great eonomist. He also has a ver imortant statistis relationshi with his name attahed to it. M friends and like to retend that he was also a multi-sort athlete in ollege, but ursued aademia rather than rofessional sorts due to an injur near the end of his senior ear.

19 EC 352: ntermediate Miroeonomis, Leture 5 nferior good Giffen good < 0 < 0 < 0, but smaller than < 0 < 0 < 0 and larger than > 0 Eamle 5.4 f we start with the utilit funtion: U(,) we get the Marshallian (unomensated) and Hiksian (omensated) demand funtions: (,, ) and (,, V) V The artial derivative of the Marshallian demand funtion with reset to is: 2 2 Now, we an think about the substitution effet and the inome effet. The substitution effet is the artial derivative of the omensated demand funtion with reset to the rie of : V 1.5 2

20 EC 352: ntermediate Miroeonomis, Leture 5 but V.5 2 0, giving us: (substitution effet) The inome effet is equal to the rodut of and, or: But is given b the demand funtion 2, so we have: And the entire Slutsk equation an be written as: As ointed out in the book, for the Cobb-Douglas utilit funtion (a utilit funtion of the form U a 1-a ) the substitution and inome effets are of the same magnitude. This is not true for all utilit funtions. Demand Elastiities Elastiities are oular in eonomis. We ll talk about three different elastiities here. Note that as the equations are written here the are based on Marshallian (normal or unomensated) demand urves.

21 EC 352: ntermediate Miroeonomis, Leture 5 1. Prie elastiit of demand desribes the relationshi between the erentage hange in the rie of a good and the resulting erentage hange in the quantit demanded. This an be eressed in a variet of was: e, % % t is also equal to the inverse of the sloe of the demand urve multilied b. 2. nome elastiit of demand eresses the relationshi between erentage hange in inome and the erentage hange in the demand for the good. This an also be eressed in several was: e, % % f this is ositive, the good is a normal good. f this is negative, the good is an inferior good. 3. Cross rie of elastiit eresses the relationshi between the erentage hange in the rie of one good and the erentage hange in demand for another good. This an also be eressed in several was: e, % % Prie elastiit of demand Prie elastiit of demand (sometimes alled own rie elastiit of demand to distinguish it from ross rie elastiit) is erent hange in quantit demand divided b erent hange in rie. This is usuall a negative number. f the elastiit is less than 1 (-1.5 or 2.0, for eamle) then demand for the good is said to be elasti. Put somewhat siml, this is onsistent with onsumers being ver fleible or elasti in their demand for the good, so that if the rie rises even slightl,

22 EC 352: ntermediate Miroeonomis, Leture 5 the an easil substitute awa from it. f elastiit is 4, for eamle, when the rie rises 5% the quantit demanded will fall b 20%. f the elastiit is greater than 1 (-0.8 or, for eamle) then demand for the good is said to be inelasti. This is onsistent with onsumers being ver infleible or inelasti in their demand for the good, so that even if the rie rises a lot the annot easil redue their onsumtion. f elastiit is 0.1, for eamle, when the rie rises b 20% the quantit demand will fall b onl 2%. f demand is elasti, when the rie rises total eenditures on the good will fall and when rie falls total eenditures on the good will rise. f demand is inelasti, when the rie rises total eenditures will rise and when rie falls total eenditures will fall. This an be established using alulus. Total eenditures are. Taking the derivative of this with reset to gives us: ( ) + 1, ( e + 1) So the hange in eenditures resulting from a hange in rie deends on the quantit onsumed (), and the sum of the elastiit and one. f demand for the good is elasti, e, when rie rises eenditures fall. f demand for the good is elasti, e, when rie rises eenditures rise. < 1 and the term in arentheses is negative, so > 1 and the term in arentheses is ositive, so f demand is unit elasti, then regardless of the rie. e, 1 and total eenditures remain the same

23 EC 352: ntermediate Miroeonomis, Leture 5 Comensated Demand Elastiities Whereas before we were talking about Marshallian (unomensated or ordinar) demand elastiities, we will now onsider Hiksian (omensated) demand elastiities. To some etent the differene is onl notational, as the ordinar demand funtions are relaed b the omensated demand funtions:, % % e, % % e How different these are from the unomensated elastiities deends on how big the inome effet is. The unomensated elastiities inlude inome effets while the omensated elastiities are based onl on substitution effets. The differene an be seen b aling the Slutsk equation to these elastiities. e, So, the differene between the omensated and unomensated rie elastiit of demand is. Put somewhat differentl, Unomensated Elastiit Comensated Elastiit - UE CE - UE CE - UE CE -, e

24 EC 352: ntermediate Miroeonomis, Leture 5 So, the unomensated elastiit is equal to the omensated elastiit minus the rodut of and e,. e, is the inome elastiit of demand. is the total eenditure on divided b inome. This is also known as the share of inome sent on good, whih goes b the name s. Thus we get: UE CE - s e, So the differene between the unomensated and omensated elastiities for a good is the rodut of the share of inome sent on the good and the inome elastiit of demand for the good. The differene between the two will be small if a small share of inome is sent on the good (like with salt or hewing gum) or if the inome elastiit of demand is small. f either or both of these are true, then the differene between omensated and unomensated elastiities will be negligible. Relationshis Between Goods and Elastiities will resent these without roof or a lot of disussion. Just sort of hold on to them for now and don t lose an slee over them. Homogeneit of elastiities for a good The sum of the own rie elastiit, ross rie elastiities and inome elastiit of demand for a good is zero: e, + e, + e, 0 Engel aggregation The sum of the roduts of inome shares of eah good and the inome elastiit of eah good is one: s e, + s e, 1

25 EC 352: ntermediate Miroeonomis, Leture 5 Cournot aggregation Cournot aggregation is a statement about how the own rie elastiit and the ross rie elastiit are onstrained b the budget onstraint: s e, + s e, s An Endorsement for Eamle 5.5 You should go through eamle 5.5 in the tet and be sure that ou an work from the original utilit funtions to get the demand funtions and elastiities disussed in this eamle. To be lear about this, ou should be able to get demand funtions for and from the following: U(,) α 1-α U(,) + U(,) The first is a Cobb-Douglas utilit funtion and the remaining two are onstant elastiit of substitution (CES) utilit funtions. As a safet ti, it might not be ossible for the third utilit funtion. Consumer Surlus Consumer surlus is the gain from trade aruing to the onsumer or onsumers in a market. n a riniles of miroeonomis sense, this is equal to the differene between a onsumer s total willingness to a for a good and their total eenditure on that good. Put somewhat differentl, the onsumer surlus from a good at a rie is the area under the Marshallian demand urve above the rie. t is the maimum amount that an individual onsumer would a for the right to make voluntar urhases at that rie. From the oint of view of indifferene urves and budget lines, there are a oule of different aroahes to onsumer surlus. n terms of eenditures, we an think about the ost to a onsumer of an inrease in the rie of one good, holding the rie of the other good and utilit level onstant. That is, when the rie of one good rises, how muh more do ou have to send to maintain our initial level of utilit? This measure of the hange in onsumer welfare or the hange in onsumer surlus is alled the omensating variation (CV). Mathematiall, this an be eressed in terms of the eenditure funtion as:

26 EC 352: ntermediate Miroeonomis, Leture 5 ( 1 ) ( ),, U E,, U CV E f the rie of good rises from to, then eenditures will rise. This inrease in eenditures is the omensating variation. a measure of the loss in onsumer welfare aused b the rie inrease. Put slightl differentl, this is the additional inome that the onsumer would need in order to reserve her initial level of utilit, U 0. n terms of a grah, this looks like the following. First, we start out with inome and ries of 0 and. Then the rie of good rises to 1, reresented b the rotation of the budget line as shown below. 0 A grah showing the imat on the budget line of an inrease in the rie of good. The roblem is that the inrease in the rie of uts the onsumer on a lower indifferene urve. The omensating variation (CV) is the additional inome that would be needed to bring the onsumer bak u to the original utilit level (U 0 ), given the new, higher rie of. This is shown below:

27 EC 352: ntermediate Miroeonomis, Leture 5 A grah showing the omensating variation assoiated with an inrease in the rie of good. This is the additional inome that would be neessar to make a erson as well off after the rie inrease as she was before the rie inrease. Now, it might hel to have this tied bak to the usual iture of onsumer surlus from the demand urve, whih sort of looks like this: A grah showing onsumer surlus on a demand grah. Now we an do better. Consumer surlus is most reisel defined using the omensated demand urve, rather than the Marshallian demand urve. We an use the

28 EC 352: ntermediate Miroeonomis, Leture 5 omensating variation from above as a measure of the hange in onsumer surlus when the rie of the good rises from 0 1 to, holding utilit onstant. This is: A grah showing omensating variation assoiated with a rie hange and a Hiksian (omensated) demand urve. Marshallian versus Comensated Consumer Surlus There s a nast mess of ideas resented in the book in Figures 5.8 and 5.9. The relate to two different measures of onsumer surlus. These are the omensating variation, whih is disussed above, and the equivalent variation, whih will attemt to elain here. Looking at Figure 5.8(a), there are two utilit levels, U 0 and U 1. U 0 reresents the higher utilit level. These an be related to two omensated demand urves for good :

29 EC 352: ntermediate Miroeonomis, Leture 5 A grah showing that there are two different omensated demand urves when the rie of a good hanges, the omensated demand urve assoiated with the initial rie of the good and the omensated demand urve assoiated with the new rie of the good. The question is, whih of the two omensated demand urves should be used in alulating the hange in onsumer surlus resulting from the rie inrease from 1? The answer reall deends on what ou believe the onsumer has a right to. f ou believe that the onsumer has a right to the initial, lower rie of, then the loss of onsumer surlus is the larger area and uses the U 0 demand urve to alulate lost onsumer surlus. This is the amount of omensation, the omensating variation, that the onsumer would need to bring her bak u to her original utilit level. f ou believe that the onsumer doesn t have a right to the initial rie, then the loss of onsumer surlus is the smaller area and uses the U 1 demand urve to alulate lost onsumer surlus. This is the amount that she would be willing to a, the equivalent variation, to fae the lower rie for instead of the higher rie, given that she starts out faing the higher rie. So, the differene between omensating variation (CV) and equivalent variation (EV) is that CV reresents a onsumers willingness to aet, or the omensation the would require to be omensated for a rie inrease, whereas the EV reresents willingness to a to revent a rie inrease. The differene between the two reall deends on the inome effet of the rie hange. Now, the Marshallian or unomensated demand urve is the quantit atuall demanded at eah of the two ries and is sort of a omromise between the two omensated 0 to

30 EC 352: ntermediate Miroeonomis, Leture 5 demand urves. The onsumer surlus alulated on the Marshallian demand urve will be in between the alulations from the two omensated demand urves. A grah showing the relationshi between the Marshallian and Hiksian (omensated) demand urves. An Eamle To see the differene between CV and EV, imagine a utilit funtion U(,). To alulate the CV, imagine that we start with 1 and 1 and 100. At a utilit maimizing bundle, we have 50, 50 and U2500. f the rie of rises so that 2, ahieving that same level of utilit, U2500, will now require a higher level of inome. This an be alulated from the eenditure funtion E(,, U), whih turns out to be given b: ( ) 5 0. E,, U 2 U So, in the original ase with 1 and 1 and U2500, we had 100. When the rie of rises to 2, to maintain the original utilit level we have 2 and 1 and U2500 and: E ( 2,1, 2500) So, the omensating variation is the etra inome needed to ahieve the original utilit level at the new ries, or $

31 EC 352: ntermediate Miroeonomis, Leture 5 To alulate the EV, we will onsider what this erson s utilit level would be if the rie of were to rise to 2, assuming their inome staed at 100. This is based on the indiret utilit funtion V(,, ), whih turns out to be given b: V (,, ) 2 4 When the rie of rises to 2 and inome is at 100 we have: V(2,1,100) At the original ries, this utilit level would be ahieve with an inome of: ( ) E 1,1,1250 The differene between this and the original inome level, So, the equivalent variation is inome that this erson would be willing to give u to avoid the rie inrease, or $ Some Eerises For a standard diagram with indifferene urves and budget lines, show the following. 1. An inrease in the rie of. 2. An inrease in the rie of. 3. An inome inrease with both goods normal. 4. An inome inrease with inferior. 5. An inome inrease with inferior. For the following rie hanges, show the substitution and inome effets. 6. nrease in the rie of with both goods normal. 7. nrease in the rie of with inferior. 8. nrease in the rie of with inferior. 9. Derease in the rie of with both goods normal. 10. Derease in the rie of with inferior. 11. Derease in the rie of with inferior. 12. nrease in the rie of with both goods normal. 13. nrease in the rie of with inferior. 14. nrease in the rie of with inferior.

32 EC 352: ntermediate Miroeonomis, Leture Derease in the rie of with both goods normal. 16. Derease in the rie of with inferior. 17. Derease in the rie of with inferior. 18. What does it mean to sa that Marshallian demand funtions are homogeneous of degree zero in ries and inome? For eah of the following, indiate whether the good desribed is a normal good, an inferior good or a Giffen good. 19. Substitution effet dominates the inome effet. 20. Substitution effet dominates the inome effet, but the inome effet of a rie inrease is ositive. 21. nome effet dominates the substitution effet, but the inome effet of a rie inrease is ositive. Calulate the unomensated (Marshallian) and omensated (Hiksian) demand funtions for and for the following Cobb-Douglas utilit funtions. 22. U(,) 23. U(,) U(,) U(,) U(,) U(,) U(,) U(,) U(,) For eah of the following utilit funtions given in questions 22-30, write out the entire Slutsk equation.