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by: Daron Kemmer IV


Daron Kemmer IV
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This 11 page Class Notes was uploaded by Daron Kemmer IV on Thursday October 22, 2015. The Class Notes belongs to SPAN 210A at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 52 views. For similar materials see /class/226852/span-210a-university-of-california-santa-barbara in Spanish at University of California Santa Barbara.




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Date Created: 10/22/15
Lecture Notes on Separable Preferences Ted Bergstrom UCSB Econ 210A When applied economists want to focus attention on a single commodity or on one commodity group they often nd it convenient to work with a two commodity model where the two commodities are the one that they plan to study and a composite commodity called other goods For example someone interested in the economics of nutrition may wish to work with a model where one commodity is an aggregate commodity food and the other is money left over for other goods To do so they need to determine which goods are foods and which are not and then de ne the quantity of the aggregate commodity food as some function of the quantities of each of the food goods In the standard economic model of intertemporal choice model of intertem poral choice commodities are distinguished not only by their physical attributes but also by the date at which they are consumed In this model if there are T time periods and n undated commodities then the total number of dated commodities is nTi Macroeconomic studies that focus on savings and invest ment decisions often assume that there is just one aggregate good consumed in each time period and that the only nontrivial consumer decisions concern the time path of consumption of this single good In these examples and in many other applications of economics the tactic of reducing the number of commodities by aggregation can make dif cult problems much more manageable In general such simpli cations can only be purchased at the cost of realismi Here we examine the separability conditions that must hold if this aggregation is legitimate To pursue the food example suppose that the set of n commodities can be partitioned into two groups foods and nonfoods Let there be m food commoditiesi We will write commodity vectors in the form I IFINF where IF is a vector listing quantities of each of the food goods and INF is a vector quantities of each of the nonfood goods If we are going to be able to aggregate it must be that consumers have preferences representable by utility functions u of the form u IFJNF U fIF7 INF f is a realvalued function of m variables and where U is a strictly increasing function of its rst argument The function f is then a measure of the amount of the aggregate commodity foodi Notice that the function u is a function of n variables while U is an function of only n 7 m 1 variables which denote quantities of each of the nonfood variables and a quantity of the food aggregate Economists7 standard general model of intertemporal choice is based on the use of dated commodities Thus if there are n undated commodities and T time periods we de ne In to be the quantity of commodity i consumed in period t We de ne I to be the nvector of commodities consumed in period t and we de ne z 11 i i i IT to be the nTvector listing consumption of each good in each period This is sometimes called a time pro le of consumption We assume that individuals have preferences over time pro les of consumption that are representable by a utility function uzli i i IT where each I is an n vectori Suppose that u takes the special form um A A A 71T U 1311 A A A 7fTIT where each ft is a realvalued function of n variables Then U is a function of T variables each of which is an aggregate of consumption in a single period Preference relations and separability We can express these ideas a little more formally in terms of preferences and consumption setsi Let M be a subset of the commodity set 1 i i i n with m lt n members and let N M be the set of commodities not in Mi Let the consumption set S be the Cartesian product SM gtlt SNM of possible consumption bundles of the goods in M and of goods in N Mi1 Where I E S we write I IMINM where MM is an m vector listing a quantity of each good in M and INM lists a quantity of each good in N Mi De nition 1 Preferences R are separable on M if whenever it is true that IMawaRIllerM for some INM it must also be that IvaNMRIllvxNM for all zNM E SNM ln words the separability condition says that if you like Mbundle 1M better than the Mbundle 15 when each Mbundle is accompanied by the nonM bundle INM then you will also like Mbundle 1M better than 1M if each Mbundle is accompanied by any other bundle from the nonM groupi Be sure to notice that in each of these comparisons with differing Mbundles the nonM bundle is held constant Let consider an example in which preferences are not separable There are 3 commodities cars bicycles and gasoline Let 11 be the number of cars that a consumer consumes in a week 12 the number of bicycles and 13 the number of gallons of gasoline Suppose that this consumer prefers to drive to work rather than ride her bicycle and that 10 gallons of gasoline will suf ce to get her car to work every day of the week Thus she prefers 1010 to 0110 If her preferences are separable between the commodity group 1 2 and commodity 3 then she must also prefer l 0 0 to 010 But unless she prefers sitting in a stopped car to commuting bicycle this latter preference does not seem likely In this example the third good gasoline is a complement for good 1 1The assumption that S is the Cartesian product of SM and SNM rules out the possibility that the possible consumptions in group M depend on what is consumed in N M or vice versa cars but not for good 2 bicyclesi As a result the commodity group 12 is not separable from the rest of the commodity bundle The main theorem that relates separable preferences to the structure of utility functions is Theorem 1 If there are n Jquot and r I are p it 7 by a utility function with M C 1 i i i n then preferences are separable on M if and only there exists a real valued aggregator function f SM A R such that uzMzNM U fzMzNM where U Rn m1 A R is afunction of n 7 m 1 variables that is increasing in its rst argument Proof of Theorem 1 First show that if utility has this form then preferences are separable on Mr Note that if 1M INM R 1M INM then U fzM INM 2 U INM Since U is an increasing function of its first argument and since the same INM appears on both sides of the inequality it must be that 2 But if 2 then it must be that U fzMzNM 2 U for all zNM E SNMi Therefore zMz NMRz Mz NM for all zNM E SNM which means that R is separable on Mr Conversely suppose that preferences are separable on Mr Select any ENM E SNMi lt doesnlt matter which one just pick one De ne u 1M ENM For all 1M INM define U fzM INM u 1M INM Now this definition is legitimate if and only if for any two vectors 1M and 1M of Mcommodities such that and for any INM E SNM it must be that uzMzNM u zMi INM This definition would be illegitimate if there were some 1M and 15 such that y but uzMzNM uz wzNM because then we would have two conflicting definitions for U y INMl It is immediate from the definition of separability and the definition of the function f that if fzM then uzMzNM uz lzNM for all INM E SNMi All that remains to be checked is that U must be an increasing function of This should be easy for the reader to verify Exercise 1 Suppose that there are four commodities and that utility can be written in the form Uzl 12 13 I4 U fzl 12 13 14 where the functions U and f are di erentiable a Show that the marginal rate of substitution between goods 1 and 2 is independent of the amount of good 5 b Construct an example of a utility function of this form such that the marginal rate of substitution between goods 5 and 4 depends on the amounts of goods 1 and 2 The notion of separability extends in the obVious way to allow more than one aggregate commodityi Suppose for example that there are six commodities and that that utility can be represented in the form U117 716 UT f1117127f213714715715i In this case we say that preferences are separable on the commodity groups 12 and 345 We could also say that preferences are separable on the singleton commodity group Exercise 2 Where utility functions are of the functional form U117 7 16 Ulltf1lt 17 12 f2ltI37 I47 15 16 a show that the marginal rate of substitution between goods 5 and 4 is in dependent of the quantities of goods 1 and 2 b Construct an example of a utility of this form where the marginal rate of substitution between goods 5 and 6 depends on the quantity of good 1 For the next two exercises suppose that there are 3 time periods and two ordinary commodities apples and bananas and consumption of apples and bananas in period t are denoted by xat and xbt Exercise 3 Where utilities of time pro les are of the functional form U f11a17 1117 f2lt1a27 1122 f3lt1a37 173 a show that a consumer s marginal rate of substitution between apples in pe riod 2 and bananas in period 2 is independent of the amount of apples consumed in period 1 b construct an example of a utility function of this type where the consumer s marginal rate of substitution between apples in period 2 and apples in period 5 depends on consumption of apples in period 1 One can also produce interesting nested structures of separable preferences Exercise 4 Where utility of time pro les of apple and banana consumption can be represented in the form U f1a17 111 v f21a27 112 f3lt1a37 113 a show that the marginal rate of substitution between apples in period 2 and apples in period 5 does not depend on the quantity of apples in period 1 b construct an example of a utility function of this form where the marginal rate of substitution goods apples in period 1 and and apples in period 2 depends on the quantity of apples in period 5 Exercise 5 Consider teams of males and females who play mixed doubles tennis matches Suppose that it is possible to assign numbers xi to each female i and yi to each male i in such a way that there is a realvalued function f of four variables such that a team consisting of male i and female i will defeat a team consisting of malej and female j and only fxiyixjyj gt 0 Explain why it might not be reasonable to assume that there exist realvalued functions F and g of two variables such that the team ofi s would beat the team ofj s and only Fgxiyigxjyj gt 09 If such functions exist how would you interpret them Exercise 6 A consumer has utility function Uzlzg 13 2112 Zr Q 13 Solve for this consumer s demand for each of the three goods as a function of prices 171102103 and income m Where there is separability consumer choice problems can be simpli ed by recasting them as twostage problems For example suppose that Sarahls util ity function can be written as Uzlulzn F fc zlmzkzk1luzn where the rst h commodities are items of clothing Suppose that Sarah faces a budget constraint ELIEzl nm We can decompose the problem into two choices For any given clothes budget how should she spend it Knowing how she would spend her clothes budget how much money should she allo cate to clothes and how much to each of the other goods Given the price vector for clothing p0 p1lupk and clothes budget y she will choose a clothing bundle zc zlluzk that maximizes fczlulzk subject to Eilpizi yl Her indirect utility for clothes vcpcy is then de ned as the maximum value of f0 that she can achieve with this budget Now Sarahls choice of how much to spend on clothing can be expressed as choos ing y to maximize F vcpcy zk1lu zn subject to the budget constraint 9 Elk PiIi m Exercise 7 A consumer has utility function Uzl12zgz4 2112 Zr Q z inQl a If the prices of goods 5 and 4 are p3 and p4 and a consumer spends a total ofy on good 5 and 4 how much good 5 and how much good 4 should he 9 uy b If the consumer spends y on goods 5 and 4 and the prices of good 1 and good 2 are p1 and p2 how much good 1 should he buy how much good 29 c Write an expression for this consumer s utility she spends her budget to maximize her utility given that she spends total of y on goods 5 and 4 d Now nd the choice ofy that leads to the highest utility and the resulting quantities of goods 12 5 and 4 Additively separable preferences Suppose that there are n quot39 and that A f can be A in the additive form uzl l l l zn where the functions vi are real valued functions of a single real variable If this function is differentiable then the marginal rate of substitution between any two commodities j and h is independent of the quantities of any other goods since it is equal to the ratio of derivatives 1 More generally notice that if preferences are representable in this additive form then for any subset M of the set of commodities preferences must be separable on the set M of commodities and on its complement N Mi lf preferences can be represented by a utility function of this additive form we say that preferences are additively separable on all commodities77i When are preferences additively separable The most useful necessary and sufficient condition for preferences to be addi tively separable is that every subset of the set of all commodities is separablei The proofs that I know of for this proposition are a bit more elaborate than seems appropriate here A somewhat more general version of this theorem can be found in a paper by Gerard Debreu Debreu7s paper seems to be the rst satisfactorily general solution to this problemi Other proofs can be found in 5 and 2 Theorem 2 Assume that preferences are representable by a utility function and that there are at least three preferencerelevant commodities where com modity i is said to be preferencerelevant there exist at least two commodity bundles z and y that di er only in the amount of commodity i and such that z is preferred to Then preferences are representable by an additively separable utility function and only every nonempty subset M of the set of commodities is separable Additive separability with two goods You might wonder whether it is always possible to write an additively separable utility function in case there are only two goods The answer is no and will show you a counterexample in a minute If there are two goods and if preferences can be represented by a utility function of the form Uzlzg v1zl 11212 then the following double cancellation condition77 must hold lf 1112Ry1y2 and y122R21zg then 1122R21y2i To see that the double cancellation property is a nec essary condition for additive separability note that if 1112Ry1y2 and y1 22R21 12 then v111 WW2 2 v1 91 v2y2 and v1y1 11222 2 v1y1 02 Adding these two inequalities and cancelling the terms that appear on both sides of the inequality we have 11111 v222 Z v121 U2y27 which means that 11 22R21y2i Debreu 1 showed that the double cancellation condition is both necessary and suf cient for preferences to be additively separable when there are only two goods Theorem 3 Assume that preferences are representable by a utility function If there are two 739quot en 1 J t p it 39l by an additively separable utility function if and only if the double cancellation condition holds It is not always easy to discern at first glance whether a given utility func tion can be converted by a monotonic transformation into additively separable formi Consider the utility function Uzl 12 11 12 1112 In this form it is not additively separable But note that 11 12 1112 l 11 l 12 The function Vzl 12 anzlzg lnl 11 lnl 12 is a monotonic transformation of U and is additively separable Therefore the preferences rep resented by U are additively separable On the other hand consider preferences represented by the utility function Uzlzg 11 12 1113 These preferences can not be represented by an additively separable utility function How do we know this We can show that these preferences violate the double cancellation condition To show this note that Ul l U30 3 and that U32 U8l 17 The double cancellation condition requires that Ul2 U80i But Ul2 7 and U80 8 and 8 f 7 So these preferences violate the double cancellation condition and hence cannot be additively separable It is sometimes not easy to see whether a given utility function can be mono tonically transformed to additively separable formi Checking that the double cancellation conditions are satisfied everywhere may take forever Fortunately there is a fairly easy calculus test We leave this as an exercise Exercise 8 Show that Uzl 12 F 11111 v2zg for some monotoni cally increasing function F then it must be that the log of the ratio of the two partial derivatives of U is the sum of two functions one of which depends only on 11 and one of which depends only on 12 Cardinality and af ne transformations There is an old debate in economics about whether utility functions are car dinal77 or ordinal The ordinalist position is that utility functions are oper ationally meaningful only up to arbitrary monotonic transformations If all that economists are able to determine are ordinal preferences then any two util ity functions that represent the same ordinal preferences are equally suitable for describing preferences and a unit of utility has no operational meaning or many applications it would be helpful to have a more complete mea surement Suppose for example that you wanted to compare the happiness of one person with that of another It would be handy to have a utility functions and Uj for persons i and j such that you could say that person i is happier consuming 11 than personj is consuming zj if gt Uj But of course if utility is only unique up to monotonic transformations one can choose different representations for i andj to make the answer to the 77who is happier77 question come out either way Even if we abandon hope of comparing one person7s utility to that of another it would still be nice to be able to say things like Lucy cares more about the difference between bundle w and bundle 1 than she cares about the difference between bundle y and bundle 2 To do this we would need a utility function such that we could make meaningful statements of the form If 7 gt 7uz then Lucy cares more about the difference between w and 1 than she cares about the difference between y and 2 If we can do arbitrary monotonic transformations on utility then we can reverse the ruling on which difference is bigger by taking a monotone transformationi For example suppose that there are two goods and uzl 12 11 12 Consider the four bundles w 111 1 00 y 33 and z 22 Then 7uz 21gt 7uz 2 The utility function 1111 12 11 122 represents the same preferences as u but vw 7 111 441 lt vy 7 vz 7 On the other hand suppose that the only utility transformations that we are willing to do are af ne transformations where we say that u is an af ne transformation of v if av b for some positive number a and some real number b then the ordering of utility differences is preserved under admissable transforationsi77 Additively separable representations turn out to be unique up to af ne transformations Theorem 4 If two di erent additively separable utility functions U ELI u and V ELI vi represent the same preferences on R then it must be that for some real numbers a gt 0 and bi auZz b for all i li i i n In this case Uz aVz b where b bi The proof of this theorem is not dif cult I plan to post a proof here when I get a bit of time Additively separable and homothetic preferences Theorem 5 pr f t are p it 39l by a utility function then they are additively separable and homothetic if and only they are repre J sentable by a utility function of one of these two forms Zaizg Z ai ln z A proof of this theorem can be found in Katzner Separability and Stationarity of Intertemporal Preferences Suppose that a consumer has preferences over intertemporal streams of con sumption Let there be T time periods and n regular commodities Then an in tertemporal consumption stream is an 7LT dimensional vector 1 111 1 i 111 1 i In where I is the n vector of commodities consumed in period t Suppose that these preferences are additively separable over time so that preferences over intertemporal consumption are represented by a utility function of the form Uzluizn 1 Prices in such a model can be written as p 01 wa 11107 where pt is the price vector in period t Demand theory in this environment would be especially easy to work with if the budget constraint is PI Zinnia M 2 21 for some measure of wealth Mi Think of intertemporal prices in the following way Let t be the vector of time t prices measured in time t dollarsi Suppose that money borrowed in period t and returned in period t 1 pays back 1 7 in period t 1 for each dollar borrowed in period t Then a dollar in period 1 will be worth 1 Hlt1ri dollars in period t Conversely a dollar in period t is worth 4 11211 Ti dollars in period 1 Then the ptls will represent intertemporal prices in a budget constraint if we de ne PL 11111 TV and if we de ne M to be the present value of all future income ows Of course if there are credit constraints and differences between borrow ing and lending rates the budget constraint becomes much more complex Economists like to ignore these complexities and very frequently work with intertemporal models where the budget is of the form in Equation 2 Note that this intertemporal budget becomes simpler if the interest rate 7 is constant over all time If preferences are representable in this way then there is are wellde ned singleperiod demand functions that depend only on prices in that period and total expenditures in that period If someone is maximizing a utility function of the form in Equation 1 subject to a budget constraint of the form in Equation 10 27 then it must be that for any t she is choosing It so as to maximize utzt subject to the constraint that ptzt e where e is her total expenditure in period t measured in the intertemporal prices Of course7 in general to nd out how much she intends to spend in period t7 we may need to know things about interest rates and prices in other periods There are some fairly plausible additional assumptions that impose interest ing special structure on the utility functions in Equation 1 These results are due to Tjalling Koopmansi Consider any two time streams 117 127 i i i 7 In and 117 127 i i i 7 that have the same consumption bundle in period 17 but possibly different bundles in other periods Preferences are said to be stationary over time if the follow ing is true 117 127 71 i 117 127 i i i 71 if and only if 127 i i i 717711 5 27Hi7zn7r1i Stationarity over time requires that mere position in time does not change the preference ordering over consumption stream Thus if one prefers the stream zg7ui7zn starting period 2 to the stream 127 i i 7177 starting in period 27 then one would feel the same way if this stream were moved up one period to start in period 1 Additively separable preferences are said to be impatient if whenever you like single period bundle I better than single period bundle yt7 you would prefer to get the nice bundle earlier rather than later That is7 if zl7zg7ui7zn gt 917 127 i i 7zn7 then zl7y27 137 i i 7In gt 917I27i i i 7 Theorem 6 If preferences over time streams are representable by a continuous utility function then they will be additively separable and stationary over time if and only they can be represented by a utility function of the form n U1171n Zatqu 3 21 for some a gt 0 Such preferences are impatient if and only ifa lt 1 Proof By assumption7 preferences are represented by a utility function of the form 1 Suppose that 117127 i i i 7177 5 11712 i i i Then 22 utzt 2 22 Stationarity implies that in this case7 127 i i 7 177 11 i 127 i i 7 1 11 and therefore 22 ut1zt 2 22 ut1z i Therefore the utility func tions 22 utzt and 22 ut1zt represent the same preferences Since two additively separable utility functions that represent the same preferences must be af ne transformations of each other7 it follows that 22 u I 122 ut1zt for some a gt 0 But if this is the case7 it must be that u2z au1z and u3z au2z and so on But this means that for all t 17Hi7n7 utz a u z D What happens if people age What happens if survival probability varies over lifetime What about an intergenerational interpretation Suppose that I is the lifetime consumption of generation ti Still more structure Suppose that in addition to assuming additive separability and stationarity one also assumes that preferences over time streams of consumption are homothetic then by Bergson s theorem we have preferences representable by one of the following forms washroom Za z lt4 21 Uzlulzn Za lnztl 5 21 Uncertainty7 separability and time preferences The assumption that decisionmakers are expected utility maximizers implies that they have additively separable preferences across events This assumption generates a von NeumannMorgenstern utility function Uz that is determined up to af ne transformations The assumption that preferences are additively separable and stationary over time generates a single period utility function that is also determined up to an af ne transformation The question arises What would it mean for the von NeumannMorgenstern utility function and the intertemporal utility function to be the same function a Another question that arises is How could they not be To answer the rst question Suppose that these two functions are the same lndividuals choosing alternative lotteries over time streams of consump tion would then choose so as to maximize an expected utility function of the form EEt a uzt E a Euztl or the second question We could consider alternative theories in which preferences were additively separable over time and stationary where utility is represented by E a fEuzt for some nonlinear function What differ ence would this make References H Gerald Debreul Topological methods in cardinal utility In Kenneth J Arrow Samuel Karlin and Patrick Suppes editors Mathematical methods in the social sciences pages 16726 Stanford University Press Stanford California 1960 E Peter Fishburnl Utility theory for decision making Wiley New York 1970 E Donald Wt Katznerl Static Demand Theory Macmillan New York 1970 E Tjalling Jl Koopmansl Stationary ordinal utility and impatience Econo metrica 28 1960


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