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ON REAL ECONOMIC FREEDOM October 2005 by Serge-Chrstophe KOLM * Abstract A pror, real economc freedom s purchasng (and sellng) power. Yet, Xu s theorem comforts rankng the freedom of choce provded by budget

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ON REAL ECONOMIC FREEDOM October 2005 by Serge-Chrstophe KOLM * Abstract A pror, real economc freedom s purchasng (and sellng) power. Yet, Xu s theorem comforts rankng the freedom of choce provded by budget sets as ther volume n dervng t from three axoms. However, one and a half of these axoms can be dscussed. In contrast, the standard measure of purchasng power leads one to order the freedom provded by budget sets as the dstance to the orgn of the ntersecton of the budget hyperplanes wth a gven ray from the orgn. Hence, equal budget freedoms correspond to pencls of budget hyperplanes. Appled to labour and earnngs of ndvduals wth dfferent wage rates, ths equal freedom yelds the dstrbutve prncple of equal labour ncome equalzaton. Keywords: freedom, freedom of choce, economc freedom, purchasng power, budget rankng, equalty. JEL classfcaton numbers: D31, D46, D63, H21, J33 1. Economc freedom Economc freedom often means freedom of exchange. Freedom of exchange s a part of basc freedoms or basc rghts, whch consttute the legal and moral bass of our socetes. It has often been reproached to these rghts that they may leave you wth lttle actual freedom f you do not have the means to make use of them. In partcular, even f all men are free and equal n rghts (the 1789 Declaraton), ths allows for very unequal actual freedom. Rch and poor are equally free to sleep under brdges (Anatole France). Marx denounces these basc rghts * CREM, IDEP, EHESS. 2 as beng only formal freedom and not provdng real freedom. 1 Yet, he thus nspred suppressons of formal freedoms wth the hstorcal dramatc consequences we know. Real freedom requres both formal freedom and other means, that s, wth free exchange, ncome or means to acqure t such as a suffcent wage rate. The budget sets are the domans of possble choce whch provde the correspondng freedom of choce. One can certanly admt that, wth gven prces, a hgher ncome provdes a hgher such freedom. At least, t provdes a possblty set that ncludes the other n addng possbltes to have more of all goods. As common language puts t: purchasng power s hgher. Yet, what can be sad when prces are not the same and the budget sets are not related by ncluson? Ths stuaton s a very common problem for economsts and economc statstcans. They face t n choosng a prce ndex for computng a purchasng power n dvdng ncome by ths ndex. The result s also real n another sense of the term, now opposed to nomnal,.e., t does not change f ncomes and prces are multpled by the same number (for whatever reason). More precsely, the prce ndex s a lnear functon of prces, wth coeffcents that represent quanttes of goods n a basket the choce of whch s the subect of the theory of prce ndexes. The result thus measures the purchasng power of an ncome facng a set of prces, hence of the correspondng budget set, as the number of such baskets that ths ncome can buy when facng these prces. That s to say, ths classcal real freedom s the dstance to the orgn of the ntersect of the budget hyperplane wth the ray from the orgn bearng the vector of the coeffcents of the prce ndex. One consequence s that the equalty of such freedoms means that the budget hyperplanes pass through the same pont whch represents the vector of the prce-ndex basket multpled by the same number (they consttute a pencl of hyperplanes). Ths comparson of purchasng powers s often done for the dfferent prces at dfferent dates or n dfferent places. Yet, there s a stll more mportant applcaton, snce freedom s a pror that of ndvduals, and dfferent ndvduals generally face dfferent prces n free exchange for the good whch can be sad to be the most mportant one, labour, whose prce, the wage rate, depends on the dfferent productve capactes and on the demands for them, and s also the market prce of the good lesure. Ths leads to the determnaton of gven ncomes, or ncome transfers, that mantan an equal purchasng power, or real economc 1 Marx also obected to the ndvdualsm of basc rghts, thus echong a reproach made by Robesperre as early as 1789. 3 freedom, for ndvduals endowed wth dfferent gven productve capactes, hence facng dfferent wage rates. The result n the smplest case turns out to be that each ndvdual receves the net transfer t =k ( w w ), where w s ndvdual s wage rate, w =(1/m)Σw s the average wage rate for m ndvduals, k s a number defnng the prce ndex, and t s a subsdy f t 0 and a tax of t f t 0 (see Secton 4 below). 2 However, a remarkable recent artcle of Yongsheng Xu (2004) proposes a strong axomatc bass for the applcaton to compettve budget sets of an old alternatve proposal, whch conssts of rankng the freedom offered by domans of choce of quanttes of goods by ther volume. Ths s at odds wth the foregong result based on standard practce. For nstance, n the case of two goods, the budget lnes correspondng to equal real freedom pass through the same pont wth the standard comparson and are tangent to the same rectangular hyperbola wth the volume rankng. In 1972, Jean-Marc Oury, then a student, proposed to me, as topc of dssertaton, the measure of the freedom of choce of a bundle of commodtes by the volume of the possblty set. I rather dscouraged hm. One of my reasons was that ths amounts to measurng real ncome wth a prce ndex whch s the geometrc mean of prces, and ths neat property has strange consequences shortly recalled and needs to be ustfed. 3 However, Xu derves hs concluson from axoms, not from consequences. 4 The volume rankng was nevertheless among the rankngs or measures of freedom of choce whose propertes I analyzed. 5 Yet, the contradcton has to be elucdated at the level of the axoms and of ther meanng. 6 Most generally, there are three knds of freedoms, whch can be labelled negatve freedom, postve freedom, and mental freedom. They are often closely nterrelated. Negatve 2 And Kolm 2004a. 3 Moreover, forbddng access to any one good makes you less free than any other knd of constrant. Yet, ths s not a budget constrant, and hence ths could be obected to Oury but cannot to Xu. 4 Classcal epstemology states that rules should be udged ontly accordng to ther statements, ther condtons (ncludng axoms), and ther consequences (see, e.g., Plato s dalectc n Republc or Rawls s reflectve equlbrum ). 5 See Kolm 2004a, Chap. 24 (also 1993). 6 Dscouraged, Jean-Marc Oury abandoned hs Ph.D. Hred by the French ndustry mogul Guy Deouany at the Compagne Générale des Eaux, he became n charge of the real-estate branch where he drove the Masons Phenx to buoyancy and then to falure. Ths dd not dscourage Deouany from hrng other brght young people from the best school. The next one was Jean-Mare Messer, who had Oury fred, took over the frm, and dd exactly the same thng n the meda busness whle renamng the frm Vvend Unversal. 4 freedom or socal freedom s freedom from the forceful nterference of other humans. It s defned by the nature of the constrants and t conssts of the classcal basc rghts or lbertes. In economcs, t conssts of free enterprse and free exchange and market. People are of course constraned not to forcefully nterfere wth others, n partcular to respect the consequences of free actons or agreements (such as rghts so created). Socal freedom rases no ssue of rvalry and can be at satety (remanng rvalry concerns the allocaton of rghts concernng gven resources). Postve freedom s defned by the domans of free choce, and t also s the real freedom of the dscussons referred to above. In a market system, t s aptly called purchasng power (and sellng power). Mental freedom s the freedom to determne one s emotons and preferences. The autonomy of Rousseau and Kant s a type of t. In economcs t would refer to ssues such as an absence of manpulaton by advertsement, delberatve choces, or choce of crtera of farness. 7 The present topc fully dscards preferences and hence mental freedom. It s concerned wth the second knd of lberty, postve freedom, n the context of the economc manfestaton of the frst, free markets. That s, t concerns the purchasng power of budget sets, or budget freedom. 8 Secton 2 consders Xu s theorem. The consequence of the alternatve of classcal prce ndexes s shown n Secton 3. Secton 4 apples ths result to the queston of macroustce Freedom as volume The freedom of choce offered by domans of choce (possblty or opportunty sets) D wll be ordered by an orderng assumed here to be representable by an ordnal freedom functon 7 One can also buld an economc model of conscously nfluencng one s preferences (see Kolm 1982, chap.23, 1985). 8 The vocabulary of freedom has both ths establshed base and fluctuatons. For nstance, what Isaah Berln (1958) calls postve freedom s largely mental freedom volated by deologes. General mental freedom, ncludng the basc ssue of masterng one s desres whch n prncple could have drastc economc consequences, s the topc of my book Happness-Freedom (Deep Buddhsm and Modernty) (1982, n French). 9 I wsh to thank Ncolas Gravel for very usefull correctons and suggestons. 5 F(D) (ths representablty s no restrcton for the present problem). 10 One property s D D F( D ) F( D). 11 We consder partcular D s whch are budget sets. Denote as one of n goods, x 0 ts quantty, x={x } R n + a bundle of these goods, p 0 a constant prce of good, p={p } the prce vector, and y an ncome. In the space of x, Σp x =y s the equaton of the budget hyperplane, and Σp x y wth x 0 for all defnes the budget set. Then, a =y/p s the x of the ntersect of the budget hyperplane wth the axs of the x, and a={a } denotes the set or vector of the a s. The budget set s defned by the par (y, p), and the freedom functon can be wrtten as F(y, p). A pror, ths functon represents a real ssue (not a nomnal one), and hence t s homogeneous of degree zero n y and p. Hence, F(y, p)=f(1, 1/a 1,..., 1/a n )=φ(a). Functon F wll be taken as ncreasng n y (and decreasng n the p ) snce an ncrease n y (and a decrease n a p ) add bundles contanng larger quanttes of all goods than prevous possble ones. Hence functon φ s ncreasng n the a. The followng remarks n ths secton are standard or obvous. If functons F and φ are such that ther orderng of the a does not change f the same coordnate of all a s multpled by the same postve number, for all coordnates and numbers, that s, φ(a) φ( a ) φ(a 1,...,λa,..., a n ) φ( a 1,...,λ a 1,..., a n ) (1) for all a,, and λ 0, then φ s of the form φ(a)=φ 1 ( Π a α ) (2) 10 Xu s orderng s so representable. He has studed varous possble general propertes of freedom orderngs and functons n Pattanak and Xu (2000). 11 Ths property seems unavodable f the varous costs of choce are consdered a dfferent ssue from freedom (ncludng materal costs, mental costs, dslkng responsblty, and angush of choce emphaszed by Kerkegaard and Sartre). where Π denotes the product, α 0 are numbers, and φ 1 s an ncreasng functon; and conversely Then, functon φ(a) wth ths form s symmetrcal f and only f all the α are equal: α =α 0 for all. Then, φ(a)=φ 2 ( Π a ) where φ 2 s an ncreasng functon. Yet, the volume of the budget set s V=(1/n!) Π a. Hence, φ(a)=φ 3 (V) where φ 3 s an ncreasng functon. Xu postulates the symmetry of the functon φ(a). Ths amounts to the symmetry of the freedom functon F(y, p) n the prces. Then, your freedom of choce s not changed by a permutaton of the prces of the goods. It s not changed f you pay a sandwch for the prce of a car and conversely. The mad store manager who assgns randomly hs prce tags to the goods leaves hs customers as free. Moreover, f the prce of one good doubles, your freedom s the same whatever the good, for nstance a good of whch you consume much or one whch has no nterest for you. Xu would not accept the argument that you could become less free because your favourte goods become more expensve, snce ths would depend on your preferences (ths s assumed to apply also to the case where the good you need most becomes more expensve). However, ths concepton does not belong to all the actual uses of the terms free and freedom, as we wll see n concluson. Yet, the man problem may be wth condton (1), Xu s nvarance of scalng effects. Ths problem s that functons F and φ are not a pror unt-nvarant. Indeed, the proposed ustfcaton of condton (1) s that the rankng of freedom should not depend on the unts n whch the quanttes of goods are measured. If the unt of good becomes λ tmes smaller, then number a should become λ tmes larger, and nothng real s changed. However, ths s 12 More explctly, condton (1) s the followng. Let vector a take four values a 1, a 2, a 3, a 4, such that 3 a =λ 1 4 a, a =λ 2 3 a, and, for all, a = 1 4 a and a = a 2. Then φ(a 1 ) φ(a 2 ) φ (a 3 ) φ (a 4 ) for any a 1, a 2,, and λ 0 f and only f (2) holds. 7 not descrbed by condton (1) because, n fact, functons F and φ should change for takng account of ths change n unt. In techncal terms, they ncur the correspondng contravarant transformaton. Precsely because they represent real ssues and not nomnal ones. If one wants to make explct ths ssue of neutralty wth respect to the choce of unts of measurement, one has to order not sets a={a } of the numbers a, but sets of numbers {a /b } where b s a gven (arbtrary) quantty of the good : when unts change, both a and b are multpled by the same number, and a /b does not change (b s a real not nomnal unt of measure of good ). An example wll be met shortly. 13 Ths can also be seen n another strong consequence of the proof under dscusson, namely, all utlty functons are Cobb-Douglas provded t does not matter whether bread s measured n klos or n grams (and the lke). Indeed, assume the a denote now the quanttes of goods consumed by an ndvdual, and assume that the orderng under consderaton s the preference orderng of ths ndvdual. Consder Xu s axoms except symmetry. The ndvdual prefers hgher a. Assume she s ndfferent to the unts of measure of the quanttes of all goods and smlarly translate ths as the scalng effect. Then, the ndvdual s ordnal utlty functon has form (2), a specfcaton of whch s Π a α, a Cobb-Douglas utlty functon. 3. Freedom as ponted dstance 13 Xu works wth orderngs, and hence hs orderngs can ncur the contravarant transformaton. Hs orderng are representable by ordnal freedom functons. The knd of ssue met s not unfrequent n normatve economcs. Another case has exactly the same structure. Suppose you want to ustfy Nash barganng soluton. Then, take φ to be a socal welfare functon, and a =u (x) u (x 0 ) where u s a cardnal utlty functon, x the socal state, and x 0 a partcular reference state. Then a s defned up to an arbtrary multplcatve factor ndependently for each. If ths s nterpreted as mplyng condton (1), ths leads to the form φ=φ 1. A appeal to symmetry may then produce φ=φ 2, and hence Πa as a specfcaton of the socal welfare functon (Nash s soluton for n=2). Yet, ths appeal to symmetry has no ustfcaton, and the appeal to condton (1) bypasses the contravarant transformaton. In another story, φ s agan a socal welfare functon, a s a cardnal utlty of ndvdual, and φ s also requred to be cardnal (for nstance, they are the correspondng von Neumann-Morgenstern specfcatons). Then, each of these functons beng defned up to an ncreasng affne functon, plus an appeal to symmetry, would requre that φ s a utltaran sum. Yet, ths omts the contravarant transformaton, and the symmetry has no ustfcaton (see Kolm 1996a, Chap. 14, and Maskn s dervaton of utltaransm).a smlar fallacy underles the argument, whch has been proposed, that an ndex of ncome nequalty should be scale-nvarant or homogeneous of degree zero because t should not change f the unt of measure of ncomes changes (measure n dollars or n cents). Yet, an other ndex need not be unt-neutral and can ncur the contravarant transformaton when the unt for measurng ncomes change. Note that the very term scale-nvarant may suggest the mstake (homogenety of degree zero as a real property, that s dependence on ratos here of ncomes only, characterzes the measures that engneers and physcsts call ntensve ). 8 Wth ths volume rankng of budget freedom, ths freedom s also ranked as Πa and hence as (Πa ) 1/n =y/(πp ) 1/n. Ths expresson s homogeneous of degree zero n y and the p. The denomnator the geometrc mean of the prces s homogeneous of degree one n the p. It s a knd of prce ndex and, then, the expresson s a measure of real ncome. More generally, a prce ndex s a lnearly homogeneous functon of prces π(p), and y/π(p) s the correspondng real ncome. Budget freedom s ranked by real ncome when the freedom functon F can be wrtten as F(y,p)=ϕ[y, π(p)], snce the homogenety of degree zero of F n y and the p mples ϕ[y, π(p)]=ϕ[y/π(p),1]=f[y/π(p)] where f s an ncreasng functon. Snce real ncome s also called purchasng power, budget freedom s ranked as the correspondng purchasng power. The volume rankng of budget freedoms takes the geometrc mean of the prces as prce ndex (Kolm 2004a). In contrast, the standard prce ndex s a lnear form π(p)=σb p wth b 0 for all and b 0 for at least some. It seems that ths s the only meanngful form of a prce ndex, because ths s the cost of buyng the set of quanttes b of the goods, the ncome necessary for ths purchase at these prces. It also seems that, for ths reason, all the prce ndexes actually used are applcatons of ths form. Wth ths ndex, budget freedom s ranked as β=y/σb p. Hence, Σβb p =y. Ths shows that the pont of coordnates βb s on the budget hyperplane of equaton Σp x =y. If b={b }denotes the vector of the coeffcents of the prce ndex b, ths pont s βb. 9 Hence, budget freedom s ranked as the dstance to zero of the ntersect of the budget hyperplane wth the ray from the orgn bearng the vector b of the coeffcents of the prce ndex (fgure 1). FIGURE 1 In partcular, budget sets wth equal budget freedom are budget sets whose budget hyperplanes pass through the same pont βb (fgure 2). FIGURE 2 Conversely, f a number of budget hyperplanes pass through the same pont of the non-negatve orthant, they can be sad to determne equal budget freedoms, relatvely to a vector of coeffcents of the prce ndex n the drecton of ths pont. That s, equal budget freedom corresponds to the pencls of budget hyperplanes. 14 Ths result also amounts to the classcal prncple of free choce from an equal (dentcal) allocaton whch s represented by the common pont of the budget hyperplanes. 15 However, the ssue and result are mportant when stuatons wth dfferent prces are compared. Ths can be dfferent tmes, or places, or ndvduals when some prces are specfc to them (such as ther wage rate whch s the market prce of ther lesure or labour). The choce of prce ndexes for comparng real ncome across tmes or places s the classcal topc of the theory of prce ndexes and of a vast lterature. The correspondng queston for ndvduals and wage rates wll shortly be noted. 14 Wth the general freedom functon F(y, p), equal-freedom hyperplanes (the hyperplanes lmtng equal-freedom budget sets) have an envelop, and these envelops for varous freedoms do not ntersect and are the graphs of quas-concave functons whch can be wrtten as E(x)=F (see Appendx B). These envelops degenerate nto ponts βb for a lnear prce ndex. They are rectangular hyperbolas for volume measures and n=2. 15 See Kolm 1971. 10 The obtaned form can llustrate the precedng remark about unt neutralty and contravarant transformatons. Indeed, y/σb p = (Σb /a ) 1. (3) Each coeffcent b has the dmenson of a quantty of good. When the unts of good are dvded by λ, both a and b are multpled by λ, and expresson (3) does not change. The rankng of budget sets by ther coeffcents β permts the comparson of economc stuatons wth one budget set for each ndvdual, say β for ndvdual. The logc s that of the comparson of profles of co-ordnal ndces (ordnal and nterpersonnaly comparable). The followng propertes are meanngful. 16 Varous countng of β that are larger or smaller than, or equal to, others. Equalty of the β (hence of the freedom of budget sets) n a stuaton, and across stuatons for the same or dfferent ndvduals. Pareto-lke domnance. 17 Permutaton of the β among ndvduals (sy

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