The Existence and Optimality of Equilibrium

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1 The Exstence and Optmalty of Equlbrum Larry Blume March 29, Introducton These notes quckly survey two approaches to the exstence. The frst approach works wth excess demand, whle the second works wth prmtves, such as preferences, endowments, and the lke, from whch excess demand could be derved. 2 Excess Demand For ths approach n some detal, see Chapter 2 of Arrow and Hahn. Consder frst an exchange economy. Each of I ndvduals has an endowment of L commodtes, ω for ndvdual. Were ndvduals utlty maxmzers, we would derve a demand functon or correspondence, d (p), for the L commodtes, whch depends upon the vector p of market prces. Excess demand for ndvdual n s z (p) = d (p) e, the excess of trader s demand over hs supply. Aggregate excess demand s then Z(p) = z (p). Equlbrum can be defned n terms of excess demand: Defnton 1. A compettve equlbrum s a prce vector p 0 such that Z(p) = 0. Another defnton one often sees s: Defnton 2. A compettve equlbrum s a prce vector p 0 such that Z(p) 0 and p Z(p) = 0. 1

2 2 The dfference between the two s a complementary slackness condton. In the frst, excess demand for each good must be 0. In the second, excess demand for a good could be negatve, but then ts prce must be 0. We are gong to buld a model n ths secton n whch the equlbrum prce wll be strctly postve, that s, all goods wll be demanded n equlbrum. It s a good exercse to consder how to relax the assumptons that are to follow n such a way as to requre the complementary slackness condton. It s mportant to understand that whle excess demand could be derved from utlty maxmzaton, t need not be. We could just thnk of excess demand as arbtrary behavor rules. If aggregate excess demand s suffcently regular, an equlbrum (ether defnton) wll exst. We wll defne demand for consumer formally as a functon The phrase "suffcently regular" means: A.1. Z(p) s homogeneous of degree 0. d : R l ++ R +. If choce s defned on budget sets, then ths axom must hold because the descrpton of a gven budget set s nvarant to changes n scale of the prces whch defne t. Because of homogenety, we can normalze prces, whch s to say that we can choose a scale for prces. It s convenent to choose a scale such that l p l = 1. Let = {p R l ++ : l p l = 1}. Ths set s the unt smplex n R l ++. Because of Axom 1, there s no loss of generalty n assumng and we shall do so from here on out. A.2. For all p R l ++, p Z(p) = 0. Z : R l, Axom 2 s known as Walras law. Why should t be true? For ndvduals, p z n (p) = 0 s merely the clam that ndvdual n consumes on her budget lne, and Walras Law follows from summng. A.3. Z s a contnuous functon on nt, or Z s a correspondence whch s compact-, convex- and non-empty valued, and upper hem-contnuous. If demand s sngle-valued, then contnuty, Axom 3, s a natural assumpton. The assumpton s not requred on the boundary snce we are not assumng demand s well-defned when the prce of a

3 3 good s 0. When excess demand s a correspondence, what does contnuty mean? For a functon, contnuty s the requrement that for every open set O, Z 1 (O) s an open set. For a correspondence, there are two natural notons of nverse: Defne Z w (O) = {p : Z(p) O} and Z s (O) = {p : Z(p) O}. These are the weak and strong nverse of Z, respectvely. The requrement that the weak nverse of an open set be open s called lower hem-contnuty, whle the requrement that the strong nverse of an open set be open s called upper hem-contnuty. (An older lterature uses the termnology sem-contnuty, whch unfortunately was already n play for somethng else.) A correspondence s contnuous f t s both lower- and upper hem-contnuous. If a correspondence s sngleton-valued, t s contnuous f and only f, as a functon, t s a contnuous functon. A.4. Z s bounded below; that s, there s a vector B such that Z(p) B for all p R n ++. If ndvduals can never demand negatve amounts of a commodty, then excess demand s bounded below by e, the aggregate endowment. Ths s Axom 4. More generally, f we assume that consumer s budget sets are all subsets of a consumpton set X and each X s bounded from below, then Axom 4 wll be satsfed. A.5. If p s a prce vector such that for some good, p = 0, then for every sequence of strctly postve prces p n convergng to p, Z(p) +. Fnally, as the prce of a commodty converges to 0, ts demand becomes arbtrarly large. Ths s Axom 5. Actually, t s stronger than Axom 5, whch only requres the demand for some commodty to dverge. Theorem 1. If excess demand Z(p) satsfes Axoms 1 5 on R n ++, then there s a p = 0 such that Z(p ) = 0. How does one prove such a theorem? The problem s a fxed pont problem. The man tool s the followng: Theorem 2 (Brouwer). If C s a compact, convex set and f : C C s a contnuous functon, then there s an x C such that f(x) = x. That s, x remans fxed under the acton of f. Theorem 3 (Kakutan). If C s a compact, convex set and F : C C s an upper hem-contnuous correspondence, then there s an x C such that x F(x).

4 4 Proof. Suppose frst that demand s a contnuous functon. Snce demand s homogeneous of degree 0, prces can be normalzed so that p = 1. Let = {p R n ++} such that p = 1. Ths set s the strctly postve unt smplex n R n. For ǫ > 0, defne ǫ = {p : p ǫ}. These are compact, convex sets. Choose n 1 > ǫ > 0. Snce Z s bounded from below on, max Z (p) < B on. f ǫ (p) = p ( 1 + B 1 Z (p) ) + δ 1 + nδ where δ = ǫ/(1 nǫ). Let f(p) = ( f (p),..., f n (p) ). The term ( 1 + B 1 Z (p) ) s postve. Thus on, f ǫ (p) > ǫ. Furthermore, f ǫ (p) = 1 ( ) 1 + nδ p 1 + B 1 p Z (p) + nδ = 1 ( ) 1 + nδ 1 + nδ = 1 because of Walras Law. Thus f ǫ : ǫ. It s contnuous, and so Brouwer s Theorem mples that f ǫ has a fxed pont p ǫ. Choose a sequence ǫ k 0. The sequence p ǫ k has a subsequence wth a lmt pont p. We wll see that p s a compettve equlbrum. Clearly p 0 and p = 1. We also have that p ǫ k + nδ k p ǫ k = p ǫ k + B 1 Z (p ǫ k) + δ k Frst we need to show that p > 0. Suppose not. Then, takng lmt superors of both sdes, 0 = lm sup B 1 Z (p ǫ k), but ths volates Axom 5. Fnally, snce p and Z s contnuous on, Z(p ) = 0. For the correspondence case, defne the correspondence F such that q F ǫ (p) ff there s a z Z(p) such that q = p ( 1 + B 1 ) z + δ 1 + nδ Ths correspondence s clearly convex- and non-empty valued. For those who are nterested, t s easy to show t s upper hem-contnuous. Thus accordng to Professor Kakutan (the father of Mchko Kakutan, for the lterate among you), the correspondence F ǫ has a fxed pont p ǫ. The proof now proceeds as before.

5 5 The relevant fact for provng that the excess demand correspondence s uhc, whch we shall not prove here, concerns the graph of uhc correspondences. Defnton 3. The graph of a correspondence F : X Y s the set {(x, y) : y F(x)} X Y. Ths s the same defnton as that of the graph of a contnuous functon. Lemma 1. If F : X Y s a correspondence and Y s compact, then F s uhc f and only f ts graph s closed. The mathematcally adept may enjoy provng ths fact. 3 Prvate Ownershp Economes The prvate ownershp economy s a general framework whch fts together producton and consumpton. It locates consumpton n ndvdual consumers and producton n ndvdual frms. Ths s the most general framework n whch we wll examne the emergent propertes of markets the exstence of prce equlbra and ther connecton to optmal allocatons. The model supposes the exstence of I consumers, J frms and L goods. Each consumer s characterzed by a preference order defned on a consumpton set X R L, an endowment bundle ω R L, and a vector θ = (θ j ) J j=1 representng the share of frm j consumer owns. Let ω = ω denote the aggregate endowment of the economy. Each frm s characterzed by a producton set Y j R L. We adopt the conventon that for any vector n a frm s producton set, negatve terms represent nputs and postve terms represent outputs. The ultmate reductonst act of economc theory s to defne an economy thus: Defnton 4. A prvate ownershp economy s a tuple ( (X,, θ, ω ) I =1, (Y j) J j=1). Ths lst contans all the data necessary to derve demand and supply functons, fnd equlbra, and so forth. The behavoral assumptons that go along wth ths are preference maxmzaton by consumers and proft maxmzaton by frms. Note already the ways n whch ths model s restrctve.

6 6 1. Frms are assumed to proft maxmze. What about alternatve, "manageral" goals of frm behavor? How would models of "home producton" ft nto ths framework. 2. There are no consumpton externaltes. Ths btes n several ways: Frst, s preferences cannot be nfluenced by j s actons. There s no way to model your demand for ar freshener on the consumpton of your roommate s cgarettes. Second, ndvduals may have preferences not over ndvdual consumpton but over entre socal states. Consumer s rankngs of her own choces may be nvarant to js consumpton, but js consumpton may nonetheless effect s utlty. Thrd, s consumpton opportuntes may be constraned by js choces. 3. There are no producton externaltes. You can magne the lst of possbltes. A basc defnton of the general compettve model s that of an allocaton: Defnton 5. An allocaton (x, y) s a specfcaton of a consumpton plan for each consumer, a vector x X, and a producton plan for each frm j, a vector y j Y j. An allocaton s feasble ff x = ω + j y j. Followng MWG, the set of feasble allocatons s denoted by A R L(I+J). We wll call x the consumpton allocaton and y the producton allocaton assocated wth the allocaton z = (x, y). 3.1 Compettve Equlbrum One dea behnd compettve equlbrum s that supply equals demand. But snce we have a theory of where supply and demand come from, we are tempted to perce the vale of Marshallan curves and understand compettve equlbrum drectly n terms of the prmtve, atomc economc concepts. Let E = ( (X,, θ, ω ) I =1, (Y j) J j=1) denote a prvate ownershp economy. I m gong to follow MWG here, but n fact one can do much better than ths. Defnton 6. A compettve equlbrum for the economy E s an allocaton (x, y ) and a prce vector p such that 1. For every frm j, y j maxmzes profts among all feasble producton plans n Y j : p y j p y j for all y j Y j.

7 7 2. For every consumer, x s preference-maxmal among all affordable consumpton plans. That s, x x for all x n the set 3. (x, y ) A. {x : x X and p x p ω + θ j p y j }. j Ths defnton s due to Debreu (see Theory of Value, ch. 5.5). Here s hs exstence theorem: Theorem 4. A compettve equlbrum for the prvate ownershp economy E exsts f for every, 1. X s closed, convex and bounded from below, 2. s non-satated n X, 3. the relaton s contnuous, 4. If x x, then for all 0 < t < 1, tx + (1 t)x x, 5. there s an x 0 n X such that ω x 0; for every consumer j, 1. 0 Y j, 2. the aggregate producton set Y = j Y j s closed and convex, 3. Y ( Y) =, 4. Y R L. Just to revew, non-sataton means that for all x X there s an x X such that x x. Contnuty means that for all x the sets {x X : x x } and {x X : x x } are closed n X. Condtons 2 and 4 together mply that preferences are locally non-satated. Condton 5 for consumers s the cheaper pont assumpton. Its purpose s to make sure that demand has the rght contnuty propertes at the boundary of the consumpton set. Suppose that a consumer s consumpton set s R L + and her endowment s the the 0 vector. Imagne a sequence of prces n whch the prce of good 1, say, s always postve, but s convergng to 0. Then demand for good 1 s always 0, but n the lmt s could be strctly postve or even empty. Contnuty mples that the so-called better than sets are open. The set {x X : x x } s the complement of the set {x X : x x }. Smlarly for the worse than sets.

8 8 For the frm, condton 3 states that producton s not reversble. If the aggregate producton plan y s feasble, then the plan y s not. Assumpton 4 s free dsposal. Another useful concept s that of a compettve equlbrum wth transfers. The dea s to fnd market clearng prces after we allow for arbtrary wealth transfers among consumers. Defnton 7. A compettve equlbrum wth transfers for the economy E s an allocaton (x, y ), a prce vector p and an assgnment of wealths (w1,..., w I ) to consumers such that 1. For every frm j, y j maxmzes profts among all feasble producton plans n Y j : p y j p y j for all y j Y j. 2. For every consumer, x s preference-maxmal among all affordable consumpton plans. That s, x x for all x n the set {x : x X and p x w }. 3. (x, y ) A. 4. w = p ω + j p y j. 3.2 Pareto Optmalty The basc noton of socal desrablty s the Pareto order: Defnton 8. A consumpton plan x s Pareto-better than consumpton plan x, wrtten x P x, ff for all, x x, and for some consumer k, x k x k. An allocaton z = (x, y) s Pareto optmal ff t s feasble, and f for no other feasble consumpton plan z = (x, y ) s t true that x P x. How do we know an optmum exsts? In exchange economes ths s not hard. The set of feasble allocatons s obvously compact, so sutable contnuty assumptons on preferences should do the trck. When producton s possble, compactness of the set of feasble allocatons s not so obvous. Debreu (Theory of Value, Ch. 6.2.) gves us an answer. Theorem 5. The prvate ownershp economy E has an optmum f 1. for all, X s closed and bounded from below,

9 9 2. for every x X, the set {x X : x x } s closed, 3. j Y j s closed, convex, and Y R L + = {0}, and 4. ω X j Y j. Proof. There are two parts to the proof. Frst, show that the set A of feasble allocatons s compact, and then to show that a Pareto optmum exsts. We wll do half of the frst part and all of the second. To see that A s closed, let M denote {z = (z 1,..., z I, z I+1,..., z I+J ) R (I+J)L : I+J k=1 z k = I =1 ω }. Ths s an affne subspace of R (I+J)L, hence closed. The set M = I =1 X J j=1 Y j R (I+J)L s the product of closed sets, hence closed. Fnally A = M M, and so s closed. Provng that A s bounded takes some apparatus that we are not gong to ntroduce here. To show that an optmum exsts, defne the relatons on M such that (x, y) (x, y ) ff x x. These relatons nhert all the propertes of. In partcular, they are preference orders, and no worse than sets are closed. Let O 1 = z A {z A : z 1 z}. Each set n the ntersecton s compact because t s the ntersecton of a closed no worse than set wth the compact set A. Any fnte ntersecton of theses sets s non-empty because 1 s transtve. If, for nstance, z 1 1 z 2, then {z A : z 1 z 1 } {z A : z 1 z 2 } = {z A : z 1 z 1 }. The fnte ntersecton property of compact sets tells us that that O 1 s non-empty (and compact). Any allocaton n O 1 s a feasble allocaton whch s best for consumer 1 among all feasble allocatons. Now repeat the same argument for consumer 2 on the set O 1. The outcome s a non-empty and compact set O 2 wth the property that consumer 2 s gettng an allocaton whch s preferencemaxmal for her among all allocatons that are preference-maxmal for consumer 1 on A. Repeatng ths successvely for all consumers generates a set of allocatons O I for whch each consumer s gettng preference-maxmal allocatons among those whch are preference maxmal for consumer 1 among those.... These allocatons are Pareto optmal. They are a pecular set of Pareto optma, I admt, but demonstraton of even one optmal allocaton suffces to prove exstence. The connecton between equlbrum and optmalty s subtle. The Frst Welfare Theorem gves condtons guaranteeng that a compettve equlbrum allocaton s Pareto optmal. The Second Welfare Theorem guarantees that a Pareto optmal allocaton s an equlbrum allocaton for some endowment allocaton. The condtons for the Second Theorem are stronger than the frst, and the Frst s not automatc.

10 10 Fgure 1: Falure of the Frst Welfare Theorem. In Fgure 1, the red polygon encloses a thck ndfference curve of consumer A, whose orgn s at the bottom left. The dot s an equlbrum allocaton, and the straght lne represents both an ndfference curve for consumer B and the equlbrum budget sets of the two consumers. The equlbrum allocaton s obvously not optmal. Theorem 6 (Frst Welfare Theorem). Let E be a prvate ownershp economy wth an equlbrum (p, x, y ). Suppose for all, s locally non-satated at x. Then (x, y ) s a Pareto-optmal allocaton. Recall that a preference order s locally non-satated at x f n every open neghborhood of x there s an x x. Provng the Frst Welfare Theorem requres that n any equlbrum, any consumpton bundle whch s better for consumer costs more. Ths s just what preference maxmzaton on the budget set means. The proof requres more; specfcally, than any bundle whch s at least as good costs at least as much. Ths s exactly what fals n the example of Fgure 1. The followng lemma shows when ths s true: Lemma 1. If s locally non-satated at bundle x whch s preference-maxmal on the set {x X : px px }, and f x x, then px px. Proof of Lemma 1. Snce s locally non-satated at x, there s a sequence of consumpton bundles x n wth lmt x such that xn x. Transtvty mples that xn x. Preference maxmalty mples that px n > px. Takng lmts, px px.

11 11 Proof of the Frst Welfare Theorem. Suppose that (x, y ) s Pareto-superor to (x, y ). Then for all, x x, and for some ndvdual ths rankng s strct. Ths means that p x p x for all, wth strct nequalty for some. Furthermore, for each j, p y j p y j snce each frm proft maxmzes n equlbrum. Thus p ω = p x p y j < p j x p y j. j The equalty s a consequence of feasblty of the equlbrum allocaton, and the nequalty follows from the relatons just establshed. Consequently, ω = x j y j. That s, the allocaton (x, y ) s not feasble. Notce that ths argument requres no convexty whatsoever. The Second Welfare Theorem requres some convexty assumptons because, along the way, one has to prove the exstence of an equlbrum. The welfare theorems are about the dualty, n some sense, between Pareto optmalty and compettve equlbrum. Unfortunately the dualty s not perfect. The natural expresson of dualty for the Pareto problem s not compettve equlbrum, but a slghtly dfferent noton known n the lterature as quas-equlbrum. Defnton 9. A quas-equlbrum for the economy E s an allocaton (x, y ) and a prce vector p such that 1. For every frm j, y j maxmzes profts among all feasble producton plans n Y j : p y j p y j for all y j Y j. 2. For every consumer, x s expendture-mnmal on the no worse than set. That s, p x x for all x n the set (x ). 3. (x, y ) A. Quas-equlbra are usually compettve equlbra because expendture mnmzaton s usually dual to preference maxmzaton; but not always, and we wll return to ths after the proof of the Second Welfare Theorem. It wll be useful to have some addtonal notaton for the statement and proof of the theorem. Let (x ) = {x X : x x } and (x ) = {x X : x x }. Theorem 7 (Second Welfare Theorem). Let (x, y ) be a Pareto Optmal allocaton for a prvate ownershp economy E wth the propertes that

12 12 1. for all, X s convex, 2. the sets (x ) are convex, 3. for some consumer k, (xk ) s convex and k s locally non-satated at xk, 4. Y s convex. Then there s a p such that (x, y, p ) s a quas-equlbrum for E. The pont of the Second Welfare Theorem s that compettve markets are unbased. There are no Pareto optmal allocatons whch are unreachable by compettve markets or cannot be sustaned by compettve markets after a sutable redstrbuton of ncome. That s to say, the only bas n the market s wealth. Proof. Defne the set G = =k (x )+ k (xk ) Y. Ths set s convex and ω s not n G because the allocaton s Pareto optmal. Thus there s a vector p such that p ω p g for all g G. Snce consumer k s locally non-satated, there s a sequence of consumpton plans xk n wth lmt x k, each element of whch s better for k than xk. Then for all n the vector gn = =k x + xk n j y j s n G, and the sequence g n converges to x j y j = ω. Thus p ω = nf{p g : g G}, and we can conclude that p x mnmzes expendture on the set (x ) for all = k. Ths s also true for consumer k because of local non-sataton. (Why?) Also, p y j s mnmal on Y j, whch s to say that y j s proft-maxmal for frm j at prce p. So far we have a feasble allocaton and a prce system such that frms proft maxmze and consumers mnmze expendture on ther no worse than sets. Ths s the natural dual equlbrum soluton to the Pareto optmzaton problem, and mportant enough that t has ts own name: quasequlbrum. However t s not yet a compettve equlbrum. We need to move from expendture mnmzaton to utlty maxmzaton. Ths s the content of the next lemma. Lemma 2. Suppose that the preference order has the property that for all x X, the set (x ) s open. Suppose at a prce p, x mnmzes expendture on (x ). Suppose too that there s an x 0 X such that px 0 < px. Then x s preference-maxmal on the set {x X : px px }. The exstence of x s referred to as the cheaper pont assumpton. Fgure 2 demonstrates what can go wrong wth the dualty between expendture mnmzaton and utlty maxmzaton when the cheaper pont assumpton does not hold. In ths fgure, the consumpton X s R 2 + n whch the open trangle wth vertces (0, 0), (1, 0) and (0, 1) has been removed. Prces and wealth are such that the

13 13 X p Fgure 2: No cheaper pont. budget set s the lower 45 degree lne. The ndcated consumpton bundle s expendture mnmzng on ts no worse than set, but t s not preference maxmal on the budget set. Proof of Lemma 2. Suppose contrary to the clam of the lemma that there was an x {x X : px px } such that x x. Snce x mnmzes expendture on the set (x ), t must be that px = px. Snce (x ) s open, there s an open heghborhood N contanng x such that for all v N, v x. Now consder the convex combnatons x(t) = tx0 + (1 t)x. For all t > 0, px(t) < px. For t postve but suffcently small, x(t) N, and so for these t, x(t) (x ). Ths contradcts the hypotheses that x s expendture-mnmzng. Fnally, we can clarfy the relatonshp between quas- and compettve equlbrum. Theorem 8. Suppose that (x, y, p ) s a quas-equlbrum for the prvate ownershp economy E. Suppose that for all consumers and for all x X, the set (x ) s open. Then (x, y, p ) s a compettve equlbrum wth transfers. Proof. Take w = p x. The theorem s then a consequence of the defnton of a quas-equlbrum and the precedng Lemma. One mght ask, s every quas-equlbrum allocaton Pareto-optmal? It clearly need not be, for exactly the reason llustrated n the pcture. What does t take to get a a quas-equlbrum allocaton to be Pareto-optmal? The cheaper pont assumpton and open better than sets; that s, precsely wwhen the quas-equlbrum s a compettve equlbrum.

14 14 4 A Calculus Approach to the Welfare Theorems It should be clear by now that the Welfare Theorems pose a dualty of sorts. Take fndng Pareto optma as a prmal problem. The shadow prces for the feasblty constrants are the compettve equlbrum prces. Ths s especally clear n the proof of the Second Welfare Theorem, whch dentfes compettve equlbra as a supportng hyperplane, part of the dual descrpton of G n the proofs of the last secton. Ths nterpretaton becomes clearer to most people when we move to the more famlar ground of constraned optmzaton and Lagrange (Kuhn-Tucker) multplers. The message s that equlbrum prces are shadow prces for the resource constrants n the Pareto optmzaton problem. Ths s most clearly seen n an exchange economy. Suppose that each of I consumers has preferences whch are represented by strctly concave, C 2 and strctly ncreasng utlty functons u 1,..., u I defned on X whch s convex and has non-empty nteror n R L. That s, D 2 u s negatve defnte and Du 0 on X. Suppose too each consumer has a strctly postve endowment. 4.1 Optmalty If x s a Pareto optmal allocaton, then there s no reallocaton that can ncrease the utlty of any consumer wthout decreasng the utlty of anyone else. let u (x ) = u. Then x solves the optmzaton problem on X : PO : max u 1 (x 1 ) s.t. u (x ) u for = 2,..., I, x = x. Snce the u are strctly ncreasng, the weak nequaltes can be assumed to be equaltes. Let us, for smplcty, consder an allocaton n whch each x s nteror to X. The frst order condtons are Du 1 (x 1 ) = λ ν Du (x ) = λ. From these condtons the usual equalty condtons for margnal rates of substtuton follow. These condtons, along wth the constrants, are suffcent for an allocaton to be Pareto optmal.

15 Equlbrum Now suppose an allocaton x 1,..., x I s a compettve equlbrum at prce vector p. Then x = ω, and for each the bundle x solves the optmzaton problem CE : max u (x ) s.t. px pω. Agan one can take the nequalty to be an equalty. The frst order condtons are Du (x ) = η p These too are suffcent because of the concavty assumptons. 4.3 The Welfare Theorems The proof of the welfare theorems amounts to showng: Frst Welfare Theorem If for all, x, η solves the frst order condtons for CE wth prces p, then x and λ = η 1 p, ν = η 1 /η solves the PO frst order condtons. Second Welfare Theorem If x, ν and λ solve PO, then takng ν 1 = 1, x, η = 1/ν and p = λ solve all the CE frst order condtons. Ths s smple algebra. Here s what one should take away from ths exercse. The shadow prce ratos on the resource constrants measure the relatve scarcty of the commodtes. And those prce ratos are exactly the compettve prce ratos. So n a compettve equlbrum, the equlbrum prce ndexes the scarcty of the commodtes. It s an nterestng exercse to carry ths exercse out wth producton.