Notes on General Equlbrum n an Exchange Economy Ted Bergstrom, Econ 20A, UCSB November 29, 206 From Demand Theory to Equlbrum Theory We have studed Marshallan demand functons for ratonal consumers, where D (p, m ) s the vector of commodtes demanded by consumer when the prce vector s p. In general, the ncomes of ndvduals depend on the prces of goods and servces that they have to sell. Therefore n the study of general equlbrum theory, we need to make ncomes depend on the prces of commodtes. Ths s ncely llustrated n the example of a pure exchange economy where there s no producton, but agents have ntal endowments of goods whch can they brng to market and trade wth each other. Each consumer ntally has some vector of endowments of goods. These goods are traded at compettve prces and n equlbrum the total demand for each good s equal to the supply of that good. A Pure Exchange Economy There are m consumers and n goods. Consumer has a utlty functon u (x ) where x s the bundle of goods consumed by consumer. In a compettve market, Consumer has an ntal endowment of goods whch s gven by the vector ω 0. Where p s the vector of prces for the n goods, consumer s budget constrant s px pω whch smply says that the value at prces p of what he consumes cannot exceed the value of hs endowment. Consumer chooses the consumpton vector D (p) that solves ths maxmzaton problem. Where x (p, m ) s s Marshallan demand curve, we
have D (p) = x (p, pω ). Let us denote s demand for good j by D j(p), whch s the jth component of the vector D (p). A pure exchange equlbrum occurs at a prce p such that total demand for each good equals total supply. Ths means that m m Dj( p) = ωj = = for all j =,... n. Ths vector equaton can be thought of as n smultaneous equatons, one for each good. Fndng a compettve equlbrum prce amounts to solvng these n equatons n n unknowns. There are two mportant facts that smplfy ths task f the number of commodtes s small. Homogenety and a numerare The frst s that the functons D (p) are all homogeneous of degree zero n prces and hence, so s D (p). To see ths, note that f you multply all prces by the same amount, you do not change the budget constrant (snce f px = pω, then t must also be that kpx = kpω for al k > 0. Therefore we can set one of our prces equal to and solve for the remanng prces. Snce any multple of ths prce vector would also be a compettve equlbrum, we lose no generalty n settng ths prce to. Walras Law and one Equalty for Free The second fact s a lttle more subtle. It turns out that f demand equals supply for all n goods other than the numerare, then demand equals supply for the numerare good as well. Ths means that to fnd equlbrum where there are n goods, we really only need to solve n equatons n n unknowns. Thus f n = 2, we only need to solve a sngle equaton. If n = 3, we stll only need to solve 2 equatons n 2 unknowns. To see why ths happens, we prove an equalty that s known as Walras Law. 2
so If all consumers are locally nonsatated, we know that pd (p) = pω and p D (p) = p ω or equvalently, m n p j (Dj(p) ωj) = 0. () = j= Ths equalty s preserved f we reverse the order of summaton, n whch case we have n m p j (Dj(p) ωj) = 0. (2) j= = Let us defne aggregate excess demand for good j as E j (p) = Then Equaton 2 can be wrtten as m ( ) D j (p) ωj. (3) = n p j E j (p) = 0. (4) j= Ths s the equaton commonly known as Walras Law. Equaton 4 mples that p j E j (p) = p k E k (p). (5) j k Let good k be the numerare. Suppose that at prce vector p, demand equals supply for all commodtes j k. Then E j ( p) = 0 for all j k. Therefore It follows from Equaton 5 that But p k =. Therefore E k ( p) = 0. p j E j ( p) = 0. (6) j k p k E k ( p) = 0. 3
Example There are m consumers and two goods. Consumer has utlty functon U (x, x 2 ) = x α x α 2 and endowment (ω, ω2). Let good be the numerare wth prce and let p be the prce of good 2. Then the demand functon of Consumer for good 2 s D 2(, p) = ( α ) p Aggregate excess demand for Good 2 s gven by ( ω + pω 2) = p ( α )ω + ( α )ω 2 (7) n ( E 2 (p) = D 2 (, p) ω2) = (8) At a compettve equlbrum prce p for good 2, t must be that E 2 ( p) = 0. From equatons 7 and 8 t follows that at a compettve equlbrum prce p, we have p m m m ( α )ω + ( α )ω2 = ω2 (9) = = = By rearrangng the terms of Equaton 9, we can solve for the equlbrum prce p whch s m= ( α )ω p = m= (0) α ω2 Usng Walras Law, we know that when excess demand s zero for good, t s also zero for good 2. Therefore where p s gven by Equaton 0, at prce vector ( p, ) we have demand equal to supply both for good and for good 2. In the specal case where preferences are dentcal, so that α = α for all we see that the soluton n Equaton 0 smplfes to p = ( α) α ω. () ω2 In ths case, the prce of good 2 s nversely proportonal to the rato of the supply of good 2 to the supply of good and s drectly proportonal to the rato of the Cobb-Douglas exponent on good 2 relatve to that on good. 4
Notce that ths would also be the soluton f there were only one consumer who had an ntal endowment of ω of good and ω 2 of good 2. In ths case, at the prce vector (, p) ths rch consumer would demand exactly the entre amount of each good that s avalable. Problems Problem There are m consumers, all of whom have dentcal homothetc utlty functons. Note that aggregate demand s the same as t would be f one consumer had all the utlty. So how can you fnd equlbrum prces? At what prces would ths consumer demand exactly the quanttes that are avalable n the endowment? Specal case. Suppose u (x,..., x n ) = n a j x α j α j= where α. Suppose that ntal endowment of consumer s gven by the vector ω = (ω,..., ωn). Fnd an explct soluton for a compettve equlbrum prce vector. Now fnd the quanttes of each good purchased by each consumer. (Hnt: Remember that ther utlty functons are dentcal and homothetc.) Problem 2 There are three commodtes n a pure exchange economy. Let good 3 be the numerare. There are m consumers. The total endowments of goods, 2, and 3, are gven respectvely by ω, ω 2, and ω 3. The aggregate demand functons for goods and 2 (when ω 3 s large enough) are gven as follows: D (p, p 2, ) = a b p + cp 2 D 2 (p, p 2, ) = a 2 + cp b 2 p 2 (2) where a > ω, a 2 > ω 2 and b b 2 > c 2. A) Compute compettve equlbrum prces. What do we mean by ω 3 s large enough? 5
B) Can you fnd ndvdual utlty functons for consumers such that aggregate demand takes ths form? Hnt: What f u (x, x 2, x 3) = x 3 + a x + a 2x 2 b 2 (x ) 2 b 2 2 (x 2) 2 + cx 2x 2 (3) Partal and General Equlbrum Comparatve Statcs Suppose that there are n commodtes n an exchange economy. For convenence, let commodty n be the numerare. Let p = ( p,..., p n ) be a compettve equlbrum prce vector. Suppose that the demand for Good s gven by the functon D ( p,... p n, α), where α s a parameter that shfts the demand curve. If the aggregate endowment of good s ω, then n compettve equlbrum t must be that D ( p,... p n, α) = ω (4) We are nterested n predctng the effects of a shft n the demand curve or of a change n the aggregate supply. Let us frst consder the partal equlbrum approach. If we knew the demand functon, what would be our predcton about the change n the equlbrum prce of good that would result from a change n the supply of good? The partal equlbrum approach s to assume that prces of all goods other than good are held constant, and to see what change n p s needed to reestablsh equlbrum f we change the supply, ω. To fnd ths, we dfferentate both sdes of equaton 4 wth respect to ω. When we do so, we fnd that D ( p,... p n, α) p dp dω = (5) From Equaton 5 we fnd our predcton of the effect of a change n endowment on the prce of good. dp dω = D ( p,... p n,α) (6) p Ths should be no surprse to anyone who has studed elementary economcs. Suppose that you draw an nverse demand curve for good curve 6
wth quantty of good on the horzontal axs, and prce of good on the vertcal axs, where the quantty correspondng to prce p s D (p, p 2... p n, α). The slope of ths curve s D ( p,... p n,α). p A vertcal supply lne drawn at ω wll ntersect ths curve at D ( p,... p n, α). If you move the supply curve by, the change n the prce wll be D ( p,... p n,α). p Now let us consder the general equlbrum soluton for the effect of a change n the supply of good. In general, a change n the prce of good wll change demand n some of the other markets. So we need to fnd changes n all n prces such that demand n market s changed by the amount of supply change and such that demand n all the other markets (where supply has not changed) s the same as t was before the prce change. Let us defne p(ω) to be the equlbrum prce vector f the vector of aggregate supples s ω. We recall that n equlbrum t must be true that for all commodtes =,..., n, D (p(ω)) = ω. Let us dfferentate both sdes of each of these n equatons wth respect to ω. We wll fnd that n j= and for = 2,..., n, n j= D (p (ω),..., p n (ω), ) p j D (p (ω),..., p n (ω), ) p j ( ) pj (ω) = (7) ( ) pj (ω) = 0 (8) We can wrte Equatons 7 and 8 as a matrx equaton of the form Mx = y where M s the n by n matrx whose jth entry s D (p (ω),..., p n (ω), ) p j, 7
x s the column vector whose elements are ( p... pn ), and y s the column vector whose frst element s and wth all other elements equal to 0. We want to solve for x whch s the vector of changes n each of the prces. If M has an nverse, ths s x = M y. But snce y s a vector whose frst element s and all other elements are zero, we see that the vector x s just the frst column of the matrx M. Example Consder a three good example lke that we consdered n Problem 2. The demand functons for goods and 2 D (p, p 2, ) = a b p + cp 2 D 2 (p, p 2, ) = a 2 + cp b 2 p 2 (9) If c > 0, the goods are substtutes. If c < 0, the goods are complements. The matrx of partal dervatves s M = ( b c c b 2 ) and so Therefore and M = ( b2 c b b 2 c 2 c b p (ω, ω 2 ) = b 2 b b 2 c 2 ) p 2 (ω, ω 2 ) = c b b 2 c 2. In the specal case where c = 0, these mply that and p (ω, ω 2 ) = b p 2 (ω, ω 2 ) = 0. 8
These are the partal equlbrum answers, whch wll dffer from the general equlbrum answers f there are cross effects. We see that whether the goods are substtutes or complements, the exstence of cross-effects amplfes the effect of supply changes. If the goods are substtutes, an ncrease n the supply of good wll drve the prce of good down, but a fall n the prce of good reduces demand for good 2 and ths wll requre a reducton n the prce of good 2 to clear market 2. But ths prce reducton n market 2 wll reduce demand n market and so the prce n market 2 wll have to be reduced further to equlbrate supply and demand n market. But ths means that the prce n c market 2 has to be reduced further. And so t goes...but f b b 2 c 2 > 0, ths process settles down. If the goods are complements, an ncrease n the supply of good drves the prce of good down, but ths ncreases the demand for ts complement, good 2. The ncrease n demand for good 2 means that the prce of good 2 has to ncrease to mantan equlbrum. n market 2. Ths reduces demand for good and so a further prce reducton s needed to acheve equlbrum n market. And so t goes... Problem 3 A pure exchange economy has N Type consumers and N 2 type 2 consumers. There are two goods, X and Y. Consumers of Type each have an ntal endowment of ω y unts of good Y and none of good X. Type 2 s each have an ntal endowment of ω x unts of good X and none of good Y. Each consumer of type has preferences over consumpton bundles (x, y) that are represented by the utlty functon Let Good X be the numerare. u (x, y) = x α y ( α ). A) Solve for the quantty of each good demanded by consumers of each type when the prce of good Y s p. Let Good X be the numerare. B) Solve for a competve equlbrum prce as a functon of the prarmeters N, N 2, α, α 2, ω and ω 2. C) Fnd an expresson for the utlty of each type of consumer n compettve equlbrum as a functon of these parameters. Comment on the qualtatve nature of the effect of these parameters on ndvdual welfare. Who gans and who loses from changes n the varous parameters? 9
Income effects and multple equlbra Ths example s nspred by a paper by Lloyd Shapley and Martn Shubk from the Journal of Poltcal Economy n 977. A more thorough treatment appears n Smple economes wth multple equlbra by Ted Bergstrom, Ken- Ich Shmomura and Takehko Yamato, found at http://escholarshp. org/uc/tem/6qv909xs#page-2 A Shapley-Shubk economy s a pure exchange economy wth two consumers and two goods X and Y. Both consumers have quas-lnear preferences, but ther preferences are lnear n dfferent goods. Each consumer s ntal endowment ncludes postve amounts only of the good n whch he has lnear utlty. Thus the utlty functons of consumers and 2 are: U (x, y ) = x + f (y ) U 2 (x 2, y 2 ) = y 2 + f 2 (x 2 ) (20) where f ( ) and f 2 ( ) are ncreasng, concave functons. The ntal endowment of consumer s (x 0, 0) and the ntal endowment of consumer 2 s (0, y 0 ). You may ask, but f there are just two consumers and each one s a monopolst of one of the goods, why should one thnk they would trade at compettve prces? Probably they would not. But suppose that the economy conssted of 000 people just lke consumer and 000 people just lke consumer 2. The compettve equlbrum prces and the equlbrum consumptons that we would fnd for each of our two consumers would be the same for each of the 2000 consumers n the bg economy. In ths economy each consumer has 999 compettors offerng the same products n the market. Let us consder a specal case of a Shapley-Shubk economy where the functons f and f 2 are quadratc and are mrror-mages of each other. In partcular, suppose that for some a > 0, for all y a. U (x, y) = x + ay 2 y2 U 2 (x, y) = y + ax 2 x2 for all x a. We can make good X the numerare by settng ts prce to one and solvng for the excess demand functons as a functon of the prce p of good Y. If 0
consumer demands postve amounts of both goods at prce p, t must be that consumer s margnal rate of substtuton between good Y and good X s equal to p. Ths gves us the equaton a y = p, where y s the amount of Y consumed by consumer. It follows that consumer s demand for good Y at prce p s D Y (p) = a p (2) If consumer 2 demands postve amounts of both goods, then t must be that consumer 2 s margnal rate of substtuton between good Y and good X s equal to p. Ths requres that a x 2 = p where x 2 s the amount of good 2 demanded by person 2. Therefore consumer 2 s demand for X s gven by D 2 X = a p. (22) To fnd consumer 2 s demand for Y, we make use of consumer 2 s budget constrant. Recall that consumer 2 s ntal holdngs vector s (0, y 0 ), so 2 s budget constrant s x 2 + py 2 = py 0. (23) From Equatons 22 and 23 t follows that D 2 Y (p) = y 0 p D2 X(p) = y 0 a p + p 2. (24) If p s a compettve equlbrum prce, excess demand for Y must be 0. Snce Consumer 2 s the only one who has a postve ntal endowment of good 2, the total supply of good 2 s y 0. Therefore 0 = E Y (p) = D Y (p) + D 2 Y (p) y 0 (25) Substtutng from equatons 2 and 24 nto equaton 25 we fnd that at a compettve equlbrum, E Y (p) = a p + y 0 a p + p 2 y0 = 0 (26)
Equaton 26 smplfes to E Y (p) = a p a p + p 2 = 0 (27) We can get some qualtatve nformaton about E Y by seeng what happens when p s small and when p s large. Note that lm p 0 E Y (p) = and that lm p E Y (p) =. Note also that E Y ( ) s a contnuous functon. Draw a dagram and you wll see that there must be at least one soluton. The symmetry of the problem suggests one possble soluton. It s easly verfed that E() = 0. Are there other solutons? (One trck that you could use s to note that f E () > 0, there must be more solutons besdes p =. Look at a graph to see why. For what values of a s E () > 0? ) For ths functon, there s a good drect way to fnd out whether there are other solutons and what they are. If we multply both sdes of equaton 27 by p 2, we see that Equaton 27 s satsfed only f p 3 ap 2 + ap = 0. (28) We see that ths s a cubc equaton, and as such wll have at most three solutons. We already know that one of the solutons s p =, so we know that equaton 28 can be factored to be expressed as p tmes a quadratc. A bt of fddlng (long dvson) shows that p 3 ap 2 + ap = (p ) ( p 2 + ( a)p + ) (29) Therefore the other two roots of the equaton p 3 ap 2 + ap = 0 are found by applyng the quadratc formula to the quadratc equaton ( p 2 + ( a)p + ) = 0. We see that ths expresson wll have two roots p and f and only f a > 3. p These are the solutons. Another trck that you could employ s to note that for p, t must be that wth ths excess demand functon, f E(p) = 0, then E(/p) = 0 and so any solutons other than p = come n recprocal pars. 2
Workng backwards s even easer. We can choose the parameter a to gve us three equlbra wth equlbrum prces p,, and / p, where p s any postve number we want. Suppose, for example, we want to have three solutons, /2,, and 2. We need to have ( p 2 + ( a)p + ) = 0 whch s equvalent to a = + p + /p. If p = /2, ths s the case when a = + 2 + 2 = 3 2 3