Chapter 12 General Equilibrium and Welfare


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1 Chater 2 General Equilibrium and Welfare U to this oint we have dealt with onl one market at a time; Partial Equilibrium Models. ogic suggests that markets are highl interconnected. What haens in one market affects man others. In the etreme, the adjustments in other markets ma come full circle and affect the original market. Nothing haens in isolation. The whole econom is simultaneousl related. We need a model that allows the interconnectedness of markets in contrast to Partial Equilibrium Models. We need a simultaneous ersective. The General Equilibrium Model outlined below hels us develo hotheses about the real world. The model is an abstraction from realit that hels us understand the real world.
2 Perfectl Cometitive Price Sstem Assumed: A set of erfectl cometitive markets eists for inuts and oututs. Each market has an equilibrium rice. Equilibrium rices clear the markets. Zero transaction costs. Perfect knowledge. No transortation costs. The aw of One Price Each good and inut obes this law, which is that the good or inut is homogeneous and sells for the same rice to everone in all markets. Arbitrage assures one rice. All consumers are rational rice takers both as buers of goods and sellers of inuts (eg., labor). Each market has a ver large number of utilitmaimizing consumers for an one good or inut. All firms are rational rice takers in both inut and outut markets. Each market has a ver large number of rofitmaimizing firms roducing each homogeneous good or inut.
3 Wage W P If all markets follow the above assumtions, we can analze general equilibrium. For eamle: P w S P More income Preference for c 3 S sent on wine and D D wine increase. less on cloth Wine Wage D D Cloth W 4 W S More wine roduced ess cloth roduced 2 D D (P c W w W ) D (P W W w W ) D Picker labor (inut for wine) Weaver labor (inut for cloth) But the stor could continue if unemloed weavers are trained to be grae ickers, increasing the sul of ickers, and because more ickers are available, the wage rate for ickers declines, increasing the sul of wine. Also a decrease in the wage rate for weavers might decrease the demand for wine and cloth, while an increase in the wage rate for ickers might increase the demand for wine and cloth. And so on, and so on, etc.
4 K Isoquant 2 X 5 B C A 2 3 O 4 K Grahical Model of General Equilibrium (GE) Demand in a GE model is reresented b societ s indifference curves for two goods (later). Sul in a GE model for two goods ( and ) and two inuts (K and ) begins with use of the Edgeworth Bo. O An oint in the bo reresents a combination of K and used to roduce each good ( and ), given technolog. Each oint reresents full emloment of K and, and the amounts of K and are fied. An isoquant shows a given quantit of a good roduced and the different combinations of K and used to roduce the given quantit. Each oint is on an isoquant for and for. Points on the line, O O, are technicall efficient in that movements awa from the line involve less of or or less of both. Point A is inefficient. Points down the 3 isoquant from oint C toward oint A give less without changing, while moving down the 2 isoquant from oint B toward oint A gives less without changing. Movements from a oint on O O between oints B and C toward oint A give less and less.
5 No Production Possibilit Frontier (PPF) On the efficient line, O O, the RTS K are equal for and. The efficient line O O where the RTS = RTS shows the maimum amount of (or ) that can be roduced for the fied amount of (or ). Thus, we can take the information from the O O line and construct a roduction ossibilities frontier (PPF). The PPF shows combinations of two RPT small O oututs that can be roduced with fied 5 B quantities of inuts if the inuts are 4 A C full and efficientl emloed. We got 3 2 this PPF b transferring oints along the O RPT large O line onto this grah. Points toward the origin from the PPF (eg., No oint A) are inefficient (off O O ) O Rate of Product Transformation (RPT ) is the negative of the sloe of the PPF and shows how can be substituted for with efficient use of full emloed inuts and total inut quantities and technolog constant. The RPT equals d/d (the negative of the sloe of the PPF). A curved PPF gives an increasing RPT ; a concave PPF.
6 The total cost function for societ might be C = C(,), which would be constant along a PPF because of full emloment and fied inut quantities. C C Thus, the total differential dc d d set to 0. Maniulating, C C d d C C d d RPT The RPT is a measure of the relative marginal costs of the two goods. For eamle, if the RPT equals 2, equals 2, so the econom must give u 2 to get. Oortunit Cost and the RPT As increases, increases and as decreases, decreases, so the RPT increases, imling that the PPF is concave! If the RPT increases, the oortunit cost of increases. As we get more, the cost of getting an additional unit of increases in terms of the amount of that must be sacrificed. Wh would the PPF be concave?. One could argue that each good is roduced under decreasing returns to scale. Then the of each good rises as its quantit rises. As increases and decreases, / increases. However, diminishing returns to scale are not ubiquitous.
7 2. If some units of K are better suited to than other units of K, then would increase as increases (as more costl K was used to roduce ) and would decrease as decreased (less costl K to roduce ) and vice versa. For eamle, if a farmer increases cotton roduction and decreases corn roduction, more of the unroductive land will be used to roduce cotton, increasing the of cotton and less unroductive land will be used to roduce corn, decreasing the of corn. However, we have assumed homogeneous inuts, but this elanation could be a factor in the real world. 3. Concavit will eist if the two goods use K and in different roortions at an oint on O O. This occurs if O O is bowed above or below the diagonal. If bowed above, is K intensive relative to. If bowed below the diagonal, is K intensive relative to.
8 K B A B B O K B K A A is more Kintensive and is more intensive. K A O A In the bo above, the ratio of K/ is greater for than for at ever oint on O O. At oint A, the K A / A for is large and the K A / A for is medium. At oint B, the K B / B for is medium and the K B / B for is small. In going from A to B, the K/ declines for both and, causing to increase and to decrease, so / = RPT increases. If O O is linear, the PPF is also linear. The increases as increases because an additional unit of roduction causes the worst suited inut () to increase at a faster rate than the best suited inut (K). The oosite argument is used to elain wh decreases as decreases. B K
9 Equilibrium Price Determination in the GE Model Combine PPF (reresenting sul) with Indifference Curves (reresenting demand). Generalize the discussion to all firms. Profitmaimizing firms in the absence of trade will choose given the rice O ratio and RPT ( / = RPT = / ). Isorevenue line, R = + Sloe =  / 0 O or RPT $ AC / = 0 The isorevenue line shown in the grah is the highest attainable given PPF,, and. Budget line; I=E= + ; sloe =  /. 0 R=I=E U Because R = +, and revenue is eenditures (=income) to consumers, this is the budget line for consumers. Also remember that consumers will maimize utilit b reaching highest indifference curve where / = MU /MU = MRS.
10 Because we assume the same / eists for roducers and consumers, we can combine the grahs to show a general equilibrium. At D, MRS Budget line and Isorevenue line Sloe =  / E D d d Ind MU MU At D, the highest U is achieved given PPF. The riceratio line is the onl one consistent with oint D. d d PPF RPT The ratio of s equals the ratio of MU s. and identif oint D along O O in the Edgeworth Bo. All inuts are used efficientl and full emloed. Q demanded = Q sulied = PPF U Q demanded = Q sulied = Point D is the equilibrium because other oints (sa E) will induce tendencies toward D.
11 ES If ' / ' eisted, firms would choose E, while consumers would choose F. MRS = RPT = ' / ', but ecess demand for ( 2 ) and ecess sul of (  2 ) eist, so these rices are not general equilibrium rices. The would not be stable; would increase and would decrease, rotating isorevenue line until it is tangent to Indifference curve U and the PPF at D. Equilibrium for this twogood, twoinut econom is established at D where Q D = Q S, Q D = Q S, and MRS = RPT = /. 2 0 E D ED F 2 U U 3 U 2 Sloe =  ' / ' This analsis is etendable to man goods where firms are maimizing rofit and consumers are maimizing utilit. Markets are cleared b changes in and. Prices are the communicators and equilibrator! The essence of equilibrium is that Q S = Q D (for both goods) and MRS = RPT = / (two conditions if no trade).
12 ED 2 3 The two basic forces determining the equilibrium rice ratio, Q, and Q are: The reference sstem of consumers (indifference curves, demand) and roduction technologies of the firms for and (isoquants, sul). Assume references shift awa from toward. U shows a R reference for. U and U 2 show a shift awa from. Point D is no longer an equilibrium oint because at R rice ratio, consumers want oint F (Q D < Q S and Q D > Q S ). More E 2 F ES This will U 2 U 3 ess D cause to decrease and to increase. Equalities of / = MRS and / = PPF will change until E is reached. The result will be more at a higher rice and less at a lower rice. Consumers R 2 desires were accommodated given the PPF U constraint. This eamle assumes no trade. With trade an econom can be at oint F, with higher utilit. We could show a change in roduction technolog as a shift in the PPF curve and trace changes to new equilibrium rices, roduction, and consumtion. For eamle, assume a change in technolog favoring roduction, would decline, so the RPT = / would decline, causing / to decline also. Prices are determined jointl b utilit functions and roduction functions. Grah this situation on our own.
13 Eistence of General Equilibrium Does a set of rices eist for which all markets are in equilibrium simultaneousl? Assume a set of goods, i =,2,3,,n Sul of each good is fied at S i and distributed in some wa among individuals Demand for each good is deendent on the rices of all goods: D i D ( i, 2,..., n ) D (P) i wherep reresents a set of n rices. * D(P ) S, i...n Walras wanted to rove that P * eists for which i i State this in terms of ecess demand because ED (+ or ) drives rice changes: ED i (P) D (P) S i i for all i, 2,...,n The question is whether a set of equilibrium P * eists for which ED (P * ) 0 i for all i,2,...,n These n ED functions are not indeendent of each other. The are related b n ED (P) 0 For a cit ark: If i = 0, ED i < 0(E. Su.), i i If i > 0, ED i = 0 (not free; Walras aw i(see Footnote, Page 350 for roof) rice rations ark use). For all other goods, i>0 and Edi<0 or i>0 and Edi>0.
14 Walras aw sas that the i th good s ED times its rice summed over all goods is zero, ie., total value of ecess demand is zero. When all goods are summed, there can be no ositive or negative value of ED. For equilibrium, Walras aw is obvious because ED i = 0 for all i, but this law is true for equilibrium and nonequilibrium rices (as in the grah two slides earlier). The sstem of equations, ED i (P) 0, i,2,...,n, for all i can be solved simultaneousl because n of these are indeendent and can be solved for n equilibrium rices (big job; some equations are nonlinear). Walras solved this sstem of equations iterativel to aroimate equilibrium rices and roved that equilibrium rices do eist. Because these equations can be solved for onl n equations, relative rices are imortant, not absolute rices. Must introduce mone to look at actual rices. Mone doesn t change the rice ratio at which goods trade. It just acts as: A medium of echange, a store of value, and a unit of account.
15 Efficienc of Perfect Cometition The free working of all connections and resulting adjustments in the erfectl cometitive economic world aear ver chaotic. It ma aear that no good end could come from so much aarent chaos. Surel, careful lanning can bring about a better allocation of resources! Adam Smith s invisible hand concet ointed out that self interest and freel cometitive markets would lead to maimum welfare for societ as a whole. Smith s ideas lead to the notion that efficient allocation of resources would result from cometitive ricing of those resources.
16 Pareto Efficienc An allocation of resources is Pareto efficient if it is not ossible (through further reallocation) to make one erson better off without making someone else worse off. Inefficienc is indicated if unambiguous imrovement in welfare can be made. A. Efficienc in Production or Technical Efficienc (no rices et) It occurs when an econom is on its PPF (not just each firm, but all firms together). In Pareto terms, roduction efficienc (or technical efficienc) occurs when no more of one good can be roduced without reducing roduction of another good. Thus, an oint inside the econom s PPF is inefficient. Technical efficienc is a necessar condition for overall economic efficienc.
17 Three conditions must hold to have technical efficienc:. Efficient allocation of inuts within each firm ( firm, 2 inuts, 2 oututs). We have shown that an efficient allocation of inuts between oututs would occur where the RTSs are the same for all oututs. Thus, a firm roducing two oututs with two inuts would be efficient when the sloes of the isoquants are the same (RTS = RTS ). All oints along the O O line in the Edgeworth bo are efficient in roduction. On this line, a firm cannot roduce more without reducing roduction. 3 2 O Doesn t sa anthing about K how much and to roduce 3 or how much K and to use to 2 roduce each outut. O K
18 Efficient allocation of inuts within each firm Mathematicall: If the firm s roduction functions are = f(k, ) and = g(k, ) and full emloment is assumed ( K and) then g(k K, ). Efficienc requires that be as large as ossible for an given, sa. Could Maimize: f(k, ) f(k, FOC: ) K ) λ( g(k K, f λg K K 0, 3) g(k K, ) 0 λ f Divide FOC 2 b to get: Allocation Rule : Efficient allocation of a fied quantit of inuts within a firm occurs where all inuts are full emloed and the RTS between the inuts is the same for all oututs. f K g g st : )) g(k, 2) f λg K K ) 0 g g 0 Ratios of are equal for all oututs. RTS K RTSK K Same as shown on 0 0 in Edgeworth Bo (roduction). The firm is on its PPF. Thus, the tangencies of the isoquants are where is maimized given.
19 2. Efficient allocation of inuts among firms (2 firms, 2 inuts, outut). Inuts should be allocated to the firms where the are used most efficientl. Suose two firms roduce the same good with roduction functions: f(k, ), f2(k, ) and total quantities of inuts are K and : f(k, ) 2 Ma f2(k,2) St: K K K Substitute the constraints into the objective function: K 2 2 Ma f(k, ) f2( K K, ) X FOC: K X f f 2 f f 2 0 f K K K K 2 K f f 2 f f 2 f 0 ( ) 2 2 and 2 f2 K2 f2 Changes in K must have the same effect as changes in K 2 ecet oosite in sign, because K + K 2 = K. Allocation Rule 2: For efficient roduction, inuts should be allocated among firms such that the of each inut in the roduction of a given good is the same no matter which firm roduces that good. In roduction of a single good, s of each inut are the same among firms. 2 K 2 K 2
20 3. Efficient choice of oututs b firms (2 firms, constant inuts, 2 oututs). This refers to both what and how much each firm should roduce (same question) secialization. The RPT for two goods, each roduced b two firms, must be equal between the two firms (RPT = RPT 2 ) Firm d d A B d d Firm 2 B 2 d.5 d A 2 d 2 d Could have a corner solution If two firms roduce equal amounts of two goods and if the RPT for Firm is d/d = at A, while for Firm 2 it is 2 at A 2, then the first firm should roduce more and the second firm should roduce more until RPT s are equal (somewhere between and 2; ossibl at.5 in grah). At oints B (B and B 2 ), RPT = RPT 2. Firms should do what the are good at doing. If both firms are roducing 50 and 50 at A and A 2 (total is 00 of each good), Firm could move to B and Firm 2 to B 2, resulting in totals of 0 and 05. 0
21 This rule also means that. 2 The s for and are not necessaril equal between firms; just the ratios are equal between firms. Either firm can have lower s for both goods and societ still gain from firms secializing in roduction. This notion is used as the concet of Comarative Advantage in international trade. Allocation Rule 3: Two firms roducing the same goods must oerate where their RPT s are equal on their PPF s. 2 These three allocation rules mean the econom is somewhere on its PPF (ie., technical efficienc). Consumers have the oortunit to get the most out of the econom s resources.
22 Efficienc in Product Mi Mathematicall Assume two goods, and, and that references are determined b societ s references: U = U(,) is societ s utilit function, T(,) = 0 is the imlicit roduction ossibilities frontier for all firms. The roblem is to maimize U subject to T. The agrangian is: U(, ) FOC U λt U λ T(,) 0 λ(t(,)) λt 0 0 Divide st FOC b 2 nd FOC to get U U T T The righthand side of this eression is T, T which is the amount of given u for a small increase in (ie., the oortunit cost of in terms of ). Negative sloe of the indifference curve (U fied) d d Ind MRS MU MU RPT d d PPF Negative sloe of PPF (resources fied)
23 0 D E PPF econom Societ s references U U 2 U 3 d d Ind MRS MU MU RPT d d These conditions will give Pareto Efficienc at oint D because at oints other than D, the consumer or roducer can be made better off without making others worse off. Because this is the GE solution roduced earlier b free markets and rice movements, this shows that the free market GE result is Pareto Efficient under conditions of erfect cometition. We did not need rices to show Pareto Efficienc. We will see, however, that erfectl cometitive ricing brings about Pareto Efficienc b causing societ to reach oint D. PPF
24 Summar of Pareto Efficienc for the Econom A. Criteria for roduction (technical) efficienc (no rices et). Efficient choice of inuts within each firm. (Rule ) firm, 2 inuts, 2 oututs K K Efficient allocation of inuts among firms. (Rule 2) 2 firms, 2 inuts, outut Efficient choice of oututs b firms. (Rule 3) 2 firms, fied inuts, 2 oututs B. Criteria for efficienc in roduct mi (no rices et). Production (technical) efficienc (Rules 3), and Ind PPF d MU d d RTS RPT K MRS K RPT RTS 2 K 2 MU K and RPT d
25 C. Cometitive Prices and Efficienc (now add rices) The First Theorem of Welfare Economics If all economic agents (roducers and consumers) are considered together, a Pareto efficient allocation of resources siml means that the rate of tradeoff between an two goods or inuts is the same for all agents. This rate of tradeoff will equal the ratio of the two goods or inuts rices in a erfectl cometitive situation. All agents are rice takers and the all face the same rices. Thus, all tradeoff rates will be equalized. This is the First Theorem of Welfare Economics.
26 A. Technical (or Production) Efficienc firm, 2 goods, 2 inuts K. Allocation Rule said that the RTS between an two inuts used b a single firm must be the same for all goods roduced b the firm. But if markets for inuts are erfectl cometitive, the RTS K for each outut will equal a common ratio of inut rices (w/v), causing the firm s RTSs of for K to be equal for all oututs. Thus, the RTSs of for K will be the same across all oututs if the firm maimizes rofit under erfect cometition. Isoquant sloes =  RTS K X Y RTS K For rofit maimization, K dk d w v dk d K RTS K Isocost sloe = w/v.
27 2. Allocation Rule 2 said that all firms that use a given inut to roduce a given outut will have the same for that inut in roducing that outut. This condition will hold because in erfect cometition all firms will face the same inut and outut rices. Thus, when each rofitmaimizing firm equates its outut P (= MR) times the of the inut (MRP) to the MIC of the inut (= w or v), each firm s of the inut must equal the for all other firms. This will be true for each inut that the firm uses to roduce the given outut because w (or v) and P are common to all firms. MR MIC MR MIC K All firms face w/, so the s ( ) w and (K ) v of labor will be equal across all rofit maimizing firms w v and roducing. Same for v/p. Firm w K Firm2 and Firm K v Firm2 K
28 3. Allocation Rule 3 said that the RPT for an two goods will be the same for all firms roducing the two goods (given inut levels). This condition holds in erfect cometition because the RPT = / and each firm will roduce where = and =. Thus, because and are common to all firms, so are and, and so are /, and so are RPT for all firms. Firms & 2 Firm Firm 2 = = Sloe = All firms face and, so RPT for all firms. Thus, self interest (rofitmaimizing behavior) combined with erfectl cometitive markets will lead to efficienc in roduction, meaning the econom is somewhere on its PPF. Prices act as signals to rofitmaimizing decisionmakers to bring about efficienc in roduction for the econom. RPT F Sloe = RPT F2
29 B. Efficienc in Product Mi Because outut rices to roducers are costs of goods to consumers (and these are the same for all roducers and consumers), the MRS common to all consumers will equal the RPT common to all roducers for an two goods. Again, equilibrium rices assure that Q D = Q S for all goods individuall. Efficienc is assured because all consumers and roducers face the same rices! MRS = P /P = RPT. D PPF Cometitive ricing brings about the GE solution for the whole econom RPT = MRS! We reviousl demonstrated that U Sloe oint D is a stable equilibrium. Tendenc toward equilibrium is assured b rices, ecess demand, and ecess sul. The GE is the result of Adam Smith s Invisible Hand.
30 Conclusions about Cometitive Pricing and Efficienc Given the assumtions of erfect cometition, it is not the ublic sirit or kindheartedness of roducers nor the directions of governments that lead to efficienc. Rather, it is self interest and cometition that lead to efficienc through rice signals that carr information about consumer references and roducer technologies, given available resources. Because erfect cometition leads to erfect efficienc, and intervention is likel to be counter efficient (sends wrong rice signals). Selfinterest (UMa and πma) and cometition make the world a better lace to live! If onl the markets of our world were erfectl cometitive! In general, the are not. Effects of Deartures from Cometition Three tes of deartures from cometition eist: ) Imerfect cometition, 2) Eternalities, and 3) Public goods.
31 Imerfect Cometition occurs when agents have ower to noticeabl affect rice. Thus, the roducer realizes that as his/her outut increases, * market rice declines causing MR <. The roducer equates to MR (< ), which causes less than otimal X to be roduced. Sloe RPT B * MR Sloe C MRS at B MR U U 0 RPT at B This means that Pareto otimizing equalit is lost. MRS * Sloe * MU RPT MRS at C MU Antime we have a divergence between MR and the rice sstem will not lead to Pareto efficienc because the communication mechanism is fault! = MR * and MR MR If has a erfectl cometitive market but does not, too much and too little will be roduced comared with the cometitive equilibrium at oint C. Rather, equilibrium B will occur and utilit will be onl U 0 rather than the U that could have been attained with the available resources and technolog (PPF). The econom is still on its PPF, so roduction is efficient, but the roduct mi is not efficient. Market rices did not lead to Pareto efficienc for the econom. *
32 Eternalities Eternalities occur when not all costs or benefits of an activit are reflected in the market rice. Pollution b a manufacturer is a common eamle. Pollution is an effect not accounted for in the rice sstem. Eternalities occur when the benefits or costs to societ diverge from the rivate benefits and costs of decisionmakers (economic agents). A cost to societ is left out of the firm s curve, so does not reflect the total cost of resources used to roduce the last unit of outut. Or a benefit to societ is left out of the good s market rice. MR or P does not reflect total satisfaction to societ of the last unit consumed. Societ s RPT the firm s RPT and/or societ s MRS the consumer s MRS.
33 Public Goods Goods for which the roduction or urchase b one agent makes the good (or benefits from it) available to others on a noneclusive basis. Thus, each otential roducer or consumer waits for someone else to a for the good so the can benefit without aing (freerider roblem). Eamles are est control, national defense, light houses, and education. These goods will be underroduced b the free market and are usuall rovided b government.
34 Distributional Problems Distribution of the benefits of efficient roduction and roduct mi ma be regarded as unfair. Nothing in our model assures fairness. Definitions of fairness var, but an econom can have Pareto efficienc where a few individuals are wealth and wasteful, while others freeze and starve. Attemts to imrove distribution have met with mied success and have had man undesirable sideeffects (eg., Y 0 A Smith food aid to less develoed countries). Contract Curve Jones 0 X At oint A on the Contract Curve, Smith is oor and Jones is rich, but MRS XY Smith = MRS XY Jones. If the econom is on its PPF at oint A, Pareto efficienc eists in the econom.
35 U J A Echange with Initial Endowments U S A 0 Smith M M 2 A Contract Curve Jones 0 In this case, initial endowments are reresented b oint A. Initial endowments favor Jones (Jones on higher indifference curve than Smith). Smith and Jones can both move to higher indifference curves if the trade goods and until the reach the contract curve between M and M2. Even though Jones is rich he/she will not trade to reduce his/her utilit b reaching the contract curve to the right of M 2 and Smith will not trade to reduce utilit b reaching the contract curve to the left of M. Therefore, because of differences in initial endowments, Smith will remain oor and Jones will remain rich even after the trade in resonse to cometitive rices (assumes Jones receives no utilit from giving to Smith). The Second Theorem of Welfare Economics states that an desired distribution of welfare among the individuals in an econom can be achieved in an efficient manner through cometitive ricing if initial endowments are adjusted aroriatel. Economists frequentl argue for cometitive ricing to make the economic ie as large as ossible, and then adjust the resulting distribution to be fair though the use of lumsum transfers.
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