Lecture #3 General equilibrium
Partial equilibrium equality of demand and supply in a single market (assumption: actions in one market do not influence, or have negligible influence on other markets) General equilibrium adjustment of demand and supply in all markets at the same time (including interactions between markets)
Example two interdependent markets; adjustment to equilibrium Goods are substitutes: DVD movies (rental stores) shows (cinema tickets) The change in price in one market will influence the other. Let: P ticket = 20 PLN (zloty) P DVD = 10 PLN The government imposes a tax of 2 PLN per cinema ticket.
Example two interdependent markets; adjustment to equilibrium Tax => decrease in supply Increase in the price for cinema tickets => Increase in demand for DVDs P P S* M S V S M 21 PLN 20 PLN 12 PLN 10 PLN D V D M D V Q k Q k Cinema tickets Q V Q V DVDs
Example two interdependent markets; adjustment to equilibrium P Increase in the price of DVDs => increase in demand for cinema tickets and in the price of tickets P Influence of one market on the other will continue to take place up to the point when general equilibrium is reached 23 PLN 22 PLN 21 PLN S* M S M 13 PLN 12 PLN S V 20 PLN D* M 10 PLN D* V D V D M D M D V Q k Q k Q* k Q k Cinema tickets Q V Q V Q* V DVDs
Example two interdependent markets; adjustment to equilibrium With no analysis of interactions between the markets (as in the limited case of partial equilibrium analysis) the influence of the tax on the market for cinema movies will be underestimated. For a similar situation but in the market for goods which are complements this influence would be overestimated.
MARGINAL UTILITY, MARGINAL RATE OF SUBSTITUTION Marginal utility (MU) change in total utility resulting from the change in consumption of a given good by 1 unit. MM X = U X Marginal rate of substitution (MRS) a measure indicating how many units of one good a consumer is willing to forego for an additional unit of a second good, maintaining his/her utility at the same level. MMM xx = Y X = MM x MM y
MRS FOR DIFFERENT UTILITY FUNCTIONS Cobb-Douglas utility function normal goods U = Ax a y b => MMM xx = a b y x Linear utility functions perfect substitutes U = ax + by => MMM xx = a b Leontief utility function complements U = min{ax,by} => no substitution between goods
UTILITY MAXIMIZATION For decreasing MRS (convex indifference curve) we have: mmm U MMSSS = P x P y MRS xy slope of the indifference curve -P x /P y slope of the budget line y m/py y* Optimum (x*,y*) tangency point of the budget line and an indifference curve x* m/px x
The pure exchange model (only final goods, only consumers) Consumers have an endowment of goods at their disposal and can exchange these goods between themselves (we omit the production process).
The pure exchange model (only final goods, only consumers) Initial endowment of consumer i (initial allocation): w i1,w i2,...,w in Gross demand of consumer i (final allocation quantity of the good, which the consumer wants to consume): x i1,x i2,...,x in Excess (net) demand of consumer i (quantity of the good, which the consumer wants to buy; the difference between the quantity that he/she wants to consume and his/her initial endowment): z i1 = x i1 - w i1, z i2 = x i2 - w i2,..., z in = x in - w in where i = (1, k) denotes consumers, n the number of markets (products); one of the goods can be labor a resource at the disposal of each consumer
Feasible allocation An allocation satisfying the following set of equations: x 11 +x 21 +...+x k1 = w 11 +w 21 +...+w k1, x 12 +x 22 +...+x k2 = w 12 +w 22 +...+w k2,... x 1n +x 2n +...+x kn = w 1n +w 2n +...+w kn
The case of 2 goods and 2 consumers To maximally simplify the analysis we consider only 2 goods. (This approach is more general than one could think the second good may represent all other goods.)
Edgeworth box Analysis of feasible allocations for the case when k = n = 2 (putting two coordinate systems used for studying consumer choice together: box width = w 11 + w 21, box height = w 12 + w 22 ; the second coordinate system is rotated by 180 )
Pure exchange Edgeworth box X B 6Y 10X 3X Consumer B Y B Y A 1Y Consumer A X A 7X P 10X 5Y 6Y 15
Pure exchange Edgeworth box 10X 6Y B Benefits from exchange In point P: MRS A MRS B Each point within the grey area is more beneficial (advantageous) for both consumers A P U A = const U B = const 10X 6Y
Pure exchange Edgeworth box 10X 6Y In point S both R MRS daje are wyższą equal uż. the ale allocation wciąż nieefektywny is efficient B T A T is also an efficient solution (bargaining power) S U B 3 R U B 2 P U B 1 U 3 A U 2 A U 1 A 10X 6Y
Pareto efficient allocation A Pareto efficient allocation it is not possible to improve the situation of any participant to the exchange without worsening the situation of another one (mutually advantageous exchange is not feasible). Within the Edgeworth box there are numerous Pareto efficient allocations (on the highest indifference curve of one consumer while on a given indifference curve of the other consumer). The set of all Pareto efficient points is called the Pareto set or the contract curve. For each point on the contract curve MRS xy A = MRS xyb.
Contract curve Y B G F E A X
Equilibrium in a purely competitive market Many potential buyers and sellers If buyers are not satisfied by the terms of exchange offered by one of the sellers, they can turn to other sellers.
Walras Law (general formulation and proof will follow) Excess demand (here: for good 1): z 1 (p 1,p 2 )=x 11 (p 1,p 2 )+x 21 (p 1,p 2 )-w 11 -w 21 Walras Law: The value of the aggregate excess demand is identically zero, p 1 z 1 (p 1,p 2 ) + p 2 z 2 (p 1,p 2 ) 0
Corollary from Walras Law If demand is equal to supply in one market, the same must be true of the other market. If p 1 z 1 (p* 1,p* 2 ) + p 2 z 2 (p* 1,p* 2 ) = 0 then from z 1 (p* 1,p* 2 ) = 0 it follows that, as long as p 2 >0, z 2 (p* 1,p* 2 ) = 0
The equilibrium In a purely competitive market with consumers characterized by convex indifference curves the equilibrium will be found in the Edgeworth box in the tangency point of the indifference curves. The slope of the tangent line in this point is equal (in absolute value) to the equilibrium price ratio p* 1 /p* 2.
The equilibrium is Pareto efficient Based on the latter we can conclude that the equilibrium is efficient: tangent convex indifference curves cannot intersect, therefore Pareto improvement is not feasible.
Adjustment to the market equilibrium Market prices of goods (P x and P y ) will change/adjust until the point when the following conditions are satisfied: A (1) X MRS XY = - (2) PY P B XY MRS = - P P X Y (3) The sum of excess demand is zero.
Equilibrium in the consumer market 10x B 6y P P x /P y S U A 2 A U A 1 A U B 2 U B 1 P 10x 6y
In equilibrium at given prices the demanded quantity of a good equals its supplied quantity in each market. Not every price set for which: -P x /P y = MRS xy A = MRS xy B guarantees an equilibrium Disequilibrium is not permanent. D x >S x => increase of P x D x <S x => decrease of P x
Algebraic example Assume the Cobb-Douglas utility function: and similarly for B. u A (x 1 A, x 2 A) = (x 1 A) α (x 2 A) 1-α This implies the demand function: x 1 A(p 1,p 2,m A ) = αm a /p 1 and similarly for the second good and for B. The monetary value of A s endowment is: m A = p 1 ω 1 A + p 2 ω 2 A This implies the aggregate excess demand: z 1 (p 1,p 2 ) = (p 1 ω 1 A + p 2 ω 2 A)α/p 1 + (p 1 ω 1 B + p 2 ω 2 B)β/p 1 - ω 1 A - ω 1 B
Algebraic example Assuming price p 2 = 1 (numeraire) and setting excess demand of one good equal to zero we obtain: aw 2 A + bw 2 B p * = 1 (1 -a) w 1 1 A + (1 - b ) w B
First Theorem of Welfare Economics If (p*,x*) is the market equilibrium point and: - all market participants are price takers - all transactions are voluntary, - preferences satisfy the condition of local nonsatiation then x* is a Pareto efficient allocation.
Proof for pure exchange Let k = n = 2. Assume that (p*, x*) is an equilibrium, however x* is not a Pareto efficient allocation. Therefore, there exists an allocation x, which is advantageous for one consumer, while not being worse (disadvantageous) for the other one. x 11 *+x 21 * = x 11 +x 21 = w 11 +w 21, x 12 *+x 22 * = x 12 +x 22 = w 12 +w 22, however: [x 11, x 12 ] [x 11 *, x 12 *], [x 21, x 22 ] [x 21 *, x 22 *]. If given the set of prices p* the second consumer did not choose the preferred bundle [x 21 ', x 22 '] but bundle [x 21 *,x 22 *] instead, this means that the first bundle was not affordable: p 1 *x 21 + p 2 *x 22 > p 1 *x 21 * + p 2 *x 22 *. For consumer 1: p 1 *x 11 + p 2 *x 12 p 1 *x 11 * + p 2 *x 12 *. Adding up the two inequalities: p 1 *x 21 + p 2 *x 22 + p 1 *x 11 + p 2 *x 12 > p 1 *x 21 * + p 2 *x 22 * + p 1 *x 11 * + p 2 *x 12 * p 1 *(x 11 + x 21 ) + p 2 *(x 12 + x 22 ) > p 1 *(x 11 * + x 21 *) + p 2 *(x 12 * + x 22 *). At the same time, however, both sides must equal: p 1 *(w 11 +w 21 )+p 2 *(w 12 +w 22 ) what contradicts the inequalities. Therefore, the equilibrium is a Pareto optimum.
Consumer s equilibrium is difficult to reach if all markets are not purely competitive. An efficient allocation may be achieved by central planning However, market solutions are preferred as then consumers and producers are able to better specify their preferences and production possibilities.
Efficiency and fairness While we may have many efficient allocations, some may be more fair than others. What is a fair allocation?
Utility (possibility) frontier U of Kate O J H E F L G O J Jan zero U O K Kate zero U E, F, G points on the contract curve H inefficient allocation (ones in the shaded area are advantageous) L unfeasible allocation O K U of Jan
Second Theorem on Welfare Economics If individual preferences are given by convex indifference curves, then each efficient allocation (each point on the contract curve / in the Pareto set) will be a competitive equilibrium for some initial endowment allocation of the goods.
When indifference curves are nonconvex, a Pareto efficient allocation will not necessarily be an equilibrium
Generalization of the model (exchange and production) Additional notation and definitions: h = 1,..., m numbering of firms y hj net supply of good j by firm h y j = y 1j +...+y mj net supply of good j z j = x j - w j - y j excess demand for good j p h = p 1 y h1 +... + p n y hn profits of firm h q ih the share of consumer i in the profits of firm h; for each firm h, q 1h +... + q kh = 1
Remarks: If y hj < 0, then firm h uses more of good j than it produces of that good. The number -y hj = y hj then gives the net quantity demanded of good j for firm h. w ij and q ih are parameters in the model, while x ij, y hj and therefore also x j, y j and z j are functions of the prices p.
Walrasian equilibrium point is (p*,x*) such that for each good j: x j * = x 1j (p*)+...+x kj (p*) w 1j +...+w kj +y 1j +...+y hj, i.e. z j (p*) 0 Budget constraint of consumer i: p 1 x i1 +...+p n x in = = p 1 w i1 +...+p n w in +q i1 j p j y 1j +...+q im j p j y mj
Walras Law With budget constraints satisfied for all consumers the value of the aggregate excess demand is 0, i.e.: j p j z j = 0
Proof of Walras Law j p j z j 1 = j p j (x j -w j -y j ) = 2 = j p j ( i x ij - i w ij - h y hj ) = 3 = j p j ( i x ij - i w ij - h ( i q ih )y hj ) = 4 = j i(p j x ij -p j w ij - h q ih p j y hj ) = 5 = i ( j p j x ij - j p j w ij - h q ih j p j y hj ) = 6 = i 0 = 0. Explanation of subsequent steps: 1 follows from the definition of excess demand 2 definitions of x j, w j and y j plugged in 3 q 1h +...+q kh = 1 inserted for each h 4 p j dragged inside the brackets 5 the sequence of adding changed 6 the formulation that is being summed is the difference between the right- and the left-hand side of the budget constraint
Remark: Walras Law holds for any price set p, not just for the equilibrium prices.
More definitions A free good is a commodity for which excess demand is negative. The equilibrium price of such good is 0. A desired good is a commodity for which excess demand for a zero price is positive. If all goods 1,...,n are desired, then in equilibrium excess demand in all markets is 0.
Corollary from Walras Law (2 goods) If demand is equal to supply in one market, the same must hold for the other market. Since p 1 z 1 (p* 1,p* 2 ) + p 2 z 2 (p* 1,p* 2 ) = 0 therefore from z 1 (p* 1,p* 2 ) = 0 it must follow that, as long as p 2 > 0, z 1 (p* 1,p* 2 ) = 0
Corollary from Walras Law More generally, if all markets except for one are in equilibrium, then the latter market must also be in equilibrium. Important application: the labor market (against keynesianism)
And now something more practical... The Australian state of Victoria was considering an optimal allocation of arcade machines Starting assumptions: budget revenues providing entertainment controlling access Proposal: an auction respecting the limits imposed on the number of machines
Arcade machines auction 4 000 machines 178 separate markets (2 types of licenses in each of the 89 local government areas (LGAs)) 600 participants First project of its kind
How does one proceed? Hire experts Allow them to design an auction mechanism Allow them to run a zillion of experiments, with an increasing number of participants Charlie Plott Tim Cason Result: revenue of $620 million!
Auction mechanism Sequential auction. In each round: 1) Participants submit their demand function in markets which they select (Walrasian Auctioneer!) 2) A computer maximizes the joint consumer surplus in all markets 3) A computer matches this allocation with a price in each market (Second Theorem of Welfare Economics!) 4) Feedback regarding prices and potential allocations, loop. Stop when activity level is low.