OPIM 952 – Market Equilibrium Ralph Ahn

Todays Lecture A general introduction to market equilibria Walras-Cassel Model The Wald corrections The Arrow-Debreu Model On the complexity of Equilibria

The problem: Individuals are endowed with factors (labor, capital, raw goods) …and they demand produced goods. Firms demand factors and produce goods with a fixed production technology. Does general equilibrium exist? What does that realistically tell us about competitive markets?

Definition of equilibrium - in english The Walras-Cassel Model: Where market demand is equal to market supply in all markets (factor or goods) Maximizes utility for each consumer Firms maximize proftis

Notation – Walras Cassel model Economy: H households, F firms, n produced commodities, m primary factors Produced commodities: X h is a vector of commodities demanded by household h X f is a vector of commodities supplied by firm f p is the vector of commodity prices

Notation continued – Walras-Cassel Factors: v h is a vector of factors supplied by household h v f is a vector of factors demanded by firm f w is the vector of factor prices Technology (fixed proportions, same tech for all firms: b ji = v j f / x i f is a unit output prod. coefficient B is n x m matrix of unit-input coefficients

Elaboration of technology b ji = v j f / x i f is a unit output prod. Coefficient Represents the amount of factor j necessary to produce a unit of output I B is m x n matrix of unit-output coefficients so that… v = Bx, where vector x is the supply of produced goods and v is the demand for factors

Setting Equilibrium assumptions Perfect competition: Entrepreneurs makes no positive profits or no losses, so… Total revenue px f equals total cost wv f for every firm f, or: px f = wv f or as b ji = v j f / x i f, then the perfect comp. assumption implies p = Bw

Objective of households (consumers) Maximize utility: U h (x h, v h ) utility increases with consumption of produced commodities x h Utility decreases with supply of factors v h Given an announced set of prices (p, v), the h th household max. the following: max U h = U h (x h, v h ) s.t. px h <= wv h (cant consumer more than you have: household income comes in the form of selling factors wv h )

Resulting Functions A result of the budget constraint, we can derive a a set of output demand functions and factor supply functions of the following general form: x i = D i (p, w) for each commodity i = 1,..,n v j = F j (p, w) for each commodity j = 1,..,n asd

Market Clearing Functions Supply must meet demand, more explicitly: Σ h x i h = Σ h x i f for each commodity i = 1, …,n Σ h v i h = Σ h v i f for each factor j = 1,…,m Given our earlier notation, this can be rewritten as D i (p, w) = x i F j (p, w) = v j = B j x

Bring it all together… Perfect competition: p i = B i w Output market equil:D i (p, w) = x i Factor market equil.:v j = B j x Market factor supplies: F j (p, w) = v j

Bring it all together… Perfect competition: p i = B i w Output market equil:D i (p, w) = x i Factor market equil.:v j = B j x Market factor supplies: F j (p, w) = v j

Bring it all together… Caveat 1: These are supposed to be vectors Caveat 2: the output demand function is also a function of w, and the factor demand function is also a function of p, so there is interaction between the diagrams which will cause the curves to shift around.

Equilibrium? The Walras proof Perfect competition: p i = B i w Output market equil:D i (p, w) = x i Factor market equil.:v j = B j x Market factor supplies: F j (p, w) = v j Unknowns: Quan. of produced goods: x i Quan. of factors: v j Output Prices: p i Factor Prices: w j

Equilibrium? The Walras proof Perfect competition: p i = B i w (n eq) Output market equil:D i (p, w) = x i (n eq) Factor market equil.:v j = B j x (m eq) Market factor supplies: F j (p, w) = v j (m eq) Unknowns: Quan. of produced goods: x i (n unk) Quan. of factors: v j (m unk) Output Prices: p i (n unk) Factor Prices: w j (m unk)

The proof: There are 2n+2m equations and 2n+2m unknowns. This must be a solvable set of equations. The problem: xy = 3 and x+y = 1, has no real solution In context: have only one factory and 2 factors s.t. p = b 1 w 1 + b 2 w 2. Has no solution since there are numerous comb. of w 1 and w 2 that can yield the same output price.

Other problems: With the imposition of the equalities v = Bx and p = Bw, prices may turn out to be negative While negative prices and quantities are completely feasible, they mean nothing in this context.

Walds fixings Idea of free goods. All factors are scarce and thus all have a price However, a factor is scarce only if there is more demand of it than is available, but if supply far exceeds the demand, the factor should have zero price. Real life examples: tap water, air, internet. Carl Menger one does not know at the outset which goods are free goods, so one should insert into the equality the poss. of an unused residual

Walds system The problem was, if introduce inequalities is existence of an equilibrium certain? v j >= B jx In equil., for a particular factor j, either the quantity supplies is equal to the quantity demanded (i.e v j = B jx) or the quantity of that factor supplied exceeds the demand for the factor v j > B jx, but then the corresponding return to factor j must be zero i.e. w j = 0

Primal and Dual Proof of Existence Primal: max px (output revenue) s.t. v>=Bx ; x>=0 Dual: min wv (factor paid) s.t. p =0 Other constraint was that we needed to make sure cost of production (Bw) did not fall below output price, if it did then the company wouldnt produce it, in other words x i = 0.

Duality Theorem of linear programming There exists a solution to the primal (x*) if and only if there exists a solution to the dual (w*) The maximized values of the objectives of the primal and dual are the same. (In our case, that means px* = w*v, which is another way of imposing pure competition at equilibrium)

Duality Theorem of linear programming The langragian multipliers which satisfy the primal problem is the solution vector to the dual (i.e. λ* = w*) and the multipliers which solve the second problem is the solution vector in the primal (i.e. μ* = x*). Implies that in equilibrium: w* [ v-Bx* ] = 0 x* [ Bw-p ] = 0 Complementary slackness conditions

These are important b/c they replace the conds for equilibrium and allow for free goods: w* [ v-Bx* ] = 0 Either v = Bx* (factor market equil) or if there is an excess of a factor, v j > B j x* then w j * = 0 (the factor is free). Thus w j * > 0 only when v j = B j x* (i.e. a factor earns a positive return only if the market for that factor clears) x* [ Bw-p ] = 0 Same as above except every produced good must equal its cost of production.

Proof of existence If we assume that v (factor supply) is inelastic i.e. quantity is fixed, then we only have to be concerned with output x*, factor return w*, and output price p*. The maximized values of the objectives of the primal and dual are the same. i.e. px* = w*v assures us if we are given p* then w* and x* just follows. More in depth proof online

Proof of existence (cont.) How do we know if p is p*? Define equilibrium as a set of inequalities that must be met. ( 1) D i (p*, w*) <= xi* for all i = 1,.., n (output market clearing) (2) If D i (p*, w*) < x i *, then p i * = 0(rule of free goods) (3) v j >= B j x* for all j = 1,.., m(factor market clearing) (4) If v j > B j x*, then w j = 0(rule of free factors) (5) p i * <= B i w*(competition) (6) If p i * < B i w*, then x i = 0(rule of viability) the solution of the duality problem meets many of these inequalities

Walds proof Not only did Wald prove existence of an equilibrium, he also proved that it was unique. A good rundown of the proof can be found here:

Arrow-Debreu Model Arrow-Debreu also proved the existence of an competitive equilibrium. They defined the four conditions necessary for a competitive equilibrium to be: Maximize profits for the company Maximize utility w/budget constraint (considered stock rights and dividends) Prices had to be non-negative and not all zero The idea of a free good

Significance They proved that markets achieve equilibrium by the price mechanism. In any such situation, there is a price vector pЄR m +, the price equilibrium s.t. if x i (p) denotes the allocation that optimizes u i (x) subject to px<=pe i, then Σ i x i (p) = Σ i e i In other words, if everyone just did their own thing, the market would clear on its own. Produces an equilibrium w/o any communication between the parties. The markets invisible hand?

Equilibrium under uncertainty Theyve also introduced the idea of states and uncertainty for a market. Let there be S states of nature, n physically- differentiated outputs, h households. The commodity space of agent h, x h is some subset of R ns defined as x h = [x 1 h,…,x s h ] where each x s h is a vector of commodities received in state s. An economy-wide allocation X is defined as a vector [x 1,…x H ] where x h is the allocation to the h th household and x h is defined as before. Agent h also has other state dependent variables similar to xs vectors, price p, endowments e h =[e 1 h,…,e s h ], In addition ownerships shares θ (this was constant)

Consumer Preference Then household h prefers commodity vector x h to another y h if exp. utility is greater i.e. Suppose agent h assigns a prob. π s to a particular state s occurring x h >=y h if and only if Σ sЄS π s u s h (x s h )>= Σ sЄS π s u s h (y s h ) Lastly there are F firms that produce state- contingent production plans y s f

Arrow-Debreu Equilibrium An Arrow-Debreu equilibrium is a set of allocations (x*, y*) and a set of prices p* such that: (i) for every fЄF, y f * satisfies p*y f *>=p*y f for all y f ЄY f (firms maximize profits given state dependency) (ii) for every hЄH, x h * is maximal in the budget set B h = {x h Є X h I p*x h <= p*e h + Σ h θ hf p*yf*} (maximize consumption w/budget constraint) (iii) Σ hЄH x h * = Σ fЄF y f * + Σ hЄH e h (market clears, i.e. supply meets demand)

On the complexity of Equilibria Given the e i s (supply) and some representation of the u i s, could we find the price equilibrium? To date, there is no known poly-time algorithm for it NP-hardness is highly unlikely due to it being guaranteed that an equilibrium exists. Although some very good approx. equil. algorithms exists.

Indivisible goods If the commodities are indivisible then an equilibrium may not exist. Possible approx. algorithms? (linear utilities) Market approx. clears: (1-Є) Σ i e i Σ i x i <=Σ i e i For all i, the utility is at least (1-Є) times the optimal solution subject to py<=pЄ I It is NP-hard to calculate or even approximate within any factor better than 1/3 the deficiency of a market with indivisible goods (even when there are only two agents, and even when it is known that equilibrium prices exist).

Polynomial time Algorithms Given linear markets with a bounded number of divisible goods, there apparently is a poly-time algorithm for finding an equilibrium. There is a poly-time algorithm for computing an Є- pareto curve in linear markets with indivisible commodities and a fixed number of agents. With a bounded number of goods, there is a poly- time algorithm which, for any linear indivisible market for which a price equilibrium exists, and for any Є>0, finds an Є-approximate equilibrium.