On Price Formation Theory

Summary: On price formation theory in Sabiou M. Inoua and Vernon L. Smith Economics of Markets

To suppose that utility maximizing individuals choose quantities to buy (sell) contingent on given prices is to pose a consumer demand and supply problem without a price-determining solution. This neoclassical problem formulation imposes (1) exogenous prices, (2) price-taking behavior, and (3) the law of one equilibrium-clearing price on markets, before prices can have formed. Hence, unexplained prices are presumed to exist before consumers arrive in the market. If conditions (2) and (3) are hypothesized to characterize markets, the theoretical challenge is to show that they follow from a theory of how markets function. Hence, neoclassical economics did not, because it could not, articulate a market price formation process.

 

The classical economists suffered none of these inconsistencies. (see for example Adam Smith, 1776, book 1, chapter VII) They articulated a coherent theory of price formation and discovery based on operational pre-market assumptions about the decentralized information that buyers and sellers brought to market, their interactive market behavior in aggregating this information, and simultaneously determining prices and contract quantities in the market’s end state.

Buyers (sellers) were postulated as having pre-market max willingness to pay, wtp (min willingness to accept, wta) value (cost) for given desired quantities to purchase (sell) that bounded the price at which each would buy (sell) as they sought to buy cheap (sell dear). We articulate a mathematical theory of this classical price formation process, its connections with

Shannan information theory, and the unexpected role of early experiments in using designs and reporting results consistent with classical theory:

  1. Individuals go to market with pre-market max wtp (min wta) reservation values (costs) for discrete (integer) units Arriving, they bring aggregate WTP (WTA) conditions governing price bounds, and motivation to buy cheap (sell dear).
  2. Any trial (bid/offer) price, P, if too low, tends to rise; if too high, tends to fall. Hence, in classical price adjustment, the “law of demand and supply”, is dynamic. Formally, price change and excess demand, e(P), have the same sign: e(P) ΔP/Δt > 0, if e≠ 0; price changes if excess demand is not
  3. Short side rationing: If any (bid or offer) trial price, P, is too low, the units bought (demanded) are limited by the supply quantity; if P is too high, the units sold (supplied) are limited by the demand quantity. Hence, quantity traded is the minimum of quantity supplied and quantity demanded, or formally, min[s(P), d(P)].

From 2., let V(P) = integral (sum) of -e(P) [namely excess supply, s(p)-d(p)]; for discrete values

𝑉𝑉(𝑝𝑝) = � |𝑣𝑣 − 𝑝𝑝 | + � |𝑐𝑐 − 𝑝𝑝 |,

𝑣𝑣≥𝑝𝑝                      𝑐𝑐≤𝑝𝑝

 

where the notation means summation over all values, v ≥ p, and all costs, c ≤ p, to assure that no goods trade at a loss. Define (the market center of value),

C = arg Min V(P),

which includes market clearing, with C = P* (“equilibrium” price), but is more general, by including important cases like constant cost industries where the exchange quantity is determined by demand.

 

Notice that V(P), a Lyapunov function, measures the distance between price and the traders’ reservation values in profit space. At any P we have, ΔV/Δt = ─e(P)ΔP/Δdt ≤ 0 where t is transaction number.

Hence, V changes non-positively as transactions increase, a parameter-free law of classical market convergence. To get convergence speed, a quantitative result, we would need institutional parameters relating transactions to calendar time.

For a smooth “large market”, where we let the number of agents increase without bound (infinite),

𝑉𝑉(𝑝𝑝) =  𝑝𝑝 𝑆𝑆(𝑥𝑥)𝑑𝑑𝑥𝑥 +   ∞ 𝐷𝐷(𝑥𝑥)𝑑𝑑𝑥𝑥

                 ∫0                                    ∫𝑝𝑝

 

and dV/dt = -e(P)dP/dt ≤ 0.

 

Here is a chart illustrating the above equations for large markets in which all “motion” is in terms of transactions, t ≥ 0 , not time, and is therefore a qualitative dynamics.

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