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Typically in market clearing, the supply and demand (ask and bid) curves meet at a certain point, and the projection of these curves creates a quantity and price. Does this price have a deeper meaning than the clearing mechanism? As in, could the price be defined in another way, so volume weighted average?

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A "volume-weighted average" is how "market price" is (theoretically) measured, when we acknowledge that transactions happen at different prices due to differentiated aspects that may, in theory, be handled in the context of monopolistic competition, but in practice, the different prices are so many that taking them into account would deprive us of any chance to understand the whole. So we (would like to) consider a volume-weighted index as a "market price" (in practice we may be content with simple arithmetic averages).

But the weighted average is the proper theoretical measure because it answers the following question: "if all transactions were condensed in one transaction with the sum of all volumes, what would be the price so as the total value would be the same?"

Denote $p_i$ the price of transaction $i$,$q_i$ the corresponding quantity, $Q_m$ total quantity of all transactions and $V = \sum_{i=1}^n p_iq_i$

We want a "market price" $P_m$, such that $$P_mQ_m = \sum_{i=1}^n p_iq_i \implies P_m = \sum_{i=1}^n \frac {q_i}{Q_m}p_i$$

which is a convex combination of individual prices, since $\sum_{i=1}^n \frac {q_i}{Q_m}=1$.

In this way we preserve the total value of transactions and we can consider the evolution of the market through time for example, using a single time series.

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In general equilibrium that combination of price and quantity determines the allocation of goods according to the shape of the utility functions and the production functions. But when you say that it could be the volume weighted average, I think you refer to stock prices, which have a different behaviour. On that you have different theories and assumptions, ranging from the assumption that prices incorporate all the past information, to the assumption that financial markets are perfect and thus prices are a result of all the future (discounted) events.

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