- Submitted By: ayeds
- Date Submitted: 12/16/2010 1:09 PM
- Category: Miscellaneous
- Words: 274
- Page: 2
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Prices and Markets

Valuation = Willingness to pay, reservation price

1. Total valuation of the buyers is the sum of the Q highest valuations; denote this by v(Q).

2. Total cost of the sellers is the sum of the Q lowest costs; denote this by c(Q).

Total surplus is v(Q) − c(Q)

For efficiency, the volume Q should maximize v(Q) − c(Q). Let’s state the marginal conditions, as described in Section P.5. First, we define marginal valuation mv(Q) and marginal cost mc(Q) of the Qth unit traded:

1. mv(Q) = v(Q) − v(Q − 1), which is the Qth-highest valuation of the buyers;

2. mc(Q) = c(Q) − c(Q − 1), which is the Qth-lowest cost of the sellers.

Then the marginal conditions for volume Q to maximize total surplus are that

1. mv(Q) ≥ mc(Q) (surplus does not go up by trading one fewer unit), and

2. mv(Q + 1) ≤ mc(Q + 1) (surplus does not go up by trading one more unit).

Stated more succinctly in a way that is approximate here but exact for the case of a smooth model, the marginal condition is mv(Q) = mc(Q)

Demand curve is the inverse of the MV curve

Supply curve is the inverse of the marginal cost curve

Demand has unit elasticity when P/(¯P −P) = 1,

that is, when P = ¯P/2.

The function has a minimum value at x = a if f '(a) = 0

and f ''(a) = a positive number.

The function has a maximum value at x = a if f '(a) = 0

and f ''(a) = a negative number.

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