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.