This paper analyzes the effects of contracts for exhaustible resources. The conclusions are as follows: (1) for a single mine, with increased risk production for the spot market falls, long-term contracting increases, total output falls, contract length generally decreases, equilibrium spot prices increase, and contract prices may rise or fall; (2) with changing economies because of past production, output (spot prices) rises (fall) and then falls (rise), and this scalloped pattern remains with entry and storage included; and (3) contracting permits pairwise gains, may create losses for others, and prices are “fuzzy” signals as they may change for several different reasons.

G. S. Maddala, R. Blaine Roberts; Statistical Cost Analysis Re-Revisited, The Quarterly Journal of Economics, Volume 96, Issue 1, 1 February 1981, Pages 177–181