Abstract
We examine an infinite horizon game in which producers’ output can be purchased by speculators for resale in a future period. The existence of speculators serves to constrain the feasible set of prices that can result from producers’ output game in each period. Absent speculation, producers play a repeated Cournot game with random demand. With speculative inventories possible, the game becomes a dynamic one in which speculative stocks are a state variable which firms can control via their influence on price. We employ collocation methods to find the unknown expected price and value functions required for computation of equilibrium quantities. We demonstrate that strategic considerations result in an incentive to sell to speculators that is non-monotonic in the number of producers: speculation has the largest effect on equilibrium prices and welfare for market structures intermediate between monopoly and perfect competition. Using a computed example, we demonstrate that the effect of speculative storage on the average price level can be substantial, even though the effects on social welfare can be ambiguous.
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Notes
- 1.
U.S. Senate (2006), p.15, Fig. 6.
- 2.
Antimony, Beryllium, Cobalt, Fluorspar, Gallium, Germanium, Graphite, Indium, Magnesium, Niobium, Platinum Group Metals, Rare earths, Tantalum, and Tungsten.
- 3.
The effects of producer storage on the equilibrium in a Cournot duopoly is examined in Thille (2006), in which, rather than speculators engaging in storage, producers themselves store in the face of random variations in demand and cost.
- 4.
In Mitraille and Thille (2014) a uniform demand is considered.
- 5.
See Başar and Olsder (1995).
- 6.
If both thresholds were positive, both (11.14) and (11.15) must hold. The left hand side of (11.15) is clearly higher than that of (11.14) while the right hand side of (11.15) is lower than that of (11.14) if the expected price function is decreasing in future stocks, so both (11.14) and (11.15) cannot hold simultaneously.
- 7.
- 8.
In order to facilitate convergence of the expected value function we replace the terminal value of zero in Mitraille and Thille (2014) with the value of an infinite stream of Cournot profit following some “terminal time” beyond which speculation is not possible. In consequence, the initial condition for the value function is that which occurs in a period in which speculators are forced to sell their inventory and unable to replenish it again.
- 9.
The code used to compute the solution uses numerical routines from NumPy and SciPy (Jones et al. 2001). The code is available from the authors on request.
- 10.
The expected price function converges much more quickly than the expected value function.
- 11.
It is demonstrated in Mitraille and Thille (2014) that the L equilibrium is unlikely to occur when there are few, but more than one, firms.
- 12.
In using this benchmark, we are examining the effects of adding speculators with a storage technology to a model in which no storage technology exists. To examine the effects of speculation alone, we would need to allow producers to store the good, which would be a substantial complication. However, Thille (2006) has shown that, in the absence of speculators, the average price level was the same in the equilibrium in which producers can store the commodity as in the equilibrium in which they could not store. Consequently, we are confident that the results we present below are predominantly due to the presence of speculation and not simply due to the addition of a storage technology.
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Mitraille, S., Thille, H. (2016). Speculative Constraints on Oligopoly. In: Thuijsman, F., Wagener, F. (eds) Advances in Dynamic and Evolutionary Games. Annals of the International Society of Dynamic Games, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28014-1_11
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