Our model solutions are described as follows. The first equilibrium constitutes as the unconstrained highest ranking one. The second group of equilibria constitutes of mid-ranked ones, which are subject to a single binding constraint. Finally, the third group of equilibria constitutes of the lowest ranked ones, which are subject to two binding constraints.
The Economically Highest Ranked Equilibrium (PCSt+1 – Solution 1, see Table)
This signifies an interior equilibrium, where the hedging or risk management constraints on agents are not binding (i.e. µt+1 = Φ't+1 = Φ''t+1 = 0). In other words, the futures contracting of all agents are in the satiation region. This equilibrium is evaluated by superimposing the demand-supply financial sector (i.e., futures) constraint (equation 14) on the respective pricing functions of various agents derived in Sections 3.2-3.4. Since this equilibrium involves four endogenous variables (ft+1, qP
t+1, qC
t+1, q
St+1), four independent equations (7), (10), (12) and (14) are sufficient to yield a unique normal backwardation solution. In this case:
ft+1 Et(p~
Here the marginal utility of each agent adjusts in such a way that no agent can extract any economic surplus from the other. Deviation of futures price from expected spot price is given in terms of a covariance term (of marginal utility of stochastic consumption with price
risk) divided by the expectation of marginal utility of consumption. The stochastic consumption parameter of all agents is impacted jointly by the operational (i.e., yield) and price risks, as illustrated in equations (5), (8) and (11). We note that the above pricing function (along with the others given below such as equations (17–22), and (23–31)) reflect the risk profile of the agents in the economy. This is different from that derived from the
″cash–and–carry″ arbitrage, which is free of risk aversion parameters.
The Economically Mid-Ranked Equilibria
In general, non-satiation of futures contracting of either agent in the economy leads to a strictly binding hedging or risk management constraint. The binding constraint endows the respective agent with market power. Since constraints lead to a reduction in welfare, the agent subject to it gains market power. S/he can take the price of futures determined by the competing agents at a favorable level.
To elaborate this point further:
(i) Equilibrium PCt+1 (Solution 5):
If the risk management constraint on the Speculator is binding (Φ''t+1 > 0; qSt+1= q
¯t+1), then the futures pricing is determined by both Producers and Consumers, while the economic surplus is retained by the Speculator. That is,
ft+1 Et(p~
Equilibrium PSt+1 (Solution 9):
Here, the Consumer’s hedging constraint binds (Φ't+1 > 0; qCt+1 = Min.(~c
t+1)).
Futures pricing is thus determined by both Producers and Speculators, while the economic surplus is retained by the Consumer. That is,
ft+1 Et(p~
ft+1 Et(p~
Equilibrium CSt+1 (Solution 13):
Here, the Producer’s hedging constraint binds (µt+1> 0; qP
t+1= Min.(y*~
t+1)).
Futures pricing is determined by both Consumers and Speculators, while the economic surplus is retained by the Producer. That is,
ft+1 Et(p~
Equilibrium P'C't+1 (Solution 2):
Here the futures pricing is determined jointly by the Producers and Consumers, while the Speculator is priced out of the market (qS
t+1 = 0). That is,
Equilibrium P'S't+1 (Solution 3):
Here, the futures pricing is determined jointly by the Producers and Speculators, while the Consumer is priced out of the market (qC
t+1 = 0). That is, Equilibrium C'S't+1 (Solution 4):
Here, the futures pricing is determined jointly by the Consumers and Speculators, while the Producer is priced out of the market (qP
t+1 = 0). That is,
ft+1 Et(p~ determined by the remaining agents’ reservation price. Here the market power is extricated by the agents facing the constraint, as their futures pricing conditions hold as strict inequalities.
(i) Equilibrium Pt+1 (Solution 17):
Here, the futures pricing is determined solely by the Producers, while the Consumers and Speculators are price-takers. That is,
ft+1 Et(p~
Equilibrium Ct+1 (Solution 21):
Here, the futures pricing is determined solely by the Consumers, while the Producers and Speculators are price-takers. That is,
ft+1 Et(p~
Equilibrium St+1 (Solution 25):
Here, the futures pricing is determined solely by the Speculators, while the Producers and Consumers are price-takers. That is,
ft+1 Et(p~ Speculators are price-takers; and (ii) the Consumers are priced out of the market (qC
Equilibrium (C')1(t+1) (Solution 8)
Here, the futures pricing is determined solely by the Consumers, while: (i) the Speculators are price-takers; and (ii) the Producers are priced out of the market (qPt+1 = 0). That is,
Equilibrium (S')1(t+1) (Solution 12)
Here, the futures pricing is determined solely by the Speculators, while: (i) the Consumers are price-takers; and (ii) the Speculators are priced out of the market (qSt+1 = 0). That is,
Equilibrium (C')2(t+1) (Solution 14)
Here, the futures pricing is determined solely by the Consumers, while: (i) the Producers are price-takers; and (ii) the Speculators are priced out of the market (qSt+1 = 0). That is,
ft+1 Et(p~
t+1) =
Covt(US'(c~S
t+1), p~
t+1) Et(US'(c~S
t+1))
>/< 0,
ft+1 Et(p~
t+1) >
Covt(UP'(c~P
t+1), p~
t+1) Et(UP'(c~P
t+1))
.
(31)Thus, the presence of the three tier equilibria in a dynamic setting can lead to the market moving between multiple equilibria, alternating between normal backwardation and contango, thereby increasing volatility and making the empirical analysis of time series data unintelligible, leading to a puzzling behavior of futures prices.
Q.E.D.
No. Hedging
t+1 with Prod., Con. and Spec. setting price of futures
2 0 0 0 > 0 > 0 0 P'C'
t+1 with Prod. and Con. setting price and Spec. not participating (qS
t+1 = 0)
3 0 0 0 > 0 0 > /< 0 P'S'
t+1 with Prod. and Spec. setting price and Con. not participating (qC
t+1 = 0)
4 0 0 0 0 > 0 > /< 0 C'S't+1 with Con. and Spec. setting price and Prod. not participating (qP
t+1 = 0)
5 0 0 > 0 > 0 > 0 > /< 0 PC
t+1 with Prod. and Con. setting price, while spec. is a price taker (extricating economic surplus)
6 0 0 > 0 > 0 > 0 0 This equilibrium does not make sense as constraint and K-T condition contradict each
other.
7 0 0 > 0 > 0 0 > / < 0 P'
1(t+1) with Prod. setting price, Spec. being a price-taker and Con. not participating (qC
t+1 = 0)
8 0 0 > 0 0 > 0 > / < 0 C'
1(t+1) with Con. setting price, Spec. being a price-taker and Prod. not participating (qP
t+1 = 0)
9 0 > 0 0 > 0 > 0 > / < 0 PS
t+1 with Prod. and Spec. setting price, while consumer is price-taker (extricating economic surplus)
10 0 > 0 0 > 0 > 0 0 P'
2(t+1) with Prod. setting price, Con. being a price-taker, and Spec. not participating (qS
t+1 = 0)
11 0 > 0 0 > 0 0 > / < 0 This equil. does not make sense as constraint and K-T condition contradict each other.
12 0 > 0 0 0 > 0 > / < 0 S'
1(t+1) with Spec. setting price, Con. being a price taker, and Prod. not participating (qP
t+1 = 0)
13 > 0 0 0 > 0 > 0 > / < 0 CS
t+1 with Con. and Spec. setting price, while Prod. is a price taker (extricating
economic surplus)
14 > 0 0 0 > 0 > 0 0 C'
2(t+1) with Con. setting price, Prod. being a price-taker and Spec. not participating (qS
t+1 = 0)
15 > 0 0 0 > 0 0 > / < 0 S'
2(t+1) with Spec. setting price, Prod. being a price-taker and Con. not participating (qC
t+1 = 0)
16 > 0 0 0 0 > 0 > / < 0 This equil. does not make sense as constraint and K-T condition contradict each other.
17 0 > 0 > 0 > 0 > 0 > / < 0 P
t+1 with Prod. setting price, while both Con. and Spec. are price-takers.
18 0 > 0 > 0 > 0 > 0 0 This equil. does not make sense as constraint and K-T condition contradict each other.
19 0 > 0 > 0 > 0 0 > / < This equil. does not make sense as constraint and K-T condition contradict each other.
20 0 > 0 > 0 0 > 0 > / < 0 This equil also does not make sense as Prod. is supposed to set prices while the other two agents are price-takers. However, qP
t+1 = 0.
21 > 0 0 > 0 > 0 > 0 > / < 0 C
t+1 with Con. setting price, while Prod. and Spec. are price-takers (extricating economic surplus).
22 > 0 0 > 0 > 0 > 0 0 This equil. does not make sense as constraint and K-T condition contradict each other.
23 > 0 0 > 0 > 0 0 > / < 0 This equil also does not make sense as Con. is supposed to set prices while the other two agents are price-takers. However, qC
t+1 = 0.
24 > 0 0 > 0 0 > 0 > / < 0 This equil. does not make sense as constraint and K-T condition contradict each other.
25 > 0 > 0 0 > 0 > 0 > / < 0 S
t+1 with Spec. setting price, while Con. and Prod. are price-takers.
26 > 0 > 0 0 > 0 > 0 0 This equil also does not make sense as Spec. is supposed to set prices while the other two agents are price-takers. However, qS
t+1 = 0.
27 > 0 > 0 0 > 0 0 > / < 0 This equil. does not make sense as constraint and K-T condition contradict each other.
28 > 0 > 0 0 0 > 0 > / < 0 This equil. does not make sense as constraint and K-T condition contradict each other.
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