## Abstract

We model an economy that alternates randomly between abundance and scarcity episodes. We characterize in detail the structure of the Markovian competitive equilibrium. Accumulation and drainage of stocks are the main focuses. Economically appealing comparative statics results are proved. We also characterize the stationary distribution of states. We extend the model to discuss price stabilization policies, injection and release costs, and limited storage capacity. Overall, the analysis delineates the notion of “flexible economy.”

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## Notes

The notion of convenience yield was introduced by the economists Kaldor and Working who studied the theory of storage. In the context of commodities, the convenience yield captures the benefit from owning a commodity minus the cost of storing it. The flow of benefits from storage (the reduction in production costs) drives a wedge between the price of a commodity today and its value in the future.

The Samuelson effect arises when, for a given commodity, forward price volatility declines with the contract horizon.

Backwardation occurs when the price of a commodity for the actual period exceeds the price for future periods.

On the role of banking in smoothing permit prices, see [1] and Godby et al. (1997), on its effect on control costs, see [18]. The study of equilibrium in [23] assumes that risk-neutral firms minimize their expected discounted costs. When firms anticipate the possibility of a permit stockout, the expected change in marginal abatement costs could be negative (for further detail, see [3]). Potential permit stockout could partially explain normal backwardation in permit prices; the same mechanism is at the core of the results in [22].

[17], generalizing previous results by [28] and [21], considers stabilization at exactly the mean price as a decision made to eliminate price fluctuations, presumably enhancing welfare. A costless stock established by an authority achieves the objective and enhances welfare. Welfare analysis of price stabilization has been extended to encompass alternative assumptions about price expectations, risk attitudes [19], and nonlinearities ([26, 27], among others). Storage in this literature is made by a public authority, which is in charge of managing a buffer stock. Reference [13] is the only model that questions the optimality of stabilization schemes. The private storage industry and arbitrage opportunities are considered, instead, in modern dynamic stochastic models with i.i.d disturbances [29].

A more general structure with injection and withdrawal costs and limited storage capacity is discussed in Sect. 6.

Graphics obtained with Mathematica 8.

In case of crisis, the stock immediately starts being used so that state (\( C,S=\overline{S}\)) does not last. This implies that \(\pi _{C}\), the price of storage services for congestion during the crisis, has no measurable economic effect.

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## Acknowledgments

The authors thank Ivar Ekeland, Thomas Mariotti, Huyên Pham, Rémi Rhodes, Stephen Salant and the referees for their suggestions. Participants to the 2013 Bonn Workshop “Stochastic Optimization - Models and Algorithms” are also thanked. The authors take full responsibility for any remaining errors.

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## Appendix

### Appendix

### 1.1 Proof of Lemma 1

The fundamental remark to prove Lemma 1 is that the axes can be reached but not crossed in a Markovian equilibrium. Indeed, according to Eqs. (4), (5), no point on the axes can be part of the solution, except for \(S=0\) since (6) becomes relevant. But no acceptable solution would connect parts from two different quadrants on an axis, even at \(S=0\). Indeed this latter case would imply that a slightly positive inventory would be associated with two different prices, depending of which part (i.e. which quadrant) of the trajectory we consider, a contradiction with the notion of Markov solution.

We leave aside for the moment the singular (one point) trajectory \((p_{C}^*,p_{A}^*)\) for which \(S^*=0\).

The North-East and the South-West quadrants can be eliminated easily. Indeed, in the North-East quadrant, if there were an \(S>0\) such that \(p_{C}[S]>p_{C}^*\) and \(p_{A}[S]>p_{A}^*\), then the trajectory would be characterized by increasing inventories in all states and times (see Figure 1), a contradiction with the transversality condition. In the South-West quadrant, inventories are decreasing; Figure 1 shows that \(S=0\) could happen only at prices \(\ne (p_{C}^*,p_{A}^*)\), a contradiction with condition (6).

The South-East quadrant can also be eliminated since all trajectories having a point in the interior of this quadrant end up on the half-line \(\{ p_{C}>p_{C}^* \, ; \,p_{A}=p_{A}^{*}\}\) without crossing it (according to the fundamental remark above). Indeed, in that quadrant, the RHS of (4) is positive and bounded away from 0, so \(p_{C}'\) is positive and is also bounded away from 0; as \(p_{C}\) increases, the RHS of (5) becomes negative. Therefore \(p_{A}'\) becomes positive: the trajectory goes to the North-East as \(S\) increases, and reaches vertically the axis, as inspection of \(p_{A}'/p_{C}'\) indicates. This provides the contradiction with the transversality condition: above a certain level, inventories never decrease.

### 1.2 Proof of Proposition 1

**1. and 2.** If \((r+\lambda _{C})p_{A}^{*}-\lambda _{C}p_{C}^{*}+c<0\), then \(S^{*}=0\), \(p_{C}[0]=p_{C}^{*}\) and \(p_{A}[0]=p_{A}^{*}\) cannot be an equilibrium. Indeed, in a competitive economy, a storer who anticipates this dynamics expects other storers not to store, but he sees a profitable opportunity: replenishing at price \(p_{A}^{*}\) when abundance returns is profitable in expectation. Similarly, if the inequality is reversed, no storage is an equilibrium.

All trajectories entering in region **II** also cross region **III**. This is due to the fact that trajectories in region **II** necessarily go North-West and cross \((CC^{\prime })\).

Trajectories in region **III**, in turn, all go North-East and end up on the vertical axis for finite prices. Indeed, using (4) and (5), we see that \(p_{A}^{\prime }/p_{C}^{\prime }\) is necessarily bounded above as \(p_{A}\) goes to infinity while \(p_{C}\) stays below \(p_{C}^{*}\). This means that the trajectories reach the vertical axes with a flat tangent at some \(S>0\), a contradiction with conditions (4,5,6).

Trajectories in **I** are of 3 types: those going in region **II** (excluded), those passing through \(\Omega \), and those reaching the horizontal axis on the right of \(\Omega \). The trajectories that do not reach \(\Omega \) are eliminated, because this case contradicts conditions (4,5,6).

If \((r+\lambda _{C})p_{A}^{*}-\lambda _{C}p_{C}^{*}+c \ge 0\), the trivial solution \((p_{C}^*,p_{A}^*)\), with no inventories works. In fact this is the only one, as previous arguments also show.

**3.** Remark that \(|dS/dp_C|\) is bounded away from 0 along the trajectory (the trajectory remains distant from \(CC'\)). The range of \(p_{C}\) being bounded, \(S\) can only vary in a bounded interval. Therefore \(S^*\) is finite.

**4.** We now show that the trajectory passing through \(\Omega \) is unique. The Cauchy-Lipschitz Theorem cannot be applied at \(\Omega \), a singular point of the system. We use the following argument: choose any starting point in the interior of **I**, denoted by \((p_{C}^{0},p_{A}^{0})\); it is necessarily nonsingular. The trajectory passing through this point is unique (Cauchy-Lipschitz). Consider the point \((p_{C}^{0},p_{A}^{0}+ \varepsilon )\) where \(\varepsilon \) is some small real. Straightforward calculations show that the slope of the trajectory passing through \( (p_{C}^{0},p_{A}^{0}+\varepsilon )\), which is positive, decreases as \( \varepsilon \) increases. To see this, one can directly reason on

This means that trajectories move apart as \(S\) increases, i.e. as they approach \(\Omega \). The consequence is that there cannot be multiple trajectories through \(\Omega \). This proves uniqueness.

### 1.3 Proof of Proposition 2

We first determine how trajectories move in the phase diagram as parameters change. Rewrite the system of ODE (4) et (5) in compact form as

Note that \(\frac{\partial P_{C}}{\partial c}=1/\Delta _{C}[p_{C}]<0\) and \(\frac{\partial P_{A}}{\partial c}=1/\Delta _{A}[p_{A}]>0\), thus \(p_{A}^{\prime }/p_{C}^{\prime }=P_{A}/P_{C}\) decreases as \(c\) increases (all trajectories in **I** are flatter). Similar observations prove that all trajectories in **I** are also flatter when \(r\) increases, when \(\lambda _{A}\) increases and when \(\lambda _{C}\) decreases.

We can now reposition equilibrium trajectories as parameters change. Increasing \(c\) or \(r\), or decreasing \(\lambda _{C}\), move \(\Omega \) to the right; increasing \(\lambda _{A}\) has not effect on \(\Omega \). In all cases, the equilibrium trajectory moves below the former one: to each \(p_{C}\) is associated a smaller \(p_{A}.\)

Remark that \(\frac{dS}{dp_{C}}=1/P_{C}<0\), thus

Since \(\Omega \) goes to the right as \(c\) increases, the range of \(p_{C}\) becomes smaller; it remains to be verified that \(1/P_{C}\), as a function of \( p_{C}\), is also smaller. For example, along the equilibrium trajectories, for a fixed \(p_{C}\)

(\(P_{C}\) grows in absolute value and thus \(1/P_{C}\) decreases in absolute value.) This proves that as \(c\) increases, a given price is associated with a smaller \(S\). Similar reasonings can be applied to the other parameters to prove the claims.

### 1.4 Proof of Proposition 3

We have

Clearly, trajectories in **I** are steeper with a higher \(\beta _{C}\) or a smaller \(\beta _{A}\). Remark that the frontier of **I** (\(\Omega \) in particular) is unchanged in this comparative statics. Remark also (this concerns point 2) that a proportional increase of \(\beta _{C}\) and \(\beta _{A}\) does not change the trajectories (but a given point corresponds to a different \(S\)). The type of reasoning used in the proof of Proposition 2 can now be applied to show the claims.

The comparative statics with respect to \(p_{C}^{*}\) and \(p_{A}^{*}\) require further precautions. In the former, remark that trajectories are steeper with a higher \(p_{C}^{*}\) (\(p_{C}<p_{C}^{*}\)) and that **I** is extended to the right (trajectories are simply going further to the right). These two effects concur to increase the price for given stocks. In the latter, trajectories are flatter with a (say) smaller \(p_{A}^{*}\) but \(\Omega \) moves along down \((AA^{\prime })\). The first effect decreases prices, hence point 1, but the second could lead to a higher \(S^{*}\) [a smaller function is integrated over a longer interval, since the range of \(p_{C}\) increases, see Eq. (49)].

### 1.5 Equivalent expressions for prices

**On the right of**
\(S=0\). From (4), we know that

thus, writing approximations on both sides we get

which yields Eq. (18).

**On the left of**
\(S=S^{*}\).

Let \(x_{A}^{\prime }\) denote \(\Delta _{A }[p_{A}] \cdot p_{A}^{\prime }\). Given that \(p_{C}^{\prime }[S^{*}]\ne 0\), we can approximate \(p_{C}[S]\) around \(S^{*}\) with \(p_{C}[S^{*}]+p_{C}^{\prime }[S^{*}](S-S^{*})+o(S-S^{*})\). We denote \(-p_{C}^{\prime }[S^{*}]\) [which can be calculated exactly using (4)] by \(M\), with

Given that

we can calculate that \(p_{A}[S]-p_{A}^{*}+o(p_{A}[S]-p_{A}^{*}) =\sqrt{\frac{2}{\Delta _{A}^{\prime }[p_{A}^{*}]}}x_{A}^{1/2}[S]\), or equivalently \(p_{A}[S]-p_{A}^{*}= \sqrt{\frac{2}{\Delta _{A}^{\prime }[p_{A}^{*}]}}x_{A}^{1/2}[S]+o(x_{A}^{1/2}[S]).\) We plug these two equivalent expressions into (5), which yields

Consider now the ODE

The unique solution to (58) is \(K_{A}^{2}(S^{*}-S)^{2}\) with

We show now that this exact solution of approximate ODE (58) is an approximation of the solution to ODE (57).

Consider the residual \(o(S-S^{*})+o(x_{A}^{1/2}[S])\) in the ODE (57). For all \(\varepsilon >0\), there is a left neighborhood of \(S^{*}\), denoted \(V_{\varepsilon }\), in which the absolute value of the residual is smaller than \(\varepsilon \times (S^{*}-S)\) and \(\varepsilon \times (x_{A}^{1/2}[S])\). Consider the ODE

The solution to this equation is smaller than \(x_{A}\) on \(V_{\varepsilon }:\) indeed, both \(x_{A}^{\prime }\) and \(y^{\prime }\) are negative, but if \( y>x_{A}\) for some \(S\) in \(V_{\varepsilon }\), it remains so for any larger stock because \(y^{\prime }>x_{A}^{\prime }\). This is in contradiction with the fact that \(y[S^{*}]=x_{A}[S^{*}]\). In other terms,

A similar reasoning shows that

These two inequalities give the approximation of \(x_{A}\) at \(S^{*}\), from which we derive that of \(p_{A}\) since \(p_{A}[S]-p_{A}^{*}=\sqrt{\frac{2}{\Delta _{A}^{\prime }[p_{A}^{*}]}}x_{A}^{1/2}[S]+o(x_{A}^{1/2}[S])\).

### 1.6 Proof of Lemma 2

See Theorem 3.10, p. 130, in [6]. The application of the theorem is quite simple here. If the process stays in state \(C\), the stocks is emptied in finite time whatever the initial inventories. Denote by \(T^{*}=T(S^{*})\) the time its takes to sell out all if the economy has \(S^{*}\) in the reserves and the state stays in \(C\). Given the simple Markov structure of the process \(\sigma _t\), whatever the initial state, the economy will stay without interruption in state \(C\) a duration larger than or equal to \(T^{*}\) in finite time with probability 1. The expected time of a passage to state \((C,0)\) is finite.

### 1.7 Proof of convergence of Algorithm 2

Remark that the ODE commanding \(\phi _{A}\) can be written

On the right of \(S=0\), \(\Delta _{C}\rightarrow 0\) so the RHS of (63) is equivalent to \(-\frac{\lambda _{A}}{\Delta _{C}}\), i.e., using (18), to \(\frac{K_{0}}{\sqrt{S}}\) where \(K_{0}\) is a nonnegative real

Thus \(\lim _{S\rightarrow 0}\phi _{A}\) is finite and strictly positive. Indeed, for all \(\varepsilon >0\), there exists \(\eta \) such that for all \( S\le \eta \),

Take \(S_{1}\) and \(S_{2}\) both smaller than \(\eta \) with \(S_{1}\le S_{2}\) and integrate the inequality above between these two reals. We find

This proves that \(\phi _{A}\) is bounded away from 0 (fix \(S_{2}\) and let \(S_{1}\) converge to \(0\)). Given that \(\phi _{A}\) is also monotonic (increasing) in a neighborhood of \(0\), the limit that we denote by \(\phi _{A}[0]\) exists and is nonnegative.

So, at 0, \(f_{A}\) is finite and nonnegative whereas \(f_{C}\sim _{0}\frac{K_{f_{C}}}{\sqrt{S}}\) where \(K_{f_{C}}\) is some nonnegative real. This implies that, though the density \(f_{C}\) diverges at 0, its integral is well defined.

### 1.8 Proof of Proposition 4

On the left of \(S^{*},\Delta _{A}\rightarrow 0\) so the RHS of (63) is equivalent to \(-\frac{\lambda _{C}}{\Delta _{A}}\), i.e. \(\frac{K_{S^{*}}}{S-S^{*}}\) where \(K_{S^{*}}\) is a nonnegative real with

For all \(\varepsilon >0\), there exists \(\eta \) such that for all \(S\ge S^{*}-\eta ,\)

Take \(S_{1}\) and \(S_{2}\) both larger than \(S^{*}-\eta \) with \(S_{1}\le S_{2}\) and integrate the inequality between these two real numbers. We find

i.e.

This implies that \(\lim _{S\rightarrow S^{*}}\phi _{A}=0\), from which we can conclude that \(\mathbb P [S^{*}]=0.\)

We can now derive a tight condition on the shape of the density function \(f_{A}\) around the upper bound \(S^{*}\). Indeed, given that \(\phi _{A}=f_{A}\cdot \Delta _{A}\),

Equation (71) together with \(\phi _{C}=-\phi _{A}\) proves Proposition 4.

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Cretì, A., Villeneuve, B. Commodity storage with durable shocks: a simple Markovian model.
*Math Finan Econ* **8**, 169–192 (2014). https://doi.org/10.1007/s11579-013-0108-7

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DOI: https://doi.org/10.1007/s11579-013-0108-7