Commodity Trade in Continuous Time, Long-Term Availability and Storage Capacity

Abstract

The canonical model of competitive storage is formulated in discrete time, and this brings about a high non-linearity and computational complexity of equilibrium equations. In this chapter, we consider a continuous-time commodity market model utilizing the advantages of stochastic calculus that make this model a more convenient analysis tool. It is shown that the argument of the equilibrium price function is the long-term availability of a commodity instead of the current availability, which does not make sense for continuous time. The equilibrium price function satisfies a second-order differential equation and is given by the saddle path under the standard boundary conditions for switching between the regimes of commodity trade. We refine these conditions for the model extension with an upper-boundary constraint on storage capacity and examine the structure of equilibrium price functions in this case.

Appendices

Appendices

1.1 A.1 Smooth-Pasting Condition (8.22)

Our elucidation of this condition replicates the one given by Dixit and Pindyck (1994, pp. 130–132). Consider a small zone around the point of pasting of the functions P(a) and p(a), a^∗ = x^∗/μ satisfying the value-matching condition (8.21), P(a^∗) = p(a^∗). Suppose that the smooth-pasting condition (8.22) does not hold, P^′(a^∗) ≠ p^′(a^∗). Then a^∗ is the kink point shown in Fig. 8.9a, b.

Fig. 8.9

(a) P^′(a^∗) > p^′(a^∗). (b) P^′(a^∗) < p^′(a^∗)

The differential for availability at threshold point a^∗ is given by Eqs. (8.17) and (8.18):

$$ da=-y\left(p\left({a}^{\ast}\right)\right) dt+\left(\sigma /\mu \right) dw $$

(8.33)

for da ≥ 0 and

$$ da=-{x}^{\ast } dt+\left(\sigma /\mu \right) dw $$

(8.34)

for da < 0. The drift rates coincide at point a^∗, because y(p(a^∗)) = x^∗.

Consider the discrete-time random walk approximation of processes (8.33), (8.34) near point a^∗ with discrete-time increments Δt and upward and downward steps of size $\pm \Delta a=\pm \left(\sigma /\mu \right)\sqrt{\Delta t}$. The upward and downward binomial probabilities for this approximation are given by:

$$ \pi =\frac{1}{2}\left(1+\frac{a^{\ast}\mu }{\sigma}\sqrt{\Delta t}\right),\kern0.5em 1-\pi =\frac{1}{2}\left(1-\frac{a^{\ast}\mu }{\sigma}\sqrt{\Delta t}\right). $$

(8.35)

The expected small change of the equilibrium price for a close to a^∗ is

$$ E\Delta p\left({a}^{\ast}\right)=\pi p\left({a}^{\ast }+\Delta a\right)+\left(1-\pi \right)P\left({a}^{\ast }-\Delta a\right)-p\left({a}^{\ast}\right)\approx \pi {p}^{\prime}\left({a}^{\ast}\right)\Delta a-\left(1-\pi \right){P}^{\prime}\left({a}^{\ast}\right)\Delta a, $$

since P(a^∗) = p(a^∗). From Eq. (8.35):

$$ {\displaystyle \begin{array}{l}\pi {p}^{\prime}\left({a}^{\ast}\right)\varDelta a=\frac{1}{2}\left(1+\frac{a^{\ast}\mu }{\sigma}\sqrt{\varDelta t}\right){p}^{\prime}\left({a}^{\ast}\right)\varDelta a=\frac{1}{2}\left(1+\frac{a^{\ast}\mu }{\sigma}\sqrt{\varDelta t}\right){p}^{\prime}\left({a}^{\ast}\right)\left(\sigma /\mu \right)\sqrt{\varDelta t}\\ {}=\frac{1}{2}{p}^{\prime}\left({a}^{\ast}\right)\left(\left(\sigma /\mu \right)\sqrt{\varDelta t}+{a}^{\ast}\varDelta t\right).\end{array}} $$

Similarly,

$$ -\left(1-\pi \right){P}^{\prime}\left({a}^{\ast}\right)\Delta a=-\frac{1}{2}{P}^{\prime}\left({a}^{\ast}\right)\left(\left(\sigma /\mu \right)\sqrt{\Delta t}-{a}^{\ast}\Delta t\right). $$

Consequently,

$$ E\Delta p\left({a}^{\ast}\right)=\frac{1}{2}\left({p}^{\prime}\left({a}^{\ast}\right)-{P}^{\prime}\left({a}^{\ast}\right)\right)\left(\sigma /\mu \right)\sqrt{\Delta t}+\frac{1}{2}\left({p}^{\prime}\left({a}^{\ast}\right)+{P}^{\prime}\left({a}^{\ast}\right)\right){a}^{\ast}\Delta t. $$

(8.36)

If p^′(a^∗) ≠ P^′(a^∗), the first term on the right-hand side of Eq. (8.36) dominates the second one for Δt close to zero, because Δt goes to zero faster than $\sqrt{\Delta t}$. For the same reason, the expected price change EΔp(a^∗) dominates in absolute value the opportunity cost of storage: |EΔp(a^∗)| > rp(a^∗)Δt. Storage is unprofitable for a small interval of time in the case p^′(a^∗) < P^′(a^∗) shown in Fig. 8.9a or it allows for arbitrage opportunities in the case p^′(a^∗) > P^′(a^∗) shown in Fig. 8.9b. As a result, the smooth-pasting condition (8.22) must hold at the threshold point a^∗, because otherwise the no-arbitrage condition (8.13) is violated, Edp(a^∗) ≠ rp(a^∗)dt.

1.2 A.2 System (8.23), (8.24)

For y(p) =  − blnp, the stationary state of this system is the origin, because it is the intersection of the two curves in the phase plane in Fig. 8.4, corresponding to zero derivatives: p^′(a) = 0 and z^′(a) = 0. The first curve is z = 0 and the second one is given by the function Z(p) = rp/blnp, which is drawn with curve Z in this figure.

The characteristic equation of linearized system (8.23), (8.24) is:

$$ \left|\begin{array}{cc}-\lambda & 1\\ {} gz{y}^{\prime }(p)+ gr& gy(p)-\lambda \end{array}\right|={\lambda}^2- gy(p)\lambda -g\left(z{y}^{\prime }(p)+r\right)= $$

$$ {\lambda}^2+g(blnp)\lambda -g\left(r- bz/p\right)=0, $$

(8.37)

where λ is the characteristic root, g = 2(μ/σ)². Near the origin, this equation has two real roots of a different sign, since −z/p ≥ 0, hence the stationary state is the saddle point.

Consider the function Z(p) = rp/bln(p). It has derivative

$$ {Z}^{\prime }(p)=\frac{r\left( lnp-1\right)}{b{(lnp)}^2}, $$

which tends to zero in the origin. One can see from Fig. 8.4 that the slope of the saddle path near the origin is above the slope of Z(p) and below zero. Consequently, the slope of the saddle path and the negative root of Eq. (8.37) tend to zero as a→ + ∞.

Consider the saddle path for a→ − ∞. Rewrite Eq. (8.24) as

$$ \frac{z^{\prime }(a)}{p^{\prime }(a)}=- gblnp(a)+ gr\frac{p(a)}{p^{\prime }(a)}. $$

The right-hand side of this equation tends toward −∞ for p→∞, because $\underset{p\to \infty }{\lim } lnp=+\infty$ and p(a)/p^′(a) < 0. Consequently, the slope of the saddle path tends toward infinity for a→ − ∞.

1.3 A.3 Inequalities (8.32)

Since p(a) > P(a) for any a ∈ (a^l, a^h), the price differential is higher for p(a) at point a^l: dp(a^l) > dP(a^l). Due to the smooth-pasting condition (8.28), this implies the inequality for second-order derivatives (for a second-order approximation):

$$ {p}^{\prime \prime}\left({a}^l\right)>{P}^{\prime \prime}\left({a}^l\right). $$

(8.38)

The value-matching condition (8.27) is equivalent to

$$ y\left(p\left({a}^l\right)\right)=\mu {a}^l, $$

(8.39)

because $y\left(p\left({a}^l\right)\right)=y\left(P\left({a}^l\right)\right)=y\left(\mathcal{P}\left(\mu {a}^l\right)\right)=\mu {a}^l$. For the logarithmic net demand function, we have: $P\left({a}^l\right)=\mathcal{P}\left(\mu {a}^l\right)={e}^{-{a}^l\mu /b}$, ${P}^{\prime}\left({a}^l\right)=-\left(\mu /b\right){e}^{-{a}^l\mu /b}$ and ${P}^{\prime \prime}\left({a}^l\right)={\left(\mu /b\right)}^2{e}^{-{a}^l\mu /b}$. From the value-matching and smooth-pasting conditions (8.27), (8.28), we find that

$$ p\left({a}^l\right)={e}^{-{a}^l\mu /b},{p}^{\prime}\left({a}^l\right)=-\left(\mu /b\right){e}^{-{a}^l\mu /b}. $$

Inserting these conditions and (8.39) into differential equations (8.20) implies

$$ \frac{1}{2}{\left(\upsigma /\upmu \right)}^2{\mathrm{p}}^{\prime \prime}\left({\mathrm{a}}^{\mathrm{l}}\right)+\left(\upmu /\mathrm{b}\right){\mathrm{e}}^{-{\mathrm{a}}^{\mathrm{l}}\upmu /\mathrm{b}}\upmu {\mathrm{a}}^{\mathrm{l}}=\mathrm{r}{\mathrm{e}}^{-{\mathrm{a}}^{\mathrm{l}}\upmu /\mathrm{b}}, $$

which is rearranged as

$$ \frac{{\left(\sigma /\mu \right)}^2b}{2{e}^{-{a}^l\mu /b}}{p}^{\prime \prime}\left({a}^l\right)+{\mu}^2{a}^l= br $$

and yields

$$ {\mu}^2{a}^l= br-{\sigma}^2{p}^{\prime \prime}\left({a}^l\right)/2b{P}^{\prime \prime}\left({a}^l\right), $$

(8.40)

because ${P}^{\prime \prime}\left({a}^l\right)={\left(\mu /b\right)}^2{e}^{-{a}^l\mu /b}$. Inequality (8.38) implies that σ²p^′′(a^l)/2bP^′′(a^l) > σ²/2b, hence Eqs. (8.31) and (8.40) imply:

$$ \overset{\sim }{a}=\frac{2{b}^2r-{\sigma}^2}{2b{\mu}^2}>{a}^l. $$

One can show similarly that $\overset{\sim }{a}<{a}^h$.