An Impulse-Regime Switching Game Model of Vertical Competition
AAn Impulse–Regime Switching Game Modelof Vertical Competition
René Aïd Luciano Campi Liangchen Li Mike LudkovskiJune 9, 2020
Abstract
We study a new kind of non-zero-sum stochastic differential game with mixed impulse/switchingcontrols, motivated by strategic competition in commodity markets. A representative upstreamfirm produces a commodity that is used by a representative downstream firm to produce a finalconsumption good. Both firms can influence the price of the commodity. By shutting down orincreasing generation capacities, the upstream firm influences the price with impulses. By switching(or not) to a substitute, the downstream firm influences the drift of the commodity price process.We study the resulting impulse–regime switching game between the two firms, focusing on explicitthreshold-type equilibria. Remarkably, this class of games naturally gives rise to multiple Nashequilibria, which we obtain via a verification based approach. We exhibit three types of equilibriadepending on the ultimate number of switches by the downstream firm (zero, one or an infinitenumber of switches). We illustrate the diversification effect provided by vertical integration in thespecific case of the crude oil market. Our analysis shows that the diversification gains stronglydepend on the pass-through from the crude price to the gasoline price.
Contents a r X i v : . [ q -f i n . M F ] J un Case study: diversification effect of vertical integration 256 Conclusion 277 Proofs 27
Since Hotelling’s (1931) [18] seminal study of commodity prices, considerable efforts have been un-dertaken to understand the dynamics of the equilibrium price of commodities and in particular, itslong–run properties. The cyclical nature of price dynamics is driven by the substitution effect, wherebyconsumers will switch to a different commodity if prices rise too high. In a deterministic setting theswitching time to the substitute is simple to analyze, but with the stochastic economic cycle consumersface a huge challenge in determining when is the appropriate moment to switch. The succession ofbooms and busts of commodity prices complicates the switching timing. In the long–run, productioncapacities adapt to demand and make the price oscillate around a long–term equilibrium. Indeed, thelong–run behaviour of commodity prices exhibits super–cycle patterns. The econometric studies inLeon and Soto (1997) [23], Erten and Ocampo (2012) [12], Jacks (2013) [19] and more recently Stue-mer (2018) [30], all find the presence of super–cycles of several decades in the price of commodities.This phenomenon makes one wonder whether it is even necessary for the consumers to ever switch andwhether it is not preferable to just wait for the prices to crash again.In this paper we design a dynamic model of competition between production and consumption ofa commodity used as an intermediate good, allowing to draw conclusions on the long–run dynamics ofthe commodity price. In our model, two factors drive the price of the commodity: on the one hand,short–term but persistent shocks of demand and/or production, and on the other hand, strategicdecisions of the (representative) upstream production firm and of the (representative) downstreamconsumer firm. The upstream producer extracts the commodity at cost c p and sells it for a price X .The downstream industry buys the commodity and converts it into a final good that has a price P ,non–decreasing in X . This framework covers a wide range of industries. One might think for example,of the agricultural sector where soy enters as an input for the food industry to produce a large rangeof consumer goods. In the aluminum industry, upstream smelters produce aluminum to be used bythe automotive and transportation industries. In the oil industry, the crude is extracted by productionfirms, then transformed into gasoline and kerosene by downstream refineries, and then consumed inthe retail market. For the sake of simplicity, we identify the downstream firm that transforms thecommodity with the final consumer and this downstream firm’s profit with the consumer’s surplus.We focus on the role of the commodity price X that intrinsically creates competition between therepresentative agents of producers and consumers. In a nutshell, producers prefer high price X , whileconsumers prefer low price X . This competition is dynamic and manifests itself through strategicprice effects actuated by the two industries. Therefore, X is (partially) jointly controlled by theproducers/consumers, leading to game–theoretic impacts.On the upstream production side, the producer needs the commodity price X to be high enoughto make a profit margin. We suppose that the dynamics of investment and disinvestment in upstreamproduction is driven by production capacity shocks that cause jumps in the price X . This assumptionis consistent with the theory of real options that predicts the existence of threshold prices triggeringthe decision of entry and the exit from the market (see MacDonald and Siegel (1986) [25] and Dixitand Pindyck (1994) [11]). It is also consistent with the observations of quick swings in investment anddisinvestement in production, see e.g. the boom and bust of commodity prices in 2008–10.2n the downstream consumer side, consumers induce a long–term effect on the commodity priceonly if they switch to a substitute, and they switch to a substitute only if they anticipate that X willremain high enough for a long time. The downstream side faces slower dynamics because it involvesthe transformation of many local installations using the commodity. To have an example in mind,one may think of the thousands of adjustments required to change heating systems in buildings, or ofthe slow effect of the energy saving programs launched by OECD after the 1970s oil shock. Thus, inour model, the downstream market for the final good can be in contraction or expansion regime. Thecontraction regime corresponds to a decreasing demand for the primary commodity, i.e. the marketis abandoning the use of the commodity for a substitute, while the expansion mode corresponds toan increasing demand for the commodity. Depending on the state of the downstream retail market,the drift of the commodity price takes either a constant positive value in the expansion mode or anegative value in the contraction mode. Because such consumer shifts are slow and expensive, the stateis persistent (i.e. piecewise constant in time) and changing the state of the final good market incursheavy switching costs. This toggling of the price trend can be interpreted as endogenous regime-switching , a common way of modeling commodity prices through the business cycle. Beyond theimpact of producers and consumers decisions, the commodity price is subject to exogenous short–termstochastic shocks, captured through a Brownian motion driving risk factor.Our aim is to construct and characterize the dynamic equilibrium in the commodity market due tothis vertical competition. Our major contribution is to provide an endogenous, game-theoretic basisfor two key stylized features of commodity markets: (i) super–cycles that manifest as long–term mean-reversion; (ii) fundamental impact of supply and demand that maintains the price in a range of valuesrather than a single equilibrium value. Furthermore, our model allows for three types of equilibriadepending on the number of demand switches undertaken by the consumer at equilibrium: zero, one,or an infinite number of switches. All equilibria exhibit the latter qualitative properties. Besides, thehigher the consumer’s switching cost, the more she is compelled to endure an unfavorable range ofprices.Along the way, we also make mathematical contributions to the literature on non-zero-sum stochas-tic games (see Martyr and Moriarty (2017) [24], Atard (2018) [4], De Angelis et al. (2018) [13], Aïdet al. (2020) [1]). To our knowledge ours is the first paper that: (i) considers a mixed impulse-control/switching-control stochastic game; (ii) explicitly constructs impulse-switching threshold-typeequilibria in non-zero-sum games; provides new verification theorems regarding best-response strategiesfor (iii) an impulsing agent in a regime-switching setting and (iv) switching agent with an impulsedstate process. While our solution is non exhaustive in the sense that we a priori focus on a special classof equilibria (leaving open the question of existence of other equilibrium families), it is highly tractable.Namely, we are able to provide closed-form description of the dynamic equilibrium, offering precisequantitative insights regarding the producer and consumer roles and their equilibrium behavior.To emphasize the latter point, beyond several synthetic examples that illustrate and visualize ourmodel features, we also present a detailed case-study of the diversification effect provided by verticalintegration in the crude oil market circa 2019, viewed as a competition between crude oil producersand oil refiners that convert crude into gasoline and other consumer goods. Indeed, the industrialorganization of upstream and downstream segments is an important concern both for the anti-trustregulators and for the firms themselves. It is reflected in the extensive economic literature on thesubject and in its persistent presence within the political debate (see Lafontaine and Slade (2007)[20] for a review of the topic). From a firm’s perspective, vertical integration brings multiple virtues,including the potential to reduce the long–term exposure to commodity price fluctuations. See forexample Helfat and Teece (1987) [16] for an empirical estimation of the hedge procured by verticalintegration in the oil business. In our case study, we consider the generic type of equilibrium and asmall downstream firm asking herself whether she has an interest in getting more vertically integrated.We show that the gains from integration are directly linked to the pass-through parameter that linksthe crude oil price to the retail gasoline price. The higher this pass-through, the higher productionactivity dominates the retail activity both in terms of expected rate of profit and the standard deviation3f rate of profit.The rest of the paper is organized as follows. Section 2 sets up the competitive producer-consumercommodity market. Section 3 then constructs the respective threshold-type impulse-switching equi-libria by considering the producer and consumer best-response strategies. Section 4 illustrates anddiscusses the different types of emergent equilibria using toy examples. Section 5 presents the abovevertical integration case study and Section 6 concludes. All the proofs, as well as additional compara-tive statics, are delegated to Section 7. We use ( X t ) to denote the (pre-equilibrium) commodity price, modeled as a continuous-time stochasticprocess. The two players are denoted as p roducer and c onsumer. In what follows sub-index p (resp. c )in the notation will always refer to the producer (resp. consumer). The market involves the originalraw commodity that is being produced and the goods market (e.g. gasoline). The producer extractsthe commodity at cost c p and sells it for price x . The consumer buys it for price x , converts it into afinal good, and sells it for price P . Profit rates:
The price x of the commodity influences the volume of trade, captured by the demandfunction D p ( x ) . A similar phenomenon plays out in the final-good market: the goods price P leads tosales volume D c ( P ) . Since the consumer is in effect the intermediary between the commodity and thegoods market, she will pass some of her input price shocks to the output price P ≡ P ( x ) .We ignore the players’ fixed costs because they can be considered to be integrated in the investmentcosts, and concentrate on the variable costs and revenues that are driven by the respective input/outputprices. Based on the above discussion, the instantaneous profit rate of the producer is π p ( x ) := ( x − c p ) D p ( x ) . (1)Let c c be the processing/conversion cost from input commodity to final good and α be the respectiveconversion factor, so that one unit of commodity becomes α units of the final good (e.g. barrels ofcrude oil, converted into barrels of gasoline). Then the instantaneous profit rate of the consumer is π c ( x ) := D c ( P ) P − D c ( P ) α ( x + c c ) . (2)We note that while the consumer has market power, he is not the only user of the commodity (e.g. crudeoil is also used by the petrochemical industry), so there is no direct link between production volume andconsumption volume. Thus, while there is a physical link between the consumer input volume D c ( P ) /α and her output volume D c ( P ) , there is no direct link between D c ( P ) /α and aggregate commoditydemand D p ( x ) .We shall consider linear inverse demand D p ( x ) = d − d x. If we further assume that P ( x ) = p + p x (the price of the final good is linearly proportional to thecommodity price), and D c ( P ) = d (cid:48) − d (cid:48) P (final good demand is linearly decreasing in its price P ), theprofit rate of the consumer becomes: π c ( x ) = D c ( P ( x )) · (cid:18) P ( x ) − ( x + c c ) α (cid:19) = (cid:0) d (cid:48) − d (cid:48) p (cid:1) (cid:16) p − c c α (cid:17) + (cid:18)(cid:0) d (cid:48) − d (cid:48) p (cid:1) (cid:18) p − α (cid:19) − d (cid:48) p (cid:16) p − c c α (cid:17)(cid:19) x + d (cid:48) p (cid:18) α − p (cid:19) x =: γ + γ x + γ x . (3)4he consumer profit is concave in the commodity price x , γ < , if and only if the pass–throughcoefficient p is higher than the conversion factor /α . It means that the final good price increasesfaster than the need of the downstream industry to produce one more good, which is a sound economiccondition for having a sustainable downstream industry. To sum up, the profit rates of both producerand consumer are concave and quadratic in x . Market conditions:
We model the commodity price process ( X t ) as a controlled Itô diffusion of theform dX t = µ t dt + σdW t − dN t . (4)The Brownian motion ( W t ) captures exogenous price shocks due to random demand or productionfluctuations, or pertinent economic shocks for the industry. In this sense, the model is agnostic inthe reasons why the commodity price fluctuates around its mean trend. The point process N t := (cid:80) i ≥ ξ i { τ i ≤ t } captures the producer interventions at times ( τ i ) i ≥ and impulses ( ξ i ) i ≥ . A positiveimpulse is triggered by an investment phase, and has a negative impact on the price. A negativeimpulse is induced by a disinvestment phase and has a positive impact on the price.The drift process ( µ t ) represents the state of the retail market for the final good. It is either inexpansion or in contraction state. When in expansion, demand is growing faster than the availableproduction capacity, hence prices tend to rise: µ t = µ + > . When in contraction, the demand isshrinking faster than the production capacity, thus the price tends to decrease, and thus µ t = µ − < .This modeling corresponds to an imperfect adjustment of the market as in a sticky price model inmacroeconomics. The drift is fully controlled by the consumer, µ t = µ + ∞ (cid:88) i =0 { σ i ≤ t<σ i +1 } + µ − ∞ (cid:88) i =1 { σ i − ≤ t<σ i } , t ≥ , where σ i is the i -th switching instance taken by the consumer in the case µ − = µ + (with the convention σ = 0 , so that σ is the first switching time) and analogously when µ − = µ − by interchanging oddand even switching times. Thus, both players influence ( X t ) , although their actions are of distincttypes, namely impulse control ( N t ) by the producer and switching-drift control ( µ t ) by the consumer.The resulting controlled price dynamics are denoted as X ( µ,N ) .The quadratic nature of the upstream and downstream profit rate functions π p ( · ) and π c ( · ) impliesthat each player has their own natural habitat given by the intervals ( x c , x c ) and ( x p , x p ) for commodityprice levels with: x p := min (cid:110) c p , d d (cid:111) , x p := max (cid:110) c p , d d (cid:111) , (5) x c := min (cid:110) p − c c /α /α − p , d (cid:48) − d (cid:48) p p d (cid:48) (cid:111) , x c := max (cid:110) p − c c /α /α − p , d (cid:48) − d (cid:48) p p d (cid:48) (cid:111) . (6)Players make a positive profit only if the price stays in the interval ( x i , x i ) , i ∈ { c, p } . The concavityof the profit functions implies that players have preferred commodity levels ¯ X p , ¯ X c that maximize theirprofit rates, namely: ¯ X p := d + c p d d , ¯ X c := − γ γ . (7)Typically, we expect that ¯ X c < ¯ X p , so that the preferred commodity price of the consumer is lowerthan that of the producer. The stochastic fluctuations coming from ( W t ) can generate three differentmarket conditions: X t < ¯ X c I: abnormally low prices ;¯ X c ≤ X t ≤ ¯ X p II: vertical competition ;¯ X p < X t III: abnormally high prices .
5n the first and last cases, both players have the same preferences to raise or decrease X t ; in theintermediate case, they compete against each other. Because both players can in principle push ( X t ) in either direction, the market organization is influenced by their relative gain of doing so, as well astheir action costs. In cases I and III, the players are in waiting mode because of the second–moveradvantage, hoping that the other will act first which allows the second to benefit from the price effectwithout paying the cost of (dis-)investment or of switching. In case II they are in preemption mode,with the player who moves first being able to increase her profits at the expense of the other. Thesedynamic shifts between waiting and preemption is an important feature of vertical competition. Objective functions and admissible strategies:
The objective functionals of the players consistof integrated profit rates π · ( x ) , discounted at constant rate β > and subtracting the control coststhat are paid at respective intervention epochs. We take the investment cost function of the producerto be some convex function K p : R → R , and of the consumer as H : { µ − , µ + } → R + . We denotethe latter as H ( µ − ) = h − , H ( µ + ) = h + . Depending on the initial drift µ being positive/negative theproducer’s objective function is given by: J ± p ( x ; N, µ ) := E (cid:104) (cid:90) ∞ e − β t (cid:0) X t − c p ) D p ( X t ) dt − (cid:88) i e − β τ i K p ( ξ i ) (cid:12)(cid:12)(cid:12) µ = µ ± , X = x (cid:105) , (8)and, similarly, the representative consumer’s objective function is: J ± c ( x ; N, µ ) := E (cid:104) (cid:90) ∞ e − β t (cid:0) γ + γ X t + γ X t ) dt − (cid:88) j e − β σ j H ( µ σ j ) (cid:12)(cid:12)(cid:12) µ = µ ± , X = x (cid:105) . (9)In order for the state variable dynamics and players’ expected payoffs to be well-defined we give thefollowing definition of admissible strategies. To this end, let (Ω , F , ( F t ) t ≥ , P ) be a probability spacewith a filtration satisfying the usual conditions and supporting an ( F t ) t ≥ -Brownian motion ( W t ) . Definition 1 (Admissible strategies) . We say that ( τ i , ξ i ) i ≥ is an admissible strategy for the producerif the following properties hold:1. ( τ i ) i ≥ is a sequence of [0 , ∞ ] -valued stopping times such that ≤ τ < τ < · · · and lim i →∞ τ i = ∞ a.s., with the convention that τ i = ∞ for some i ≥ implies τ k = ∞ for all k ≥ i ;2. ( ξ i ) i ≥ is a sequence of real-valued F τ i -measurable random variables;3. the sequence ( τ i , ξ i ) i ≥ satisfies (cid:80) i ≥ e − βτ i ξ i ∈ L ( P ) .Similarly, we say that the sequence ( σ j ) j ≥ is an admissible strategy for the consumer if4. each σ j is a [0 , ∞ ] -valued stopping time, ≤ σ < σ < · · · , with the convention that σ j = ∞ for some j ≥ implies σ k = ∞ for all k ≥ j ;5. (cid:80) j ≥ e − βσ j ∈ L ( P ) .The set of all producer’s (resp. consumer’s) admissible strategies is denoted by A p (resp. A c ). Remark 1.
Observe that the property 1 above implies that the producer intervention times do notaccumulate in finite time, so that for all t > the process N t = (cid:80) i ≥ ξ i { τ i ≤ t } , t ≥ , is well-defined,adapted and finite-valued. Moreover, the integrability condition in 5 gives that σ j → ∞ (as j → ∞ ),i.e. the switching times of the consumer do not accumulate in finite time either, so that the dynamicsof the controlled state variable (4) is well-defined too. Regarding the expected profits of the players,they are both finite due to integrability properties in 3 and 5 above. Remark 2.
According to the definition of admissibility above, neither player can intervene more thanonce at a time. However, simultaneous interventions coming from both of them are not excluded. Asdiscussed, the dynamics of the intervention in upstream production is much faster than the switchingof the consumption regime for final good. Thus, in case both players try to act simultaneously, weassume that the producer has priority. This avoids unnecessary technicalities and allows for a consistentmodeling of the vertical competition. 6 .2 Equilibrium
Using this notion of admissible strategies, we give the definition of Nash equilibrium.
Definition 2 (Nash equilibrium) . A Nash equilibrium is any pair (( ξ i , τ i ) i ≥ , ( σ j ) j ≥ ) ∈ A p × A c satisfying the following property: J ± p ( x ; N (cid:48) , µ ) ≤ J ± p ( x ; N, µ ) , J ± c ( x ; N, µ (cid:48) ) ≤ J ± c ( x ; N, µ ) , ∀ x ∈ R , for any other pair of strategies (( ξ (cid:48) i , τ (cid:48) i ) i ≥ , ( σ (cid:48) j ) j ≥ ) ∈ A p × A c , where in the payoffs J − r ( x ; · ) , r ∈ { c, p } ,above we have N (cid:48) t = (cid:80) i ≥ ξ (cid:48) i { τ (cid:48) i ≤ t } and µ (cid:48) t = µ + (cid:80) ∞ i =0 { σ (cid:48) i ≤ t<σ (cid:48) i +1 } + µ − (cid:80) ∞ i =1 { σ (cid:48) i − ≤ t<σ (cid:48) i } for t ≥ , µ (cid:48) − = µ + and the convention σ (cid:48) = 0 (analogously in the other case µ (cid:48) − = µ − by interchanging oddand even switching times). In line with the envisioned Markovian structure and in order to maximize tractability, we concen-trate on a specific class of dynamic equilibria. Namely, we aim to construct threshold-type FeedbackNash Equilibria which are of the form τ = 0 , τ i = inf { t > τ i − : X t ∈ Γ p ( t − ) } , i ≥ , ξ i = δ ( X τ i , µ τ i − ) , (10)and σ = 0 , σ j = inf { t > σ j − : X t ∈ Γ c ( t ) } , j ≥ , (11)where Γ r ( t ) = Γ + r { µ t = µ + } + Γ − r { µ t = µ − } , r ∈ { c, p } , for some measurable function δ : R → R and some suitable Borel sets Γ ± p , Γ ± c ⊂ R . Thus, (10)-(11) imply that players act based solely on the current price ( X t ) and demand regime ( µ t ) , ruling outhistory-dependent strategies, and moreover the strategies are characterized through fixed action regions Γ ± p , Γ ± c and impulse maps δ ( · ) . We will denote by τ (cid:48) k the aggregated intervention times coming jointlyfrom the two players. The fact that in (10) producer’s intervention times τ i are defined via Γ p ( t − ) translates the assumption that in case of simultaneous interventions, the producer plays first and soher thresholds naturally depend on the drift µ t − just before her and consumer’s actions (compare toRemark 2).The action regions Γ ± p , Γ ± c are expected to be as follows. The impulse intervention region of theupstream production Γ ± p = ( x ± (cid:96) , x ± h ) is two-sided: the producer will act whenever X t reaches x ± h from below or drops to x ± (cid:96) from above. Note that these thresholds x ± (cid:96) , x ± h are µ -dependent. On theconsumption side, when µ t = µ + (expansion regime), the consumer will switch to µ − if X t gets toohigh: Γ + c = ( y h , ∞ ) . Similarly when µ t = µ − (contraction regime), she will switch to µ + if X t getstoo low Γ − c = ( −∞ , y (cid:96) ) . Finally, when the producer intervenes, he will bring X t to her impulse level x ±∗ r so that the impulse amount is ξ ± r = x ± r − x ±∗ r . The natural ordering we expect is the producerimpulses towards ¯ X p x ± (cid:96) < x ±∗ (cid:96) and x ±∗ h < x ± h , (12)and the consumer switches towards ¯ X c , y (cid:96) < ¯ X c < y h , (13)so that when acting both players try to move X towards their preferred levels. However, the preciseordering between the impulse thresholds x ± r and the switching thresholds y ’s is not clear a priori andwill emerge as part of the overall equilibrium construction.7 .3 Illustration of Competitive Dynamics To further understand the market evolution under competition of the producer and consumer, we focuson the case where both players are active. The producer’s strategy is summarized via a × matrix C p which lists the thresholds x ± (cid:96) , x ± h and the target levels x ±∗ (cid:96) , x ±∗ h . Thus, the no-intervention regionsare [ x ± (cid:96) , x ± h ] and impulse amounts are x ± h − x ±∗ h , x ±∗ (cid:96) − x ± (cid:96) : C p = (cid:20) x + (cid:96) , x + ∗ (cid:96) , x + ∗ h , x + h x − (cid:96) , x −∗ (cid:96) , x −∗ h , x − h (cid:21) . (14)The consumer has two switching thresholds y (cid:96) , y h ; in a typical setup we expect them to satisfy thefollowing ordering x − (cid:96) < y (cid:96) < y h < x + h . (15)Note that in the expansion regime (drift µ + ), we assume that y h < x + h . Therefore, coming frombelow, X ∗ t hits y h first, causing the consumer to switch into the contraction regime with drift µ − . Asa result, the impulse threshold x + h is not effective, i.e. it will never get triggered along an equilibriumpath of ( X t ) . Similar argument implies that x − (cid:96) is not effective either if x − (cid:96) < y (cid:96) . In the left panel ofFig. 1 we illustrate such threshold-based vertical competition among the two players. t Figure 1:
Left panel:
Dynamic competition between producer and consumer. The blue arrows representdrift-switching controls exercised by the consumer at levels y (cid:96) and y h , while the red curved arrowsrepresent impulse controls exercised by the producer at levels x + (cid:96) , x − h that instantaneously push X t to x + ∗ (cid:96) and x −∗ h respectively. Right : A sample path of the controlled commodity price ( X ∗ t ) undercompetitive equilibrium. Observe that X ∗ t ∈ [1 , for all t .To illustrate competitive dynamics, the right panel of Fig. 1 shows a sample trajectory of ( X ∗ t ) (thesuperscript emphasizing the fact that we are now looking at equilibrium) with producer and consumerstrategies C p = (cid:20) . , . , . , . . , . , . , . (cid:21) , ( y (cid:96) , y h ) = (1 . , . . According to the above discussion, the effective thresholds are ( x + (cid:96) , y h ) when µ t = µ + , or ( y (cid:96) , x − h ) when µ t = µ − . In other words, in the expansion regime, ( X t ) will be between [1 . , . and in thecontraction regime it will be between [1 . , . . In Fig. 1 (Right), we start in the contraction regimewith X = 1 . and µ = µ − . On this trajectory, ( X ∗ t ) moves down until it touches the consumer’sthreshold y (cid:96) , where the consumer switches to a positive drift to draw the price up. Nevertheless, theprice keeps decreasing and hits x + (cid:96) = 1 . , whereby the producer intervenes and pushes it to x + ∗ (cid:96) = 1 . .Prices then continue to rise up to y h = 1 . at which point the consumer switches again and startspushing them back down (supposedly she wishes to keep them somewhere around 1.5). This cyclicbehavior continues ad infinitum, yielding a stationary distribution for the pair ( X ∗ t , µ ∗ t ) . Note that8he consumer uses her switching control to keep X ∗ t from going too high or too low, essentially cyclingbetween y (cid:96) and y h . Indeed, starting at X ∗ t = y (cid:96) , the consumer switches to expansion which causesprices to trend up; once they hit y h the consumer switches to contraction, causing prices to trenddown. As a result, µ t alternates between µ + , µ − generating a mean-reverting behavior. Throughout,the producer acts as a “back-up”, explicitly forcing prices from becoming extreme (namely from fallingin the expansion regime, or rising in the contraction regime). These additional interventions by theproducer make the domain of ( X ∗ t ) bounded.It is also possible that, say, x + h < y h so that in the expansion regime the producer will act firstboth when ( X ∗ t ) falls (impulse threshold x + (cid:96) ) and when ( X ∗ t ) rises ( x + h ), making the consumer inactive .In that case it is plain to see that the drift µ t ≡ µ + will stay positive forever; ( X t ) will be forced to abounded domain but will not have mean-reverting dynamics since the drift is constant. Instead, it willexperience repeated impulses downward to counteract the upward trend due to ongoing consumptiongrowth. To obtain a threshold-type Feedback Nash Equilibrium we view it as a fixed point of the producerand consumer best–response maps . Therefore, our overall strategy is to (i) characterize threshold–typeswitching strategies for the consumer given a pre-specified, threshold–type behavior by the producer;(ii) characterize threshold–type impulse strategies for the producer who faces a pre-specified regime–switching behavior of ( X t ) ; (iii) employ tâtonnement, i.e. iteratively apply the best–response controlsalternating between the two players to construct an interior, non-preemptive equilibrium satisfying theordering (15).To analyze best–response strategies, we utilize stochastic control theory, rephrasing the relateddynamic optimization objectives through variational inequalities (VI) for the jump–diffusion dynamics(4). The competitor thresholds then act as boundary conditions in the VIs. To establish the desiredequilibrium we need to verify that the best response is also of threshold-type and solves the expectedsystems of equations. We note that all three pieces above are new and we have not been able to findprecise analogues of the needed verification theorems in the extant literature. Nevertheless, they dobuild upon similar single–agent control formulations, so the overall technique is conceptually clear. Fixing impulse thresholds x ± r ( r = h, l ), the consumer faces a two–state switching control problem onthe bounded domain ( x ± (cid:96) , x ± h ) . Namely, given a producer’s impulse strategy ( τ i , ξ i ) i ≥ with τ = inf { t : X t / ∈ [ x ± (cid:96) , x ± h ] } , we expect the following stochastic representation for her value functions w ± ( x ) with x ∈ [ x ± (cid:96) , x ± h ] w ± ( x ) = sup σ ∈T E x, ± (cid:34) (cid:90) τ ∧ σ e − βt π c ( X t ) dt + e − βτ { τ<σ } (cid:16) w ± ( X τ − ξ ) (cid:17) + e − βτ { τ>σ } (cid:16) w ∓ ( X σ ) − h ± (cid:17)(cid:35) , (16)where E x, ± denotes expectation with respect to µ t ∈ { µ − , µ + } and h ± are the fixed intervention costsof the consumer. The above is a system of two coupled equations, which locally resembles an optimalstopping problem with running payoff π c ( · ) , reward w ∓ ( · ) (last term), and stop–loss payoff (middleterm) w ∓ ( · ) due to the producer impulse at τ . This is almost the formulation as considered in [3]except with two modifications: • The domain is bounded on both sides (previously there was a one–sided stop–loss region). • The boundary condition w + ( x (cid:96) ) = w + ( x + ∗ (cid:96) ) is autonomous but nonlocal. Therefore, the twostopping–type VIs for the consumer are coupled only through the free boundaries, not throughthe stop–loss thresholds as in [3]. 9ow, given a producer strategy C p , if the consumer’s response is such that y (cid:96) < x − (cid:96) and x + h < y h ,the consumer will be stuck forever in the initial regime because the price touches x − (cid:96) before y (cid:96) in thecontraction regime and x + h before y h in the expansion regime. In this case, the price will oscillatebetween x − (cid:96) and x − h if the initial market is in the contraction regime, and between x + (cid:96) and x + h in theexpansion regime.In the case where the consumer’s response satisfies y (cid:96) < x − (cid:96) and y h < x + h , depending on the initialstate, the consumer will switch once to the expansion regime or will be stuck in the initial expansionregime. If the initial regime is µ + , the price will touch y h , the regime will switch to contraction, theprice will never touch y (cid:96) and will oscillate between x − (cid:96) and x − h . If the initial state is already µ + , noswitch of regime will ever occur. The same reasoning applies for the symmetric case where x − (cid:96) < y (cid:96) and x + h < y h .Finally, if the consumer’s response satisfies x − (cid:96) < y (cid:96) and y h < x + h , then whatever the initial regime,the state ( µ t ) will switch many times between the two regimes.The best–response of the consumer consists in picking the best response amongst the three possibleones above. Thus, we distinguish three cases: (a) No-Switch: The consumer is completely inactive and simply collects her payoff based on thestrategy ( x ± (cid:96),h ) . (b) Single-Switch: The consumer always prefers one regime to the other. Then she is inactive (likein case (a) above) in the preferred regime and faces an optimal stopping (since there is only asingle switch to consider) problem in the other regime. (c) Multiple-Switch:
The consumer switches back and forth between both regimes: the continuationregion is ( y (cid:96) , y h ) .Proposition 1 provides the value function of the consumer in case (a). The system (24) characterizesthe game payoff in case (b), and Proposition 2 provides the value function of the consumer in case (c). Regardless of the consumer strategy, in the continuation region, a direct application of the Feynman–Kac formula on (16) shows that her value function solves the following ordinary differential equation(ODE) − βw + µ ± w x + 12 σ w xx + π c ( x ) = 0 . (17)Solving this inhomogeneous second-order ODE, we obtain w ± ( x ) = (cid:98) ω ± ( x ) + u ± ( x ) , where letting θ ± < < θ ± be the two real roots of the quadratic equation − β + µ ± z + σ z = 0 , • u ± ( x ) = λ ± e θ ± x + λ ± e θ ± x solves the homogeneous ODE − βu + µ ± u x + σ u xx = 0 and λ ± i, , i = 1 , are to be determined from appropriate boundary conditions; • (cid:98) ω ± ( x ) is a particular solution to (17), given by (cid:98) ω ± ( x ) = Ex + F ± x + G ± where E = γ β , F ± = 1 β (cid:16) γ + 2 µ ± γ β (cid:17) , G ± = 1 β (cid:0) γ + σ γ β + µ ± F ± (cid:1) . (18)When the consumer is inactive (denoted by w ± ), the continuation region is [ x ± (cid:96) , x ± h ] with theboundary conditions at the impulse levels w ± ( x ± r ) = w ± ( x ±∗ r ) , r ∈ { (cid:96), h } . (19)10rom (19) the respective coefficients λ ± , , λ ± , are solved from the following uncoupled linear system: λ ± , · (cid:2) e θ ± x ± (cid:96) − e θ ± x ±∗ (cid:96) (cid:3) + λ ± , · (cid:2) e θ ± x ± (cid:96) − e θ ± x ±∗ (cid:96) (cid:3) = (cid:98) ω ± ( x ±∗ (cid:96) ) − (cid:98) ω ± ( x ± (cid:96) ) , (20) λ ± , · (cid:2) e θ ± x ± h − e θ ± x ±∗ h (cid:3) + λ ± , · (cid:2) e θ ± x ± h − e θ ± x ±∗ h (cid:3) = (cid:98) ω ± ( x ±∗ h ) − (cid:98) ω ± ( x ± h ) . (21)For x > x ± h we take w ± ( x ) = w ± ( x ±∗ h ) and similarly in the contraction regime, we take w ± ( x ) = w ± ( x ±∗ (cid:96) ) for x < x ± (cid:96) . Proposition 1.
Let ( λ ± , , λ ± , ) ∈ R be the solution to the system (20) - (21) . Then the functions w ± ( x ) , x ∈ [ x ± (cid:96) , x ± h ] , are the value functions for an inactive consumer, i.e. w ± ( x ) = J ± c ( x ; N, µ ± ) , where N isthe producer impulse strategy associated with the thresholds ( x ± (cid:96) , x ±∗ (cid:96) ; x ± h , x ±∗ h ) with x ± (cid:96) < x ± h . The role of w ± ( · ) is important for judging the other two cases, and moreover for deciding whetherthe best–response ought to be of threshold–type. We next consider the situation where the payoff in the expansion regime is higher than the contractionone for any price x , so that the consumer is never incentivized to switch to the contraction regime. Wethen expect the consumer’s corresponding best–response to be either a single–switch strategy (to thepreferred regime) or no–switch (if already there). Economically, this corresponds to y h > x + h so thatas the price rises, the producer impulses ( X t ) down, and the consumer is not intervening to decreaseher demand. As a result, the consumer never switches (except perhaps the first time from negative topositive drift) and lim t →∞ µ t = µ + . This can be observed when demand switching is very expensive,so that the producer has full market power and is able to keep prices consistently low. The consumeris forced to be in the expansion regime forever and she is not able to influence ( X t ) .Suppose that the consumer prefers expansion regime ( µ t = µ + ) and adopts threshold-type strate-gies. Given C p , her strategy is summarized by y (cid:96) > x − (cid:96) , y h = + ∞ , and the resulting contraction–regime value function w − should be a solution to the variational inequal-ity sup (cid:8) − βw − + µ − w − x + 12 σ w − xx + π c ; w +0 − h − − w − (cid:9) = 0 , (22)where w +0 is from Proposition 1 and the continuation region is [ y (cid:96) , x − h ] . This is a standard optimalstopping problem. Note that while the above equation for w − depends on w +0 , the equation for w +0 isautonomous—the system of equations becomes decoupled because the two regimes of ( µ t ) no longercommunicate.To solve (22) we posit that her best–response is of the form w − ( x ) = w +0 ( x ) − h − , x ≤ y (cid:96) , (cid:98) ω − ( x ) + λ − e θ − x + λ − e θ − x , y (cid:96) < x < x − h ,w − ( x −∗ h ) , x − h ≤ x, (23)with the smooth pasting and boundary conditions: (cid:98) ω − ( y (cid:96) ) + λ − e θ − y (cid:96) + λ − e θ − y (cid:96) = (cid:98) ω + ( y (cid:96) ) + λ +1 , e θ +1 y (cid:96) + λ +2 , e θ +2 y (cid:96) − h − , ( C at y (cid:96) ) (cid:98) ω − ( x − h ) + λ − e θ − x − h + λ − e θ − x − h = (cid:98) ω − ( x −∗ h ) + λ − e θ − x −∗ h + λ − e θ − x −∗ h , ( C at x − h ) (cid:98) ω − x ( y (cid:96) ) + λ − θ − e θ − y (cid:96) + λ − θ − e θ − y (cid:96) = (cid:98) ω + x ( y (cid:96) ) + λ +1 , θ +1 e θ +1 y (cid:96) + λ +2 , θ +2 e θ +2 y (cid:96) . ( C at y (cid:96) ) (24)11he system (24) is to be solved for the three unknowns y (cid:96) , λ − , λ − , while λ +1 , , λ +2 , are the coefficientsof the consumer’s payoff associated to the no-switch strategy in the µ + regime, see previous subsection.We can re-write it as first solving for λ − , from the linear system (cid:34) e θ − y (cid:96) e θ − y (cid:96) e θ − x − h − e θ − x −∗ h e θ − x − h − e θ − x −∗ h (cid:35) · (cid:20) λ − λ − (cid:21) = (cid:20) w +0 ( y (cid:96) ) − (cid:98) ω − ( y (cid:96) ) − h − (cid:98) ω − ( x −∗ h ) − (cid:98) ω − ( x − h ) (cid:21) (25)and then determining y (cid:96) from the smooth pasting C -regularity w − x ( y (cid:96) ) = w +0 ,x ( y (cid:96) ) . (26)The case of a single–switch from expansion to contraction regime can be treated analogously in asymmetric way. Finally, we consider the main case where the consumer adopts threshold–type switches, i.e. the orderingin (15) holds. Given C p , the w ± are then supposed to be a solution to the coupled variational inequalities sup (cid:8) − βw + + µ + w + x + 12 σ w + xx + π c ; max { w − − h + , w + } − w + (cid:9) = 0 , (27) sup (cid:8) − βw − + µ − w − x + 12 σ w − xx + π c ; max { w + − h − , w − } − w − (cid:9) = 0 , (28)where we expect continuation regions of the form ( x + (cid:96) , y h ) and ( y (cid:96) , x − h ) . To set up a verificationargument for the consumer’s best–response we make the ansatz w + ( x ) = w + ( x + ∗ (cid:96) ) , x ≤ x + (cid:96) , (cid:98) ω + ( x ) + λ +1 e θ +1 x + λ +2 e θ +2 x , x + (cid:96) < x < y h ,w − ( x ) − h + , x ≥ y h , (29a) w − ( x ) = w + ( x ) − h − , x ≤ y (cid:96) , (cid:98) ω − ( x ) + λ − e θ − x + λ − e θ − x , y (cid:96) < x < x − h ,w − ( x −∗ h ) , x ≥ x − h . (29b)This yields 6 equations: (cid:98) ω + ( y (cid:96) ) + λ +1 e θ +1 y (cid:96) + λ +2 e θ +2 y (cid:96) − h − = (cid:98) ω − ( y (cid:96) ) + λ − e θ − y (cid:96) + λ − e θ − y (cid:96) , ( C at y (cid:96) ) (cid:98) ω + ( x + (cid:96) ) + λ +1 e θ +1 x + (cid:96) + λ +2 e θ +2 x + (cid:96) = (cid:98) ω + ( x + ∗ (cid:96) ) + λ +1 e θ +1 x + ∗ (cid:96) + λ +2 e θ +2 x + ∗ (cid:96) , ( C at x + (cid:96) ) (cid:98) ω − ( y h ) + λ − e θ − y h + λ − e θ − y h − h + = (cid:98) ω + ( y h ) + λ +1 e θ +1 y h + λ +2 e θ +2 y h , ( C at y h ) (cid:98) ω − ( x − h ) + λ − e θ − x − h + λ − e θ − x − h = (cid:98) ω − ( x −∗ h ) + λ − e θ − x −∗ h + λ − e θ − x −∗ h , ( C at x − h ) (cid:98) ω + x ( y (cid:96) ) + λ +1 θ +1 e θ +1 y (cid:96) + λ +2 θ +2 e θ +2 y (cid:96) = (cid:98) ω − x ( y (cid:96) ) + λ − θ − e θ − y (cid:96) + λ − θ − e θ − y (cid:96) , ( C at y (cid:96) ) (cid:98) ω − x ( y h ) + λ − θ − e θ − y h + λ − θ − e θ − y h = (cid:98) ω + x ( y h ) + λ +1 θ +1 e θ +1 y h + λ +2 θ +2 e θ +2 y h . ( C at y h ) (30)The six equations can be split into a linear system for the four coefficients λ ± , ’s e θ +1 y (cid:96) e θ +2 y (cid:96) − e θ − y (cid:96) − e θ − y (cid:96) e θ +1 x + (cid:96) − e θ +1 x + ∗ (cid:96) e θ +2 x + (cid:96) − e θ +2 x + ∗ (cid:96) − e θ +1 y h − e θ +2 y h e θ − y h e θ − y h e θ − x − h − e θ − x −∗ h e θ − x − h − e θ − x −∗ h · λ +1 λ +2 λ − λ − = (cid:98) ω − ( y (cid:96) ) − (cid:98) ω + ( y (cid:96) ) − h + (cid:98) ω + ( x + ∗ (cid:96) ) − (cid:98) ω + ( x + (cid:96) ) (cid:98) ω + ( y h ) − (cid:98) ω − ( y h ) − h − (cid:98) ω − ( x −∗ h ) − (cid:98) ω − ( x − h ) (31)and the smooth-pasting conditions determining the two switching thresholds y (cid:96),h (viewed as free bound-aries) w + x ( y r ) = w − x ( y r ) , r ∈ { (cid:96), h } . (32)12 roposition 2. Let the -tuple ( λ ± , λ ± , y h , y (cid:96) ) be a solution to the system (31) - (32) such that the orderin (15) is fulfilled. Then, the functions defined in (29) give the best–response payoffs of consumer, anda best–response strategy is given by (ˆ σ i ) i ≥ , where ˆ σ = 0 , ˆ σ i = inf { t > ˆ σ i − : X t ∈ Γ c ( t ) } , i ≥ , with Γ + c = [ y (cid:96) , + ∞ ) and Γ − c = ( −∞ , y h ] . Figure 2 illustrates the shapes of the consumer’s value function in the different case of best–response.For the strategy given, we have a dominant function in the contraction regime ( w − ) and a dominantfunction in the expansion regime ( w +0 ). x x (a) C p = (cid:20) . , . , . , . . , . , . , . (cid:21) Figure 2: Value functions w ± , w ± and w ± of the consumer given the producer’s strategy (a). Remark : For comparison purposes, it is also useful to know the continuation region of the consumerwhen she alone controls the market price ( X t ) . As usual, this region is ( −∞ , y h ) in the expansionregime and ( y (cid:96) , + ∞ ) in the contraction regime, with the natural ordering y (cid:96) < y h . The value functions w ± satisfy: sup (cid:8) − βw + + µ + w + x + 12 σ w + xx + π c ; w − − h + (cid:9) = 0 , (33) sup (cid:8) − βw − + µ − w − x + 12 σ w − xx + π c ; w + − h − (cid:9) = 0 . (34)To set up a verification argument for the consumer’s best–response we make the ansatz w + ( x ) = (cid:40) w − ( y h ) − h + , x ≥ y h , (cid:98) ω + ( x ) + λ +1 , e θ +1 x + λ +2 , e θ +2 x , x < y h , (35a) w − ( x ) = (cid:40)(cid:98) ω − ( x ) + λ − , e θ − x + λ − , e θ − x , x > y (cid:96) ,w + ( y (cid:96) ) − h − , x ≤ y (cid:96) . (35b)Furthermore, in the expansion regime, to keep w + ( x ) bounded as x → −∞ we must have λ +2 , = 0 because θ +2 < . In the contraction regime, a similar argument gives λ − , = 0 . We are left with thefour unknowns y (cid:96) , y h and λ p , and λ − , determined from the following smooth pasting conditions: (cid:98) ω − ( y (cid:96) ) + λ − , e θ − y (cid:96) = (cid:98) ω + ( y (cid:96) ) + λ +1 , e θ +1 y (cid:96) − h − , ( C at y (cid:96) ) (cid:98) ω + ( y h ) + λ +1 , e θ +1 y h = (cid:98) ω − ( y h ) + λ − , e θ − y h − h + , ( C at y h ) (cid:98) ω − x ( y (cid:96) ) + λ − , θ − e θ − y (cid:96) = (cid:98) ω + x ( y (cid:96) ) + λ +1 , θ +1 e θ +1 y (cid:96) , ( C at y (cid:96) ) (cid:98) ω + x ( y h ) + λ +1 , θ +1 e θ +1 y h = (cid:98) ω − x ( y h ) + λ − , θ − e θ − y h . ( C at y h ) (36) (cid:50) .2 Producer Best Response We now consider the best–response of the producer, given the consumer’s switching strategy denotedby C c := [ y (cid:96) , y h ] . Once again, we face three Cases:1. The producer is a monopolist, i.e. the consumer is completely inactive;2. The consumer adopts a single–switch strategy;3. The consumer adopts a double–switch strategy. To begin with, we determine the monopoly-like strategy of the producer assuming the consumer adoptsa no–switch strategy. In that case µ t is constant throughout and the functions v ± of the producersatisfy the variational inequality (VI): sup (cid:110) − βv ± + µ ± v ± x + 12 σ v ± xx + π p , sup ξ (cid:8) v ± ( · + ξ ) − v ± ( · ) − K p ( ξ ) (cid:9)(cid:111) = 0 . (37)Note that the two VIs for v + and v − are autonomous, hence uncoupled from each other. In thecontinuation region, the general solution of the ODE − βv + µ ± v x + 12 σ v xx + π p ( x ) = 0 is of the form v ± ( x ) = (cid:98) v ± ( x ) + u ± ( x ) , where u ± = ν ± e θ ± x + ν ± e θ ± x , with θ ± , θ ± as before, satisfiesthe homogenous ODE − βu + µ ± u x + σ u xx = 0 , and (cid:98) v ± ( x ) is a particular solution given by (cid:98) v ± ( x ) = Ax + B ± x + C ± , (38)where the coefficients A, B ± , C ± are identified as: A = − d β , B ± = 1 β (cid:16) d − µ ± d β + c p d (cid:17) , C ± = 1 β (cid:0) µ ± B ± + Aσ − c p d (cid:1) . Assuming the producer adopts threshold–type impulse strategies defined by ξ ∗ ( x ) in the interven-tion region, her expected payoff is of the form: v ± ( x ) = v ± ( x ±∗ h ) − K p ( ξ ∗ ( x )) x ≥ x ± h , (cid:98) v ± ( x ) + ν ± e θ ± x + ν ± e θ ± x , x ± (cid:96) < x < x ± h ,v ± ( x ±∗ (cid:96) ) − K p ( ξ ∗ ( x )) x ≤ x ± (cid:96) . (39)When applying the optimal impulse ξ ±∗ ( x ) at the threshold x ± r , r = (cid:96), h , the producer brings X t back to the price level x ±∗ r := x ± r − ξ ±∗ ( x ± r ) .For optimality, the respective impulse amounts satisfy thefirst order conditions v ± x ( x ±∗ h ) = − ∂ ξ K p ( ξ ∗ ( x ± h )) , v ± x ( x ±∗ (cid:96) ) = − ∂ ξ K p ( ξ ∗ ( x ± (cid:96) )) . (40)We reinterpret the above as the equation to be satisfied by ξ ∗ ( x ± r ) which are treated temporarily asunknowns and plugged into further equations. To ensure that the value function is continuous at x ± r we further need v ± ( x ± r ) = v ± ( x ±∗ r ) − K p ( ξ ±∗ r ) . (41)14inally, making the hypothesis that the value function is differentiable at the borders of the interventionregion, we have: v ± x ( x ± (cid:96) ) = v ± x ( x ±∗ (cid:96) ) − ∂ ξ K p ( ξ ∗ ( x ± (cid:96) )) , (42a) v ± x ( x ± h ) = v ± x ( x ±∗ h ) − ∂ ξ K p ( ξ ∗ ( x ± h )) . (42b)We consider two cases of impulse costs: (i) constant K p ( ξ ) = κ and (ii) linear K p ( ξ ) = κ + κ | ξ | .In case (i), because the impulse cost is independent of the intervention amount there will be an optimalimpulse level x ±∗ r so that for any x in the intervention region the strategy is to impulse back to x ±∗ r which is the same at the two thresholds. In case (ii), ∂ ξ K p = ± κ and all the smooth pasting andboundary conditions can be gathered in the following system: (cid:98) v ± ( x ± h ) + ν ± e θ ± x ± h + ν ± e θ ± x ± h = (cid:98) v ± ( x ±∗ h ) + ν ± e θ ± x ±∗ h + ν ± e θ ± x ±∗ h − κ − κ ( x ± h − x ±∗ h ) , ( C at x ± h ) (cid:98) v ± ( x ± (cid:96) ) + ν ± e θ ± x ± (cid:96) + ν ± e θ ± x ± (cid:96) = (cid:98) v ± ( x ±∗ (cid:96) ) + ν ± e θ ± x ±∗ (cid:96) + ν ± e θ ± x ±∗ (cid:96) − κ − κ ( x ±∗ (cid:96) − x ± (cid:96) ) , ( C at x ± (cid:96) ) (cid:98) v ± x ( x ±∗ h ) + ν ± θ ± e θ ± x ±∗ h + ν ± θ ± e θ ± x ±∗ h = − κ ( C at x ±∗ h ) (cid:98) v ± x ( x ±∗ (cid:96) ) + ν ± θ ± e θ ± x ±∗ (cid:96) + ν ± θ ± e θ ± x ±∗ (cid:96) = κ , ( C at x ±∗ (cid:96) ) (cid:98) v ± x ( x ± h ) + ν ± θ ± e θ ± x ± h + ν ± θ ± e θ ± x ± h = (cid:98) v ± x ( x ±∗ h ) + ν ± θ ± e θ ± x ±∗ h + ν ± θ ± e θ ± x ±∗ h − κ , ( C at x ± h ) (cid:98) v ± x ( x ± (cid:96) ) + ν ± θ ± e θ ± x ± (cid:96) + ν ± θ ± e θ ± x ± (cid:96) = (cid:98) v ± x ( x ±∗ (cid:96) ) + ν ± θ ± e θ ± x ±∗ (cid:96) + ν ± θ ± e θ ± x ±∗ (cid:96) + κ . ( C at x ± (cid:96) ) (43)Note that there are two uncoupled linear systems for v + and v − . The C conditions are from (41), thefirst two C conditions are from (40) which determines the optimal impulse destination, and the lasttwo C conditions are from (42).By a standard verification argument, one can show that if both systems above admit solutions ν ± , and x ± (cid:96),h , where the latter satisfy the order condition x ± (cid:96) < x ± h , then the functions v ± ( x ) as in (39)are the value functions of the producer and his optimal strategies are given by the thresholds x ± (cid:96),h andimpulse amounts ξ ∗ ( x ±∗ (cid:96),h ) . This can be done by following exactly the arguments in, e.g., [7] (see alsotheir Remark 2.1), which are very standard in the literature of impulse control problems. Therefore,details are omitted. Suppose the following ordering, which is similar to (15), holds: x ± (cid:96) < y l < y h < x ± h . (44)We then expect v ± to solve the VIs (cid:40) sup (cid:8) − βv + + µ + v + x + σ v + xx + π p ; sup ξ ( v + ( · − ξ ) − v + − K p ( ξ )) (cid:9) = 0 , sup (cid:8) − βv − + µ − v − x + σ v − xx + π p ; sup ξ ( v − ( · − ξ ) − v − − K p ( ξ )) (cid:9) = 0 . (45)To obtain the producer best–response it suffices to identify the two active impulse thresholds x + (cid:96) , x − h and the respective target levels x + ∗ (cid:96) , x −∗ h . The other two boundary conditions take place at the con-sumer thresholds y (cid:96) , y h , so that the strategy (see (47) below) is C p = (cid:20) x + (cid:96) , x + ∗ (cid:96) , − , + ∞−∞ , − , x −∗ h , x − h (cid:21) . Thegame coupling shows up in the additional boundary condition that when the consumer switches, theproducer’s value is unaffected: v + ( y ) = v − ( y ) , y ∈ ( −∞ , y (cid:96) ] ∪ [ y h , + ∞ ) . (46)15ccordingly, our ansatz is v − ( x ) = v − ( x −∗ h ) − K p ( ξ ∗ ( x )) , x ≥ x − h , (cid:98) v − ( x ) + ν − e θ − x + ν − e θ − x , y (cid:96) < x < x − h ,v + ( x ) , x ≤ y (cid:96) , (47a) v + ( x ) = v − ( x ) , x ≥ y h , (cid:98) v + ( x ) + ν +1 e θ +1 x + ν +2 e θ +2 x , x + (cid:96) < x < y h ,v + ( x + ∗ (cid:96) ) − K p ( ξ ∗ ( x )) , x ≤ x + (cid:96) . (47b)To simplify the presentation, let us concentrate on the proportional impulse costs K p ( ξ ) = κ + κ | ξ | .We have the smooth pasting C and boundary conditions: (cid:98) v + ( y (cid:96) ) + ν +1 e θ +1 y (cid:96) + ν +2 e θ +2 y (cid:96) = (cid:98) v − ( y (cid:96) ) + ν − e θ − y (cid:96) + ν − e θ − y (cid:96) , ( C at y (cid:96) ) (cid:98) v − ( y h ) + ν − e θ − y h + ν − e θ − y h = (cid:98) v + ( y h ) + ν +1 e θ +1 y h + ν +2 e θ +2 y h , ( C at y h ) (cid:98) v + ( x + (cid:96) ) + ν +1 e θ +1 x + (cid:96) + ν +2 e θ +2 x + (cid:96) = (cid:98) v + ( x + ∗ (cid:96) ) + ν +1 e θ +1 x + ∗ (cid:96) + ν +2 e θ +2 x + ∗ (cid:96) − K p ( ξ ∗ ( x + (cid:96) )) , ( C at x + (cid:96) ) (cid:98) v − ( x − h ) + ν − e θ − x − h + ν − e θ − x − h = (cid:98) v − ( x −∗ h ) + ν − e θ − x −∗ h + ν − e θ − x −∗ h − K p ( ξ ∗ ( x − h )) , ( C at x − h ) (cid:98) v + x ( x + (cid:96) ) + ν +1 θ +1 e θ +1 x + (cid:96) + ν +2 θ +2 e θ +2 x + (cid:96) = (cid:98) v + x ( x + ∗ (cid:96) ) + ν +1 θ +1 e θ +1 x + ∗ (cid:96) + ν +2 θ +2 e θ +2 x + ∗ (cid:96) − κ , ( C at x + (cid:96) ) (cid:98) v − x ( x − h ) + ν − θ − e θ − x − h + ν − θ − e θ − x − h = (cid:98) v − x ( x −∗ h ) + ν − θ − e θ − x −∗ h + ν − θ − e θ − x −∗ h + κ . ( C at x − h ) (cid:98) v + x ( x + ∗ (cid:96) ) + ν +1 θ +1 e θ +1 x + ∗ (cid:96) + ν +2 θ +2 e θ +2 x + ∗ (cid:96) = − κ ( C at x + ∗ (cid:96) ) (cid:98) v − x ( x −∗ h ) + ν − θ − e θ − x −∗ h + ν − θ − e θ − x −∗ h = κ , ( C at x −∗ h ) (48)Unlike the single–agent setting (43), the equations (48) are coupled. The coefficients ν ± , are thesolution to the linear system e θ +1 y (cid:96) e θ +2 y (cid:96) − e θ − y (cid:96) − e θ − y (cid:96) e θ +1 x + (cid:96) − e θ +1 x + ∗ (cid:96) e θ +2 x + (cid:96) − e θ +2 x + ∗ (cid:96) − e θ +1 y h − e θ +2 y h e θ − y h e θ − y h e θ − x − h − e θ − x −∗ h e θ − x − h − e θ − x −∗ h · ν +1 ν +2 ν − ν − = (cid:98) v − ( y (cid:96) ) − (cid:98) v + ( y (cid:96) ) (cid:98) v + ( x + ∗ (cid:96) ) − (cid:98) v + ( x + (cid:96) ) − K p (cid:98) v + ( y h ) − (cid:98) v − ( y h ) (cid:98) v − ( x −∗ h ) − (cid:98) v − ( x − h ) − K p (49)and the thresholds x + h , x − (cid:96) are determined by the C smooth–pasting (recall that x −∗ h = x − h − ξ ∗ ( x − h ) , x + ∗ (cid:96) = x + (cid:96) − ξ ∗ ( x + (cid:96) ) ): (cid:40) v − x ( x − h ) = v − x ( x −∗ h ) ,v + x ( x + (cid:96) ) = v + x ( x + ∗ (cid:96) ) , (50)and the first order conditions (FOCs) giving the optimal impulses: v − x ( x −∗ h ) = − ∂ ξ K p ( ξ ∗ ( x − h )) v + x ( x + ∗ (cid:96) ) = − ∂ ξ K p ( ξ ∗ ( x + (cid:96) )) . (51) Proposition 3.
Let the -tuple ( ν ± , ν ± , x + h , x − (cid:96) , x + ∗ h , x −∗ (cid:96) ) be a solution to the system (48) , such thatthe order in (44) is fulfilled and x + (cid:96) < x + ∗ (cid:96) , x −∗ h < x − h . Let v ± be defined in (47) and assume v + xx ( x + ∗ (cid:96) ) < , v − xx ( x −∗ h ) < . (52) Then the functions v ± are the best–response payoffs of the producer, and a best–response strategy isgiven by τ ∗ = 0 , τ ∗ i = inf (cid:8) t > τ ∗ i − : X ∗ t ∈ Γ p ( t − ) (cid:9) , (53) ξ ∗ i ( x + (cid:96) ) = x + ∗ (cid:96) − x + (cid:96) , ξ ∗ i ( x − h ) = x − h − x −∗ h , i ≥ , (54) with Γ p ( t ) = Γ + p { µ t = µ + } + Γ − p { µ t = µ − } , where Γ + p = ( −∞ , x + (cid:96) ] and Γ − p = [ x − h , + ∞ ) , while ( X ∗ t ) followsthe dynamics corresponding to the consumer’s strategy ( σ i ) i ≥ and the producer’s impulse strategy ( τ ∗ i , ξ ∗ i ) i ≥ . .2.3 Preemptive Response It is possible that the static discounted future profit of the producer satisfies, say, v + ( x ) ≥ v − ( x ) forany x , so that he always prefers expansion regime to contraction regime.In that case, the consumer switching at y h from expansion to contraction hurts the producer andone possible strategy for him is to preempt in order to prevent the consumer from switching the driftto µ − . This situation could be viewed as looking for best x + h < y h , given y h . In the latter casethe constrained solution could be x + h = y h − , whereby the system (43) does not hold and the best–response is to impulse ( X t ) right before it hits y h , x + h = y h − . This strategy is not well-defined (i.e. thesupremum is not achieved on the open interval ( x + (cid:96) , y h ) ), but the resulting preemptive best–responsevalue in the µ + regime can be obtained by using the ansatz (where we slightly abuse the notation towrite x + ∗ h = y h − ξ ∗ ( y h ) for the target impulse level at y h ) v + ( x ) = v + ( x + ∗ h ) − K p ( ξ ∗ ( x )) , x ≥ y h , (cid:98) v + ( x ) + ν +1 e θ +1 x + ν +2 e θ +2 x , x + (cid:96) < x < y h ,v + ( x + ∗ (cid:96) ) − K p ( ξ ∗ ( x )) , x ≤ x + (cid:96) , (55)and the boundary conditions for determining the target impulse levels v + x ( x + ∗ h ) = − κ , v + x ( x + ∗ (cid:96) ) = + κ . (56)Note that we now have 5 unknowns, ν +1 , , x + (cid:96) , x + ∗ (cid:96) , x + ∗ h rather than six as we “fixed” x + h = y h . Thisyields the following system (cid:98) v + ( y h ) + ν +1 e θ +1 y h + ν +2 e θ +2 y h = (cid:98) v + ( x + ∗ h ) + ν +1 e θ +1 ( x + ∗ h ) + ν +2 e θ +2 ( x + ∗ h ) − K p ( ξ ∗ ( y h )) ( C at y h ) (cid:98) v + ( x + (cid:96) ) + ν +1 e θ +1 x + (cid:96) + ν +2 e θ +2 x + (cid:96) = (cid:98) v + ( x + ∗ (cid:96) ) + ν +1 e θ +1 x + ∗ (cid:96) + ν +2 e θ +2 x + ∗ (cid:96) − K p ( ξ ∗ ( x + (cid:96) )) ( C at x + (cid:96) ) (cid:98) v + x ( x + (cid:96) ) + ν +1 θ +1 e θ +1 x + (cid:96) + ν +2 θ +2 e θ +2 x + (cid:96) = (cid:98) v + x ( x + ∗ (cid:96) ) + ν +1 θ +1 e θ +1 x + ∗ (cid:96) + ν +2 θ +2 e θ +2 x + ∗ (cid:96) + κ ( C at x + (cid:96) ) (cid:98) v + x ( x + ∗ h ) + ν +1 θ +1 e θ +1 x + ∗ h + ν +2 θ +2 e θ +2 x + ∗ h = − κ ( C at x + ∗ h ) (cid:98) v + x ( x + ∗ (cid:96) ) + ν +1 θ +1 e θ +1 x + ∗ (cid:96) + ν +2 θ +2 e θ +2 x + ∗ (cid:96) = κ . ( C at x + ∗ (cid:96) ) (57)Preemption in the contraction regime writes in a symmetric way.In general, we need to manually verify whether x + h > y h (the “normal” case) or x + h = y h (thepreemptive case) whenever we consider the producer best–response. The two situations lead to differentboundary conditions at the upper threshold, and hence cannot be directly compared. Considering theoptimization problem for x + h , we expect his value function to increase in x + h on ( x + (cid:96) , y h ) and experiencea positive jump at y h , i.e. conditional on someone acting, the producer prefers the consumer’s switchto applying his impulse. However, if this is not the case, the consumer action hurts the producer andassuming the impulse costs are low, the best-response is x + h = y h . This corner solution arises due tothe underlying discontinuity: on ( x + (cid:96) , y h ) the producer compares the value of waiting to the value ofdoing an optimal impulse, but at y h he compares the value of switching to that of doing an optimalimpulse. So it could be that “waiting” > impulsing > switching at y h , leading to pre-emptive impulseto prevent the worst (for the producer) outcome. Proposition 4.
Assume µ − = µ + . Let the -tuple ( ν +1 , ν +2 , x + (cid:96) , x + ∗ (cid:96) , x + ∗ h ) be a solution to the system (57) , such that the order in (44) is fulfilled and x + (cid:96) < x + ∗ (cid:96) , x + ∗ h < y h . Let v + be defined as in (55) andassume v + xx ( x + ∗ (cid:96) ) < , v + xx ( x + ∗ h ) < . (58) Then, the function v + is the best–response payoff of the producer in the expansion regime, and a best–response strategy is given by τ ∗ = 0 , τ ∗ i = inf { t > τ ∗ i − : X t ∈ Γ p ( t − ) } , (59) ξ ∗ i ( x + (cid:96) ) = x + ∗ (cid:96) − x + (cid:96) , ξ ∗ i ( y h ) = y h − x + ∗ h , i ≥ , (60)17 ith Γ p ( t ) = Γ + p ( t ) = ( −∞ , x + (cid:96) ] ∪ [ y h , + ∞ ) , while X ∗ follows the dynamics corresponding to theproducer’s impulse strategy ( τ ∗ i , ξ ∗ i ) i ≥ . x x Figure 3: Value functions v ± , v ± , and v ± pre of the producer given the consumer’s strategy C c = (3 , . Figure 3 illustrates the shapes of the producer’s value function in the different cases of best–response. For the given consumer strategy, we have a dominant function in the contraction regime( v − ) and a dominant function in the expansion regime ( v +1 ). The best–response functions defined in Section 3 lead to three types of potential market equilibria,depending on the equilibrium behaviour of the consumer and characterized by the relative positions ofthe consumerand producer thresholds: • Type I – generic: y (cid:96) ≤ x − h and x + (cid:96) ≤ y h . • Type II – transitory: −∞ = y (cid:96) ≤ x − (cid:96) and x + (cid:96) ≤ y h ; or y (cid:96) ≤ x − h and y h = + ∞ . • Type III – preemptive: y (cid:96) ≤ x − h and x + (cid:96) = y h ; or y (cid:96) = x − h and x + (cid:96) ≤ y h .In equilibrium Type I, the consumer switches back and forth forever between the two expansion andcontraction regimes. The optimal policy of the consumer is given by the threshold y (cid:96) in the contractionregime and y h in the expansion regime, while the optimal policy of the producer is formed by the pair ( x + (cid:96) , x + , ∗ (cid:96) ) in the expansion regime and symmetrically by the pair ( x + , ∗ ,h x − h ) in the contraction regime.We anticipate this to be the most common equilibrium type; it was precisely described and illustratedin Section 4.1.In equilibrium Type II, the consumer and the producer both prefer a given regime and thus, theconsumer switches at most once when the market is initialized in the opposite regime. Afterwards,only the producer acts to maintain the price between ( x (cid:96) , x h ) . Consider the case of a single switchfrom expansion to contraction; the consumer’s optimal policy consists then in only one threshold, y h .The optimal policy of the producer is more complicated: in the expansion regime, it consists of thepair ( x + (cid:96) , x + , ∗ (cid:96) ) and in the contraction regime, it consists of a quadruplet ( x − (cid:96) , x − , ∗ (cid:96) , x − , ∗ h , x − h ) . The samereasoning applies in the other single switch case. This equilibrium is described in Section 4.2.The last type of equilibrium, named Type III, resembles the preceding one in the sense that atmost one switch can be observed. But it differs because here the consumer is stuck forever in a stateshe wishes to leave. In that case described in Section 4.3, only the producer acts. Starting in theexpansion regime, for instance, the consumer would like to switch to the contraction regime when theprice reaches a threshold y h . But the producer, who prefers perpetual expansion, preempts the switchby acting at the threshold y − h , just before the action of the consumer.Threshold–type equilibria offer analytical tractability to describe the long–run market behavior.The latter can be summarized by the stationary distribution of the commodity price ( X ∗ t ) and the18onsumer regimes ( µ ∗ t ) as induced by the equilibrium strategies ( N ∗ t , µ ∗ t ) . To quantify these effects, wedefine an auxiliary discrete–time jump chain ( M ∗ n ) ∞ n =0 which takes values in the state space E := (cid:8) S + , S − , I − (cid:96) , I − h , I + (cid:96) , I + h (cid:9) . (61)The chain M ∗ keeps track of the sequential actions of the players, where S ± represents the switchesof the consumer (“ S + ” stands for the switch µ − → µ + and “ S − ” for µ + → µ − ) and I ± r the impulses(up/down at the two impulse boundaries) of the producer. Thus, M ∗ summarizes the sequence ofmarket interventions stored within τ i , σ i stopping times. Note that states M ∗ n ∈ { S + , I + (cid:96)h } imply apositive drift µ + of X ∗ , while the rest imply a negative drift µ − . Moreover, if the consumer adoptsa double–switch strategy and the producer adopts a non-preemptive strategy as discussed in Section4.1, then the thresholds x + h and x − (cid:96) will be hit at most once by X ∗ and therefore the correspondingstates I + h and I − (cid:96) of M ∗ n are transient.Because the dynamics of X ∗ between interventions are always Brownian motion (BM) with drift,the transition probabilities of M ∗ can be described in terms of hitting probabilities of a BM. Thisoffers closed-form expressions for the the transition probability matrix P of M ∗ , and its invariantdistribution denoted by (cid:126) Π . Moreover, the sojourn times of M ∗ correspond to ( X ∗ t ) hitting the variousthresholds (in terms of the original continuous-time “ t ”) and are similarly linked to BM first passagetimes. Combining the above ideas, we can then derive a complete description of ( µ ∗ t ) , namely thelong-run proportion of time that the commodity demand is in expansion/contraction regimes and therespective expected switching time, see (71).In the following section otherwise stated, we use the parameter values in Table 1, such that π i ( x ) = a i ( x − x i )( x i − x ) , i ∈ { c, p } . This yields consumer and producer preferred price levels of ¯ X c = 3 , ¯ X p = 4 .The same set of parameters yields an equilibrium of each type, showing the non-uniqueness of equilibriain this model. x Figure 4:
Producer’s and consumer’s profit rates as afunction of x . Market Consumer Producer β . x c x p σ . x c x p µ + . a c a p µ − − . h ± κ κ Model parameters for Section 4.1.
We look for an interior, non-preemptive equilibrium satisfying the ordering (15), i.e. a pair of consumerand producer strategies of the form ( y ∗ (cid:96) , y ∗ h ) and ( x + (cid:96) , x + ∗ (cid:96) , x −∗ h , x − h ) . To construct this equilibrium, weemploy tâtonnement, i.e. iteratively apply the best–response controls alternating between the twoplayers. This corresponds to the interpretation of Nash equilibrium as a fixed point of best–responsemaps BR . The equilibrium is obtained using two different fixed–point algorithms. Given strategies C p
50 100 150 200 t y h y l x h- x l+ t y h x h- x l- t y h x l+ Figure 5: A sample path of the controlled market price ( X ∗ t ) under a Type I equilibrium (Left), Type II(Center) and Type III (Right) together with E (cid:2) X ∗ t (cid:3) (black solid curve). The colors along the x -axisindicate the corresponding µ ∗ t . We start with µ ∗ = µ + .and C c , we have either an asynchronous or synchronous algorithm, namely C k +1 p = BR ( C kc ) , C k +1 p = BR ( C kc ) , C k +1 c = BR ( C k +1 p ) , C k +1 c = BR ( C kp ) , asynchronous synchronous . The resulting equilibrium found using both algorithms is the same and is C I , ∗ p = (cid:20) . , . , − , + ∞−∞ , − , . , . (cid:21) , C I , ∗ c = [2 . , . . (62)The dynamic equilibrium of the commodity price X ∗ is illustrated in Figure 5 (Left). The marketstarts in the expansion regime, µ ∗ = µ + . We observe that x −∗ h is close to y ∗ h , implying that once theprice has reached the switching level y ∗ h , it is likely to touch soon thereafter the threshold x −∗ h , makingthe price drop to x −∗ h . The producer “backs up” this mean-reversion by impulsing down if prices risetoo much and impulsing down if they drop too much. Otherwise, she lets the consumer be in chargevia switching control that benefits him as well.At equilibrium, the price X ∗ fluctuates in a range of values where neither the producer nor theconsumer have negative profit rate. If alone in the market, the optimal monopolistic strategies of theproducer and the consumer are C mp := (cid:20) . , . , . , . . , . , . , . (cid:21) , C mc := [1 . , . . (63)We see that the equilibrium strategy of the producer C I , ∗ p is quite close to what he would have doneif alone in the market. On his side, the consumer-induced equilibrium price range is wider than hewould prefer ( . against . if alone). In equilibrium, it is as if the producer lets the consumer do thejob of bringing back the price to his preferred level ¯ X p . The producer intervenes only if X ∗ t drops toolow or gets too high, after the regime switching has occurred. But, in the long–run, the average price lim t →∞ E [ X ∗ t ] is close to . , which is the mid–value between ¯ X p and ¯ X c .The players’ equilibrium strategy profile yields a stationary distribution for the pair ( X ∗ t , µ ∗ t ) . Themacro market µ ∗ switches between the expansion and the contraction regimes back and forth, whilethe jointly controlled price ( X ∗ t ) is bounded in the range [ x + (cid:96) , x − h ] and fluctuates in a mean-reverting pattern due to alternating signs of its drift. These stylized features can be broadly traced in the worldcommodity markets which undergo cyclical Expansion/Contraction patterns. Dynamics of ( X ∗ t ) in the equilibrium : The dynamics of the commodity price ( X ∗ t ) are less tractabledue to the impulses applied by the producer. Let φ ∗ ( · ) denote the long-run (i.e. stationary) distributionof ( X ∗ t ) . In Figure 6, we show φ ∗ obtained from an empirical density based on a long trajectory of ( X ∗ t ) , relying on Monte Carlo simulations and the ergodicity of the recurrent, bounded process ( X ∗ t ) .For additional interpretability, we also plot the invariant distributions φ ∗± conditional on µ t = µ ± .20 Price (a)
Price (b)
Figure 6: Estimated long-run densities φ ∗ of X ∗ in a double–switch and one–sided impulse equilibrium.Panel (a): overall kernel smoothed φ ∗ ( · ) . (b): long–run distributions φ ∗± of X ∗ conditional on µ ( t ) = µ + (resp. µ ( t ) = µ − ). Recall that the domain of X ∗ t depends on the current regime, so the support of thetwo densities differs. In another type of equilibrium the consumer switches only in one regime, with the other being absorb-ing. For this reason, we name it transitory . To fix ideas, suppose that the consumer only switchesfrom expansion regime to contraction regime. In that case, the producer effectively acts like a profitmaximizing monopoly in the contraction regime with two–sided impulses; in the expansion regime shewill apply a one–sided impulse as in the equilibrium type I.To solve for such equilibrium, we first compute the producer strategy in the contraction regimewhich is a decoupled VI as in (39) leading to the 6 equations in (43) but only for v − , x − r , x −∗ r , r ∈ { (cid:96), h } .This solution induces the corresponding no–switch solution ω − of the consumer as in (20)-(21). Both v − , ω − are then fixed and act as source terms to solve for the equilibrium in the expansion regime.For the latter, we need to compute v + ( · ) and the associated thresholds x + (cid:96) , x + ∗ (cid:96) (only one threshold),as well as ω + ( x ) and the switching threshold y h (note that there is no y (cid:96) ). The boundary conditionsare v + ( y h ) = v − ( y h ) , ω + ( y h ) = ω − ( y h ) − h .This reasoning leads to the following algorithm. If x + (cid:96) and x + , ∗ (cid:96) are fixed, we compute the best–response of the consumer by solving the variational problem for the consumer value function ω + suchthat: w + ( x ) = w − ( x ) − h , y h ≤ x, (cid:98) ω + ( x ) + λ +1 e θ +1 x + λ +2 e θ +2 x , x + (cid:96) < x < y h ,w + ( x + , ∗ (cid:96) ) , x ≤ x + (cid:96) . This is exactly the best–response in the single–switch case with the solution given by the system (24)and which provides the consumer’s threshold y h . Now, if we consider that y h is fixed, we can computethe best–response of the producer by solving a VI for the value function v + that satisfies v + ( x ) = v − ( x ) , y h ≤ x, (cid:98) v + ( x ) + ν +1 e θ +1 x + ν +2 e θ +2 x , x + (cid:96) < x < y h ,v + ( x + ∗ (cid:96) ) − κ − κ ( x + , ∗ (cid:96) − x ) , x ≤ x + (cid:96) . ( x + (cid:96) , x + , ∗ (cid:96) , ν +1 , ν +2 ) are: v − ( y h ) = (cid:98) v + ( y h ) + ν +1 e θ +1 y h + ν +2 e θ +2 y h , ( C at y h ) (cid:98) v + ( x + (cid:96) ) + ν +1 e θ +1 x + (cid:96) + ν +2 e θ +2 x + (cid:96) = (cid:98) v + ( x + ∗ (cid:96) ) + ν +1 e θ +1 x + ∗ (cid:96) + ν +2 e θ +2 x + ∗ (cid:96) − κ − κ ( x + , ∗ (cid:96) − x + (cid:96) ) , ( C at x + (cid:96) ) (cid:98) v + x ( x + (cid:96) ) + ν +1 θ +1 e θ +1 x + (cid:96) + ν +2 θ +2 e θ +2 x + (cid:96) = (cid:98) v + x ( x + ∗ (cid:96) ) + ν +1 θ +1 e θ +1 x + ∗ (cid:96) + ν +2 θ +2 e θ +2 x + ∗ (cid:96) + κ , ( C at x + (cid:96) ) (cid:98) v + x ( x + ∗ (cid:96) ) + ν +1 θ +1 e θ +1 x + ∗ (cid:96) + ν +2 θ +2 e θ +2 x + ∗ (cid:96) = κ . ( C at x + ∗ (cid:96) ) (64)Now, we can perform the iterations y h → ( x + , (0) (cid:96) , x + , ∗ (0) (cid:96) ) → y h → ( x + , (1) (cid:96) , x + , ∗ (1) (cid:96) ) . . . .We find the following fixed–point of the best–response functions of the producer and the consumer: C II , ∗ p = (cid:20) . , . , − , + ∞ . , . , . , . (cid:21) , C II , ∗ c = [ −∞ , . , (65)The system starts in the expansion regime, and once the price reaches level X ∗ t = 4 . , the consumerswitches to contraction and the systems remains in that state for ever. After that, she relies on theproducer to impulse ( X ∗ t ) up/down when prices get too low/too high but never reverts to the Expansionregime. Thus, in the long-run ( X ∗ t ) is simply a Brownian motion with negative drift µ − that has twoimpulse boundaries x − (cid:96) = 2 . , x − h = 6 . .Compared to the double–switch equilibrium of the previous section, the above market equilibriumin (65) has two important differences. First, as t → ∞ we have that µ ∗ t → µ − so that in the long–runthe market will be in the contraction regime and the consumer becomes inactive. Second, because theproducer eventually “takes over”, she will intervene much more frequently (see center panel of Figure 5),benefiting himself and reducing consumer value. The producer may have an interest to preempt the switch, say, from the expansion regime to thecontraction regime to avoid decline in the consumption of the commodity he produces. In this case,the equilibrium is a fixed point of the best–response function of the producer described in Section 3.2.3and the best–response function of the consumer described in Section 3.1.2. We look for an equilibriumwhere the consumer would like to switch at y h in the expansion regime, but where the producer makes y h his own intervention threshold to impulse the price down.Using the same protocol as in Type I and Type II equilibrium research, we find the followingthreshold strategy for the producer and the consumer: C III , ∗ p := (cid:20) . , . , . , . − , − , − , − (cid:21) , C III , ∗ c := [ − , . . (66)In the preemptive equilibrium, the price fluctuates in a narrower range than the other two equilibria.Here, ( X ∗ t ) oscillates between x + (cid:96) = 1 . and x + h = y h = 4 . . There are at least three potential equilibria. A natural question is thus whether one of them ispreferable to the others. Figure 7 shows the value functions of the producer and of the consumer inthe two market regimes (expansion and contraction) and for the different equilibria from type I totype III. We observe that the producer would prefer in both regimes to live in a type I equilibrium.The function v ± dominates all the other ones (note that there is no v − because in equilibrium type IIIcontraction never happens). However, the consumer would rather be in the preemptive equilibriumtype III: her value function w +3 dominates the other two. Intuitively, we may think that the switchingcosts she saves by letting the producer do all the work of maintaining the price around its long termaverage value compensate for the inconvenience of having prices that are higher than preferred.22 Figure 7: Producer and consumer value functions in the different equilibria.
As one example of comparative statistics that are possible in our model, we investigate the impact ofvolatility parameter σ of X on the equilibrium profits and behavior of X ∗ . In Table 2 we list statisticsof φ ∗ for a range of market volatilities σ . We also quantify the profitability of the two players throughtheir average percentage of optimality (APOO) APOO := (cid:82) D π r ( x ) φ ∗ ( dx ) π r ( ¯ X r ) , which is the ratio between average profit rate π r ( X ∗ t ) in equilibrium and the maximum profit thatcould be hypothetically obtained at the first–best level π r ( ¯ X r ) , r ∈ { c, p } .In all types of equilibria, both players are worse off in terms of expected profit rate as σ increase.This occurs even though in type I and in type II equilibria the average price E [ X ∗ ] increases. However,that gain is dominated by the losses due to higher Var( X ∗ ) which implies that prices tend to be furtherfrom their preferred levels ¯ X r decreasing E φ ∗ [ π r ] . A key parameter that controls which equilibrium type we face is the consumer’s switching cost h .Starting from the double–switch situation, as h increases ( > . ), the consumer is less incentivisedto switch from µ + to µ − and we enter the single–switch scenario of Section 3.1.2. Consequently, shereceives the No–Switch payoff ω +0 ( x ) when µ t = µ + and solving for her best–response boils down tosolve for y h only. Once h gets very large, her best–response is simply the No–Switch response ω ± .Conversely, as h ↓ her actions become free. In that situation, we can reduce the producer problem23 E φ ∗ [ X ∗ ] Var φ ∗ ( X ∗ ) E φ ∗ (cid:2) π p (cid:3) E φ ∗ (cid:2) π c (cid:3) Switch (per yr)Type I 0.25 3.52 0.73 0.81 (80%) 2.4 (80%) 0.0210.3 3.62 0.80 0.80 (79%) 2.2 (73%) 0.0210.4 3.77 0.94 0.76 (75%) 1.90 (62%) 0.020Type II 0.25 3.73 0.68 0.87 (85%) 2.3 (74%) 0.0200.3 3.76 0.74 0.85 (84%) 2.2 (71%) 0.0200.4 3.81 0.85 0.81 (80%) 1.95 (64%) 0.020Type III 0.25 3.41 0.45 0.86 (85%) 2.7 (90%) 0.00.3 3.35 0.51 0.83 (82%) 2.7 (90%) 0.00.4 3.28 0.61 0.78 (77%) 2.7 (88%) 0.0Table 2: Long–run mean and variance of X ∗ , long–run profit rates, and frequency of regime switchesas market volatility σ changes (APOO in parentheses).to a single, piecewise VI with a free boundary ˜ X c : sup (cid:110) − βv ( x ) + µ − v x + 12 σ v xx + π p ( x ) ; sup ξ (cid:8) v ( x + ξ ) − v ( x ) − K p ( ξ ) (cid:9)(cid:111) = 0 x > ˜ X c , (67) sup (cid:110) − βv ( x ) + µ + v x + 12 σ v xx + π p ( x ) ; sup ξ (cid:8) v ( x + ξ ) − v ( x ) − K p ( ξ ) (cid:9)(cid:111) = 0 x < ˜ X c , (68)with the C regularity at ˜ X c : lim x ↑ ˜ X c v ( x ) = lim x ↓ ˜ X c v ( x ) .Fig. 8 shows that for low h , x + ∗ h < y (cid:96) and x + h is greater but close to y h . Thus, when consumptionswitches from expansion to contraction, it is very likely that the price touches x + h soon thereafter andis impulsed back to x + ∗ h and thus, the regime rapidly switches back to expansion again. When theswitching cost increases, this solution disappears. h Figure 8: Equilibrium thresholds as a function of consumer switching cost h given Table 1 parametervalues. We show the consumer thresholds y (cid:96) , y h , the producer thresholds x + (cid:96) , x − h and respective impulsetarget levels x ∗ , + (cid:96) , x ∗ , − h . Remark 3.
It is possible for the impulse amounts to be so large as to lead to a double simultaneouscontrol: producer’s impulse instantaneously followed by the consumer switching. In this setting, theproducer effectively forces the consumer to switch the regime by impulsing X ∗ hard enough. Thissituation corresponds to x −∗ h < y (cid:96) , so that the impulse in the contraction regime moves X ∗ intothe respective switching region ( −∞ , y (cid:96) ) , and as a result the consumer immediately switches to theexpansion regime. This situation occurs if, for instance, the drifts are µ − = 0 . , µ + = 0 . , sothat the consumer is not able to ever efficiently lower prices. Consequently, the producer is forcedto fully control price reduction. We observe in the above situation the equilibrium thresholds of x −∗ h = 3 . < y (cid:96) = 3 . . (cid:50) Case study: diversification effect of vertical integration
The industrial organization of upstream and downstream segments is linked to anti-trust regulations.From a regulatory perspective, vertical integration could be used to increase market power and foreclosecompetitors. For example, see De Fontenay and Gans (2005) [10] who develop a game theoretic modelfor the foreclosure effect of vertical concentration and Hasting and Gilbert (2005) [15] for the relatedempirical facts in the context of the US retail gasoline market. At the same time, consumers can benefitfrom vertical integration of commodity producers; we refer to related analysis of electricity marketsfrom a market equilibrium point of view (Aïd et al. (2011) [2]) and an empirical point of view (Mansur(2007) [26]). Another virtue of vertical integration is its potential to reduce the long-term exposure ofthe firm to commodity price fluctuations. See Helfat and Teece (1987) [16] for an empirical estimationof the hedge procured by vertical integration in the oil business.In this section, we accordingly study whether or not downstream or upstream firms have an interestin being vertically integrated. To this end we consider a small firm that has no market power regardingthe commodity price X and focus on the case of the market equilibrium type I (generic case). Wethen investigate whether the firm can benefit from a diversification effect by having activity both inthe downstream consumer side and the upstream conversion side.To make the case study concrete, we consider a simplified version of the crude oil and gasolinemarkets, the latter a shorthand for refined products, calibrated to the ballpark of the 2019 state of theworld. Currently, world oil consumption is about 100 Mb/d (millions of barrels per day), normalizedto 1 "barrel" per day. We take as a nominal initial price X = 50 USD/b and a nominal volatility ofcrude σ =
10 USD/b. To calibrate our model, we consider that crude oil producers have a preferredrange of prices that goes from x p = 30 USD/b to x p = 100 USD/b and that the average cost of oilextraction is c p = 30 USD/b. This leads to a demand function D p ( x ) = 1 − . x , which captures thelow sensitivity of the demand of crude to prices. The crude is transformed into gasoline with a smallamount of losses 5%, so that the conversion factor is α = 0 . .We set the transfer function of crude oil price to average price of gasoline to P ( x ) = 10 + 1 . x ,where P ( x ) is also expressed in USD/b. There is evidence that the (pre-tax) price of gasoline is alinear function of the crude. For instance, using monthly data of the Energy Information Agency ofthe US Department of Energy on refined products prices from January 1983 to November 2019 , weregressed the US Total gasoline Retail sales by refineries ˆ P to the monthly crude oil price ˆ X and founda linear relation ˆ P m = 1 . X m + 14 + (cid:15) m with a regression R = 95% . Considering that the basket of refined products includes not just gasoline(even if it accounts for the largest share), we simplified the relation. Note that the condition p ≥ α forhaving a downstream convex profit function holds. Furthermore, refinery costs c c are highly variablebetween 4 to 10 USD/b. We take the higher value of c c = 10 . Finally, we consider that the demandfunction for refined products, D c ( P ) = d (cid:48) − d (cid:48) P is such that d (cid:48) = 5 b/d of crude equivalent refinedproducts and d (cid:48) = 0 . . With these parameters, the preferred range of crude prices for the consumeris between x c = 11 and x c = 82 USD/b. We consider fixed action costs both for the production firmand the downstream firm. We consider that the producer and the downstream firm lose two yearsof profit at optimal price to change state making κ = 2 π p ( ¯ X p ) and h = 2 π c ( ¯ X c ) . Finally, we take µ ± = ± . per year, which implies that it takes 10 years for the crude price to increase by 1.5 USD.The resulting equilibrium type I producer impulse strategy C I , ∗ p and consumer’s switching strategy C I , ∗ c associated to the calibration summarized in Table 3 are given by C I , ∗ p = (cid:20) , , − , + ∞−∞ , − , , (cid:21) , C I , ∗ c = (cid:2) . , (cid:3) . (69)Thus, in equilibrium, the crude price X ∗ fluctuates between . and USD/b, with potential excur-sions up to or down to USD/b at which point producers intervene. Data available at . β . Discount rate %/year X Initial oil price USD/b d Demand function for oil: intercept Mb/d d . Demand function for oil: slope Mb/d/(USD/b) d (cid:48) Demand function for gasoline: intercept Mb/d d (cid:48) . Demand function for gasoline: slope Mb/d/(USD/b) α . Transformation rate dimensionless p Crude – gasoline price function: intercept USD/b p . Crude – gasoline transfer price function: slope USD/b/(USD/b) c p Oil production cost USD/b c c Refining cost USD/b µ ± ± . Annualized crude drift parameters USD/b σ Annualized crude volatility USD/b h π c ( ¯ X c ) = 29 Consumption switching cost USD κ π p ( ¯ X p ) = 24 . Production switching cost: fixed USD κ Production switching cost: proportional USD/bTable 3: Nominal values for model parameters for the crude oil case study. x Figure 9: Left: Profit rate function of the consumer π c ( x ) (red) as the pass-through parameter p isvaried, as well as the fixed producer profit rate π p ( x ) (blue). Middle: Respective equilibrium strategythresholds y (cid:96) , y h (red) and x + (cid:96) , x − h (blue) as a function of p . We also plot ¯ X c and E φ ∗ [ X ∗ ] , shadingthe typical commodity price range [ E φ ∗ [ X ∗ ] ± σ φ ∗ ( X ∗ )] . Right: Risk-minimizing integration level λ ∗ as a function of p .
15 20 25 30020406080100 0 1 24 26 28 30 32 3420406080100 01 30 40 50 6030405060708090100 01
Figure 10: The curve λ (cid:55)→ ( σ ( π λ ) , E (cid:2) π λ (cid:3) ) as a function of the pass-through parameter p .Now let us consider a small firm engaging in a fraction λ ∈ (0 , of activity in the downstreamsector and − λ in the upstream sector. Her profit rate is thus π λ := λπ c + (cid:0) − λ (cid:1) π p . The firm isvertically integrated when < λ < . Denote by σ ( π λ ) the standard deviation of her profit rate π λ ( · ) integrated against the stationary distribution φ ∗ of X ∗ , and by E (cid:2) π λ (cid:3) = (cid:82) π λ ( x ) φ ∗ ( dx ) the respective26xpected profit rate. To fix ideas and because the analysis is symmetric, we are interested in situationswhere a pure downstream firm ( λ = 0 ) would be better off having part of her activity in the upstreamsector. This will take place when the upstream activity provides a higher expected profit rate and/or alower risk as measured by σ ( π λ ) . Figure 10 presents the risk–return curves λ (cid:55)→ ( σ ( π λ ) , E (cid:2) π λ (cid:3) ) as thepass-through parameter p increases from the nominal value of . to . . We observe that for lowvalues of p diversification gains are limited: expected profit rate goes up but risk also increases. Formoderate p a pure downstream firm unambiguously benefits from some upstream activity: she canachieve the same level of risk with a higher expected profit. For high p the upstream sector dominatescompletely with lower risk and higher average profit. Figure 9 (Right) shows the critical integrationlevel λ ∗ that minimizes the risk σ ( π λ ) and captures the “variance–minimal” business model.We observe that for high enough pass-through values, being a producer ( λ = 1) dominates anyother combination of activity. This phenomenon happens even though the maximum profit rate of thedownstream firm π c ( ¯ X c ) increases and gets higher than the producer’s maximum profit rate function π p ( ¯ X p ) as shown in the left panel of Figure 9. As shown by the evolution of equilibrium price rangein Figure 9 (Middle), as p increases, the equilibrium is getting more and more detrimental to thedownstream firm. The shaded salmon area represents the interval [ E φ ∗ [ X ∗ ] − σ φ ∗ ( X ∗ ) , E φ ∗ [ X ∗ ] + σ φ ∗ ( X ∗ )] where commodity prices tend to reside. The average commodity price remains stable around USD/b, and its the standard deviation is not affected much by p either, while ¯ X c is steadilydecreasing. Thus, since the expected profit rate of the integrated firm is a function of the expectedprice and its standard deviation, it does not change much. But its variance grows as a function of p andthus increases significantly. To conclude, in our model we do observe a diversification effect obtainedby mixing upstream and downstream activities, however the integration gains depend closely on thepass-through parameter p which serves as a transmission channel of the volatility of the commodityprice to the retail price. We showed how a simple model of competition between upstream and downstream representative firmshaving different pace of intervention can lead to a rich variety of equilibria, potentially non–unique.The fact that the upstream firm can impact the price more rapidly than the downstream firm givesthe producer a significant advantage, enabling him to lock the consumer in the producer’s preferredrange of prices. Further, in the case of the crude oil market and its refinery products, we stressedhow the pass-through parameter p plays a key role for the diversification effect induced by verticalintegration. Vertical integration is beneficial for low values of p while for higher values, productiondominates downstream activity both in terms of expected profit rate and profit standard deviation. Proof.
The proof is standard, nonetheless we give some detail for the reader’s convenience. To easethe notation, let us consider only the case µ = µ + , the other case being identical. Let w +0 ( x ) = (cid:98) ω + ( x ) + u + ( x ) , where the parameters ( λ +1 , , λ +2 , ) ∈ R solve the system (20)-(21). By construction thefunction w +0 is of class C everywhere. Hence we can apply Itô’s formula to e − βs w +0 ( X s ) on the timeinterval [0 , t ∧ ζ n ) , yielding e − β ( t ∧ ζ n ) w +0 ( X t ∧ ζ n ) = w +0 ( x ) + (cid:90) t ∧ ζ n e − βs (cid:2) w + (cid:48) ( X s − )( µ + ds + σdW s − dN s ) − βw +0 ( X s − ) ds (cid:3) + σ (cid:90) t ∧ ζ n e − βs w + (cid:48)(cid:48) ( X s − ) ds + (cid:88) w + ( x ) − h and it is optimal to switch to µ − atany x (therefore µ + would never be observed in the resulting game evolution). At the same time, wesee that if h is moderate (the solid line), then the region where w − ( x ) > w +0 ( x ) − h is disconnected ,so it is likely that a two–threshold switching strategy is an optimal response. x w w (a) C p = (cid:20) . . . . . . . . (cid:21) x w w w - h (b) C p = (cid:20) . . . . . . . . (cid:21) Figure 11: No–Switch payoffs ω ± ( x ) of the consumer given the producer’s strategy C p . Proof of Proposition 2.
By construction, the functions w ± ( x ) in (29) solve the system of VIs in (27)-(28) and satisfy w + ∈ C (( x + (cid:96) , x + h ) \ { y (cid:96) } ) ∩ C (( x + (cid:96) , x + h )) ∩ C ( R ) and w − ∈ C (( x − (cid:96) , x − h ) \ { y h } ) ∩C (( x − (cid:96) , x − h )) ∩ C ( R ) . Let N denote the pure jump component in X ’s dynamics associated to theproducer’s strategy with thresholds ( x ± (cid:96) , x ±∗ (cid:96) ; x ± h , x ±∗ h ) . The proof is structured in two steps. – Step 1: optimality. The following verification argument proves that such functions coincide with thebest–response payoffs of the consumer and that the switching times ˆ σ i as in the statement are optimal28rovided they are admissible. First, by an approximation procedure as in the first part of the proofin [1, Theorem 3.3], we can assume without loss of generality that w + ∈ C (( x + (cid:96) , x + h )) ∩ C ( R ) . Let µ − = µ + . Consider two consecutive switching times of any consumer admissible strategy, say σ i and σ i +1 , for i ≥ with the convention σ = 0 , and recall that over [ σ i , σ i +1 ) the state process X hasdrift µ + . Applying Itô’s formula to e − βt w + ( X t ) over the interval [ σ i ∧ T, σ i +1 ∧ T ) , for some finite T > , we obtain e − β ( σ i +1 ∧ T ) w + ( X σ i +1 ∧ T ) = e − β ( σ i ∧ T ) w + ( X σ i ∧ T )+ (cid:90) σ i +1 ∧ Tσ i ∧ T e − βs (cid:26) w + x ( X s ) dX s + σ ω + xx ( X s ) ds − βX s ds (cid:27) + (cid:88) σ i ∧ T and the fact that ζ z,µ > almost surely for all ( z, µ ) ∈ Z ± . This shows that sequence of switching times ˆ σ i is an admissible consumer’s strategy andconcludes the proof. Proof of Proposition 3.
Let v : { µ − , µ + } × R → R be the function defined as v ( µ ± , x ) = v ± ( x ) ,with v ± as in (47). By construction, the functions ( v + , v − ) solve the system of VIs in (45) andmoreover v ± ∈ C (( x + (cid:96) , x − h ) \ { y (cid:96) , y h } ) ∩ C ( R ) , hence not necessarily C at the points y (cid:96) , y h . Recall that µ t = µ + (cid:80) ∞ i =0 { σ i ≤ t<σ i +1 } + µ − (cid:80) ∞ i =1 { σ i − ≤ t<σ i } , t ≥ , where without loss of generality we canassume σ i is the i -th switching instance taken by the consumer in the case µ − = µ + (remember theconvention σ = 0 ). The other case µ − = µ − can be treated in a similar way, it is therefore omitted.We split the rest of the proof in two steps. – Step 1: optimality. The following verification argument proves that such functions coincide with thebest–response payoffs of the producer and that the impulse strategy as in the statement is optimalprovided it is admissible. First, by an approximation procedure as in the first part of the proof in [1,Theorem 3.3], we can assume without loss of generality that v ± ∈ C (( x + (cid:96) , x − h )) ∩ C ( R ) . Consider any30roducer admissible strategy ( τ i , ξ i ) i ≥ as in the first part of Definition 1. Applying Itô’s formula to e − βt v ( µ t , X t ) over the interval [ σ i ∧ T, σ i +1 ∧ T ) , for some finite T > , we obtain e − β ( σ i +1 ∧ T ) v ( µ σ i +1 ∧ T , X σ i +1 ∧ T ) = e − β ( σ i +1 ∧ T ) v + ( X σ i +1 ∧ T )= e − β ( σ i ∧ T ) v + ( X σ i ∧ T )+ (cid:90) σ i +1 ∧ Tσ i ∧ T e − βs (cid:26) v + x ( X s ) dX s + σ v + xx ( X s ) ds − βv + ( X s ) ds (cid:27) + (cid:88) σ i ∧ T and the fact that ζ z,µ > almost surely for all ( z, µ ) ∈ Z ± . This shows that ( τ ∗ i , ξ ∗ i ) i ≥ is an admissible producer’s impulse strategy and concludesthe proof. Proof of Proposition 4.
Here, notice that x + h = y h so that, given producer’s priority in case of simul-taneous interventions (cf. Remark 2), the drift is always equal to µ + (recall that we are in the case µ − = µ + ). Hence, this proof can be performed as the one of Proposition 3, by ignoring the intervalswhere the drift is µ − so that the second half in the RHS of inequality (70) is zero. The admissibilityis proved in the same way. The details are therefore omitted.32 .4 Equilibrium Dynamics Computation Let ( a, b ) be an arbitrary interval and x ∈ ( a, b ) be an interior starting location. We define δ + ( x ; a, b ) to be the first passage time associated to the interval ( a, b ) of a Brownian Motion with drift µ + startingfrom x and P + ( x ; a, b ) to be the probability that this BM hits a before b (similarly for δ − ( x ; a, b ) and P − ( x ; a, b ) associated to drift µ − ). These quantities admit explicit expressions, see [6].The expected time τ − := inf { t : µ t = µ − } for µ t to switch from µ + to µ − within a double–switchand one–sided impulse equilibrium is then E (cid:2) τ − (cid:3) = E (cid:2) δ + ( x ; x + (cid:96) , y h ) (cid:3) + P + ( x ; x + (cid:96) , y h ) P I + h ,S − E + (cid:2) δ ( x + ∗ (cid:96) ; x + (cid:96) , y h ) (cid:3) , (71)where P is the transition matrix of M ∗ n . Above, the first term denotes the time to either reach x + (cid:96) (producer impulses up) or y h (switch to contraction); the second term counts the additional time if x + (cid:96) is reached first multiplied by the respective probability P + ( x ; x + (cid:96) , y h ) . Let (cid:126)ζ be the resulting vector ofexpected sojourn times. Then the long-run proportion of time that X ∗ carries a positive drift ( µ + ) is ρ + = Π S + ζ S + + Π I + (cid:96) ζ I + (cid:96) + Π I + h ζ I + h (cid:126) Π · (cid:126)ζ † , (72)and similarly the long-run proportion associated to a negative drift ( µ − ) is ρ − = 1 − ρ + . References [1] Aïd, R., Basei, M., Callegaro, G., Campi, L., and Vargiolu, T. (2020). Nonzero-sum stochasticdifferential games with impulse controls: a verification theorem with applications.
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