# Dynamic Oligopoly with Sticky Prices: Off-Steady-state Analysis

## Abstract

In this paper we carry out a comprehensive analysis of the model of oligopoly with sticky prices with full analysis of prices’ behaviour outside their steady-state level in the infinite horizon case. An exhaustive proof of optimality is presented in both open loop and closed loop cases.

## Keywords

Cournot oligopoly Sticky prices Nash equilibrium Dynamic oligopoly model Open loop Feedback Closed loop## Mathematics Subject Classification

91A10 91A23 91A40 91B24 91B55## JEL Classification

L13 C72 C73 C61## 1 Introduction

Dynamic oligopoly models have a long history, starting from Clemhout et al. [12] and encompassing many issues, including, among other things, advertisement (e.g. Cellini and Lambertini [8]), adjustment costs (e.g. Kamp and Perloff [24], Jun and Viwes [23]), goodwill (e.g. Benchekroun [7]), pricing (see e.g. Jørgensen [22]), hierarchical structures (e.g. Chutani and Sethi [11]), nonstandard demand structure (e.g. Wiszniewska-Matyszkiel [29] with demand derived from dynamic optimization of consumers at a specific market) or a combination of several of these aspects (e.g. De Cesare and Di Liddo [15], Dockner and Feichtinger [16] and some papers cited previously). One of important issues considered in such models is price stickiness.

In this paper we do a comprehensive analysis of the model of oligopoly with sticky prices, first proposed by Simaan and Takayama [27]. The analysis of the model, using game theoretic tools, was afterwards continued by Fershtman and Kamien [19, 20], Cellini and Lambertini [9] and further generalised in many directions by, among others, Benchekroun [6], Cellini and Lambertini [10], Dockner and Gaunersdorfer [17] and Tsutsui and Mino [28].

Comprehensive reviews of differential games modelling oligopolies including models with sticky prices can be found in, among others, Dockner et al. [13] and Esfahani [18].

Both open loop and feedback information structures in the infinite horizon case were considered in [19] and [9]. In both papers, the analysis of the open-loop problem was restricted to the calculation of steady states only. When the feedback case is considered, off-steady-state behaviour was considered only in [19] and only for 2 players, but even in this case the focus was on the steady state. Consequently, the only results that could be compared in the \(N\) players model with infinite horizon were steady states for the open loop and feedback cases. This seems very partial solution of the problem.

First, the optimal stationary pair (costate variable, price) and, consequently, optimal stationary pair (production, price) for the open-loop case in all the previous papers was proved to be a saddle point, so unstable in the sense of Lapunov.

In such a case, an obvious expectation for almost all initial conditions is that solutions would not converge to this steady state. However, we shall prove that this is not going to happen and that the lack of stability is only apparent. Moreover, previous calculations are correct. These statements may seem a contradiction and, therefore, we shall return to this issue and give more emphasis on it in Remark 1 in Sect. 3.1. Here we only emphasise that stability results could not be proved without precise off-steady-state analysis, which we do in this paper.

Our calculations allow us to compare solutions of both types for the same initial price. Note that comparing trajectories of the open loop and feedback case solutions originating from the same initial price is impossible if we have only information about their steady states, which differ, since at least one trajectory in such a case is not stationary. Therefore, an analysis of the off-steady-state behaviour is really needed.

Another issue, important to obtain completeness of reasoning, is an appropriate infinite horizon terminal condition. As we can see from this paper, in order to have the standard terminal condition for the Bellman equation fulfilled in the feedback case, we have to impose additional constraints on the initial problem. Also in the open-loop case, applying an appropriate form of Pontryagin maximum principle (as it is well known, the standard maximum principle does not have to be fulfilled in the infinite horizon case), is nontrivial, even using the latest findings in infinite horizon optimal control theory.

To address these two issues, in this paper we concentrate on the off-steady-state analysis of the model, both in the open loop and the feedback information structure cases, and we give a rigorous proof, including applicability of a generalisation of a Pontryagin maximum principle and checking terminal conditions in both.

When the feedback case is considered, we use the standard Bellman equation stated in e.g. Zabczyk [30], since the value function is proved to be smooth.^{1}

## 2 The Model

We consider a model of an oligopoly consisting of \(N\ge 2\) identical firms, each of them with a cost function \(C_i(q)=cq +\frac{q^2}{2}\). At this stage we only assume that amounts, \(q\), are nonnegative.

We assume that the inverse demand function is \(P(q_1,\dots ,q_N)=A-\sum _{i=1}^N q_i\), with \(q_i\) being the amount of production of \(i\)-th firm and \(A\) a positive constant substantially greater than \(c\). This defines how the price would react to firms’ production levels if the adjustment was immediate.

However, since prices are sticky, they adjust according to an equation of the form \(\dot{p}=s(P-p)\), where \(s>0\) denotes the speed of adjustment. This equation will be fully specified in Sect. 2.2.

### 2.1 The Static Case

If we consider the static case, with prices adjusting immediately, then each firm maximises over its own strategy \(q_i\) the instantaneous payoff \(pq_i -C(q_i)\), where the price \(p\) can be treated in two ways.

First, we can look for a standard Nash equilibrium solution. Generally, a profile of players’ strategies is a Nash equilibrium if no player can increase his payoff by changing strategy unless the remaining players change their strategies. By applying the concept of Nash equilibrium to the static oligopoly model with strategies being production levels, we obtain the standard Cournot equilibrium, which is often referred to as the Cournot-Nash equilibrium.

At the Cournot-Nash equilibrium, each of the firms knows its influence on the price, therefore, it maximises \(P(q_1,\dots ,q_N)q_i-C(q_i)\). The resulting simultaneous optimization of each firm, as it can be easily calculated, results in the equilibrium production of each firm \(q_i^{\text {CN}}=\frac{A-c}{N+2}\) and the price level \(p^{\text {CN}}=\frac{2A+Nc}{N+2}\).

For comparison, at the competitive equilibrium, in which firms are price takers, and maximise with \(p\) treated as a constant, production of each firm \(q_i^{\text {Comp}}=\frac{A-c}{N+1}\) and the price level \(p^{\text {Comp}}=\frac{A+Nc}{N+1}\).

### 2.2 Dynamics Resulting from Sticky Prices

Now, we introduce dynamics reflecting price stickiness.

Given a measurable \(q_i\), the corresponding trajectory of price is absolutely continuous. At this stage we do not have to assume that prices are nonnegative. Obviously, we shall prove in Proposition 1 that at every equilibrium they are positive.

## 3 Open-Loop Nash Equilibria

It is well known that generally in the infinite time horizon the standard transversality condition \(\lambda (t) {{\mathrm{e}}}^{-\rho t}\rightarrow 0\) (where \(\lambda \) is a costate variable) is not necessary. There are many papers with counterexamples to this transversality condition, see e.g. Halkin [21] and Michel [25].

For the specific case considered in this paper, we can prove that the standard transversality condition is necessary. As our problem is nonautonomous, the well known Aseev and Kryazhimskiy [1, 2] results cannot be directly applied. However, we use a very general result of Aseev and Veliov [3, 4], which extends the Pontryagin maximum principle and can be applied to nonautonomous infinite horizon optimal control problems.

As we prove, applying these necessary conditions to the optimal control arising from calculation of the open-loop Nash equilibrium, and given the initial price, restricts the set of possible solutions to a singleton, so it is enough to check that the optimal solution exists to prove sufficiency of the condition. To this end, we use existence of an optimal solution proven by Balder [5].

This issue is tackled in Appendix. Necessary conditions generalising standard Pontryagin maximum principle from Aseev and Veliov [3, 4] and Balder’s [5] existence theorem are stated in sections “Aseev and Veliov Extension of the Pontryagin Maximum Principle” and “Existence of Optimal Solution”, while we prove our model fulfils the assumptions of those theorems in sections “Checking Assumptions for Theorem 5 for the Model Described in Sect. 2” and “Checking Assumptions for Theorem 6 for the Model Described in Sect. 2”, respectively.

This also allows us to determine, what is not so obvious in infinite horizon optimal control problems and differential games, not only the terminal condition for the costate variable at infinity, but also the initial condition for it, which, in turn, determines uniquely what is the trajectory of the state variable and the optimal control for every initial value of the state variable—the initial price.

### 3.1 Application of the Necessary Conditions

In this section we are going to use the necessary condition given by Theorem 5 to derive the optimal production and price at Nash equilibria for the open-loop case.

However, to simplify further calculations, instead of the usual hamiltonian we use present value hamiltonian \(H^{\text {PV}}(t,x,u,\lambda )=g(t,x,u)+<f(t,x,u),\lambda >\) (using notation of section “Aseev and Veliov Extension of the Pontryagin Maximum Principle”), where the new costate variable \(\lambda (t)=e^{\rho t}\Psi (t)\). We rewrite the maximum principle formulated in Theorem 5 for optimization of payoff by player \(i\) with fixed strategies of the remaining players \(q_j\in \fancyscript{S}^{\mathrm {OL}}\).

### **Lemma 1**

Let \(i\in \{1,2,\cdots ,N\}\) be an arbitrary number and let \(q_j\in \fancyscript{S}^{\mathrm {OL}}\) for \(j\ne i\) be any strategies such that the trajectory corresponding to the strategy profile \((q_1,\dots ,0,\dots ,q_N)\), where \(0\) is on the \(i\)-th coordinate, is nonnegative.

### *Proof*

As it is proved in section “Checking Assumptions for Theorem 5 for the Model Described in Sect. 2” the assumptions of the Aseev-Veliov maximum principle—Theorem 5 are fulfilled.

### **Lemma 2**

Under the assumptions of Lemma 1, the necessary condition for \(q_i\) to be an optimal control for optimization problem given by Eq. (2) with dynamics of \(p\) given by Eq. (1) is as follows.

### *Proof*

It is immediate as a result of application of maximum principle to our problem, given by Lemma 1. \(\square \)

### **Lemma 3**

At every open-loop Nash equilibrium there exist costate variables \(\lambda _i \) such that for every \(t\ge 0\), \(\lambda _i (t)>0\).

### *Proof*

By Lemma 1 the adjoint variable \(\lambda _i(t)\) is nonnegative.

Assume that \(\lambda _i(\bar{t})=0\) for some \(\bar{t}>0\). It implies \(\forall w\ge \bar{t}\), \(q_i(w)=0\). Suppose \(p(w)>c\) for some \(w\ge \bar{t}\). In this case we can increase payoff by increasing \(q_i\) to \(\epsilon \) on some small interval \([w,w+\delta ]\). This contradicts the assumption that we are at the Nash equilibrium.

Now we shall concentrate on symmetric Nash equilibria. Note that if \(p_0 \ge c\), then at equilibrium it is impossible to have \(p(w)< c\) for any \(w\), since in such a case instantaneous profit at some interval is negative for positive \(q_i\) for \(p(w)\le c\), which implies that optimization of profit results in \(q_i(w)=0\) for \(p(w)\le c\). Since the same analysis holds for all players, \(p(w)=c\) implies \(\dot{p} (w)>0\).

So the only case left is \(p(w)=c\) for all \(w\ge \bar{t}\), which we have just excluded. \(\square \)

### **Theorem 1**

Let \((\lambda (t),p(t))\) be a solution to Eq. (13) with initial value \((\lambda _0,p_0)\). Then \(\lambda (t){{\mathrm{e}}}^{-\rho t}>0\) and converges to \(0\) as \(t\rightarrow \infty \) if and only if \((\lambda _0,p_0)\in \Gamma \), where \(\Gamma \) is a stable manifold of the steady-state \((p^{\mathrm {OL},*},\lambda ^*)\) of Eq. (13).

### *Proof*

The phase diagram of Eq. (13) is presented in Fig. 1.

Let \(\Gamma _1\) be a part of the stable manifold of the steady state for \(p<p^{\mathrm {OL},*}\) and \(\Gamma _2\) be a part of the stable manifold of the steady state for \(p>p^{\mathrm {OL},*}\). Looking at the phase portrait we can deduce the following behaviour of the solution to Eq. (13). If the initial point \((\lambda _0,p_0)\) lies left to the stable manifold of the steady state (a thick dark brown line in Fig. 1), then there exists \(\bar{t}>0\) such that \(\lambda (\bar{t} )\le 0\).

### **Corollary 1**

Now we are ready to formulate theorem considering the symmetric open loop Nash equilibrium.

### **Theorem 2**

### *Proof*

At a Nash equilibrium every player maximises his payoff treating strategies of the other players as given. Let us consider a player \(i\). The formula for his optimal strategy \(q_i\) and costate variable \(\lambda _i\) is derived in Lemma 2.

If we consider a symmetric open-loop Nash equilibrium, then all \(q_i\) are identical, and, therefore, all the costate variables of players are identical, so at this stage we can skip the subscript \(i\) in \(\lambda _i\) and substitute \(q_j(t)=q_i(t)\), which implies the state equation becomes \(\dot{p}(t)=s(A-Nq_i(t)-p(t))\).

The condition appearing in Eq. (10) splits the first quadrant of \((\lambda ,p)\) plane into two sets: \(\Omega _1\) defined by Eq. (11), and \(\Omega _2\) defined by Eq. (12). Using them, and Theorem 1 we can deduce that \(p\) and \(q\) fulfil Eq. (13). \(\square \)

### *Remark 1*

We have proved that the steady-state \((\lambda ^*, p^{\mathrm {OL},*})\) from the necessary conditions of each player’s optimization problem is globally asymptotically stable on the set of possible initial conditions (fulfilling these necessary conditions).

We emphasise this fact, since the previous papers on dynamic oligopolies with infinite horizon and sticky prices, which considered only steady-state analysis in the open-loop case, ended the analysis by conclusion that the steady state is a saddle point, which means lack of stability in the sense of Lapunov. This lack of Lapunov stability was on the whole space \((\lambda , p)\) or \((q,p)\). Our calculations, mainly Theorem 1, prove that there is a unique costate variable \(\lambda _0\) for given \(p_0\) for which the necessary conditions are fulfilled and this \(\lambda _0\) places the pair \((\lambda _0, p_0)\) on the stable saddle path, which implies that whatever the initial \(p_0\) is, \(\lambda , p\), and consequently, \(q\) converges to their steady states.

Now, we prove the following Lemma that we will use instead of checking sufficiency condition for the candidate for optimal control.

### **Lemma 4**

Let us consider any player \(i\) and fixed strategies of the remaining players \(q_j\in \fancyscript{S}^{\mathrm {OL}}\) for \(j\ne i\) such that the trajectory corresponding to the strategy profile \((q_1,\dots ,0,\dots ,q_N)\) (with \(0\) in the \(i\)-th coordinate) is nonnegative. Then the set of optimal solutions of the optimization problem of player \(i\) is a singleton.

### *Proof*

The optimal control exists by Balder’s existence Theorem 6 (see section “Existence of Optimal Solution”).

### **Theorem 3**

There is a unique open-loop Nash equilibrium and it is symmetric.

### *Proof*

Further, we calculate the equilibrium. By Corollary 1, for \(p_0<\bar{p}\), \(p\) is given by Eq. (21), while in the opposite case by Eq. (23). Thus, to derive formula for \(q^{\mathrm {OL}}\) we consider two cases.

So we have obtained a candidate for optimal control with the corresponding state and costate variables trajectories fulfilling the necessary condition. In Lemma 4, we have proved that there exists a unique optimal control for any strategies of the remaining players such that the trajectory corresponding to the strategy profile \((q_1,\dots ,0,\dots ,q_N)\) is nonnegative. Therefore, a strategy which fulfils the necessary condition is the only optimal control. Applying this to all players, together with the fact that there is only one symmetric profile fulfilling the necessary condition, implies that there is exactly one symmetric Nash equilibrium. \(\square \)

## 4 Feedback Nash Equilibria

It is worth emphasising that assuming even only continuity is in many cases too restrictive, since it excludes, among others, so called bang-bang solutions, which are often optimal. Therefore, the only thing we assume a priori besides measurability is that a solution to Eq. (29) exists for initial conditions in \([c,+\infty )\) and it is unique (if a solution is not continuous at certain point we assume Eq. (29) holds almost everywhere). As the symbol of all feedback strategies we use \(\fancyscript{S}^{\text {F}}\).

Here, we want to mention that in some works this form of information structure and, consequently, strategies is called closed loop. In this paper, as in most papers, we shall understand by closed-loop information structure as consisting of both time and state variable, so using the notation of closed-loop strategy \(q_i(t,p)\) we can encompass, as trivial cases, both open loop and feedback strategies.

We can formulate the following result.

### **Theorem 4**

To prove Theorem 4, we need the following sequence of lemmata.

### **Lemma 5**

- (a)Consider a dynamic optimization problem of a player \(i\) with the feedback information structure, with dynamics of prices described by (29) and the objective function (30), but with the set of possible control parameters available at a price \(p\) extended to some interval \([-B|p|-b,B|p|+b]\), with some constants \(B,b>0\). Assume that the strategies of the remaining players are described by \(\hat{q}(p)=p(1-sk)-c-sh\) for constants \(k\), \(h\) given by (33) and (34), respectively, and \(g\) given byThen, there exists \(B\), \(b>0\) such that the quadratic function$$\begin{aligned} g=\frac{c^{2}-sh(sh-2A-2N(c+sh))}{2\rho }. \end{aligned}$$(37)is the value function of this optimization problem, while \(\hat{q}\) is the optimal control.$$\begin{aligned} V_{i}(p)=\frac{kp^{2}}{2}+hp+g \end{aligned}$$
- (b)For these \(B\), \(b\), \(q_i\equiv \hat{q}\) for \(k_i=k\), \(h_i=h\) defines a symmetric feedback Nash equilibrium. Moreover, equality \(\hat{q}(\tilde{p})=0\) holds. The corresponding price level fulfils$$\begin{aligned} p^{\mathrm {F}}(t)=\left[ p_{0}-\frac{A+N(c+sh)}{N(1-sk)+1}\right] {{\mathrm{e}}}^{s(Nsk-N-1)t}+\frac{A+N(c+sh)}{N(1-sk)+1}. \end{aligned}$$(38)

### *Proof*

Note that \(2s(Nsk-N-1)t\le \rho \), which implies that \({{\mathrm{e}}}^{\rho t}V(p^{\text {cand}}(t))\rightarrow 0\). Therefore, we can take \(B=1-sk+\epsilon \) for a small \(\epsilon \) (we can easily check that \(1-sk>0\)) and an arbitrary \(b>|c+sh|\), then we have both \({{\mathrm{e}}}^{\rho t}V(p(t))\rightarrow 0\) for every admissible trajectory, which means that the terminal condition (40) is fulfilled and \(q^{\text {cand}}\) belongs to the set of admissible controls \(|q_i|\le B|p|+b\). This ends the proof of (a).

The point (b) follows immediately. \(\square \)

### **Lemma 6**

- (a)Consider a dynamic optimization problem of a player \(i\) with feedback information structure. Assume that the strategies of the remaining players are described by \( \hat{q}(p)=p(1-sk)-c-sh\) for \(k\), \(h\) defined by (33) and (34), respectively. For \(p\ge \tilde{p}\), where \(\tilde{p}\) is defined by (31) the value function fulfils \(V_{i}(p)=V_i^+(p)\) for \(V_i^+=\frac{k p^2}{2}+hp+q\) defined in Lemma 5, while the optimal control is$$\begin{aligned} q_i(p)=q^{\text {cand}}(p) \end{aligned}$$
- (b)The equationholds at a symmetric Nash equilibrium.$$\begin{aligned} q_i(p)=q^{\text {cand}}(p) \text { for } p\ge \tilde{p}, \end{aligned}$$

### *Proof*

- (a)
If the remaining players choose \(q^{\text {cand}}\), then \(q^{\text {cand}}\) is the optimal control over a larger class of controls, since \([-B|p|-b,B|p|+b]\) contains \([0,q_{\max }]\) (by Proposition 1, this set of control parameters leads to results equivalent to the initial case with the set of control parameters \(\mathbb {R}_+\)). By (38), if \(p_0\ge \tilde{p}\), then for all \(t\ge 0\), the corresponding trajectory fulfils \(p(t)\ge \tilde{p}\). The price \(\tilde{p}\) is the threshold price such that for \(p\ge \tilde{p}\), \(q^{\text {cand}}\ge 0\), otherwise it is negative. To conclude, the optimal control for analogous optimization problem with larger set of controls is in this case contained in our set of controls, so it is optimal in our optimization problem. The point (b) follows immediately.\(\square \)

### **Lemma 7**

- (a)Consider a dynamic optimization problem of a player \(i\) with feedback information structure. Assume that the strategies of the remaining players are \(q^{\text {F}}_i\) defined by Eq. (35). If the optimal control of a player \(i\) is also given by (35) , then the value function fulfils$$\begin{aligned} V_{i}(p)={\left\{ \begin{array}{ll} \frac{kp^{2}}{2}+hp+g &{} \text { for } p\ge \tilde{p},\\ (A-p)^{-\frac{\rho }{s}}(A-\tilde{p})^{\frac{\rho }{s}}\left( \frac{k\tilde{p}^{2}}{2}+h\tilde{p}+g\right) &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$(43)
- (b)
The function \(V_i\) defined this way is continuous and continuously differentiable.

### *Proof*

- (a)In this case the set of \(q_i\) for which the maximum of right-hand side of the Bellman Eq. (39) is calculated is \([0,q_{\max }]\). By Lemma 6, for \(p\ge \tilde{p}\), the value function fulfils \(V_{i}(p)=V_i^+(p)=\frac{k\tilde{p}^{2}}{2}+h\tilde{p}+g\) and the optimal control for these \(p\) coincides with \(q_i^{\text {F}}\). If we take \(p<\tilde{p}\) and we substitute \(q_i^{\text {F}}\), we get thatfor \(\tilde{t}\) defined by (32). Let us introduce another auxiliary function \(V_i^-\) defined on \([c,A)\) by$$\begin{aligned} V_i(p)&= \int _0^{\tilde{t}}{{\mathrm{e}}}^{-\rho t}\left( (p^{\text {F}}(t)-c)q_i^{\text {F}}(p^{\text {F}}(t))- \frac{q_i^{\text {F}}(p^{\text {F}}(t))^2}{2}\right) dt\\&\quad +\int _{\tilde{t}}^{\infty }{{\mathrm{e}}}^{-\rho t}\left( (p^{\text {F}}(t)-c)q_i^{\text {F}}(p^{\text {F}}(t))- \frac{q_i^{\text {F}}(p^{\text {F}}(t))^2}{2}\right) dt\\&= \int _0^{\tilde{t}}0dt+\int _{\tilde{t}}^{\infty }{{\mathrm{e}}}^{-\rho t}\left( (p^{\text {F}}(t)-c)q_i^{\text {F}}(p^{\text {F}}(t)) -\frac{q_i^{\text {F}}(p^{\text {F}}(t))^2}{2}\right) dt\\&= {{\mathrm{e}}}^{-\rho \tilde{t}}V_i(\tilde{p}), \end{aligned}$$and consider also \(V_i^+ \) as a function defined on \([c,A)\). The fact that proposed \(q_i\) is optimal, results in the value function$$\begin{aligned} V_i^-(p)=(A-p)^{-\frac{\rho }{s}}(A-\tilde{p})^{\frac{\rho }{s}}\left( V_i^+(\tilde{p}) \right) \end{aligned}$$(44)$$\begin{aligned} V_{i}(p)={\left\{ \begin{array}{ll} V_i^+ &{} \text { if }p\ge \tilde{p}\\ V_i^- &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$(45)
- (b)Continuity is immediate, while to prove continuous differentiability we have to check whether the function \(V_i\) defined by (43) is \(C^1\) at \(\tilde{p}\), which is equivalent to equality of derivatives of \(V_i^+\) and \(V_i^-\) at \(\tilde{p}\):Checking it does not require a dull substitution of predefined constants, since by the Bellman equation derived in Lemmas 5 and 6 and the fact that the maximum in the Bellman equation for \(p=\tilde{p}\) is attained at \(0\), \(\rho V_i^+(\tilde{p})=(V_i^+(\tilde{p}))'s(A-\tilde{p})\). Thus, Eq. (46) is equivalent to \((V_i^+(\tilde{p}))'=\frac{1}{s}\frac{(V_i^+(\tilde{p}))'s(A-\tilde{p})}{A-\tilde{p}}\), which reduces to the required equality.\(\square \)$$\begin{aligned} (V_i^+(\tilde{p}))'=(V_i^-(\tilde{p}))'=\frac{\rho }{s}\frac{V_i^+(\tilde{p})}{A-\tilde{p}}. \end{aligned}$$(46)

### *Proof*

First, we have to prove that the Bellman equation is fulfilled and \(q_{i}^{\text {F}}(p)\) maximises the right-hand side of the Bellman equation. For \(p\ge \tilde{p}\) it has been already proved in Lemma 6.

The terminal condition for the optimization problem is also fulfilled. First, the set of prices is bounded from above, therefore \( \mathrm limsup _{t\rightarrow \infty } V_i(p(t)){{\mathrm{e}}}^{-\rho t}\le 0\). On the other hand, \(q_i\in [0,q_{\max }]\), therefore, \(\dot{p}\ge s(A-Nq_{\max }-p)\). Hence, \(p(t)\ge c_1+c_2 {{\mathrm{e}}}^{-st}\), which implies that \( \mathrm limsup _{t\rightarrow \infty } V_i(p(t)){{\mathrm{e}}}^{-\rho t}\ge 0\).

Now, let us calculate the trajectory of the price corresponding to the symmetric Nash equilibrium we have just determined. We start from the first case in Eq. (36): \(p(0)=p_{0}\) is such that \(p_{0}\ge \tilde{p}\).

## 5 Relations and Comparison

In this section we compare two classes of equilibria for various values of parameters.

First it is obvious that whatever the initial condition is, the open loop Nash equilibrium is not a degenerate feedback equilibrium. We obtain it immediately by the fact that the steady states of the state variable—price—for open loop and feedback equilibrium are different and globally asymptotically stable.

### 5.1 Asymptotic Values of the Nash Equilibria with Very Slow and Very Fast Price Adjustment

Consider the asymptotic of the Nash equilibria for \(s\rightarrow 0\) and \(s\rightarrow +\infty \).

### 5.2 Graphical Illustration

#### 5.2.1 Relations Between Open Loop and Feedback Equilibria

It implies that the feedback Nash equilibrium ensures higher utility to the consumers, while open-loop Nash equilibrium yields higher profits to the producers. In other words, the feedback Nash equilibrium is more competitive, which is a result consistent with the previous literature, among others, Fershtman and Kamien [19] and Cellini and Lambertini [9].

The reason for the difference between these two types of equilibria results from the way in which players perceive the influence of their current decisions about production level on future trajectory of prices. In the feedback case, every player has to take into account the fact that his/her current decision affects future trajectory of prices not only directly, but also indirectly, since it affects future decisions of the remaining players, which in the feedback case is dependent on the market price. Therefore, there are two contradictory effects: negative effect that increase of his/her production has on future prices, and indirect inverse effect, resulting from the fact that the other players’ production level is an increasing function of prices. In this second effect an increase of player’s production, by resulting in a decrease of future price, indirectly decreases the other players’ future production decisions. Consequently, as a sum of two effects of opposite signs, the resulting decrease of prices is smaller. In the open-loop case, in which only direct influence on future prices is considered in players’ optimization problems, such an inverse effect does not take place.

#### 5.2.2 Dependence on the Number of Firms

#### 5.2.3 Dependence on the Speed of Adjustment

## 6 Conclusions

In this paper we study a model of oligopoly with sticky prices performing a complete analysis of trajectories of production and prices at symmetric open loop and feedback Nash equilibria. We consider not only constant trajectories, resulting from assuming that initial values are steady states of these equilibria, respectively, but also all admissible Nash equilibrium trajectories. This allows us to compare two approaches in a way similar to comparisons observed in the real life, in which it makes sense to compare only trajectories of both kinds of equilibria starting from the same initial value. It also allows us to find interesting properties which cannot be observed when only the steady-state behaviour is analysed, like intersection of trajectories of production level for various number of firms or speed of adjustment. In both cases for larger value of the parameter considered, there was first faster increase of production and afterwards convergence to a lower steady state.

We also proved, by refining previous results, that the steady state in the open-loop case, is, as in the feedback case, globally asymptotically stable.

## Footnotes

- 1.
Generally, value functions obtained in similar dynamic game theoretic problems may have a point at which they are not differentiable. To solve such problems, there are generalisations of the standard Bellman equation for continuous but nonsmooth value function, using viscosity solution approach. This approach was already used in dynamic games with applications in dynamic economic problems, e.g. by Dockner at al. [13], Dockner and Wagener [14] or Rowat [26].

- 2.

## Notes

### Acknowledgments

This research of A. Wiszniewska-Matyszkiel supported by Polish National Science Centre grant 2011/01/D/ST6/06981.

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