Pedro Jara-Moroni
email: pjara@dim.uchile.cl

The Cournot Outcome as the Result of Price Competition *

Keywords: Subgame Perfect Nash Equilibrium, Price, Quantity, Pure Strategies, Cournot. JEL Classification: C73, D43, L13

In a homogeneous product duopoly with concave revenue and convex costs we study a two stage game in which, first, firms engage simultaneously in capacity (production) and, after production levels are made public, there is sequential price competition in the second stage. Randomizing the order of play in the price subgame, we can find: (i) that the Cournot outcome can be sustained as a pure strategy subgame perfect Nash equilibrium (SPNE) of the whole game, (ii) a SPNE in which firms produce strictly more than the Cournot outcome.

Introduction

In their paper of 1983, Kreps and Scheinkman state that the difference between Cournot and Bertrand competition is more than just the strategy space, but that timing of decisions is also relevant. To illustrate this, they study and solve a Bertrand like duopoly model of competition where timing of decision is inverted. Capacity decision is made simultaneously and before price decision (as opposed to Bertrand-Edgeworth models where the choice of capacity and price is interpreted as being simultaneous), and the low priced firm may not serve all the demand at her price (as it is in the Bertrand approach) due to capacity constraints. In a two stage game where firms first set simultaneously capacity and then engage in simultaneous price competition with demand rationed following the efficient rationing rule, the unique Subgame Perfect Nash Equilibrium (SPNE) has as outcome the Cournot quantities and prices. This result has been subject to criticism. [START_REF] Davidson | Long-run competition in capacity, short-run competition in price, and the cournot model[END_REF] argue that the Kreps and Scheinkman result depends strongly on the chosen rationing rule and that in fact, it is not likely that the Cournot outcome may raise as the result of the price game in most of the cases. However, their proof assumes value zero for the cost in the first (capacity-setting) stage. Moreover, for an important range of capacities the unique Nash Equilibrium of the price-setting subgame in [START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF] is in non degenerate mixed strategies, rendering the result hard to interpret and motivating a literature that argues that the only possible pure strategy Nash Equilibrium of price games must be the competitive outcome (see [START_REF] Allen | Price-setting firms and the oligopolistic foundations of perfect competition[END_REF] or [START_REF] Dixon | The competitve outcome as the equilibrium in an edgeworthian price-quantity model[END_REF]).

Related to the first issue [START_REF] Madden | Elastic demand, sunk costs and the kreps-scheinkman extension of the cournot model[END_REF] shows, in a slightly different framework, that for uniformly elastic demands the Kreps and Scheinkman result holds, even if proportional rationing is used. Related to the second one, we have that departing from the original constant marginal cost and no capacity constraints hypothesis [START_REF] Bertrand | Théorie mathématique de la richesse sociale[END_REF], we pass to models with voluntary trading constraint at the competitive supply or strictly increasing marginal costs [START_REF] Edgeworth | The pure theory of monopoly[END_REF], which lead to problems with existence of equilibrium in pure-strategies. Liberating the supply constraint to any possible quantity in order to restore pure-strategy equilibria existence when convex costs are present, we end up with the competitive outcome. These Bertrand-Edgeworth approaches to oligopolistic competition, however, still feature the main difference between Bertrand and Cournot models that Kreps and Scheinkman were assessing. Namely, the production decision is actually done after prices and demands are revealed. For instance, in [START_REF] Dixon | The competitve outcome as the equilibrium in an edgeworthian price-quantity model[END_REF] the equilibrium-statedintention of serving the whole demand can be interpreted as a non credible threat or a unsustainable engagement since once a firm realizes that it has the lowest price it is not rational to serve more than the competitive supply at the announced price. Thus we are back at the timing of decisions issue.

The objective of this note is then threefold: without changing any assumptions on primitives and maintaining the timing of decisions (prices after quantities), provide a setting where the Cournot outcome can be sustained as a SPNE in pure strategies of the duopoly game, give space for the possibility of finding non-Cournot outcomes sustained as SPNE in pure strategies and give a clear condition on rationing rules that rejects Cournot as the outcome of equilibrium.

We explore a setting where in a first stage firms set capacity simultaneously and in a second stage engage in sequential price competition where the order of play is assigned randomly. The possibility of obtaining clear pure strategy subgame perfect equilibria for each subgame will help to clearly understand the strategic interaction in the duopoly framework that results in the Cournot outcome, although competition in prices is assumed. We will see that under the same assumptions on primitives as in the seting of Kreps and Scheinkman the Cournot outcome is sustained as a subgame perfect equilibrium. We will also see a simple condition on rationing rules to reject the Cournot outcome in both the Kreps and Scheinkman and the current setting. A last new feature of the approach presented in this paper is the possibility of finding non Cournot outcomes sustained as SPNE without changing any of the assumptions on demand rationing [START_REF] Davidson | Long-run competition in capacity, short-run competition in price, and the cournot model[END_REF][START_REF] Dixon | The competitve outcome as the equilibrium in an edgeworthian price-quantity model[END_REF], nor on concavity or continuity of involved functions nor cost structure [START_REF] Vives | Commitment, flexibility and market outcomes[END_REF]Deneckere andKovenock, 1992, 1996), but only based on uncertainty of leadership and the dynamic nature of the price subgame. We explore this issue and give conditions that allow for it to occur.

The paper provides a clear and more tractable story for the Cournot outcome as the result of price competition.

Preliminaries

Consider a market with two identical firms a and b that produce a homogeneous good. Aggregate demand for the firms outputs as a function of price is given by the function D : R + → R + . We assume that 0 < D(0) < +∞, that there is an upper bound for the set where D(p) > 0, and that D is continuous and strictly decreasing in this set. With this we can consider the Price function P : R + → [0, P (0)] defined by P (q) := D -1 (q) for q ∈ [0, D(0)] and P (q) := 0 for q > D(0). The cost of producing is given by a function C : R + → R + .

We make the following assumptions:

1. The function p → pD(p) is strictly concave.

2. Function C is strictly increasing, convex, and C(0) = 0;

P satisfies P (0) > lim k→0 C (k) > 0.
With these elements we can define the Cournot best response mapping r C : R + → [0, D (0)] as:

r C (k) := argmax x∈[0,D(0)-k] {P (k + x) x -C (x)} if k ≤ D (0) 0 if k > D (0)
For simplicity we note r(k) ≡ r 0 (k). Assumptions 1 and 2 imply that r C is strictly decreasing when it is positive, that the function k → r (k) + k is non-decreasing (strictly increassing in the interval where r (k) > 0) and that if we have two cost functions C 1 and C 2 with C 1 > C 2 then r C1 (y) < r C2 (y) ∀ y ∈ [0, D(0)] such that 0 < r C2 (y). Thus, for each function C as above there is a unique Cournot equilibrium outcome with price and production quantities (p * C , k * C , k * C ) defined by:

r C (k * C ) = k * C D(p * C ) = 2k * C or equivalently p * C = P (2k * C ) And if C 1 > C 2 then k * C1 < k * C2 and p * C1 > p * C2 .
We will note k * and p * for k * 0 and p * 0 respectively. Finally we note R(k) the value:

R(k) := P (r(k) + k) r(k),
the revenue associated with the best response to a quantity k when there is no production cost and p m the zero cost monopoly price1 .

We will study a game in which these firms engage in a two stage competitive situation. In a first stage, firms simultaneously and independently build capacity (which can be understood as a decision in production quantities). A vector of capacity quantities k = (k a , k b ) is understood as an available supply on the market of k a + k b to be traded after the second stage. In the second stage, after learning how much capacity their opponent installed, firms set prices sequentially.

Net payoffs in the setting of [START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF] are given by the function [START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF] show that in this setting, the only subgame perfect Nash equilibrium outcome of a two stage game with simultaneous price competition in the second stage, is the Cournot outcome. Note that we are using, as in their work, the surplus maximizing rationing rule. The choice of rationing rule is not obvious. For instance, [START_REF] Davidson | Long-run competition in capacity, short-run competition in price, and the cournot model[END_REF] show that not all standard rationing rules lead to obtain Cournot price and quantity as the outcome of two stage game with simultaneous price setting. We address this issue as well in this work in a further section.

π i : R 2 + × R 2 + → R i ∈ {a, b}: π i (k, p) :=        p i min {k i , D(p i )} -C(k i ) if p i < p j p i min k i , max D(pi) 2 , D(p i ) -k j -C(k i ) if p i = p j p i min {k i , max {0, D(p i ) -k j }} -C(k i ) if p i > p j

Capacity Constrained Price Competition

In this section we analyze subgame perfect equilibria of the sequential price game with capacity constraint. In this stage, each firm i has installed a capacity k i i ∈ {a, b} and firms know the capacity installed by the rival. In this stage, since cost has already been sunk, the only relevant part of payoff is the revenue associated to sells. Firms name a price in order to maximize this revenue. Note that in capacity constrained price competition, prices smaller than P (k a + k b ) are strictly dominated and will therefore not be best replies to the rivals price nor be announced in equilibrium. This is true in sequential as well as in simultaneous price competition.

To continue, we first need to define certain elements.

Definition 3.1. For k i > 0, (i, j) ∈ {(a, b), (b, a)}. We define p(k i , k j ) as follows:

(i) If k i ≥ r(k j ) then let p (k i , k j ) be the smallest solution2 of the equation:

p min{k i , D(p)} = R(k j ) (3.1) (ii) If k i < r(k j ) then p (k i , k j ) := P (k a + k b )
R(k j ), the revenue for firm i when playing the best response to j in the zero cost Cournot game, is the maximum revenue that i can receive by being the high priced firm when having enough capacity. Indeed, revenue for i being the high priced firm is equal to p i min {k i , max {0, D(p i ) -k j }} which, if k i ≥ r(k j ) turns into p i max {0, D(p i ) -k j } and is maximized at p i = P (r (k j ) + k j ) when k j ≤ D (0) with the value P (r (k j ) + k j ) r (k j ) ≡ R (k j ). 3When k i ≥ r (k j ), price p(k i , k j ) equals the payoff of being the low priced firm to R(k j ), that is, price p (k i , k j ) makes firm i indifferent between being the high priced firm and the low priced firm. If k i < r (k j ), then the previous analysis does not make sense, but in this case payoff is bounded from above by pk i for any

p ≤ P (k a + k b ) and if p > P (k a + k b ) then it is in the decreasing part of p max {0, D(p) -k j }.
Roughly, if firm j sets a price lower than p (k i , k j ) then firm i would rather set the highest between the residual demand monopoly price P (r(k j ) + k j ) and the competitive price P (k a + k b ), and if j sets a higher price then firm i would rather undercut or match. Finally note that if

k i ≥ r(k j ) then p (k i , k j ) > P (k a + k b ).
A very useful result regarding the limit prices p (k i , k j ) defined in (3.1) is:

Lemma 3.2. min {D (0) , k i } > k j k j > r(k i ) =⇒ k i R(k i ) < k j R(k j ) Moreover if k i , k j > 0, p (k i , k j ) > p (k j , k i )
We will name by firm 1 the firm that sets the price first and firm 2 the one who goes second. We obtain a capacity constrained Stackelberg price game. This kind of game has been analyzed in [START_REF] Deneckere | Price leadership[END_REF], and more recently in [START_REF] Dastidar | On stackelberg games in a homogeneous product market[END_REF], where capacity is assumed to be the competitive supply at the announced price. In both approaches, following [START_REF] Deneckere | Price leadership[END_REF], demand is assigned first to the follower and then to the leader in order to have no problems with upper semi-continuity of payoffs at their suppremum. We will make this same assumption on tied prices: all demand goes first to the follower (and is not equally shared as in [START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF]). Thus when matching a price p 1 , firm 2 will net π 2 (k, p 1 , p 1 ) = lim p p1 π 2 (k, p 1 , p). Formally, tied prices revenues become,

π 1 (k 1 , k 2 , p, p) := p min {k 1 , max {0, D(p) -k 2 }} π 2 (k 1 , k 2 , p, p) := p min {k 2 , , D(p)}
That is, given capacities (k a , k b ) and price p 1 , for firm 2 we use the upper semi-continuous hull of the payoff function of the Kreps and Scheinkman setting, and for firm 1, given p 2 we use the lower semi-continuous hull of the payoff function of the Kreps and Scheinkman setting.

With this "slight" modification the sequential price-setting subgame has always (meaning for all capacity pairs) a subgame perfect equilibrium in pure strategies. The main implication of the modification is that now player two has always a best response (since for all p 1 it's payoff as a function of price is sup-compact and upper semi-continuous). The strategy for firm 2 will simply be to match the price of firm 1 if it is greater or equal to p (k 2 , k 1 ) and take the residual monopoly price if not. Given the capacities set in the first stage, the best response mapping p * 2 ( • | k 1 , k 2 ) : R + ⇒ R + for firm 2 as a function of price p 1 , set by firm 1, is:

p * 2 (p 1 | k 1 , k 2 ) =      R + if p 1 = p (k 2 , k 1 ) = 0 max {P (k a + k b ) , P (r (k 1 ) + k 1 )} if p 1 ≤ p (k 2 , k 1 ) and p (k 2 , k 1 ) > 0 min {p 1 , max {P (k 2 ) , p m }} if p 1 ≥ p (k 2 , k 1 ) (3.2)
The outcome of the game will depend on the action of firm 1, since it's profit maximizing price, given the strategy of firm 2, may depend on the first stage choice of capacity through the relative values of the prices p (k 1 , k 2 ) and p (k 2 , k 1 ). Proposition 3.3 below is similar to Theorems 2 and 3 of [START_REF] Deneckere | Price leadership[END_REF]. The difference is that here we explicitly give the payoff of the low priced firm in the subgame where capacities are in the range where the simultaneous price-setting game has equilibrium in mixed strategies [START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF].

6

- Proposition 3.3. With respect to the sequential price-setting subgame we can distinguish five regions of interest:

D (0) D (0) 3 k 2 = r(k 1 ) 2 k 1 = r(k 2 ) 1 5 ? k 1 k 2 -4
1. If k 1 ≤ r(k 2 ) and k 2 ≤ r(k 1 )
, there is a unique pure strategy subgame perfect Nash Equilibrium (SPNE) in which both firms set price

P (k 1 + k 2 ). Payoffs are π 1 (k 1 , k 2 ) = P (k 1 + k 2 )k 1 and π 2 (k 1 , k 2 ) = P (k 1 + k 2 )k 2 . 2. If k 1 < k 2 , k 1 < D (0) and k 2 > r(k 1 )
then there is a unique SPNE where firm 1 sets price p (k 2 , k 1 ) and firm 2 responds by setting

P (r(k 1 ) + k 1 ). Payoffs are π 1 (k 1 , k 2 ) = p (k 2 , k 1 ) k 1 and π 2 (k 1 , k 2 ) = R(k 1 ) 3. If k 1 > k 2 , k 2 < D (0) and k 1 > r(k 2 )
then there there is a unique SPNE where both firms set price P (r(k

2 ) + k 2 ). Payoffs are π 1 (k 1 , k 2 ) = P (r(k 2 ) + k 2 )r(k 2 ) = R(k 2 ) and π 2 (k 1 , k 2 ) = P (r(k 2 ) + k 2 )k 2 . 4. If k 1 = k 2 = k, k < D (0)
and k > r(k), then there are two SPNE in pure strategies. One where firm 1 sets p (k, k) and firm 2 responds by setting

P (r(k) + k) with payoffs π 1 (k, k) = p (k, k) k = R (k) and π 2 (k, k) = R(k).
And another where both firms set P (r(k

) + k) and net π 1 (k, k) = P (r(k) + k)r(k) = R(k) and π 2 (k, k) = P (r(k) + k)k. 5. If min {k 1 , k 2 } ≥ D (0),
∈ R + if p * 1 = 0. Payoffs are π 1 (k 1 , k 2 ) = 0 for firm 1 and π 2 (k 1 , k 2 ) = min {p * 1 , p m } D (min {p * 1 , p m })
Suppose we have k a > k b and k a > r (k b ), so that we stand in the case were in the simultaneous price game the equilibrium is in non degenerate mixed strategies and the residual monopoly profit for firm a is greater than the market clearing profit4 . If firm b is the leader, then it knows that if it sets a price higher than p (k a , k b ) it will be undercut and then it would get the residual demand, but since p (k b , k a ) < p (k a , k b ) firm b would net strictly more by setting a price p (k b , k a ) < p b ≤ p (k a , k b ) since then firm a would rather take the residual demand. On the other hand, if firm b is the follower, since it is the small firm, in order to sell it's whole production by being the low priced firm, it can set a bigger price than firm a (P (k b ) > P (k a )). Firm a then assumes it will be undercut and thus maximizes it's payoff, as the high priced firm, setting the residual demand monopoly price P (r (k b ) + k b ). Then, firm b will set the same price (this is also true in the case k a = k b = k > r (k) since the rule assigns first the demand to the follower and thus it will sell all it's capacity). Then firm a will obtain a payoff R (k b ) with certainty and so it is indifferent between being leader or follower in the price setting game, while firm b would rather go second since it nets [START_REF] Deneckere | Price leadership[END_REF] they address the "choice of roles" issue rising from this observation, since we could say that if leadership were endogenized, then the large firm would become leader. The small firm would wait for the announcement of the large firm, which would then set the corresponding price first. The authors focus then on games of timing of price announcement.

p (k a , k b ) k b if it is leader and P (r (k b ) + k b ) k b if it is follower and in this capacity region we have P (r (k b ) + k b ) > p (k a , k b ). In
For the special case k a = k b = k > r(k) we have two SPNE, one where firms set different prices and one where both firms set the residual demand monopoly price, but only the second one allows us to have upper semi-continuity of reduced payoff in the capacity setting game. Since the first firm is indifferent between the two equilibria, we turn to upper semi-continuity of payoffs to find possible non Cournot symmetric Nash equilibria of the reduced capacity setting game in the next section. Regarding the second of these equilibria, because of the rationing rule, the firm that announces it's price in second place will net a strictly greater payoff than the leader, therefore considering this SPNE we could say that both firms would prefer to wait for the announcement of the rival before announcing it's own price. The rationing rule assigns the demand first to the follower, which is the sole difference between roles for this strategy profile.

It is well known that when splitting demand on tied prices there is no subgame perfect equilibrium on pure strategies in the price setting subgame, since for many prices p set by the leader, the follower does not have a well defined best response. For many of these cases we can argue that firm 2 infinitesimally undercuts firm 1 when out of the equilibrium path, allowing us to find pure strategy Nash Equilibria of the subgame. That is when the high capacity firm acts as follower. However, we still have the non existence problem when this firm is leader. There is a whole set of pairs (k a , k b ) for which no equilibrium exists. The problem rises when we analyze the game that begins at the information set described by k 1 ≥ k 2 and k 2 > r(k 2 ) (this region of the capacity vector plane falls into items 3 and 4 of Proposition 3.3, see Figure 2). In this price-setting subgame with equal sharing at tied prices there is no pure strategy NE, since it is not optimal for firm 2 to set the price P (r(k 2 ) + k 2 ) when firm 1 sets this price, because of the downward jump in its revenue function. Nevertheless, under a subgame perfect equilibrium logic, we can think of the strategy where firm 2 undercuts if firm 1's price is strictly greater than p (k 2 , k 1 ) and takes the residual demand monopoly price if not. Given this strategy for firm two, it is optimal for firm 1 to set the price P (r(k 2 ) + k 2 ). Then ∀ ε > 0, there would exist δ > 0 such that firm 2 could announce a price P (r(k 2 ) + k 2 ) -δ in order to secure a revenue u δ 2 (k 2 ) := (P (r(k

2 ) + k 2 ) -δ)k 2 that satisfies P (r(k 2 )+k 2 )k 2 -ε < u δ 2 (k 2 ) < P (r(k 2 )+k 2 )k 2 and π 1 (k 1 , k 2 ) = P (r(k 2 )+k 2 )r(k 2 ) = R(k 2 ).
That is, an "intuitive" revenue outcome for this subgame would be π

1 (k 1 , k 2 ) = P (r (k 2 ) + k 2 ) r (k 2 ) = 6 - D (0) D (0) 3.a k 2 = r(k 1 ) 2 k 1 = r(k 2 ) 1 3.b ? @ @ R ? k 1 k 2 -
no price set by 1 can be sustained as a NE Figure 2: Equilibria with demand splitting at tied prices. In region 3.b there is no (pure strategy) Nash Equilibrium in the sequential price subgame.

R (k 2 ) and π 2 (k 1 , k 2 ) = u δ 2 (k 2 ) ---→ ε→0 P (r(k 2 ) + k 2 )k 2 .
Proposition 3.3 shows that changing the splitting rule on tied prices not only helps to solve the mathematical problem of upper semicontinuity of payoffs, but also does not depart from the intuitive outcome (nor in payoffs nor in strategies) of the price game in the original setting, allowing to formally obtain these intuitive results over price competition with capacity constraints when the small firm is follower. We see that the continuity problem being solved by the Deneckere and Kovenock assumption on demand splitting on tied prices, does not affect the economic conclusions of the model and effectively helps to clarify the behavior of the agents.

Capacity Setting Reduced Game

Now that we have obtained all the SPNE of the prices-setting subgame, we can study the capacity setting stage of the game. The usual approach would be to use reduced payoff functions obtained from the outcomes of the subgame for each possible capacity pairs of the first stage generating a capacity setting reduced game. The problem is that for the case of k a = k b = k > r (k) we don't have a unique payoff for the follower. So situations regarding one of the firms setting a capacity k i greater than k * have to be studied carefully. Nevertheless, let's introduce the first stage capacity-setting game and the price-setting SPNE associated payoffs. In this stage firms set capacities simultaneously. They maximize their expected payoff of setting capacity k i knowing that the choice of the opponent will have an effect on the final profit. When setting capacity, firms do not know whether they will set their price first or second in the price-setting stage. This uncertainty may come, for instance, from the nature of the variable under scrutiny. Capacity building or stock production may occur long before price competition. Even more, once we enter the stage of price competition, and thinking in some kind of endogenous timing or role in the price game, we have that the only preference on roles is for the large firm being follower5 . As we have said, the small firm is indifferent between announcing its price first or second. Thus it is not clear how long firms can wait before being forced to announce a price. We will assume then that there is an exogenous factor that determines the role in the price setting stage. Firms know, however, in the first stage of the game, that there is a probability λ i of being leader in the price game for firm i, i ∈ {a, b}. In an incomplete information type of game, these distributions could represent the probability of being impatient in the second stage of the game. They can be interpreted as well as priors or beliefs of the firms about their role in the second stage price game. Similarly, they induce priors onto the possible payoff outcomes (not necessarily roles) of this subgame. Although we allow for the probabilities to be different for each firm and they could well depend on the installed capacities, it is better, to keep track of the exercise, to think of them as being equal to 1/2. As we will see below, the values of the probabilities λ i are irrelevant for the purpose of telling a neat economic story behind the Cournot outcome as a result of price competition. These values become important when seeking for other equilibria of the game, an issue that we address as well in what follows. The interpretation of these distribution as priors may raise as well questions about their genesis. Questions for instance, of endogenous generation of priors, which would inspire an exercise closely related to the choice of roles [START_REF] Hamilton | Endogenous timing in duopoly games: Stackelberg or cournot equilibria[END_REF][START_REF] Deneckere | Price leadership[END_REF][START_REF] Van Damme | Endogenous price leadership[END_REF].

The capacity setting reduced game has as players the two firms, with strategy space R + and payoff given by the function

π i : R + × R + → R, π i (k i , k j ) := λ i π 1 (k i , k j ) + (1 -λ i ) π 2 (k j , k i ) -C (k i ) (i, j) ∈ {(a, b) , (b, a)}
where π 1 and π 2 are the profit functions obtained from the equilibria of the sequential price-setting subgame described in Proposition 3.3.

Cournot Outcome

Regarding capacity setting competition, we first notice that the Cournot outcome is a Subgame Perfect Nash equilibrium of the two-stage game with sequential price-setting, as it was in the two-stage game with simultaneous price-setting stage case. If a firm sets the Cournot capacity, then the opponent gets the Cournot payoff up to a certain capacity, namely the Cournot best response for zero cost, and for greater capacities, since the higher capacity firm nets the residual demand monopoly payoff no matter being leader or follower in the second stage game, income remains fixed while cost increases. So now we have a clear neat story behind the Cournot outcome as the result of price competition. If firms set small capacities, then price competition produces clear-market prices. Facing Cournot capacity of the rival, a firm that sets a high capacity will net an income that depends only on the capacity set by it's rival. The large firm if being leader in the price game knows it will be undercut. If being follower, the small firm will set the highest price possible at which it can sell it's whole capacity, that is, p (k i , k j ) with k i > k j , the price that leaves the high firm indifferent between matching and taking the residual demand, so the large firm nets R (k j ).

The Cournot outcome can be sustained as a subgame perfect equilibrium in pure strategies at all stages of the game.

Other Equilibria

Considering the possibility of other equilibria, note that setting the same capacity and it being smaller than the value of the zero cost Cournot equilibrium k * , can not be an equilibrium since deviating to the Cournot best response maximizes payoff given the capacity of the rival. It is the same argument as in the previous Proposition. This rules out any pure-strategy Nash equilibrium with both capacities smaller than k * .

However it could be possible to find symmetric Nash equilibria where firms engage in higher capacities. Suppose that the expected reduced payoff (expectation taken over the possibility of being leader or follower) in the first stage game, given that the rival plays a capacity k strictly greater than the zero cost Cournot equilibrium k * , is increasing in the interval where there is a difference between being leader or follower, and that the expected reduced payoff of matching this capacity is greater (or equal) than the maximum payoff that could be attained by setting a capacity in the indifference (between being leader or follower) interval (if not void); and consider the following strategies:

In the first stage set k. In the second stage, once roles are assigned, follow the subgame perfect strategies given in Proposition 3.3. For the special case where k a = k b = k let the leader set the price P (r (k) + k).

Then, using price leadership as a coordination device, the outcome k, k can be sustained as a SPNE of the game.

Indeed, once capacities are set, what follows has already been discussed. So the question is: why would a firm set this capacity given that the other one did? The answer comes from the possibility that the firm was follower in the second stage game because the follower can set a relatively high price:

P 2 k < P r k + k .
First, notice that setting a higher capacity can not be a best response since payoff is decreasing in that zone. Given the strategies in the price-setting subgame, upper semi-continuity of reduced payoff gives the existence of the best response in the interval 0, k . Second, note that reduced payoff for the small firm is of the form

k → E p | k, k k -C(k) where E p | k,
k is the expected price faced by a small firm. Expected price E p | k, k for firm a against capacity k is given by:

E p | k a , k = P k a + k if k a ≤ r -1 k λ a p k, k a + (1 -λ a ) P (r (k a ) + k a ) if r -1 k ≤ k a < k
so that payoff for the small firm is continuous in 0, k and strictly greater than the Cournot payoff for capacities in the interval r -1 k , k .

For the small firm as a second mover, when comparing6 the maximum profit from setting a capacity k such that k < r -1 k and setting k, the fall in price (P k + k P r k + k > P k + k ) and the increase of the cost C(k) C k may be compensated by the increase in the volume of sales k k . The second mover price function, P (r (k) + k), is less steep than the standard Cournot price function, allowing the possibility that payoff was increasing in the interval r -1 k , k .

Second mover advantage has to be such that it drags sufficiently upwards the expected payoff

k → E p | k, k k -C(k) and the functions k → E p | k, k and k → -C(k)
need not be too strongly decreasing.

These two conditions can be written as find k such that:

P r k + k λ i r k + (1 -λ i ) k -C k ≥ max k∈[0,r -1 ( k)] P k + k k -C(k) (4.1) lim k k ∂ ∂k E p | k, k k -C(k) ≥ 0 (4.2)
Condition (4.1) states that the payoff of matching is greater than the payoff of setting a capacity in the zone where there is no difference between being leader or follower, and condition (4.2) states that the left hand derivative7 of the payoff function at the point k is positive which, along with concavity of payoff in the region where there is difference between being leader and follower and continuity at the point k = r -1 ( k), assures that payoff is maximized at k = k. We illustrate this situation with the following example:

Example 4.2. Consider the duopoly with price function P (q) = 1 -q, and cost C (q) = cq for both firms. The Cournot best response function is then:

r C (k) = max 1 -c -k 2 , 0 .
The zero-cost Cournot equilibrium is given by k * ≡ 1/3 and k * C = (1 -c) /3. If λ a = λ b = 1/2 and c is sufficiently small then we can find a continuum of equilibria along the line k a = k b . For instance for c ≡ 0.05, ka = kb = k = 0.3375 can be sustained by a SPNE of the game and there is a continuum of Nash Equilibria along the line k a = k b in a neighborhood of k. Of course, k * C is not in this neighborhood.

Cournot Outcome and Contingent Demands

In [START_REF] Davidson | Long-run competition in capacity, short-run competition in price, and the cournot model[END_REF] Theorem 1 argues that the Cournot outcome can not emerge as an equilibrium of the two stage game discussed by Kreps and Scheinkman considering almost any rationing rule, in a certain class, different from the surplus maximizing rationing rule. This is relevant as well for this work since the central argument states that if we change the rationing rule then, given that in the first stage the set capacities were the Cournot quantities, the Cournot price is not a best response to itself, ruling out as well the Cournot outcome as the equilibrium of the sequential subgame. However the proof is made only for the case of zero cost of capacity. As we will see below their result, as stated in their work, is in fact only valid for this case and once there are costs a finer treatment becomes necessary. [START_REF] Madden | Elastic demand, sunk costs and the kreps-scheinkman extension of the cournot model[END_REF] addresses this matter for a special setting that is not completely compatible with the Kreps and Scheinkman or Davidson and Deneckere assumptions on the demand function, but that enlightens what really is relevant when trying to answer the question about how dependent are the results on the chosen rationing rule.

Consider the class of contingent demands that satisfy:

1. D i (p i |p j ) downward-sloping twice differentiable except in p i = p j .

D

i (p i |p j ) = D(p i ) for p i < p j 3. D i (p i |p j ) = max D(pi) 2 , D(p i ) -k j when p i = p j 4. max {0, D (p i ) -k j } ≤ D i (p i |p j ) ≤ max {0, min {D(p j ) -k j , D(p i )}} for p i > p j
Both rationing rules: proportional and surplus maximizing satisfy points 1 to 4. Suppose that contingent demand for the high priced firm is given by a C 1 function CD ( • |p j , k i , k j ) : R + → R + such that:

D i (p i |p j ) =        D(p i ) if p i < p j max D(pi) 2 , D(p i ) -k j if p i = p j max {0, CD (p i |p j , k j , k j )} if p i > p j
and D i satisfies conditions 1 to 4. Note that for D i to satisfy conditions 1 to 4, we must have that CD (p j |p j , k j , k j ) = D (p j ) -k j . Using the notation e CD q,p for the elasticity of demand

CD ( • |p * C , k * C , k * C )
we can state the following proposition.

Proposition 4.3. If:

e CD q,p (p * C ) > -1 (4.3)
then the Cournot outcome does not rise as the equilibrium of the two stage game (sequential or simultaneous).

The proof follows the same ideas behind Davidson and Deneckere's proof. The condition implies that payoff as a function of price, is increasing at the Cournot price. In their result, the condition of D i (p i |p j ) being locally distinct to the right from the surplus maximizing contingent demand at the point p i = p j = p * C when k i = k j = k * C , is not sufficient when costs are not equal to zero and is a special case of our condition (4.3). Condition (4.3) is also in the line of [START_REF] Madden | Elastic demand, sunk costs and the kreps-scheinkman extension of the cournot model[END_REF], if the residual demand is not sufficiently elastic at the Cournot price, then the Cournot outcome can not emerge as an equilibrium of the two stage game.

Concluding Remarks

We have studied a three stage duopoly game where firms first set simultaneously capacities and then engage in sequential price competition. We have related the outcome of the sequential price game [START_REF] Deneckere | Price leadership[END_REF] to the simultaneous price game [START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF] and we have obtained that the Cournot outcome of the market can be sustained as a SPNE in pure strategies Mixed strategies are a solution to the problem of existence in simultaneous price games. However, there is still not a consensus about what a mixed strategy outcome represents in a price game. Usual justifications for mixed strategy equilibria such as a population distribution from where individuals are randomly selected to play the game, as in evolutionary game theory do note seem appealing. Moreover, the fact that the pure strategies that are in the support of the mixed strategy distributions are not necessarily best responses to the pure strategies that are in the support of the rival's mixed strategy distribution, gives rise to the regret property of mixed strategies equilibria: once a realization of the distribution is played, players may not be satisfied with the outcome, which is somehow incoherent with Nash behavior or perfectness of equilibria (see for instance [START_REF] Vives | Oligopoly Pricing: Old Ideas and New Tools[END_REF]; [START_REF] Friedman | On the strategic importance of prices versus quantities[END_REF] for a discussion on Mixed strategies Nash equilibria in oligopoly games).

We have proposed in this note an alternative approach to the problem of existence of equilibrium in pure strategies that is more in the line of [START_REF] Harsanyi | Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points[END_REF]. Firms are uncertain of their role in the price stage of the game. They know that sequential price competition will take place, but since this competition is so far in the future, price leadership is unknown. As in an incomplete information game, players know a probability distribution over their roles in the price game and maximize expected payoff in the first stage. Price leadership serves then as a coordination device that allows to obtain equilibrium outcomes in pure strategies. It is important to notice that this result does not depend on the value of the probability distribution, but only in the characteristics of the involved functions. That is, on the structure of the market: cost of production, elasticity of demand. We have seen that suitably and slightly modifying the sharing rule on tied prices it is possible to formally reproduce the intuitively expected behavior of firms in the original setting (equal share at tied prices).

Finally we have made space for the possibility of finding other symmetric quantity outcomes sustained as SPNE depending only on primitives and on the probabilities of being leader in the price game. No changes with respect to the [START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF] setting have been made on cost or demand functions. The discussion is opened then on how to treat such probabilities (endogenously generated, parameters of the model, firm independent, etc.). Many lines of exploration can be taken: general n firm oligopoly, non symmetric cost structures of firms [START_REF] Deneckere | Bertrand-edgeworth duopoly with unit cost asymmetry[END_REF] and effects of changes on contingent demand [START_REF] Davidson | Long-run competition in capacity, short-run competition in price, and the cournot model[END_REF], to give some examples.

A Proofs

Proof of Proposition 3.3. Consider the best response mapping for firm 2 presented in 3.2 and solve by backward induction.

For 1 p (k 2 , k 1 ) = P (k a + k b ) so we know that for any possible price set by firm 1 in equilibrium, firm 2 will match or undercut. Thus firm 1 gets the residual demand payoff which is maximized at p * 1 = P (k a + k b ).

For 2 given the strategy of firm 2, firm 1 nets the low priced firm payoff when setting p 1 < p (k 2 , k 1 ) and the residual demand payoff when setting p 1 > p (k 2 , k 1 ). Since p (k 1 , k 2 ) < p (k 2 , k 1 ), the only possibility for firm 1 to have a maximizer is to have upper semi-continuity at p 1 = p (k 2 , k 1 ). And this only obtains if firm 2 sets P (r (k 1 ) + k 1 ).

In 3 we have p (k 1 , k 2 ) > p (k 2 , k 1 ) and thus firm 1 maximizes it's payoff by setting P (r (k 2 ) + k 2 ) at which firm 2 responds by matching.

In 5, p (k 2 , k 1 ) = 0, so if p 1 > 0 firm two will match or undercut and firm 1 will net 0. Then, firm 1 can set, in equilibrium, any price p * 1 > 0, and firm 2 will match or undercut depending on the value of p * 1 . For the special case of p 1 = 0, note that then firm 1 still nets 0 (so it is a possible strategy in equilibrium) and firm 2 nets 0 as well, so then firm 2 can actually set any price in the price set. Thus any pair (0, p * 2 ), p * 2 ∈ R + can be sustained as a SPNE of the price game.

Figure 1 :

 1 Figure 1: The five regions of interest in Proposition 3.3. The arrows indicate at what region belong the boundaries.

  Proposition 4.1. The Cournot outcome defined by k a = k b = k * C and p a = p b = p * C can be sustained as a SPNE of the two-stage game.

C 0 8

 0 Proof of Proposition 4.1. Set k * b = k * C . Then k * b < k * and thus Proposition 3.3 gives a unique reduced payoff for firm a:π a (k a , k * C ) = P (k a + k * C ) k a -C (k a ) if k a ≤ r (k * C ) R (k * C ) -C (k a ) if k a > r (k * C ) Clearly π a ( • , k * C ) is continuous and if k a > r (k * C ) it is strictly decreasing. Moreover, from what we said in section 2, the function k a → k a P (k a + k * C ) -C (k a ), the Cournot payoff function, attains it's maximum at k * a = r C (k * C ) < r (k * C ) and thus k a = k * b is the best response to k * b = k * Cin the reduced form capacity-setting game. Since the (unique) best response to k * C in the reduced form game is to set the same capacity and the game is symmetric, we conclude.Proof of Proposition 4.3. Suppose the two players set k * C in the first stage. If player j sets price p * C the payoff for player i as a function of price is:π i (p, p * C , k * C , k * C ) = pk * C p ≤ p * C p max {0, CD (p|p * C , k * C , k * C )} p > p * CThis function is then continuous 8 and increasing to the left of p * C . The derivative to the right ofp * C is ∂ ∂p π i (p, p * C , k * C , k * C ) = p CD (p|p * C , k * C , k * C ) + CD (p|p * C CD (p * C |p * C , k * C , k * C ) + 1 > Since CD p|p * C , k * C , k* C satisfies 2, lim p↓p * C CD p|p * C , k * C , k * C = k * C . Then p * C / ∈ argmax p {π i (p, p * C , k * C , k * C )}.

  then there is a continuum of SPNE indexed by p * 1 where firm 1 sets p * 1 ∈ R + and firm 2 sets min {p * 1 , p m } if p * 1 > 0 and sets any price p * 2

p m = argmax pD (p)

Concavity of pD(p) and the fact that it takes value 0 in p = 0 and p = P (0), assure that (3.1) has at most two solutions.

When k j > D (0), the function p i → p i max {0, D(p i ) -k j } is constant equal to zero and so it is maximized in R + with value 0 (≡ R (k j ) since both P (r (k j ) + k j ) and r (k j ) equal 0).

The simultaneous price capacity constrained subgame has been analyzed by[START_REF] Levitan | Price duopoly and capacity constraints[END_REF] for the case of linear demand, no cost and equal capacities; and with more generality by[START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF] and[START_REF] Osborne | Price competition in a capacity-constrained duopoly[END_REF].

In the absence of naturally generated roles in sequential play[START_REF] Gal-Or | First mover and second mover advantages[END_REF][START_REF] Amir | Second-mover advantage and price leadership in bertrand duopoly[END_REF],[START_REF] Hamilton | Endogenous timing in duopoly games: Stackelberg or cournot equilibria[END_REF] and van[START_REF] Van Damme | Endogenous price leadership[END_REF] study game forms where players chose when to play in a sequential game. When choosing the same timing it is assumed that the simultaneous subgame equilibrium payoffs are obtained, which we prefer to avoid in this note, as[START_REF] Deneckere | Price leadership[END_REF] do in their work.

This comparison is relevant only if k ≤ r(0).

From the definition of p (k i , k j ) when k j k i .


Andrew E Clark
email: author.andrew.clark@ens.fr.

Nicolai Kristensen

Niels Westergård

Niels Westergård-Nielsen

Tor Eriksson

Takao Kato

ÉCOLE NATIONALE DES PONTS ET CHAUSSÉES -ÉCOLE NORMALE SUPÉRIEURE Job Satisfaction and Co-worker Wages: Status or Signal?

Keywords: C23, C25, D84, J28, J31, J33 Job satisfaction, co-workers, comparison income, wage expectations, tournaments Job Satisfaction, Co-workers, Comparison Income, Wage Expectations, Tournaments. JEL codes: C23, C25, D84, J28, J31, J33

This paper uses matched employer-employee panel data to show that individual job satisfaction is higher when other workers in the same establishment are better-paid. This runs contrary to a large literature which has found evidence of income comparisons in subjective well-being. We argue that the difference hinges on the nature of the reference group. We here use co-workers. Their wages not only induce jealousy, but also provide a signal about the worker's own future earnings. Our positive estimated coefficient on others' wages shows that this positive future earnings signal outweighs any negative status effect. This phenomenon is stronger for men, and in the private sector.

Introduction

A significant amount of work in the burgeoning literature on subjective wellbeing has focused on the role of relative income in determining satisfaction or happiness. Some labour-market examples are [START_REF] Capelli | Satisfaction, Market Wages and Labor Relations: An Airline Study[END_REF], [START_REF] Pfeffer | The Effect of Wage Dispersion on Satisfaction, Productivity, and Working Collaboratively: Evidence from College and University Faculty[END_REF], Clark and Oswald (1996), [START_REF] Law | Relative importance of referents on pay satisfaction: A review and test of a new policy-capturing approach[END_REF], [START_REF] Bygren | Pay reference standards and pay satisfaction: what do workers evaluate their pay against?[END_REF][START_REF] Ferrer-I-Carbonell | Income and Well-Being: An Empirical Analysis of the Comparison Income Effect[END_REF], and [START_REF] Brown | Does Wage Rank Affect Employees' Wellbeing? Industrial Relations[END_REF], using survey data, and [START_REF] Shafir | Money Illusion[END_REF] in experimental work. 2 This work has generally concluded that relative wages are important in determining workers' job or pay satisfaction. One implication is that the simple neoclassical utility model, where utility depends only on the individual's own income or consumption, should probably be extended to incorporate relative income or consumption terms.

In parallel, the literature on establishment wage policies has highlighted the potential importance of wage compression. One prominent example is the fair wageeffort hypothesis formulated by [START_REF] Akerlof | The Fair Wage-Effort Hypothesis and Unemployment[END_REF], which largely corresponds to [START_REF] Adams | Toward an Understanding of Inequity[END_REF] theory of equity, in which effort depends on the relationship between fair and actual wages. In this theory, higher wages for some groups of workers -perhaps because they are in short supply -will raise wages for all of the workers in the establishment through the demand for pay equity.

The link between worker well-being and the establishment wage distribution is important for human resource managers, whose choice of pay policy will take into account the impact of worker dissatisfaction on profits and worker turnover (for empirical evidence, see [START_REF] Patterson | Organizational Climate and Company Productivity: The Role of Employee Affect and Employee Level[END_REF]. More broadly, wage comparisons may have important consequences for the functioning of the entire labor market, explaining women's labor force participation [START_REF] Neumark | Relative Income Concerns and the Rise in Married Women's Employment[END_REF], unionization [START_REF] Farber | Why Workers Want Unions: The Role of Relative Wages and Job Characteristics[END_REF], money illusion [START_REF] Shafir | Money Illusion[END_REF], hysteresis in unemployment [START_REF] Summers | Relative Wages, Efficiency Wages, and Keynesian Unemployment[END_REF][START_REF] Bewley | Why not cut pay?[END_REF], and wage rigidity [START_REF] Levine | Fairness, Markets and Ability to Pay: Evidence from Compensation Executives[END_REF][START_REF] Campbell | The Reasons for Wage Rigidity: Evidence from a Survey of Establishments[END_REF][START_REF] Campbell | The Reasons for Wage Rigidity: Evidence from a Survey of Establishments[END_REF].

The above literature appeals to the general area of preference interactions, as termed by [START_REF] Manski | Economic Analysis of Social Interactions[END_REF], where what others do, or what happens to them, directly affects my own utility. While evidence of such income interactions has been steadily accumulating for a number of years, a smaller number of recent papers have uncovered empirical results of the opposite sign, with some measure of individual well-being being positively correlated with reference group income: the more others earn, the happier I am.

This finding has been interpreted as demonstrating Hirschman's tunnel effect [START_REF] Hirschman | The Changing Tolerance for Income Inequality in the Course of Economic Development[END_REF]: while others' good fortune might make me jealous, it may also provide information about my own future prospects. [START_REF] Manski | Economic Analysis of Social Interactions[END_REF] calls these phenomena expectations interactions, where what happens to others allows me to update my information set. The associated empirical work refers to information effects or signals.

In this paper we provide some of the first evidence that information effects may be stronger than comparison effects (i.e. that signal outweighs status) in the context of developed Western economies. Individuals may therefore be better off as others earn more, and consequently may not object to some degree of income inequality. We emphasize that the key parameter on which the balance between status and signal rests is the strength of the correlation between current reference group income and my own future earnings. At the peer group or geographical level, this correlation is arguably small. In the context of [START_REF] Luttmer | Neighbors as Negatives: Relative Earnings and Well-Being[END_REF], it is not because my neighbor receives a wage raise that my own future income prospects may necessarily look any brighter.

The signal effect is arguably far greater within the same establishment. In this paper we thus appeal to employer-employee panel data, and model individual job satisfaction as a function of the earnings of all other workers within the same establishment. This unusually rich data set results from the matching of survey panel data (over the period 1994-2001) to administrative longitudinal records of employer-employee data.

We show that workers are indeed more satisfied when their co-workers are betterpaid. The "Hirschmanian establishment" or signal interpretation is that others' wages provide sufficient information about my own future prospects to outweigh any jealousy I might feel towards my colleagues. This Hirschman effect is stronger for men than for women, and in the private sector.

We provide some further structure to this result by considering the "high-paid" and "low-paid", those whose wages are respectively above and below the establishment mean wage. The correlation between satisfaction and the establishment mean wage for the high-paid is very insignificant. However, the satisfaction of the low-paid is strongly positively correlated with the establishment mean wage, which is consistent with the latter playing more of an information role for those with relatively low wages. These two results together yield the perhaps unpleasant implication that raising salaries towards the top of the wage distribution can make everyone happier: because their own wage has risen for the high-paid and for information reasons for the less well-off.

These results are broadly supportive of Tournament theory [START_REF] Lazear | Rank-Order Tournaments as Optimum Labor Contracts[END_REF], where (some of) my colleagues' current wages reflect my opportunities in the establishment's internal labor market. This paper is organized as follows. Section 2 presents a simple model of status and signal effects from others' wages. Section 3 then describes the data that we use, and Section 4 presents the main empirical results. Last, Section 5 concludes.

Status or Signal?

There has been substantial interest across most of social science in the notion of status or comparisons to others. The very broad idea here is of negative externalities emanating from the consumption or income of others within the reference group: the more others earn, the lower is my utility, ceteris paribus. Empirically, the majority of work in this area has appealed to either measures of individual behaviour (such as labour supply or consumption), or measures of subjective well-being. In this latter case, a variable such as life satisfaction is shown to be positively correlated with own income, but negatively correlated with reference group income. 3 The negative correlation is consistent with the presence of income comparison terms in the utility function.

Personnel Economics has arguably not paid much attention to such income comparison effects. However, it has underlined the incentive role played by the income that certain others within the same establishment may receive. In particular, in the tournament model [START_REF] Lazear | Rank-Order Tournaments as Optimum Labor Contracts[END_REF] employees within a given establishment are seen as contestants for promotion. Relative worker performance determines the winner, who receives a fixed prize set in advance. The level of individual effort then increases with the wage difference between winning and losing the tournament. High wages at the top of the establishment's hierarchy are incentives for workers at lower job levels.

These two literatures confront each other when we consider individuals within the same establishment. In this case, one viable reference group is co-workers. As such, coworkers' wages may have two opposing effects on individual utility. The first is a comparison or status effect, whereby co-workers' higher wages make me feel relatively deprived, and the second is a signal effect, where higher co-worker wages provide me with information about my own future income prospects.

To illustrate this tension, we develop a simple model encompassing both status and signal effects. Imagine a simple linear utility function for individual i at time t:

* it it it U = + w w α β (1)
Here w it denotes the individual's own wage and denotes the level of reference group earnings, which in our model is the within-establishment average wage. We imagine that α>0 and a standard comparison story would have β<0; the latter reflects the importance of others' income in the individual utility function. For expositional purposes, assume that there are two time periods, 1 and 2.

* i1 i1 i1 U = + w w α β ( ) ( ) * * i2 1 2 2 U = + (1-) (1 ) (1 ) i i i i i w w p w p w w δ α β δ α θ β ⎡ ⎤ + + + - ⎣ ⎦ * + So that ( ) ( ) ( ) ( ) 2 * * * 1 1 1 2 2 1 * * 1 1 2 (1 ) (1 ) (1 )
(1 ) (1 ) (1 ) (1 ) (1 )

it i i i i i i i t i i i i i U w w w w p w p w w w p w w w pw α β δ α β δ α θ β * * 2 i w α β δ α θ δ α β δ α δ β = ⎡ ⎤ = + + + + - + + - + ⎣ ⎦ = + + - + + + + - - + - ∑
It is assumed that individuals take their future into account, so that their satisfaction response today includes information on how they expect their job to be in the future4 (otherwise the information element plays no role).

A standard regression in the field of income comparisons models job/life satisfaction at time t as a function of both and . The δ(.) term, the third above, represents the outside options (in terms of both income and reference group income) should the match come to an end. This can be considered to be picked up by demographic variables, or by the individual effect in panel analyses. Most empirical estimation does not control for the levels of future income and reference group income ( and ) that pertain when the individual does not accede to the current reference group wage (although we can argue that w i2 will be closely correlated with ).

1 i w * 1 i w 2 i w * 12 w 1 i w
The key implication of this model is that the coefficient on in the estimation but also the information that the establishment average wage (or whatever the measure of reference group income is) provides about the worker's future prospects. In our model, instead of estimating β, we in fact obtain an estimate of

= + (1-) p(1+ ) β β δ α % θ (2)
This estimated coefficient, β % , will be positive, setting θ equal to zero for simplicity, if

(1 ) p

β δ α - > -
.

Proposition 1:

The signal effect is more likely to dominate the status effect, so that others' wages are positively correlated with my own well-being, as:

1) the probability of acceding to the reference group (p) is higher;

2) the jealousy parameter (β) is lower;

3) the match destruction rate (δ) is lower; and 4) the marginal utility of own income (α) is higher.

The empirical literature on income comparisons has taken the estimated value of β in (1) as an indicator of the strength of status effects. However, the simple model above highlights that this interpretation fails when there is also a signal component; in this case there is no clean test of comparisons as the estimated coefficient on picks up two opposing phenomena. In general, any estimated value of * w β % will be consistent with the presence of income comparisons in the utility function. From (2), the strength of the comparison term can only be estimated in three distinct cases:

(i) α = 0, so that a priori only others' income matters in the utility function, with no role for one's own income. This prior is obviously unattractive.

(ii) p=0, so that there is no chance of acceding to the reference group job. It might be argued that a geographical definition of a reference group in Western countries, as in [START_REF] Luttmer | Neighbors as Negatives: Relative Earnings and Well-Being[END_REF] or [START_REF] Blanchflower | Well-being over time in Britain and the USA[END_REF], goes some way to meeting this condition -I am perhaps relatively unlikely to end up with my neighbor's job. This would likely be a worse assumption in the case of [START_REF] Knight | Subjective well-being and its determinants in rural China[END_REF], where the reference group (others in the same rural Chinese village) is more homogeneous.

(iii) δ=1. All matches are destroyed, so that there is no chance of staying in the same job.

This is unlikely in field data, but can easily be engineered in experimental tests of comparison income, such as [START_REF] Mcbride | Money, Happiness, and Aspirations: An Experimental Study[END_REF].

Our empirical work uses matched employer-employee data and considers a reference group of other workers within the same establishment. We therefore expect a non-zero information effect from others' wages, especially for those who have a greater chance of moving up the establishment's wage ladder, and for those who expect to stay in the establishment longer. This kind of data provides a good setting in which to test for the relative strength of status and signal effects. Respondents answer the satisfaction question using an ordered scale from 1 (not at all satisfied) to 6 (fully satisfied). Figure 1 shows the distribution of job satisfaction in this sample. As is usual, there is bunching towards the right-hand side of the satisfaction scale.

Empirical Approach and Data

The Data

[Figure 1 By matching the two datasets we can thus obtain information on survey respondents' job satisfaction, their own wage, and the entire establishment wage distribution. In order for the concepts of relative wage and wage distribution to make sense, we limit the sample to establishments with a minimum of 5 colleagues. 7 Our key regressions will model individual job satisfaction as a function of both own wages and a measure of others' wages within the same establishment, as well as a set of standard demographic control variables. 8

Econometric Specification

We are interested here in the determinants of overall job satisfaction, and in particular the role of the individuals' position in the establishment's wage distribution.

Job satisfaction is an ordered variable, reported on a scale from 1 to 6. In comparison to previous work that has considered the role of the establishment's wage structure, our use of panel data means that we are able to control for unobserved individual characteristics. [START_REF] Lykken | Happiness is a stochastic phenomenon[END_REF] estimate that between 50% and 80% of the variation in individuals' reported well-being results from genes and upbringing, underlining the importance of controlling for individual-specific fixed effects. However, introducing fixed effects in this particular case is problematic since the establishment's average wage changes only little over time (and relatively few individuals change establishments), producing too little variation in the data to identify accurately the impact of reference 7 Using thresholds of 4 and 6 colleagues makes little difference to the results. 8 The means and standard deviations of the key variables are presented in Appendix , where i w denotes the mean wage of individual i over all the waves in which she is observed. This term is included as an application of [START_REF] Mundlak | On the pooling of time-series and cross-section data[END_REF] method,

where the individual effect is parameterized as a function of the mean values of some of the (time-varying) regressors that are thought to be correlated with the individual random effects.

Wages are likely to be correlated with the individual random effect whereas, e.g., age is not a candidate since individuals grow older independently of personal traits.

The income-related terms, it IT , consist of own wage, , and the average wage of all other employees at the same workplace, . To test for asymmetry in the way in

Results

In this section we first present the main results, and then a number of extensions.

Finally, we discuss the results and their implications.

Basic Results

The first set of results, in Table 1, refers to the baseline model specification including the individuals' own wage and the establishment average wage, as well as a number of other covariates. The key estimated parameters in these regressions refer to own wage and to the wage of the reference group (the establishment average wage).

[Table 1 about here]

9 Note that the specification of wage surplus and wage deficit actually yields the value function from [START_REF] Kahneman | Prospect Theory: An Analysis of Decision Under Risk[END_REF] Prospect Theory, valued around = .

it w * it w 13

There are two columns in Table 1. The first refers to a model without reference group information, but including both own income and the Mundlak income term; the second column adds the establishment average wage. The estimated coefficient on own income is positive and significant in both of these columns (at the five per cent level in column 1, and just outside that level in column 2). The estimated coefficient on the Mundlak term is positive in both columns, but only significant in column 1. The broad positive relationship between own income and job satisfaction, conditional on the other right-hand side variables, is unsurprising, and is consistent with most results in the literature.

More unusually, the estimated coefficient on the establishment average wage in column 2 is positive and significant at the 5 per cent level. 10 In terms of our model, the signal effect thus dominates the status effect, yielding a net positive estimated coefficient.

We therefore conclude that something akin to Hirschman's tunnel effect appears to exist in these data. We discuss the implications of this finding in section 4.3 below.

The results for the other control variables are standard. We find a U-shape between job satisfaction and age (Clark et al., 1996), which minimizes here at age 36. We also find a strong effect of health (although we cannot say anything about causality), and that employees in relatively small establishments (fewer than 100 employees) report statistically significant higher job satisfaction levels than do employees in establishments with 100 employees or more. 10 In terms of the model in Section 2 we therefore have

(1 ) p β δ α - > - .
We are interested in identifying certain groups for whom the signal effect of others' wages may be stronger. To this end, we replace the establishment average wage used in Table 1 with the wage surplus and wage deficit variables defined in Section 3 above, both for the whole sample and by gender. These two new variables allow others' wages to have asymmetric effects on job satisfaction, with a kink at = . We might in particular wonder whether the signal effect is stronger for those who earn less than the mean wage in the establishment. This asymmetry turns out to be important in explaining job satisfaction. The results for the whole sample, and separately by gender, are shown in Table 2. 2 about here]

The parameter estimate on wage surplus is insignificant in all of these regressions: the establishment average wage does not affect the job satisfaction of those who earn more than the mean wage in the establishment. On the contrary, over the whole sample the estimated coefficient on wage deficit is positive and significant at the 1 per cent level (in Table 2): those who earn less than the establishment mean wage are more satisfied when they work in high-wage establishments. It is worth emphasizing that this correlation holds controlling for the individual's own wage, so that the establishment wage is not acting as a proxy for own wages in these regressions.

This asymmetric effect of co-workers' wages, depending on whether the individual earns more or less than the establishment average wage, is of interest for a number of reasons. In the first instance, it allows us to at least partially discount the hypothesis that high-wage establishments are in some general sense pleasant places to work (for non-wage reasons): they could provide better non-monetary rewards or larger offices, for example. This explanation can only hold if any such benefits are specifically targeted towards lower-paid rather than higher-paid workers (which seems unlikely).

Dividing the sample by gender in Table 2, wage deficit remains positive and highly significant for males, but is insignificant for women. This result might be thought of as a large-sample survey data counterpart to the well-known findings in the experimental literature that men react more strongly in competitive environments (Gneezy et al., 2003, and Niederle and Vesterlund, 2007).

Extensions

Tables 1 and2 have shown that, at a given level of own wage, workers seem more satisfied in high-wage establishments. Our interpretation is that the signal effect of others' wages (in terms of what I myself might hope to earn in the future) outweighs any status effect from wage comparisons. This sub-section presents a number of extensions to these basic results. The first two refer to different specifications of the establishment wage, while the last draws some links to tournament theory by showing that the signal effect from others' wages is stronger for certain groups in the labor market.

Comparison to the 75 th percentile

The first issue we address is whether the linear-in-means specification is the most appropriate in the context of tournaments. In particular, this latter would suggest that only the wages of those above you in the hierarchy matter; as such the establishment average wage may not be a particularly good measure. We thus replace the average establishment wage in the baseline specification by the 75 th percentile wage. The results in the first column of Table 3 show that, again, workers are more satisfied in high-wage firms. The coefficient on the 75 th percentile of the establishment wage is positive and significant.

[Table 3 about here]

We also suspect, in line with tournament theory, that the signal effect here will only apply to those who earn less than the 75 th percentile wage, and that the size of this signal will increase the closer the individual is to the 75 th percentile (from below). The last three columns of Table 3 explore this possibility, by running separate estimations according to whether the individual is in the bottom half of the wage distribution, the third quartile (i.e. the 50 th to the 75 th percentile) or the top quartile. The results accord with our prior: the effect of the 75 th percentile wage is positive and significant for those in the first three quartiles, and especially for those in the third quartile (i.e. those who are closest to the establishment wage measure). By way of contrast, the 75 th percentile wage is very insignificant for those in the top quartile of the establishment wage distribution.11

Additional Wage Comparisons by Occupation

A second specification issue refers to the sample used to calculate the establishment average wage. Up until now, we have used all workers in the establishment. One topic of discussion in the income comparisons literature has been to whom individuals actually compare. Within the establishment, it is perhaps likely that workers compare their wages more to those of others who are doing the same kind of job.

With this in mind, Table 4 shows the results of estimations with two establishment wage terms. The first is the 75 th percentile establishment wage, as in Table 3: we expect this to reflect the signal component of others' wages. The second is the mean wage by occupation, split up into three categories (Managers and equivalent, Middle-ranking positions, and Lower-level jobs). 12 We calculate the 75 th percentile establishment wage at the establishment level (and not by occupation within the establishment) as the whole ethos of tournaments is that the winners climb up the establishment hierarchy.

[Table 4 about here]

The results in the first column of Table 4 show the estimation results over the whole sample. These reveal a positive effect from the 75 th percentile establishment wage, as before, but a negative significant effect from mean wage by occupation (within the establishment). We interpret the first of these as a signal effect, but the second as reflecting wage comparisons to similar others, as emphasized in much of the existing empirical literature.

Comparisons Differ between Labor Market Groups

The first column of Table 4 is consistent with the presence of both signal and status effects from others' wages. The remaining columns of this table show that these effects are not homogeneous across different labor market groups; in particular, both are stronger for men than for women and for private-sector workers than for public-sector workers.

Columns 2 and 3 of Table 4 show that, as already suggested in Table 2, signal effects from establishment wages are stronger for men than for women. The results here also show that the putative status wage effect is very significant and negative for men, but totally insignificant for women.

Finally, columns 4 and 5 of Table 4 show that the signal vs. status distinction in others' wages is much sharper in the private than in the public sector. This is perhaps unsurprising, as wages in the Danish public sector are determined by centralized collective bargaining, with relatively little scope for individual establishments to set up wage tournaments.

Discussion and Implications

The results presented above are consistent with the interpretation that some individuals are inherently "Hirschmanian" within the establishment, in the sense that the signal effect from others' wages dominates any jealousy or status effect. Consistent with tournament theory, the signal effect is always insignificant for those who earn more than the indicator of establishment wage that we employ.

The finding of a positive well-being effect from others' wages differs from those in the majority of the published literature. We think that the key distinction lies in the composition of the reference group. As previously noted, the fact that the Joneses living next door earn more than I do, as in [START_REF] Luttmer | Neighbors as Negatives: Relative Earnings and Well-Being[END_REF], may only reveal little information about my future pay prospects: the entire effect of the Joneses' pay thus passes by a comparison or status effect, reducing relative income and job satisfaction. Things may well be very different when work colleagues serve as the comparison group. In this case a high reference wage serves as a signal regarding one's own future pay check.

The published work that has found a positive well-being effect from others' income, [START_REF] Senik | When Information Dominates Comparison: A Panel Data Analysis Using Russian Subjective Data[END_REF]2007) and [START_REF] Kingdon | Community, comparisons and subjective well-being in a divided society[END_REF], can also be interpreted in terms of signal effects. [START_REF] Senik | When Information Dominates Comparison: A Panel Data Analysis Using Russian Subjective Data[END_REF] uses Russian RLMS data to establish a positive correlation between life satisfaction and (geographical) reference group income, especially for younger workers. In general, [START_REF] Senik | Ambition and Jealousy. Income Interactions in the "Old Europe" versus the[END_REF] makes the point that most evidence of comparison or jealousy effects comes from stable Western countries. In Senik's work, the reference group consists of other people who are similar to you. In a very unstable labor market, what happens to similar others today may well be thought of as providing a signal about your own future labor market outcomes. [START_REF] Kingdon | Community, comparisons and subjective well-being in a divided society[END_REF] use South African data to show that the average income of others in the local residential area is positively correlated with household utility (while average income by district or province is negatively correlated with well-being). Again, this can be interpreted as showing that individuals are more likely to end up with their close neighbor's job than with their more distant neighbor's job.

This paper has uncovered this kind of signal effect using a natural reference group (colleagues within the same establishment) in an OECD country. Denmark has one of the most equal income distributions in the world, as well as very high income and wage mobility by international standards (OECD, 1997). These two facts together with our results suggest that: i) even in a stable economic environment there can be substantial income mobility, and in this context it is not surprising that signal effects exist even in the affluent Danish economy; and ii) perhaps Danish income re-distribution has gone too far. This latter point is arguably supported by the fact that increasing the pay of well-paid employees increases everyone's job satisfaction. In Table 2, the well-paid's job satisfaction rises because their own pay has gone up, and the low-paid's job satisfaction increases via the establishment's average wage. The provocative implication is that greater wage inequality within an establishment (via higher pay for the well-paid) may represent a Pareto improvement. This anti wage-compression result corresponds to some earlier findings in economics and psychology underlining a positive relationship between income inequality and measures of subjective well-being (see [START_REF] Clark | Inequality-Aversion and Income Mobility: A Direct Test[END_REF], and the references therein).

Conclusion

A common theme in the subjective well-being literature has been comparisons to others, whereby low income relative to a reference group reduces well-being. We argue that this correlation is conditional on reference group income being uninformative about the individual's own future income prospects. In much of the existing literature, this condition is satisfied (it is not necessarily because my neighbors or my cohort receive a pay rise that my own future prospects look brighter). We here analyze a data set where this condition probably does not hold, using wage information on all other workers within the same establishment. We do so by matching Danish ECHP data to administrative records.

The results are unambiguous. Job satisfaction is positively correlated with own wages, but it is also positively correlated with the average wage of all other workers within the same establishment. This Hirschman effect is stronger for men than for women, and in the private sector relative to the public sector. These findings are arguably consistent with the signal effect dominating the status effect for the subgroups that are most likely to be promoted. We also find that the signal effect is larger for those who earn less than the establishment wage measure.

Although we have not presented any direct tests of tournament theory, our results are nonetheless consistent with this model. When my colleagues earn higher wages, I learn about my future opportunities within the establishment. In line with the Tournament model this effect is more pronounced for men (who are more likely to be promoted), and in the private sector (where wage-setting is more individualized). Our results corroborate the findings of one of the few empirical studies of Tournament theory, [START_REF] Eriksson | Executive Compensation and Tournament Theory: Empirical Tests on Danish Data[END_REF], who finds support for this theory using Danish data.

Taking these results at face value, workers may not oppose wage inequality -at least not in the context of the firm's pay scale. Indeed, in the current data, higher wages for better-paid workers will improve everyone's job satisfaction. There are however likely limits beyond which this result will no longer hold. Future research should attempt to identify more accurately the relationship between worker well-being, on the one hand, and both own and others' wages on the other, while explicitly recognizing that my own current relative wage misfortune may contain the promise of a brighter future.

22

B. Descriptive Statistics

  not only represent the comparison part of the utility function,

  about here] The Danish component of the ECHP was sampled randomly from the central administrative database, the Central Personal Register (CPR). The CPR contains an entry for each individual in Denmark; each individual has a unique CPR number. This number can then be matched to the administrative IDA 6 database, maintained by Statistics Denmark, containing labour market information on all individuals aged 15 to 74 (demographic characteristics, education, labor market experience, tenure and earnings) and employees in all workplaces in Denmark over the period 1980-2001. This database includes, amongst many other things, identifiers for both the firm and the establishment where the individual works, and the individual's gross annual income. We therefore have administrative information on the income of all of the individual's colleagues at their place of work. Our use of administrative data likely reduces problems associated with measurement error regarding income.

γ

  is the individual-specific timeinvariant random component and it ε is the individual and time-varying idiosyncratic disturbance term. Following the discussion above, we parameterize the individual

  wages affect job satisfaction, we follow[START_REF] Ferrer-I-Carbonell | Income and Well-Being: An Empirical Analysis of the Comparison Income Effect[END_REF] and define two variables, wage surplus and wage deficit, as follows:i) If >, then wage surplus = ln( ) -ln( ) and wage deficit = 0. wage deficit = ln ( ) -ln( ) and wage surplus = 0.Both wage surplus and wage deficit are thus weakly positive. If there is no asymmetry in the effect of others' wages, we expect the estimated coefficients on wage surplus and wage deficit to be equal and opposite in sign.

Figure

  Figure 1. Distribution of Job Satisfaction. Danish ECHP Sample, 1994-2001.

  This paper is based on data of unusual richness. Eight waves of survey data from the Danish sample of the European Community Household Panel (ECHP) 5 have been merged with administrative records. The ECHP survey data, which constitute a panel spanning 1994-2001, cover about 7,000 individuals in the first few years. Due to sample attrition this falls to about 5,000 individuals by 2001. Here we only consider employees, so that our effective sample size is reduced to about 16,000 observations on around 4,100 individuals over the whole eight-year period. Our dependent variable results from an overall job satisfaction question as follows:

	How satisfied are you with your work or other main activity?

Table B1 .

 B1 group wages. In addition, individual random effects are likely to be inconsistent, as own wage is very likely endogenous in a model with job satisfaction as the dependent variable. As an 'intermediate' solution we therefore introduce a Mundlak correction term,

	as described below.
			We assume a latent unobserved continuous measure of reported job satisfaction,
	JS	*	, which is assumed to be a function of individual covariates such as age, education
	and other background characteristics,	it X and income-related terms, it IT (which will also
	be discussed below). The empirical model is then:
			JS	it	=	α	it IT	+	β	X	it	i + +ε γ	it	,	i=1,...,N; t=1,...,T i .
			The error term is denoted by it	i	it

* '

'

Table 1 .

 1 Baseline Specification.

	Model (1)	Model (2)

Table 2 . Asymmetric Comparisons: Wage Surplus and Wage Deficit.

 2 Standard errors in parentheses + significant at 10%, * significant at 5%; ** significant at 1%. Other controls as in Table1.

		All	Males	Females
	Ln(own wage)	0.218**	0.247**	0.182*
		(0.053)	(0.075)	(0.076)
	Wage surplus	0.038	0.063	-0.050
		(0.065)	(0.087)	(0.100)
	Wage deficit	0.211**	0.282**	0.122
		(0.060)	(0.086)	(0.084)
	No. of Observations	16,031	8,594	7,437
	Note:

Table 3 . Comparisons to the 75 th percentile. All Below Avg. Above Avg. but below 75 th pctile Above 75 th pctile

 3 Standard errors in parentheses, + significant at 10%; * significant at 5%; ** significant at 1%. Other controls as in Table1. Sample restricted to establishments with 10 or more employees.

	Ln(own wage)	0.043	0.023	-0.405	0.222
		(0.026)	(0.033)	(0.263)	(0.146)
	Ln(75 th percentile wage)	0.138**	0.207**	0.750**	0.0003
		(0.048)	(0.076)	(0.269)	(0.114)
	Observations	16,031	5,918	4,686	4,962
	Notes:

Table 4 . Comparisons to the 75 th percentile and Mean Occupational Wage.

 4 Standard errors in parentheses, + significant at 10%; * significant at 5%; ** significant at 1%. Sample restricted to establishments with 10 or more employees.

		All	Male	Female	Private	Public sector
					sector
	Ln(own wage)	0.042	0.043	0.061	0.049	0.050
		(0.033)	(0.045)	(0.048)	(0.043)	(0.055)
	Ln(75 th percentile wage)	0.314**	0.402**	0.164	0.307**	0.183
		(0.090)	(0.127)	(0.126)	(0.109)	(0.183)
	Ln(avg. wage by
	establishment &	-0.165*	-0.311**	0.017	-0.189+	-0.144
	occupation)
		(0.078)	(0.110)	(0.109)	(0.099)	(0.131)
	Observations	12,188	6,676	5,621	6,818	5,370
	Notes:

Table B1 . Means and Standard Deviations of Key Variables

 B1 Note: The PPP exchange rate between the Danish Kroner (DKK) and the US dollar was 8.66 in 1994.

	Variable	Mean	Std. Dev.
	Real wage/1000 (1994 DKK)	227.49	109.10
	Age	40.29	10.66
	Female	0.46	0.50
	Occupation
	Managers	0.19	0.39
	White collar	0.17	0.38
	Blue collar	0.43	0.50
	Health problems	0.20	0.40
	Single	0.23	0.42
	Hours work/week	37.28	6.99
	Subordinates (Yes=1)	0.31	0.46
	Establishment size
	5-19	0.21	0.41
	20-49	0.19	0.39
	50-99	0.18	0.38
	100 or more	0.26	0.44
	Education
	Primary/Secondary	0.21	0.40
	High School	0.07	0.25
	Vocational	0.39	0.49
	College-short	0.06	0.23
	College-long	0.20	0.40
	University	0.08	0.26

See Clark et al. (2007) for a recent survey.

Where the reference group might be the individual's peer group (those who share the same characteristics), others in the same household, the spouse/partner, friends, neighbors, work colleagues, or the individual herself in the past.

This interpretation is explicitly tested in the context of worker quitting by[START_REF] Lévy-Garboua | Job Satisfaction and Quits[END_REF]. A second piece of supporting evidence is that promotion opportunities attract a positive estimated coefficient in job satisfaction regressions.

See http://epp.eurostat.cec.eu.int for details of the ECHP data.

Integreret Database for Arbejdsmarkedsstatistik (Integrated Database for Labour Market Statistics).

We can produce the same flavor of results with the 90 th percentile of the establishment wage. In this case the size of what we call the signal effect increases monotonically up to the 90 th percentile, with the largest effect being found for workers between the 75 th and the 90 th percentile; the effect for those in the top decile is very insignificant.

We only use three occupation categories to avoid problems with small cell sizes.