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How asymmetrically increasing joint strike costs need not lead to fewer strikes.

Introduction

This paper revisits the joint costs theory of strikes (Reder and Neumann 1980, Kennan 1980). The purpose is to reconcile the model's failure to explain contradictory empirical findings in the strike literature. For example, whereas Reder and Neumann (1980), Barlow and Buckley (1998), Nicolitsas (2000) and Geraghty and Wiseman (2008) find strong support, Cousineau and Lacroix (1986), Maki (1986), and Ahmed (1989) do not, and Sopher (1990) finds only moderate validation. In addition, the theory is problematic in explaining cyclical strike incidence, as Hirsch and Addison (1986) observe "... joint strike costs seem likely to increase with the level of economic activity, leading to the incorrect prediction of counter-cyclical strike activity (incidence)" (p. 104).

We show how higher worker or firm strike costs need not always lead to fewer strikes, as the joint cost model predicts. What drives the result is asymmetrically increasing strike costs, a consideration not explored in depictions of the joint-strike model (e.g. Sopher 1990).

Model Outline and Predictions

We build as simple a model as we can to illustrate why the joint costs theory could yield ambiguous results with respect to strike incidence. Our approach begins with standard Hicksian concession curves modified by Mauro (1982) to account for imperfect information. From these curves we derive a payoff matrix under alternative union and firm strategies. The resulting payoff matrix is consistent with the game of chicken. A strike occurs when both unions and firms hold out.

The solution to this game indicates no one pure Nash-equilibrium strategy. Instead, each player must adopt a mixed strategy so that choices become probabilistic depending on the payoff matrices, which in turn depend on union and firm concession curve parameters. This mixed strategy implies that each player occasionally holds out, which is perfectly rational and consistent with Hicks's (1963) assertion that "a union which never strikes may lose its ability to organize a formidable strike" (p. 146).

The results indicate that each party's strike costs reflected, in rates of concession, have an ambiguous effect on strike incidence. What drive the results are asymmetric changes in relative costs. For example, as union strike costs rise, the union holdout probability falls. But if unions hold out less (i.e., concede more), the firm's expected profit from conceding decreases (because by conceding firms have to pay higher wages). Lower expected profits from conceding causes the firm to hold out more. In turn, holding out more potentially increases strike incidence. Whether strikes actually increase depends on both union and firm holdout probabilities. We show that under plausible circumstances the firm's holdout probability increases more quickly than the union's holdout probability diminishes, which can lead to greater strike incidence. This asymmetric rise in employee-employer strike costs implies that strike probabilities can rise. (1) Therefore, higher strike costs need not lead to fewer strikes.

The Game

According to Hicks (1963, first published in 1932), if strikes were costly, both a firm and its workers would pay to renegotiate a contract rather than strike. To sec this, we present the main elements of Hicks's model and the assumptions we make in simplifying his model for purposes of this paper.

For firms, higher wages are associated with greater costs of giving in to union demands because future profits will be affected adversely. However, because a strike of a given length will bear a direct output cost (and hence foregone profits), an employer will be indifferent between a set of maximum wage offers and expected strike durations that will define his concession curve. According to Hicks, the employer concession curve will start off from some offer wage independent of the union effect, possibly at the pre-negotiation wage, then slope upwards, since profit losses for the firm are proportional to expected strike duration, but flatten out at some point beyond which the firm would not be willing to concede any farther, as such high wages could drive the firm to bankruptcy. The slope of the employer concession curve denotes the rate at which the firm will trade offering higher wages to avoid longer strike durations.

Similarly for unions, a strike of a given length will bear the direct cost of lost work, and foregone lost earnings, to the union. The union will be indifferent between a set of minimum wage demands and expected strike durations that will define its resistance curve. According to Hicks, the union resistance curve will start at some union reservation wage, then slope downward, having a horizontal portion at some range for which wage union members "consider themselves entitled" (Hicks 1963, p. 143), and finally cut the horizontal axis at a point that indicates the "maximum time beyond which the union cannot last out whatever be the terms offered" (Hicks 1963, p. 143). The slope of the union resistance curve denotes the rate at which the union will trade accepting lower wages to avoid longer strike durations.

If the firm knows the resistance curve of the union, and the union knows the concession curve of the firm, both sides can avoid a strike by compromising on a wage defined by the intersection of the union resistance and firm concession curves. This compromise wage [W.sup.*] avoids a strike of duration [S.sup.*], thus benefiting both sides. To derive an expression for [W.sup.*], this paper presents a special case of the Hicksian analysis where [W.sup.*] is more likely to be observed. In that region the employer concession curve slopes upward and the union resistance curve slopes downward. For simplicity and without loss of generality we denote these two curves to be linear since all nonlinear surfaces can be approximated by linear functions (Jennings 1964, p. 13). The simplified version is presented in Fig. 1 where the employer concession curve [W.sub.f] is upward-sloping and linear, and the union resistance curve Wu is downward-sloping and linear.

Let [W.sub.u]([S.sub.u]) be the wage a union would accept now to avoid a strike of expected duration Su. This wage is determined by balancing a strike's cost with the expected gain from striking. To illustrate analytically, take the simplest possible model in which all strike costs are foregone wages and no permanent layoffs result from the strike. (2) As such, strike costs are [NW.sub.c][S.sub.u] where N is the number of union worker and [W.sub.c] is the current wage before the strike. Assuming a long-lived union and firm, as well as a discount rate d, expected gains are N/d {([W.sub.r] - [W.sub.u]([S.sub.u])) - [W.sub.c]] where [W.sub.r] is the union's reservation wage (i.e., the wage that the union will accept now in order to avoid a strike of zero expected duration). Equating costs and benefits yields

[W.sub.u] = [W.sub.r] - [bS.sub.u], b > 0 (1)

where [W.sub.u] is now normalized to reflect the wage gain [[W.sub.u]([S.sub.u]) - [W.sub.c]] from a strike of duration [S.sub.u], and b is d[W.sub.c]. The slope, b, of the resistance curve reflects the union's cost of extending the strike one time period.

Let [W.sub.f] be the wage a firm would offer now to avoid a strike of expected duration [S.sub.f]. Then:

[W.sub.f] = [W.sub.0] + [cS.sub.f], c > 0 (2)

where [W.sub.0] is the maximum wage that the firm would offer now in order to avoid a strike of zero expected duration, and c is the slope of [W.sub.f] which reflects the firm's costs of prolonging a strike one additional time period. The greater the firm strike costs, the higher the wages the firm is willing to offer in order to avoid a strike, and hence the greater the magnitude of coefficient c.

Setting [S.sub.u] = [S.sub.f] and solving for [W.sup.*], one obtains

[W.sup.*] = ([cW.sub.r] + [bW.sub.0])/(b + c) (3)

which is the wage that the union would accept and the firm would offer now, if they both expected a strike of length [S.sup.*] as in Fig. 1. Note that if b = c, i.e. if the concession curves have equal slopes, then

[W.sup.*] = 1/2([W.sub.r] + [W.sub.0]) (4)

which is the special case when the two parties split the difference.

Hicks (1963, p. 146) argues that incomplete or asymmetric information can make W initially unattainable, thereby leading to a strike. This can be illustrated easily. As in Mauro (1982), suppose the firm underestimates the minimum union asking wage and overestimates union strike costs. Such misperceptions lead the firm to offer a wage unacceptable to the union. Graphically, this implies that the firm perceives the union resistance curve to be lower and steeper (e.g. [W.sub.u.sup.p]) than it really is. This leads the firm to offer a wage no higher than [W.sub.0.sup.*], the intersection of its perceived union resistance and its own actual concession curve, thinking it acceptable to the union (Fig. 1). Algebraically, the firm's perceived union resistance curve [W.sub.u.sup.p] can be written as

[W.sub.u.sup.p] = [[theta].sub.1] - [[theta].sub.2] [S.sub.u], [[theta].sub.1] < [W.sub.r], and [[theta].sub.2] > b.

The degree to which [[theta].sub.1] < [W.sub.r] reflects the firm's misperception of union reservation wages. The degree to which 02 > b reflects the firm's overestimate of union strike costs.

Similarly the union would accept a wage no lower than Wr (the intersection of perceived firm concession and its own actual resistance curve) if it wrongly perceives the firm concession curve to be higher and steeper (e.g. [W.sup.p.sub.f]) than it is. Algebraically, the union's perceived firm concession curve ([W.sup.p.sub.f]) can be depicted as

[W.sub.f.sup.p] = [[phi].sub.1] + [[phi].sub.2][S.sub.f], [[phi].sub.1] > [W.sub.0] and [[phi].sub.2] > c (6)

where [[phi].sub.1] reflects its overestimate of the firm's offer wage and [[phi].sub.2] its overestimate of the firm's strike costs. Such misperceptions lead to an impasse, because the union now is willing to concede as low a wage as [W.sub.r.sup.*] (the intersection of its resistance curve, [W.sub.u], and what it perceives to be the firm's concession curve, [W.sup.p.sub.f]), while the firm now is willing to concede as much as [W.sub.0.sup.*] (the intersection of its concession curve, and what it perceives to be the union resistance curve, [W.sub.u.sup.p]). Since [W.sub.r.sup.*] > [W.sub.0.sup.*], an impasse is reached (Fig. 1). We believe that it makes sense to model such an impasse as a game, though we emphasize that with perfect information the two sides would agree on W without a strike and no game would result.

In the game, each player-participant has a choice: concede to the other party's offer (demand) or hold out. If the union concedes while the firm holds out, the union obtains a wage W., for its workers, the highest wage the firm is willing to offer given its expectations about the union's resistance curve. If the firm concedes while the union holds out, the union obtains [W.sup.*], which is the lowest wage that the union would accept given its expectations about the firm's concession curve. When both sides concede, it is reasonable to assume a wage in between, e.g.

[W.sup.*]=f([W.sub.0.sup.*],Wr*,b,c) (7)

such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and partial derivative]f/[partial derivative]. However, when both sides cooperate by conceding, as Hicks observed, the result yields the original settlement obtained by equating [W.sub.u] and [W.sub.f] depicted by eq. (3). Lastly, when both sides hold out, a strike ensues yielding a zero wage.

The Payoff Matrix

Payoffs for the two parties are recorded in Table 1. Union payoffs are denoted as [U.sub.ij], where i depicts the union's action (concede or hold out) and j the firm's. Thus [U.sub.cc] is the union's payoff when both the union and the firm concede; (7/,t, the union payoff when the union holds out and the firm concedes, [U.sub.ch], the union's payoff when the union concedes, and the firm holds out, and [U.sub.hh], the union's payoff when both parties hold out.

We assume union objectives are identified with those of the median union voter, an assumption for which there is ample precedent (e.g. Hirsch and Addison 1986) and empirical support (e.g. Kaufman and Martinez-Vasquez 1990; 1988). To satisfy the "median" union voter we assume these demands will mostly concern the wage. From Fig. 1, union welfare levels [U.sub.ij] are [U.sub.cc] = [W.sup.*], [U.sub.hc] = [W.sup.*], [U.sub.ch] = [W.sup.*] 0, and [U.sub.hh] = 0, where [W.sub.r.sup.*] > [W.sup.*] > [W.sub.0.sup.*] > 0.

The firm's objectives are identified with the profit function 77 = R(L) - wL, where L is employment level and R(L) is the revenue function resulting from an output generated by L workers. Firm payoffs are denoted as [[product].sub.ij] where again i depicts the union's action and j the firm's. Thus, [[product].sub.cc] is the firm's payoff when both sides concede, [[product].sub.hc], the firm's payoff when the union holds out and the firm concedes, [[product].sub.ch], the firm's payoff when the union concedes but the firm holds out and, [[product].sub.ch], the firm's payoff when both parties hold out. These payoffs are computed by substituting wage and employment levels associated with the union's and firm's action (concede and hold out) into the profit function so that [[product].sub.cc] = R([L.sub.cc]) - [W.sup.*][L.sub.cc], [[[product].sub.ch] = R([[product].sub.ch]) - [W.sub.0][L.sub.ch], [[product].sub.hc] = R([L.sub.hc]) - [W.sub.r.sup.*][L.sub.ch], and [[product].sub.hh] = 0. Clearly, if the firm holds out while the union concedes, wages are lower than if the firm concedes and the union holds out, and as a result profits are higher, so [[product].sub.ch] > [[product].sub.hc]. Both the union and firm conceding yields a wage in between ([W.sub.r.sup.*] [greater than or equal to] [W.sup.*] [greater than or equal to] [W.sub.0.sup.*]) so that [[product].sub.ch] [greater than or equal to] [[product].sub.cc] [greater than or equal to] [[product].sub.hc]. Clearly, a strike yields no production and hence lower profits or even losses, which we simply denote as zero profits. Thus one would expect [[product].sub.ch] [greater than or equal to] [[product].sub.cc] [greater than or equal to] [[product].sub.hc] >[[product].sub.hh].

A strike depends on the outcome of playing this game. If at most one side holds out, no strike results. Hence, finding the conditions under which the hold out - hold out (H-H) outcome is more likely, will lead us to an explanation of strike incidence the first time the game is played. If no strike results, the game is not played again until a new contract is up for re-negotiation. If, however, a strike results the game is repeated. The duration of the strike will depend on how many H-H solutions one gets as the game is repeated (with adjusted payoffs) during the course of the strike.

Implications and Derivations of Optimal Strategies

After examining both parties' possible options from the previous matrix, we find no dominant strategy. From the point of view of the firm, it is best to hold out ([[product].sub.ch] [greater than or equal to] [[product].sub.cc]) if the union concedes, while it is best to concede ([[product].sub.hc] [greater than or equal to] 0) if the union holds out. From the point of view of the union, it is best to hold out ([U.sub.hc] > [U.sub.cc]) if the firm concedes, while it is best to concede ([U.sub.ch] [greater than or equal to] 0) if the firm holds out. This leads to two possible pure Nash equilibria, the first being concede-hold out (C-H) and the second being hold out-concede (H-C), and a mixed Nash equilibrium strategy where each side chooses either to concede or hold out with an optimally determined probability.

It is easy to show that the other two pure strategies, i.e. concede-concede (C-C) and hold out-hold out (H-H) are not stable. If both sides concede and settle at [W.sup.*], an agreement is reached without a strike. This solution, however, is not stable since each party has an incentive to cheat by holding out. However, a double holdout strategy spells trouble for both, since a strike occurs and both lose, making this strategy also unstable.

The payoff matrix in Table 1 is consistent with the game of chicken. All entries are Pareto superior to the H-H (strike) outcome, but because C-H dominates concede-concede for the firm, and hold out-concede dominates C-C for the union, it is in each party's best interest to threaten to take a holdout strategy, hoping that its rival will be scared into conceding. Thus, as in the game of chicken, each party has an incentive to display toughness even if each party has no intentions of holding out all the way. To display this toughness, each side adopts a mixed strategy by choosing to hold out with a probability determined by each side maximizing its expected payoff.

Derivation of the Firm's and Union's Optimal Strategies

The risk-neutral union maximizes its expected utility by maximizing its expected payoff. If the union concedes, the expected wage will be:

[W.sub.c] = [W.sup.*] [P.sub.f] + [W.sub.0.sup.*] (1 - [P.sub.f]) (8)

where [P.sub.f] is the probability that the firm will concede, [W.sub.0.sup.*] = (c[[theta].sub.1] + c[[theta].sub.1][W.sub.0]) / ([[theta].sub.2] + c), and [W.sup.*] is as defined previously. Similarly, if the union holds out, the expected union wage will be:

[W.sub.h] = [W.sub.r.sup.*] [P.sub.f] + 0(1-[P.sub.f]) = [W.sub.r.sup.*][P.sub.f] (9)

where [W.sub.r.sup.*] = ([[phi].sub.2][W.sub.r] + b[[phi].sub.1])/(b + [[phi].sub.2]). Then, the union will maximize its expected payoff U:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [P.sub.u] is the probability that the union will concede.

In a mixed strategy equilibrium, [P.sub.u] can assume values ranging from 0 to 1, while [P.sub.u] can be either 0 or 1 if only pure strategies are allowed. To maximize U one derives the following first order condition by differentiating it with respect to the choice variable [P.sub.u]:

[partial derivative]U/[partial derivative][P.sub.u] = [W.sub.c] - [W.sub.h] = [P.sub.f] [W.sup.*] + (1 - [P.sub.f]) [W.sub.0.sup.*] - [P.sub.f] [W.sub.r.sup.*] = 0 (11)

which is an expression that does not contain [P.sub.u] as an argument. This implies that an interior solution (i.e. 0 < [P.sub.u] < 1) for the union depends on the firm's strategy. Solving for the optimal value of [P.sub.f], [P.sup.*.sub.f], we then get:

[P.sup.*] = [W.sub.0.sup.*]/([W.sub.0.sup.*] + [W.sub.r.sup.*] - [W.sup.*]) (12)

implying that in equilibrium the firm will choose to concede exactly [P.sup.*.sub.f] percent of the time. If the firm chose to concede more than [P.sup.*.sub.f] percent of the time, the union's expected wage from conceding would be less than the union's expected wage from holding out (i.e. [W.sub.c] < [W.sub.h] or [W.sub.c] - [W.sub.h] > 0), and hence the union will always choose to hold out; if the firm chose to concede less than [P.sup.*.sub.f] percent of the time, the union's expected wage from conceding would exceed the union's expected wage from holding out (i.e. [W.sub.c] > [W.sub.h], or [W.sub.c] - [W.sub.h] > 0), and hence the union will never choose to hold out. Therefore, in a mixed strategy equilibrium, the firm must choose to concede exactly Pt percent of the time, or equivalently the firm must choose to hold out (1 - [P.sub.f]) = (IV,.* [W.sup.*]) / ([W.sub.0.sup.*] + [W.sup.*] - [W.sup.*]) percent of the time.

In order to get the effects of b and c on (1 - [P.sup.*.sub.f]), and hence the effect of each party's strike costs as reflected through their concession rates on the likelihood of holding out for the union, we differentiate (1 - [P.sup.*.sub.f]) with respect to b and c. The result is: (3)

[partial derivative](1 - [P.sup.*.sub.f])/[partial derivative]c < 0 and [partial derivative](1 - [P.sup.*.sub.f])/ [partial derivative]b > or < 0.

Holding the union's strike costs b constant, higher firm costs c have an unambiguous negative effect on the firm's holdout probability. However, holding firm costs c constant, higher union costs b have an ambiguous effect on the firm's holdout probability. Higher union strike costs can increase the firm's probability of holding out when initial union strike costs are low. However, as the union's strike costs increase, after a point the probability of the firm holding out decreases. As such, initially higher union costs decrease the return to conceding now and can lead to a higher likelihood of holdingout. This result suggests that higher costs do not necessarily lead to a reduced holdout probability.

Assume the firm maximizes expected profits [PI]. If the firm concedes, expected profits will be:

[[product].sub.c] = [P.sub.u][[product].sub.cc] + (1 - [P.sub.u])[[product].sub.hc]. (13)

If the firm holds out, its expected profits will be:

[product].sub.h] = [P.sub.u] [product].sub.ch] + (1 - [P.sub.u])0 = [P.sub.u] [product].sub.ch]. (14)

Then the firm maximizes expected profits [PI]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

To maximize the above function, one must derive the first order condition:

[partial derivative][PI]/[partial derivative][P.sub.f] = [P.sub.u][[product].sub.cc] + {1 - [P.sub.u]) [product].sub.hc] - [P.sub.u][[product].sub.ch] = 0 (16)

and solve for the optimal [P.sub.u], [P.sup.*.sub.u]. We get:

[P.sup.*.sub.u] = [[product].sub.hc]/([[product].sub.hc] + [[product].sub.ch] - [[product].sub.cc]). (17)

Hence, in a mixed strategy equilibrium, the union must choose to concede exactly [P.sup.*.sub.u] percent of the time, or equivalently it must choose to hold out exactly (1 - [P.sup.*.sub.u]) = ([[product].sub.ch] - [[product].sub.cc])/([[product].sub.hc] - [[product].sub.ch] - [[product].sub.cc] percent of the time.

To get the effects of b and c on (1 - [P.sup.*.sub.u]) and hence on the union's likelihood of holding out, we differentiate (1 - [P.sup.*.sub.u]) with respect to each of these variables and we get: (4)

[partial derivative](1 - [P.sup.*.sub.u])/[partial derivative]b < 0 and [partial derivative] (1 - [P.sup.*.sub.u])/[partial derivative]c > or < 0.

Hence, higher union costs b, holding firm costs c constant have an unambiguous negative effect on the union's probability of holding out. However, higher firm strike costs c, holding union costs b constant have an ambiguous effect on the union's likelihood of holding out. At low cost levels, higher costs increase the holdout probability, but at higher cost levels, the effect on holding out becomes negative. It is that ambiguous result that may lead to a lower strike probability as strike costs c increase.

Joint Union or Firm Equilibrium Strike Behavior

We now combine optimal behavior for both sides. A strike results only when both sides hold out. Hence, the probability of a strike is given by the product of each side's holdout probabilities so that P(strike) = P(firm holdout) * P(union holdout) = (1 - [P.sup.*.sub.f])(1 - [P.sup.*.sub.u]), or

P(strike) = [([W.sub.r.sup.*] - [W.sup.*])/([W.sub.0.sup.*] + [W.sub.r.sup.*] - [W.sup.*])] [([[product].sub.ch] - [[product].sub.cc])/([[product].sub.hc] + [[product].sub.hc] + [[product].sub.cc])]. (18)

The joint costs theory of strikes predicts how the strike probability changes when union strike costs b and or firm strike costs c rise. We now show that higher strike costs need not always lead to a lower strike probability.

First look at how higher union costs b affect strike probability, holding firm costs c constant. To do that we differentiate the strike probability [(1 - [P.sup.*.sub.f]) (1 - [P.sup.*.sub.u])] with respect to b. The result is (5):

[partial derivative][{1 - [P.sup.*.sub.f])(1 - [P.sup.*sub.u])]/[partial derivative]b > or < 0.

This result indicates holding firm costs c constant, higher union strike costs b, have an ambiguous effect on the strike probability.

Next, we examine the impact of increasing firm strike costs c, holding union strike costs b constant. To do that we differentiate the strike probability [(1 - [P.sup.*.sub.f])(1 - [P.sup.*.sub.u])] with respect to c. The result is: (6)

[partial derivative][(1 - [P.sup.*.sub.f])(1 - [P.sup.*.sub.u])]/[partial derivative]c < or > 0.

Concession and resistance curve estimates are not common, so it is difficult to guess parameters of the relevant curves like [W.sub.r.sup.*] and [W.sub.0.sup.*]. Nevertheless, given our concern only with illustrating that the joint costs model need not hold, we merely take reasonable parameter values (based in part on Farber 1978 and Siebert et al. 1985) and try to simulate strike probabilities. These simulations are presented in Table 2 and in Figs. 2 and 3, and they confirm our models' predictions. Two examples are given: one in which union strike costs b increase, holding constant firm strike costs (left panel of Table 2 and Fig. 2); and one in which firm strike costs c increase, holding constant union strike costs (right panel of Table 2 and Fig. 3). Presented are firm and union holdout probabilities (1 - [P.sup.*.sub.f]) and (1 - [P.sup.*.sub.u]), as well as the strike probability P(str). As we increase union strike costs b, the union's holdout probability continually declines. On the other hand, higher values for b first increase then decrease the firm's holdout probability. The joint effect is that strike probability first increases (up to 4%) then decreases. Similarly, as we increase firm strike costs c, the firm's holdout probability continually declines. However, higher values for c first increase then decrease the union's holdout probability. The joint effect as firm strike costs c increase is that strike probability first rises (up to 5%) then declines. It is interesting to note that the British strike probability is between 0.8 and 4.9% (Ingram et al. 1993). though at least in the past somewhat higher in the U.S. (Gramm 1987).

Concluding Remarks

Motivated by the mixed success of the joint costs model of strike activity, we analyzed union-firm bargaining behavior in the context of Hicksian concession curves. We found both unions and firms to fare best when both concede. They fare worst when both hold out. The union does best when it holds out and when the firm concedes. Firms do best when they hold out while the union concedes. This reward structure yields a payoff matrix comparable to the game of chicken. In chicken, no single pure Nash equilibrium solution emerges. Instead, there exist two pure Nash equilibria and a mixed Nash equilibrium. The perfectly rational firm and union follow a mixed strategy so that they hold out occasionally to preserve credibility, even if both sides could see a better deal by jointly conceding. Hence, the union or the firm may choose to hold out even if the expected payoff from holding out or conceding in a mixed-strategy equilibrium is less than the payoff when both concede. This result is consistent with Hicks (1963), who reached the conclusion that "the trade union leadership will embark on strikes occasionally, not so much to secure greater gains upon that occasion (which are not very likely to result) but in order to keep the weapon burnished ... (p. 146)."

The results show that increasing strike costs asymmetrically can have ambiguous effects on the strike probability. Increasing one side's strike costs decreases its incentive to strike. However, in response, the other side's incentive can increase, since under many circumstances it bargains harder to collect relatively larger expected rents. As such, the probability of a strike can rise even as joint strike costs increase. This result accounts for the mixed success of the joint cost model in explaining strike activity.

Acknowledgments The authors would like to thank John Abowd. Steven Brams, Charlie Brown, Ronald Ehrenberg, Jose Galdon-Sanchcz, Carlos Sciglic, Maia Giicll-Rotlan. Thomas Head. Phillip Nelson, and Stanley Siebert for valuable insights and discussion. Special thanks to an anonymous referee for helpful comments. Any remaining errors are our own.

Electronic supplementary material The online version of this article (doi:10.1007/s11293-017-9539-5) contains supplementary material, which is available to authorized users.

Published online: 5 May 2017

DOI 10.1007/s11293-017-9539-5

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Reder, M. W., & Neumann, G. R. (1980). Conflict and contract: The case of strikes. Journal of Political Economy, 88(5), 867-886.

Siebert, W. S., et al. (1985). The political model of strikes: A new twist. Southern Economic Journal, 52(1), 23-33.

Sopher, B. (1990). Bargaining and the joint-cost theory of strikes: An experimental study. Journal of Labor Economics, 88(1), 48-74.

(1) The same holds when firm costs rise more quickly than union costs.

(2) Obviously one can complicate this by introducing other costs. Examples include the costs of negotiation, the distaste of temporarily being out of work, the cost of setting up picket lines, the possible termination of workers, and more. But additional costs are not necessary given our only objective is to prove that increasing strike costs need not always decrease the probability of a strike. Thus we take the simplest possible model.

(3) The mathematical calculations are available upon request from the authors.

(4) The mathematical calculations are available upon request from the authors.

(5) The mathematical calculations are available upon request from the authors.

(6) The mathematical calculations are available upon request from the authors.

Archontis L. Pantsios [1] * Solomon W. Polachek [2]

[mail] Solomon W. Polachek

polachek@binghamton.edu

Archontis L. Pantsios pantsia@hope.ac.uk

[1] School of Business, Liverpool Hope University, Liverpool, UK

[2] Department of Economics, State University of New York at Binghamton, Binghamton, NY. USA

Caption: Fig. 1 Mauro-modified Hicks concession and resistance curves; [W.sub.f] and [W.sub.u] are employer concession and union resistance curves respectively; [W.sup.p.sub.f] and [W.sup.p.sub.u] are the perceived employer concession and union resistance curves. Source: based on Mauro (1982. Fig. 2, p. 525)

Caption: Fig. 2 Simulated strike probabilities as union strike costs rise. For union strike costs, the additional parameters are: [W.sub.0] = 15, [[phi].sub.1] = 16, [[phi].sub.2] = 30, Wr = 25, [[theta].sub.1] = 24, [[theta].sub.2] = 30, and c = 5

Caption: Fig. 3 Simulated strike probabilities as firm strike costs rise. For firm strike costs, the additional parameters are: [W.sub.0] = 15, [[phi].sub.1] = 16, [[phi].sub.2] = 30, Wr = 25, [[theta].sub.1] = 24, [[theta].sub.2] = 30, and b = 5
Table 1 Payoff matrix

                   Concede (firm)             Hold out (firm)

Concede (union)    [U.sub.cc] = [W.sup.*]     [U.sub.ch] =
                                              [W.sub.0.sup.*]

                   [[product].sub.cc] = R     [[product].sub.ch] =
                   ([L.sub.cc]) -             R([L.sub.ch]) -
                   [W.sup.*][L.sub.cc]        [W.sup.p.sub.0]
                                              [L.sub.ch]

Hold Out (union)   [U.sub.hc] =               [U.sub.hh] = 0
                   [W.sub.r.sup.*]

                   [[product].sub.hc] =        [[product].sub.hh] = 0
                   R([L.sub.hc]) -
                   [W.sup.p.sub.r][L.sub.hc]

Table 2 The impact of changing costs on holdout and strike probabilities

Changing union strike cost

Union       Firm holdout prob.    Union holdout prob.     Strike prob.
strike
costs

(b)         (1-[P.sup.*.sub.f])   (1 - [P.sup.*.sub.u])   P(str)
0           0                     0.35                    0
0.1         0.01                  0.34                    0.003
0.2         0.02                  0.34                    0.01
0.5         0.04                  0.33                    0.015
1           0.08                  0.31                    0.02
1.5         0.1                   0.3                     0.03
2           0.12                  0.28                    0.04
3           0.15                  0.26                    0.04
4           0.17                  0.23                    0.04
5           0.18                  0.21                    0.04
10          0.21                  0.13                    0.03
15          0.22                  0.09                    0.02
20          0.21                  0.05                    0.01
23          0.21                  0.04                    0.008
25          0.207                 0.03                    0.006
28          0.203                 0.02                    0.004
30          0.2                   0.01                    0.002
32          0.197                 0.005                   0.0001

Changing firm strike costs

Firm        Firm holdout prob.    Union holdout prob.     Strike prob.
strike
costs

(c)         (1 - [P.sup.*.sub.f]) (1-[P.sup.*.sub.u])     P(str)
0           0.37                  0                       0
0.1         0.36                  0.02                    0.006
0.2         0.36                  0.03                    0.01
0.5         0.34                  0.07                    0.02
1           0.32                  0.11                    0.04
1.5         0.29                  0.14                    0.04
2           0.27                  0.16                    0.04
3           0.24                  0.19                    0.05
5           0.19                  0.21                    0.04
10          0.11                  0.22                    0.02
15          0.06                  0.21                    0.01
20          0.04                  0.2                     0.007
25          0.02                  0.18                    0.004
28          0.01                  0.178                   0.002
30          0.007                 0.174                   0.001
31          0.006                 0.172                   0.001
33          0.002                 0.168                   0.0003
34          0                     0.166                   0

Source: Own calculations based on the following parameters: For
union strike costs, the additional parameters are [W.sub.0] - 15,
[[phi].sub.1], = 16, [[phi].sub.2] = 30, [W.sub.r] = 25,
[[theta].sub.1] = 24, [[theta].sub.2] = 30, and c = 5. For firm
strike costs, the additional parameters are: [W.sub.0] = 15,
[[phi].sub.1] = 16, [[phi].sub.2] = 30, [W.sub.r] = 25,
[[theta].sub.1] = 24, [[theta].sub.2] = 30, and b = 5
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Author:Pantsios, Archontis L.; Polachek, Solomon W.
Publication:Atlantic Economic Journal
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Date:Jun 1, 2017
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