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Contract renewal under uncertainty
Journal of Economic Dynamics & Control 26 (2002) 637}652
Contract renewal under uncertainty夽 Torben M. Andersen*, Morten Stampe Christensen CEPR, Centre for Dynamic Modeling in Economics, Department of Economics, University of Aarhus, Building 350, DK-8000, Aarhus C, Denmark Received 1 September 1997; accepted 10 August 2000
Abstract The incentive to adjust prices is considered from a bilateral perspective in a setting where changes in outside opportunities drive the incentive to renew contracts and costs preclude continuous renewal. A model encompassing several contract forms is formulated, and the existence of an equilibrium to the bilateral renewal game is established. Prices display inertia, and the incumbent contract is found to be more resistant to changes in outside opportunities, the larger the costs of contract renewal, the variability of outside opportunities and the lower the discount rate. The model is shown to match a number of empirical observations on contracts. 2002 Elsevier Science B.V. All rights reserved. JEL classixcation: C72; D81; E31 Keywords: Contract renewal; Stopping game; Price adjustment
1. Introduction Many transactions take place within a long-term relationship. This trading relationship may be formalized in an explicit long-term contract, but often the contractional relationship is implicit. One important question is how the terms 夽 We gratefully acknowledge comments from an anonymous referee, Morten Hviid and participants at the workshop &Imperfect Markets and Macroeconomics', University of Copenhagen. Financial support from the Danish Social Science Research Council is gratefully acknowledged. * Corresponding author. Tel.: #45-8942-1609; fax: #45-8613-6334. E-mail address: [email protected] (T.M. Andersen).
0165-1889/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 8 8 9 ( 0 0 ) 0 0 0 7 5 - 0
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of trade are adjusted in such relationships since continuous trading over long periods of time at unchanged prices is seldom observed, but neither is continuous price adjustment. The bilateral character of the adjustment problem is essential, since both parties may have an incentive to change the price although not necessarily in the same circumstances. Both the &seller' and the &buyer' may thus take the initiative to change the terms at which transactions take place. If adjustment is costless the problem is trivial, but if adjustment entails costs in an uncertain environment, the problem is complex. The unilateral adjustment problem has been extensively studied for both product and labour markets. The typical setting for the product market is a monopolist "rm quoting a nominal price (implicit contract with customers) (Sheshinski and Weiss, 1977, 1983; Danziger, 1983, 1984, 1987; Caplin and Spulber, 1987; Caplin and Leahy, 1991). Adjustment costs (menu costs) imply that nominal prices remain unchanged to &small' shocks while there is an adjustment to &large' shocks. The adjustment process may also display asymmetries and hysteresis (see e.g. Tsiddon, 1993; Dixit, 1991). For the labour market the focus has been on how uncertainty a!ects the optimal contract length assuming that either the "rm or the workers determine the terms of the contract (see e.g. Gray, 1978; Dye, 1985) The possibility of reopening contracts in case of &large' shocks has been considered by Danziger (1995). The present paper di!ers from the above-mentioned literature by taking an explicit bilateral approach in which the incentives of both the &buyer' and the &seller' side to adjust the terms of the contract (price) are considered. This is clearly relevant in many cases and brings forth the interesting problem of how the incentives to call for contract renewal are mutually dependent for the two parties given that they usually have opposite interests. To see the opposing interests of the two parties, it is useful to start by considering the case of price adjustment in contracts where quantities cannot be changed (in the short run). In this case price adjustment may be perceived to be a question of pecuniary redistribution and therefore essentially a zero sum game (Williamson, 1979). The incentive underlying price adjustment should thus be symmetric in the sense that what one party gains, the other loses. We show that this perception is misleading due to the frictions created when reopening of contracts is costly in an uncertain environment. Contract renewal demanded by one party thus has a transaction cost externality on top of the pecuniary redistribution to the other Even when transactions are settled by long-term contracts with a duration as long as 50 years, there is usually an option for changing the price, see e.g. Crocker and Masten (1991), Goldberg and Erickson (1987) and Joskow (1988)). Also for labour contracts with a typical duration of 1}5 years (Carlton, 1986) one "nds reopening clauses allowing for changes in the wage (Vroman, 1989). The problem is formulated such that neither party to the contract has any market power except what is created by adjustment cost. In this way we also disregard a possible lock in e!ect from e.g. speci"c investments, etc. See e.g. McLeod and Malcomson (1993) and Holden (1995).
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party of the contract. This a!ects the incentives underlying contract renewal and gives the terms of the incumbent contract a special role, and may cause asymmetry in the incentive to adjust prices. Moreover, the noncooperative contract renewal game may have ine$ciencies since an action by one party to call for contract renewal does not take into account the consequences for the opponent in terms of contract renewal costs. The paper is organised as follows: Section 2 sets up the basic problem of contract renewal under uncertainty when fully contingent contracts cannot be signed, while the renewal game is solved in Section 3. Section 4 o!ers a few concluding remarks.
2. Contract renewal under uncertainty Consider a contract between a principal (P) and an agent (A) stipulating a given #ow of services or actions to be taken by the agent who in turn is compensated by a #ow payment from the principal. At any point in time there is the possibility that either the principal or the agent may want to suggest a contract renewal } at a cost } because outside opportunities have changed, that is, the agent "nds that he can receive a better compensation by shifting to another principal, or the principal perceives that he can replace the agent by another agent willing to accept a lower payment. Examples of contracts "tting this description are legio including employment contracts, tenant contracts, delivery contracts, etc. Let q denote the payment according to the incumbent contract and let w denote the outside opportunity (alternative price). If the contract is renewed R at time t, the new payment will be w . By specifying an exogenous outside R opportunity, we avoid having to go into details about the bargaining procedure which allows us to focus on the implications of uncertainty and adjustment costs. The problem faced by the agent (principal) is when a costly contract renewal should be undertaken to change the current payment q (!q) to w R (!w ). A similar problem will arise if the incentive to call for contract renewal is R driven by internal factors like, e.g. changes in productivity. To focus on the We express payments in present value terms, i.e. if the #ow payment is q then q"e\HQq ds"q/, where is the time-invariant discount rate. Similarly for costs. Thereby we also leave out problems arising from attempts to exploit the market power arising from &switching costs', see Klemperer (1995). The way the problem is formulated precludes the possibility of negotiating a "xed price for the entire contract period, it is not time consistent. An alternative interpretation of the model is to interpret w as a process driving the value of the R output produced by the agent. In this case the analysis carries through if wages are settled in negotiations between the agent and principal such that the wage is a share of productivity (w ), R leaving a share (1!) of output as pro"ts ((1!)w ). R
640 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652
incentives underlying price adjustment, the quantities transacted are assumed given (see Andersen and Christensen (1997) on endogenous quantity determination). If the outside opportunity was given deterministically, the contract renewal problem would be trivial and the timing of contract renewal could easily be determined. However, if the outside opportunities evolve stochastically, the question of contract renewal becomes nontrivial. Speci"cally, we assume that w evolves according to a geometric Brownian motion process, i.e. R dw "w dt#w dz , R R R R where dz is the increment of a Wiener process, i.e. E(dz )"0 and Var(dz )"dt. R R R Obviously, if contract renewal is costless the contract would be renewed continuously with changes in outside opportunities.This is counterfactual. Contract renewal is costly and usually involves both "xed and variable costs. Speci"cally, it is assumed that renewal costs are proportional to the new payment #ow, and equal to w for both the agent and the principal. In R Andersen and Christensen (1997) we show how the analysis can be extended to allow for asymmetric costs including both a "xed and a variable component. Both the principal and the agent are risk-neutral and have an in"nite horizon with a discount rate . We assume ' to rule out the trivial case where the parties to the contract are always better o! by waiting and therefore never exercise the option to call a contract renewal (see e.g. Pindyck, 1991). To see the basic mechanisms involved, we start by considering the special case where the contract can only be renewed once (see Andersen and Christensen, 1997) for a generalization to an in"nite number of renewals). The option of being able to renew the contract has a value to the contract parties. Assume that a contract renewal is called by the agent if c q"w and by the principal if R c q"w . Denote the point in time where the agent will call for contract renewal R by ¹ and similarly ¹ for the principal. . Consider now the consequences to the agent of a contract renewal. The possibility of a contract renewal is an asset to the agent if the outside opportunity improves and the agent has the possibility of raising the payment. Oppositely, the possibility that the principal can call for contract renewal if outside The contract problem is thus when to replace the deterministic payment q with a stochastic outside payment w. In Andersen and Christensen (1997) it is shown how the framework can be adopted to deal with the case where the pay-o! is also stochastic under the incumbent contract. These costs may arise through di!erent channels: lawyers' fees are dependent on the contract sum, stamp duties are often based on the contract sum, the value of the time used to settle the contract (evaluated at the opportunity wage). Assuming that costs are independent of the payment #ow causes problems in an in"nite horizon model with drift in the payment #ow since it implies counterfactually that the costs of contract renewal relative to the gain from renewal may approach zero.
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opportunities deteriorate constitutes a liability to the agent as he will be worse o! in this case. The expected value to the agent of a contract renewal when the outside opportunity is w , x (w ), can be expressed in terms of the sum of the R R expected value of the asset (a) and the liability (l) component of the option, i.e. x (w )"x (w )#x (w ) R R R "E(e\H2 (c q!q!c q)w ¹ (¹ ) R . #E(e\H2. (c q!q!c q)w ¹ (¹ ). . . R . The value to the principal of the contract renewal opportunity can similarly be written as x (w )"x (w )#x (w ) . R . R . R "E(e\H2. (q!c q!c q)w ¹ (¹ ) . . R . #E(e\H2 (q!c q!c q)w ¹ (¹ ). R . The expected value of the contract renewal option satis"es (cf. Karlin and Taylor, 1980, Section 15.3) the following second-order di!erential equation: x x H #x H "x , j"A, P. x H w H 2 H w This has a straightforward interpretation since the left-hand side according to Ito's lemma gives the expected change in the value of the option of being able to renew the contract while the right-hand side gives the deterministic pay-o! if the contract renewal is exercised immediately. Clearly, these two forces have to balance to have a nontrivial solution to the contract renewal problem. Solving this second-order di!erential equation yields x (w )"m w?#n w@, j"A, P, H R H R H R where
(1)
1 2 ! # '1, 2
1
, ! # 2
1 , ! ! 2
1 2 ! # (0. 2
The parameters m and n are determined by the boundary conditions stating H H that at the time of contract renewal, the value of the contract renewal option is equal to the value of contract renewal (value matching condition). For the agent
642 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652
we have x (c q)"[c (1!)!1]q, , x (c q)"[c (1!)!1]q, . . and for the principal x (c q)"[1!c (1#)]q, , x (c q)"[1!c (1#)]q, . . . . . Imposing these boundary conditions, (1) can be rewritten as 1 x (w )" [[ c@ ! c@ ]q\?w?#[ c? ! c? ]q\@w@], R . R . R 1 x (w )" [[ c@ ! c@ ]q\?w?#[ c? ! c? ]q\@w@], . R . R . R where ,c? c@ !c? c@ '0. . . 3. The contract renewal game There is a strategic interaction in contract renewal between the principal and the agent, since the action to demand contract renewal has consequences to the other part. We look for Nash equilibria to this contract renewal game where each party decides on its optimal critical value c (j"A, P) given the critical H value of the opponent. To this end we need Lemma 1. For any c 3[0, 1], there exists a unique best response c 3[1, R[ for . the agent, where
c 3 max 1, , . ( !1)( !1)(1!) ( !1)(1!) For any c 3[1,R[, there exists a unique best response c 3[0, 1] where
c 3 , min 1, . ( !1)(1#) ( !1)( !1)(1#)
.
Proof. See the appendix. The strategic interaction in contract renewal implies that the more reluctant the agent is to demand contract renewal (the larger c ), the more reluctant the principal will be in calling a contract renewal (the smaller c ). This shows . a strategic complementarity in contract renewal decisions.
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Fig. 1. Nash equilibrium to the noncooperative renewal game.
Existence of a Nash equilibrium to the contract renewal game is ensured by (See Fig. 1). Proposition 1. A Nash equilibrium (c* , c*) exists to the contract renewal game. . Moreover, for an open set of parameters (, , , ) the equilibrium is nontrivial, i.e. c* '1 and c*(1. A suzcient condition for uniqueness is absence of drift in the . outside payment w ("0). R Proof. See the appendix. The Nash equilibrium to the contract renewal game implies an interval [c q, . c q] of inaction in the sense that as long as the outside opportunity payment remains in this interval, none of the parties to the contract have an incentive to call a contract renewal. This interval of inaction resembles the [s, S]-rules imposed on unilateral adjustment problems as in, e.g. the menu cost models. Although the implications are the same, the region of inaction follows here from a game between parties having opposite incentives concerning payment revisions. It is noteworthy that the critical values for contract renewal are path independent as We disregard the case where either c 41 or c 51, as this implies immediate renewal. This . may, of course, be optimal with a "nite number of renewals, but in the case of an in"nite number of contract renewals, it is not possible for c "c "1 to be a solution. .
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Corollary 1. The equilibrium values of c* and c* are invariant to the realizations of . the stochastic variable w , and the size of the incumbent payment yow q. R Proof. Follows from proof of Proposition 1. The actual payment displays, however, path-dependence. In the absence of contracts the spot market payment would equal the outside opportunity, while with a "xed payment long-term contract the payment would be constant over time. In the present setting the payment is determined by a high (low) outside payment in the past if the payment has been revised upwards (downwards). In this sense extreme market conditions in the past come to determine payments due to the lock-in e!ect caused by costly contract renewal. In a study of labour contracts Beaudry and DiWardo (1991) "nd empirical support for wages being positively correlated with the best labour market conditions observed since the worker was hired. We also "nd that adjustment is asymmetric as Corollary 2. The interval supporting the incumbent payment yow [c q, c q] is not . geometrically symmetric around q, i.e. c c O1. In the case of no drift in the . payment yow w ("0), we have c c "(1!)\'1 and hence the nonadjustR . ment region is rightward-skew, i.e. c 'c\. . Proof. Follows from Lemma 2 in the appendix. Although the contract renewal problem is set up to be symmetric (payment changes are a zero-sum game and contract renewal costs are symmetric), the region of no-adjustment is (geometrically) asymmetric. As a point of reference it is noted that in the case of certainty the critical values are c "(1!)\, c "(1#)\. Comparing to the case of uncertainty without . drift ("0) we "nd that the product of the critical levels are the same, i.e. c c "c c (compared to the unilateral case, cf. e.g. Dixit, 1991), but the e!ect . . of uncertainty is to expand the range implying price inertia, i.e. c 'c , c (c . . . The intuition is simply that uncertainty adds to the costs of adjusting prices since there is an option value of waiting. Proposition 2. The cooperative renewal game (c! , c!) has a larger interval support . ing the incumbent contract than the non-cooperative renewal game (c* , c*), i.e. . c! 'c* , c!(c*. . . Proof. See the appendix. The intuition is straightforward as there is a transaction cost externality of calling a contract renewal on top of the pecuniary redistribution. The party calling a contract renewal imposes transaction costs on the other party. When this externality is internalized, the region of inaction expands. Even though costs
T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652 645
Fig. 2. (A) c and c as a function of . (B) c and c as a function of . (C) c and c as a function of . . . . (D) c and c as a function of the cost parameters . .
imply price inertia, prices may still be adjusted too frequently due to the interplay between uncertainty and renewal costs. Due to the complexity of the model it is not possible to obtain analytical results on how the contract renewal problem is a!ected by the drift parameter (), the variance (), the discount rate () and the cost parameter (). Accordingly, numerical simulations have been undertaken, and they are reported in Fig. 2 which is based on a benchmark case where ("0, "0.04, "0.1, "0.05), and then allowing one parameter to change in each experiment. The simulations show that an increase in the cost parameter enlarges the band supporting the existing contract payment as the critical level increases for the agent, and decreases for the principal. It is worth pointing out that although both the agent and principal are assumed to be risk-neutral, the contract renewal problem is a!ected by risk. The reason is simply that the value of waiting to have a contract renewal depends on the variability of outside opportunities. It is found here that an increase in the variance enlarges the interval supporting the initial price. This re#ects that the possibility of extreme values becomes larger, i.e. the value of waiting increases and therefore the interval supporting the initial contract payment expands. A higher discount
646 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652
factor reduces the gains from waiting and therefore the critical level decreases for the agents and increases for the principal. Both critical values are increasing in the drift parameter. The contract renewal problem implies that the point in time at which contract renewal will be called by one of the parties is stochastic. This allows us to consider the length of the contract in terms of the expected time to contract renewal which can be written >(w )"E[¹w ], R R where ¹"¹ if w "c q and ¹"¹ if w "c q. Following Karlin and R . R . Taylor (1980) the expected time to contract renewal satis"es w >#w >#1"0 R R and the boundary conditions are given by >(c q)">(c q)"0. . Fig. 3 shows how the expected contract length depends on the parameters of the contract renewal problem. The expected contract length is decreasing in the
Fig. 3. (A) The expected length of a contract as a function of . (B) The expected length of a contract as a function of . (C) The expected length of a contract as a function of . (D) The expected length of a contract as a function of .
T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652 647
discount rate (), increasing in the cost parameter () and decreasing in the variance (). While the two "rst results are straightforward implications of the results found above on the critical values, the latter e!ect is not. A larger variability induces a larger interval supporting the initial payment, cf. above, and this tends to lengthen the time to contract renewal. However, the likelihood of having a contract renewal increases as the variance increases, and this tends to reduce the time to contract renewal. The latter e!ect dominates such that the expected contract length is decreasing in the variance. For the drift parameter we "nd a nonmonotone relationship with the longest expected duration in the case of zero drift. This is intuitive as the drift term implies an underlying deterministic trend in outside opportunities which is bound to release a contract renewal. These predictions are in accordance with empirical analysis of contract duration in labour markets in which it is found that contract length is decreasing in uncertainty, decreasing in in#ation and increasing in contracting costs (Vroman, 1989; Murphy, 1992).
4. Concluding remarks In a bilateral contract renewal problem in which the incentive to call for contract renewal is driven by (stochastic) changes in outside opportunities, it has been shown that payments display inertia. Contract renewal costs prevent continuous payment revisions and the incumbent contract payment comes to play a crucial role. This implies among other things path dependence in payment and price rigidities. Although the model-structure and "ndings can be shown to be robust to various modi"cations of the contract renewal problem (see Andersen and Christensen, 1997), the model remains in a number of respects stylized. An important issue for future research would be to combine the question of payment adjustment with the problem of long-term investment and thus to explain why there is an incentive to enter contracts in the "rst place.
Appendix. Proof of Lemma 1 Deriving the optimal c (c ) can be done either by solving the "rst-order . condition for the maximization of x (x ) or by imposing a &smooth pasting' . condition, see Dixit (1988). As both methods yield the same results, we stick to the "rst one as this is most intuitive. Denote by FOC the derivative of x (w ) wrt. c and by FOC the derivative of x (w ) wrt c . The agent's R . . R . (principal's) choice of c (c ) is found by equating FOC (FOC ) to zero . .
648 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652
which yields FOC c [q\?\@(c? q?w@!c@ q@w?)]\ . R . R "!c? c? [(1! )(1!)c # ]!c@ c? [( !1)(1!)c ! ] . . !c?>@[( ! )(1!c (1!))]"0, . FOC c [q\?\@(c? q?w@!c@ q@w?)]\ . . R R "c@ c? [(1! )(1#)c # ]#c? c@ [( !1)(1#)c ! ] . . . . #c?>@[( ! )(1!c (1#))]"0. . The second-order conditions evaluated at the points where the "rst-order conditions are satis"ed are given by SOC "!c\ [ #c (1!)( !1)(1! )](0, SOC "c\ [ #c (1#)( !1)(1! )](0. . . . It is now straightforward to show that FOC
FOC
(0, ? A ?\\O
51. ( !1)(1!)
'0, A
?@ ?\@\\O
51. ( !1)( !1)(1!)
(A.1)
(A.2)
Hence, either c* "1 or
cH 3 , ( !1)( !1)(1!) ( !1)(1!)
in which case SOC assures us that we have a maximum. Furthermore, this is the unique value satisfying FOC as SOC (0 for all values of c belonging to the above interval. Similarly, it is easily shown that FOC .
(0, A.
@ @\>O
41, ( !1)(1#)
(A.3)
T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652 649
FOC .
41. ( !1)( !1)(1#)
'0, ?@ A ?\@\>O
(A.4)
Hence, either c*"1 or .
cH3 , . ( !1)(1#) ( !1)( !1)(1#)
and the second-order conditions ensure a unique maximum value. Proof of Proposition 1. The proof proceeds by considering what happens with the optimal choice of c (c ) for extreme values of c (c ). From Lemma 1 and . . the "rst-order conditions derived in the proof of Lemma 1, it follows
, Lim c ,c( 5Max 1, ( !1)( !1)(1!) A.
Lim c ,c " , ( !1)(1!) A.
Lim c ,c( 4Min 1, , . . ( !1)( !1)(1#) A Lim c ,c " . . . ( !1)(1#) A By drawing the reaction curves in a (c , c ) diagram using the above end points, . it follows from the continuity of the curves that they must have at least one intersection. Hence, at least one Nash equilibrium exists. Uniqueness can be proven in the case of zero drift, i.e. "0 by using the following lemma: Proof of Lemma 2. If "0, and FOC "0 for
1 (c , c )" c , , . c (1!)(1#) then FOC also equals zero. .
650 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652
Proof. Insert c "1/c (1!)(1#) and c (1!)"1/c (1#) in FOC to . . obtain FOC "0 implying 1 !c?\(1!)?\(1#)?\ (1! ) #
c (1#) . 1 !c\?(1!)\?(1#)\? (! ) !(1! ) c (1#) . 1 !c (1!2 ) 1! "0. c (1#)
Rearranging and multiplying with c (1#) we get . #c? c\?[(1! )(1#)c !(1! )]#c\?c? [(1! )(1#)c # ] . . . . #c [(1!2 )(c (1#)!1)]"0 . It follows that FOC "0. .
䊐
To prove a unique Nash equilibrium for "0 we observe that: (i) at least one Nash equilibrium exists, (ii) furthermore it follows from the end points of the reaction curves that an odd number of equilibria exists, (iii) Lemma 2 implies that whenever the reaction curve for c (or c ) intersects the hyperbola . c "1/c (1!)(1#), a Nash equilibrium exists. We now show that the . reaction curve for c has at most two intersections with the hyperbola c "1/c (1!)(1#), and it then follows from (i) and (ii) that there is only one . intersection, and hence only one Nash equilibrium for
1
1
!1 1 1 c 3 , and c 3 , . 1! !1 1! .
1# 1# The proof proceeds by inserting the following in FOC 1 1 c " , . c (1!)(1#) c Di!erentiating twice wrt. c we obtain 2 (1!3 #2 )c\\?\\?(c #c??)(1!c (1!))(0. We conclude that FOC is concave in c along the hyperbola c "1/c (1!)(1#) and therefore is zero at most twice. .
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Proof of Proposition 2. Follows from the following facts: X X '0 (0, c c . and the fact that SOC , SOC (0 at the points where FOC , FOC "0. . . Hence, a cooperative renewal game must have c 'cH , c (cH. . . References Andersen, T.M., Christensen, M.S., 1997. Contact renewal under uncertainty. Memo 1997-5, Department of Economics, University of Aarhus. Beaudry, P., DiWardo, J., 1991. The e!ect of implicit contracts on the movement of wages over the business cycle: evidence from micro data. Journal of Political Economy 99, 665}688. Caplin, A., Leahy, J., 1991. State-dependent pricing and the dynamics of money and output. Quarterly Journal of Economics CVI, 683}799. Caplin, A.S., Spulber, D.F., 1987. Menu costs and the neutrality of money. Quarterly Journal of Economics CII, 703}725. Carlton, D.W., 1986. The rigidity of prices. American Economic Review 76, 637}658. Crocker, K.J., Masten, S.E., 1991. Pretia ex machina? Prices and process in long-term contracts. Journal of Law and Economics XXXIV, 69}99. Danziger, L., 1983. Price adjustments with stochastic in#ation. International Economic Review 24, 699}707. Danziger, L., 1984. Stochastic in#ation, and the optimal policy of price adjustments. Economic Inquiry XXII, 98}108. Danziger, L., 1987. On in#ation and real price variability. Economic Inquiry 25, 285}299. Danziger, L., 1995. Contract reopeners. Journal of Labor Economics 13, 62}87. Dixit, A.K., 1988. A heuristic derivation of the conditions for optimal stopping of brownian motion. Working paper, Princeton University. Dixit, A.K., 1991. Analytical approximations in models of hysteresis. Review of Economic Studies 58, 141}145. Dye, R.A., 1985. Optimal length of labor contracts. International Economic Review 26, 251}270. Gray, J.A., 1978. On indexation and contract length. Journal of Political Economy 86, 1}18. Goldberg, V.P., Erickson, J.R., 1987. Quantity and price adjustment in long-term contracts: a case study of petroleum coke. Journal of Law and Economics XXX, 369}398. Holden, S., 1995. Renegotiation and the e$ciency of investments. Working paper, Department of Economics, University of Oslo. Joskow, P.L., 1988. Price adjustment in long-term contracts: the case of coal. Journal of Law and Economics XXXI, 47}83. Karlin, S., Taylor, H.M., 1980. A second course in stochastic processes. Academic Press, New York. Klemperer, P., 1995. Competition when consumers have switching costs } an overview with applications to industrial organization, macroeconomics and international trade. Review of Economic Studies 62, 515}539. McLeod, W.B., Malcomson, J., 1993. Investments, holdup and the form of market contracts. American Economic Review 83, 811}837. Murphy, K.J., 1992. Determinants of contract durations in collective bargaining agreements. Industrial and Labor Relations Review 45, 352}365.
652 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652 Pindyck, R.S., 1991. Irreversibility, uncertainty and investment. Journal of Economic Literature XXIX, 1110}1148. Sheshinski, E., Weiss, Y., 1977. In#ation and costs of price adjustment. Review of Economic Studies 44, 287}303. Sheshinski, E., Weiss, Y., 1983. In#ation and costs of price adjustment. Review of Economic Studies 50, 513}529. Tsiddon, D., 1993. The (mis) behaviour of the aggregate price level. Review of Economic Studies 60, 889}902. Vroman, S.B., 1989. In#ation uncertainty and contract duration. Review of Economics and Statistics 71, 677}681. Williamson, O.E., 1979. Transactions-cost economics: the governance of contractual relations. Journal of Law and Economics XXII, 233}261.
Contract renewal under uncertainty夽 Torben M. Andersen*, Morten Stampe Christensen CEPR, Centre for Dynamic Modeling in Economics, Department of Economics, University of Aarhus, Building 350, DK-8000, Aarhus C, Denmark Received 1 September 1997; accepted 10 August 2000
Abstract The incentive to adjust prices is considered from a bilateral perspective in a setting where changes in outside opportunities drive the incentive to renew contracts and costs preclude continuous renewal. A model encompassing several contract forms is formulated, and the existence of an equilibrium to the bilateral renewal game is established. Prices display inertia, and the incumbent contract is found to be more resistant to changes in outside opportunities, the larger the costs of contract renewal, the variability of outside opportunities and the lower the discount rate. The model is shown to match a number of empirical observations on contracts. 2002 Elsevier Science B.V. All rights reserved. JEL classixcation: C72; D81; E31 Keywords: Contract renewal; Stopping game; Price adjustment
1. Introduction Many transactions take place within a long-term relationship. This trading relationship may be formalized in an explicit long-term contract, but often the contractional relationship is implicit. One important question is how the terms 夽 We gratefully acknowledge comments from an anonymous referee, Morten Hviid and participants at the workshop &Imperfect Markets and Macroeconomics', University of Copenhagen. Financial support from the Danish Social Science Research Council is gratefully acknowledged. * Corresponding author. Tel.: #45-8942-1609; fax: #45-8613-6334. E-mail address: [email protected] (T.M. Andersen).
0165-1889/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 8 8 9 ( 0 0 ) 0 0 0 7 5 - 0
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of trade are adjusted in such relationships since continuous trading over long periods of time at unchanged prices is seldom observed, but neither is continuous price adjustment. The bilateral character of the adjustment problem is essential, since both parties may have an incentive to change the price although not necessarily in the same circumstances. Both the &seller' and the &buyer' may thus take the initiative to change the terms at which transactions take place. If adjustment is costless the problem is trivial, but if adjustment entails costs in an uncertain environment, the problem is complex. The unilateral adjustment problem has been extensively studied for both product and labour markets. The typical setting for the product market is a monopolist "rm quoting a nominal price (implicit contract with customers) (Sheshinski and Weiss, 1977, 1983; Danziger, 1983, 1984, 1987; Caplin and Spulber, 1987; Caplin and Leahy, 1991). Adjustment costs (menu costs) imply that nominal prices remain unchanged to &small' shocks while there is an adjustment to &large' shocks. The adjustment process may also display asymmetries and hysteresis (see e.g. Tsiddon, 1993; Dixit, 1991). For the labour market the focus has been on how uncertainty a!ects the optimal contract length assuming that either the "rm or the workers determine the terms of the contract (see e.g. Gray, 1978; Dye, 1985) The possibility of reopening contracts in case of &large' shocks has been considered by Danziger (1995). The present paper di!ers from the above-mentioned literature by taking an explicit bilateral approach in which the incentives of both the &buyer' and the &seller' side to adjust the terms of the contract (price) are considered. This is clearly relevant in many cases and brings forth the interesting problem of how the incentives to call for contract renewal are mutually dependent for the two parties given that they usually have opposite interests. To see the opposing interests of the two parties, it is useful to start by considering the case of price adjustment in contracts where quantities cannot be changed (in the short run). In this case price adjustment may be perceived to be a question of pecuniary redistribution and therefore essentially a zero sum game (Williamson, 1979). The incentive underlying price adjustment should thus be symmetric in the sense that what one party gains, the other loses. We show that this perception is misleading due to the frictions created when reopening of contracts is costly in an uncertain environment. Contract renewal demanded by one party thus has a transaction cost externality on top of the pecuniary redistribution to the other Even when transactions are settled by long-term contracts with a duration as long as 50 years, there is usually an option for changing the price, see e.g. Crocker and Masten (1991), Goldberg and Erickson (1987) and Joskow (1988)). Also for labour contracts with a typical duration of 1}5 years (Carlton, 1986) one "nds reopening clauses allowing for changes in the wage (Vroman, 1989). The problem is formulated such that neither party to the contract has any market power except what is created by adjustment cost. In this way we also disregard a possible lock in e!ect from e.g. speci"c investments, etc. See e.g. McLeod and Malcomson (1993) and Holden (1995).
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party of the contract. This a!ects the incentives underlying contract renewal and gives the terms of the incumbent contract a special role, and may cause asymmetry in the incentive to adjust prices. Moreover, the noncooperative contract renewal game may have ine$ciencies since an action by one party to call for contract renewal does not take into account the consequences for the opponent in terms of contract renewal costs. The paper is organised as follows: Section 2 sets up the basic problem of contract renewal under uncertainty when fully contingent contracts cannot be signed, while the renewal game is solved in Section 3. Section 4 o!ers a few concluding remarks.
2. Contract renewal under uncertainty Consider a contract between a principal (P) and an agent (A) stipulating a given #ow of services or actions to be taken by the agent who in turn is compensated by a #ow payment from the principal. At any point in time there is the possibility that either the principal or the agent may want to suggest a contract renewal } at a cost } because outside opportunities have changed, that is, the agent "nds that he can receive a better compensation by shifting to another principal, or the principal perceives that he can replace the agent by another agent willing to accept a lower payment. Examples of contracts "tting this description are legio including employment contracts, tenant contracts, delivery contracts, etc. Let q denote the payment according to the incumbent contract and let w denote the outside opportunity (alternative price). If the contract is renewed R at time t, the new payment will be w . By specifying an exogenous outside R opportunity, we avoid having to go into details about the bargaining procedure which allows us to focus on the implications of uncertainty and adjustment costs. The problem faced by the agent (principal) is when a costly contract renewal should be undertaken to change the current payment q (!q) to w R (!w ). A similar problem will arise if the incentive to call for contract renewal is R driven by internal factors like, e.g. changes in productivity. To focus on the We express payments in present value terms, i.e. if the #ow payment is q then q"e\HQq ds"q/, where is the time-invariant discount rate. Similarly for costs. Thereby we also leave out problems arising from attempts to exploit the market power arising from &switching costs', see Klemperer (1995). The way the problem is formulated precludes the possibility of negotiating a "xed price for the entire contract period, it is not time consistent. An alternative interpretation of the model is to interpret w as a process driving the value of the R output produced by the agent. In this case the analysis carries through if wages are settled in negotiations between the agent and principal such that the wage is a share of productivity (w ), R leaving a share (1!) of output as pro"ts ((1!)w ). R
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incentives underlying price adjustment, the quantities transacted are assumed given (see Andersen and Christensen (1997) on endogenous quantity determination). If the outside opportunity was given deterministically, the contract renewal problem would be trivial and the timing of contract renewal could easily be determined. However, if the outside opportunities evolve stochastically, the question of contract renewal becomes nontrivial. Speci"cally, we assume that w evolves according to a geometric Brownian motion process, i.e. R dw "w dt#w dz , R R R R where dz is the increment of a Wiener process, i.e. E(dz )"0 and Var(dz )"dt. R R R Obviously, if contract renewal is costless the contract would be renewed continuously with changes in outside opportunities.This is counterfactual. Contract renewal is costly and usually involves both "xed and variable costs. Speci"cally, it is assumed that renewal costs are proportional to the new payment #ow, and equal to w for both the agent and the principal. In R Andersen and Christensen (1997) we show how the analysis can be extended to allow for asymmetric costs including both a "xed and a variable component. Both the principal and the agent are risk-neutral and have an in"nite horizon with a discount rate . We assume ' to rule out the trivial case where the parties to the contract are always better o! by waiting and therefore never exercise the option to call a contract renewal (see e.g. Pindyck, 1991). To see the basic mechanisms involved, we start by considering the special case where the contract can only be renewed once (see Andersen and Christensen, 1997) for a generalization to an in"nite number of renewals). The option of being able to renew the contract has a value to the contract parties. Assume that a contract renewal is called by the agent if c q"w and by the principal if R c q"w . Denote the point in time where the agent will call for contract renewal R by ¹ and similarly ¹ for the principal. . Consider now the consequences to the agent of a contract renewal. The possibility of a contract renewal is an asset to the agent if the outside opportunity improves and the agent has the possibility of raising the payment. Oppositely, the possibility that the principal can call for contract renewal if outside The contract problem is thus when to replace the deterministic payment q with a stochastic outside payment w. In Andersen and Christensen (1997) it is shown how the framework can be adopted to deal with the case where the pay-o! is also stochastic under the incumbent contract. These costs may arise through di!erent channels: lawyers' fees are dependent on the contract sum, stamp duties are often based on the contract sum, the value of the time used to settle the contract (evaluated at the opportunity wage). Assuming that costs are independent of the payment #ow causes problems in an in"nite horizon model with drift in the payment #ow since it implies counterfactually that the costs of contract renewal relative to the gain from renewal may approach zero.
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opportunities deteriorate constitutes a liability to the agent as he will be worse o! in this case. The expected value to the agent of a contract renewal when the outside opportunity is w , x (w ), can be expressed in terms of the sum of the R R expected value of the asset (a) and the liability (l) component of the option, i.e. x (w )"x (w )#x (w ) R R R "E(e\H2 (c q!q!c q)w ¹ (¹ ) R . #E(e\H2. (c q!q!c q)w ¹ (¹ ). . . R . The value to the principal of the contract renewal opportunity can similarly be written as x (w )"x (w )#x (w ) . R . R . R "E(e\H2. (q!c q!c q)w ¹ (¹ ) . . R . #E(e\H2 (q!c q!c q)w ¹ (¹ ). R . The expected value of the contract renewal option satis"es (cf. Karlin and Taylor, 1980, Section 15.3) the following second-order di!erential equation: x x H #x H "x , j"A, P. x H w H 2 H w This has a straightforward interpretation since the left-hand side according to Ito's lemma gives the expected change in the value of the option of being able to renew the contract while the right-hand side gives the deterministic pay-o! if the contract renewal is exercised immediately. Clearly, these two forces have to balance to have a nontrivial solution to the contract renewal problem. Solving this second-order di!erential equation yields x (w )"m w?#n w@, j"A, P, H R H R H R where
(1)
1 2 ! # '1, 2
1
, ! # 2
1 , ! ! 2
1 2 ! # (0. 2
The parameters m and n are determined by the boundary conditions stating H H that at the time of contract renewal, the value of the contract renewal option is equal to the value of contract renewal (value matching condition). For the agent
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we have x (c q)"[c (1!)!1]q, , x (c q)"[c (1!)!1]q, . . and for the principal x (c q)"[1!c (1#)]q, , x (c q)"[1!c (1#)]q, . . . . . Imposing these boundary conditions, (1) can be rewritten as 1 x (w )" [[ c@ ! c@ ]q\?w?#[ c? ! c? ]q\@w@], R . R . R 1 x (w )" [[ c@ ! c@ ]q\?w?#[ c? ! c? ]q\@w@], . R . R . R where ,c? c@ !c? c@ '0. . . 3. The contract renewal game There is a strategic interaction in contract renewal between the principal and the agent, since the action to demand contract renewal has consequences to the other part. We look for Nash equilibria to this contract renewal game where each party decides on its optimal critical value c (j"A, P) given the critical H value of the opponent. To this end we need Lemma 1. For any c 3[0, 1], there exists a unique best response c 3[1, R[ for . the agent, where
c 3 max 1, , . ( !1)( !1)(1!) ( !1)(1!) For any c 3[1,R[, there exists a unique best response c 3[0, 1] where
c 3 , min 1, . ( !1)(1#) ( !1)( !1)(1#)
.
Proof. See the appendix. The strategic interaction in contract renewal implies that the more reluctant the agent is to demand contract renewal (the larger c ), the more reluctant the principal will be in calling a contract renewal (the smaller c ). This shows . a strategic complementarity in contract renewal decisions.
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Fig. 1. Nash equilibrium to the noncooperative renewal game.
Existence of a Nash equilibrium to the contract renewal game is ensured by (See Fig. 1). Proposition 1. A Nash equilibrium (c* , c*) exists to the contract renewal game. . Moreover, for an open set of parameters (, , , ) the equilibrium is nontrivial, i.e. c* '1 and c*(1. A suzcient condition for uniqueness is absence of drift in the . outside payment w ("0). R Proof. See the appendix. The Nash equilibrium to the contract renewal game implies an interval [c q, . c q] of inaction in the sense that as long as the outside opportunity payment remains in this interval, none of the parties to the contract have an incentive to call a contract renewal. This interval of inaction resembles the [s, S]-rules imposed on unilateral adjustment problems as in, e.g. the menu cost models. Although the implications are the same, the region of inaction follows here from a game between parties having opposite incentives concerning payment revisions. It is noteworthy that the critical values for contract renewal are path independent as We disregard the case where either c 41 or c 51, as this implies immediate renewal. This . may, of course, be optimal with a "nite number of renewals, but in the case of an in"nite number of contract renewals, it is not possible for c "c "1 to be a solution. .
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Corollary 1. The equilibrium values of c* and c* are invariant to the realizations of . the stochastic variable w , and the size of the incumbent payment yow q. R Proof. Follows from proof of Proposition 1. The actual payment displays, however, path-dependence. In the absence of contracts the spot market payment would equal the outside opportunity, while with a "xed payment long-term contract the payment would be constant over time. In the present setting the payment is determined by a high (low) outside payment in the past if the payment has been revised upwards (downwards). In this sense extreme market conditions in the past come to determine payments due to the lock-in e!ect caused by costly contract renewal. In a study of labour contracts Beaudry and DiWardo (1991) "nd empirical support for wages being positively correlated with the best labour market conditions observed since the worker was hired. We also "nd that adjustment is asymmetric as Corollary 2. The interval supporting the incumbent payment yow [c q, c q] is not . geometrically symmetric around q, i.e. c c O1. In the case of no drift in the . payment yow w ("0), we have c c "(1!)\'1 and hence the nonadjustR . ment region is rightward-skew, i.e. c 'c\. . Proof. Follows from Lemma 2 in the appendix. Although the contract renewal problem is set up to be symmetric (payment changes are a zero-sum game and contract renewal costs are symmetric), the region of no-adjustment is (geometrically) asymmetric. As a point of reference it is noted that in the case of certainty the critical values are c "(1!)\, c "(1#)\. Comparing to the case of uncertainty without . drift ("0) we "nd that the product of the critical levels are the same, i.e. c c "c c (compared to the unilateral case, cf. e.g. Dixit, 1991), but the e!ect . . of uncertainty is to expand the range implying price inertia, i.e. c 'c , c (c . . . The intuition is simply that uncertainty adds to the costs of adjusting prices since there is an option value of waiting. Proposition 2. The cooperative renewal game (c! , c!) has a larger interval support . ing the incumbent contract than the non-cooperative renewal game (c* , c*), i.e. . c! 'c* , c!(c*. . . Proof. See the appendix. The intuition is straightforward as there is a transaction cost externality of calling a contract renewal on top of the pecuniary redistribution. The party calling a contract renewal imposes transaction costs on the other party. When this externality is internalized, the region of inaction expands. Even though costs
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Fig. 2. (A) c and c as a function of . (B) c and c as a function of . (C) c and c as a function of . . . . (D) c and c as a function of the cost parameters . .
imply price inertia, prices may still be adjusted too frequently due to the interplay between uncertainty and renewal costs. Due to the complexity of the model it is not possible to obtain analytical results on how the contract renewal problem is a!ected by the drift parameter (), the variance (), the discount rate () and the cost parameter (). Accordingly, numerical simulations have been undertaken, and they are reported in Fig. 2 which is based on a benchmark case where ("0, "0.04, "0.1, "0.05), and then allowing one parameter to change in each experiment. The simulations show that an increase in the cost parameter enlarges the band supporting the existing contract payment as the critical level increases for the agent, and decreases for the principal. It is worth pointing out that although both the agent and principal are assumed to be risk-neutral, the contract renewal problem is a!ected by risk. The reason is simply that the value of waiting to have a contract renewal depends on the variability of outside opportunities. It is found here that an increase in the variance enlarges the interval supporting the initial price. This re#ects that the possibility of extreme values becomes larger, i.e. the value of waiting increases and therefore the interval supporting the initial contract payment expands. A higher discount
646 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652
factor reduces the gains from waiting and therefore the critical level decreases for the agents and increases for the principal. Both critical values are increasing in the drift parameter. The contract renewal problem implies that the point in time at which contract renewal will be called by one of the parties is stochastic. This allows us to consider the length of the contract in terms of the expected time to contract renewal which can be written >(w )"E[¹w ], R R where ¹"¹ if w "c q and ¹"¹ if w "c q. Following Karlin and R . R . Taylor (1980) the expected time to contract renewal satis"es w >#w >#1"0 R R and the boundary conditions are given by >(c q)">(c q)"0. . Fig. 3 shows how the expected contract length depends on the parameters of the contract renewal problem. The expected contract length is decreasing in the
Fig. 3. (A) The expected length of a contract as a function of . (B) The expected length of a contract as a function of . (C) The expected length of a contract as a function of . (D) The expected length of a contract as a function of .
T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652 647
discount rate (), increasing in the cost parameter () and decreasing in the variance (). While the two "rst results are straightforward implications of the results found above on the critical values, the latter e!ect is not. A larger variability induces a larger interval supporting the initial payment, cf. above, and this tends to lengthen the time to contract renewal. However, the likelihood of having a contract renewal increases as the variance increases, and this tends to reduce the time to contract renewal. The latter e!ect dominates such that the expected contract length is decreasing in the variance. For the drift parameter we "nd a nonmonotone relationship with the longest expected duration in the case of zero drift. This is intuitive as the drift term implies an underlying deterministic trend in outside opportunities which is bound to release a contract renewal. These predictions are in accordance with empirical analysis of contract duration in labour markets in which it is found that contract length is decreasing in uncertainty, decreasing in in#ation and increasing in contracting costs (Vroman, 1989; Murphy, 1992).
4. Concluding remarks In a bilateral contract renewal problem in which the incentive to call for contract renewal is driven by (stochastic) changes in outside opportunities, it has been shown that payments display inertia. Contract renewal costs prevent continuous payment revisions and the incumbent contract payment comes to play a crucial role. This implies among other things path dependence in payment and price rigidities. Although the model-structure and "ndings can be shown to be robust to various modi"cations of the contract renewal problem (see Andersen and Christensen, 1997), the model remains in a number of respects stylized. An important issue for future research would be to combine the question of payment adjustment with the problem of long-term investment and thus to explain why there is an incentive to enter contracts in the "rst place.
Appendix. Proof of Lemma 1 Deriving the optimal c (c ) can be done either by solving the "rst-order . condition for the maximization of x (x ) or by imposing a &smooth pasting' . condition, see Dixit (1988). As both methods yield the same results, we stick to the "rst one as this is most intuitive. Denote by FOC the derivative of x (w ) wrt. c and by FOC the derivative of x (w ) wrt c . The agent's R . . R . (principal's) choice of c (c ) is found by equating FOC (FOC ) to zero . .
648 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652
which yields FOC c [q\?\@(c? q?w@!c@ q@w?)]\ . R . R "!c? c? [(1! )(1!)c # ]!c@ c? [( !1)(1!)c ! ] . . !c?>@[( ! )(1!c (1!))]"0, . FOC c [q\?\@(c? q?w@!c@ q@w?)]\ . . R R "c@ c? [(1! )(1#)c # ]#c? c@ [( !1)(1#)c ! ] . . . . #c?>@[( ! )(1!c (1#))]"0. . The second-order conditions evaluated at the points where the "rst-order conditions are satis"ed are given by SOC "!c\ [ #c (1!)( !1)(1! )](0, SOC "c\ [ #c (1#)( !1)(1! )](0. . . . It is now straightforward to show that FOC
FOC
(0, ? A ?\\O
51. ( !1)(1!)
'0, A
?@ ?\@\\O
51. ( !1)( !1)(1!)
(A.1)
(A.2)
Hence, either c* "1 or
cH 3 , ( !1)( !1)(1!) ( !1)(1!)
in which case SOC assures us that we have a maximum. Furthermore, this is the unique value satisfying FOC as SOC (0 for all values of c belonging to the above interval. Similarly, it is easily shown that FOC .
(0, A.
@ @\>O
41, ( !1)(1#)
(A.3)
T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652 649
FOC .
41. ( !1)( !1)(1#)
'0, ?@ A ?\@\>O
(A.4)
Hence, either c*"1 or .
cH3 , . ( !1)(1#) ( !1)( !1)(1#)
and the second-order conditions ensure a unique maximum value. Proof of Proposition 1. The proof proceeds by considering what happens with the optimal choice of c (c ) for extreme values of c (c ). From Lemma 1 and . . the "rst-order conditions derived in the proof of Lemma 1, it follows
, Lim c ,c( 5Max 1, ( !1)( !1)(1!) A.
Lim c ,c " , ( !1)(1!) A.
Lim c ,c( 4Min 1, , . . ( !1)( !1)(1#) A Lim c ,c " . . . ( !1)(1#) A By drawing the reaction curves in a (c , c ) diagram using the above end points, . it follows from the continuity of the curves that they must have at least one intersection. Hence, at least one Nash equilibrium exists. Uniqueness can be proven in the case of zero drift, i.e. "0 by using the following lemma: Proof of Lemma 2. If "0, and FOC "0 for
1 (c , c )" c , , . c (1!)(1#) then FOC also equals zero. .
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Proof. Insert c "1/c (1!)(1#) and c (1!)"1/c (1#) in FOC to . . obtain FOC "0 implying 1 !c?\(1!)?\(1#)?\ (1! ) #
c (1#) . 1 !c\?(1!)\?(1#)\? (! ) !(1! ) c (1#) . 1 !c (1!2 ) 1! "0. c (1#)
Rearranging and multiplying with c (1#) we get . #c? c\?[(1! )(1#)c !(1! )]#c\?c? [(1! )(1#)c # ] . . . . #c [(1!2 )(c (1#)!1)]"0 . It follows that FOC "0. .
䊐
To prove a unique Nash equilibrium for "0 we observe that: (i) at least one Nash equilibrium exists, (ii) furthermore it follows from the end points of the reaction curves that an odd number of equilibria exists, (iii) Lemma 2 implies that whenever the reaction curve for c (or c ) intersects the hyperbola . c "1/c (1!)(1#), a Nash equilibrium exists. We now show that the . reaction curve for c has at most two intersections with the hyperbola c "1/c (1!)(1#), and it then follows from (i) and (ii) that there is only one . intersection, and hence only one Nash equilibrium for
1
1
!1 1 1 c 3 , and c 3 , . 1! !1 1! .
1# 1# The proof proceeds by inserting the following in FOC 1 1 c " , . c (1!)(1#) c Di!erentiating twice wrt. c we obtain 2 (1!3 #2 )c\\?\\?(c #c??)(1!c (1!))(0. We conclude that FOC is concave in c along the hyperbola c "1/c (1!)(1#) and therefore is zero at most twice. .
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Proof of Proposition 2. Follows from the following facts: X X '0 (0, c c . and the fact that SOC , SOC (0 at the points where FOC , FOC "0. . . Hence, a cooperative renewal game must have c 'cH , c (cH. . . References Andersen, T.M., Christensen, M.S., 1997. Contact renewal under uncertainty. Memo 1997-5, Department of Economics, University of Aarhus. Beaudry, P., DiWardo, J., 1991. The e!ect of implicit contracts on the movement of wages over the business cycle: evidence from micro data. Journal of Political Economy 99, 665}688. Caplin, A., Leahy, J., 1991. State-dependent pricing and the dynamics of money and output. Quarterly Journal of Economics CVI, 683}799. Caplin, A.S., Spulber, D.F., 1987. Menu costs and the neutrality of money. Quarterly Journal of Economics CII, 703}725. Carlton, D.W., 1986. The rigidity of prices. American Economic Review 76, 637}658. Crocker, K.J., Masten, S.E., 1991. Pretia ex machina? Prices and process in long-term contracts. Journal of Law and Economics XXXIV, 69}99. Danziger, L., 1983. Price adjustments with stochastic in#ation. International Economic Review 24, 699}707. Danziger, L., 1984. Stochastic in#ation, and the optimal policy of price adjustments. Economic Inquiry XXII, 98}108. Danziger, L., 1987. On in#ation and real price variability. Economic Inquiry 25, 285}299. Danziger, L., 1995. Contract reopeners. Journal of Labor Economics 13, 62}87. Dixit, A.K., 1988. A heuristic derivation of the conditions for optimal stopping of brownian motion. Working paper, Princeton University. Dixit, A.K., 1991. Analytical approximations in models of hysteresis. Review of Economic Studies 58, 141}145. Dye, R.A., 1985. Optimal length of labor contracts. International Economic Review 26, 251}270. Gray, J.A., 1978. On indexation and contract length. Journal of Political Economy 86, 1}18. Goldberg, V.P., Erickson, J.R., 1987. Quantity and price adjustment in long-term contracts: a case study of petroleum coke. Journal of Law and Economics XXX, 369}398. Holden, S., 1995. Renegotiation and the e$ciency of investments. Working paper, Department of Economics, University of Oslo. Joskow, P.L., 1988. Price adjustment in long-term contracts: the case of coal. Journal of Law and Economics XXXI, 47}83. Karlin, S., Taylor, H.M., 1980. A second course in stochastic processes. Academic Press, New York. Klemperer, P., 1995. Competition when consumers have switching costs } an overview with applications to industrial organization, macroeconomics and international trade. Review of Economic Studies 62, 515}539. McLeod, W.B., Malcomson, J., 1993. Investments, holdup and the form of market contracts. American Economic Review 83, 811}837. Murphy, K.J., 1992. Determinants of contract durations in collective bargaining agreements. Industrial and Labor Relations Review 45, 352}365.
652 T.M. Andersen, M.S. Christensen/Journal of Economic Dynamics & Control 26 (2002) 637}652 Pindyck, R.S., 1991. Irreversibility, uncertainty and investment. Journal of Economic Literature XXIX, 1110}1148. Sheshinski, E., Weiss, Y., 1977. In#ation and costs of price adjustment. Review of Economic Studies 44, 287}303. Sheshinski, E., Weiss, Y., 1983. In#ation and costs of price adjustment. Review of Economic Studies 50, 513}529. Tsiddon, D., 1993. The (mis) behaviour of the aggregate price level. Review of Economic Studies 60, 889}902. Vroman, S.B., 1989. In#ation uncertainty and contract duration. Review of Economics and Statistics 71, 677}681. Williamson, O.E., 1979. Transactions-cost economics: the governance of contractual relations. Journal of Law and Economics XXII, 233}261.