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Comparative statics with adjustment costs

and the le Chatelier principle

Eddie Dekel

†John K.-H. Quah‡Ludvig Sinander§

22 February 2023

Abstract:We develop a theory of monotone comparative stat- ics for models with adjustment costs. We show that comparative- statics conclusions may be drawn under the usual ordinal comple- mentarity assumptions on the objective function, assuming very little about costs: only a mild monotonicity condition is required. We use this insight to prove a general le Chatelier principle: under the ordinal complementarity assumptions, if short-run adjustment is subject to a monotone cost, then the long-run response to a shock is greater than the short-run response. We extend these results to a fully dynamic model of adjustment over time: the le Chatelier principle remains valid, and under slightly stronger assumptions, optimal adjustment follows a monotone path. We apply our results to models of capital investment and of sticky prices. Keywords:Adjustment costs, comparative statics, le Chatelier.

JEL classication numbers:C02, C7, D01, D2, D8.?

We are grateful to Gregorio Curello for close reading and comments, and to Sara Ne, Simon Quinn, Jakub Steiner, Quitze Valenzuela-Stookey, Nathaniel Ver Steeg, JanZemlicka and audiences at Berlin, Bonn, CERGE-EI, Oxford, the 8th TUS conference and the 5th

Bergamo IO Winter Symposium for helpful comments.

†Department of Economics, Northwestern University and Department of Economics, Tel Aviv University. Email address:. ‡Department of Economics, Johns Hopkins University and Department of Economics, National University of Singapore. Email address:. §Department of Economics and Nueld College, University of Oxford. Email address: .

1arXiv:2206.00347v2 [econ.TH] 22 Feb 2023

1 Introduction

Adjustment costs play a major role in explaining a wide range of economic phenomena. Examples include the investment behavior of rms (e.g. Jor- genson, 1963
; Hayashi, 1982
; Cooper & Haltiwanger, 2006
), price stickiness (e.g. Mankiw, 1985
; Caplin & Spulber, 1987
; Golosov & Lucas, 2007
; Midri- gan, 2011
), aggregate consumption dynamics (e.g. Kaplan & Violante, 2014

Berger & Vavra,

2015
) and housing consumption and asset pricing (Grossman & Laroque, 1990
In this paper, we develop a theory of monotone comparative statics with adjustment costs. Our fundamental insight is that, surprisingly, very little needs to be assumed about the cost function: comparative statics requires only thatnotadjusting be cheaper than adjusting, plus the usual ordinal complementarity assumptions on the objective function. We use this insight to show that Samuelson's ( 1947
)le Chatelier principleis far more general than previously claimed: it holds whenever adjustment is costly, given only minimal structure on costs. We extend our comparative-statics and le Chatelier results to a fully dynamic model of adjustment. We apply our results to models of factor demand, capital investment, and pricing. These models are typically studied only under strong functional-form assumptions, and the cases of convex and nonconvex costs are considered sep- arately and handled very dierently. Our general results yield robust compar- ative statics for these standard models, dispensing with auxiliary assumptions and handling convex and nonconvex costs in a unied fashion. The abstract setting is as follows. An agent chooses an actionxfrom a sublatticeLRn. Her objectiveF(x;) depends on a parameter. At the initial parameter, the agent chosex2argmaxx2LF(x;). The parameter now increases to > , and the agent may adjust her choice. Adjusting the action by=xxcostsC()>0, and the agent's new choice maximizes

G(x;) =F(x;)C(xx).

Our only assumption on the cost functionCismonotonicity: C(1;:::;i1;0i;i+1;:::;n)6C() whenever 060i6ior 0>0i>i. This means that cost falls whenever an adjustment vectoris modied by 2 shifting one of its entries closer to zero (\no adjustment"). An additively sepa- rable cost functionC() =Pn i=1Ci(i) is monotone exactly if each dimension's cost functionCiis single-dipped and minimized at zero. By relying on monotonicity alone, we eschew restrictive assumptions such as convexity; even quasiconvexity is not needed. Some adjustmentsmay be infeasible, as captured by a prohibitive costC() =1. In some of our results, the monotonicity assumption may be weakened tominimal monotonicity:cost falls whenever an adjustment vectoris modied by replacing all of its positive entries with zero (C(^0)6C()), and similarly for the negative entries (C(_0)6C()). In the additively separable case, this means that each dimension's costCiis minimized at zero. Our basic question is under what assumptions on the objectiveFand costCthe agent's choice increases, in the sense thatbx>xfor somebx2 argmax x2LG(x;) (provided the argmax is not empty; such qualiers are omitted throughout this introduction). Our fundamental result, Theorem 1 answers this question: nothing need be assumed about the costCexcept mini- mal monotonicity, whileFneed only satisfy the ordinal complementarity con- ditions ofquasi-supermodularityandsingle-crossing dierencesthat feature in similar comparative-statics results absent adjustment costs (see Milgrom & Shannon, 1994
). Thus costs need not even be monotone, and the objective need not satisfy any cardinal properties, such as supermodularity or increas- ing dierences. We also give a \8" variant (Proposition1 ): adding either of two mild assumptions yields the stronger conclusion thatbx>xforevery bx2argmaxx2LG(x;). We use our fundamental result to re-think Samuelson's ( 1947
)le Chatelier principle,which asserts that the response to a parameter shift is greater at longer horizons. Our Theorem 2 pro videsthat the le Chatelier principle holds whenever short-run adjustment is subject to a monotone adjustment costC, long-run adjustment is frictionless, and the objectiveFsatises the ordinal complementarity conditions. Formally, the theorem states that under these as- sumptions, given any long-run choice x2argmaxx2LF(x;) satisfying x>x, we have x>bx>xfor some optimal short-run choicebx2argmaxx2LG(x;).11 Furthermore, if xis the largest element of argmaxx2LF(x;x), then x>bxforany short-run choicebx2argmaxx2LG(x;). 3 This substantially generalizes Milgrom and Roberts's (1996) le Chatelier prin- ciple, in which short-run adjustment is assumed to be impossible for some dimensionsiand costless for the rest: that is,C() =Pn i=1Ci(i), where some dimensionsihaveCi(i) =1for alli6= 0, and the rest haveCi0. We show that our le Chatelier principle remains valid if long-run adjustment is also costly (Proposition 2 We then extend our comparative-statics and le Chatelier theorems to a fully dynamic, forward-looking model of costly adjustment over time. The pa- rametertevolves over timet2 f1;2;3;:::g, and the adjustment cost function C tmay also vary between periods. Starting atx0=x2argmaxx2LF(x;), the agent chooses a path (xt)1t=1to maximize the discounted sum of her pe- riod payosF(xt;t)Ct(xtxt1). Theorem3 v alidatesthe le Chatelier principle: under the same assumptions (ordinal complementarity ofFand monotonicity of eachCt), if6t6in every periodt, then given any x2argmaxx2LF(x;) such thatx6x, the agent's choices satisfyx6xt6x along some optimal path (xt)1t=1. If the parameter and cost are time-invariant (t=andCt=Cfor all periodst), then a stronger le Chatelier principle holds (Theorem 4 ): under additional assumptions, the agent adjusts more at longer horizons, in the sense thatx6xt6xT6xholds at any datest < T along some optimal path (xt)1t=1. The le Chatelier principle remains valid if decisions are instead made by a sequence of short-lived agents (Theorem 5 ): under similar assumptions, if6 t(6t+1)6in every periodt, thenx6ext(6ext+1)6xin every periodt along some equilibrium path (ext)1t=1. Thus short-lived agents adjust in the same direction as a long-lived agent would. They do so more sluggishly, however:

Theorem

6 asserts that under stronger assumptions (including con vexityof each costCt),x6ext6xt6xholds along some short-lived equilibrium path (ext)1t=1and some long-lived optimal path (xt)1t=1. The rest of this paper is arranged as follows. In the next section, we de- scribe the environment. We present our fundamental comparative-statics in- sight (Theorem 1 ) in section 3 . In section 4 , we develop a general le Chatelier principle (Theorem 2 ), and apply it to pricing and factor demand. In sec- tion 5 , we introduce a dynamic, forward-looking adjustment model, derive two dynamic le Chatelier principles (Theorems 3 and 4 ), and apply them to pricing 4 and investment. We conclude in section6 b yderiving the le Chatelier princi - ple for short-lived agents (Theorem 5 ) and comparing their behavior to that of a long-lived agent (Theorem 6 ). The app endix c ontainsdenitio nsof some standard terms, an extension to allow for uncertain adjustment costs, and all proofs omitted from the text.

2 Setting

The agent's objective isF(x;), wherexis the choice variable and2 is a parameter. The choice variablexbelongs to a subsetLofRn. At the initial parameter=, an optimal choicexwas made: x2argmaxx2LF(x;): (Note that we allow for a multiplicity of optimal actions.) This is the agent's \starting point," and we shall consider how she responds in the short and long run to a change in the parameter fromto , where< Adjustment is costly: adjusting fromxtoxcostsC(xx). The cost function Cis a map L![0;1], where L=fxy:x; y2Lg. Note that we allow some adjustments2Lto have innite costC() =1, meaning that they are infeasible. We assume throughout thatC(0)<1.

The agent adjusts her actionx2Lto maximize

G(x;) =F(x;)C(xx):

2.1 Order assumptions

Throughout,Rn(and thusL) is endowed with the usual \product" order>, so \x>y" means \xi>yifor every dimensioni." We write \x > y" whenever x>yandx6=y. We assume that the choice setLis asublatticeofRn: for anyx;y2L, the following two vectors also belong toL: x^y= (minfx1;y1g;:::;minfxn;yng) andx_y= (maxfx1;y1g;:::;maxfxn;yng): 5 Examples of sublattices includeL=Rn, \boxes"L=fx2Rn:x6x6x+g for givenx;x+2Rn, and \grids" such asL=Zn(whereZdenotes the integers). Another example is the half-planeL=f(x1;x2)2R2:x1+x2> kgfor given606andk2R. The parameterbelongs to a partially ordered set . We use the symbol> also for this partial order. In applications, the parameteris often a vector of real numbers, in which case is a subset ofRnand>is the usual \product" order. More elaborate applications are possible; for example, could be a set of distributions, with>being rst-order stochastic dominance.

2.2 Monotonicity assumptions on costs

Our results feature dierent assumptions about the cost functionC. In partic- ular, most of our results requiremonotonicity,but our rst theorem requires onlyminimal monotonicity.We now dene these two properties. The cost functionCismonotoneexactly if for any adjustment vector

2Land any dimensioni,

C(1;:::;i1;0i;i+1;:::;n)6C() whenever 060i6ior 0>0i>i. In other words, modifying an adjustment vector by shifting one dimension's adjustment toward zero always reduces cost. An equivalent denition of mono- tonicity is thatC(0)6C() holds whenever0is \between 0 and" in the sense that in each dimensioni, we have either 060i6ior 0>0i>i.

IfChas theadditively separableformC() =Pn

i=1Ci(i), then it is mono- tone exactly if each of the cost functionsCiis single-dipped and minimized at zero.

2Thus in particular, if the choice variable is one-dimensional (LR),

then monotonicity requires thatCbe single-dipped and minimized at zero. Example 1.For a one-dimensional choice variable (LR), the following cost functions are monotone, for any values of the parametersk;a2(0;1): (a) Fixed cost:C() =kfor6= 0andC(0) = 0. (b) Quadratic cost:C() = a

2. (c) Quadratic cost with free disposal:C() =a2if>0andC() = 02

GivenXR, a function:X![0;1] issingle-dippedexactly if there is anx2X such thatis decreasing onfy2X:y6xgand increasing onfy2X:y>xg. 6 otherwise. (d) Quadratic cost with a constraint:C() =a2if2 Eand C() =1otherwise, where the constraint setE Ris convex and contains0. Example 2.The following cost functions are monotone: (a) Additively sepa- rable:C() =Pn i=1Ci(i), where eachCiis of one of the types in Example1 . (b) Euclidean:C() =pP n i=12i. (c) Cobb{Douglas:C() =Qn i=1jijai, where a

1;:::;an2(0;1).

Monotonicity is consistent with quite general nonconvexities, and even with failures of quasiconvexity: the cost function in Example 2 (c) is monotone, but not quasiconvex (except if the choice variable is one-dimensional, i.e.LR). A cost functionCis called 0-monotoneif and only if

C(1;:::;i1;0;i+1;:::;n)6C()

for any adjustment vector2Land any dimensioni. This property weak- ens monotonicity by requiring cost to fall only when an adjustment vector is modied bycompletely cancellingthe adjustment in one dimension. An even weaker property isminimal monotonicity,which requires that

C(^0)6C()>C(_0) for any adjustment vector2L.

In other words, simultaneously cancelling allupwardadjustments, by replacing all of the positive entries of an adjustment vectorwith zeroes, reduces cost; similarly, cancelling alldownwardadjustments reduces cost. Although minimal monotonicity is weaker than 0-monotonicity, the two properties coincide for additively separable cost functions: both require that eachCibe minimized at zero. Thus in particular, minimal monotonicity and

0-monotonicity coincide when the choice variable is one-dimensional (LR):

both require merely that cost be minimized at zero. Example 3.Consider the cost function in Example1 (d), with a constraint setE Rthat that contains0but isnotconvex. For instance,E= (1;0][ [I;1)for someI >0, as in the recent literature on the investment behavior of entrepreneurs in developing countries (see section 5.4 ). OrE=Zdue to an integer constraint. Such a cost function is0-monotone, but not monotone. 7 Example 4.ForLR2, the (nonseparable) cost functionC(1;2) =j12j is minimally monotone, but not0-monotone (hence not monotone).3This cap- tures situations in which it is costly to adjust the gap betweenx1andx2. Monotonicity, 0-monotonicity and minimal monotonicity are all preserved by strictly increasing transformations: ifChas one of these properties, then so doesCfor any strictly increasing map: [0;1]![0;1]. In other words, these properties areordinal.That makes them easier to check in practice.4

2.3 Complementarity assumptions on the objective

We assume throughout that the objective functionFsatises the standardor- dinal complementarity conditionsofquasi-supermodularityandsingle-crossing dierences(see Milgrom & Shannon,1994 ), dened as follows. The objectiveF(x;) hassingle-crossing dierences in(x;) exactly if F(y;0)F(x;0)>(>) 0 impliesF(y;00)F(x;00)>(>) 0 wheneverx6y and0600. Economically, this means that a higher parameter implies a greater liking for higher actions: whenever a higher action is (strictly) pre- ferred to a lower one, this remains true if the parameter increases. A sucient condition isincreasing dierences,which requires thatF(y;)F(x;) be increasing inwheneverx6y. Related concepts, such aslogincreasing dif- ferences, are dened in the app endix A function:L!Ris calledquasi-supermodularif(x)(x^y)>(>) 0 implies(x_y)(y)>(>) 0. A sucient condition issupermodularity,which requires that(x)(x^y)6(x_y)(y) for anyx;y2L. IfLR, then every function:L!Ris automatically supermodular. In caseLis open and convex andis continuously dierentiable, supermodularity demands precisely that@=@xibe increasing inxj, for all dimensionsi6=j. See theapp endixfor denitions of related concepts, such assubmodularity.3

0-monotonicity fails since, for example,C(0;1) = 1>0 =C(1;1). For minimal mono-

tonicity, if1>062thenC(^0) =C(0;0) = 06C() =C(_0), and similarly if1<

0> 2; if1<062thenC(^0) =C(1;0) =j1j> C(0;0)6j2j6C(0;2) =C(_0),

and similarly if1>0> 2.

4For instance, the cost function in Example2 (b) is monotone because it is a strictly

increasing transformation of the additively separable cost functionCy() =Pn i=12i, which is monotone sincei7!2iis single-dipped and minimized at zero. 8 We say thatF(x;) is(quasi-)supermodular inxif for each parameter

2, the functionF(;) :L!Ris (quasi-)supermodular. This captures

complementarity between the dierent dimensions of the action.

3 Comparative statics

Recall that the agent choosesx2Lto maximizeG(x;) =F(x;)C(xx). Our fundamental comparative-statics result is the following. Theorem 1.Suppose that the objectiveF(x;)is quasi-supermodular inx and has single-crossing dierences in(x;), and that the adjustment costCis minimally monotone. If > , thenbx>xfor somebx2argmaxx2LG(x;), provided the argmax is nonempty. In words, an increased parameter leads to a higher action (modulo tie- breaking). This parallels the basic comparative-statics result for costless ad- justment (see Milgrom & Shannon, 1994
, Theorem 4), one version of which states that under the same ordinal complementarity conditions on the objec- tiveF, we have x>xfor some x2argmaxx2LF(x;), provided the argmax is nonempty. Theorem 1 sho wsthat this basic result is strikingly robu stto adjustment costs: the objectiveFneed not satisfy any additional property, and the costCneed only be minimally monotone.

Theorem

1 do esnotfollow from applying the basic comparative-statics re- sult to the objective functionG(x;), because its assumptions do not guarantee thatG(;) is quasi-supermodular.5A dierent argument is required. Proof.Letx02argmaxx2LG(x;). We claim thatbx=x_x0also maximizes G(;); obviouslybx>x. We haveF(x;)>F(x^x0;) by denition ofx. ThusF(x_x0;)>F(x0;) by quasi-supermodularity, whenceF(x_x0;)> F(x0;) by single-crossing dierences. Furthermore, by minimal monotonicity,

C(x_x0x) =C((x0x)_0)6C(x0x). Thus

G(bx;) =F(x_x0;)C(x_x0x)>F(x0;)C(x0x) =G(x0;):5

Its second termx7! C(xx) need not be quasi-supermodular, and in any case, the sum of two quasi-supermodular functions is not quasi-supermodular in general. 9 Sincex0maximizesG(;) onL, it follows thatbxdoes, too.QED The minimal-monotonicity assumption on the costCis essential for The- orem 1 : without this assumption, it may be thatbxx. 6

In applications, it is often useful that Theorem

1 require sFto satisfy only the ordinal complementarity conditions, rather than the strongercardinal complementarity conditionsof supermodularity and increasing dierences. In monopoly pricing, for example, the objectiveFhas single-crossing dierences, but not increasing dierences|see section 4.2 b elow.

Theorem

1 has a coun terpartfor pa rameterdecreases:under the same assumptions, if < , thenbx6xfor somebx2argmaxx2LG(x;), provided the argmax is nonempty. The proof is exactly analogous.

7All of the results in

this paper have such counterparts for parameter decreases; we will not discuss them explicitly.

Theorem

1 also extends str aightforwardlyth ecase in w hichadjustmen t costs are uncertain, even if the agent is risk-averse. Several subsequent results also generalize in this way. This is shown in appendix B Remark 1.Theorem1 is phrased dieren tlythan the usual statemen tof the basic result (see Milgrom & Shannon, 1994
, Theorem 4), which asserts that whenF(x;) is quasi-supermodular inxand has single-crossing dierences in (x;), if > then x

002argmaxx2LF(x;) andx02argmaxx2LF(x;)

=)x00^x02argmaxx2LF(x;) andx00_x02argmaxx2LF(x;):

A version of Theorem

1 with this form also holds: under the same h ypotheses, if > then x2argmaxx2LF(x;) andx02argmaxx2LG(x;) =)x^x02argmaxx2LF(x;) andx_x02argmaxx2LG(x;):6 For example, ifC() =1for all6=bandC(b) = 0, whereb0, thenbx=x+bx.

7The proof of Theorem1 uses only one-half of the minimal m onotonicityas sumption:

thatC(_0)6C() for every2L. The proof of its parameter-decrease counterpart uses (only) the other half, namely thatC(^0)6C() for every2L. 10 The latter property (concerningx_x0) is exactly what is shown in the proof of Theorem 1 . For the former property (concerningx^x0), suppose it were to fail; thenF(x;)> F(x^x0;), so that replicating the steps in the proof of

Theorem

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