Comparative Statics With Adjustment Costs: Dekel, Quah & Sinander (2025)

What this is. The paper’s core theorems, the model structure, and the proof approach: a distilled skeleton sufficient to understand what was established and how, without reading all 34 pages. To replicate or extend the results, read the original at https://doi.org/10.3982/ECTA22841.

TL;DR

The paper develops a general theory of monotone comparative statics for models with costly adjustment. The key insight is that the standard comparative-statics conclusion (an increase in a parameter leads to a higher optimal action) holds under the ordinal complementarity conditions of quasi-supermodularity and single-crossing differences on the objective, with only minimal monotonicity of the cost function: canceling all increases or all decreases in the adjustment vector must weakly lower cost. The Le Chatelier result, originating with Samuelson (1947), requires monotone adjustment costs and bounds the short-run response to a shock by the long-run response. Both results are extended to a fully dynamic model with long-lived forward-looking agents (and, separately, short-lived agents). Applications include saving by wishful thinkers, factor demand following Milgrom and Roberts (1996), pricing, labor supply, and capital investment. A key feature is that convex and nonconvex adjustment costs are handled in a unified framework.

Core results

Theorems are qualitative; the “magnitude” column gives the precise conclusion. All results require quasi-supermodularity of F in x and single-crossing differences in (x, θ).

#	Result	Locator	Conclusion
R1	Theorem 1: comparative statics with adjustment costs	Theorem 1, p. 666	If C is minimally monotone and θ̄ ≥ θ̲, then x̂ ≥ x̲ for some x̂ ∈ arg max G(x, θ̄) (provided the argmax is nonempty)
R2	Theorem 2: Le Chatelier principle	Theorem 2, §4, p. 671	If C is monotone and x̄ ∈ arg max F(x, θ̄) satisfies x̄ ≥ x̲, then x̄ ≥ x̂ ≥ x̲ for some short-run x̂; if x̄ is the largest long-run optimum then x̄ ≥ x̂ for every short-run x̂
R3	Theorem 3: dynamic Le Chatelier (long-lived agents)	Theorem 3, §5.2, p. 675	Given a frictionless optimum x̄ ≥ x̲ and θ̲ ≤ θ_t ≤ θ̄ in every period, if the long-lived agent’s problem has a solution, some optimal path satisfies x̲ ≤ x_t ≤ x̄ for every t
R4	Theorem 4: strong dynamic Le Chatelier	Theorem 4, §5.2, p. 675	With F supermodular and BCS in x, single-crossing differences, a time-invariant monotone additively separable cost C, and a permanent shift θ_t = θ̄, some optimal path satisfies x̲ ≤ x_t ≤ x_{t+1} ≤ x̄ for every t
R5	Theorem 5: short-lived dynamic Le Chatelier	Theorem 5, §6, p. 679	With monotone C_t and a permanent upper bound x̄, some equilibrium path satisfies x̲ ≤ x_t ≤ x̄ when θ̲ ≤ θ_t ≤ θ̄; if θ_t is also nondecreasing, it satisfies x̲ ≤ x_t ≤ x_{t+1} ≤ x̄
R6	Theorem 6: short- vs. long-lived agents	Theorem 6, §6, pp. 679-680	With F supermodular, monotone additively separable convex C_t, equi-BCS conditions, and nondecreasing θ_t, a suitably bounded short-lived equilibrium path and some optimal long-lived path satisfy x̲ ≤ x̃_t ≤ x_t ≤ x̄
R7	Theorem 1*: comparative statics with a shifting constraint set	Theorem 1*, §3.1, p. 668	If θ̄ ≥ θ and L̄ ≥ss L, minimally monotone costs imply some maximizer x̂ on L̄ satisfies x̂ ≥ x
R8	Proposition 1: every optimum moves up under strictness	Proposition 1, §3.2, pp. 668-669	Strict single-crossing differences with minimally monotone costs, or single-crossing with strictly minimally monotone costs, implies x̂ ≥ x for every maximizer x̂
R9	Proposition 2: wishful thinking reduces saving	Proposition 2, §3.3, pp. 670-671	For some costly-adjustment optimum (ĉ, Ĝ), c̄ ≥ ĉ ≥ c₀ and Ḡ ≥₁ Ĝ ≥₁ G₀; the optimistic believer consumes at least as much as the wishful thinker
R10	Proposition 3: Le Chatelier with costly long-run adjustment	Proposition 3, §4.1, p. 672	With monotone C₁ and C₂, choices can be selected so x₂ ≥ x₁ ≥ x across sequential short- and long-run adjustments
R11	Proposition 4: every short-run optimum stays below the long-run optimum	Proposition 4, §4.1, p. 672	Strictly monotone costs imply x̄ ≥ x̂ ≥ x for every short-run maximizer x̂
R12	Theorem 1†: necessity of minimal monotonicity	Theorem 1†, §7, pp. 680-681	Minimal monotonicity is equivalent to the stated universal comparative-statics property for both parameter increases and decreases
R13	Theorem 2†: necessity of weak monotonicity	Theorem 2†, §7, p. 681	Weak monotonicity is necessary and sufficient for the Le Chatelier bracketing property to hold across objectives
R14	Theorem 3†: necessity of monotonicity in dynamic Le Chatelier	Theorem 3†, §7, pp. 681-682	For the stated admissible cost set and parameter conditions, monotonicity is necessary and sufficient for dynamic Le Chatelier bracketing

Overall (paper’s conclusion). Comparative statics and the Le Chatelier principle are robust to adjustment costs under ordinal (not cardinal) complementarity conditions on the objective. Theorem 1 requires only minimal monotonicity of costs, while the classical Le Chatelier theorem requires monotonicity; Section 7 establishes converses, with weak monotonicity necessary and sufficient for the classical Le Chatelier result and monotonicity necessary and sufficient for the dynamic result under the stated conditions. The prior literature, including Milgrom and Roberts (1996), required that short-run adjustment be completely infeasible in some dimensions. Theorem 4 has no converse in the paper.

Theory / model

Setting

An agent chooses an action x from a sublattice $L \subseteq \mathbb{R}^n$. Her objective $F(x, \theta)$ depends on a parameter $\theta \in \Theta$. At the initial parameter $\theta = \underline{\theta}$, the optimal choice is (p. 664):

$$\underline{x} \in \arg\max_{x \in L} F(x, \underline{\theta}).$$

When the parameter rises to $\bar{\theta} \geq \underline{\theta}$, adjusting from $\underline{x}$ to $x$ costs $C(x - \underline{x}) \geq 0$, where $C : \Delta L \to [0, \infty]$ and $\Delta L = \{x - y : x, y \in L\}$. The agent’s new choice maximizes (p. 664):

$$G(x, \bar{\theta}) = F(x, \bar{\theta}) - C(x - \underline{x}).$$

Infinite cost captures infeasibility. Since $\underline{x}$ is held fixed, the cost depends on the adjustment vector $\varepsilon = x - \underline{x}$ only.

Monotonicity conditions on C

Two conditions on C appear in the results (pp. 664-665):

Monotone: Shifting any one dimension’s adjustment closer to zero reduces cost:

$$C(\varepsilon_1, \ldots, \varepsilon_{i-1}, \varepsilon'_i, \varepsilon_{i+1}, \ldots, \varepsilon_n) \leq C(\varepsilon) \quad \text{whenever } 0 \leq \varepsilon'_i \leq \varepsilon_i \text{ or } 0 \geq \varepsilon'_i \geq \varepsilon_i. \tag{M}$$

An additively separable $C(\varepsilon) = \sum_{i=1}^n C_i(\varepsilon_i)$ is monotone if and only if each $C_i$ is single-dipped at zero. Monotonicity is an ordinal property, preserved by strictly increasing transformations.

Minimally monotone: Simultaneously canceling all upward (or all downward) adjustments reduces cost:

$$C(\varepsilon \wedge 0) \leq C(\varepsilon) \geq C(\varepsilon \vee 0) \quad \text{for any adjustment vector } \varepsilon \in \Delta L, \tag{MM}$$

where $\varepsilon \wedge 0$ replaces all positive entries of $\varepsilon$ with zero and $\varepsilon \vee 0$ replaces all negative entries with zero. Monotonicity implies minimal monotonicity; the converse fails. In the additively separable case, minimal monotonicity requires that each $C_i$ is minimized at zero.

Ordinal complementarity conditions on F

Throughout, F satisfies (pp. 666-667, following Milgrom and Shannon (1994)):

• Single-crossing differences in (x, θ): $F(y, \theta') - F(x, \theta') \geq 0$ implies $F(y, \theta'') - F(x, \theta'') \geq 0$ whenever $x \leq y$ and $\theta' \leq \theta''$.
• Quasi-supermodularity in x: $F(x, \theta) - F(x \wedge y, \theta) \geq(>) 0$ implies $F(x \vee y, \theta) - F(y, \theta) \geq(>) 0$ for all $x, y \in L$.

These are ordinal properties, strictly weaker than the cardinal conditions of increasing differences and supermodularity.

Dynamic model

In Section 5, the agent is long-lived and forward-looking. In each period $t \in \mathbb{N} = \{1, 2, 3, \ldots\}$, she takes action $x_t \in L$ and earns $F(x_t, \theta_t)$. Adjusting from $x_{t-1}$ to $x_t$ costs $C_t(x_t - x_{t-1})$. Given initial choice $x_0 = \underline{x}$, the agent maximizes (p. 674):

$$\mathcal{G}\!\left((x_t)_{t=1}^\infty, x_0\right) = \mathcal{F}\!\left((x_t)_{t=1}^\infty\right) - \mathcal{C}\!\left(x_0, (x_t)_{t=1}^\infty\right),$$

where

$$\mathcal{F}\!\left((x_t)_{t=1}^\infty\right) = \sum_{t=1}^\infty \delta^{t-1} F(x_t, \theta_t) \qquad \text{and} \qquad \mathcal{C}\!\left(x_0, (x_t)_{t=1}^\infty\right) = \sum_{t=1}^\infty \delta^{t-1} C_t(x_t - x_{t-1}).$$

Section 6 considers the alternative in which each period’s action is chosen by a short-lived (or myopic) agent who takes $x_{t-1}$ as given and maximizes the period-t payoff $G_t(x, x_{t-1}) = F(x, \theta_t) - C_t(x - x_{t-1})$.

Method

The paper uses lattice-theoretic methods, building on the lattice-comparative-statics framework of Topkis, and Milgrom and Shannon (1994). The key tools are:

Sublattice operations. For $x, y \in L$, the meet $x \wedge y = (\min\{x_1, y_1\}, \ldots, \min\{x_n, y_n\})$ and join $x \vee y = (\max\{x_1, y_1\}, \ldots, \max\{x_n, y_n\})$ both belong to L. The proof of Theorem 1 (p. 667) constructs $\hat{x} = \underline{x} \vee x'$ for any $x' \in \arg\max G(x, \bar{\theta})$. The key step uses minimal monotonicity: $C(\underline{x} \vee x' - \underline{x}) = C((x' - \underline{x}) \vee 0) \leq C(x' - \underline{x})$, so $G(\hat{x}, \bar{\theta}) \geq G(x', \bar{\theta})$, confirming that $\hat{x}$ also maximizes G and satisfies $\hat{x} \geq \underline{x}$.

Monotonization argument (Theorem 4, Appendix I, pp. 686-688). The strong dynamic result is proved by showing that any solution $(x_t)_{t=1}^\infty$ to the forward-looking problem can be replaced by the running-maximum sequence $X_t = x_1 \vee x_2 \vee \cdots \vee x_t$ without reducing optimality. The key inequality (eq. (5) in Appendix I) is, for each dimension $i$:

$$C_i(y \vee z - x \vee y) + C_i(y \wedge z - x \wedge y) \leq C_i(y - x) + C_i(z - y) \quad \text{for all } x, y, z, \tag{5}$$

which holds because each $C_i$ is single-dipped at zero. Combined with supermodularity of $F(\cdot, \bar{\theta})$, this ensures monotonization preserves optimality.

Necessity results (Section 7, Theorems 1†, 2†, 3†, pp. 680-682). The paper proves converses by explicit counterexample construction. For example, Theorem 1† (p. 680) shows that minimal monotonicity is equivalent to: for all quasi-supermodular F with single-crossing differences, $\theta \geq \underline{\theta}$ implies $\hat{x} \geq \underline{x}$ for some $\hat{x} \in \arg\max G(x, \bar{\theta})$, and $\theta \leq \underline{\theta}$ implies $\hat{x} \leq \underline{x}$. The counterexample when C fails minimal monotonicity is constructed on a sublattice $X = \{\underline{x} \wedge \hat{x}, \underline{x}, \hat{x}, \underline{x} \vee \hat{x}\}$ with explicitly specified F values (pp. 680-681).

Empirical specifications

This paper contains no empirical analysis and therefore has no estimating specification. Its numbered equations (1)-(8) appear in appendices, not in the main text; they are proof steps. The formal results are applied to standard economic models, deriving comparative statics without imposing the auxiliary functional-form assumptions often used in those literatures.

Saving by wishful thinkers (§3.3, pp. 669-671, Proposition 2). Following Caplin and Leahy (2019), an agent consumes $c \in [0, w]$ and chooses a belief $G$ (a CDF over future income) from a set $\mathcal{G}$ ordered by first-order stochastic dominance. The lifetime payoff is (p. 669):

$$U(c, G) = u_1(c) + \int_{\mathcal{Y}} u_2\!\left((1+r)(w - c) + y\right) G(\mathrm{d}y),$$

where $u_1, u_2$ are continuous, concave, and strictly increasing. A wishful thinker chooses:

$$(\hat{c}, \hat{G}) \in \arg\max_{(c, G) \in [0, w] \times \mathcal{G}} \left[U(c, G) - C(G - G_0)\right],$$

where $C$ is minimally monotone and $G_0$ is the realist’s belief. Proposition 2 (p. 670) establishes $\hat{c} \geq c_0$ (wishful thinkers over-consume) and $\hat{G} \geq_1 G_0$ (wishful thinkers adopt more optimistic beliefs). The proof applies Theorem 1*, the constraint-shift variant of Theorem 1, since $[0, w] \times \mathcal{G}$ is a sublattice. Notably, Caplin and Leahy (2019) assumed the Kullback-Leibler functional form for C; this assumption is not needed here.

Factor demand (§4.2, p. 673). A firm uses capital k and labor $\ell$ to produce output $f(k, \ell)$. Profit at factor prices $(r, w)$ is $F(k, \ell, -w) = f(k, \ell) - rk - w\ell$. With monotone adjustment costs, Theorem 2 implies that a drop in the wage w precipitates a short-run increase in both k and $\ell$ when $f$ is supermodular (complements), with a further increase in the long run. When $f$ is submodular (substitutes), rewriting the choice variable as $(x_1, x_2) = (-k, \ell)$ restores supermodularity; $\ell$ still increases in both runs while k now decreases.

Pricing (§4.3 and §5.3, pp. 673-674, 677). A monopolist with constant marginal cost $c$ faces demand $D(p, \eta)$ where $\eta$ is an elasticity shifter. Profit $F(p, (c, -\eta)) = (p - c)D(p, \eta)$ has single-crossing differences in $(p, (c, -\eta))$ (using the “log increasing differences” condition, p. 673) and is quasi-supermodular since $p \in \mathbb{R}$. Theorem 1 implies a price increase when marginal cost rises or demand becomes less elastic ($\eta$ falls) if the adjustment cost is minimized at zero. If the cost is single-dipped (and hence monotone in this one-dimensional setting), Theorem 2 bounds the short-run price below the long-run price, and Theorem 4 implies monotone upward price adjustment over time after a permanent shock.

Labor supply (§5.4, pp. 677-678). A worker chooses labor supply $x \in L \subseteq \mathbb{R}_+$ with per-period utility $F(x, T) = wx - T(wx) - \kappa(x)$, where T is the tax schedule and $\kappa$ is effort disutility. A tax reform from T to $\tilde{T}$ with lower marginal rates ($\tilde{T} \geq_{\text{flat}} T$) yields F with single-crossing differences in $(x, T)$. With single-dipped adjustment costs, the Le Chatelier results imply that labor supply rises at every horizon and adjusts monotonically upward over time under a one-off permanent rate cut.

Capital investment (§5.5, pp. 678-679). A firm adjusts capital $k_t \in \mathbb{R}_+$ with per-period profit $F(k, (p, \eta, -r)) = pf(k, \eta) - rk$ where $f$ has increasing differences (so F has increasing differences in $(k, \theta)$ for $\theta = (p, \eta, -r)$). Investing $i_t = k_t - k_{t-1}$ costs $C(i_t) \geq 0$, assumed only to be single-dipped at zero. Theorem 4 delivers monotone upward adjustment of capital over time whenever the marginal product of capital rises (fall in r, rise in p, or rise in $\eta$). The analysis covers both convex and nonconvex (lumpy) investment; when adjustment costs are not even single-dipped (e.g., a minimum investment threshold creates a region $C(\varepsilon) = \infty$), Theorem 1 still applies since C remains minimally monotone.

Datasets used

This is a pure-theory paper. No datasets are used.

When to read the full paper

Read the original at https://doi.org/10.3982/ECTA22841 if you:

• are applying these results to a new economic model and need the precise conditions and all proof details (Appendices A-N, pp. 682-693);
• are working on dynamic-adjustment models with nonconvex or general cost functions and need the full statements of Theorems 3-6;
• need the extension to uncertain adjustment costs (Appendix B, pp. 683-684, Theorems 1’-3’);
• need the constraint-shift variant (Theorem 1*, §3.1, pp. 668-669) or the necessity results of Section 7 to understand when the conditions can be relaxed.

Attribution and rights

Source: peer-reviewed, Econometrica 93(2), March 2025. This distillation was extracted by an LLM and re-checked against the source PDF; its results have not been independently reproduced.

Attribution (CC BY 4.0). Dekel, Eddie, John K.-H. Quah, and Ludvig Sinander. “Comparative Statics With Adjustment Costs and the Le Chatelier Principle.” Econometrica 93, no. 2 (March 2025): 661-694. DOI: 10.3982/ECTA22841. © 2025 The Authors. Licensed under the Creative Commons Attribution License (CC BY 4.0). This page is an adaptation by the Institute for Automated Research: core results extracted and re-expressed; changes were made.