The Dynamic Distribution in the Fixed Cost Model

An Analytical Solution

Jonathan J. Adams

Federal Reserve Bank of Kansas City

KC Fed Brown Bag

August 4 2026

The views expressed herein are those of the author and do not necessarily represent the views of the Federal Reserve Bank of Kansas City or the Federal Reserve System.

Motivation

  • Macroeconomics with fixed costs of adjustment depends on the distribution of agents in the inaction region

  • This requires solving a PDE: the Kolmogorov Forward Equation (KFE)

  • Analytical solutions aid tractability and understanding (e.g. Alvarez–Lippi sufficient statistics)

  • But an analytical solution is difficult because the PDE is endogenous: its evolution depends on the flow of resets

  • Existing methods require shortcuts, such as symmetry or small shocks

This Paper’s Contributions

  • Derive an analytical solution to a generic fixed cost model (KFE + boundary conditions)

  • Key insight: in the frequency domain, you can solve for the endogenous flow of resets without first finding the entire distribution

  • Reset flow is like magic! All aggregate IRFs are linear functions of the reset flow alone: entire distribution is not needed

  • Opens up new theoretical results!

    • Example: functional sufficient statistic for the entire IRF to a macro shock.

  • Application: the advent of AI with investment adjustment costs

    • Analytical solution reveals aggregation tricks: can derive aggregate IRFs without joint distribution
    • If AI increases productivity growth rate, fixed adjustment costs imply boom-lull pattern.
    • Size matters: bigger shock = deeper lull

The Microeconomics of Fixed Costs are Simple

  • Examples: sticky prices, investment, inventories, rational inattention, hiring and firing, wage negotiation, etc.

  • Agent’s state \(x\) follows a diffusion inside inaction region; agent pays a cost to adjust \(x\) only occasionally

    \[ dx=\mu\,dt+\sigma\,dW, \qquad x\in[0,\bar x], \qquad x\longrightarrow x^* \]

  • Agent’s only decision is the inaction region + optimal reset. Well-understood problem!

  • Aggregation (macroeconomics) is the challenge

Individual Behavior Illustrated

The simulated individual state follows a diffusion and resets to x* at either boundary.
Diffusion path Boundary hit and reset to x*

Aggregate Behavior: The Dynamic Distribution

  • The density \(h(x,t)\) follows the Kolmogorov Forward Equation (KFE): \[ \partial_t h(x,t) =\frac{\sigma^2}{2}\,\partial_x^2 h(x,t)-\mu\,\partial_x h(x,t)-\eta h(x,t) +\delta(x-x^*)F(t) \] \[ h(0,t)=h(\bar x,t)=0 \]

  • Lemma 1: flow \(F(t)\) of agents resetting at time \(t\) is

\[ F(t) =\frac{\sigma^2}{2}\,\partial_x h(0,t) -\frac{\sigma^2}{2}\,\partial_x h(\bar x,t) +\eta \]

  • Challenge: \(F(t)\) is itself determined by \(h(x,t)\)

Know the Flow \(\longrightarrow\) Solve the Model

  • The known Green’s function \(G(x,y,t)\) is the response of density over \(x\) at \(t\) from impulse at \(y\)

  • Textbook formula for \(h(x,t)\) conditional on \(F(t)\):

\[ h(x,t) = \underbrace{\int_0^{\bar x}h(y,0)G(x,y,t)\,dy}_{\text{initial distribution}} + \underbrace{\int_0^tF(\tau)G(x,x^*,t-\tau)\,d\tau}_{\text{earlier reset cohorts}} \]

  • If you know the flow \(F(t)\), you know everything.

We Can Find the Flow!

Theorem 1: Expression for \(\widehat{F}(s)\) (Laplace transform) in terms of parameters, initial condition: \[ \widehat F(s)=\frac{\widehat A(s)}{1-\widehat B(s)} \] \[ F(t)=A(t)+\int_0^t B(t-\tau)F(\tau)\,d\tau \]

  • \(A(t)\): resets generated by the initial distribution
  • \(B(t)\): subsequent resets generated by a unit mass entering at \(x^*\)
  • Once \(F(t)\) is known, \(h(x,t)\) can be recovered if it is needed (previous slide)
  • Closed form expressions!

Know the Flow \(\longrightarrow\) Aggregate IRFs

  • The reset flow is magic; you do not need to solve the distribution to study aggregates!

\[ Z(t)=\int_0^{\bar x}f_Z(x)h(x,t)\,dx \]

  • Theorem 2: Aggregate variable \(Z(t)\) satisfies:

\[ \widehat Z(s)=\widehat A^Z(s)+\widehat B^Z(s)\widehat F(s) \]

\[ Z(t)=A^Z(t)+\int_0^tB^Z(t-\tau)F(\tau)\,d\tau \]

  • Closed-form expressions for \(A^Z,B^Z\) (for common \(f_Z\))

Generalization

  • Closed form solutions apply to the (common) baseline

  • But insights (importance/utility of reset flow) are more general

\[ \partial_t h(x,t) = \partial_x^2\!\left(\frac{\sigma^2(x)}{2}h(x,t)\right) -\partial_x\!\left(\mu(x)h(x,t)\right) -\eta(x)h(x,t) +\delta(x-x^*)F(t) \] \[ h(0,t)=h(\bar x,t)=0 \]

\[ F(t) = - \left.\partial_x\!\left(\frac{\sigma^2(x)}{2}h(x,t)\right)\right|_{x=0}^{x=\bar x} + \int_0^{\bar x}\eta(x)h(x,t)\,dx \]

  • Still hold: Theorems 1, 2, and 3…

Sufficient Statistics in Fixed Cost Models

  • Analytical solution opens possibility of useful new theoretical results

  • Example: a new sufficient statistic approach for the effects of macro shocks

  • Existing results map steady-state adjustment moments into the cumulative impulse response (e.g. Alvarez-Lippi-Le Bihan 2016)

  • Specific to some specialized settings, particular aggregates (e.g. the avg. state)

  • Can we learn more?

Steady-State Adjustment Objects

  • \(S(t\mid y)\) denotes the probability that an agent beginning at \(y\) has not reset by \(t\)

\[ S(t\mid y)=\int_0^{\bar x}G(x,y,t)\,dx \qquad m(t\mid y)=-\partial_tS(t\mid y) \] \[ \acute{\mathbf S}(t) \equiv\int_0^{\bar x}\partial_yS(t\mid y)\,\bar h(y)\,dy \qquad \mathbf m(t)\equiv m(t\mid x^*) \]

  • \(\acute{\mathbf S}(t)\) is the average cross-sectional slope of conditional survival
  • \(\mathbf m(t)\) is the reset time density, conditional on an adjustment
  • Can be measured in panel data on states+adjustments (e.g. Alvarez et al 2024, Baley and Blanco 2026)

A Functional Sufficient Statistic for the Reset Flow IRF

  • Consider a small shock to the steady state that increases all states by \(\Delta\).
  • Theorem 3: The marginal reset-flow IRF is

\[ \widehat F^\Delta(s) = \frac{-s\,\widehat{\acute{\mathbf S}}(s)} {1-\widehat{\mathbf m}(s)} \]

\[ F^\Delta(t) =-\partial_t\acute{\mathbf S}(t) +\int_0^t\mathbf m(t-\tau)F^\Delta(\tau)\,d\tau \]

  • Steady state objects \(\mathbf m\) and \(\acute{\mathbf S}\) identify the entire marginal response path

  • \(F^\Delta(t)\) + Theorem 2 \(\longrightarrow\) get the marginal IRF of other aggregates

Example: The Advent of AI with Investment Fixed Costs

  • Firms use productivity \(P\) and capital \(K\) to produce \[ Y=P^{1-\alpha}K^\alpha \]
  • Idiosyncratic log productivity \(p \equiv \log P\) follows \[ dp=g_p\,dt+\sigma_p\,dW \]
  • Firms must pay a fixed cost \(\kappa P\) to adjust capital at price \(q\)
  • Capital depreciates at rate \(\zeta\)
  • PE: firms discount by fixed interest rate
  • State variable is the capital gap: \[x = \log k - \log p - \text{constant}\]

A special case of Baley and Blanco (2021); KFE is:

\[ \partial_t h(x,t) = \frac{\sigma_p^2}{2}\,\partial_x^2h(x,t) +(\zeta+g_p)\,\partial_xh(x,t) +\delta(x-x^*)F(t) \qquad h(0,t)=h(\bar x,t)=0 \]

Illustration: A Positive Productivity Shock

Stationary distribution Post-shock initial condition Reset flow