The Dynamic Distribution in the Fixed Cost Model

This Paper’s Contributions

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

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…

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

Illustration: A Positive Productivity Shock

The Productivity-Weighted Density is Enough

• Aggregate productivity is \(\bar P(t)\equiv\int_0^{\bar x}\int_{-\infty}^{\infty}e^p f(p,x,t)\,dp\,dx\)
• The productivity-weighted density is \(h_P(x,t) \equiv \frac{\int_{-\infty}^{\infty}e^p f(p,x,t)\,dp}{\bar P(t)}\)
• gives detrended aggregate capital, output:

\[ \begin{aligned} K_P(t)\equiv\frac{K(t)}{\bar P(t)} &=\int_0^{\bar x}e^{\underline{\widetilde k}+x}h_P(x,t)\,dx\\ Y_P(t)\equiv\frac{Y(t)}{\bar P(t)} &=\int_0^{\bar x}e^{\alpha(\underline{\widetilde k}+x)}h_P(x,t)\,dx \end{aligned} \]

Crucial: Weighted Density is Just Another Special Case

\[ \partial_t h_P =\frac{\sigma_p^2}{2}\partial_x^2h_P + \left(\zeta+g_p + \sigma^2\right)\partial_xh_P +\delta(x-x^*)F_P(t) \]

\[ F_P(t)=\frac{\sigma_p^2}{2} \left(\partial_xh_P(0,t)-\partial_xh_P(\bar x,t)\right) \]

• \(h_P(x,t)\) inherits absorbing boundaries, integrates to one, all density re-enters at \(x^*\)

• The reset-flow solution therefore applies to \(F_P(t)\) and \(h_P(x,t)\)

• Can also use Theorem 2 to derive aggregates, Theorem 3 to calculate the suff. stat.

The AI Shock: Increase in TFP Growth Rate

Investment Transition Follows a Boom-Lull Pattern

Size Matters: Large Shock Creates Deeper but Shorter Lulls

• Lull depth rises nonlinearly with the increase in TFP growth
• More severe lulls return to the new steady state sooner

Backup

Derivations, validation, and robustness

The main results extend beyond constant coefficients

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

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

• The reset-flow, aggregation, and functional sufficient-statistic results continue to hold
• Constant coefficients deliver the closed-form eigenfunctions used for computation
• Truly dynamic boundaries require an extension of the method

Eigenfunctions reduce the reset-flow problem to scalar sums

\[ G(x,y,t) =\sum_{n=1}^{\infty} \psi_n^r(x)\psi_n^\ell(y)e^{-\lambda_nt} \]

\[ \widehat A(s) =\sum_{n=1}^{\infty}\frac{\theta_na_n}{s+\lambda_n}, \qquad \widehat B(s) =\sum_{n=1}^{\infty}\frac{\theta_nb_n}{s+\lambda_n} \]

\[ a_n=\int_0^{\bar x}\psi_n^\ell(y)\phi(y)\,dy, \qquad b_n=\psi_n^\ell(x^*) \]

• The theorem is general; the constant-coefficient model gives closed-form expressions for every term

The functional statistic is accurate for small level shocks

Free adjustments alter magnitudes, not the boom-lull pattern

Capital and output adjust much less than investment

• Detrended capital and output move much less than detrended investment
• Their IRFs require only the one-dimensional productivity-weighted gap density