Pricing models in the sequence space¶

NBER Heterogeneous-Agent Macro Workshop

Matthew Rognlie

Spring 2023

(based on Auclert, Rigato, Rognlie, Straub 2023, forthcoming in QJE)

Imports¶

Generalized time-dependent (TD) models¶

• Exogenous survival function $\Phi_s$ probability a price still in force $s$ periods after being set, starts at $\Phi_0=1$

• Hazard rate $\lambda_s \equiv (\Phi_{s-1} - \Phi_s)/\Phi_s$ of price being reset at $s$
  • Visit from "generalized Calvo fairy" with probability $\lambda_s$

• Objective when resetting in simple problem: minimize discounted quadratic loss function given shifter to log nominal marginal cost (normalized mean zero)
  • weighted by probability price still in effect!

$$\hat{P}_t^* \equiv \arg \min_{\hat{P}} \sum_{s=0}^\infty \beta^s\Phi_s\frac{1}{2}(\hat{P}-\widehat{MC}_{t+s})^2$$

Examples of time-dependent models¶

• Calvo: $\Phi_s = \theta^s$ for some Calvo adjustment rate $1-\theta$
  • Constant hazard rate $\lambda_s = 1-\theta$

• Taylor: $\Phi_s = 1$ for $s< N$ and $\Phi_s=0$ for $s \geq N$
  • Hazard rate is 0 until we reach $N$th period, where it becomes 1 (extreme example of increasing hazard rate)
  • Prices only last for $N$ periods, for some determinate $N$

First-order condition for log reset price¶

• Want to set log prices equal to weighted average of future log prices:

$$\hat{P}_t^* = \frac{\sum_{s=0}^\infty \beta^s\Phi_s\widehat{MC}_{t+s}}{\sum_{s=0}^\infty \beta^s\Phi_s}$$

• In vectorized form:

$$\hat{\mathbf{P}}^*= \frac{1}{\sum_{s=0}^\infty \beta^s \Phi_s}~ \begin{pmatrix} \Phi_0 & \beta \Phi_1 & \beta^2 \Phi_2 & \cdots \\ 0 & \Phi_0 & \beta \Phi_1 & \cdots \\ 0 & 0 & \Phi_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix} \widehat{\mathbf{MC}}$$

Aggregate prices¶

• Assume aggregate price index is average of prices currently in effect
  • (to first-order approximation around zero-inflation steady state)

• The distribution of ages of prices is proportional to survival function, so we have

$$\hat{P}_t = \frac{\sum_{s=0}^\infty \Phi_s \hat{P}_{t-s}^*}{\sum_{s=0}^\infty \Phi_s}$$

• In vectorized form:

$$\hat{\mathbf{P}} = \frac{1}{\sum_{s=0}^\infty \Phi_s}~ \begin{pmatrix} \Phi_0 & 0 & 0 & \cdots \\ \Phi_1 & \Phi_0 & 0 & \cdots \\ \Phi_2 & \Phi_1 & \Phi_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}~\hat{\mathbf{P}}^*$$

Combining everything¶

Substituting first matrix equation into second to get matrix mapping $\widehat{\mathbf{MC}}$ to $\hat{\mathbf{P}}$:

$$\hat{\mathbf{P}} = \underbrace{\frac{1}{\left(\sum_{s=0}^\infty \Phi_s\right)\left(\sum_{s=0}^\infty \beta^s \Phi_s\right)}~ \begin{pmatrix} \Phi_0 & 0 & 0 & \cdots \\ \Phi_1 & \Phi_0 & 0 & \cdots \\ \Phi_2 & \Phi_1 & \Phi_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix} \begin{pmatrix} \Phi_0 & \beta \Phi_1 & \beta^2 \Phi_2 & \cdots \\ 0 & \Phi_0 & \beta \Phi_1 & \cdots \\ 0 & 0 & \Phi_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}}_{\equiv \Psi}~ \widehat{\mathbf{MC}}$$

We call this matrix the pass-through matrix, mapping shocks to log nominal marginal cost to changes in price

Let's calculate this!¶

$$\Psi \equiv \frac{1}{\left(\sum_{s=0}^\infty \Phi_s\right)\left(\sum_{s=0}^\infty \beta^s \Phi_s\right)}~ \begin{pmatrix} \Phi_0 & 0 & 0 & \cdots \\ \Phi_1 & \Phi_0 & 0 & \cdots \\ \Phi_2 & \Phi_1 & \Phi_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix} \begin{pmatrix} \Phi_0 & \beta \Phi_1 & \beta^2 \Phi_2 & \cdots \\ 0 & \Phi_0 & \beta \Phi_1 & \cdots \\ 0 & 0 & \Phi_0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}$$

Pass-through matrix for Calvo¶

Pass-through matrix for Taylor¶

General points about pass-through matrices¶

• "Tent-shaped": an MIT shock to nominal marginal cost at some date passes through to aggregate shocks partly at that date...
  • but partly in anticipation
  • and partly afterward
  • as sticky prices are stuck after shock, and set in anticipation of shock

• Perfectly flexible prices correspond to identity matrix $\Psi=\mathbf{I}$
  • we're assuming that firms ideally want 100% pass-through if flexible
  • more generally, $\Psi$ is mapping from statically optimal prices to actual aggregate prices

• Impulse response to permanent nominal marginal cost shock given by $\Psi\cdot \mathbf{1}$, i.e. sum of columns
  • converges to 1, i.e full long-run pass-through

New Keynesian Phillips curve: real marginal cost and inflation¶

The New Keynesian Phillips curve is derived in terms of real marginal cost $\widehat{mc}_t = \widehat{MC}_t - \hat{P}_t$:

$$\pi_t = \kappa \widehat{mc}_t + \beta\pi_{t+1}$$

which can then be related, in models, to the output gap, employment, etc.

How can we go from the pass-through matrix, relating nominal marginal cost and prices, to something like this, relating real marginal cost and inflation?

Solving fixed point¶

We get a fixed-point equation—conditional on real marginal cost, prices themselves affect the price-setting decision:

$$\hat{\mathbf{P}} = \Psi \cdot\widehat{\mathbf{MC}} = \Psi \cdot(\widehat{\mathbf{mc}} + \hat{\mathbf{P}})$$

Solving this gives:

$$\hat{\mathbf{P}} = (\mathbf{I}-\Psi)^{-1}\cdot \Psi \cdot \widehat{\mathbf{mc}}$$

Equivalent to $\hat{\mathbf{P}} = (\Psi + \Psi ^2 + \Psi^3 + \ldots)\cdot \widehat{\mathbf{mc}}$, representing "rounds" of feedback

Finally, first differences to obtain inflation:

$$\boldsymbol{\pi} = \underbrace{(\mathbf{I}-\mathbf{L})\cdot (\mathbf{I}-\Psi)^{-1}\cdot \Psi}_{\equiv \mathbf{K}} \cdot \widehat{\mathbf{mc}}$$

Generalized Phillips curve¶

We have

$$\boldsymbol{\pi} = \underbrace{(\mathbf{I}-\mathbf{L})\cdot (\mathbf{I}-\Psi)^{-1}\cdot \Psi}_{\equiv \mathbf{K}} \cdot \widehat{\mathbf{mc}}$$

and call matrix $\mathbf{K}$ mapping real marginal cost to inflation the generalized Phillips curve

Generalized Phillips curve for Calvo is matrix representation of NKPC:

$$\mathbf{K} = \begin{pmatrix} \kappa & \beta\kappa & \beta^2\kappa & \cdots \\ 0 & \kappa & \beta\kappa & \cdots \\ 0 & 0 & \kappa & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}$$

Calculate generalized Phillips curve¶

Need high $T$ to avoid artifacts from truncation close to $T$, but can implement formula directly:

Get what we expect for Calvo!¶

Something much uglier for Taylor, with a bit of inertia¶

Note: this looks like numerical error, but isn't—the generalized Phillips curve for Taylor pricing is really this ugly-looking!

Inertia: inflation persists a bit after real marginal cost shocks (at dates 0, 5, 10, 15):

Smoother version of increasing hazard¶

Plot hazard rate of price adjustment:

Generalized Phillips curve for this case: smoother, still with inertia¶

But this case looks a lot like Calvo in the long run...¶

Same $\beta$ rate of decay in anticipation (can prove this!), mostly forward-looking, inertia small in comparison!

Opposite case: decreasing hazard, anti-inertia¶

Also looks more like Calvo in long run...¶

Long-run decay of anticipation at rate $\beta$, distinct behavior confined to neighborhood of shock (and date 0)

Summary of what we've learned for time-dependent¶

• No generalized Phillips curves look too different from Calvo
  • they're all more forward-looking than backward-looking, with anticipation decaying at $\beta$

• Increasing hazards lead to a bit of inertia (and vice versa for decreasing hazards)

• Intuition: with increasing hazards, people resetting prices today are more likely not to have reset prices recently...

• ... so they missed any recent inflation, and need to set higher prices to "catch up", resulting in inertial inflation

Nerdy sequence-space Jacobian side note¶

The pass-through matrix is a sequence-space Jacobian, and TD $\Psi$ has fake news matrix $F_{t,s} = J_{t,s} - J_{t-1,s-1}$ equal to a simple rank-one matrix:

$$F \equiv \frac{1}{\left(\sum_{s=0}^\infty \Phi_s\right)\left(\sum_{s=0}^\infty \beta^s \Phi_s\right)}~ \begin{pmatrix} \Phi_0 \\ \Phi_1 \\ \Phi_2 \\ \vdots \end{pmatrix} \begin{pmatrix} \Phi_0 & \beta \Phi_1 & \beta^2 \Phi_2 & \cdots \end{pmatrix}$$

A "fake news" shock about future marginal costs at $s$ leads to an increase in reset prices at date 0 in proportion to $\beta^s \Phi_s$, which has a persistent effect at date $t$ in proportion to $\Phi_t$

Nerdy side note continued: recovering survival function¶

Only one nonzero singular value, i.e. this is rank one:

Corresponding singular vectors are proportional to $\Phi_s$ and $\beta^s \Phi_s$, respectively, can rescale so first entry is 1 and see exponential decay:

Other detour: mixture models¶

Can easily assume fraction of firms follow one TD rule, fraction follow another, just by combining pass-through matrices.

For instance, let's try simple Calvo mixture, with some more flexible and some less, averaging out to same 0.25 frequency of price change:

Generalized Phillips curve shows anti-inertia¶

Same intuition as decreasing hazards: price-setters are more likely to have recently reset, so they lower prices after inflation to go closer to sticky counterparts

Menu cost models¶

Basic menu cost model¶

Continuum of firms, assume their ideal log price $p_{it}^*$ follows random walk $p_{it}^* = p_{it-1}^* + \epsilon_{it}$ with symmetric, mean-zero shocks $\epsilon_{it}$.

Defining idiosyncratic price gap as $x_{it}\equiv p_{it} - p_{it}^*$, firm objective in simple model becomes

$$\min_{\{x_{it}\}} \mathbb{E}_0 \sum_{t=0}^\infty \beta^t \left[ \frac{1}{2}(x_{it} - \widehat{MC}_t)^2 + \xi 1_{x_{it}\neq x_{it-1}-\epsilon_{it}} \right]$$

i.e. pick a state-contingent plan for price gaps $x_{it}$ to minimize quadratic loss, subject to time-varying aggregate marginal cost shocks, and a menu cost $\xi$ for adjusting your price (i.e. not letting price gap drift with shocks)

Optimal policy is "Ss" rule¶

Optimal policy: adjust to reset point $x_t^*$ if shock takes you outside adjustment bands $[\underline{x}_t, \overline{x}_t]$

$$x_{it} = \begin{cases} x_t ^* & \text{if }x_{it-1} - \epsilon_{it} \notin [\underline{x}_t, \overline{x}_t] \\ x_{it-1} - \epsilon_{it} & \text{otherwise} \end{cases}$$

Note policies $(x_t ^*, \underline{x}_t, \overline{x}_t)$ common across all firms $i$

Symmetry of shocks implies $x_t^* = 0$ and $\underline{x}_t = -\overline{x}_t$ in steady state

Law of motion for density of price gaps¶

Let $g(x)$ be steady-state density of price gaps $x_{it-1} - \epsilon_{it}$ in any period, "before" firms outside bands have adjusted to $x_t^*$, and $\text{freq}$ be steady-state fraction of firms adjusting:

$$\text{freq} = 1 - \int_{-\bar{x}}^{\bar{x}} g(x)dx$$ $$g(x) = \text{freq}\cdot f(x) + \int_{-\bar{x}}^{\bar{x}} f(x-x')g(x')dx'$$

Calculate steady-state density $g$¶

Assume shocks to ideal log price normal, define first mean-zero normal pdf:

We'll assume quarterly calibration where standard deviation of shocks is 0.05 (roughly calibrated value from our paper):

Steady-state density continued¶

Assume we're given adjustment band $\overline{x}$ (and by symmetry $-\overline{x}$ on other side), iterate on law of motion for $g$:

Plot density for arbitrary $\overline{x}$:¶

Calibrate $\bar{x}$ to hit average frequency of 0.25¶

Don't need to know $\xi$ itself to do this!

Actual density looks similar to what we already saw¶

Defining expectation functions in our context (cf fake news algorithm)¶

If end-of-period price gap is $x$ today, what (following steady-state policy) will it be on average in $t$ periods?

Law of iterated expectations, plus symmetry (reset to zero, where expectations are zero), imply:

$$E^t(x) = \int_{-\overline{x}}^{\overline{x}} f(x'-x)E^{t-1}(x')dx'$$

Plot some expectation functions¶

Decay much faster than probability of resetting prices, due to "selection effects"

Or ignore the first to see typical shape¶

Exactly how fast do these decay?¶

Let's look at "persistence" of expectation function starting from reset point $x^*=0$ or adjustment band $\overline{x}=0$:

$$\Phi_t^e \equiv \frac{E^t(\overline{x})}{\overline{x}}~~~~~~~~~~~~~ \Phi_t^i \equiv E^{t \prime}(0) = \lim_{x\rightarrow 0} \frac{E^t(x)}{x}$$

Very fast decay¶

Initially persistence at $\bar{x}$ is less, because people are more likely to adjust there:

But "hazard" very quickly approaches same constant¶

Can also iterate to track actual "survival probability" of prices¶

Decays much more slowly than persistences—"selection effect"!¶

Proposition 1 from Auclert, Rigato, Rognlie, Straub (2023)¶

Assume aggregate price is average of price gaps, $\hat{P}_t = \int x_{it}di$ (idiosyncratic shocks average out to zero, so price gaps give price level)

The pass-through matrix from nominal marginal cost to price for the menu cost model is given by a mixture of time-dependent models

$$\Psi = \alpha \Psi^{\Phi^e} + (1-\alpha)\Psi^{\Phi^i}$$

where $\Psi^{\Phi^e}$ and $\Psi^{\Phi^i}$ are pass-through matrices with survivals $\Phi^e_t$ and $\Phi^i_t$

First term gives effect of extensive margin, second term gives effect of intensive margin, with weight on extensive margin of $\alpha \equiv 2g(\overline{x})\overline{x}\sum_{t=0}^\infty \Phi_t^e$

Partial intuition: "persistence" at $\overline{x}$ gives persistent effect of extensive margin decisions¶

• Suppose I'm right at the threshold $\overline{x}$ and decide not to adjust today

  • effect on price level today: $+\overline{x}$

  • effect tomorrow: $+E^1(\overline{x})$

  • ... effect in $t$ periods: $+E^t(\overline{x})$

• Expectation function summarizes effect of my decision today on future price level
  • it incorporates fact that if I don't adjust today, I'll probably adjust soon anyway
  • "persistence" of effect is $\Phi^e_t = E^t(\overline{x}) / \overline{x}$

Re-plot expectation functions to visualize this¶

Partial intuition, continued: "persistence" at $x^*=0$ gives persistent effect of intensive margin decisions¶

• Suppose I'm adjusting and I decide to raise my price by $dx^*$, slightly above zero

  • effect on price level today: $+dx^*$

  • effect tomorrow: $+E^1(dx^*) = +E^{1\prime}(0)dx^*$

  • ... effect in $t$ periods: $+E^{t\prime}(0)dx^*$

• Again, expectation function summarizes effect decision today on future price level
  • incorporates fact that if I set higher price today, I'm more likely to adjust downward in future
  • "persistence" of higher price is $\Phi^i_t +E^{t\prime}(0)$

Taking stock¶

• We've seen that extensive margin decisions "persist" at $\Phi_t^e$, intensive margin decisions "persist" at $\Phi_t^i$, each like in a properly calibrated time-dependent model

• But that's only one side of time-dependent models!

• Key to time-dependent models is that future matters proportional to $\beta^s \Phi_s$!
  • Recall reset-price equation for time-dependent case:

$$\hat{P}_t^* = \frac{\sum_{s=0}^\infty \beta^s\Phi_s\widehat{MC}_{t+s}}{\sum_{s=0}^\infty \beta^s\Phi_s}$$

What's intuition for why future shocks matter in proportion to $\Phi_s$?¶

• Loose intuition: when making a decision at extensive or intensive margin, you care about future marginal costs only insofar as your decision today actually affects your future price
  • and that's exactly what these persistences $\Phi_t^e$ and $\Phi_t^i$ are capturing!

• More formal intuition: envelope theorem argument implies that if there's some perturbation $\widehat{MC}_s$, then for $t<s$, we have same recursion as for expectation functions, with added $\beta$:

$$dV_t(x) = \beta \mathbb{E}_{x'|x} dV_{t+1}(x')$$

• Since $dV_s(x) = -x = -E^0(x)$, follows that $dV_t(x) = -\beta^s E^{s-t}(x)$
  • future marginal costs affect value function in proportion to expectation function!

Let's implement!¶

Pad with zeros to get $T$-length, build pass-through matrices for extensive and intensive margin:

Weight extensive by $\alpha$, intensive by $1-\alpha$:

Visualize pass-through matrix¶

Very sharp spikes, corresponding to near-flexible prices (rapidly declining "virtual survival", i.e. low persistence of decisions):

What about generalized Phillips curve?¶

Insanely close to the Calvo New Keynesian Phillips curve, but a lot more flexible (slope of 2, not 0.1)!

Why is this?¶

Both extensive and intensive very quickly converge to the same constant hazard (corresponding to leading odd eigenvalue—cf Alvarez and Lippi Ecta 2022 and our paper), much higher than actual probability of adjusting (near 0.25)...

... so they behave like a Calvo, but one with more like a 0.7 quarterly adjustment frequency!

Adding "free resets"¶

Modified menu cost model with free resets¶

• Suppose that there's a $\lambda$ chance of a "free reset", where menu cost is zero
  • Then always adjust back to $x_t^*$
  • Higher $\lambda$ as share of total resets means adjustment is less state-dependent, more Calvo-like
  • Close to "Calvo-Plus" model of Nakamura and Steinsson, also others in literature

• We'll redo steps quickly modifying the model to account for this, not going into much detail
  • We'll calibrate to $\lambda=0.2$ and still adjustment frequency of 0.25, so that four-fifths of adjustments are free
  • Same equivalence result holds with suitably redefined expectations functions and weights!

Modified steady-state density function (almost same as before)¶

Need wider adjustment bands to match same frequency¶

Now that so many adjustments are free, bands have to be wider to hit same total frequency of adjustments:

Much less density near bands as well:

Also rewrite expectation iteration and functions¶

Same as before, but adding an extra $(1-\lambda)$ factor to account for free resets to zero:

Rewrite code that calculates virtual and actual survival:

What do hazards look like now?¶

Still converging to a constant, different from actual hazard, but it takes longer ($\overline{x}$ higher, so bigger difference between starting at $\overline{x}$ and $x^*=0$) and less of a gap:

Calculate pass-through matrix now¶

Weight on extensive margin is smaller, because so many adjustments are free that it's less important:

Visualizing pass-through matrix¶

Less sharp spikes, corresponding to less flexible prices:

Generalized Phillips curve a bit less Calvo-like, but still extremely close¶

(If you run the NKPC as a regression on data that follows this, it still holds almost exactly.)

Paradox: making model more Calvo-like gives it a slightly less Calvo-like shape?¶

• Might seem like a paradox: why is the basic menu cost model a more perfect fit to the Calvo NKPC than a model with Calvo-like free-adjustments?

• Explanation of paradox: wider bands means that extensive and intensive virtual hazards more different, take longer to converge to constant hazard characteristic of Calvo
  • In limit of only free adjustments, converges to Calvo, but takes a while to get there...

• Paradox is only about shape rather than scale: slope is much closer to the Calvo case (but still higher)

Generalizing the approach¶

• More time-dependent models needed with more distinct margins
  • asymmetries or drift from inflation: 3 models to represent separate lower and upper extensive margins
  • mean-reverting shocks: even more models
  • different objective functions or aggregation: slightly generalize TD model

• Go all the way to generalized hazard functions (cf Alvarez Lippi Oskolkov) in paper, requiring a continuum of TD models representing extensive margin

• Similar results should hold for similar models with fixed costs

• Near-equivalence to Calvo remarkably robust!

Big picture for pricing models¶

• Hard to escape New Keynesian Phillips curve functional form!

• Menu cost models better-microfounded but worsen "slope puzzle" (gap between micro and macro) for the NKPC

• Some promising other routes
  • multisector (e.g. Rubbo; Afrouzi and Bhattarai)
  • departures from FIRE (e.g. Angeletos and Lian; also Ludwig's lecture for techniques!)

Big picture for our techniques¶

• Generalizing sequence-space Jacobians to new kinds of models

• Example of simple discrete choice, in this case without taste shocks
  • Needed continuous shocks and to be careful about integration

• Equivalence result is really just the fake news algorithm in disguise
  • If you try to figure out what the algorithm means, you'll find it!