Are State- and Time-Dependent Models Really Different? — handwritten notes

These are my older hand-written notes on the paper, kept as they were apart from tidying. The main note is a newer, fuller digest.

What?

• Paco’s wustl 502’s reading list for basic New Keynesian model
• This paper (title) is similar to Klenow and Kryvtsov (2008). However, the former mostly discuss the empirical evidence that drive td and sd models.

Why?

Why do we want to compare state- and time-dependent models?

time-dependent (td) and state-dependent (sd) describes how the underlying price frictions are modelled in nk. The canonical nk models assert assumptions on price adjustments in order to model sticky prices. Two major assumptions are

• Menu-cost models à la Golosov and Lucas Jr. (2007)
• Limited information-gathering and information-processing ability (rational inattentiveness)
  • Some argue (how?) that information frictions will generate stronger real effects of monetary policy shocks (Mankiw and Reis 2002; Klenow and Kryvtsov 2008)

(?) Make clear the distinctions on the papers that outlined the two approaches….

How?

Study the propagation of a permanent unexpected shock $\delta$, starting from the steady state of an economy with an inflation rate $\pi$ and idiosyncratic shocks with variance $\sigma^{2}$.

Data

• Panel of monthly data on cpi inflation and nominal exchange rate
• 70 countries in periods of moderate inflation, post-Bretton Woods

How are shocks to the price level described?

This statistic is first introduced in Caballero and Engel (2007). A monetary shock $\delta$ has an effect on price level is

where

• $\Theta(\delta; \pi))$ describes the instantaneous impact adjustment at the time of the shock
• $\theta(\delta, t)$ describes the continuous flow of adjustment in $t > 0$ after the shock
  • For td models, $\theta(\delta,t) = \theta(1, t)\delta$, thus $\mathcal{P}(\delta,t) = \mathcal{P}(1, t) \delta$ for all $\delta$. In other words,$\mathcal{P}(\cdot, t)$ is linear with a zero intercept.
• flexibility index $\Theta'(\delta)$ describes the impact effect of a monetary shock on inflation.
  • By this definition then $\Theta'(\delta) = 0$ in td models.
  • Alvarez, Lippi, and Passadore (2017) eventually show that first-order impacts in sd are zero too: $\Theta'(\delta) = 0$.

$\mathcal{P}$ is estimated monthly.

How is the shock to output modelled?

The cumulative value-added output $\mathcal{M}$ after a shock $\delta$ is

where $\varepsilon$ is the elasticity of labor supply.

For small $\delta$, we can approximate $\mathcal{M}$ by

(not sure where the proof is documented.) Intuition: think of

• $N(\Delta p_i)$ as measuring the degree of flexibility of the economy.
• $\operatorname{Kur}(\Delta p_i)$ measures the selection effect – a term introduced by Golosov and Lucas Jr. (2007) that basically says that firms who change their prices after shocks are firms that are most in need of adjustment.
  • Although the selection effect is a “sd hallmark”, the authors show that it holds for both sd and td.
  • This happens because the selection effect operate equally via two mechanisms:
    • Through the size distribution of price changes (sd)
    • Through the distribution of time between adjustments (td)

$\mathcal{M}$ is briefly described in Nakamura and Steinsson (2010, n. 21). It’s not the main measure they used (output variance $\operatorname{Var}(C_{t})$), but they do included $\mathcal{M}$ as extended results.

Empirical specifications

Is short-term pass-through bigger for large exchange rate movements?

where

• $\pi_{i, (t, t+h)} = \frac{p_{i,t}}{p_{i,t+h}}\times100$ is inflation rate of country $i$ in period from $t$ to $t+h$, with $h = 1,3,6,12,24$
• $\Delta e_{i,t} = \frac{e_{i,t}}{e_{i,t-1}} \times 100$ is the devaluation in the previous month to $t$ of country $i$ to the us
• An operator $\operatorname{sign}(\cdot)$ is used to impose “symmetry” between devaluation and appreciation

In theory

• For sd: larger shocks, larger response
• For td: impulse response should be independent from the size of shock

How are state-dependency and time-dependency modelled in this paper (or the literature in general)?

Firms’ price-setting problem

Production

Firm $k$ sells quantity $y_{ki}$ of goods $i \in (1,2\dots n)$:

where $Z_{ki}(t) = \exp (\sigma \mathcal{W}_{ki}(t))$. Specifically:

• $\ell_{ki}(t)$ is labor input
• The log of productivity $\mathcal{W}_{ki}(t)$ follows a brownian motion with variance $\sigma^{2}$: $$\mathcal{W}_{ki}(t) = \frac{\overline{\sigma}}{\sqrt{\overline{\sigma}^{2} + \sigma^{2}}}\overline{\mathcal{W}}_{k}(t) + \frac{\sigma}{\sqrt{\overline{\sigma}^{2} + \sigma^{2}}} \tilde{\mathcal{W}}_{ki}(t)$$
  • $\overline{\mathcal{W}}_k$ is productivity shock that is common across all products of firm $k$
  • $\tilde{W}_{ki}$ is idiosyncratic productivity shock
  • The processes $\left\{ \overline{\mathcal{W}}_{k}(t), \tilde{\mathcal{W}}_{ki}(t) \right\}$ are independent across $k$ and $i$. Thus $\{\mathcal{W}_{ki}\}$ are independent across $k$.

Price gaps

Define price gap $g_{ki}(t)$ as the log difference between nominal price and the static profit-maximizing price for good $i$, firm $k$:

where

• $P_{ki}(t)$ is the nominal price
• $W(t)Z_{ki}(t)$ is the nominal marginal cost, with $W(t)$ be the nominal wage
• $\eta$ is the demand elasticity for the firms’ bundle of $n$ products
• The bundle has an elasticity $\rho$ between each of its $n$ varieties. (?) Look at eq (A1) and (A2)

(?) Verify the nominal marginal cost $W(t)Z_{ki}(t)$

With $\pi$ constant induces a constant drift in $W(t)$, the dynamics of price gaps $g_{ki}(t)$ can be described as

Profit function

The nominal profit for firm $k$ can be approximated by

where $c(t)$ short hand for any variables that enter the profit function in a weakly separable way (e.g. aggregate demand in an equilibrium). See Kreps (2012, 40) for the definition. Basically this implies the effect of $c(t)$ is diminished up to the second order.

State-dependent pricing rules

This section discuss the representative firm, so $k$ is dropped for brevity. Let $g = (g_1, g_2, \dots, g_n)$ be the vector of price gaps.

Microfoundation of state-dependent rules

Microfounding by solving this generalized problem: consider a firm that chooses a stopping time to change price $\{\tau_{i}\}$ and by how much $\Delta P_j(\tau_i)$:

where

• The two main parameters here are $\psi_m$ (the menu cost) and $n$ (the number of product).
• Once $\psi_m$ is paid, firms can change the prices of all goods with no extra costs.

Challenge: nonconcave convex stochastic sequence

Classic menu cost à la Golosov and Lucas Jr. (2007)

In this model $n=1$ and $\psi$ is constant. If $\pi=0$ then we derive the $Ss$ rule: price changes when actual price and $g$ reaches the threshold $\pm \overline{g}$. A treatment can be found in Romer (2018, sec. 7.5). Let’s stop for a moment and gather a summary about this literature.

The literature of menu-cost models

This summary is based on Nakamura and Steinsson (2013, sec. 7).

• A good deep dive would start from the stylized Caplin and Spulber (1987):
  • Time is continuous. Nominal output follows a brownian motion with drift. Shocks always drive aggregate demand upwards.
  • As first formulated by Sheshinski and Weiss (1977, 1983), firms wait until price level drop to $s$, at when they increase it to $S$.
  • There’s no effects on real output in this model (no nk yet). So consider a short interval of time where nominal output increase by $\Delta m$. It must be the case that $\Delta m = \Delta p$.
  • This is in contrast with the Calvo setting where firms are chosen at random. The contrasts in conclusions between these two models are also striking. However, both models are extremes:
    • Calvo: aggregate shock has no effect on which and how many firms change prices
    • Caplin-Spulber: shock is the only determinant as to why firms change prices
• Klenow and Kryvtsov (2008) point out the empirical contradictions with these models:
  • Price changes are larger than the models would suggest: about 10%
  • Also 40% are price decreases
• Golosov and Lucas Jr. (2007) revisit this model:
  • They coined the term selection effect – firms who change prices are those who have the most pent-up desire to do so.
  • They interpret the Klenow and Kryvtsov (2008) findings as evidence for the existence of large, transitory, idiosyncratic shocks to firms’ costs.
  • Their calibrated model (with idiosyncratic shocks) reduces the degree of money non-neutrality to 1/6 that of Calvo.
  • They concluded that price rigidity yields “small and transient” monetary non-neutrality
• Midrigan (2011) then argues that the distribution of the idiosyncratic shocks matters
  • If idiosyncratic shocks are either zero or very large, the selection effect is eliminated and we have the Calvo results
  • Propose: shock distributions are leptokurtic (more dispersed than Golosov-Lucas would suggest)
  • Propose: returns to scale in price adjustments – the second onward menu costs are free
• We should always be warned that all empirical data shows large heterogeneity between products
• Also, the existence of temporary $\vee$-shaped price changes somewhat undermine the arguments of menu-costs.
• Interesting approaches include Head et al. (2012), where they provide a search-based argument as to why micro price rigidity may be divorced from the aggregate price level

More generalized case with $n > 1$ (maybe later)

Time-dependent pricing rules

Definitions

$\tau_{i+1} - \tau_i$ is a random variable with cumulative distribution (hazard function):

For time-dependent models, the realization for $\tau_{i+1}$ is independent of price gaps for $t \geq \tau_{i}$.

Some examples:

• In Taylor (1980) staggered pricing: $H(t) = 1$ for $t < T$ and $H(t) = 0$ otherwise.
• In Mankiw and Reis (2002): $H(t) = e^{-\lambda t}$

Microfoundation for time-dependent rules

Alvarez, Lippi, and Paciello (2016) described explicit microfoundation with rationally inattentive firms. In this setting, firms must pay a observation cost $\psi_{o}$ in order to gather information about production costs $(Z_1, \dots, Z_n)$.

Firms solve:

with

where

• $s_i$ is an exponentially distributed random variable
• $T_i$ and $P_i(t)$ for $t \in [\tau_i, \tau_{i+1})$only depends on the information up until $\tau_{i}$
• States $(Z_n)$ become known to firms exogenously and with exponentially distribution with time $\lambda$
• $(Z_n)$ and $\psi_o$ and $\lambda$ are independent
• $Z_i$ follows a martingale
  • This implies that there are no incentives for price plans

Variants of time-dependent pricing in literature

• Caballero (1989) and Reis (2006) analyzed the case where $\lambda = 0$ and $\psi_0$ constant
• Calvo model: two setups
  • $\psi_o$ very large
  • $\lambda = 0$, but $\psi_o$ follows a distribution
• Markovian time: The times between observations and price changes form a first-order Markovian process (see Alvarez, Lippi, and Paciello 2016)

Cool variants (maybe later)

And?

Under what circumstances does the nature of underlying friction matter for monetary shocks?

Three results:

1. For small shocks the nature of friction is irrelevant (Alvarez, Lippi, and Passadore 2017).
   • For td models the flexibility index is always zero
   • For sd models the shock (following a diffusion) doesn’t have first-order effects on aggregated prices (provided that firms follow an $Ss$ decision rule)
2. Small shocks have the same effects on total cumulated output, regardless of models
3. For large shocks there is differences:
   • For td models the response function of prices is proportional to the size of the shock
   • For sd models the response is nonlinear

Reduce the analysis to the price gaps

First, can show that after a monetary shock of size $\delta$:

where

• $R(t)$ is the real interest rate
• $\overline{W}(t)$ is the wage rate in the steady state before the shock
• $W(t)$ is the wage rate after the shock

Deviation of prices from the ss relate one-to-one to deviation on output:

where

• $\overline{c}$ is the constant flexible price equilibrium output
• $P(t)$ is the ideal price index at time $t \geq 0$
• $\overline{P}(t)$ is the path of the price level in the ss before the shock
  • $\overline{P}(t) = e^{\pi t} \overline{P}$ for all $t \geq 0$

Following Alvarez and Lippi (2014), the price level after the shock can be approximated as a function of the price gaps:

where

• $\tilde{g}_{ki}$ are the price gaps in ss before the shock
• $o(x)$ is of order smaller than $x$

Monetary shock increases the desired prices of all firms by $\delta$, implying price gaps to decrease by $\delta$ for all price gaps that are not adjusted on impact $\tilde{g}_{ki}$.

Impulse responses

Prices impulse responses

Impulse responses on prices is written as

Some properties

• The impact effect is bounded by $\delta$ $$0 \leq \mathcal{P}(\cdot) \equiv \Theta(\cdot) \leq \delta$$
• Long-term shocks are completely pass-through to prices $$\lim_{t \rightarrow \infty} \mathcal{P}(\cdot) = \delta$$
• In the flexible price case, prices jump on impact $$\mathcal{P}^{\text{flex}}(\cdot) = \delta$$

Output impulse responses

Impulse responses on nominal output response one-to-one with prices

where $\frac{1}{\varepsilon}$ is the uncompensated labor supply elasticity.

The measure of impulse response $\mathcal{M}$ is defined as the cumulation:

Some properties

• Boundedness $$0 \leq \mathcal{Y}(\cdot) \equiv \frac{1}{\varepsilon}[\delta - \Theta(\cdot)] \leq \frac{\delta}{\varepsilon}$$
• Diminished long-term $$\lim_{t \rightarrow \infty} \mathcal{Y}(\cdot) = 0 \qquad \forall \delta$$
• Diminished with flexible prices $$\mathcal{Y}^{\text{flex}}(\cdot) = 0 \quad \forall t \Rightarrow \mathcal{M}^{\text{flex}}(\cdot) = 0 \quad \forall \delta$$

What kind of empirical evidence can be used to identify the nature of the underlying frictions?

• Follows international economics literature that studies the pass-through of exchange rate shocks
• Uncover evidence of nonlinear pass-through of devaluation on inflation: $\gamma_{h}$ is strongly significant at least for $h = 3,6$
• Regression: inflation $\pi$ ~ exogenous nominal exchange rate innovation
  • Assumption: The shock is orthogonal to other regressors (exclusion) and is unanticipated (ci)
  • One potential problem is when exchange rate innovation occurs in response to domestic variable (e.g. $\pi$)

References