Most macroeconomic models embed some form of sticky prices, i.e. inertia in price-setting decisions. The main motivation for sticky prices is the empirical observation that individual good prices tend to remain fixed for long periods of time, typically a few months for CPI items. The sticky prices assumption is central in understanding how monetary policy actions propagate the macro economy; absent sticky prices, standard models predict that monetary policy is irrelevant for the dynamics of output and other real variables.

## Modelling sticky prices

Yet, how to best model price stickiness is an unresolved issue with relevant macroeconomic consequences. Depending on the details of how stickiness is modelled, the effects of monetary shocks range from negligible to substantial. Examples include:

- The Taylor model, in which firms keep prices unchanged for a fixed period of time (say six months);
- The menu cost model, according to which firms explicitly incur a cost for changing prices (for instance, the physical cost of changing the price sticker), and select the timing of their changes in prices weighing the costs and benefits of inaction.
- The Calvo model, in which there is a constant fixed probability of changing the price, independent of the economic environment, and of time since the last price change.

Figure 1 below, taken from an influential paper by Golosov and Lucas (2007), illustrates that different modelling assumptions, consistent with the same firm-level price stickiness (say two price changes per year), imply large differences in the response of the aggregate economy to monetary shocks. Each curve plots the response of GDP as a function of time after a monetary shock -- a 1% increase in money supply in their analysis. The solid blue curve corresponds to a menu cost model; the red dashed curve corresponds to a Calvo model. We stress that both models produce a frequency of price adjustment, so they have the same ‘price stickiness’. Yet, the effects on GDP are very different. The total amount of extra GDP that the economy produces after the monetary shock, as given by the area under the curve, illustrates this point. In the case of Calvo, the order of magnitude of the cumulative response is around 0.5% of GDP. In the menu cost model, the cumulative output effect is smaller (about 6 times) than in the Calvo model.

**Figure 1**. Output response to monetary shock in the menu cost versus the Calvo model

*Source*: Golosov and Lucas (2007).

Understanding how to model sticky prices is important given the implications for macro outcomes. The issue is relevant for central banks and other policy institutions that want to improve their macroeconomic models. Our recent research (Alvarez et al. 2014) focuses on selecting models which cross-sectional predictions comply with the new and growing microeconomic evidence on price-setting behaviour. The empirical documentation of price-setting patterns using large micro datasets has grown enormously over the past decade, following an initial contribution of Bils and Klenow (2004). Most interestingly, the recent studies provide information on the size of the distribution of price changes, such as the proportion of small price changes, the variance of price changes, the proportion of increases and decreases, and their frequency. The last generation of sticky price models takes this evidence seriously and uses it as a model selection device; models that are inconsistent with these cross-sectional data are discarded.

One striking feature of the empirical evidence on price setting is the simultaneous presence of very large and very small price changes. This pattern is illustrated in Figure 2, a histogram of price changes based on the price records underlying the French CPI -- one of the datasets we analyse in the paper. Similar evidence emerges using US data. The distribution is very peaked. A convenient summary statistic to capture the peakedness of a distribution -- that is, the extent to which ‘large’ and ‘small’ observations (in absolute value) appear relative to intermediate values – is its Kurtosis. There are several concerns with measuring Kurtosis. One is heterogeneity across types of goods, e.g. price changes in clothing and apparel are very volatile, while those in haircuts are not. This heterogeneity tends to create spurious Kurtosis. We correct for this in Figure 2 plotting a distribution of standardised prices changes; each price change underlying Figure 2 is demeaned and divided by the standard error in a type of good and type of outlet category. Another concern is that CPI data may contain measurement error, an issue emphasised by Eichenbaum et al. (2014). Our analysis also introduces a correction for measurement errors. Overall, after taking into consideration measurement error and cross-sectional heterogeneity, we estimate that the shape of the size-distribution of price changes is between a Normal (the green curve, with Kurtosis equal to 3) and a Laplace distribution (the red curve with Kurtosis equal to 6), with a Kurtosis that is about 4 for the US and about 5 for France.

**Figure 2**. Distribution of standardised price changes

*Source*: French CPI data.

## Modelling the output effect of a small monetary shock

Our recent work aims at constructing models that are able to reproduce such cross-sectional evidence. Key features of the theoretical models we build are:

- They embed several classic models as special cases; and
- They retain enough tractability to allow studying the output effects of a monetary shock analytically.

A main result of our analysis is a simple formula, which applies to a large class of models (such as those listed at the beginning) and is useful to approximate the total cumulative output effect of a small unexpected monetary shock. The formula states that the cumulative output effect of a monetary shock depends on the ratio between two steady-state statistics: The Kurtosis of the distribution of price changes Kur(∆p) and the average number of price changes per year N(∆p). Formally, given the labour supply elasticity 1/ε and a small monetary shock δ, the cumulative output effect M, namely the area under the output impulse response function, is given by

M = δ Kur(∆p) / ( 6 ε N(∆p) ) (1)

The impact of the frequency N(∆p) on the real output effect echoes a standard result in the ‘New Keynesian’ literature -- the stickier prices are, the more the inflation response is muted after a monetary policy shock, and the larger is the output response. It is precisely this result which motivated a large body of empirical literature on measuring the frequency of price changes.

The formula in (1) captures this in a stark fashion. Halving the frequency of price change doubles the output response.

The main novelty of the formula is that it shows that the effect of Kur(∆p) is equally important. The statistic on Kurtosis turns out to embody the extent to which ‘selection’ of price changes occurs in a model. The selection effect, a terminology introduced by Golosov and Lucas (2007), indicates that the size of price changes observed after a monetary shock is not taken at random in the menu cost model; the price changes immediately after a shock are also the largest, so that the CPI response is fast. Our model generalises the standard menu cost model and allows for ‘random’ menu costs. Increasing the randomness of menu costs weakens that selection effect, so both small and large price changes are observed after the shock. This results in a larger Kurtosis of price changes, along with a lower aggregate CPI response, and a larger output response. The Calvo set-up is an extreme case of our model. It features lots of very small and very large price changes (hence, Kurtosis is highest) and real effects six times larger than in the standard menu cost model, confirming the quantitative analysis of Golosov and Lucas (2007).

Surprisingly, the formula in (1) is also able to explain models that differ mainly in the timing (as opposed to the size) of price adjustments. For instance, the average size of price adjustments in the Calvo and Taylor models are identical, but the timing of actions is different. Calibrating these models to the same average frequency of price adjustment implies that the cumulative output effect in Calvo is twice the one in Taylor. The source of this difference is that in the Calvo model a fraction of firms waits for an arbitrarily long time before adjusting prices, while in Taylor no price will ever last more than its expected duration. Indeed, the distribution of the timing of adjustment differs in the two set-ups, even controlling for the mean duration of prices. This emphasises again the role of the specifics of micro-assumptions which, as we announced, are efficiently encoded in the cross -sectional Kurtosis of price changes.

Our empirical analysis of the datasets for the US and France suggests the value of Kurtosis, and hence the effect of a monetary shock, is between that predicted by a Calvo and a menu cost model, and overall close to that predicted by a Taylor model. If one were to select a simple model matching the distribution of price changes, the Taylor model would fare well.

## References

Alvarez, F, H Le Bihan, and F Lippi (2014), “Small and Large Price Changes and the Propagation of Monetary Shocks”, NBER Working Papers 20155.

Bils, M, and P J Klenow (2004), “Some Evidence on the Importance of Sticky Prices”, *Journal of Political Economy*, Vol. 112(5), 947-985.

Eichenbaum, M, N Jaimovich, S Rebelo, and J Smith (2014), “How Frequent Are Small Price Changes?” *American Economic Journal: Macroeconomics*, American Economic Association, Vol. 6(2), 137-55.

Golosov, M and R E Lucas J (2007), “Menu Costs and Phillips Curves”, *Journal of Political Economy*, Vol. 115, 171-199.

P J Klenow and O Kryvtsov (2008), “State-Dependent or Time-Dependent Pricing: Does It Matter for Recent U.S. Inflation?” *The Quarterly Journal of Economics*, Vol. 123(3), 863-904.