Price Setting with Menu Cost for Multi-Product Firms
Social Science Research Network
We model the decisions of a multi-product firm that faces a fixed "menu" cost: once it is paid, the firm can adjust the price of all its products. We characterize analytically the steady state firm's decisions in terms of the structural parameters: the variability of the flexible prices, the curvature of the profit function, the size of the menu cost, and the number of products sold. We provide expressions for the steady state frequency of adjustment, the hazard rate of price adjustments, and
... adjustments, and the size distribution of price changes, all in terms of the structural parameters. We study analytically the impulse response of aggregate prices and output to a monetary shock. The size of the output response and its duration increase with the number of products, they more than double as the number of products goes from 1 to ten, quickly converging to the ones of Taylor's staggered price model. JEL Classification Numbers: E3, E5 Economics Meeting in NY for their comments. Alvarez thanks the ECB for the Wim Duisenberg fellowship. We are grateful to Katka Borovickova for her excellent assistance. an elongated S shape, with a finite asymptote. Comparing across values of n, while keeping the expected number of adjustment constant, we show that the asymptote of the hazard rate is increasing in n. As n increases, an adjustment becomes more likely later on. As n → ∞ the hazard rate function converges to an inverted L shape, i.e. to the one of models with deterministic adjustments as in Taylor's (1980) or in Reis's (2006) models. 4 Third, we characterize the shape of the distribution of price changes. While price changes occur simultaneously for the n products, we characterize the marginal distribution of prices, i.e. the statistic that is usually computed in actual data sets. A closed form expression for the density of the marginal distribution of price changes as a function ofȳ and n is given. Using this density we compute several statistics that have been computed in the data, such as the standard deviation of price changes Std(∆p), and other moments which are only functions of n, such as the coefficient of variation and the excess kurtosis of the absolute value of price changes. We show that, as the number of products increases, the size of the adjustments decreases monotonically, i.e. with more products the typical adjustment is smaller in each product. These cross-section predictions can be used to identify the parameters of the model and test its implications. When compared to the tabulations for US data by Bhattarai and Schoenle (2010) we find matching patterns: higher values of n imply higher dispersion, smaller average price changes, and higher kurtosis. We show that once the size of price changes is controlled for, the shape of the sizedistribution is exclusively a function of the number of products n. For n = 2 the distribution is bimodal, with modes at the absolute value of √ȳ , for n = 3 it is uniform, for n = 4 it peaks at zero and it is concave, and for larger n it is bell-shaped. As n → ∞, the density of price changes converges to a Normal. We find the sensitivity of the shape of price changes with respect to n an interesting result to identify different models of price adjustments. In particular bimodality is only predicted for n = 1 or n = 2, as in the models of Golosov and Lucas (2007) and Midrigan (2011) respectively. 5 Furthermore, this helps to discriminate 4 Midrigan's (2011) comments that "Economies of scope flatten the adjustment hazard and thus weaken the strength of the selection effect even further " (pp. 1167). Figure 1 shows that, without fat tailed shocks, the hazard rate steepens with n, and indeed the economy converges to Taylors' staggered adjustment model, not Calvo's random adjustment model. 5 Midrigan's (2011) writes that "economies of scope generate many price changes that are very small" (pp. 1167) referring to his model with n = 2. While a model with n = 2 does generate some small price changes, we show that this is actually a minimal amount, as the density of the price changes is U shaped, with a minimum at 0 and the modes are the extreme values. We also show that, in order to get a bell shaped density for price changes, it is required that n ≥ 4. 7 Midrigan's (2011) interpretation of the selection effect in his model with n = 2 (Section 4.B, pages 1165-1168) uses the techniques developed by Caballero and Engel for the case of n = 1. This is incorrect for the multi-product case (n > 1) because the threshold condition for price adjustments involves a vector of price gaps, not just one. We discuss the appropriate extension in Section 5. American Mathematical Society :72-74.