موجودی و انعطاف ناپذیری واقعی در مدل های جدید چرخه کسب و کار کینزی

Kryvtsov and Midrigan (2008) study the behavior of inventories in an economy with menu costs, fixed ordering costs and the possibility of stockouts. This paper extends their analysis to a richer setting that is capable of more closely accounting for the dynamics of the US business cycle. We find that the original conclusion survives in this setting: namely, the model requires an elasticity of real marginal cost to output approximately equal to the inverse intertemporal elasticity of substitution in consumption in order to account for the countercyclicality of the aggregate inventory-to-sales ratio in the data.

Real rigidities are factors that dampen the responsiveness of a firm’s desired price to a monetary disturbance. Recent work with New Keynesian sticky price models1 has argued that real rigidites are a key ingredient necessary to reconcile the apparently slow response of prices to nominal disturbances at the aggregate level2 with the fairly rapid rate at which individual price setters update their nominal prices.3 Models with real rigidities can be broadly categorized into two classes.4 The first class of models is characterized by assumptions on preferences or technology that make it costly for firms to charge prices that are too different from those of their competitors. Those firms that choose to reset their nominal prices in time of a monetary disturbance thus choose to not fully respond to this disturbance in order to avoid the losses associated with deviating from their competitor’s prices.5 Thus even though prices change frequently in nominal terms, they initially respond little to the monetary injection because of the pricing complementarity arising from non-constant demand elasticities and/or upward sloping marginal cost at the individual producer’s level. Although measuring price elasticities or scale returns in the production function is difficult in practice, recent work using micro-price data has argued that simple versions of models that feature this first class of real rigidities are difficult to reconcile with the observed dispersion in relative prices in very narrowly defined product groups within outlets.6 In this paper, we focus on a second class of real rigidities that lower the elasticity of economy-wide real marginal cost to output. In this second class of models, assumptions on preferences, the degree to which factor utilization can vary, or frictions in the labor market or in the market for intermediate inputs generate slow adjustment of (nominal) factor prices to a monetary shock. As a result, real marginal costs of production respond little to a monetary disturbance, thus amplifying real effect of the shock.7 Notice that this second class of real rigidities is, in effect, a set of assumptions on aggregate quantities, and in particular, on the firms’ (collective) ability to hire additional labor during booms (or hoard labor during recessions), purchase intermediate inputs, and vary capital’s work-week. Even when real rigidities take the form of sticky wages or intermediate good’s prices, as in much of the recent work, an important assumption made is that these sticky prices are allocative and quantities are demand-determined. These assumptions, that quantities can be relatively costlessly varied during the cycle, and that factor adjustment costs are small, are clearly other key ingredients that are necessary to lower the elasticity of economy-wide marginal cost of production to output. The discussion above suggests that inferring the elasticity of real marginal cost to output, a measure of the strength of real rigidities in this second class of models, is difficult in practice. In particular, the researcher must be able to measure the relative importance of factor adjustment costs, the degree to which factor prices are allocative, the cost of varying the work-week of capital and labor, as well as the degree of frictions in the labor and intermediate goods market. Bils and Kahn (2000) show that the behavior of inventories over the cycle is informative about the cyclicality of costs. In Kryvtsov and Midrigan (2008) we use Bils and Kahn’s insights to gauge the implications of models of the second class of real rigidities for the behavior of inventories. If the marginal cost of acquiring and holding inventories is indeed lower in times of monetary expansions, we should see this lower cost reflected not only in a slow adjustment of prices to a monetary shock, but also in an increase in the firm’s inventory holdings. In fact, models with inventories predict that a firm’s price is proportional to its shadow valuation of its inventories. In turn, when the firm’s cost of buying and holding inventories decreases (as it does in times of a monetary expansion), the firm purchases more inventories so as to equalize its shadow valuation of inventories to their marginal cost. Thus real rigidities of this second class must operate through inventories: an increase in the stock of inventories held by the firm is necessary for the shadow valuation of inventories (given concavity of the value function) to decrease and thus for the firm’s real price (relative to the money stock) to fall. If the firm is unable to purchase more inventories, either because of the quantity restrictions by suppliers, or because of other costs of adjusting the stock of inventories, the relatively lower factor prices do not translate into a lower shadow valuation of inventories, and the firm finds it optimal to keep its real price high. We thus argue that a model’s ability to account for the behavior of inventories in the data (and in particular the strong countercyclicality of the inventory-to-sales ratio) is an important empirical test of this class of models. Our earlier paper studies a menu cost sticky price model in which firms hold inventories of goods from one period to another in order to: (a) avoid stockouts given demand uncertainty and a delay between orders and deliveries and (b) economize on fixed ordering costs. The model is sufficiently rich in that it incorporates these two most widely studied inventory motives in recent work, and yet very parsimonious in that only two additional parameters are added to otherwise standard menu cost model (the size of the fixed ordering costs and the volatility of demand shocks). These parameters are calibrated to match several micro-economic moments in the inventory data. We then study the model’s responses to monetary disturbances under a number of assumptions regarding the strength of real rigidities (modeled there as a wedge in the consumer’s labor-leisure tradeoff). In that paper we find that even small departures of the elasticity of real marginal cost to consumption away from the inverse of the intertemporal elasticity of substitution (IES) give rise to large differences in the inventory-to-sales ratio’s simulated response to a monetary shock. When the elasticity of real marginal cost to consumption is equal to the inverse IES, the cost of purchasing and holding inventories (which depends on both the wholesale price and the real interest rate) does not change with a monetary shock and the firm does not substitute intertemporally. In contrast, when the elasticity of real marginal cost to consumption is lower than the inverse IES, the combined cost of acquiring and holding inventories decreases and the firm finds it optimal to raise its inventory stock by a large amount. We thus conclude that in that simple setup it is difficult to reconcile strong real rigidities (low elasticities of real marginal cost to ouput) with the behavior of inventories in the data, unless one also assumes a high elasticity of intertemporal substitution. In this paper we extend the analysis in Kryvtsov and Midrigan (2008) along several dimensions in order to gauge the robustness of those results. In particular, highly non-linear nature of firm decision rules in our earlier paper precluded us from embedding our micro-economic model of inventories into a full-blown medium-scale equilibrium model of the type studied in Christiano et al. (2005) or Smets and Wouters (2007). One objection to our earlier analysis is that the simplicity of our original setup precludes it from accounting for the dynamic responses of output, interest rates, costs and inflation to the monetary disturbance. Given that the dynamic paths of interest rates, inflation and costs are crucial for firm’s optimal inventory holdings, the concern is that our earlier results are in part driven by our model’s simplicity and inability to match the dynamics of key macroeconomic variables in the data. A second concern that arises is that our results stem from the non-linear firm policy rules in our earlier setup with fixed price and inventory adjustment costs. Finally, we are able to investigate whether perturbations of the model (allowing for higher depreciation rates and adding adjustment costs on factors of production) help alleviate the counterfactual implications of models with inventories and real rigidities. To address these concerns, we study in this paper a Calvo sticky price model in which firms adjust nominal prices with a constant hazard and in which demand uncertainty and a lag between orders and deliveries of goods give rise to a stockout-avoidance motive that makes it optimal for firms to carry inventories across periods. To build intuition for the mechanism at work, we first start with a simple cash-in-advance version of the model that is very similar to that in our earlier paper. We show that our earlier results are robust in this alternative setup. In particular, if the elasticity of the real marginal cost to consumption is much lower than the inverse IES, the model predicts a sharp increase in the inventory-sales ratio during monetary expansions. Similarly, if the elasticity of the real marginal cost to consumption is much greater than the inverse IES, the model predicts a sharp decrease in the inventory/sales ratio during monetary expansions. Given that one component of output is inventory investment, this large spike in the inventory-sales ratio in models with strong real rigidities implies a counterfactually large spike in output (almost ten times larger than the increase in consumption after the monetary expansion). Further more, allowing higher rates of depreciation (8% per month) does resolve this counterfactual implication of the model but now provides counterfactual micro-economic implications. In particular, with such high depreciation rates, firms hold the 1.4 monthly inventory-to-sales ratio observed in the US data only if faced with considerable demand uncertainty (237% standard deviation of shocks when the elasticity of substitution is equal to 5). Similarly, adding convex adjustment costs does help bring the response of inventories to a monetary shock in line with the data; however, they do so by raising firm’s costs of replenishing its stock of inventories. In this case, relatively low factor prices during booms do not feed into lower retail prices as the firm’s effective marginal cost of purchasing more inventories increases because of the adjustment costs. Finally, we ask in this simple setup: what are the consequences of allowing wedges in a consumer’s savings–consumption and consumption–leisure decision that allow the model to exactly match the impulse responses of wages and real interest rates to a monetary policy shock. We find that absent additional frictions on the firm’s ability to purchase inventories, the inventory-sales ratio increases strongly during a boom, as in the simpler versions of the model without wedges. In a final set of exercises we embed the stockout-avoidance inventory holding motive into a medium-scale macroeconomic model with a richer set of shocks and frictions of the type studied in Smets and Wouters (2007). We find that our original results extend to this setting. In particular, a version of the model in which we borrow all parameter estimates from Smets and Wouters (2007) but allow for inventories, predicts a counterfactually large initial increase in output and inventory-to-sales ratios as well as hours worked. Only by allowing for adjustment costs on inventory investment which increase the firm’s shadow valuation of inventories and thus neutralize the effect of real rigidities can the model account simultaneously for the behavior of inventories and factor prices in the data. However in this case, as in the simple Calvo model, the response of the aggregate price level after a monetary shock is faster than in the Smets and Wouters’ model, implying smaller real effects of monetary policy. Our work is related to a number of recent papers studying the behavior of inventories, costs and markups over the business cycle. Our starting point is the observation by Bils and Kahn that inventories are closely linked to markups and marginal costs and thus may provide important information about the cyclicality of the latter. Khan and Thomas, 2007 and Wen, 2008 study real business cycle models in which inventories arise due to fixed ordering costs, and a stockout-avoidance motive, respectively. Both of these papers find that the model is capable of accounting for the countercyclicality of the inventory-sales ratio in the data. Our conjecture is that they do so because investment in capital in times of expansions in these models drives up the cost of purchasing (through a higher elasticity of real marginal cost to output) and holding (through higher interest rates) inventories. Most closely related to our analysis is a paper by Jung and Yun (2005) who also study a sticky price model in which firms invest in inventories because these act as a taste shifter in consumer’s preferences. They estimate their model by matching impulse responses of aggregate time-series to a monetary shock and find that high rates of depreciation and/or convex costs of deviations from a target inventory-to-sales ratio is necessary to reconcile the model with the data. Finally, Chang et al. (2009) study the responses to a productivity shock in a sticky price models with inventories. They find that whether an industry expands or contracts employment depends on the carrying costs of inventories: higher carrying costs prevent firms from responding to the productivity shock by investing in inventories and as a result cut employment given that prices are sticky and quantities demand-determined. This paper proceeds as follows. In Section 2 we briefly review the evidence of the cyclical properties of the inventory-sales ratio and the response of this ratio to identified monetary policy shocks, thus reproducing the facts discussed in Bils and Kahn, 2000 and Jung and Yun, 2005 and in our earlier paper. To build the intuition for our results, in Section 3 we present a simple model of Calvo sticky prices and inventories. Section 4 studies this model’s quantitative implications. Section 5 embeds inventories into a richer Smets and Wouters (2007)-type framework and studies its implications. Section 6 concludes by suggesting several potential resolutions to the challenge of accounting for the behavior of inventories in a model with real rigidities: financing frictions that disconnect fluctuations in the real interest implied by the consumer’s pricing kernel from the rate of interest faced by the inventory-carrying firms; additional sources of countercyclical markups; additional frictions that reduce firms’ ability to purchase and carry inventories and hence the sensitivity of inventories to costs.

In this paper we have considered a number of extensions to our analysis in Kryvtsov and Midrigan (2008) in which we study the behavior of inventories in a model with sticky prices and real rigidities. We find that, consistent with our earlier results and the results in Jung and Yun (2005), inventories are highly sensitive to cyclical fluctuations in the cost of acquiring and holding them. This is true both in a simple Calvo model with a stockout-avoidance motive for holding inventories, as well as in a richer Smets and Wouters (2007)-type model. Thus even small amounts of real rigidities predict a counterfactually high increase in the inventory-to-sales ratio in response to monetary expansions. In contrast, in the data this ratio persistently declines in times of booms. We have shown that adding adjustment costs on output (or more generally factors of production) allows the model to simultaneously match the behavior of real wages and other time-series in the data. This modification implies, however, that a firm’s shadow valuation of inventories increases sharply during booms despite the sluggishness of factor prices. As a result, the implications for inflation behavior in the model that is capable of accounting for the observed behavior of inventories, resemble those in a model with little real rigidities. We conclude that standard models of inventories pose a challenge for New Keynesian sticky price models in which real rigidities take the form of slow responsiveness of real marginal cost to output (e.g. sticky wages or intermediate good’s prices, variable factor utilization, etc.). Potential resolutions to this challenge include: (a) allowing for financing frictions that disconnect fluctuations in the real interest implied by the consumer’s pricing kernel from the rate of interest faced by inventory-carrying firms, (b) allowing additional sources of countercyclical markups (other than nominal price rigidities) that would decrease the benefits of carrying inventories during booms and (c) additional frictions on the firms’ ability to purchase and carry inventories (e.g., non-linear rates of depreciation, capacity constraints) that reduce the sensitivity of inventories to costs. Exploring these alternatives is an interesting avenue for future research.