Inventory investment and aggregate fluctuations with idiosyncratic shocks

Transcription

1 Inventory investment and aggregate fluctuations with idiosyncratic shocks Aubhik Khan The Ohio State University Julia K. Thomas The Ohio State University and NBER February 14, 21 ABSTRACT We develop a DSGE model to explain salient empirical regularities regarding fluctuations in aggregate inventory accumulation over the business cycle while simultaneously explaining a distinct set of observations on the behavior of this series at higher frequencies. We begin with a generalized (S,s) model of inventory investment wherein firms fixed costs of ordering inputs that lead them to economize on the frequency of their orders by accumulating inventories. To this environment, we add idiosyncratic shocks to firms total factor productivities. We discipline order costs using average aggregate inventory data, and firm productivity differences using observed sales and output growth volatility in firm-level data. As only a fraction of firms place orders in any period, our model yields a nontrivial distribution of firms over productivity and inventory levels. When a firm chooses not to place an order, its existing inventory determines the level of inputs available for its production. This prompts firms to raise their average stocks in times of persistently high aggregate productivity. A second set of factors influencing firms sales and inventory decisions are the risks associated with the potential for high fixed order costs in future, alongside the possibility of sharp changes in relative productivity. We find that these idiosyncratic risks can alter firms decisions sufficiently to yield important changes in aggregate dynamics. Our micro-founded inventory model succeeds in reproducing important empirical regularities involving inventory investment and the business cycle. In the model, as in the data, inventory investment is procyclical and positively correlated with final sales, and GDP is more variable than sales. Furthermore the inventory-to-sales ratio is countercyclical. Across a variety of cases, we find that the model explains between 4 and 6 percent of the measured volatility in inventory investment. While these cyclical successes are qualitatively robust across cases, we show that changes in micro-level parameters affecting the shape of the order cost distribution and the degree of firm-level productivity risk imply quantitative changes to this set of aggregate results. We show that an aggregate shock to total factor productivity increases total production and final sales,as well as both the number of firms placing orders and the average size of their orders. Because the gradual accumulation of capital slows the rise in intermediate goods supply, however, there is a trade-off between firms increased inventory accumulation and their increased production of final goods. Each activity comes at some expense to the other, so the rise in final sales is dampened by, and dampens, the rise in inventory investment. In this sense, the frictions that induce inventories also

2 offset much of the volatility in GDP that might otherwise arise from procyclical fluctuations in inventory investment. An important insight of our analysis is that changes in the persistence and variability of idiosyncratic order costs and productivities alter the distribution of firms over inventory levels. This, in turn, affects the extent and speed of firms responses to aggregate shocks, and thus the model s ability to reproduce high-frequency aspects of the aggregate data. When firms have greater certainty about their order costs, and when shifts in their relative productivities are transitory, they adjust their average inventory holdings faster following an aggregate shock. In such cases, the model succeeds not only with respect to the business cycle facts mentioned above, but also in reproducing two essential high-frequency observations, the negative correlation between sales and inventory investment and the greater volatility in sales relative to production. KEYWORDS: Inventories, (S,s) policies, business cycles, dynamic stochastic general equilibrium.

3 1 Introduction The advance estimate of annualized real GDP growth for 29Q4 was 5.7 percent, with nonfarm inventory investment contributing 3.61 percentage points of that growth. To understand whether the quarter s sharp inventory accumulation promises rapid recovery from the last recession over coming quarters, or the converse, a quantitative dynamic stochastic general equilibrium business cycle model capable of reproducing the salient empirical regularities involving inventories is required. One immediate difficulty lies in the fact that no single model has thus far proved successful with respect to the cyclical facts regarding inventory investment while simultaneously generating appropriate high-frequency behavior in this series. Because inventory investment exhibits sizeable fluctuations within both frequency ranges, we see this as an important omission. The business cycle relationships between inventory investment, GDP, and final sales stand in sharp contrast to their high frequency counterparts. Over the business cycle, inventory investment is procyclical and co-moves with final sales, so that final sales is less cyclically volatile than GDP. As shown in Khan and Thomas (27), these relationships can be reproduced in a business cycle model wherein inventories derive from an (S,s) motive. However, recent empirical work by Wen (25) documents precisely the opposite regularities in the high frequencies. There, inventory investment is negatively correlated with final sales, and the volatility of sales exceeds that of total production. Each of those observations is easily reproduced by a model wherein inventories derive from a production-smoothing motive, precisely because that model fails with respect to the business cycle facts involving inventories. In this paper, we develop a DSGE model capable of reproducing both sets of empirical regularities regarding the movements in aggregate production, sales and inventory investment. Despite its reliance on only a single (S,s) inventory motive, our model is consistent with the prevailing view that, at high frequencies, inventories act as a buffer against shocks to sales, while it is also consistent with the business cycle data wherein inventory investment rises and falls alongside sales. Beyond these aggregate successes, our model also generates empirically valid variability in sales and output growth at the level of the firm. Production occurs across two sets of firms in our model. The first set combines intermediate goods with labor to produce a final good used for consumption and investment in capital. The second set supplies the first with intermediate goods, producing these using capital and labor which are, in turn, supplied by households. Inventories arise in our economy from the assumption 1

4 that final goods firms face nonconvex costs associated with shipments of intermediate goods. These costs lead firms to place infrequent orders with their suppliers and maintain stocks of intermediate goods on hand. Firms follow generalized (S,s) policies with respect to their inventories; they place orders when their existing stock falls below a lower threshold, and their order level is determined by an upper threshold. So long as a firm s existing stock remains between these thresholds, it avoids paying order costs; hence its beginning-of-period inventory constrains the quantity of intermediate goods available for use in current production. As such, when a firm does order intermediate goods, it orders in sufficient quantity to carry it through several periods production. If the firm places an order in a time when aggregate productivity is unusually high, expecting the boom to persist, it increases the quantity it orders beyond what is necessary to cover the rise in its current production (sales), because it recognizes that continued high demand for its output in coming periods will exhaust its inventory more rapidly than usual. It is in large part for this reason that our model succeeds in generating the salient empirical regularities characterizing the movements of GDP, final sales and inventory investment over the business cycle. In particular, the model explains the procyclicality of inventory investment, its positive correlation with final sales, and thus the higher variability of GDP relative to final sales. Moreover, depending upon the specification of microlevel shocks affecting firms inventory and production decisions, the model reproduces between 4 and 6 percent of the observed cyclical variability of net inventory investment. Our current environment builds upon the (S,s) inventory model of Khan and Thomas (27). 1 To that setting, we introduce idiosyncratic shocks to firm-level productivity, as well as variation in firm-level uncertainty regarding order costs. These generalizations allow us to address the high frequency fluctuations in aggregate production, sales and inventory investment, as well as firm-level observations on the volatility of output and sales. We find that high variability in order costs implies considerable variance in the length of an inventory episode, the number of periods elapsing between one order of intermediate goods and the next. Over such episodes, firms inventories decline monotonically, while, because they draw upon these stocks for production, their sales also tend to fall. For this reason, high variance in the length of inventory episodes tends 1 For a survey of related literature, see the discussion in that paper. In addition, Wen (29) studies a DSGE model with both input and output inventories, based on a stockout avoidance motive. Kryvtsov and Midrigan (29) embed both (S,s) and stockout avoidance motives in a model with monetary policy shocks to measure the extent of real rigidities. 2

5 to imply high variability in sales and output across firms. Thus, absent idiosyncratic shocks to firms productivities, and given order costs distributed uniformly between and an upper bound dictated by the average aggregate inventory-to-sales ratio in postwar U.S. data, the model predicts excessive variation in sales and output growth relative to that observed in the firm-level data. We correct this inconsistency between model and data by reducing the uncertainty in order costs. More specifically, we find that the model can deliver empirically plausible micro-level variability when firms order costs come from a right-skewed distribution with substantial probability mass associated with relatively high costs. This change reduces the variance of inventory episodes and, in turn, reshapes the typical distribution of inventories across firms. Aside from changes in the uncertainty assumed in order costs, the introduction of shocks to firm-level total factor productivity also changes the qualitative nature of inventory adjustment. This, in turn, has nontrivial implications for the high-frequency dynamics of our model. Absent idiosyncratic shocks, firms inventory policies are effectively one-sided (S,s) rules. By this we mean that, under ordinary aggregate conditions, firms stocks never exceed the threshold level they select in times of active inventory adjustment. However, when we introduce idiosyncratic productivity shocks, each firm s shock history since the time of its last order determines its history of sales over the episode, and thus its current inventory holding. Some firms find themselves with excess inventories but, given nonconvex shipment costs, choose not to adjust their stocks. In this environment, a positive aggregate shock implies an increase in the expected real interest rate, and a fall in the relative price of intermediate goods. Both reduce the value of inventories and increase the returns to current production. Firms holding excess inventories find it worthwhile to increase production, and are able to do so without ordering. In the aggregate, the result is a more rapid response in the production of final goods, and a somewhat muted response in net inventory investment. By contrast, when we add to this setting greater certainty regarding future order costs, those firms that respond to an aggregate shock by ordering intermediate goods order far more than they otherwise would and retain a large fraction of their raised stock, because they place high probability weight on a lengthy delay before their next order. This tends to produce very sharp adjustments in the aggregate stock of inventories that weaken the comovement between sales and inventory investment, particularly in the high frequencies. Our model s ability to address the business cycle facts is robust to alterations in the specification of fixed order costs and firm-specific productivity shocks. While there are quantitative 3

6 changes, inventory investment remains procyclical and positively correlated with sales, and production remains more volatile than sales. However, at higher frequencies, aggregate dynamics are qualitatively altered with changes in the degree of firm-level risk associated with future order costs and the potential for sharp changes in relative productivity. Under specifications with less uncertainty in order costs and less persistence in firm-level productivity changes, we find that the model succeeds in reproducing the negative correlation between inventory investment and sales at high frequencies, as well as the greater volatility in sales relative to production. Moreover, as noted above, the very specifications that succeed in this regard are also the ones that are empirically plausible in light of the firm-level data. 2 Model Production in the economy occurs across two sets of firms. Final goods, which may be consumed or used for investment in capital, are produced by a unit measure of firms that operate a decreasing returns to scale technology using labour and intermediate goods. Intermediate goods are produced by a representative firm that uses a constant returns to scale technology using capital and labour. Final goods firms face fixed costs of receiving a delivery of the intermediate input, and, in response, tend to hold inventories of the good. All firms are perfectly competitive, in both factor and output markets. Final goods firms are subject to idiosyncratic shocks to their total factor productivity, ε, which follows a Markov Chain. Each period, ε ª ε 1,...,ε N(ε) with Pr {ε = ε j ε = ε i } = π ε ij, j =1,...,N(ε), and P N(ε) j=1 πε ij =1for i =1,...,N(ε). Lets denote the stock of the intermediate good held by a final good firm at the start of a period. This firm s state is (ε, s). The intermediate good firm is subject to a shock to its total factor productivity, z, which follows a Markov Chain. Each period, z ª z 1,...,z N(z) with Pr {z = z j z = z i } = π z ij, j =1,...,N(z), and P N(z) j=1 πz ij =1for i =1,...,N(z). Let k denote the capital stock of the firm at the start of a period, the firm s state is (z, k). The aggregate state of the economy will include a measure of final goods firms, μ, overε s, the capital stock, k, and total factor productivity of intermediate goods, z. We develop the model as a recursive competitive equilibrium, and all households and firms take the evolution of the aggregate state, z,k, μ, as given. Further, they assume that the endogenous component evolves 4

7 according to k = Γ k z,k, μ (1) μ = Γ μ z,k, μ. (2) These laws of motion are, in equilibrium, determined by individuals actions. 2.1 Final good firms Beginning with the firms that produce final goods, at the start of any period any such firm may choose to pay a fixed cost and order intermediate goods. After any possible delivery, the firm decide what level, m, to use in current production, where, defining s as the stock of intermediate goods held by the firm at the time of production, m s. Givenm, and employment n, production will be y = εf (m, n). Lete denote the mid-period, production time value of a firm with (ε, s), e ε, s; z, k, μ ³ = max εf (m, n) wn σs (3) m,n,s N(z) X + s=1 N(ε) X d s subject to j=1 π ε ijv ³ε j,s ; z s, k,μ s s m, and (1) - (2). Inventories held at the end of the period are s,andσ is a storage cost per unit inventory. The value of the firm, as a function of s and future productivity, is v. Furthermore, the real wage, w, is a function of the aggregate state, as are d n, n =1,...,N(z). These stochastic discount factors are contingent on the future value of z, and, in equilibrium, will equal the price of Arrow securities for the representative household. At the start of the period, firms must decide whether or not to adjust inventories. We assume that inventory adjustment involves the payment of a fixed labour cost, ξ. While these delivery costs are independent of the size of an order, and in this sense fixed, they may vary across firms and over time. Let ξ [ξ l,ξ u ] be drawn from a distribution, H (ξ). If H is degenerate, then there is no uncertainty in inventory adjustment costs. Given its productivity and its beginning of period inventory, as well as the aggregate state, a final goods firm chooses whether or not to pay the delivery cost and adjust its stock of intermediate goods. If it does adjust its stock before 5

8 production, it will choose the size of its order. Let q denote the relative price of intermediate goods, which is a function of the aggregate state. The order decision solves e εi ; z,k, μ =max s e ε i,s ; z, k, μ q (s s). (4) Notice that as gross and net adjustment are equivalent, we allow the firm to sell its existing stock at this time, s, which allows the description of a gross value of adjustment that is independent of existing inventories. Fixed delivery costs, and a constant unit price, implies that the level of inventories an adjusting firm will take into production, s, is independent of it s beginning of period level, s. Should the firm choose to not pay the current delivery organization cost, then its expected value is e (ε i,s; z,μ,k). The choice of whether or not to adjust current inventories, v ξ,εi,s; z, k, μ =max ξw + e εi ; z,k, μ,e ε i,s; z,k, μ ª, (5) determines the firm s value at the start of the period, after its current organization cost draw, ξ, v (ξ,ε i,s, ). Having described the inventory adjustment decision, we are now able to define the value function in the right-hand side of (3) using (5), v ε, s; z, k, μ = Z ξr ξ l v ξ,εi,s; z, k, μ H (dξ). (6) Let s ξ,ε,s; z,k, μ describe the decision rule for a firm with current adjustment cost, ξ, idiosyncratic shock ε and beginning of period inventories s. This production time stock of intermediate goods will equal the beginning of period level, s, ifthefirm does not pay ξ and adjust its stock. Let χ ξ,ε,s; z, k, μ =1if the firm pays ξ, otherwise. Further, let n ξ,ε,s; z, k, μ and m ξ,ε,s; z,k, μ denote the decisions for current employment and the level of the intermediate good used in production, with y ξ,ε,s; z,k, μ = εf m ξ,ε,s; z,k, μ,n ξ,ε,s; z,k, μ. (7) The end of period inventory of the firm is s = s ξ,ε,s; z, k, μ m ξ,ε,s; z, k, μ. 2.2 Intermediate good firm As mentioned above, a representative firm produces intermediate goods using capital and labour. Given our focus on inventory investment, we assume that this firm is able to frictionlessly adjust both factors of production subject to capital being predetermined. chooses n and k to solve Given k, the firm 6

9 w z, k; k, μ ³ = max qzk α n 1 α wn k (1 δ) k N(z) X ³ + d j w z j,k ; k,μ (8) n,k subject to (1) and (2). j=1 Aggregate shocks to total factor productivity, z, directly affect the intermediate good producer and only indirectly affect final good firms through movements in relative prices. This ensures that the relative price of inventories is countercyclical, which is a feature of the data (see Khan and Thomas (27)). Let n 1 z,k; k, μ describe the level of employment chosen by the intermediate producer, and k 1 z, k; k, μ the choice of capital for next period. Next, define production as x 1 z, k; k, μ = zk1 z,k; k, μ α n1 z,k; k, μ 1 α. (9) 2.3 Households Households own all firms in the economy, and, as they are identical, we do not model a market where shares in firms are traded. Instead we assume that a representative household owns all firms through a mutual fund, and this asset yields aggregate dividends which are the sum of all profits generated by final and intermediate good firms. Given their constant returns to scale production function, the latter earn no profits. However, we will assume decreasing returns to scale in F, which will lead to profits earned by final good firms. These profits allow them to pay their fixed costs, and remaining profits are redistributed to the household. Let a represent the household s beginning of period stock of Arrow securities. The household s problem solves h a, z, k, μ ³ = max u (c, 1 n)+β (a j) N(z) j=1 subject to N(z) X c + d j z,k, μ a j a + wn + p z,k, μ j=1 c, a j a (1) and (2). N(z) X j=1 ³ π ij h a j; z j, k,μ (1) Where c is current consumption and n is hours of work, p z, k, μ represents the aggregate dividend and a is a borrowing limit. Let c a, z, k, μ definethedecisionruleforcurrentcon- 7

10 sumption, a j a, z, k, μ the level of contingent claims purchased for z = z j, j =1,...,N(z), and n a, z, k, μ total hours worked. 2.4 Equilibrium In equilibrium, households maximize utility and firms of both types maximize profits. All profits, which are net of inventory adjustment and storage costs, are paid to the household in the form of dividends; the markets for final goods, intermediate goods and labour clear. As there is a representative household, the equilibrium supply of contingent claims, a, is. Hence consumption and hours worked are functions only of the aggregate state. This leads to an aggregate resource constraint for final goods, c (,z,k,μ)+(k 1 (z, k, k,μ) (1 δ) k) = Z ξu ξ l Z S {ε 1,...,ε N(ε) } y (ξ,ε,s; z, k, μ) μ (d [s ε]).(11) In the above, we have also imposed the equilibrium condition that the capital stock of the intermediate good producer is equal to the aggregate capital stock, k = k. The market-clearing condition for intermediate goods is Z ξu Z x 1 (z, k, k, μ) = ξ l S {ε 1,...,ε N(ε) } [s (ξ,ε,s; z,k,μ) s] μ (d [s ε]), (12) where s ξ,ε,s; z,k, μ 6= s if a firm of type (ξ,ε,s) adjusts it s stock of intermediate goods before production. The labor market clears when n,z,k, μ Z ξu Z = n 1 (z, k, k, μ) ξ l S {ε 1,...,ε N(ε) } [n (ξ,ε,s; z, k, μ)+χ (ξ, ε, s; z, k,μ) ξ] μ (d [s ε]), (13) where, recall χ ξ,ε,s; z,k, μ =1if a firm of type (ξ,ε,s) pays ξ and receives a delivery of the intermediate good. Given the inventory and production decisions of final good firms, total profits are p (z, k,μ) = Z ξu ξ l Z S {ε 1,...,ε N(ε) } µ y (ξ,ε,s; z,k,μ) (14) w (z, k, μ)[n (ξ,ε,s; z, k, μ)+χ (ξ,ε,s; z, k,μ) ξ] q (z,k,μ)[s (ξ,ε,s; z,k,μ) s] σ [s (ξ,ε,s; z,k,μ) m (ξ,ε,s; z,k,μ)] μ (d [s ε]). 8

11 The evolution of the distribution of final good firms, as described by Γ μ in (2), is given by μ S E = X Z π ij ε j E {(ξ,s) s (ξ,ε i,s; ) m (ξ,ε i,s; ) S } μ (ds, ε i ) H (dξ), (15) for all S E measurable. The evolution of the aggregate capital stock, Γ k in (1) is given by k = k 1 (z, k; k, μ). (16) A recursive competitive equilibrium is (i) a value function v ε, s; z, k, μ that solves the final good firm problem described in (3) - (6) with associated decision rules (s ξ,ε,s; z, k, μ,n ξ,ε,s; z,k, μ,m ξ,ε,s; z,k, μ ), (ii) a value function w z, k; k, μ that solves the intermediate good firm problem (8) and associated decision rules (n 1 z,k; k, μ, k1 z,k; k, μ ), (iii) a value function h a, z, k, μ that solves the household problem (1) with associated decision rules (c a, z, k, μ, n a, z, k, μ, N(z) a j a, z, k, μ j=1 )and(iv) relative prices (w z, k, μ,q z,k, μ, N(z) d j z, k, μ ) such that (a) (11), (12) and (13) hold, (b) j=1 a j,z,k, μ =for j =1,...,N(z), (c) p z,k, μ satisfies (14), (d) y (ξ,ε,s; z,k,μ) is given by (7), (e) x 1 (z, k; k, μ) is given by (9) and (f) Γ μ and Γ k are defined by (15) and (16). 3 Firm-level decisions We illustrate several properties of the model. The first is that all firms that place an order, and share the same idiosyncratic shock value, will have the same end-of period holdings. other words, the beginning of period stock, s, is irrelevant for production, s,andendofperiod inventories, s, for all firms that pay ξ and either buy or sell intermediate gods. We now derive this property. First, it s useful to define the conditional expectation of discounted future value, as a function of current productivity ε i and z r, v εi,s ; z r,μ,k Next, using (4) and (3), we have N(z) X s=1 N(ε) X d s j=1 π ε ijv ³ε j,s ; z s, k,μ. (17) In 9

12 ³ e (ε i ; z m,μ,k) = max εf s s,n wn qs σs s,n,s +v εi,s ; z m,μ,k subject to s s. The first-order conditions for s and n are εd 1 F s s,n = q (18) εd 2 F s s,n = w. Together, they imply that the sum s s = f (ε, q, w) where the relative price of the intermediate goods, and the real wage, are of course functions of the aggregate state. As m = s s,this shows that in periods when the firm decides to adjust its inventory, it s choice of intermediate good for current production is determined purely by its current relative price. There are no dynamic considerations. Moreover, the form of inventory adjustment costs implies that when a delivery is received the past stock s has no influence on production or the end of period stock. In this sense individual firms may be viewed as experiencing inventory episodes, with each such episode beginning with an adjustment and ending with the next adjustment. The usefulness of this view arises from the result that firm s decision rules in each inventory episode are independent of past levels. The first-order condition for s, once we use (18), this simplifies to ³ εd 1 F s s,n + σ + D 2 v εi,s ; z m,μ,k = ³ q + σ + D 2 v εi,s ; z m,μ,k = (19) Thus the problem of s solves (19) and we describe the solution as g ε i ; z,k, μ. The history independence of inventory levels, whenever there is an order, leads to the target stock adopted by adjusting firms depending upon only ε as well as aggregate state variables. Once g ε i ; z,k, μ is 1

13 known, (18) immediately yields s = g εi ; z, k, μ, the actual production level of inventories for a firm that pays it s order cost. Our next results shows that the decision to order intermediate goods is determined by a threshold order cost that varies as a function of firms productivity and inventories. Consider the set of all firms with common ε and s. Eachwillpayξ [ξ l,ξ u ] iff ξw + e εi ; z, k, μ e ε i,s; z, k, μ. This implies a threshold adjustment cost level, ξ ε, s; z,k, μ, such that the production time level of inventory holdings is g εi ; z, k, μ if ξ ξ ε, s; z,k, μ s = s otherwise. The fraction of firms of type (ε, s)that order intermediate goods is given by H ξ ε, s; z, k, μ. Notice that this fraction evolves over time as a function of the aggregate state of the economy. Our last result involve intermediate good firms. As they are perfectly competitive, and operate a constant returns to scale technology, these firms earn no profits. This makes them indifferent to any level of future capital k.wefirst derive this familiar result in the context of our model. We begin by showing that profits are linear in capital. Given capital, the optimal choice of employment is solves max qzk α n 1 α wn. n This leads to the decision rule n = (1 α) q w z 1 α k and a supply of intermediate goods x = z (1 α) q 1 α w z α k. Current profits are µ 1 α (1 α) α 1 α (qz) α k. w The linearity of the profit function implies a value function that has the same property. Once we use the optimal choice of employment to simplify (8), the first-order condition for k is 1= N(z) X s=1 µ (1 α) d s µα w s 1 α α (qs z s ) 1 α +1 δ. As long as this condition holds, the intermediate good firm is indifferent to the choice of k. 11

14 We now turn to the household s problem, in order to derive the consumption and firms discount factors implied by equilibrium in asset markets. First, notice that, in equilibrium, consumption and hours worked are functions of the aggregate state, and the real wage is given as w z,k, μ = D 2u c,z,k, μ, 1 n,z,k, μ D 1 u c,z,k, μ, 1 n,z,k, μ. In addition, the equilibrium price of an Arrow security for state z j is d j z,k, μ = π z ij D 1 u (c (,z j, Γ k (z i,k,μ), Γ μ (z i,k,μ)), 1 n (,z j, Γ k (z i,k,μ), Γ μ (z i,k,μ))) D 1 u c,z,k, μ, 1 n,z,k, μ. 4 Numerical solution The aggregate state of the model includes a distribution of firms over idiosyncratic productivity levels and inventories, as well as capital and total factor productivity. Furthermore, firm-level decisions involve discrete choices. Both the size of the aggregate state and final good firms decisions imply that a standard linear solution method infeasible. We solve the model using the method of Krusell and Smith (1999), assuming that all equilibrium prices are functions of m, a vector of moments drawn from μ. Weuse(z,k,m) as a proxy for the aggregate state, and replace Γ μ with a law of motion for Γ m. The solution of the model iterates on these laws of motion, let Γ i k, Γi m represent the i th iteration. Given these forecasting rules, we reformulate the problem of final good firms by weighting current profits by p (z,k,m), which in equilibrium is the household s marginal utility of consumption, and discounting future returns by β. This allows us to eliminate Arrow securities from the analysis, but does not affect firm s decision rules (see Khan and Thomas (28) for details). The solution algorithm iterates between a first step, where we solve the analogue to (3) - (6) and (8) given equilibrium price functions p i (z, k,m) and q i (z,k,m), and a second step where given these value functions, and the forecasting rules Γ i k, Γi m, we simulate the economy. 2 The simulation step determines equilibrium prices p and q as a function of the actual aggregate state (z, k, μ). Given p, q is determined to clear the market for intermediate goods. The equilibrium 2 A separate price function for the real wage follows immediately from p i (z, k, m) as we assume a representative household whose period utility function has constant marginal utility of leisure. This is the familar result of an environment where households face indivisible labor supply decisions and have access to employment lotteries (Rogerson (1988), Hansen (1985)). 12

15 level of p is that which implies a level of consumption that leads to a (k,m ) such that the firstorder condition for capital, derived above, holds. The moments m are derived from the actual distribution, μ, which, over the simulation, is stored using a two-dimensional grid. We use a weighted grid approximation method which, alongside multivariate spline approximation of firms value function, does not constrain individual decisions to grid values. Using simulation data, new laws of motion Γ i+1 k, Γ i+1 m are derived. 5 Calibration We evaluate the plant-level and aggregate implications of inventory investment using several numerical examples across which we vary the stochastic process for idiosyncratic shocks to plants total factor productivity and the distribution of order costs. All other production parameters, as well as preferences, are held constant throughout. Below, we discuss functional forms and parameter values for technology and preferences that are identical across models. Thereafter, in section 5.2, we explain the choice of idiosyncratic shocks and the distribution of capital adjustment costs. 5.1 Common parameters Across our model economies, we assume that the representative household s period utility is the result of indivisible labor (Hansen (1985), Rogerson (1988)): u(c, 1 n) = logc + η (1 n), and the final good firm production function takes a Cobb-Douglas form, zεf(m, n) =zεm θ m n θ n. Model parameters, other than those involving idiosyncratic productivity shocks and order costs, are selected to ensure agreement with observed long-run values for key postwar U.S. aggregates in a nested control version of our model without order costs. We assume that the length of a period is one-quarter, and set the discount factor, β, to imply an average annual real interest rate of 4 percent. The depreciation rate, δ, is selected to match an average investment-to-capital ratio of 1 percent, corresponding to the average value for the private capital stock between 1954 and 22 in the U.S. Fixed Asset Tables. This implies an annual depreciation rate of.69. Intermediate good share, θ m =.4991 (see Khan and Thomas (27)), and α and θ n are jointly chosen to imply a labor s share of.6 and an average aggregate capital-to-output ratio of as in the data. Next, the parameter governing the preference for leisure, ϕ, is taken to imply an average of one-third of available time spent in market work. 13

16 In specifying our exogenous stochastic process for aggregate productivity, we begin by assuming a continuous shock following a mean zero AR(1) process in logs: log z = ρ z log z + η z with η z N ³,σηz 2. Next, we estimate the values of ρ z and σ ηz from Solow residuals measured using NIPA data on US real GDP and private capital, together with the total employment hours series constructed by Prescott, Ueberfeldt, and Cociuba (25) from CPS household survey data, over the years Finally, we discretize the resulting productivity process using a grid with 3 shock realizations; N z = Firm-level productivity shocks and order costs The remaining parameters involve the distribution of firm-specific productivity and the fixed costs incurred with shipments of intermediate goods. To illustrate the effect of idiosyncratic shocks and uncertainty in order costs, we consider several examples. Across these, our goal is to match the aggregate inventory-to-sales ratio and to examine each model s ability to reproduce firm-level measures of the standard deviation of output and sales growth. The average real quarterly nonfarm inventory-to-sales ratio in the U.S. over the period 1954Q1-29Q3 is.85. In each of our examples, we select an upper bound on the distribution of order costs, ξ u,to reproduce this level of inventories. Comin and Phillipon (25) use COMPUSTAT data to calculate the median standard deviation of 1-year centered moving averages of output growth, and find that this measure has been rising over time. At the beginning of their sample, in 1955 the standard deviation was approximately.1; by 2, it was above.2. Bloom uses the COMPUSTAT over the period and reports an average standard deviation of annual sales growth of.165. We consider two distributions of order costs. The first is uniform between and.295. The second is a Beta distribution with parameters a =2and b = 1 2, with realizations concentrated near an upper bound of.15. This second distribution sharply reduces uncertainty in order costs. There is little agreement about the persistence of the idiosyncratic shock process, ρ ε.(compare, for example, the values in Comin and Phillipon (25) to those of Cooper and Haltiwanger (26).) In most of our exercises, we set the persistence of the idiosyncratic shock at and the standard deviation at.2. An example without firm-level shocks to total factor productivity serves as a baseline, with which we contrast our remaining cases. Our leading example adopts the Beta cost distribution alongside ρ ε =and σ ε =.2. This case implies annual sales and output 14

17 growth volatility at the firm-level close to the measures reported by Bloom (29) and Comin and Phillipon (25), at.17 and.185, respectively. Finally, we also consider a Beta distribution example with ρ ε =.5, which implies slightly higher firm-level volatility. We implement firm-level productivity shocks using a Markov Chain (ε i ) N ε i=1 ³π and ε ij i,j=1 with N ε =5to discretize the log-normal process, log ε = ρ ε log ε + η. The parameters (ρ ε,σ η ) are always chosen to imply σ ε =.2. As explained in Khan and Thomas (28), equilibrium movements in relative prices imply that the elasticity of response of firms to idiosyncratic shocks is roughly 12 times greater than that to aggregate shocks. Thus, while the standard deviation of idiosyncratic shocks is only 4 percent that of aggregate shocks in our examples, idiosyncratic shocks have a much larger impact on the production and inventory decisions of individual firms. Nε 6 Results After discussing our control model without inventories below, we study two versions of our model with uniformly distributed order costs. Both predict too much volatility in firm-level sales and output growth. The first omits productivity differences across firms; it exhibits a.33 standard deviation of annual sales growth and a.29 counterpart to the Comin and Phillipon measure of output volatility. Corresponding moments from the second case, which includes i.i.d. firm productivity shocks, are 1.78 and.52, respectively. Next, we consider the leading case of our model wherein order costs are drawn from the Beta distribution and firms face i.i.d. productivity shocks. As noted above, this case delivers a standard deviation of annual sales growth at.17 and an output volatility measure at.185. Ourfinal case retains the Beta distribution, but allows for persistence in firm productivity shocks; there, the corresponding moments rise to.2 and Business cycle dynamics Table 2 displays key features of the dynamics of HP-filtered inventory investment, final sales and GDP in the postwar U.S. alongside the corresponding moments from the baseline inventory model with uniformly distributed fixed order costs and no idiosyncratic differences in firm-level productivity. Here, we see that net inventory investment is strongly procyclical in the model, as in the data, while the model explains roughly 59 percent of the empirical relative volatility of this series. Next, given its success in generating a positive correlation between final sales and inventory investment (.49), the model also reproduces the fact that sales is less volatile than GDP over 15

18 the business cycle. Finally, given that the aggregate inventory stock responds to shocks quite gradually relative to the flow of final sales, the model also succeeds in generating a countercyclical inventory-to-sales ratio, though this is over-stated relative to the data control versus baseline inventory model We begin a more detailed examination of dynamic results with a survey of the business cycle moments arising under various specifications of our model differing in either idiosyncratic delivery costs, or idiosyncratic productivity shocks, or both. Row 1 of Table 3 focuses on the model we have termed the control. Recall that this model is identified by its absence of fixed delivery costs, and thus the lack of any inventory motive. Because this setting has no nonconvexity, its aggregate dynamics are impervious to the presence of idiosyncratic productivity shocks (see Khan and Thomas (28)). Thus, a single row suffices to outline the control model business cycle. The control model bears several familiar features of a real business cycle. First, given household preferences for smooth consumption profiles, the volatility of consumption is exceeded by that of final sales (a series identical to GDP in this model), which is, in turn, exceeded by that of capital investment. Investment is by far the most volatile series, both because it facilitates household consumption smoothing and because the persistence of aggregate shocks implies that an unanticipated rise (fall) in the marginal products of labor and capital arising from a shock leads households to anticipate a protracted rise (fall) in the returns to investment, causing strong substitution effects on savings. Moreover, the second mechanism encouraging procyclical investment changes compounds the substitution effect of the procyclical wage on labor supply. Thus, at the impact of a positive aggregate productivity shock, consumption, investment and hours worked rise alongside GDP, explaining the strong contemporaneous correlations with GDP in panel B of the table. Finally, because there is no friction in the flow of intermediate goods to final goods producers in this model, there is effectively no separate intermediate goods sector. As such, the dynamics of the labor input in final versus intermediate good production are indistinguishable. Next, we consider the dynamics of a baseline inventory model analogous to that examined in Khan and Thomas (27). In this model, the fixed delivery costs causing inventories are distributed uniformly on the interval from to ξ u (with ξ u set to yield an average annual inventorysales ratio matching that in the postwar U.S.), and there are no idiosyncratic productivity shocks. Aside from the obvious distinction arising between final sales and GDP, the basic features of the 16

19 business cycle in the control model discussed above remain largely unchanged. One difference deserves comment. Despite its positive correlation between inventory investment and final sales (.49), the baseline inventory model actually exhibits lower cyclical volatility in GDP than does the model without inventories. As was pointed out in Khan and Thomas (27), the frictions that cause inventories need not necessarily amplify business cycles, because procyclical inventory accumulation causes a dampening of fluctuations in final goods production. This happens due to atrade-off between the use of intermediate goods in production (raising final sales) versus their storage in inventory, as we explain below. Figures 1-3 complement this discussion; these figures present impulse responses from our baseline inventory economy following a persistent 1-standard deviation rise in aggregate productivity. 3 When a positive aggregate productivity shock hits the inventory economy, the marginal cost of intermediate goods production is reduced relative to normal, which in turn reduces the relative price of intermediate goods. In response, an increased fraction of final goods firms place orders, as seen in the top panel of Figure 3. The resulting rise in total demand for intermediates is compounded by the fact that those final goods firms that do place orders also order more than theywouldordinarily,whichraisesthestockthey have available at production time (Figure 3, middle panel). However, only part of their extra stock goes into current production. Because these firms anticipate that the raised demand for their goods used for consumption and investment will persist, they expect that any given stock of intermediate goods will be exhausted more quickly than usual. To avoid more frequent payments of fixed delivery costs, these firms raise their end-ofperiod inventories above normal levels (Figure 3, bottom panel). Indeed, they do so in sufficient quantity as to more than offset the raised use of intermediate goods among non-ordering firms that are constrained by their existing stocks. Thus, in the aggregate, total orders of intermediate goods rises by more than total use (Figure 1, middle panel), so that inventory investment is procyclical (its contemporaneous correlation with GDP, at.61, is near that in the data,.67). Given any fixed stock of intermediate goods, there must be a trade-off between their use and storage. The more responsive is the production of these goods to a positive productivity shock, the weaker will be this trade-off. However, whenever the predetermined capital input 3 This tradeoff is more prominent in our current baseline model relative to the analogous model in Khan and Thomas (27), because the capital share in intermediate goods production is larger here (.51 versus.37). For this reason, the dampening of final sales volatility outweighs the volatility arising from the inventory investment series, despite a quite similar contemporaneous correlation between final sales and inventory investment. 17

20 has an empirically plausible share in production (as it does here), the fall in the marginal cost of intermediate goods production following a positive productivity shock does not happen at once. Instead, this fall happens gradually as the capital stock is increased following the shock, so that the initial rise in total supply of intermediate goods is gradualized. This has several related implications. First, despite a strong rise in the demand for intermediate goods, any rise in inventory investment comes at the expense of rises in the production of final goods, as is reflected in the final columns of Table 3 and in the top and bottom panels of Figure 1. While a considerably greater rise in the demand for intermediate goods leads to a much sharper rise in employment and production in that sector relative to the control model, employment in final goods production rises by less. In other words, the use of intermediate goods rises by less in the inventory model. This implies a dampened rise in final sales, which reduces both the rise in consumption and in capital investment, explaining the reduced volatility in both series relative to the control model. Moreover, that dampened response in capital investment protracts the episode over which the rise in intermediate goods production is insufficient to prevent a tradeoff between final goods production and inventory accumulation, because it gradualizes the rise in the capital stock (Figure 2, top panel). Throughout this entire episode, the same trade-off that reduces volatility in final sales also limits the volatility of inventory investment. Thus, the rise in aggregate inventory holdings (Figure 2, middle panel) does not happen all at once, but instead takes roughly 1 periods. That, in turn implies that the ratio of the aggregate stock of inventories relative to the flow of final sales, which naturally drops sharply at the impact of the shock, recovers very slowly thereafter. On balance, the fluctuations in final sales are sufficiently reduced by the frictions causing inventories in our model as to more than offset the procyclical movements in inventory investment, despite the comovement of sales and inventory investment effects of firm-level productivity differences In the third row of Table 3, we next consider an inventory model that is otherwise identical to the baseline case, but where final goods firms experience temporary idiosyncratic shocks to their productivities. In this setting, the same basic trade-off between the use and accumulation of intermediate continues to apply, and we see similar features in the resulting second moments as before. The main difference here lies in a reduced relative weight placed on increased inventory accumulation following a positive aggregate productivity shock. Figures 4-6 revisit the 18

21 one-standard deviation positive shock considered above, this time in the model with firm-level productivity differences, and serve as useful reference as we discuss the differences in Table 3. In contrast the baseline inventory model, final goods firmsinthismodeldiffer in their optimal production-time stocks, given differences in ε. Following a positive aggregate shock, Figure 6 reveals a slightly larger rise in the fraction of firms undertaking orders in its top panel (in comparison with Figure 3). In the middle panel, we see larger rises in the production-time stocks among ordering firms currently experiencing high relative productivity, while those firms with relatively low productivities that undertake orders choose significantly smaller rises in their production-time stocks. As there is no persistence in ε, however, all ordering firms have the same expectations regarding the future value of stocks, and hence exit the period with a common inventory holding. The rise in end-of-period inventories among ordering firms in response to the aggregate shock is reduced relative to the model with common productivity, because each firm faces the possibility of a low productivity draw in a nearby coming period. This partly explains the reduced procyclical volatility in inventory investment and the raised variability in final sales in the rows of Table 3 corresponding to this model. Consider next the trade-off between accumulation and use of intermediate goods among nonordering firms. In an ordinary period, a firm transiting from a relatively high productivity to a relatively low one finds itself holding an excessive inventory of intermediate goods relative to the target inventory-sales ratio adopted by an ordering firm with the same current productivity. Rather than actively shedding stock by boosting production to a level inconsistent with its current productivity (or paying a fixed cost to send the excess back to its supplier), however, the firm will generally carry the extra inventory forward to the next period. This decision is prompted by the possibility that the firm s relative productivity may rise in the next period, alongside the possibility that it might in the same period face a high fixed cost of undertaking an order for more intermediate goods. Such inventory decisions change in a time of raised aggregate productivity. When a positive aggregate shock hits the economy, the value of any given stock is pushed downward by the fall in the relative price of intermediate goods, while the procyclical real interest rate further discourages the storage of these goods. Viewed another way, firms with relatively low individual productivities enjoy higher total factor productivity than would ordinarily be the case, as well as an unusually high demand for their goods; thus, they are encouraged to use up in current production that portion of their stocks they would ordinarily carry forward to 19

22 await higher relative productivity. On balance, while procyclical rises in orders of intermediate goods among ordering firms continue to outweigh both their increased use of these goods in production and that among the remaining firms in good times, that difference is reduced in the model with idiosyncratic productivities (Figure 4, middle panel), yielding a lesser rise in the aggregate stock and hence a larger drop in the aggregate inventory-sales ratio, particularly in the early periods following the aggregate shock (Figure 5, middle and bottom panels). However, because procyclical movements in inventory investment are dampened almost entirely through greater responsiveness in overall use of intermediate goods, with little change in that of intermediate goods production, the bottom panel of Figure 4 reveals that there are greater rises in both final sales and GDP following the positive aggregate shock relative to the baseline inventory model (Figure 1). Returning to Table 3, this explains why, despite reduced cyclical volatility in inventory investment, the current model has sufficient volatility in final goods production (final sales) as to raise the volatility of total production (GDP) effects of more certainty regarding order costs In rows 4 and 5 of Table 3, we consider alternative inventory models wherein fixed order costs are drawn from a right-skewed beta distribution. As with the models above, the upper support on the distribution is selected to reproduce the average annual inventory-sales ratio in the data. However, unlike the uniform distribution common to those models, this distribution implies more certainty about the timing of firms orders, because it concentrates more probability mass at the right-tail in the range of relatively high order costs. In this case, the average delivery rate (fraction of firms placing orders) drops from roughly 23 percent to roughly 21 percent. Beginning with the rows corresponding to model 4, we see that the shift to less random order costs (from model 3) raises the cyclical volatility not only in final sales, but also inventory investment, and it spreads the volatility in production more evenly between the intermediate and final goods sectors. While not shown here, the rise in the number of firms placing orders in response to a positive aggregate shock (roughly.3 at impact) is far smaller than it was in model 3(.8 percent in Figure 6), as far fewer firms draw relatively low order costs. However, those firms that do place orders raise their production-time stock by a considerably higher fraction (impact-date rises range between 5.5 and 6.5 percent, rather than 3 4 percent). Moreover, given 2