Bils and Kahn [Bils, M., and J. A. Kahn. 2000. “What Inventory Behavior Tells us about Business Cycles.” American Economic Review 90, 458–481.] conjectured that a competitive technology-driven business cycle model could not generate countercyclical inventory-sales ratios. Khan and Thomas [Khan, A., and J. Thomas. 2007a. “Explaining Inventories: A Business Cycle Assessment of the Stockout Avoidance and (s,s) Motives.” Macroeconomic Dynamics 11(5): 638–664. Khan, A., and J. K. Thomas. 2007b. “Inventories and the Business Cycle: An Equilibrium Analysis of (s,s) Policies.” American Economic Review 94(4): 1165–1188.] developed a model that disproved this conjecture. However, as this paper shows, that model underperforms a baseline model without inventories for many important moments. However, when variable utilization is added to the model, many of these moments perform better in the full model than in the baseline. The results suggest important interactions between variable utilization and inventory dynamics.
The purpose of this appendix is to provide additional details regarding the computational methods used in solving of the model.
After calibrating the model using the targets described in the paper, the model had to be moved into a stochastic world. In broad terms, solving the model involved an iteration of three steps:
Determining expectations functions
Approximating value functions based on those expectations
Simulating the model based on those value functions
These three steps repeated until the simulated time series for the capital stock converged. The capital stock was determined to have converged when the maximum absolute value difference between the capital stock series in one simulation and in the following simulation was less than a particular tolerance. Here, a tolerance of 10–4 was used. The following subsections provide additional detail.
Expectations functions were calculated using ordinary least squares, with a separate equation for each value of the technology shock z. The natural logarithm of each data series was used. The equation that was estimated was:
Where hats indicate natural logarithms, and “x” is replaced with the next period’s capital stock (K′), next period’s mean inventory holdings (), the marginal utility of consumption (p), or the relative price of intermediate goods (q). The details of the final estimation for each of these series is included below. In versions of the model without inventory holdings, the mean inventory term is dropped. Tables 4–7 show the values of the regression coefficients in each version of the model
Approximating value functions
This step is only required when inventories are held. In the models without inventories, nonconvexities drop out of the model, so approximating the value functions is unnecessary.
The only value functions that require an approximation are those for the final good producers, as they are the only set of agents with a serious nonconvexity. We approximate the value functions Va, E(V0) (which is equivalent to the last term in V1), and V1. Each of these was solved on a grid for the relevant state variables. The range for the grid values was chosen to encompass all values that were observed in simulations – which prevents the need for extrapolation when using the approximated value functions in simulations. In detail: There were six nodes for the aggregate capital stock K, ranging from 1 to 2.25. There were six nodes chosen for the mean inventory level m1, ranging from 0.25 to 0.75. There were nine nodes chosen for the productivity shock z, which were the same as the nine values in the Markov process. For the individual firm’s inventory holdings s, there were 50 nodes – ranging from 0 to 2.5. The number of nodes for s was significantly larger than for the other variables, as s is, in some stages of the decision making process, a choice variable for the final good firms, so finer detail was desirable. After solving on the nodes, quadratic splines were used to interpolate between the nodes. Cubic splines are a more common option, but created serious inaccuracies in the early iterations, as the cubic splines created false local maxima in the value functions. Using quadratic splines prevented that problem. Splines were interpolated using Miranda and Fackler’s CompEcon toolbox for Matlab.
The value functions were solved through iterations of the following loop.
Solve Va on the grid, using the current approximation of V1.
Approximate Va using splines.
Solve E(V0) on the grid, using the current appoximations of Va and V1.
Approximate E(V0) using splines.
Solve V1 on the grid, using the current approximation of E(V0) .
Approximate V1 using splines.
In this process, there were two points where levels of precision had to be chosen. First, the solution of each function required a maximization routine with the level of inventories being the choice variable. Golden Search was used for this, with convergence being declared when the values were within 10–11 units. (When labor choice was relevant, that was simply calculated using the first order condition.) The second point where a level of precision had to be chosen is deciding when convergence of the loop has been reached. For that purpose, the values of V1 and E(V0) could not have changed by more than 10–5 at any point on the grid. Once that point had been reached, the loop stopped after step six.
Simulating the model
The model was simulated using the following method:
A series of productivity shocks was simulated as a Markov chain.
The appropriate productivity shock was read from the series, and the marginal utility of consumption p was calculated using equation 12.
The equilibrium intermediate goods price q was found by bisection (with a precision of 10–8), using the intermediate goods firms’ first order conditions and final goods firms’ approximated value functions to determine the quantities of intermediate goods supplied and demanded.
The other variables were determined by solving the first order conditions and various constraints.
Steps 2–4 were repeated for 10,050 periods. The first 50 were discarded as an initialization period.
Correlation between detrended inventories and GDP are 0.65 for manufacturing inventories, which is roughly twice the correlation for retail and wholesale inventories.
This is the most significant difference between the model presented in this paper and that in Trupkin (2008). In Trupkin (2008), inventories are of final goods. In this paper, as in Khan and Thomas (2007b), inventories are held because of a fixed cost of delivery of intermediate goods to final goods producers. In Trupkin (2008), inventories provide direct utility to households, as a proxy for decreased shopping time and increased variety.
The approximation is formed using the technique described in Tauchen (1987).
a is the quantity of those Arrow securities that paid off in the current period.
It is theoretically possible for firms to want to sell off some of their inventories. For example, if they adjust their inventory level in one period, and then the optimal level of inventories falls faster than the amount of intermediate goods used in production. In that case, s is above s* – and the further above, the more likely the firm will adjust its holding of inventories. The main text here describes the typical case where actual inventory levels fall faster than the optimum level does.
These restrictions are present in Christiano, Eichenbaum, and Evans (2005) which finds δ(κ) to be nearly linear, and also in Altig et al. (2011) which finds δ to be approximately cubic. The estimate of Justiniano and Primiceri (2008) suggests a much more convex function, with this coefficient in the neighborhood of 7 or 8.
The more general form as used in Trupkin (2008), which includes both wear-and-tear and rust-and-dust depreciation was considered, but calibrating such a form resulted in a degree of convexity that was not plausible, as it was far beyond the levels found in Christiano, Eichenbaum, and Evans (2005) or Altig et al. (2011). The chosen form has a precedent in Smith (1970).
I used 2501 nodes spread evenly between 0 and 2.5, which span the range that inventories take in simulations.
The term s′–s can be broken down into the components of intermediate good purchases and use of intermediate goods in production. s1–s gives the purchase of new intermediate goods. s1–s′ gives the goods that get used in production. The change in inventories then would subtract those used in production from those purchased: (s1–s)–(s1–s′), which simplifies to (s′–s) , which is then weighted by the price of intermediate goods to get the change in value of those goods.
Because Table 3 reports standard deviations in relative terms, it appears that the relative price of intermediate goods q has gotten less volatile. However, this is not entirely accurate. The relative price of intermediate goods has gotten less volatile compared to the volatility of output.
As δ1 goes to infinity, this ratio goes toward the ratio for a fixed cost. That is because δ1=∞ is equivalent to the marginal cost of changing utilization being infinite – so that utilization is no longer variable.
This point was noted by Khan and Thomas (2007b).
Wang, Wen, and Xu (2011) suggests that when variable utilization is added to a model in the style of Khan and Thomas (2007b) that inventories play a significant destabilizing role during business cycles. Khan and Thomas (2007b) also make this point about the version of their model where the capital share is decreased.
Thank you to an anonymous referee for recommending this possibility.
The author gratefully acknowledges the advice and encouragement of Paul Evans, Aubhik Khan, Julia Thomas, Pok-Sang Lam, John Leahy, and members of The Ohio State University’s Macroeconomics lunch workshops. Any remaining errors are the sole responsibility of the author.
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