A flexible time-to-build model of supply chain disruptions

Abstract

1. Introduction

Globalization, technological advance and the increased complexity of goods have lengthened supply chains and extended their geographical reach.¹ This development has brought large benefits from lower costs of production and improved living standards in many countries. At the same time, events over recent decades have exposed the global economy’s heightened vulnerability to disruptions at any point along the chain. The 2020 Covid pandemic is only the most dramatic in a long list that includes Hurricane Katrina in 2005, the Fukushima earthquake in 2011 and, most recently, disruptions in shipping through the Red Sea and Suez Canal due to military conflict in the region, and through the Panama Canal as a result of drought.

This paper employs a flexible “time-to-build” technology to model the impact of supply chain disruptions on production, inventories and ultimately on final sales of investment and consumption goods. Data from the onset of the Covid-19 pandemic in 2020 suggest that the model can capture key features of that experience, and that, by extension, the model should prove useful for analysis of other disruptions, as well as for fluctuations in economic activity more generally.

The Covid pandemic, and even more so the public policy and private responses to it, had economic and social impacts far beyond the simple disruptions to the supply chain that we consider. There were significant interruptions in the availability of a range of consumer products and services: travel, live entertainment, restaurants and even medical services, for example. At the same time, there were large shifts towards spending in other categories, particularly in the direction of goods that could be delivered to and consumed in the home, such as food, entertainment and a range of consumer products. International trade was disrupted, and there were unprecedented fiscal and monetary policy responses.

Notwithstanding its complexity, the pandemic represents a natural experiment regarding supply chain disruptions. The paper focuses narrowly on production, shipments and orders within manufacturing sectors that are associated with the production of capital goods. The results of this analysis illustrate the potential of the paper’s approach to modelling such disruptions as they arise in other contexts such as strikes or natural disasters.

Mathematical models can enhance our understanding of the nature of these risks and how best to manage them. While the ultimate goal of managerial mathematics is to provide useful tools for decision-making, a logical step in developing a mathematical model is first to test its ability to describe and explain the data. The model draws on a larger literature that has focused primarily on business cycle fluctuations, in which the approach has provided useful insights. Such cross-validation gives extra credence to the model’s success in accounting for data in the entirely different context of pandemic disruptions. In the conclusions we discuss potential extensions and applications of the model to aid in risk management.

1.1 Related literature

There is a large operations research literature on supply chain disruptions that goes back decades before the Covid-19 pandemic, which we do not attempt to survey exhaustively. See Katsaliaki et al. (2022) for a partial list. Much of this is a reaction to the development of so-called “lean inventory” policies in the 1980s and 1990s that subsequently proved vulnerable to disruptions, whether from natural disasters, military conflicts, strikes or disruptions in international trade and transportation. Much of this literature is normative, analysing the efficacy of alternatives for the ex ante mitigation of supply chain risk, or the ex post management of disruptions once they occur.

The Covid-19 pandemic and widespread lockdowns inspired new research on the economic impact of both the disease itself and of the preventive measures taken, also too large to survey here. Much of this work relies on modelling the spread of the disease and the efficacy of countermeasures, often with the goal of obtaining optimal policies. Examples of this literature include Eichenbaum et al. (2021), Alvarez et al. (2021), Farboodi et al. (2021) and Vasilakis & Nikolopoulos (2024).

There is also a substantial related literature on time-to-build, dating back to Kydland & Prescott (1982). Our model draws in particular on Sarte et al. (2015). Papers focusing on investment lags include Bar-Ilan & Strange (1996), Lucca (2007) and Kahn & Maccini (2024). Also relevant to this paper is an older literature on the role of inventories, backorders and delivery lags in the response of economic activity to economic disturbances. This includes Maccini (1973) and Carlton (1983) on delivery lags as an equilibrium phenomenon, Reagan & Sheehan (1985) on inventories by stage of fabrication and Kahn (1987) on the bullwhip effect in production with finished goods inventories. West (1989) and Nalewaik & Pinto (2015) examine the role of orders, shipments and backlogs in business cycle fluctuations.

2. A flexible time-to-build model

We consider an economy with an upstream sector (indicated with a superscript |$u$|⁠) that produces capital goods, and a downstream sector (⁠|$d$|⁠) that produces a homogeneous consumption good. The production of capital goods involves |$S+1$| stages of production, where each stage takes one period and involves the production of a specific input that gets combined sequentially with the subsequent |$S$| inputs to yield the investment good. Specifically, investment in sector |$j$|  |$\left ( j=u,d\right ) $| at date |$t$|⁠, denoted |$ i_{t}^{j}$|⁠, uses inputs chosen at dates |$t-S$| through |$t$|⁠. Let |$ z_{t,t-s}^{j}$| denote the quantity of the input to |$i_{t}^{j},$| chosen at date |$t-s$|⁠, for |$s=0,...,S.$| Following Sarte et al. (2015), we use a flexible time-to-build (FTTB) technology:

where |$\sigma ,$|  |$\alpha _{s}\geq 0$| and |$\sum _{s}\alpha _{s}=1$|⁠. Note that the first subscript on |$z$| is always the period at which the final investment good containing it is produced, and the second indicates the period in which the input |$z$| is produced. Stages are counted in reverse order, the first stage denoted by |$S,$| and the final stage in which |$ i_{t}^{j}$| is assembled is denoted by |$s=0$|⁠.

Flexibility comes from the fact that although the inputs to |$i_{t+S}$| chosen from |$t$| to |$t+S-1$| are fixed as of the time of their production (that is, they become state variables), there is still opportunity to revise future investment as new information comes in. Specifically, if new information arrives at |$t+1$|⁠, any of the remaining inputs for investment at |$t+1$| or later are still flexible, at least to a degree that depends on the elasticity parameter |$\sigma $|⁠. In the extreme cases (1) represents either a fixed-coefficient (⁠|$\sigma =0$|⁠) or completely flexible (additive) technology, so that timing does not matter (⁠|$\sigma \rightarrow \infty $|⁠). For intermediate values of |$\sigma $| the inputs are chosen optimally at each point in time, taking as given the inputs already chosen at previous stages, as a smooth function of prices and other fundamentals. A non-zero elasticity of substitution between the inputs represents flexibility to alter the mix of inputs as new information arises, so that the quantity of investment at a given date is not predetermined |$S$| periods ahead.² Because only the final period input |$z_{t,t}$| is chosen contemporaneously with |$i_{t}$|⁠, the prior inputs represent state variables of the system once they are chosen.³

Production of the |$S+1$| inputs by upstream firms uses technology

Here |$z_{t+s,t}$| is the production of the input chosen at time |$t$| that goes into the production process for |$i_{t+s}^{u}$| and |$i_{t+s}^{d}$|⁠, |$ k_{t}^{s,u}$| is the beginning of the period capital stock for upstream firms, and |$n_{t}^{s,u}$| is the labour employed by upstream firms at stage |$s$|⁠.

Production of the consumption good |$c_{t}$| by downstream firms is

where |$c_{t}$| is consumption, |$k_{t}^{d}$| is the initial capital stock for downstream firms and |$n_{t}^{d}$| is labour for downstream firms. Unlike the investment good, which involves production over S+1 periods, the consumption good is produced entirely within one period.

We solve a planner’s problem to obtain the competitive equilibrium. An advantage of a general equilibrium approach is that it can capture the broader economic constraints on the economy’s dynamic response to a large disturbance. For example, an individual firm might perceive itself to be able to hire as much labour as needed to reduce order backlogs after a temporary production disruption, but with a widespread disruption as considered here, however, it would encounter higher wages from a less than perfectly elastic labour supply. We highlight manufacturing as a sector in which, as opposed to services, production is not amenable to remote work. Within manufacturing, we focus on capital goods both because their production is more subject to the lengthy supply chain, and they would have been less likely to be designated as “essential” in the short term.

Regarding investment, the planner, given an inherited capital stock from the previous period, |$k_{t}^{j},$| for |$j=u,d,$| chooses |$i_{t}^{j},$| to add to the capital stocks for in each of the |$S+2$| sectors: |$i_{t}^{d}$| for the consumption good section, and |$i_{t}^{u}$| to be allocated across the |$S+1$| upstream subsectors that produce the inputs to capital goods. We allow for adjustment costs that slow the accumulation of capital. The capital accumulation equations are

where |$\delta $| is the depreciation rate on capital and |$\varPsi \left ( i_{t}^{j},k_{t}^{j}\right ) $| represents investment in sector |$j$| net of adjustment costs. The |$\varPsi $| function is assumed to be homogeneous of degree one with |$\varPsi _{i}\leq 0$| and |$\varPsi _{k}\leq 0$|⁠, so that we can express capital accumulation in sector |$j$| as

where |$\psi \left ( i/k\right ) k\equiv \varPsi \left ( i,k\right ) $|⁠. The adjustment costs involve an amount |$i^{j}-\psi \left ( i^{j}/k^{j}\right ) k^{j}$| of capital that “disappears” in the process of investment. The functional form of the adjustment costs is standard in the literature on investment (see, for example, Jermann (1998) and Boldrin et al. (2001), among others):

Here |$\eta $| is a curvature parameter indicating the severity of adjustment costs. As |$\eta \rightarrow \infty $|⁠, we get the case of no adjustment costs. |$C$| and |$D$| are chosen so that both the level and marginal adjustment costs are zero in the steady state.

Let |$\mathbf{Z}_{t}^{u}=$|  |$\left \{ z_{t,t}^{u},z_{t+1,t}^{u},...,z_{t+S,t}^{u}\right \} $| and |$\mathbf{Z} _{t}^{d}=$|  |$\left \{ z_{t,t}^{d},z_{t+1,t}^{d},...,z_{t+S,t}^{d}\right \}.$| These are the inputs chosen at date |$t$| that go into each of the stages of production for |$i_{t}^{j},i_{t+1}^{j},...,i_{t+S}^{j}$|  |$\left ( j=u,d\right ) . $| The planner chooses |$~c_{t},~k_{t+1}^{u},~k_{t+1}^{d},~i_{t}^{u},$|  |$ ~i_{t}^{d},~\left \{ k_{t}^{s,u}\right \},\left \{ n_{t}^{s,u}\right \} ,~n_{t}^{d},~n_{t}^{u},~n_{t},~\mathbf{Z}_{t}^{u}$| and |$\ \mathbf{Z} _{t}^{d},$| to maximize

subject to (1)–(4), along with the constraints on sectoral inputs:

and given |$k_{t}^{d},k_{t}^{u}$| and inputs from past choices of |$ \mathbf{Z}^{u}$| and |$\mathbf{Z}^{d}$| already in the pipeline.

We provide the first-order conditions for the maximization problem in the Appendix. We set |$U\left ( c_{t}\right ) =\ln c_{t}$|⁠. Parameters are either based on estimates in Kahn & Maccini (2024) or calibrated to match moments in the data. A key parameter is |$S$|⁠, the number of stages of the time-to-build technology prior to production of the final capital good. The September 2020 report on manufacturing of the Institute for Supply Management (September (2020) Manufacturing ISM Report on Business (2020)) shows an “average commitment lead time for Capital Expenditures” of 135 days, with a fairly flat distribution over periods ranging from “Hand-to-mouth” (25 percent) all the way to “1 year +” (18 percent). The “average lead time for Production Materials” was 64 days, with approximately 90 percent of the distribution below 6 months. So planning time and delivery lags are important. But we know that time-to-build is also relevant for many types of complex capital equipment, as well as for structures (20–25 percent of overall business fixed investment). Output for structures, however, is measured in the data by value added each time period (“construction put in place”), so we focus on equipment in our analysis. It is also worth noting that durable goods manufacturing is primarily “production-to-order”, consistent with the lengthy lead times documented, while the manufacturing of nondurable goods is predominantly “production-to-stock,” with a much higher proportion of finished goods inventories relative to work-in-process compared with durable manufacturing. In our model simulations we assume |$S=4$| (months), implying that the overall time to build, including the final assembly, is 5 months. We assume that |$\alpha _{s}$| is uniform across stages.

Table 1 provides the rest of the parameters and their sources. The solution method is to simulate the perfect foresight path from an initial position in which some of the state variables are away from their steady-state values because of the unexpected disruption to the supply chain. The simulation tracks the transition back to the steady state.

3. Orders, shipments, unfilled orders

Our assumption is that the |$\left \{ z_{t+s,t}^{u}\right \} $| and |$\left \{ z_{t+s,t}^{d}\right \} $| are produced and shipped at date |$t$|⁠, but in general will have been ordered at a previous date. As the model does not provide explicit guidance on the correspondence between the |$\left \{ z_{t+s,t}\right \} $| variables (suppressing for now the |$j$| and |$d$| superscripts on |$z_{t+s,t}$| and |$i_{t}$|⁠) and shipments, inventories and orders, it is necessary to add some structure. We assume that in anticipation of investment |$i_{t}$|⁠, |$z_{t,t-s}$| is produced in period |$t-s$|⁠, and combined with products from previous stages in the form of a composite good we call |$x_{t,t-s}.$| The process begins at stage |$S$| with the production of |$ z_{t,t-S}$|⁠, the only component of |$x_{t,t-S}.$| Producers at stage |$S-1$| receive |$x_{t,t-S}$| and combine it with |$z_{t,t-S-1}$| to produce |$ x_{t,t-S-1.}$| The process continues until date |$t$|⁠, when producers at stage |$S$| receive |$x_{t,t-1}$| and combine it with |$z_{t,t}$| according to 1 to produce |$i_{t}$|⁠.⁴

We first consider orders. The Census Bureau measures orders received by the producer (net of cancellations).⁵ We assume that orders are placed at each stage based on the time required to produce the goods at previous stages. With |$S=4$|⁠, at date |$t$| producers of the final investment good (stage |$0)$| would receive orders for |$i_{t+4}$| from both upstream and downstream producers based on expectations as of time |$t$|⁠. This in turn would generate orders from these producers to the stage |$1$| producers for |$x_{t+4,t+3}$|⁠. Stage |$1$| producers would in turn order |$ x_{t+4,t+2}$| from stage |$2$| producers. Stage |$2$| producers would order |$ x_{t+4,t+1}$| from stage |$3$| producers, who in turn would order |$ x_{t+4,t}=z_{t+4,t}$| from stage |$4$| producers.

All of these orders would have to occur at date |$t$| because of the time involved in building up each composite good from the previous stage. Our assumption about shipments, detailed below, is that goods produced at date |$ t $| are shipped at date |$t$|⁠, and then combined at date |$t+1$| at the next stage. So, for example, the producer at stage |$0$| will order |$x_{t+4,t+3}$| at date |$t$|⁠, knowing that it requires three periods to produce because it includes |$z_{t+4,t},z_{t+4,t+1},z_{t+4,t+2}$| and |$z_{t+4,t+3}.$|

Finally, although the explicit orders are for |$\left \{ x_{t+S,t},x_{t+S,t+1},...,x_{t+S,t+S-1}\right \} $|⁠, the value of the orders is from the chain of intermediate inputs each order comprises. Table 2 shows the relationship between the orders received by each sector and their corresponding value (the sum of the column beneath each order). Therefore we have

Note that we do not include |$z_{t+S,t+S},$| the input added at the final stage |$0$|⁠, in orders, as the Census does not count as orders goods that are used within the same period. The multiplication by |$s$| corresponds to the multiple counting of goods from earlier stages as they get used at higher stages.

Shipments are measured by the value of deliveries of the intermediate goods, so there is a similar over-counting relative to value added. We assume that these goods are delivered in the period they are produced, except in the final stage, in which |$z_{t+S,t+S}$| is not shipped but added to |$ x_{t+S,t+S-1}$| to produce |$i_{t+S}$|⁠, which is then shipped by the end of the period.

Shipments at date |$t$|⁠, denoted |$\mathfrak{s}_{t}$|⁠, therefore consist of all of the composite |$x$| goods produced at date |$t$|⁠. For example, the order for |$ i_{t+2}$| will result in the production of |$x_{t+2,t+1},x_{t+2,t},...$|⁠. Of these, only |$x_{t+2,t}$| is produced and shipped at date |$t$|⁠. An order for |$i_{t+1}$| will result in the production of |$x_{t+1,t},x_{t+1,t-1},...$|⁠, with only |$ x_{t+1,t}$| shipped at date |$t$|⁠. The value of these shipments includes all of the inputs from earlier stages. The value of each of the |$x$| goods shipped at date |$t$| is the sum of its components as shown in each column of Table 3.

Note that the last column is only counted as |$i_{t},$| but reflects the value of all of the |$S+1$| inputs. So, for example, the bottom row of Table 3 consists of outputs from stage |$S$|⁠, but at |$S+1$| different dates from |$t-S$| to |$t$|⁠. The second from the bottom row is from sector |$S-1$| and repeats |$S$| times, and so on. We can think of the elements in each row as different vintages of the same product, each part of a process that began on the date indicated by the second element of the subscript, and will end as part of the final investment good at the date indicated by the first element of the subscript.⁶

So we have

Overall shipments will be approximately |$\left ( S+2\right ) /2$| times larger than value added. So |$S=4$| would imply that shipments are three times value added. This is somewhat larger than observed in durable goods manufacturing, where the value of shipments is typically two to three times larger than value added, though the ratio exceeds four in some industries.⁷

To derive the normal level of unfilled orders, denoted |$u_{t}$|⁠, we compare orders over the dates from |$t-S$| to |$t$| with shipments over the same dates. For |$S=4$|⁠, orders not fulfilled as of date |$t$| are

In our calibrated model this works out to be approximately |$2.6$| times the level of shipments in the steady state. This is in the vicinity of the aggregate ratio for industries that carry unfilled orders if we exclude the aircraft, ships, and boats (ASB) industry, which because of much longer time-to-build is an outlier.

Table 4 provides the ratios for the seven industries, along with the aggregate ratio and the aggregate excluding transportation. The large aggregate ratio is because the transportation industry includes ASB, which presumably has a time-to-build of much longer than 5 months. Note also that the ratio is increasing from industries 31 (Primary metals) to 32 (Fabricated metals) to 33 (Machinery), as would be expected given the input–output structure between those three industries.⁸

3.1 Production disruptions

Orders as specified in (9) are unfilled only because of the time-to-build feature of the model, not because of any shocks, as the equation presumes perfect foresight. Since we are considering unexpected temporary production disruptions, we need to distinguish between outstanding orders, as a stock, and new orders. With production disruptions, (9) may include some orders carried forward from previous periods; in addition, some orders placed prior to |$t$| for goods to be delivered |$t+1$| or later will be revised up or down as a consequence of the shocks.

We assume that for a disruption at date |$t$|⁠, normal orders, as well as sectoral allocations of labour and capital, are in place as of the start of the period in which the shock occurs. A disruption in production will result initially in a jump in unfilled orders. We assume that the initial unmet orders due to the disruption are carried over as part of subsequent orders. That is, if the disruption occurs at date |$t$|⁠, new orders at date |$t+1$| (denoted |$no_{t+1}$|⁠) are reduced from the quantity in (9) by the shortfall in shipments at |$t$|⁠, product by product. This includes products that had been ordered in prior periods to be produced and shipped at date |$t$|⁠.

In addition, orders at date |$t$| and earlier for goods to be shipped at date |$ t+1$| or later can be modified as a consequence of the shock.⁹ For example, an order for |$i_{t+S}$| made prior to the shock would be modified to reflect the updated expectation of |$i_{t+S}$| following the shock. If lower, part of the initial order would be cancelled; if higher, the initial order would be carried forward, and the increment would be a new order. We assume all of these adjustments are costless, so they are just a matter of measurement and accounting.

Consequently, we modify our formula for orders to take into account pre-existing orders that were unexpectedly not filled, as well as those that need to be modified. We will call the stock of unexpectedly unfilled orders “outstanding orders,” denoted by |$\omega _{t}.$| Suppose production of some of the intermediate goods is disrupted at date |$t$|⁠. The first correction applies to orders up to and including date |$t$| expected to be shipped at date |$t$|⁠. In the case |$S=4,$| we have the following potential shortfalls, assuming a production disruption at date |$t$| in sectors |$1-4$|⁠:

• Orders at |$t$| for |$z_{t+4,t}$|

• Orders at |$t-1$| for |$z_{t+3,t}$|

• Orders at |$t-2$| for |$z_{t+2,t}$|

• Orders at |$t-3$| for |$z_{t+1,t}$|

We assume these shortfalls get carried over, good by good, and are applied to subsequent orders. For example, a shortfall at |$t$| in the order for |$ z_{t+2,t}$| (which was placed at |$t-1$|⁠) gets carried forward and reduces the new order at |$t+1$| from sector |$2$| for |$z_{t+4,t+2}$|⁠. Letting |$\hat{z}$| denote the original orders for any |$z,$| total outstanding orders as of the end of period |$t$| are

New orders at date |$t+1$| will be reduced (relative to the quantity in (9)) by |$\omega _{t}$|⁠.

As noted above, the shock at date |$t$| will also result in updates to orders made at |$t$| or earlier for goods to be shipped after |$t,$| including

• Orders at |$t$| for |$z_{t+4,t+1},z_{t+4,t+2},z_{t+4,t+3},i_{t+4}$|

• Orders at |$t-1$| for |$z_{t+3,t+1},z_{t+3,t+2},i_{t+3}$|

• Orders at |$t-2$| for |$z_{t+2,t+1},i_{t+2}$|

• Orders at |$t-3$| for |$i_{t+1}$|

Updates to previous orders, which may be positive or negative, will occur at |$t+1,$| and are denoted by |$\upsilon _{t+1}$|⁠. Therefore

In practice, the |$\hat{z}$| and |$\hat{\imath }$| variables involved in both |$ \omega $| and |$\upsilon $| are their steady state values. Sticking with the example of orders from sector |$2$|⁠, |$\hat{z}_{t+3,t+1}$| was ordered at |$t$| prior to the shock. Once the shock occurs, the model calls for a different value of |$z_{t+3,t+1}$| from what had been ordered. This difference, and other similar revisions to previous orders, will get included in |$\upsilon _{t+1}$|⁠.

With these adjustments, we have

Note that if the only shock is at date |$t$|⁠, this is a one-time adjustment at date |$t+1.$| Both shipments and orders from date |$t+1$| forward reflect knowledge of the shock at |$t$|⁠, and with no subsequent shocks there will be no further unexpected shortfalls or revisions.

Figure 1 depicts the simulation of one particular supply chain disruption on the ratio of unfilled orders to shipments, and in the level of shipments, in comparison with the data from the manufacturing sector (the seven industries for which unfilled orders are reported, excluding ASB).

Fig. 1.

Shipments and unfilled orders with a supply chain disruption. Source: US Census Bureau and author’s calculations.

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The disruption in the model is an unexpected one-period 80 percent reduction in the labour inputs at stages |$2-4$| only. The magnitude of the shock is chosen to produce initial responses similar to those in the data, so the “test” for the model is how well its propagation mechanism in the model matches the data in the months after the disruption. The data series on the ratio of unfilled orders to shipments is distorted slightly due to an upward drift, visible even in this 1-year window, apparently unrelated to the pandemic disruptions.¹⁰

The similarity of the initial jumps in the model and the data is no coincidence, as the shock in the model was chosen to approximately match the initial movements in the data. More striking is the similarity of the trajectories in subsequent months, given that the shock in the model occurs only in 1 month. The remainder of the path is due to the endogenous dynamics from the time-to-build structure of the model. The propagation of the shock in the model—in particular the slow reversion to the previous steady state—closely replicates the data. In both the data and the model the ratio reverts approximately 80 percent of the way to its earlier level within 4–5 months, and subsequently very slowly, despite the fact that the shock itself has no persistence.¹¹ The only notable difference is that the model has a sharper initial recovery in the month immediately after the shock, which likely reflects, at least in part, the fact that the actual disruptions in these industries had a duration somewhat longer than 1 month.

Bear in mind that the model’s implied shipments and orders, as described in (10) and (9), apply to the aggregate, not to the individual industries. Some of the industries, such as 31 and 32, would correspond to production at the early stages only, for delivery to later stages of the process. As an example, suppose the final investment good is machinery. The first stage could be mining, then primary metals, followed by fabricated metals, which then get assembled at the fourth and final stage into machinery. Primary metals would effectively show up three times in total shipments for the aggregate of these four industries: First, when they get shipped to the fabricated metals sector, second when they are part of the value of the fabricated metals shipped to the machinery sector and then a third time as part of the value of the final product.¹²

To understand the magnitude of the drop in shipments and the slow recovery, note that an 80 percent reduction in labour reduces shipments in that period from the affected firms by 67 percent, as their capital remains in place. Since only three of the five stages are affected, value added in that period falls by 45 percent. Total shipments across all stages, however, will include the value of shipments from earlier periods, as in (10). This implies that the immediate impact on total shipments will be much smaller than the reduction in value added, and, moreover, that subsequent total shipments will be reduced relative to value added as the disruptions pass through the subsequent stages of production. Thus aggregate shipments fall initially by approximately 13 percent. Value added in the period following the disruption increases to a level more than 13 percent above the steady state, but shipments only increase about one-third of the way back to the steady state because the totals include the depressed quantities from the period of the disruptions.

3.2 Inventories

With the time-to-build structure of the model, and the focus on capital-goods-producing industries, disruptions to production will also have implications for inventories, particularly work in process. As with orders and shipments, the model does not explicitly incorporate inventories, but we can add additional structure and assumptions, consistent with the approach taken above.

Here we focus on work-in-process inventories (WIP). The Census Bureau data on these inventories are levels reported by businesses as of the end of each time period. If production simply occurred discretely and completely within each time period as implicitly assumed in the model, there would be no WIP. Production at each stage would be started and completed within the period and then shipped to the next stage. In fact, though, discrete time is just an artifact of the data. We can gain an understanding of WIP by allowing for the fact that some time elapses between the start and completion of any unit of production. Shipments represent the value of goods completed within the period, but as of the end of any period there will in general be some goods that are still in process, to be completed the next period.

We model this by maintaining the assumption that production of each good takes one period, and dividing each time period into |$h>1$| subperiods. At any point there are |$h$| overlapping production processes underway, each producing at a rate of |$1/h$| per period, or |$1/h^{2}$| per subinterval. If total production from that stage is |$z$|⁠, each process produces |$z/h$| units. At each subinterval, production of |$z/h$| unit starts, and is completed one period (⁠|$h$| subperiods) later. In the steady state, each of the |$h$| processes will complete |$z/h$| units by the end of each period, and all but one of them will have work in process. If we assume value is added at a constant rate, then total work in process will be |$\left ( h-1\right ) /\left ( 2h\right ).$| As |$h\rightarrow \infty $|⁠, the stock of WIP inventories is, in a steady state, half of the value of shipments in any period.

It follows, however, that WIP technically becomes a state variable that in general must be taken into account if there are fluctuations in output. In particular, variation in shipments over time implies that production (value added) generally differs from shipments. In the Appendix, we first show that in a steady state, value added equals shipments. But shipments in one period are completions of production begun in the previous period. As a consequence, shipments in a given period are constrained by the quantity of work in process at the end of the previous period. For example, if a producer unexpectedly wants to increase shipments as of date |$t$|⁠, say from |$ 1 $| to |$2,$| the increase would be delayed by one period, because as of the beginning of period |$t$| there is only enough WIP to complete the originally planned shipment of one unit. Production would increase in period |$t$| to |$ 1.5 $|⁠, but only to contribute an extra |$0.5$| to work in process so that shipments could increase to |$2$| in period |$t+1$|⁠.

The difference between production and shipments also depends importantly on the persistence of changes in shipments. If an increase in shipments is transitory, in this example returning to |$1$| at date |$t+2$|⁠, then value added will be |$1.5$| in |$t$| and |$t+1$|⁠, accumulating WIP during period |$t$|⁠, and drawing it back down in |$t+1$| by producing less than is shipped. Thus the increase in shipments at |$t+1$| of one unit will be matched by an increase in valued added of |$0.5$| in periods |$t$| and |$t+1$|⁠, and production is smoother than shipments. On the other hand, if the increase is permanent, then value added will increase to |$2$| at |$t+1$|⁠, and the cumulative increase in value added will exceed that of shipments because of the new higher level of WIP. This is an example of the well-known “bullwhip”effect of inventories: A persistent increase in shipments requires a more than one-for-one increase in production to permit a higher stock of inventories.¹³

The FTTB model as formulated in Section 1 ignores the distinction between value added and shipments. In effect, while allowing for stages of production of different inputs, it assumes the production of each input occurs within each time period. The question is whether, for the purpose at hand, incorporating WIP formally into the FTTB model would substantially change the results of our simulations. We argue that for the disturbances under consideration in this paper, this would have little impact on the results at the cost of considerable complexity. First, the increases in output above the previous steady state only occur at some stages, and represent a small fraction of overall shipments. In fact, as Fig. 1 shows, the overall value of shipments remains below the previous steady state throughout the adjustment to the shock. Second, even for the stages that do increase production above the previous steady state, the increases are transitory, and any change in inventories will also be transitory, and the bullwhip effect will be negligible. Hence for the exercise in this paper looking at the impact of a temporary disruption in production, we can dispense with the complication of incorporating the distinction between production and shipments.¹⁴ The Appendix does, however, provide a framework for doing so in other contexts where it may be important, but this is left for future research.

Even so, some basic implications of this approach to modelling WIP can be examined and compared with the data. As explained in the Appendix, the model of WIP implies that if one unit of production requires |$M$| periods, the steady-state ratio of WIP to shipments will be |$M/2$|⁠. Table 5 shows the average ratio over the year prior to the onset of the pandemic for the same two-digit level industries considered above. Data for these inventories are only disaggregated to the two-digit level. As a consequence, although with unfilled orders we excluded only the ASB subcategory of industry 36 (Transportation), here we only can include or exclude all of industry 36. Note that as with unfilled orders, it is an outlier in the inventory data as well.¹⁵

Four of the seven industries have a ratio that exceeds |$0.5$|⁠, indicating an average time-to-build somewhat longer than one period. Transportation shows an average time to build of more than two periods, presumably because it includes aircraft and ships.

We assume that a disruption that reduces production activity to a fraction |$ f $| of its normal rate occurs by stopping production completely at a point |$ f $| , |$0<f<1$|⁠, in the period. It is straightforward to show (as detailed in the Appendix for the limiting case of |$h\rightarrow \infty $|⁠) that beginning in the steady state, the level of WIP at the end of the period will be unaffected by such a disruption: At any point during the period there will be |$h$| processes underway at each of |$h$| different degrees of completion as there would be at the end of any period. As noted above, the disruption will, however, reduce shipments within the period, because at a point |$f$| through the period, only approximately |$fz$| units will have been completed at the affected stage of production. Note, however, that total shipments across all stages will include the value of shipments from earlier periods, as in (10). This implies that the immediate impact on total shipments will be much smaller than the direct reduction in the affected firms, and moreover that subsequent total shipments will be reduced as those reductions pass through the stages of production.

Qualitatively at least, this is what we observe in the data. Figure 2 shows the path of WIP and shipments in the Ex36 aggregate over the period from October 2019 to September 2020. WIP is nearly flat, while shipments drop approximately 20 percent from March to April 2020.

Fig. 2.

Response of work in process and shipments. Source: Census M3 Reports: manufacturing industries 31–37 excluding 36.

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To summarize, this section describes an extension to the FTTB model that incorporates WIP (details in the Appendix). We argue that for the exercise considered in this paper, a one-time transitory disruption to production, inventories would not exert a quantitatively significant impact, and the inventory data bear this out. Nonetheless, we see this extension as a useful contribution in its own right, and may be useful for future research in contexts where there are more persistent fluctuations in shipments and production.

4. Conclusions and future research

The FTTB framework described in the paper, extended to model shipments, orders, unfilled orders and work in process, produces realistic dynamic responses to production disruptions. Shocks of a magnitude calibrated to match the immediate impact on shipments and unfilled orders result in transition paths with a gradual return to pre-shock levels similar to those observed in manufacturing data. The calibrated model also is shown to match other facts in the data, including the average unfilled orders to shipments and work in process to shipments ratios.

This study is a purely after-the-fact assessment of the model’s ability to capture the impact of pandemic-related production disruptions in manufacturing. As noted in the introduction, such model validation is a natural first step, which, if successful, warrants the use of the model to address other more normative questions, such as how best to manage disruptions when they arise, or plan ahead to mitigate the adverse impact. The pandemic was unprecedented in both the size and breadth of its impact. It would have been difficult to anticipate, but it is now part of the landscape facing decision-makers. And while quantitatively larger, the disruptions considered here would be similar to those from other sources such as international trade, military conflict and natural or weather-related disasters.

The features of the model that enable it to generate realistic propagation of transitory production disruptions may prove useful in applications to more persistent or permanent shocks. In addition to the propagation of such shocks through the stages of production, extending the model to incorporate work in process would yield both inertial responses to desired increases in shipments and a “bullwhip” effect that increases the volatility of production relative to shipments in response to persistent shocks.

Extending the model to account for inventories at all stages of production would be straightforward, and useful both for additional validation and for analysis of decision-making. While there is little management can do to avoid a forced disruption from an external source, a model incorporating inventories could more readily investigate ways to mitigate the damage. For firms whose production could be directly disrupted, larger stocks of materials or work in process would facilitate a more rapid recovery following a shutdown. For downstream firms indirectly affected by upstream disruptions, holding larger inventories of goods they require from upstream firms, or having multiple sources of such goods (to the extent disruptions are idiosyncratic) would help to insulate them from temporary interruptions in the supply chain. Where diversification of sources is not feasible, downstream firms could benefit from vertical integration.

5. Acknowledgement

I would like to thank the editors for their helpful comments to improve the paper.

Data availability

The datasets were derived from sources in the public domain: The United States Census Bureau’s M3 reports on Manufacturer’s Shipments, Inventories and Orders (Historical Time Series – NAICS).

Transformations of the data were performed in Excel. These files, along with the data, are available on request from the author. Model solution and simulation was done in Matlab (R2021b), using Dynare version 5.4. Code is available on request from the author. Dynare is open-source, and may also be run with Octave (open-source).

Footnotes

References

A. Appendix

A.1 Model solution

For the planner’s problem, we can construct the Lagrangian

first-order conditions (letting |$\theta _{t}^{n}=\theta _{t}^{u,n}=\gamma n_{t}^{1/\nu },\theta _{t}^{u,k}=\varGamma _{t}^{s}\phi \left ( \frac{k_{t}^{s,u} }{B_{t}^{u}n_{t}^{s,u}}\right )^{\phi -1}~~\ ~s=0,1,...,S$|⁠)

Note that the |$k_{t}^{s,u},k_{t}^{s^{\prime },u}$| and |$ n_{t}^{s,u},n_{t}^{s^{\prime },u}$| conditions imply that |$ k_{t}^{s,u}/n_{t}^{s,u}$| is the same for all |$s$|⁠, and therefore equal to the aggregate ratio |$k_{t}^{u}/n_{t}^{u}$|

But if we divide one by the other we get

Consequently, |$\varGamma _{t}^{s}$| and |$~k_{t}^{s,u}/n_{t}^{s,u}$| do not depend on |$s$|⁠.

The steady state for this model has an analytical solution. The simulation involves starting from an initial condition (⁠|$t=0$|⁠) with |$z_{2,0},z_{3,0}$| and |$z_{4,0}$| at levels below their steady-state values, reflecting the disruption of production at that date, and numerically finding the perfect foresight path back to the steady state from that point on.

A.2 Work in process

This section outlines the calculations of work-in-process, shipments and value added based on a framework in which there are |$h$| overlapping processes each period, each process producing |$1/h$| of one period’s value added over one unit of time, adding |$1/h^{2}$| value added at each of |$h$| “stations” along the line. Completed production contains one unit of value added per unit of shipments. Here we let |$h\rightarrow \infty ,$| so that WIP/shipments |$\rightarrow 0.5.$|¹⁶ Note that while it would be feasible to have parallel production lines beginning and ending within the period, resulting in no WIP at the end of the period, this would require |$h$| production lines for one unit of output, with all but |$1/h$| of the stations along the line idle at any point in time. The overlapping structure allows one unit of output to be completed within the period with only one production line. In a steady state the inherited WIP would be completed during the period, and the same quantity of WIP would remain at the end of the period.

Here we give a graphical representation of how this would work under several scenarios. Figure A.1 depicts a steady state with value added and shipments at a rate of one per time period. The figure presumes no WIP from before |$t=-1$|⁠. Production begins at that point on a staggered basis and builds up WIP of |$0.5$| by date |$0$|⁠, denoted by the area of triangle |$W\left ( 0\right ) $|⁠. It produces one unit of value added during period |$1$|⁠, half of which completes the production begun one period earlier (triangle |$C\left ( 1\right ) $|⁠). The other half, area |$W\left ( 1\right ),$| is new production starting at |$0$|⁠, which will get carried into period |$2$| as WIP. The value of shipments each period is |$C\left ( t\right ) +W\left ( t-1\right ) $|⁠. The height of the triangle labelled |$W\left ( -1\right ) $| at any point in the interval |$\left [ -1,0\right ] $| indicates the state of progress of production as of |$t=0$| that began at that point. The height at any point in |$\left [ 0,1\right ] $| of triangle labelled |$C\left ( 1\right ) $| indicates the amount of production needed to complete a process started at the corresponding point one period earlier. So, for example, production begun at |$t=-0.8$| is |$80$| percent complete by time |$0$|⁠, and therefore is completed by |$t=0.2.$|

Fig. A.1.

Production, shipments and inventories in the steady state

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Now suppose that production is interrupted at |$t=f, f\in (0,1)$|⁠. Production resumes at |$t=1$|⁠, but now at a level |$1+x$|⁠, due to elevated unfilled orders.¹⁷  Figure A.2 shows that as of |$t=f$|⁠, there will still be WIP of |$0.5$| representing production begun over the previous unit time interval (⁠|$t=-0.8$| to |$t=0.2$|⁠). During period |$1$|⁠, however, only |$f$| units will be completed and shipped. Upon resumption, the WIP of |$0.5$| can be completed, resulting in shipment of one unit, and |$0.5$| WIP is created. So the interruption during period |$1$| does not prevent immediate return to the previous level of shipments. We assume, however, that the |$x$| additional units started at |$t=1$| , in keeping with the “assembly line” spirit of the model, that the |$x$| processes get underway during period |$2$| with the same staggered starting point as in previous periods.¹⁸ The difference is that there is no additional WIP to complete. This implies that the additional production cannot increase shipments in the current period. Thus total shipments remain at |$1$| in period 2, and WIP carried into period 3 will be |$0.5\left ( 1+x\right ) $|⁠.

Fig. A.2.

Work in process with disruption and subsequent increase in output

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The extra inventory at the end of period 2 will require extra work in period 3 to complete, partially or fully. Here we assume that just as the extra production in period 2 has staggered starting points, we assume in the subsequent period that the extra production required to complete the additional WIP can wind down as units are completed. With this assumption, inventories could end period |$3$| back at the steady state level of |$0.5.$| The entire sequence is depicted in Fig. A.2:

Table A.1 provides totals of various quantities for each period.

Note that the change in WIP equals the difference between VA and shipments. Also, given the transitory nature of the shock, there is no “bullwhip” effect: The target level of WIP is unchanged, and as a consequence, production (VA) is smoother than shipments. WIP rises above its steady state by |$0.5x$| for only one period.

The implication is that an increase in shipments to a level above the steady state after a disruption, if work in process is factored in, gets delayed by one period, while the increase in value added is smoothed over two periods. The model as laid out neglects these differences by equating value added and completed production at each stage. But the differences are small and transitory, and represent only a small part of the industry aggregates, as only some stages of production are disrupted, and, for example, aggregate shipments include production from all stages at four different time periods as described in equation (10). Hence we argue that explicitly including WIP and the distinction between value added and shipments at each stage, while feasible, would have a negligible impact on the results, and can be left for future research.

A.3 Generalization to longer or shorter production times

Suppose for one part of the chain, production of one unit takes |$M$| periods, where |$M$| could be less than or greater than one, and that there are |$h\geq M $| processes beginning every |$M/h$| periods. Each process completes |$1/M$| units per period, so |$h/M$| units are completed per period. The ratio of WIP to shipments will be |$0.5Mh/\left ( h-1\right ) $|⁠. For example, suppose |$M=2$| and |$h=4$|⁠. Work in process at the end of any one period would reflect 1/8, 1/4 and 3/8 of the value of the finished product, for processes that began one-half, one and one-and-a-half periods earlier. Shipments during each period would consist of two units, one completed halfway through the period (having begun halfway through two periods earlier), and the second (begun at the beginning of the previous period) completed at the end of the period. The WIP to shipments ratio would be 0.75. In the limiting case for any positive value of |$M$|⁠, as |$h\rightarrow \infty $|⁠, the ratio of WIP to shipments would be |$M/2.$|

In addition, it is possible to allow some processes to require longer production than others. As an example, assume that some stages have |$M=2$| and others have |$M\leq 1$|⁠, with |$S=4$|⁠. Specifically, assume that the first two stages, |$s=4$| and |$s=3$|⁠, require two periods. At time |$t,$| an order is placed for |$i_{t+4}.$| This in turn would generate an order for |$ x_{t+4,t}=z_{t+4,t},$| the first input in the chain. Because it takes two periods to produce, it will combined with |$z_{t+4,t+2}$| in |$x_{t+4,t+2},$| and then one period later with |$x_{t+4,t+3}$|⁠, which will include inputs from |$s=2$| and |$s=1$|⁠. A second process begins at |$s=3$| at date |$t+1$| with |$ x_{t+4,t+1}=z_{t+4,t+1}$|⁠. This becomes part of |$x_{t+4,t+3}$| along with the other three inputs. Table A.2 shows the value of orders given this alternative assumption:

Relative to the baseline case of one-period production throughout, this structure reduces the multiple counting in shipments, as the output from |$ s=4 $| is only counted three times, and the output from |$s=3$| is only counted twice. So we now have

if there are no shocks.¹⁹

As described above, production disruptions will modify subsequent orders. We have

These are the same as before with the exception that |$\omega _{t}$| is reduced by |$\hat{z}_{t+4,t}-z_{t+4,t}$|⁠, and |$\upsilon _{t+1}$| is reduced by |$ z_{t+4,t+1}-\hat{z}_{t+4,t+1}$|⁠.

Shipments are similarly modified. The value of each of the |$x$| goods shipped at date |$t$| is the sum of the components as shown in each column of Table A.3:

and we have

This results in a ratio of shipments to value added of approximately 2.6, versus 3 in the baseline with one-period production. Thus in addition to increasing the ratio of work-in-process to shipments, this structure also reduces the ratio of shipments to value added, which we saw in the baseline case was too high.

Finally, the stock of unfilled orders in the absence of production disruptions is now

This implies a ratio to shipments of approximately 2.77, which is very close to the ratio of 2.8 observed in the industries excluding ASB (and closer than for the baseline model that implied a ratio of 2.6).

Author notes