تاخیر تولید و موجودی مواد اولیه تحت عدم قطعیت

Firms producing seasonal goods often make order and production choices prior to highly uncertain sales, thus lending an investment quality to their decisions. Using specialized inputs imposes a delay in receiving them and linked with a long production period the firm makes its order, production and pricing choices under successively reduced uncertainty. The model shows that input and production costs have distinctive effects on the firm's order size with implications for the stock of raw materials inventory. Firms facing relatively low input costs are willing to risk leaving inputs unused as a bet for a good state of nature. Further, we investigate situations of greater uncertainty and find a more nuanced explanation of firm behavior than previous research. Firms with relatively inexpensive inputs facing equal odds of good and bad states of nature will increase their order size (a known result). However, a firm with a low relative cost of completing production may either raise or reduce its order size depending on the demand elasticity and the relative demand uncertainty. Intuitively, firms with expensive inputs facing highly uncertain demand and with many substitute output goods are inhibited by the high cost of insuring against stock-outs.

It is well documented that inventories play an important role in the business cycle. “…the drop in inventory investment has accounted for 87 percent of the drop in GNP during the average postwar recession…” Blinder and Maccini (1991). This role has motivated much research in inventories. However, Blinder and Maccini (1991), Bivin (1989) and Humphreys et al. (2001) note that most of this research focuses on finished goods inventory while input (i.e. raw materials and work-in-process) inventories comprise a larger part of aggregate inventories and are also more volatile. Humphreys et al. (2001) extend the linear-quadratic model of output inventories by adding the joint determination of input inventories. Researchers now more frequently include input inventories in their analyses, e.g. Herrera et al. (2008). Our model motivates firms’ use of raw materials inventories as a secondary defense to finished goods inventory against lost sales, following e.g. Crowston et al. (1973), Bivin (1989), and Emery and Marques (2011). In this paper we focus exclusively on raw materials inventory; the full flexibility of output prices eliminate the possibility of finished goods inventories. Specifically, we examine those inputs the firm orders which are slated to be later used in production only in the good state of nature. Should the firm find itself in the bad state of nature, it will optimally cancel the production of some goods which then become unused raw materials. We interpret these set-aside inputs as an insurance payment that would have protected the firm from lower sales and profits had the good state of nature occurred. One of our main goals is to determine the influences on the amount of such raw materials. In our model, a risk-neutral single product firm makes a decision in each of three periods. It first chooses an input quantity to order from its supplier. The state of nature is then revealed, informing the firm about whether the demand distribution is favorable or unfavorable. The second decision follows, a choice of the quantity of inputs to submit to the production process. Finally, after the demand curve is realized, the firm chooses a price. A simple pair of distributions permits to derive reduced form solutions using standard backward induction. The firm we study has price setting ability; an example of such a firm might be a style goods producer (e.g. those making fashion goods and other apparel or bicycles), goods which are the epitome of a differentiated good. Sen (2008) cites US Department of Commerce that shows that the domestic apparel market is dominated by 12 major retail groups. Further, we assume that the firm makes a single input order per season, justified for example by a long production lead time at the upstream supplier of specialized inputs. Style goods producing firms must thus commit to purchase inputs prior to production. They also have a short selling season of high demand uncertainty (Silver et al., 1998 and Sen, 2008; Choi, 2007; Hammond and Raman, 1996). Style goods often require a long production time. Bivin (1999) [using a method derived by Carlson (1973)] estimates US industries’ production lag for durable industries to “…cover a wide range of values” but that “…the lags are generally in excess of two months”.1Sen (2008) cites a Deloitte and Touche estimate of a 26 week average concept-to-production cycle time for the apparel industry. The firm in our model explicitly decomposes its order quantity into two parts; a fixed baseline part, F, which the firm will set to production whatever state of nature occurs. In addition it chooses a discretionary part, D, it aims to produce only in the good state of nature. These are the inputs to become unused raw materials inventories if the bad state of nature occurs. We note that quantity F is not riskless. If the demand realization is lower than expected, the firm will set a low price on its output good to reduce profits. With a lower probability of a good state of nature, the firm views it more likely that part D of the input order will be wasted. Accordingly the firm tries to reduce D and indeed at some low probability to choose a negative D. As this is not reasonable behavior, we restrict D to be non-negative. We also define a threshold probability p0 at which D is zero and find it increasing in input cost (C) relative to production cost (W). Our model predicts that firms will have different threshold probabilities and thus distinct behavior concerning raw materials. Those firms with low input costs relative to completing products (low p0) are likelier to sacrifice inputs as unsold raw materials inventory (even at zero scrap value) compared to firms facing higher relative input costs. This is intuitive. Rossana (1990) checked distinct input and production costs in three durable goods and the apparel industries in the only empirical analysis we are aware of. He found only in the (seasonal) apparel industry did raw materials prices have a significantly negative influence on raw materials stocks and that only in this industry did wage costs have a positive (albeit insignificant) effect on raw materials stocks. While this is consistent with our model, a more complete empirical study would clarify behavior of firms exposited in this paper. We also investigate firm behavior after a rise in uncertainty; it affects firms differently based on their relative costs. At equal probabilities of a good and bad state of nature, a firm with relatively cheap inputs (a low threshold probability p0) increases its order size while a firm with a low relative cost of completing production (a high threshold probability p0) may reduce its order size. Intuitively, the latter firm may be inhibited by the cost of insuring against lost sales. As mentioned in Gerchak and Mossman (1992), the former result is standard in the literature of cost-minimizing single product order decisions while the latter result is intuitive and novel. We have used backward induction to determine the firm's correct decisions over time. However, we have not imposed the firm's subsequent choices in each prior decision. And interestingly the firm in early choices acts to restrict its later self. Indeed when the firm faces a probability less than the threshold probability it will choose an order amount less than the amount it will optimally produce at time 1. It does this of course because at time 1 its input costs are sunk while at time 0, the order time, they are not. We note the firm often ties its own hands in its various choices. The remainder of the paper is structured as follows: in Section 2 we briefly review related literature and then in Section 3 we introduce the basics of the model. In the subsequent section after specifying the uncertainty in the model, we analyze an optimizing firm facing constant elasticity and stochastic demand at its single main sales date. In Section 5 we examine the effects on order choice of an increase in uncertainty. In Section 6 we conclude.

The firm in our model has solved its pricing, production and order problem in an uncertain environment through a dynamic programming approach. Since production and order decisions were already completed at time 2 the firm ignores both unit input cost C and unit production cost W so that profit maximization is equivalent to revenue maximization. Further, finished goods inventories do not arise in the model because prices adjust supply to demand and so all production is sold. The model allows us to be precise about the level of input safety stock ordered. In the good state of nature these stocks are used up. In the bad state of nature these stocks (if the probability of a good state exceeds the firm's threshold probability) are accounted as raw materials inventory. Yet in some sense they might be considered an insurance premium expended to ensure safety stocks now proven unneeded. If the firm in our model could both choose the price and production quantity after uncertainty is resolved, the 3 periods would collapse to 2, thus entailing an analysis similar to the Van Mieghem and Dada (1999) model of price and production postponement. It is our firm's production duration that interferes with the flexibility enjoyed by the firm in Van Mieghem and Dada. In their model, the firm can choose optimal price and quantity up to order quantity K. Should demand be higher, it then raises prices at that K. In our model, the firm is less well informed in its choice of production quantity and consequently when uncertainty is resolved, it prices all demand levels at the chosen production quantity. Related to (raw materials) inventories, a firm having postponed production can be said to choose these inventories optimally, while in our model, the firm chooses raw materials inventories when significant uncertainty remains which implies that an ex-post optimal choice of raw materials inventory is highly unlikely ex-ante. Despite our use of backward induction, the firm sensibly often acts to restrict its future decisions, since it understands that in the future its costs will be sunk and lead to non-optimal decisions when viewed ex-ante. For example, if the firm has a low threshold probability and it turns out to be in a good state of nature, its order will have been less than what is optimal to produce (optimal at least ignoring sunk costs). The threshold probability has important implications. For example, a firm with low threshold probability in the bad state of nature produces R L (or quantity F I), a quantity ignoring sunk costs. This is because a larger quantity (D +F )I of inputs is on hand so using the remaining inputs entails no extra cost. A firm with a higher threshold probability orders and produces a smaller quantity than R L. That is, the firm produces View the MathML sourceFII⁎<RL due to condition (16) and accordingly ties its future hands by including costs considered sunk at time 2. As mentioned in the Introduction, much previous empirical inventory research has lumped all inventory sub-categories into aggregate inventory and all production costs into one measure of cost. Rossana (1990) checked distinct input and production costs in three durable goods and the apparel industries in the only empirical analysis we are aware of. He found only in the (seasonal) apparel industry did raw materials prices have a significant negative influence on raw materials stocks and that only in this industry did wage costs have a positive (albeit insignificant) effect on raw materials stocks. While this is consistent with our model, a more systematic empirical study is required for greater confidence in the results. This model focuses on a firm's trade-off in making an input order to meet demand in the good state of the world. Two conditions reflect a firm’s choice to hold zero raw materials inventories. First, if its probability of a good state is less than its threshold probability, the firm finds unattractive the gamble for the good state of nature. Second, if its probability of a good state is greater than the threshold probability and the good state of nature occurs, the firm will have won the gamble and it will use all its raw materials in production. This situation is more likely true for a one-product firm in a fashion industry. It would be fruitful for future research to investigate the results with alternative initial distributions. Concerning the firm's behavior with an increase in uncertainty, and equal odds of a good and bad state of nature there are two cases. A firm with low threshold probability will order a greater amount due to the low cost of insuring against a loss of profits in the good state of nature. A firm with a high threshold probability may either order more or less depending on the elasticity of demand and the relative uncertainty. Intuitively, firms with expensive inputs may be inhibited by the high cost of insuring against a lack of inputs in the good state of nature.