Abstract
In this chapter, we will describe the basic framework and some key concepts of the economic theory of quantity adjustment. Section 3.1 characterizes the capitalist system as a demand-constrained economy in which firms are almost always competing to capture the demand for their products. This aspect of capitalism has a profound relevance with the long-term changes in technologies and products through incessant innovations. Section 3.2 formulates production rules for stockout avoidance behaviors by individual firms facing uncertain demand under sales competition. Here “stockout avoidance” means that firms seek to suppress their expected stockout occurrence ratio to a considerably low level by holding a certain amount of inventory. Section 3.3 outlines quantity adjustment in the capitalist economy as a dynamic process generated by the interactions of firms each of which follows production and ordering rules for stockout avoidance. An emphasis will be given to roles of inventories as buffer and information sustaining the loose stationarity of the economy. Section 3.4 provides a historical overview of preceding contributions to the analysis of quantity adjustment. After the appearance of General Theory by Keynes, the development in this field has been attained mainly through attempts to build dynamic and multi-sector models of the multiplier process and to clarify factors affecting the stability of this process.
Keywords
A part of the research on this chapter was supported by JSPS Kakenhi no. 15K03386.
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Notes
- 1.
- 2.
Agents of the competing side can partially compensate their disadvantageous position by some explicit or implicit agreement limiting mutual competition. However, such cooperation is not so easy, especially when the competing side consists of many agents.
- 3.
The “demand-constrained economy” is an economy wherein, usually, a producer’s efforts for increasing production are constrained by the buyer’s demand, not by the available amount of various inputs. See Kornai (1980, pp. 26–30).
- 4.
As Kornai puts it, “the sellers court the buyers” in sales competition (Kornai 1971, p. 293).
- 5.
The firm might intentionally withhold to expand its supply if queuing and long waiting time serve as a signal of high reputation. Here we do not consider such a case.
- 6.
The same can be said of surplus (unused) production capacity. Under sales competition, “every seller … makes efforts to create surplus capacity” and thus intensifies sales competition still more “by increasing the selection possibilities of the buyer” (Kornai 1971, p. 322).
- 7.
Kornai (1971, p. 179) remarks that “the economic system cannot operate with an empty, ‘cleared’ market, without stocks and reserves.”
- 8.
In this internal demand-supply relationship, the demander and the supplier, respectively, correspond to the production and purchase departments of the same firm.
- 9.
This book does not deal with the labor market. However, it should be noted that the above argument on sales competition can be, at least partly, applied to the competition among workers (suppliers of labor forces) in the labor market. Sales competition in the labor market is inseparable from the continuous existence of a surplus labor population which Marx called “the industrial reserve army” (Marx 1990, pp. 781–794). As Marx depicted, this reserve army enables firms flexible expansion of production and effective control over workers in workplaces.
- 10.
- 11.
- 12.
The hoarding of raw material inventories within firms was one of the notable features of the socialist economy. See Table 3.1 in Sect. 3.3. Socialist firms also had to compete to ensure labor forces. From the viewpoint of workers, chronic labor shortage partly compensated for the lack of independent trade unions (Kornai 1980, 1992).
- 13.
This coexistence of surplus and shortage is emphasized by Kornai (1980).
- 14.
On this aspect of the capitalist economy, see also Kornai (2014).
- 15.
In the field of scientific and technological discoveries, socialist countries (especially the Soviet Union) have made many excellent records. Nevertheless, economic systems in these countries lacked the mechanism bridging these discoveries with the introduction of new commodities in the market. Concerning this point, refer to Kornai (1971, pp. 288–289).
- 16.
Keynes (1940) vividly describes this conversion in the United Kingdom that was caused by the start of the Second World War.
- 17.
It should be noted, furthermore, that the property right is a bundle of partial rights each of which takes various forms. This complexity of private property is itself one of the causes of the regional and historical variety of capitalism. On this point, see Hodgson (1988).
- 18.
Say’s assertion can be correct only when there is no saving and all the production factors are fully employed.
- 19.
Just before this passage, Sraffa (1926, p. 543) describes the thinking of entrepreneurs as follows: “Business men, who regard themselves as being subject to competitive conditions, would consider absurd the assertion that the limit to their production is to be found in the internal conditions of production in their firm, which do not permit of the production of a greater quantity without an increase in cost.”
- 20.
As Arrow (1959, p. 46) points out, “Any estimate of the demand curve to a single entrepreneur involves a guess as to both the supply conditions and prices of other sellers, as well as some idea of the demand curve to the industry as a whole.”
- 21.
Upon the modeling of Keynes’s theory, several economists replaced the assumption of perfect competition with the assumption that the firm attempts to adapt productions to demands under constant prices. Development in this direction will be described in Sect. 3.4.
- 22.
“Itayose” was adopted by the Tokyo Grain Exchange until 2013 (after which it was dissolved). “Itaawase” is carried out at the opening and the closing time of the market in most Japanese exchanges. However, it is not a way of attaining market equilibrium. In this method, the price is determined to ensure that the amount of contract satisfying the presented orders is maximized.
- 23.
Since the demand is nonnegative, normal distribution can be used only as an approximation.
- 24.
According to Blinder (1994), out of approximately 200 American firms representing various industries, 50% change prices once or less than once a year, while less than 15% change the price at least once a month. According to Mizuho Sogo Kenkyusyo (2011), out of 120 Japanese firms, 68% consider the revision of prices once or less than once a year.
- 25.
This so-called short-side rule was introduced by Arrow et al. (1951) in a research of the optimal inventory policy.
- 26.
If the planning periods are finite, then the firm must evaluate the inventory that will remain until the end of the planning period in a certain arbitrary manner.
- 27.
- 28.
Quantity adjustment by this method will be examined in Chap. 6.
- 29.
Regarding its proof, see Morioka (2005, pp. 271–272).
- 30.
If , then this maximization problem does not have a solution. This is because the firm can infinitely prolong production without any loss as long as the cost functions of production and storage are both linear.
- 31.
Today, retail sellers of agricultural products adjust their supplies to avoid frequent occurrences of a stockout.
- 32.
“The scarce resource is computational capacity – the mind. The ability of man to solve complex problems and the magnitude of the resources that have to be allocated to solving them, depend on the efficiency with which this resource, mind, is deployed.” (Simon 1978, pp. 12–13).
- 33.
“The decision maker’s model of the world encompasses only a minute fraction of all the relevant characteristics of the real environment.” (Simon 1959, p. 272).
- 34.
“We must surrender the illusion that programmed decision-making is a process of discovering the ‘optimal’ course of action in the real, complex world. … We should view programmed decision-making as a process making choices within the framework set by highly simplified models of real-world problems” (Simon 1958, pp. 57–58).
- 35.
Concerning the concept of autonomous and higher functions of the economy, refer to Kornai (1971). He wrote in that book that “the features of the autonomous functions do not depend on the political and ownership relations of the system” (Kornai 1971, p. 185). However, Kornai (1980) withdrew this view and admitted that the autonomous functions have significantly different features under the socialist and capitalist system.
- 36.
Loose stationarity of the economy might be temporarily lost by serious events like a financial crisis, hyperinflation, shortage of a critical fuel, large-scale war, and natural disaster. The turmoil caused by these events obstructs the normal progression of the autonomous economic process on the established orbit. However, sooner or later, the looped relation between routine behaviors and stationary process can be reconstructed on a new orbit corresponding to the changed conditions.
- 37.
This point is related to the following question raised by Simon: “what is it that maintain the stability of the pattern of behavior in groups of interacting persons?” He adroitly notes that “We do not need a theory of revolution so much we need a theory of the absence of revolution” (Simon 1958, p. 60, emphasis added).
- 38.
As noted above, unused capacities of capital equipment also enhance the flexibility of production. Since it takes a significantly longer time to construct and install equipment than to purchase raw materials, the smooth progress of the quantity adjustment process would require that the production capacity of equipment be set in advance. It would also be essential to consider a certain slack over the expected average demand.
- 39.
Concerning a formal analysis of this point, see Morioka (2005, pp. 83–86).
- 40.
“In markets, where overdemand prevails, inventories are mainly held in order to protect the firm against shortages of raw materials or merchandise. In such a market which is typical in Hungary, a firm’s operations require a relatively high level of input inventories and a low level of work-in-process and finished goods inventories” (Hunyadi 1988, p. 183).
- 41.
Concerning the level, change, and trends of inventories in countries other than Japan, refer to a comprehensive comparative study by Chikán et al. (2018).
- 42.
As Kornai puts it, “changes in stocks yields outstandingly important information of non-price character. They are signals that are most economical of information and they can be observed within the firm” (Kornai 1971, p. 179).
- 43.
Stiglitz also makes a similar comment. According to him, the conventional understanding that “economic relations in capitalist economies are governed primarily by prices” is a “myth.” One of the reasons why this is a myth is that “it ignores the many non-price sources of information used by firms.” Actually, “firms look at quantitative data—like what is happening to their inventories and inventories of other firms” (Stiglitz 1994, pp. 249–250).
- 44.
A possible reason behind this weak theoretical interest in inventory is the lack of statistical data about quantity variables (including inventories) in the nineteenth and early twentieth century.
- 45.
This problem was raised by Morishima (1977). In the latter half of the 1970s, Morishima turned from an admirer of Walras to a radical Keynesian and began to emphasize the significance of quantity adjustment in industrial sectors.
- 46.
In the formal model of General Theory, the equilibrium of the aggregate product market is attained through a change in the real wage rate. Therefore, it cannot be regarded as a model of quantity adjustment.
- 47.
“If we start from a level of output very greatly below capacity, so that even the most efficient plant and labor are only partially employed, marginal real cost may be expected to decline with increasing output or, at the worst, remain constant” (Keynes 1939, p. 44).
- 48.
Lundberg (1964) made a distinction between the consumption goods sector and the investment goods sector. However, since the activities of the former are assumed to be constant except for inventory investment we can regard his model as a single-sector macro model.
- 49.
Like Hawtrey, Lundberg had a clear idea about the roles of inventories. In a reference to activities by retail traders, he writes that inventories held by these traders “act as a buffer that take up the discrepancies between supply and demand” (Lundberg 1964, p. 106, emphasis added).
- 50.
In his numerical calculation, Lundberg assumes that c = 0.9 and k = 0.5. Although this pair of values does not satisfy c(1 + k) < 1, his calculation shows the convergence of the process (Lundberg 1964, p. 201). The reason behind this convergence lies in Lundberg’s implicit assumption that a planned inventory investment cannot be negative. This implies that Y(t) is determined Y(t) = S e(t) + max {kS e(t) − Z(t − 1), 0}. However, it is difficult to find a reason justifying this assumption.
- 51.
The characteristic equation corresponding to system (3.20) is ξ 2 − c(2 + k)ξ + c(1 + k) = 0. It can be easily shown that the dominant root of this equation is less than unity if and only if c(1 + k) < 1. Furthermore, c(1 + k) < 1 implies that this equation has a pair of conjugate complex roots.
- 52.
- 53.
Concerning the derivation of this condition, see Appendix 9, Chap. 4.
- 54.
According to Arrow, the penalty cost is “the loss of the customer’s goodwill and his possible future unwillingness to do business with the firm.” He notes, furthermore, that “Such a penalty cost is real but may be very hard to measure in any precise case” (Arrow et al. 1958, p. 21).
- 55.
Concerning prices, Hicks (1956, p. 145) observed that “both the manufactures and the retailer are, for the most part, ‘price makers’ rather than ‘price takers’; they fix prices and let the quantities they sell be determined by demand.”
- 56.
Although nearly half a century has passed since then, this situation has barely changed. The buffer function of product and raw material inventory cannot be integrated into any economic theory that disregards the demand constraint on production and the successive nature of adjustment.
- 57.
Interestingly, Aoki (1978) attempted to bridge the gap between these different concepts of quantity adjustment. Having built a model of centralized successive allocation in which the central organization assigns each firm the gross output that it should produce, he transformed it into a model of decentralized quantity adjustment, in our sense, by interpreting this constraint as the perception (or forecast) of demand by the firm. See Morioka (2018).
- 58.
In Goodwin (1950), input coefficients are defined in monetary terms, that is, they denotes the value of the good required per production of the unit value of the commodity. However, this does not affect the following argument.
- 59.
- 60.
Lovell’s investigation of inventory cycle was supervised by Leontief (Lovell 1962, p. 267). This fact suggests that there was an interesting link between the multi-sector analysis of quantity adjustment and Leontief’s input-output analysis.
- 61.
Similar to the models of Goodwin and Chipman, the system (33) presupposes that there are enough raw material inventories.
- 62.
Concerning the stability of (3.34) and (3.36), see Sect. 5.1.2, Chap. 5. As will be shown there, if has as its eigenvalue, the necessary and sufficient conditions for the stability of (3.34) and (3.36) in the case of and are and , respectively. Both conditions are not necessary if A does not have negative or complex eigenvalues.
- 63.
While Foster also assumes a method of sales expectation , which he calls “customers demand’s forecast,” here we pick up only the case of the Metzler type expectation.
- 64.
Let λi = μeiθ and assume 0 < θ < 2π, |η| ≤ 1, then \( \overline{\xi}\left({\lambda}_i,\eta \right)<1 \) holds if and only if
$$ \frac{{\left(1-\eta \mu \right)}^2\left(1-\mu \right)\left(1+\mu \left(2\eta +1\right)\right)}{2\eta {\left(1+\eta \right)}^2{\mu}^3}+\cos \theta <1. $$The value of μ equalizing both sides of this inequality in 0 < μ < 1 decreases from 1 to 1/(2η + 1) as θ increases from 0 to π. Concerning the proof of this proposition, see Appendix 5 in Chap. 4.
- 65.
Especially, for negative, is equivalent to .
- 66.
In his attempt to treat final consumption demands as endogenous variables, Foster assumed that these demands are derived from the productions in the same period (Foster 1963, p. 405). Similar to Lovell, by this assumption, Foster introduced a loop of the sequence of events in his model. Foster thought that a simultaneous determination of two sets of variables can be attained, “presumably a re-contracting process,” in which “every industry keeps modifying its orders for inputs, and in turn receives changed order for outputs, until mutual consistency is achieved” (Foster 1963, p. 417). However, the obligation to participate in such an organized adjustment significantly impairs the merit of the quantity adjustment process in which every transaction can be carried out simply by the consent between the buyer and the seller.
- 67.
“The models in this volume illustrate … at micro level, ‘quantitative adaptation’ in an economic system with n participants, with decentralized decision and information” (Kornai and Martos 1981, p. 44).
- 68.
- 69.
The model by Dancs et al. (1981) contained this kind of inconsistency.
- 70.
This result was implicitly anticipated by Metzler (1941). This is because, for η = − 0.5, (3.22) represents \( {S}^e(t)=\frac{1}{2}\left(S\left(t-1\right)+S\left(t-2\right)\right) \), namely, a forecast by the simple moving average of sales in the past two periods. Let k = 0.4, then the upper bound of c for stability corresponding to η = − 0.5 is 0.771, while this bound corresponding to η = 0 (namely, to the static expectation) is 0.714. However, Metzler did not explicitly point out this stabilizing effect of averaging.
- 71.
Simonovits was one of Kornai’s collaborators in Non-Price Control.
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Appendix
Appendix
1. Proof of Theorem 1
First, we consider the case of the finite planning period. Let
and z(0) = z, y = x(1) + z, q = p + l. Then, from (3.1) to (3.3), we have
Let E n, z(y) be the expected values of Π n and \( {R}_n(y) \equiv {E}_{n,z}^{\prime }(y)+c\ \left(y\ge z\right) \), then
Let \( {\eta}^{\ast } \equiv {F}^{-1}\left(\frac{q-c}{q+h-\alpha c}\right) \), R(y) ≡ P′(y) + αcF(y), then we obtain
Therefore, there exists a unique \( {\eta}_1^{\ast } \) such that \( 0<{\eta}_1^{\ast }<{\eta}^{\ast } \), \( {R_{1}}\left({\eta}_1^{\ast}\right)=c \).
Let η n(z) be the optimal y and g n(z) ≡ E n, z(η n(z)) be the corresponding expected profit when the number of planning period is n. From the above results, we have
for \( z\le{\eta}_1^{\ast } \), and
for \( {\eta}_1^{\ast}< z \).
Next, we shall prove that the same results hold for any n and \( {\eta}_n^{\ast } \) increases with n, by mathematical induction. Assume that, for a certain n, there exists a unique \( {\eta}_n^{\ast } \) such that
Since the maximal expected future profit obtained from periods 2 onwards is equal to the expected value of αg n(z), we have
By definition, \( {R}_{n+1}(y)={P}^{\prime }(y)+\alpha {Q}_n^{\prime }(y) \). If \( y\le {\eta}_n^{\ast } \), then the assumption \( {g}_n^{\prime}\left(y-\xi \right)=c \) for 0 < ξ < y leads to \( {Q}_n^{\prime }(y)= cF(y) \) and
Since R′(y) = − (q + h − αc)f(y) < 0 and \( {\eta}_n^{\ast }<{\eta}^{\ast } \), we obtain
If \( y\ge {\eta}_n^{\ast } \), then again by the assumption, we have
This implies R n+1(y) < R(y), therefore
From \( {g}_n^{\prime }(0)=c \) and \( {g}_n^{{\prime\prime} }(z)\le 0 \), it follows
and consequently, \( {R}_{n+1}^{\prime }(y)\le -\left(q+h-\alpha c\right)f(y)<0 \). Therefore, there exists a unique \( {\eta}_{n+1}^{\ast } \) such that
It can easily be confirmed that \( {\eta}_{n+1}^{\ast } \) satisfies
Finally, let us move to the case of the infinite planning period. Since \( {\eta}_n^{\ast } \) is smaller than η ∗ and strictly increases with n, it converges to a certain limit value y ∗ as n infinitely increases. y ∗ is the optimal supply in the case of a positive production. Since E ∞, z(y) = E 1, z(y) + αQ ∞(y), we have
for z < y ≤ η ∗. Therefore, \( {E}_{\infty, z}^{\prime}\left({\eta}^{\ast}\right)=R\left({\eta}^{\ast}\right)-c=0 \), hence η ∗ = y ∗, this implies that y ∗ satisfies (3.5).
2. Proof of Theorem 2
It is sufficient to describe the points differing from the proof of Theorem 1. Let η n(z) be the optimal supply when the number of planning periods is n. Since
there exist unique \( {\overline{y}}_1 \) such that P′(\( {\overline{y}}_1 \)) = C′(0) ≡ c. When \( z<{\overline{y}}_1 \), η 1(z) is a function satisfying P′(η 1(z)) = C′(η 1(z) − z). In this interval, η 1(z) also satisfies \( z<{\eta}_1(z)<{\overline{y}}_1 \), \( 0<{\eta}_1^{\prime }(z)<1 \) because
Since \( {g}_1(z)=-C\left({{\eta}_1(z)}-z\right)+P\left({\eta}_1(z)\right) \), we have
When \( z>{\overline{y}}_1 \), then the firm halts production, that is, η 1(z) = z; hence g 1(z) = P(z), and consequently,
Assume that, for a certain n, the following propositions hold: (i) there exists a unique \( {\overline{y}}_n \) such that \( {R}_n\left({\overline{y}}_n\right)=c \). (ii) η n(z) satisfies
\( \mathrm{for}\ z<{\overline{y}}_n, \) and η n(z) = z for \( z\ge {\overline{y}}_n \). (iii) \( {g}_n^{\prime }(z)<q,\kern0.5em {g}_n^{{\prime\prime} }(z)<0 \). (iv) \( {\overline{y}}_{n-1}<{\overline{y}}_n \), \( {\eta}_{n-1}(z)<{\eta}_n(z)\ \mathrm{for}\ z<{\overline{y}}_n \). (v) g n − 1(z) < g n(z) for \( z<{\overline{y}}_n \) (\( {\overline{y}}_0=0,{g}_0=0 \)).
Because of (iii), it follows \( {Q}_n^{\prime\prime}(y)< qf(y) \), hence we have
Using this, (i) to (iv) are easily confirmed for n + 1. When \( y\le {\overline{y}}_n, \) then (v) leads to \( {Q}_n^{\prime }(y)<{Q}_{n+1}^{\prime }(y) \), and therefore,
These imply \( {\overline{y}}_n<{\overline{y}}_{n+1} \) and \( {\eta}_n(z)<{\eta}_{n+1}(z)\ \mathrm{for}\ z\le {\overline{y}}_n \), respectively.
Using the latter inequality, and considering \( {g}_{n+1}^{\prime }(z)={C}^{\prime}\left({\eta}_{n+1}(z)-z\right) \) for z < η n+1(z), we have
Since \( {R}_n^{\prime }(y)<{P}^{\prime }(y)+\alpha F(y) \), it follows \( {\overline{y}}_n<{F}^{-1}\left(\frac{q-c}{q+h-\alpha q}\right) \) for any n; hence \( {\overline{y}}_n \) and η n(z) converge to \( \overline{y} \) and η(z) satisfying
respectively, where ζ(z) ≡ η(z) − z. By observing that \( {C}^{\prime}\left(\zeta \left(\overline{y}-\xi \right)\right)>{C}^{\prime }(0)=c\kern0.5em \)for \( \xi <\overline{y} \) and C′(ζ(η(0) − ξ)) < C′(η(0)) for ξ < η(0), we have
q − (q + h − αC′(η(0)))F(η(0)) > C′(η(0)),
which implies \( \eta (0)<{y}^{\ast }<\overline{y}. \)
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Shiozawa, Y., Morioka, M., Taniguchi, K. (2019). The Basic Theory of Quantity Adjustment. In: Microfoundations of Evolutionary Economics. Evolutionary Economics and Social Complexity Science, vol 15. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55267-3_3
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DOI: https://doi.org/10.1007/978-4-431-55267-3_3
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