Abstract
In this chapter, we will extend the model of the quantity adjustment process presented in Chap. 4 in several directions. Section 5.1 examines the models considering work-in-process inventory, partial adjustment of production volume, and heterogeneity of firms within a sector. It will be shown that, under certain conditions, the stability conditions of these models are given in similar forms to those in Chap. 4, and thus, the introduction of the above factors into the model does not change the basic dynamic properties of the process. Section 5.2 traces the process with stockout, rationing, and bottleneck. A stockout of product inventory leads to a rationing of sales volume among buyers, whereas a stockout of raw material inventory leads to a reduction in the production volume due to bottleneck. Numerical computations will highlight the buffer role of inventories in the adaptations of the whole economy to a one-time increase or random fluctuations in final demand. Finally, Sect. 5.3 investigates the effects of mid- and long-term changes in final demand. It will be confirmed that, while quantity adjustment can follow the gradual movement of final demand accompanied by the inducement of consumption demand from past income, it cannot suppress the oscillations caused by unstable movements of final demand itself.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The model described below was first presented in Morioka (1992).
- 2.
If additional raw materials are necessary in the second stage, input matrix A is replaced by \( A + A^{\prime} \), where \( A^{\prime} \) is the matrix of inputs additionally required to produce a unit of product i in the second stage. This point does not affect the essence of the following argument.
- 3.
See the argument on the production and ordering decisions in Subsect. 4.1.2, Chap. 4.
- 4.
In other words, \( {s}_i^e(t) \) is used to forecast three variables: sales in week t, (i) production in weekt + 1, and production of works-in-process in week t + 1.
- 5.
In Lovell (1962), x i(t) is determined so that x i(t) + z i(t − 1) covers s i(t), and therefore, partial adjustment takes the form \( {x}_i(t)={s}_i^e(t)+{r}_i\left({k}_i{s}_i^e(t)-{z}_i\left(t-1\right)\right)\!. \) Meanwhile, here we assume that x i(t) is determined so that x i(t) + z i(t) covers s i (t + 1). Note that, in our model, both s i (t + 1) and z i (t) are unknown at the beginning of week t.
- 6.
Let ω ≥ 1 be an eigenvalue of Ω and ξ ≡ [ξ 1, ξ 2] (both ξ 1 and ξ n are 1 × n vectors) be the corresponding eigenvector. Then, it follows
$$ {\omega \xi}_1={\xi}_1 AQ\left(\omega \right),\kern0.75em Q\left(\omega \right)=\left(\omega \left(I+ BR\right)-\left(I-R+ BR\right)\right){\left(\omega I-\left(I-R\right)\right)}^{-1}, $$hence ω is also an eigenvalue of AQ(ω), thus ρ(AQ(ω)) ≥ 1. Let q i be the i-th diagonal element of Q(ω), then
$$ \left|{q}_i\right|=\left|\frac{\omega \left(1+{b}_i{r}_i\right)-\left(1-{r}_i+{b}_i{r}_i\right)}{\omega -\left(1-{r}_i\right)}\right|=\left(1+{b}_i{r}_i\right)\left|\frac{\omega -\frac{1-{r}_i+{b}_i{r}_i}{1+{b}_i{r}_i}}{\omega -\left(1-{r}_i\right)}\right|\le \frac{2-{r}_i+2{b}_i{r}_i}{2-{r}_i}. $$Hence, 1 ≤ ρ(AQ) ≤ λ F(AQ +) ≤ λ F(A(I+2BR(2I − R)−1). Therefore, (5.23) implies ρ(Ω) < 1. The same result can be directly obtained by rewriting (4.30) in Chap. 4 as
$$ X(t) = X(t-1)\Theta +\left[d,0\right], \Theta \equiv\left[\begin{array}{cc}\left(I+\Gamma B\right)A+\left(I-\Gamma \right)& I\\ {}-\left(I-\Gamma +\Gamma B\right)A& \mathrm{O}\end{array}\right]. $$where X(t) ≡ [ξ(t), ξ(\( t{-}1 \))], ξ(t) ≡ s e(t)Γ−1. If R = Γ, then Θ and Ω are different only in the direction of muttiplying A.
- 7.
This similarity in the stabilizing effect between partial inventory investment and forecast formation by the geometric moving average was first shown in Morioka (2005). While the partialization of inventory investment is related to increasing marginal cost (see the argument in Sect. 3.2.3, Chap. 3), averaging of past demands is not directly related to the conditions of production.
- 8.
Blinder and Maccini (1991) reported that estimates of the reaction coefficient ( in our model) obtained by econometric analyses were usually very small. For example, the estimate concerning manufactures that Lovell (1961) gave based on US economic data for 1948–1955 was 0.1521. However, an estimate of the reaction coefficient in partial adjustment can also be interpreted as that of the smoothing constant in demand forecast by the geometric moving average. If firms make their demand forecasts by the geometric moving average, it is not surprising that this rate takes a small value around 0.15 to 0.3. The observed slowness of adjustment likely reflects the gradual nature of the revision of demand forecast.
- 9.
In this numerical example, firm (1,1) is more technologically efficient than firm (1,2). Thus, the larger \( {\delta}_{21}^{11} \) and \( {\delta}_{22}^{11} \) are, the more efficient the economy becomes. Similarly, since firm (2,1) is more technologically efficient than firm (2,2), the larger \( {\delta}_{11}^{21} \) and \( {\delta}_{12}^{21} \) are, the more efficient the economy becomes. On the Frobenius root of A, it holds
$$ \frac{1}{\sqrt{5}}={\lambda}^{\mathrm{F}}\left(\left[\begin{array}{cc}0& 0.5\\ {}0.4& 0\end{array}\right]\right)<{\lambda}^{\mathrm{F}}\left(A\right)<{\lambda}^{\mathrm{F}}\left(\left[\begin{array}{cc}0& 0.6\\ {}0.5& 0\end{array}\right]\right)=\sqrt{\frac{3}{10}}. $$However, it should be noted that, in more general cases, the relative efficiency of each technique depends on the relative prices.
- 10.
Concerning this point, refer to Nikaido (1961, pp. 85–89). Here we assume that technologies of firms producing the same product are not significantly different. If this is not the case, then it would be meaningless to think about matrix A *.
- 11.
As we have argued in Chap. 3, this priority often plays a significant role under the predominance of purchase competition. In a market characterized by chronic shortage, buyers must make various efforts to win the favor of sellers.
- 12.
- 13.
In order to describe the short-term adjustment process in the socialist system, Kornai and Simonovits (1981), Martos (1990) and Morioka (1991–1992) developed models in which each firm adjusts productions and sales based on the deviations of the backlog of unfulfilled orders from their normal levels.
- 14.
As in the previous chapter, \( {\tilde{s}}_i(t) \) is assumed to be delivered at the beginning of week t + 1.
- 15.
Since in this case the increases rates of final demand are uniform among sectors, patterns of changes of the variables belonging to different sectors do not show any considerable differences.
- 16.
Here the term “linear process”is an abridged expression of a process following a linear difference equation.
- 17.
If the final demand is highly prioritized over the intermediate demand in the rationing at the time of stockout, then the economy continues to shrink because of the shortage of raw materials.
- 18.
A decline in the demand for capital investment requires a raise in real interest rates, while a decline in the demand for consumption goods requires a reduction in real wage rates.
- 19.
This gross income includes (s e(t) − s(t))p of the (aggregated) unintended product inventory investment and (s e(t) − s(t))Ap of the (aggregated) unintended raw material inventory investment.
- 20.
Here, we do not consider the effect that income distribution between wage and profit might exert on consumption demand.
- 21.
Let \( \overline{x} \) be the vector of production capacities, then constraint \( {x}^{\ast}\le \overline{x} \) can be rewritten as (β i − d i)p i − ε i(β 1 p 1+⋯+β n p n) ≥ 0 for any i, where β i is the i-the element of vector \( \overline{x}\left(I-A\right)\!.\) These linear inequalities restrict the range of relative prices.
- 22.
In the following analysis, prices and production conditions are assumed to remain unchanged. Certainly, the longer the considered period is, the less plausible the assumption of the constancy of these factors becomes. Nevertheless, our analysis would have some significance for understanding the way through which the multiplier process proceeds under variable consumption demand, as long as prices and production conditions changes gradually.
- 23.
Equation (5.51) presupposes that there is a 1-week lag from gaining of income to its spending. If income is spent instantaneously, it is replaced by c(t) = y(t)ɛ, and the process corresponding to this consumption function is unstable under the above parameters.
- 24.
Concerning this point, refer to Duesenberry (1949).
- 25.
In an attempt to estimate the consumption function, Friedman (1957) defined “permanent income” as a moving weighted average of incomes in past periods.
References
Blinder, A. S., & Maccini, L. J. (1991). Taking stock: A critical assessment of recent research on inventories. Journal of Economic Perspective, 5(1), 73–98.
Duesenberry, J. S. (1949). Incomes, saving, and the theory of consumer behavior. Harvard: Harvard University Press.
Friedman, M. (1957). A theory of the consumption function. Princeton: Princeton University Press.
Kornai, J. (1980). Economics of shortage. Amsterdam: North-Holland.
Kornai, J. (1992). The socialist system: The political economy of communism. Princeton: Princeton University Press.
Kornai, J., & Simonovits, A. (1981). Control by order signals. In Kornai & B. Martos (Eds.), Non-price control (pp. 267–280). Amsterdam: North-Holland.
Lovell, M. C. (1961). Manufacturers’ inventories, sales expectations, and acceleration principle. Econometrica, 29(3), 293–314.
Lovell, M. C. (1962). Buffer stocks, sales expectations and stability: A multi-sector analysis of the inventory cycle. Econometrica, 30(2), 267–296.
Martos, B. (1990). Economic control structures: A non Walrasian approach. Amsterdam: North-Holland.
Morioka, M. (1991–1992). Two types of the short-term adjustment process (in Japanese: Tanki choseikatei no niruikei). Keizai Ronso (Kyoto University), 148(4–6):140–141, 149(1–3):79–96.
Morioka, M. (1992). “The autonomous control by inventory stocks” (In Japanese: Zaiko stokku ni motozuku jiritsuteki seigyo). Kyoto Daigaku Keizai Ronsyu, 5, 45–59.
Morioka, M. (2005). The economic theory of quantity adjustment: Dynamic analysis of stockout avoidance behavior (in Japanese: Suryo chosei no keizai riron). Tokyo: Nihon Keizai Hyoronsya.
Nikaido, H. (1961). Linear mathematics for economics (In Japanese: Keizai no tameno Senkei Sugaku). Tokyo: Baihukan.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Japan KK, part of Springer Nature
About this chapter
Cite this chapter
Shiozawa, Y., Morioka, M., Taniguchi, K. (2019). Extensions of Model Analysis of the Quantity Adjustment Process in Several Directions. 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_5
Download citation
DOI: https://doi.org/10.1007/978-4-431-55267-3_5
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-55266-6
Online ISBN: 978-4-431-55267-3
eBook Packages: Economics and FinanceEconomics and Finance (R0)