The Basic Theory of Quantity Adjustment

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.

A part of the research on this chapter was supported by JSPS Kakenhi no. 15K03386.

Appendix

1. Proof of Theorem 1

First, we consider the case of the finite planning period. Let

$$ {\Pi}_n \equiv \sum \limits_{t=1}^n{\alpha}^{t-1}\left( ps(t)- cx(t)- hz(t)-l\left(d(t)-s(t)\right)\right), $$

and z(0) = z, y = x(1) + z, q = p + l. Then, from (3.1) to (3.3), we have

$$ {\Pi}_1=\left\{\begin{array}{l} (p + h)d(1)-c\left(y-z\right)- hy\kern1.5em d(1)\le y\\ {} qy-c\left(y-z\right)- ld(1)\kern1.75em d(1)>y\end{array}\right. $$

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

$$ {{{E}}}_{1,z}(y)=P(y)-c\left(y-z\right), \vspace*{-6pt}$$

$$ P(y)\equiv\underset{0}{\overset{y}{\int }}\left( (p + h)\xi - hy\right)f\left(\xi \right)\mathrm{d}\xi +\underset{y}{\overset{\infty }{\int }}\left( q y- l\xi \right)f\left(\xi \right)\mathrm{d}\xi, \vspace*{-6pt}$$

$$ {R}_{1}(y)={P}^{\prime }(y)=q-\left(q+h\right)F(y). $$

Let ${\eta}^{\ast } \equiv {F}^{-1}\left(\frac{q-c}{q+h-\alpha c}\right)$, R(y) ≡ P′(y) + αcF(y), then we obtain

$$ R\left({\eta}^{\ast}\right)={P}^{\prime}\left({\eta}^{\ast}\right)+\alpha c F\left({\eta}^{\ast}\right)=q-\left(q+h-\alpha c\right)F\left({\eta}^{\ast}\right)=c,\vspace*{-1pc} $$

$$ {R_{1}}(0)-c=q-c>0,\kern0.5em {R_{1}} \left({\eta}^{\ast}\right)-c=-\alpha cF\left({\eta}^{\ast}\right)<0. $$

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

$$ {\eta}_1(z)={\eta}_1^{\ast },\kern0.75em {g}_1(z)=-c\left({\eta}_1^{\ast }-z\right)+P\left({\eta}_1^{\ast}\right),\kern0.75em {g}_1^{\prime }(z)=c,\kern0.75em {g}_1^{{\prime\prime} }(z)=0. $$

for $z\le{\eta}_1^{\ast }$, and

$$ {\eta}_1(z)=z,\kern0.75em {g}_1(z)=P(z),\kern0.75em {g}_1^{\prime }(z)={P}^{\prime }(z)<c,\kern0.75em {g}_1^{{\prime\prime} }(z)={P}^{{\prime\prime} }(z)<0. $$

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

$$ {R}_n\left({\eta}_n^{\ast}\right)=c,\kern0.75em {\eta}_{n-1}^{\ast }<{\eta}_n^{\ast }<{\eta}^{\ast}\left({\eta}_0^{\ast }=0\right),\vspace*{-12pt} $$

$$ {\eta}_n(z)=\left\{\begin{array}{l}{\eta}_n^{\ast}\kern0.5em z\le{\eta}_n^{\ast}\\ {}z\kern1em {\eta}_n^{\ast}< z\end{array}\right.,\kern0.5em {g}_n^{\prime }(z)\left\{\begin{array}{l}=c\kern0.5em z\le{\eta}_n^{\ast}\\ \!<c\kern0.5em {\eta}_n^{\ast}< z\end{array}\right.,\kern0.5em {g}_n^{\prime \prime }(z)\le 0. $$

Since the maximal expected future profit obtained from periods 2 onwards is equal to the expected value of αg _n(z), we have

$$ {{{E}}}_{n+1,{z}}(y)=P(y)-c\left(y-z\right)+\alpha {Q}_n(y), \vspace*{-12pt} $$

$$ {Q}_n(y)\equiv{\int_{0}^{y}}{g}_n\left(y-\xi \right)f\left(\xi \right)\mathrm{d}\xi +{g}_n(0)\left(1-F(y)\right)\kern0.5em \left({Q}_0=0\right), \vspace*{-10pt} $$

$$ {Q}_n^{\prime }(y)={\int_{0}^{y}}{g}_n^{\prime}\left(y-\xi \right)f\left(\xi \right)\mathrm{d}\xi $$

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

$$ {R}_{n+1}(y)={P}^{\prime }(y)+\alpha cF(y)=R(y). $$

Since R′(y) =  − (q + h − αc)f(y) < 0 and ${\eta}_n^{\ast }<{\eta}^{\ast }$, we obtain

$$ {R}_{n+1}\left({\eta}_n^{\ast}\right)=R\left({\eta}_n^{\ast}\right)>R\left({\eta}^{\ast}\right)=c. $$

If $y\ge {\eta}_n^{\ast }$, then again by the assumption, we have

$$ {Q}_n^{\prime }(y)={\int_{0}^{y-{\eta}_n^{\ast } }}{g}_n^{\prime}\left(y-\xi \right)f\left(\xi \right)\mathrm{d}\xi +c{{\int_{y-{\eta}_n^{\ast }}^{y}}}g\left(\xi \right)\mathrm{d}\xi < cF(y). $$

This implies R _n+1(y) < R(y), therefore

$$ {R}_{n+1}\left({\eta}^{\ast}\right)<R\left({\eta}^{\ast}\right)=c. $$

From ${g}_n^{\prime }(0)=c$ and ${g}_n^{{\prime\prime} }(z)\le 0$, it follows

$$ {Q}_n^{{\prime\prime} }(y)= cf(y)+{\int^{y}_{0}}{g}_n^{{\prime\prime}}\left(y-\xi \right)\varphi \left(\xi \right)\mathrm{d}\xi \le cf(y), $$

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

$$ {\eta}_n^{\ast }<{\eta}_{n+1}^{\ast }<{\eta}^{\ast },\kern0.75em {R}_{n+1}\left({\eta}_{n+1}^{\ast}\right)=c $$

It can easily be confirmed that ${\eta}_{n+1}^{\ast }$ satisfies

$$ {\eta}_{n+1}(z)=\left\{\begin{array}{cc}{\eta}_{n+1}^{\ast }& \,\,\,\,\,\,z\le{\eta}_{n+1}^{\ast}\\ {}\!\!\!\!\!\!z& \,\,\,\,\,\,{\eta}_{n+1}^{\ast}< z\end{array}\right.,\kern0.75em {g}_{n+1}^{\prime }(z)\left\{\begin{array}{ccc}=c& & \,\,\,\,\,\,z\le{\eta}_{n+1}^{\ast}\\ {}<c& & \,\,\,\,\,\, {\eta}_{n+1}^{\ast}< z\end{array}\right.,\kern1em {g}_{n+1}^{{\prime\prime} }(z)\le 0. $$

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

$$ {E}_{\infty, z}^{\prime }(y)={R}_1(y)-c+\alpha {\int}_{\!\!\!0}^{y}{g}_{\infty}^{\prime}\left(y-\xi \right)f\left(\xi \right)\mathrm{d}\xi =R(y)-c $$

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

$$ {E}_{1,z}(y)=P(y)-C\left(y-z\right), $$

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

$$ {P^{\prime}}(0)=q>c,\kern0.5em {\eta}_1^{\prime }(z)=\frac{C^{{\prime\prime}}\left(y-z\right)}{C^{{\prime\prime}}\left(y-z\right)-{P}^{{\prime\prime} }(y)}. $$

Since ${g}_1(z)=-C\left({{\eta}_1(z)}-z\right)+P\left({\eta}_1(z)\right)$, we have

$$ \kern0.5em {g}_1^{\prime }(z)={C}^{\prime}\left({{\eta}_1(z)}-z\right)<q,\kern0.75em {g}_1^{{\prime\prime} }(z)<0. $$

When $z>{\overline{y}}_1$, then the firm halts production, that is, η ₁(z) = z; hence g ₁(z) = P(z), and consequently,

$$ {g}_1^{\prime }(z)={P}^{\prime }(z)={R}_1(z)<{c}<q,\kern0.75em {g}_1^{{\prime\prime} }(z)<0. $$

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

$$ {R}_n\left({\eta}_n(z)\right)={C}^{\prime}\left({\eta}_n(z)-z\right),\kern0.5em z<{\eta}_n(z)<{\overline{y}}_n,\,\,{} 0<{\eta}_n^{\prime }(z)<1 $$

$\mathrm{for}\ z<{\overline{y}}_n,$ 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

$$ {R}^{\prime}_{n+1}(y)=-\left(q\left(1-\alpha \right)+h\right)F(y)<0 $$

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,

$$ c={R}_n\left({\overline{y}}_n\right)<{R}_{n+1}\left({\overline{y}}_n\right), \vspace*{-18pt} $$

$$ {C}^{\prime}\left({\eta}_n(z)-z\right)={R}_n\left({\eta}_n(z)\right)<{R}_{n+1}\left({\eta}_n(z)\right)\kern0.5em \mathrm{for}\ z\le {\overline{y}}_n. $$

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

$$ {g}_{n+1}^{\prime }(z)\left\{\begin{array}{ll}>{C}^{\prime}\left({\eta}_n(z)-z\right)={g}_n^{\prime }(z)&\quad z<{\eta}_n(z)\\ {}\!\!>c\ge {R}_n^{\prime }(z)={g}_n^{\prime }(z)\ & \quad{\eta}_n(z)\le z<{\eta}_{n+1}(z)\end{array}\right.. $$

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

$$ q-\left(q+h\right)F\left(\overline{y}\right)+\alpha {\int}_0^{\overline{y}}{C}^{\prime}\big(\zeta \left(\overline{y}-\xi \right)\big)f\left(\xi \right)\mathrm{d}\xi =c, $$

$$ q-\left(q+h\right)F\big(\eta (z)\big)+\alpha {\int}_0^{\eta (z)}{C}^{\prime}\big(\zeta \left(\eta (z)-\xi \right)\big)f\left(\xi \right)\mathrm{d}\xi ={C}^{\prime}\left(\zeta (z)\right)\!, $$

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

$$ q-\left(q+h-\alpha c\right)F\left(\overline{y}\right)<c, $$

q − (q + h − αC′(η(0)))F(η(0)) > C′(η(0)),

which implies $\eta (0)<{y}^{\ast }<\overline{y}.$