T. W. Swan: “The Theory of Suppressed Inflation”

Abstract

“When a further increase in the quantity of effective demand produces no further increase in output and entirely spends itself on an increase in the cost-unit fully proportionate to the increase in effective demand, we have reached a condition… of true inflation”. This is Keynes’ definition of the point at which “inflation” begins; it corresponds with his definition of the point of “full employment”.

Appendix to The Theory of Suppressed Inflation

• (a) The Demand Equations (see Part II, paras. 11–14).

1. It has been explained in the text that the form of the equations, expressing the demands for A, B and C as functions of disposable income and “shortages” shifts with certain changes in the status of the commodities. In order to investigate these shifts, we need to define three types of shortage of supplies against demands. These three types are applicable to the case of three commodities; with more commodities the number of types of shortage to be distinguished, and the variety of shifts in the form of the equations would increase. The following definitions are formulated with commodity A as their subjects:

• (i) Primary shortage: A is in “primary shortage” if its supply is less than its normal demand a(Y − T).

• (ii) Secondary shortage: A is in “secondary shortage” if its supply is greater than its “normal” demand a(Y − T) but less than the total of its normal demand plus any demand re-directed to it from B (alone) or alternatively from C (alone). If its supply is less than either of these sums, A is in “double secondary shortage”.

• (iii) Tertiary shortage: A is in “tertiary shortage” if its supply is greater than its normal demand a(Y − T) plus either demand re-directed from B or demand re-directed from C, but less than its normal demand plus the sum of demand re-directed from B and from C.

2. The following is an illustration of the derivation of a set of demand equations, incorporating the (marginal and average) “propensities to consume” a, b and c and the (marginal and average) “re-directive propensities to consume” u, v and w:

Let A be in primary shortage, B in secondary shortage and C in tertiary shortage. Since A is the only primary shortage, the other shortages are traceable ultimately to the shortage of A, which is therefore the sources of all re-directed demand and the recipient of none. Hence

$$A + X_{a} = a(Y - T)$$

(a)

The excess demand for A is re-directed initially to B, C, and savings in amounts—

$$\frac{v}{1 - u}X_{a} \;{\text{to}}\;B,$$

$$\frac{v}{1 - u}X_{a} \;{\text{to}}\;C$$

$${\text{and}}\;\;\left( {1 - \frac{v + w}{{1 - u}}} \right)X_{a} \;{\text{to}}\;S,$$

as the denominator, (1 − u), in these expressions preserves the appropriate proportions in the re-direction of demand, having regard to the exclusion of A from the avenues available for the re-direction of its own demand.

Since C is in tertiary shortage, the outstanding excess demand for C is traceable immediately to the shortage of B, which accordingly receives no demand re-directed from C. The demand for B is therefore B's own normal demand plus the demand initially re-directed from A:

$$B + X_{b} = b(Y - T) + \frac{v}{1 - u}X_{a}$$

(b)

The supply of B being unable to take up all this demand, $X_{b}$ is re-directed to C and savings in the amounts—

$$\frac{1}{1 - u - v}X_{b} \;{\text{to}}\;C,$$

$${\text{and}}\;1 - \frac{w}{1 - u - v}X_{b} \;{\text{to}}\;S.$$

(1 − u − v) in the denominators indicates the closure of the both A and B as avenues available for this excess demand, all of which originated in A (primarily) and B (secondarily) and cannot be re-directed to these commodities.

Hence C receives its share of extra demand from the original re-direction of the excess demand for A, together with its share of the re-direction of the excess demand for B:

$$C + X_{c} = c(Y - T) + \frac{w}{1 - u}X_{a} + \frac{w}{1 - u - v}X_{b}$$

(c)

Now $X_{a} = a(Y - T) - A$ and $X_{b} = b(Y - T) + \frac{v}{1 - u}X_{a} - B$. Substituting the equations can be re-written—

$$A + X_{a} = a(Y - T)$$

(a)

$$B + X_{b} = b(Y - T) + \frac{v}{1 - u}\left[ {a(Y - T) - A} \right]$$

(b)

$$C + X_{c} = c(Y - T) + \frac{w}{1 - u - v}\left[ {(a + b)(Y - T) - A - B} \right]$$

(c)

3. If a term incorporating b(Y − T) or c(Y − T) − C were included in the demand equation for A, the demand shown for A would be anomalously reduced by the inclusion of these negative items which are negative only because the re-direction of the excess demand for A has pushed B above b(Y − T) and C above c(Y − T). This appears to be a stumbling-block to the formulation of general equations which would serve as a common framework irrespective of changes in the status of each commodity. The expression b(Y − T) − B is algebraically the same, whether its sum is a positive quantity (primary shortage) or negative quantity (excess spending). But it is not merely a matter of excluding from the equation from (c) the term b(Y − T) − B or its counterpart whenever it is negative. Equation (c) already contains such a negative item, which is legitimate because in this case the excess of B over b(Y − T) is not due to any shortage of C, but plays a part in determining the amount of demand re-directed from A which will be passed on to C.

4. The general principle is that the second term of the demand equation for any commodity Z should be of the form (c) varied wherever necessary to exclude any reference to a commodity the actual output of which is greater than could be supported by the demand which would remain if the excess demand for Z were zero. The demand for a commodity in free supply or in tertiary shortage may therefore always be written in the form (c). The demand for a commodity in primary shortage can only be so written if the other two commodities are also in primary shortage, or if one is in primary shortage and the other either in double secondary shortage or in secondary shortage in relation to the second commodity in primary shortage. The demand for a commodity in secondary shortage can be written in the form (c) provided that neither of the other commodities is in tertiary shortage or free supply. Similar principles apply to the use of the form (b).

5. As a process of deflation or inflation proceeds, one or all commodities will move into a different status in relation to this classification, and the form of the equations will change. We will get the wrong answer, for instance, if we solve the equation system for a particular variable on the assumption that $X_{a}$ = zero, if the form of the demand equations used excludes the possibility that A may be in free supply. The following is a typical sequence of shifts in the form of the equations, taking as an example a process of deflation:

Stage (1). A, B and C are all in primary shortage:

$$A + X_{a} = a(Y - T) + \frac{u}{1 - v - w}\left[ {(b + c)(Y - T) - B - C} \right]$$

(a)

$$B + X_{b} = b(Y - T) + \frac{v}{1 - u - w}\left[ {(a + c)(Y - T) - A - C} \right]$$

(b)

$$C + X_{c} = c(Y - T) + \frac{w}{1 - u - v}\left[ {(a + b)(Y - T) - A - B} \right]$$

(c)

Stage (2). Deflation reduces (Y − T) to the point at which C is in double secondary shortage: No change in the form of equations.

Stage (3). C reaches a state of secondary shortage in relation to the shortage of A, but would be in free supply if A were in free supply:

$$A + X_{a} = a(Y - T) + \frac{u}{1 - v}\left[ {b(Y - T) - B} \right]$$

(a)

$$B + X_{b} = b(Y - T) + \frac{v}{1 - u - w}\left[ {(a + c)(Y - T) - A - C} \right]$$

(b)

$$C + X_{c} = c(Y - T) + \frac{w}{1 - u - v}\left[ {(a + b)(Y - T) - A - B} \right]$$

(c)

Stage (4). C reaches a state of tertiary shortage, i.e. C would be in free supply if either A or B were in free supply, but is in short supply taking into account the demand re-directed from both together:

$$A + X_{a} = a(Y - T) + \frac{u}{1 - v}\left[ {b(Y - T) - B} \right]$$

(a)

$$B + X_{b} = b(Y - T) + \frac{v}{1 - u}\left[ {a(Y - T) - A} \right]$$

(b)

$$C + X_{c} = c(Y - T) + \frac{w}{1 - u - v}\left[ {(a + b)(Y - T) - A - B} \right]$$

(c)

With this set of equations, it is legitimate to assume $X_{c}$ = zero. Previously, none of the three excess demands could have been assumed = zero.

Stage (5). Further deflation reduces (Y − T) to the point at which B is in secondary shortages:

$$A + X_{a} = a(Y - T)$$

(a)

$$B + X_{b} = b(Y - T) + \frac{v}{1 - u}\left[ {a(Y - T) - A} \right]$$

(b)

$$C + X_{c} = c(Y - T) + \frac{w}{1 - u - v}\left[ {(a + b)(Y - T) - A - B} \right]$$

(c)

It is now legitimate to assume also $X_{b}$ = zero. The demand for B, with $X_{b}$ zero, could be written alternatively in the form (c). But $X_{b}$ cannot be put = zero unless $X_{c}$ is put = zero, in view of the assumption at Stage (4) and C is in tertiary shortage.

Stage (6). Finally, the primary shortage of A is eliminated. At this stage $X_{a}$, $X_{b}$ and $X_{c}$ are all necessarily = zero, and the supply of each commodity is in equilibrium with its normal demand. The three equations may be written in any of the forms considered, since the second term on the R.H.S. will always be zero, however it is written.

6. All this is very confusing and complicated. No doubt a real mathematician would manage to find some general form which would avoid these shifts. In Part III of the text, only stages (4), (5) and (6) are considered, C being assumed to be in tertiary shortage from the beginning, and B in primary shortage initially but later reaching a state of secondary shortage, while A remains in primary shortage.

• (b) The Solution of the “Model” Equations for Certain Variables in Stage (4)

7. With the demand equations as in Stage (4), and “model” is:

$$A + B + C = Y{-}I$$

(1)

$$S = Y - T - A - B - C$$

(2)

$$A + X_{a} = a(Y - T) + \frac{u}{1 - v}\left[ {b(Y - T) - B} \right]$$

(3)

$$B + X_{b} = b(Y - T) + \frac{v}{1 - u}\left[ {a(Y - T) - A} \right]$$

(4)

$$C + X_{c} = c(Y - T) + \frac{w}{1 - u - v}\left[ {(a + b)(Y - T) - A - B} \right]$$

(5)

$$S - X_{s} = (1 - a - b - c)(Y - T)$$

(6)

8. Taking $A^{*}$, $B^{*}$, $I^{*}$, $Y^{*}$, and $T^{*}$ as the independent variables, (i.e. assuming full employment and excess demand for each of the three commodities), the following solutions can be obtained by simple substitution or re-arrangement:

$$\begin{aligned} X_{c} & = \left( {A^{*} + B^{*} } \right)\left( {1 - \frac{w}{1 - u - v}} \right) + I^{*} - T^{*} \left[ {\frac{w}{1 - u - v}(a + b) + c} \right] \\ & \quad \quad \quad - Y^{*} \left[ {1 - \frac{w}{1 - u - v}(a + b) - c} \right] \\ \end{aligned}$$

(7)

The values of the independent variables must be such that $X_{c}$ is not less than zero; otherwise, the assumption of full employment ($Y^{*}$) must be dropped.

$$X_{b} = \left( {Y^{*} - T^{*} } \right)\left( {a\frac{v}{1 - u} + b} \right) - A^{*} \frac{v}{1 - u} - B^{*}$$

(8)

$$X_{a} = \left( {Y^{*} - T^{*} } \right)\left( {a + b\frac{u}{1 - v}} \right) - A^{*} - B^{*} \frac{u}{1 - v}$$

(9)

$$X_{s} = I^{*} - Y^{*} (1 - a - b - c) - T^{*} (a + b + c)$$

(10)

9. If $X_{c}$ is taken as the fifth independent variable (= zero), and T as unknown, the value of T which will eliminate the excess demand for C, given the levels of $A^{*}$, $B^{*}$, $I^{*}$ and $Y^{*}$ is obtainable directly from (7). But a corresponding value of T cannot be obtained from (8), (9) and (10), because with $Y^{*}$ fixed, $X_{a}$, $X_{b}$, or $X_{s}$ cannot be zero without making $X_{c}$ negative.

10. If the possibility of less than full employment output is admitted by taking Y as unknown, $X_{c}$ must be taken as zero and T may be restored to independence in place of Y. From (7), with $X_{c}$ = zero—

$$Y = \frac{{\left( {A^{*} + B^{*} } \right)\left( {1 - \frac{w}{1 - u - v}} \right) + I^{*} - T^{*} \left[ {\frac{w}{1 - u - v}(a + b) + c} \right]}}{{1 - \frac{w}{1 - u - v}(a + b) - c}}$$

(11)

This is the familiar “multiplier” equation for Y with the “bottlenecks” of A and B recognized in the numerator and the effective marginal saving rate in the denominator.

From (11), substituting for Y in (8), (9) and (10), the levels of excess demand for B and A, and of excess savings, are given by—

$$X_{b} = \frac{\begin{gathered} \left( {I^{*} - T^{*} } \right)\left( {a\frac{v}{{1 - u}} + b} \right) - A^{*} \frac{v}{{1 - u}}\left( {1 - a - b\frac{{1 - u - w}}{v} - c} \right) \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - B^{*} \left( {1 - a\frac{{v + w}}{{1 - u}} - b - c} \right) \hfill \\ \end{gathered} }{{1 - \frac{w}{{1 - u - v}}(a + b) - c}}$$

(12)

$$X_{a} = \frac{\begin{gathered} \left( {I^{*} - T^{*} } \right)\left( {a + b\frac{u}{{1 - v}}} \right) - A^{*} \left( {1 - a - b\frac{{u + w}}{{1 - v}} - c} \right) \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad - B^{*} \frac{u}{{1 - v}}\left( {1 - a\frac{{1 - v - w}}{u} - b - c} \right) \hfill \\ \end{gathered} }{{1 - \frac{w}{{1 - u - v}}(a + b) - c}}$$

(13)

$$X_{s} = \frac{{\left( {I^{*} - T^{*} } \right)\left( {a + b} \right)\left( {1 - \frac{w}{1 - u - v}} \right) - \left( {A^{*} + B^{*} } \right)(1 - a - b - c)\left( {1 - \frac{w}{1 - u - v}} \right)}}{{1 - \frac{w}{1 - u - v}(a + b) - c}}$$

(14)

At this point, we may explore the effect of changes in I, T, A and B on Y, $X_{s}$$X_{b}$, $X_{a}$, and $X_{s}$; but we must not make these changes such that an excess demand for C would emerge or (if we are considering the effect on $X_{a}$) such that B would cease to be a primary shortage. Nor can we assume that $X_{b}$, $X_{a}$, or $X_{s}$ is zero.

• (c) The Solution of the “Model” Equations for Certain Variables in Stages (5) and (6)

11. If the changes in $I^{*}$, $T^{*}$, $A^{*}$ and $B^{*}$ are such that B becomes a secondary shortage, the demand equation for A (Eq. 3) changes its form and becomes—

$$A + X_{a} = a(Y - T)$$

This causes no change in Eqs. (11), (12) or (14), in which Eq. (3) plays no part, but Eq. (13) becomes—

$$X_{a} = \frac{{a\left( {I^{*} - T^{*} } \right) + aB^{*} \left( {1 - \frac{w}{1 - u - v}} \right) - A^{*} \left( {1 - a - b\frac{w}{1 - u - v} - c} \right)}}{{1 - \frac{w}{1 - u - v}(a + b) - c}}$$

(15)

12. In this stage $X_{b}$ may be assumed = zero, provided that I or T takes its place as an unknown. From (12) the value of I or T which will eliminate excess demand for B may be discovered, and the effect of this value on $Y$, $X_{s}$ and $X_{a}$ may be seen in (11), (14) and (15).

13. Alternatively $X_{b}$ may be assumed = zero and B taken as an unknown (of value less than its “bottleneck” level $B^{*}$). In this case Eqs. (11), (14) and (15) must be re-worked to eliminate their dependence on $B^{*}$. Substituting for B and C in (1) from (4) and (5),

$$Y = \frac{{A^{*} \left( {1 - \frac{v + w}{{1 - u}}} \right) + I^{*} - T^{*} \left( {a\frac{v + w}{{1 - u}} + b + c} \right)}}{{1 - a\frac{v + w}{{1 - u}} - b - c}}$$

(16)

Substituting for Y in (10),

$$X_{a} = \frac{{a\left( {I^{*} - T^{*} } \right) - A^{*} (1 - a - b - c)}}{{1 - a\frac{v + w}{{1 - u}} - b - c}}$$

(17)

$$X_{s} = \frac{{a\left( {I^{*} - T^{*} } \right)\left( {1 - \frac{v + w}{{1 - u}}} \right) - A^{*} (1 - a - b - c)\left( {1 - \frac{v + w}{{1 - u}}} \right)}}{{1 - a\frac{v + w}{{1 - u}} - b - c}}$$

(18)

14. Now at last it may legitimately be assumed that $X_{a}$ or $X_{s}$ is zero. The value of T, I or A which will achieve this result may be learned from (17) or (18), and the consequence for Y from (16). It is apparent that any set of values which makes $X_{a}$ zero will make $X_{s}$ zero, and vice versa.