THE RELATION between the quantity of money and the level of prices and incomes is generally treated by modern monetary theorists in terms of a “closed” economy, that is, without allowing for international trade as an integral part of the theory. In his “Monetary Analysis of Income Formation and Payments Problems,”1 J. J. Polak has shown how the existence of international trade, and the consequent possibility of balance of payments deficits, modifies in an essential way the theoretical analysis of the monetary system.

THE RELATION between the quantity of money and the level of prices and incomes is generally treated by modern monetary theorists in terms of a “closed” economy, that is, without allowing for international trade as an integral part of the theory. In his “Monetary Analysis of Income Formation and Payments Problems,”1 J. J. Polak has shown how the existence of international trade, and the consequent possibility of balance of payments deficits, modifies in an essential way the theoretical analysis of the monetary system.

The central element in his theory is that a given deficit in the balance of payments has cumulative effects on the monetary supply: it directly reduces the quantity of money until the consequent income and import changes are of sufficient size to correct the initial disequilibrium. The phenomenon is thus part of economic dynamics; that is, it relates the present state of certain economic variables to the rate of change of those and other variables. The use of mathematics seems to be unavoidable in analysis of this kind; indeed, the use of several alternative mathematical representations is occasionally of value in bringing out the specific role of the various relationships in the system.

The first part of this paper reformulates Polak’s system in terms of infinitesimal and continuous changes, thus replacing his difference equation by a differential equation. As a consequence, certain relations affecting liquidity that were implicit in his model—in the sense that they influenced movements within each period—have necessarily to be made explicit, so casting further light on the essential elements of the adjustment process. This generalization also allows another version of the model, of the kind used in the work of the Netherlands Bank, to be treated as a special case.

The implications of the somewhat heavy mathematics used in the first part of the paper are discussed in the second part in terms of a numerical example which brings out the differences between the various models.

The third part of the paper examines the effects of varying income and price elasticities of foreign trade in a model of this type, so relaxing the unitary elasticities assumed in this work hitherto. It is shown that elasticities greater than unity lead to a more rapid adjustment of the system, and hence that current incomes and imports depend to a larger extent on current exogenous shocks (and to a correspondingly smaller extent on the earlier state of the system).

I. A Continuous Formulation

The theory may be set out in terms of the following six equations, which are the simplest that it has been found possible to devise to represent the system in continuous terms. The first is the familiar Cambridge equation relating a desired level of liquidity L^ (distinguished here from actual liquidity, L) to the level of income:
Secondly, there is an equation relating the change in actual liquidity to the balance of payments which, in the differential form, may be written as

where X is the value of exports and M that of imports.

Thirdly, domestic expenditure, C,2 may be supposed to be equal to income but to be depressed by any excess of desired over actual liquidity:


and, fourthly, imports may be taken as a constant fraction of income:


As an alternative, imports may be taken as a fraction of C, so as to be proportionately influenced by the liquidity situation. However, this and other variants lead to rather similar results, apart from changes in the constants.

Fifthly, it is assumed that exports are exogenously determined, so that we write


Finally, we define national income as the sum of domestic expenditure, C, plus exports less imports:


It will be apparent that (4) is equivalent to assuming a unitary income elasticity of demand for imports and a unitary elasticity with respect to prices; similarly, (5) may be regarded as implying a unitary price elasticity for exports (these assumptions are relaxed in Section III below). An additive term to represent any given rate of credit creation can easily be introduced on the right-hand side of (2) without altering the basic mathematics.

In this system a disequilibrium, say a deficit, in the balance of payments is corrected by a direct fall in the money supply via (2), followed by a fall in domestic expenditure via (3), a fall in income via (6), and a fall in imports via (4). The reductions continue until the deficit in (2) is eliminated.3 The adjustment process is thus close to that envisaged under classical price-specie-flow theory, except that the role played by price-and-income adjustments under the classical theory is played by income adjustments alone under the present theory; adjustments that are due to higher prices are taken into account only insofar as the elasticities are assumed to be unity.

The relations (1)-(6) may be combined to yield a differential equation in any of the variables concerned. Thus, for liquidity, there results


where v=1k. For imports a more complicated equation, involving also the rate of change in ƒ, is found:


where α=mv/(a+mv)(9a)

and β=amv/(a+mv).(9b)

For income, an expression similar to (8) holds, since income and imports are proportional to one another. It will be noted that only the productthe rate of change in of m and v enter into the equations, a property which was found to hold approximately in the earlier treatment by Polak and Boissonneault.4

Equation (7) may be solved for the moment by treating ƒ as a constant (if ƒ varies over time, the more complex result below holds), in which case the path of liquidity is given by


where L0 is the arbitrary value of liquidity at t = 0. With the passage of time the exponential term tends to zero, and liquidity tends to its equilibrium value, ƒ/mv (that is, a fraction, l/v of income, where income is equal to the level of imports—equal to exports—divided by the import propensity, m). Further, any exogenous disturbance of L from that level, say by an amount xo, leads to subsequent divergences of size which, again, die out exponentially.

The solution for the more complicated equation (8), when/is allowed to vary with time, is given by


as may be verified by differentiating to yield


as required by (8).

Since empirical observations on the variables are available only at discrete intervals of time, it is necessary to take the argument a stage further (for empirical application) for variables, such as imports or income, which are flows over time (but note that for liquidity, which is a stock measured at a point of time, it is necessary to go only part of the way, as noted below). Accordingly we define a variable


as being imports during the year θ, that is imports summed from the end of year (θ—1) to the end of year θ

Further, we assume that the exogenous variable ƒ(t) remains at a constant level during each calendar year, and then rises instantaneously at the end of the year to a new value. Thus, if the point t in (11) occurs during the calendar year θ, ƒ(θ) will be the constant value of exports that has ruled and influenced M(t) since the end of year (θ—1), a period which in general is less than a year; however ƒ(θ—1) will have exerted its influence for a complete calendar year and so will all earlier values of ƒ(t).

Hence, considering the first component of the integral (11), we have, say,


Integration with respect to t according to (12) then yields this component of imports summed over any calendar year; thus imports during the year θ=0 are


To evaluate the second component of the integral of (11), say


we note that ƒ′(t) = 0 at all points except at year-ends. At the latter points, f{f(θ)f(θ1)}/ε,, where ε is the arbitrarily small time interval during which the rise in ƒ takes place; consequently,




And, as before,


It may be noted in passing that the coefficients in (13) sum to unity, and those in (14) sum to zero.

The final result, with imports in the current year expressed in terms of current and earlier exogenous impulses, is then


Apart from the first term, the coefficients decline exponentially with time. Similar coefficients apply for the income equation, apart from the factor of proportionality, m ; the coefficients for the liquidity equation are clea’rly given by (13) on setting θ= t = 0.

Expressions of the cumbersome type just derived are necessary for a precise numerical analysis of empirical data. However, for other purposes it is convenient to summarize the coefficients in terms of an average weighted lag, showing the length of time by which movements in the dependent variable lag behind movements in the exogenous variable. Such an average lag may be defined as


where x(t), the weight attached to t in calculating its average, is the divergence of a variable from equilibrium t periods after it has been disturbed.

In the case of the liquidity variable, considered above in (10), the lag is of a particularly simple form. The divergence from equilibrium, as argued above, is given by




the lag thus depends only on the coefficient β in the differential equation (7).5 On substituting for β from (9b), we find


so that liquidity adjusts more rapidly, the larger are α and mv.

As an alternative definition of the adjustment lag, we may consider the following approach. Between t and t + εt, the variable considered moves from x(t) to x + εx; if we consider the whole range of movement of x from its initial value x0 to its equilibrium level x = 0, an average lag is given by


This is identically equivalent to


which is the total discrepancy from equilibrium of the variable summed over all future time (for example, if imports rise as a result of some exogenous impulse, it is the excess of what imports would have been if they had risen immediately to their equilibrium value over what they actually were) divided by the ultimate change in the variable. For the case of exponential adjustment (18), the average lag so calculated will be found to be identical with that just derived in (19) (though for other paths of adjustment this equivalence does not generally hold); since the solutions here are always of an exponential form, however, either definition may be used, according to convenience, in what follows.

The lag for imports behind the exogenous impulses can be derived most simply by using symbolical methods and the operator D=t.

Equation (8) is first rewritten in the form


so that the solution for M in terms of f is symbolically given by


M is thus seen to be the weighted average of two components; the component of weight α has a zero lag on ƒ, and that of weight (1—α) has a lag of 1/β onƒ.6 The total lag is therefore


which from (9a) and (9b) reduces to


Hence the mean lag of imports (and similarly of income) is independent of the liquidity adjustment parameter a and depends only on the product mv.

This somewhat surprising finding may be explained by considering a system originally in equilibrium, with imports equal to exports, disturbed by an exogenous rise in the latter. Equilibrium is re-attained when imports rise to the new level of exports.7 As long as imports are below this level, the country is accumulating liquidity; furthermore, imports will remain below equilibrium until the additional liquidity accumulated in this way is equal to the amount desired at the higher level of income. This amount depends only on the parameters mv, but not on a since this does not affect the equilibrium values of the system. The time-lag as calculated from (17b) is thus seen to be equal to the additional desired liquidity divided by the rise in imports.

It may be worth emphasizing that, although the mean time-lag is independent of α, the distribution of the weighted response function still depends on α (subject to the restriction of a constant mean). From the derivation of (23) it will be clear that the greater is α, the greater will be the immediate response of imports; from (9a) it appears that, for a given value of mv α rises when α falls, so that a higher value of α leads to a lower immediate response of imports.

II. Some Numerical Comparisons

The implications of the previous section may be grasped more easily with the help of the numerical examples set out in Tables 1 and 2. These are based on illustrative values of an import propensity, m=0.3, and a velocity of circulation of v = 3.3 (that is, k = 0.3).

It is convenient to consider first the theoretically lower extreme at which α = 0, so that domestic expenditure is independent of liquidity. The system of equations, as far as they concern the components of the national expenditure, then reduces to


and X=f(t).

This, of course, is reminiscent of rudimentary multiplier models; we find, not surprisingly, that the current values of the exogenous variables (here, exports) fully determine imports and income, and no weight is to be attached to preceding exogenous values. There is no mechanism in operation here which would ever bring liquidity to its desired level.

At the other extreme, where α = ∞, actual liquidity is immediately adjusted to desired liquidity without lag (as follows from equation (3)). Consequently, the system reduces to the following extreme position (reminiscent of the writings of Hume and Ricardo), in which the volume of money has the crucial role:


and dLdt=XM=mY+f(t).

The level of income depends on the stock of money, and that depends on how much gold has been imported in the past; the latter, in turn, depends negatively upon the level of income. The outcome is shown in the series of coefficients in the rows labeled ∞ in the tables; 63 per cent of the current level of (actual) liquidity is influenced by the current value of exports, and the remaining 37 per cent is due to exports in earlier years, in the proportions shown in Table 1. For imports, there

Table 1.

Coefficients for the Determination of Current Liquidity in TERMS of Previous Exogenous Impulses

(m = 0.S: v = S.S for alternative values of a)

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is an obverse tendency; 37 per cent of current imports is attributable to the current level of exports and 63 per cent to earlier years’ exports, in the proportions shown in Table 2.

Table 2.

Coefficients for the Determination of Current Imports in Terms of Previous Exogenous Impulses

(m = 0.S: v = S.S for alternative values of a)

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The values shown in this row are taken from the corresponding rows, with interpolation, of Table 3 in the study by J. J. Polak and Lorette Boissonneault, Staff Payers, Vol. VII (1959-60), p. 358.

It will be noted that there is a simple relation between the/(O) columns in Tables 1 and 2. Their sum is unity for all values of a, reflecting the fact that a rise in export receipts must either result in higher liquid balances or be spent on imports.

Intermediate values of a between zero and infinity correspond, no doubt, more closely to the real world; when income falls and it is con-venient to hold lower average balances, people adjust to their lower receipts partly by running down their excess liquid balances and partly by lower spending. Since normal balances turn over v times a year, excess liquid balances may be expected to be eliminated at a rate which in general exceeds this; in other words, α presumably exceeds v.

The case α = 2v(= 6.7 in the present example) deserves special attention. In this case, our coefficients for imports are very close to those found in the finite difference model used by Polak. That model contains fewer relations than the model set out here and appears to be simpler; further comment is therefore in order.

Polak works in terms of income periods (so that v ≡ k ≡ 1); the central equation of his model (instead of our equations (1), (3), and (4)) is then simply


where L represents liquid balances at the end of the period and Y is income earned during the period. Liquid balances are equal to one period’s income; if income rises, liquid balances move so that by the end of the period they are again in equilibrium. How they move within the period is not specified; but by following the approach of this paper, it is found that there is an exponential movement toward the new equilibrium.

The existence of the implicit lag (or lead8) in ((1a)) is not a necessary part of the finite difference approach; it can be eliminated, for example, by supposing that income is related to average balances (rather than end-of-period balances), where average balances are defined as the average of liquid balances at the beginning and end of the period. Thus ((1a)) would be replaced by


On solving this system it is found, not surprisingly, that the coefficients obtained are, for practical purposes, equal to those in the “∞” rows of the tables; that is, this system corresponds to the differential case with α = ∞.

However, the lag in (la) cannot be eliminated by defining L and Y to relate to the same instant of time without leading to difficulties elsewhere in the system. The reason for this is that the relations


and ΔLXM

require that M, X, and Y relate to a period of time, but ΔL is required to measure the change between two instants—the end of last year and the end of the present year. An inconsistency would therefore arise between the definitions used in the various relations.9

We may conveniently consider here a further alternative version of the finite difference approach which is formulated in terms of “natural” time periods, namely, calendar years. Hence (1) is replaced by


where Y is income during the calendar year and L represents liquid balances at the end of the year. This formulation is similar to that used by Holtrop and the Netherlands Bank.10 It will be clear from the above discussion that the difference between ((1c)) and ((1a) lies in the length of lag before adjustment is completed; on alternative (1c), liquid balances are in equilibrium with income at the end of each calendar year, instead of at the end of each income period. There is thus a mean lag of half a calendar year in (1c), compared with a mean lag in (1a) of half an income-period which is equal to 12v calendar years.

The numerical solution of a system incorporating (1c) yields the results shown in the final row of Table 2, and these are, for practical purposes, indistinguishable from those in the third row of the table, namely for α = 2. The slower adjustment of liquidity has the consequence of giving greater weight to current exogenous elements in determining current imports, at the expense of preceding values.

The correspondence between the various difference equation formulations and our differential equation approach is now seen to lie in a correspondence between the implied time-lag in the former and the parameter a in the latter. It would appear from the above examples that a is to be interpreted as the reciprocal of the implicit mean time-lag between liquidity and income in the difference equation. More rigorously, from (20) we have the mean lag between liquidity and the exogenous impulses:


This consists of two parts: first, a lag of liquidity behind income (1/α) and, secondly, a lag of income behind the exogenous impulses (1/mv); the latter agrees with the result derived in (24) above.

Consequently, the case α= 2 always corresponds to Holtrop’s model, and the case α = 2v corresponds to that of Polak.

The argument can perhaps be summarized as follows: Both the finite and the infinitesimal formulations envisage a temporary discrepancy in the proportionality of money to income. Under the finite formulation, the proportionality is reachieved at the end of a certain period of time (an income-period, a calendar year, etc.); under the infinitesimal formulation, a mean period of discrepancy has to be calculated. If these periods correspond, the two models will yield similar results.11

III. Varying Price Elasticities

In this section, the effects of price elasticities for exports and imports different from unity are examined, the basic model being thus generalized. The quantity of exports is assumed to depend only on the ratio of home to foreign prices; foreign prices are, for simplicity, taken as fixed, so that variations in home prices only need be considered—a rise in them leading to lower exports and to higher imports.

For ease of manipulation it is also necessary to take linear approximations. Thus, instead of (4) and (5), we now write




where P represents an index of domestic prices (with a mean of unity) and b, c, and h are constant propensities related in an obvious way to the various income and price elasticities; A and B may be regarded as constants or as exogenous functions of time.

In view of the expansion of the model, an assumption is also necessary on the relation between changes in the income level and changes in the domestic price level; it is assumed here that a unit rise in money incomes is always accompanied by a rise of w units in the price level. If both Y and P are index numbers (with a mean of unity), we may then write simply


By combining these equations with (1), (2), (3), and (6), and eliminating in the usual way, a differential equation for liquidity is obtained:






This is clearly similar to (7 above. The weighted mean lag of liquidity on the exogenous impulses is now


Similarlv. it is found that the mean las for imports and income is


As a preliminary step, we note that with unitary elasticities the price coefficients in (4a) and (5a) become c = h = 0, and b = m; hence g = m, and (29) reduces reassuringly to the results found at the end of the preceding section.

Next, suppose the elasticity of imports with respect to income is 7, and with respect to relative prices is δ; and the elasticity of exports with respect to relative prices is ε. Then, if exports and imports are approximately equal, it is found, on substituting the appropriate combinations of elasticities for propensities in (28), that


where m is the ratio of imports to income as before. Thus the greater the elasticities, the greater is g and the smaller are the average lags in the system. Elasticities in excess of unity thus have a similar effect on the lags, as does a higher value of mv in the simpler system. As an example, assume that all the elasticities are 2 and that w=1/2;

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from (31) it is found that g = 5m/2. Hence, compared with the unitary elasticity assumption, the weighted mean is reduced by 1/mv-2/mv; when m = 0.3 and v = 3.3, as in the example above, the reduction is 0.6 of a year. Such a reduction is clearly important, and could lead to a substantial quickening in the system’s reactions to current disturbances.

IV. Conclusion

This paper has analyzed, with the help of alternative tools, the dynamic adjustment of incomes, money, and changes in the balance of payments. The use of the present continuous model has emphasized the role of a disequilibrium in liquidity (in the sense of a discrepancy between actual liquid balances and those that it would be desirable to hold at the current income level) as the link between the real and the monetary sides of the economy; an explicit analysis of this type for a closed economy would also be of interest, but must be left for another occasion.

From an empirical point of view, it is important to note that the theoretical apparatus of this paper does not consider the effects of compensatory short-term capital flows from abroad, which form another part of the process of adjustment. Where these are important, the economy would readjust to exogenous disturbances even more rapidly than the present theory indicates; but even in such cases it remains a matter of interest to know how much of the observed movements in incomes and imports is attributable to the process of adjustment discussed here.


In addition to indebtedness to his colleagues at the Fund in the preparation of this paper, the author acknowledges some very helpful discussions with Mr. B. B. King, of the Economic Development Institute.


Staff Papers, Vol. VI (1957-58), pp. 1-50; this was followed by an empirical study by J. J. Polak and Lorette Boissonneault in Staff Papers, Vol. VII (1959-60), pp. 349-415, which also develops further the underlying theory.


In Polak and Boissonneault, op. cit., the symbol C is used to denote capital movements, but no confusion should result.


In attempting to explain the working of the system, it was found helpful to trace through the effect of a rise in exports to a new level from an original position of equilibrium, as follows: The rise in exports leads immediately to a higher income (6) and hence to an immediate demand for higher liquidity (1). Actual liquidity rises slowly during the year (2); but at the end of the first day, say, it will have barely changed, since (2) defines a rate of accumulation per unit of time. On the other hand, desired liquidity—which is equal to a certain number of days’ income—rises at once by a substantial step; consequently, a gap arises between desired and actual liquidity which depresses domestic expenditure below income (3). The additional actual liquidity that results from the new level of exports is thus partly accumulated; it is also partly spent on imports (4). As time proceeds, actual liquidity rises toward desired liquidity; consequently, domestic expenditure need no longer be depressed below income, and as it rises so income (6), imports, and desired liquidity also rise somewhat further. Eventually, equilibrium is reached when actual and desired liquidity are once again equal, at which point imports equal exports, and domestic expenditure equals income.


Op. dt., pp. 350-57.


Compare the treatment of the exponential lag by R. G. D. Allen in Mathematical Economics (London, 1956), pp. 23-28 and 166-70.


See Allen, loc. dt. Note that α=1, corresponding to the case α= 0 (see (9a)), leads immediately to a zero lag according to this argument; (24) is thus valid only for non-zero values of a.


See footnote 3.


Liquidity is lagged with respect to income; but since exogenous impulses in this system influence it through liquidity, the reader may prefer to think in terms of a lead of income on liquidity. The longer is this lead, the more are income and imports influenced by current exogenous impulses (see the discussion below).


The treatment in Polak and Boissonneault, op. cit., pp. 355 and 364-65, is thus technically inconsistent, though in practice the error is probably negligible in size.


See, for example, “The Relative Responsibilities of Governments and Central Banks in Controlling Inflation,” by M. W. Holtrop, a paper read to the International Economic Association Round Table on Inflation (Elsinore, 1959). No formal solution is given in that paper, but one can easily be derived; the coefficients for the determination of current imports in terms of previous exogenous impulses are m(k + m)-1rt, where r = k/(k + m), and t = 0, 1, 2, . . . .


The fact that all the rows in Table 2 have the same mean period may still appear surprising in view of the very different time patterns they represent. For the benefit of the reader who wishes to check that this is in fact so, it should be observed that the mean lag in Table 2 has to be calculated by assigning to column ƒ(0) a zero lag (since both imports and exports are flows during the year), then a lag of one year to column ƒ(—I), etc. For Table 1, however, a lag of half a year has to be assigned to column ƒ(0) since liquidity is measured at the end of the year, whereas ƒ(0) represents a flow during the year; mean lags calculated in this way will, of course, differ slightly (in the present example—in the second decimal place) from the theoretical values derived from (20).