¹ Archibald, G. R. and Lipsey, R. G., “Monetary and Value Theory: A Critique of Lange and Patinkin,” Review of Economic Studies, no. 69, Oct., 1958, 1–23.Google Scholar

² Time preference in Irving Fisher's sense is a property of the excess demand functions for present over future goods; zero time preference would be shown by indifference curves between this period's and next period's consumptions which are negatively sloped 45-degree lines. Time preference, in the sense in which I am using it, is a property of the utility function which makes the level of total utility derived from a given collection of goods depend on the dates at which they are consumed. In a stationary state, zero time preference, in my sense, implies that the indifference curves are symmetrical around a 45-degree line.

³ This analysis bears on the question, whether, in a production economy with borrowing and lending, the demand curve for real cash balances is infinitely elastic at a zero rate of interest. Define the pure rate of interest as the rate of discount on a perfectly secure loan. If cash balances have positive carrying costs, individuals would, when the marginal net productivity of the last dollar has fallen to zero, be willing to lend at a negative rate of interest. This would be logically equivalent to a service charge for insuring against such risks as theft or destruction by fire. The demand curve for real balances measured up against the money rate of interest would dip below the origin with perhaps only a slight negative incline since there are large economies of scale to insuring these risks.

⁴ In our model, cash balances are the only form of wealth and the desire for cash reflects primarily the precautionary motive (and perhaps, incidentally, pride of possession), cash balances serving to reduce the risk of default in case of unforeseen expenditures. Since an increase in the permanent flow of manna is not associated with a reduction in its variance, the desired stock of balances will rise as consumption rises. These considerations apply to Patinkin's probity motive for holding cash and a good case can be made out for supposing that satiety is reached when cash balances held are equal to expenditures, thereby providing 100 per cent reserves.

When cash is one of many forms of holding wealth, there is no a priori reason for cash holdings to increase as income increases.

⁵ Ramsey, Frank, “A Mathematical Theory of Saving,” Economic Journal, 1928.CrossRefGoogle Scholar

⁶ Let c be the current daily rate of consumption, U, its total utility, and s be the current daily rate of savings. If our man slightly increases his rate of savings to s + 1, he will save in s days the amount he would otherwise have saved in s + 1 days. He will therefore enjoy the satiety level of real balances one day earlier at the expense of one day's less consumption. Let K be the total utility of consumption when real balances are the satiety level. By savings at the rate s + 1, he gains K − U at a sacrifice measured by the disutility involved in saving an extra unit for s days. Call the marginal utility of consumption m; then the cost ms, must in equilibrium just balance the gain. Therefore, ms = K − U, or M = (K − U)/s. Note that this is true only under the assumption that the date at which our man enjoys a given stock of cash balances is immaterial—i.e., he has no time preference in our sense. Cf. Meade, J., Trade and Welfare (London, 1955), vol. II, chap, vi, 93–101 Google Scholar for a more detailed account of this problem.

⁷ Archibald and Lipsey properly reached this conclusion, but their technique was question-begging. Following Patinkin, they split time up into weeks; an individual starts each week afresh without memory of the past or anticipation of the future, and this despite the fact that he carries over from one week to the next a stock of real balances. Their myopic man gets his satisfaction entirely from holding money in this period; irrationally, he takes no account of the utility to be enjoyed in the future from the money stock held over to the end of the period and finds, at the beginning of each period, that the cash balances he has come as a gift just like the manna he receives. He is not allowed to consider the entire path to equilibrium, but adjusts week by week working himself blindly into a long-period equilibrium. This explains a curious paradox of the Archibald-Lipsey diagrammatics—viz., that the individual appears to move from a higher to a lower indifference curve.

The matter at issue is not simply one of technique such as would be involved in the use of difference rather than differential equations, but involves the economic implications of the assumptions. For example, Patinkin's concept of inferiority of real balances implies that an individual whose equilibrium balances are cut will so reduce his consumption that the stock of balances is larger next week than the equilibrium stock despite the fact that in the following weeks he dissaves to re-attain equilibrium. In the Archibald-Lipsey model this can happen if the indifference curves have appropriate slopes so that the expansion path traces out a cobweb. But, in this case, if an individual were to be faced with a reduction in his permanent flow of manna, he would commence saving in order to accumulate more wealth—a result which is economically meaningless since it implies that the short-run marginal propensity to save is negative. If one grants myopia, a cobweb is certainly possible and the fact that its implications are senseless only serves to highlight the sterility of the basic model which posits a man who, unlike the Ramsey man, has neither memory of the past nor anticipation of the future.

⁸ To use these short-run hyperbolas implies that a given percentage change in prices on the one hand and in nominal balances on the other hand enters symmetrically in determining an individual's estimated real cash balances. This is the case since we are assuming unitary elasticity of expectations.

Suppose, instead, that an individual's expected permanent price level is formed from a weighted average of current and past price with the weights declining exponentially as prices recede into the past. For example, let the weight attached to present prices be ½ and let the current price level rise by 1 per cent. Our man will ultimately desire 1 per cent more nominal cash balances in order to re-attain long-period equilibrium. However, at the present moment, his estimate of permanent prices has been revised upwards by only of 1 per cent. As time marches on, his estimate of permanent prices is revised upwards, thereby still further increasing the gap between his desired and current stock of real balances. If we compare this case with a 1 per cent reduction in his nominal balances at a constant price level, his estimated permanent cash balances are immediately reduced by 1 per cent and we might plausibly infer a different path of saving since his immediate estimate of the gap between his permanent real cash balances and the long-period equilibrium level is higher. It follows that a given change in nominal balances and in the current price level leading to the same change in current real balances may imply different adjustment paths since they lead to different estimates by an individual of his permanent real cash balances.

On either assumption about expectations, the convergence to equihbrium is smooth and no cobweb can occur.

⁹ As for the dynamic adjustment process, we might simply assume that savings is proportional to the difference between the actual and the desired stock of real balances. This leads to the following differential equation: where is the desired long-run stock, m(t) the actual stock and c is a negative adjustment coefficient. The solution is , so that the actual stock converges smoothly on the desired stock.

¹⁰ A cobweb is possible in a market experiment, when expected permanent prices are formed from a weighted average of current and past prices. A rise in prices of 1 per cent is sufficient to induce people to hold 1 per cent more nominal balances, in the long run. However, in the short run, current prices must rise by 3 per cent if expected permanent prices are to rise by 1 per cent. As time marches on, expected permanent prices are revised upwards and as larger balances are demanded, the price level falls towards its long-run equihbrium level. The initial increase in real balances is then followed by a temporary decrease below the long-period level; this overshooting results in a cobweb approach to equilibrium.

At no point do people respond to an increase in real balances by planning to decrease their wealth below the long-run level. They do so involuntarily because of an expectational pattern which leads to an incorrect estimate of the permanent increase in prices.

¹¹ When the additional cash is not distributed equi-proportionally, relative prices will be altered in the transition, and the equilibrium price ratios cannot be completely determined by the real equations of the system. The resultant breakdown of dichotomy was the basis of Patinkin's attack on “classical” theory. If, however, we assume that tastes are the same or that tastes and the redistribution of cash balances are uncorrelated, dichotomy will hold in the short run. The real demand for commodities is then invariant with respect to a redistribution of cash, an assumption, incidentally, which is no less reasonable than Patinkin's assumption that global demand is invariant with respect to a redistribution of wealth resulting from a change in the price level.

¹² It is important to remember, despite the widespread impression, due to Patinkin, that only in a pure exchange economy with neither borrowing nor lending does price level stability require a real balance term in the demand function for commodities; in a production economy, with a modern banking system, the real balance term is logically unnecessary and, if it exists, is empirically unimportant.

A case in point is ProfessorBaumol's, book, Economic Theory and Operations Research (Englewood Cliffs, 1961), chap. 12Google Scholar, which leaves the student unaware that, when we leave the confines of a pure exchange economy, stability of the price level can rest on interest rate changes caused by a substitution between cash and bonds.

¹³ For example, we might suppose that utility is some function of consumption and cash balances held, and choose as a simple function U = C ^α M ^η, which is linear in the logs (C is consumption, M is real balances). Since Y_t = C_t + S_t, S_t = M_t − M_t−1 , and Y_t = C_t + M_t − M_t−1 . Maximizing the utility function subject to the budget constraint yields λ = M^ηαC^1−α and λ = _ηMⁿ⁻¹C^α where λ is the Lagrangian multiplier. The first order condition for a maximum is that wealth be a constant proportion of consumption, the factor of proportionality (K) being α/η. This, by a suitable transformation yields the function C_t = (1/1 + K) Y_t , + (K/1 + K)C_t−1 which can be written C_t = (1 − γ) Y_t + γC_t−u where K/K + 1 = γ. This is a consumption function of the form

and since

C_t = βY_t + jC_t−1 . This is identical to C_t = (1 − γ) Y_t + γC_t− , if β + j = 1.

In long-run equilibrium, when income is constant, Y_t = Y _t−1 = Y _t−2 etc., C_t = Y_t , β(1/1 − y). If j + β = 1, the average and marginal propensity to consume is unity.

Since Y = C + S and C = 1/KW, Ċ = 1/KṠ, Ẏ/Y = 1/K.S/Y + Ṡ/S. S/Y. When S/F is constant, Ṡ/S = Ẏ/Y and S/Y = λ/(λ + 1/K) where λ = Ẏ/Y. It follows that a constant rate of growth of income leads to a constant savings ratio; when the rate of growth falls to zero so does the savings ratio.

¹⁴ Cf. Eisner, Robert, “On Growth Models and the Neo-Classical Resurgence,” Economic Journal, Dec., 1958.CrossRefGoogle Scholar

¹⁵ This consumption function has been tested by Alan Spiro with good, but not fully conclusive, results. Cf. Spiro, A., “Wealth and the Consumption Function,” Journal of Political Economy, Aug., 1962.CrossRefGoogle Scholar