Liquidity Preference as Behavior Towards Risk

Tobin opens by pointing out that “nearly two decades of drawing downward-sloping liquidity preference curves in textbooks and on classroom blackboards should not blind us to the basic implausibility of the behavior they describe”: why would anyone hold a non-interest-bearing government obligation (cash) instead of an interest-bearing one, at any interest rate, however small? What needs to be explained, he insists, is not only the mere existence of a demand for zero-yield cash but specifically an INVERSE relationship between the aggregate demand for cash and the size of the yield differential between cash and other assets (Introduction, p. 65). He distinguishes at the outset between transactions balances (needed because receipts and expenditures are not perfectly synchronized, and roughly proportionate to the volume of transactions) and investment balances (the portion of wealth that survives all such seasonal needs), and confines the paper’s analysis to the latter – the portfolio choice at the heart of Keynes’s “speculative motive” (Section 1, pp. 65-67).

In Section 2, Tobin assumes an investor holds a single, certain expectation r_e of the future interest rate on a hypothetical perpetuity (“consols”), independent of the current rate r; this generates an expected capital gain or loss g = r/r_e - 1 on holding consols, and the investor’s optimal choice is a pure corner solution: all consols if the current rate r exceeds a critical rate r_c = r_e/(1+r_e), all cash if r is below it (Section 2.2, equations 2.1-2.2, p. 67). Because this “all-or-nothing” individual behavior cannot by itself generate a smooth aggregate relationship, Tobin shows in Section 2.4 that the familiar continuous, downward-sloping Keynesian liquidity preference curve emerges only when the model is aggregated across many investors who hold DIFFERENT critical rates r_c, so that “the demand for cash at r is the total of investment balances controlled by investors whose critical rates r_c exceed r” (Section 2.4, p. 69, Figure 2.3).

Tobin recounts two major criticisms: Leontief argued that liquidity preference “must necessarily be zero IN EQUILIBRIUM, regardless of the rate of interest,” since divergence between current and expected rates should vanish as investors learn from experience – any sufficiently persistent rate can come to be accepted as “normal”; Fellner separately asked why interest rates alone should be subject to such inelastic expectations, rather than, say, pre-depression price levels (Section 2.6, pp. 70-71, citing Leontief 1947 and Fellner 1946). Tobin treats these as posing “the question whether it is possible to dispense with the assumption of stickiness in interest rate expectations without losing the implication” of an inverse cash-demand/interest-rate relationship – the question that motivates Section 3 (p. 71).

Section 3 assumes the investor is uncertain, not falsely certain, about the future capital gain or loss g on consols, with a subjective probability distribution of g that has an expected value of zero regardless of the current rate r. A portfolio holds a proportion A1 of cash and A2 of consols (A1 + A2 = 1); its return is R = A2(r+g), so its expected return is mu_R = A2r (equation 3.2) and its risk, measured by the standard deviation of R, is sigma_R = A2sigma_g, where sigma_g is the investor’s subjective standard deviation of g (Section 3.1, equations 3.1-3.3, pp. 71-72). Combining these gives the opportunity locus mu_R = sigma_R*(r/sigma_g): a straight line through the origin whose slope r/sigma_g describes the terms on which the investor can trade more risk for more expected return (equation 3.4, p. 72).

Tobin offers two alternative rationalizations. The first (Section 3.3.1) assumes the investor’s subjective probability beliefs about g belong to some two-parameter family of distributions (uniform, normal, or otherwise), so that specifying the mean and standard deviation fully determines the whole distribution, making mu_R-sigma_R indifference curves a valid summary of choice among distributions. The second (Section 3.3.2) instead restricts the utility-of-return function itself to be quadratic, U(R) = (1+b)R + bR^2, valid over a bounded range of R; on either rationalization, Tobin proves algebraically that a risk-averter’s indifference curve between expected return and risk must be concave upward, so that “all risk-averters are diversifiers; plungers do not exist” (Sections 3.3.1-3.3.2, equations 3.5-3.14, pp. 74-77, quotation p. 76).

A “diversifier” is a risk-averse investor whose indifference curves are concave upward; such an investor’s optimum is generally an interior tangency (Type I) between the opportunity locus and an indifference curve, holding positive amounts of both cash and consols (Figure 3.1, points T1, T2, T3, Section 3.4, p. 77). A “plunger” is an investor whose indifference curves, though upward-sloping, are linear or convex (not derivable from either of Tobin’s two rationalizations for a risk-averter), and who may instead choose a corner solution – all cash or all consols – as illustrated in Figure 3.3 (Section 3.4, Type II and III, pp. 77-79). Tobin’s key result is that genuine risk-aversion, properly derived, rules out plunging: only diversifiers exist among risk-averters.

A rise in the interest rate rotates the opportunity locus counter-clockwise (to a steeper slope), and Tobin notes this creates a genuine ambiguity analogous to the classic income/substitution-effect problem in the theory of saving: the substitution effect favors more consols (yield has become more attractive relative to safety), but the income effect works the other way, since a higher rate lets the investor enjoy more safety alongside more yield (Section 3.4, p. 79). For the special case of the quadratic utility rationalization, however, Tobin shows the ambiguity is “virtually excluded”: differentiating the tangency condition (equations 3.15-3.16) shows that the share of consols in the portfolio, A2, RISES with the interest rate r for all r less than sigma_g (the investor’s risk estimate), and a reversal can occur only if r also exceeds the largest capital gain the investor considers possible, a case a co-author (cited as Arthur Okun, fn. 1, p. 79) shows is incompatible with a symmetric probability distribution of g (Section 3.4, equations 3.15-3.16, p. 79).

Tobin derives a formal relationship (equation 3.17) linking the elasticity of consol demand with respect to risk sigma_g to its elasticity with respect to the interest rate r, and uses it to analyze taxation: a 50 percent tax on both interest income and capital gains, with full loss offset, simultaneously halves both the expected net return and the risk per dollar of consols, which Tobin shows leaves the investor’s chosen (mu_R, sigma_R) combination unchanged but requires him to DOUBLE his holding of consols to reach it – so such a tax reduces the demand for cash at any given market interest rate (Section 3.5, equation 3.17, pp. 80-81, Figure 3.5). By contrast, a tax on interest income alone, without any offset for capital losses, would increase the demand for cash and shift the liquidity preference curve outward (Section 3.5, p. 82).

When there are several non-cash assets, Tobin defines a “dominant” combination as one that minimizes portfolio risk (variance) for a given level of expected return, characterized by a system of linear equations (equations 3.21-3.22); he proves that all dominant combinations of the risky assets lie along a single ray from the origin – meaning the PROPORTIONATE composition of an investor’s risky-asset holdings is independent of how large a share of the total portfolio is allocated to risky assets versus cash (Section 3.6, pp. 82-84). This “makes it possible to describe the investor’s decisions as if there were a single non-cash asset, a composite formed by combining the multitude of actual non-cash assets in fixed proportions” (p. 84) – allowing the paper’s two-asset (cash vs. one risky asset) apparatus to stand in for the general many-asset case, so long as cash remains a riskless residual asset (Section 3.6, p. 85, citing Markowitz’s contemporaneous portfolio-selection work in a footnote, p. 85).

Tobin argues the risk-aversion theory is “a logically more satisfactory foundation for liquidity preference” because it does not depend on inelasticity of interest-rate expectations, only on the weaker (and in his view more defensible) assumption that the expected capital gain or loss on interest-bearing assets is always zero; it also has “the empirical advantage of explaining diversification – the same individual holds both cash and ‘consols’ – while the Keynesian theory implies that each investor will hold only one asset” (Section 4, p. 85). He also argues it partially, though not completely, answers Leontief’s objection: Tobin concedes that in a truly stationary state investors would eventually come to estimate sigma_g as zero, making such a state “of very little interest,” but maintains that comparative statics need not be confined to comparisons of states that would each take a generation to reach (Section 4, p. 85).

Tobin is explicit that the theory is “somewhat ambiguous concerning the direction of relationship between the rate of interest and the demand for cash”: at low interest rates the theory implies the usual negative elasticity of cash demand (growing more negative as the rate approaches zero), but at high interest rates – especially for investors with low risk estimates sigma_g – the demand for cash could in principle be an INCREASING function of the interest rate (Section 4, p. 86). He notes this potential reversal is “diluted” by the fact (established in Section 2.5) that the size of investment balances itself is not independent of the current interest rate, since a rate rise reduces the market value of existing consol holdings and so partly satisfies any desire to shift toward cash automatically (Section 4, p. 86). He also stresses that the assumption of a zero expected capital gain was adopted “for the theoretical reasons explained in section 2.6 rather than for reasons of realism,” and that the stickiness-of-expectations theory of Section 2 and the risk-aversion theory of Section 3 are “complementary rather than competitive” for purposes of dynamic analysis of actual market situations (Section 4, p. 86).