Price-Level Determinacy Without Control of a Monetary Aggregate

“Policies that target nominal interest rates, rather than targeting a monetary aggregate and leaving interest rates to be determined in the money markets, have…been criticized” on the ground that they leave both the money supply and the price level “completely indeterminate” (Patinkin 1961, 1965; Sargent and Wallace 1975), and quantity theorists have similarly opposed “financial innovation or deregulation of the activities of financial intermediaries” that would create money substitutes (Introduction, p. 1). Woodford states his aim directly: “I wish to challenge both of these conclusions,” arguing that these standard equilibrium conditions “often fail to uniquely determine an equilibrium price level” even under conventional exogenous-money regimes, and that a properly completed account – incorporating the government’s fiscal position – restores determinacy under interest-rate pegs and free banking alike (pp. 1-2).

“An increase in the price level reduces the real value of the net (outside) assets of the private sector, or equivalently, the real value of net government liabilities…This reduction of private sector wealth naturally reduces private sector demand for goods and services, through a straightforward wealth effect. As a result, there will typically be only one price level that results in aggregate demand that equals aggregate supply” (Section 1.A, p. 2). This wealth-effect channel “continues to be operative even if the central bank varies the money supply in response to a price level change” and “even if the government’s ‘monetary’ liabilities are perfect substitutes for other kinds of assets” – so the mere fact that money supply adjusts endogenously does not, by itself, eliminate the mechanism that pins down the price level (pp. 2-3).

Equation (1.10) states that at every date t, the real value of net government liabilities must satisfy W_t/p_t equal to the discounted sum, over all future dates s >= t, of (tax revenue minus purchases) plus the interest-savings term Delta_s m_s, discounted at the real bond return (Section 1, p. 9). Woodford calls this “an equilibrium condition that determines the price level p_t at date t, given the predetermined nominal value of net government liabilities W_t, and given expectations…regarding the current and future values of the real quantities and relative prices” on the right-hand side. He derives it as a consequence of the private household’s own present-value budget constraint (1.6) holding with equality in equilibrium, since optimizing households “plan to exhaust their own budget constraints” (p. 9).

He argues it is “often more useful to think of (1.10), rather than (1.8a), as the equilibrium condition that determines the equilibrium price level,” because equation (1.8a) “may more usefully be viewed as determining the money supply” (in an endogenous-money regime) or “the equilibrium interest-rate differential” (in an exogenous-money regime), while (1.10) has “the directness of the connection between a discrepancy between the values of the left- and right-hand sides…and economic forces that should cause the price level p_t to adjust” (Section 1.B, pp. 10-11). He further connects this to Sargent’s (1982) suggestion that the value of money, like other government debt, depends on “private agents’ expectations about the revenue streams backing it,” arguing his analysis makes this “a coherent view of the determinants of the value of money, even in the case of inconvertible fiat money” (p. 10).

Under a policy regime in which net tax collections T_t are set according to a rule that rebates the government’s interest savings on money lump-sum (equation 2.4), a change in the money supply M_t, holding government purchases and this net-tax rule fixed, “will not imply any change in W_{t+1}/R^b_t,” and hence no change in the expected value of net government liabilities in real terms next period (Section 2.A, pp. 13-14). Woodford is careful to state the precise scope of the result: “the irrelevance result states only that the announcement of a new path for the money supply at some date t has no effect upon the price level at that date. It does not imply that there is no eventual effect upon the price level” – any eventual effect operates only “as a result of the eventual effects of monetary policy upon the size of total government liabilities,” i.e., through fiscal, not quantity-theoretic, channels (Section 2.B, p. 16).

Under the traditional account, equation (2.3) alone admits “a continuum of solutions” for the price level absent an initial condition, including paths where real balances diverge, conventionally excluded as “self-fulfilling inflations” or “bubbles” driven by arbitrary expectations (McCallum 1989; Blanchard and Fischer 1989) (Section 2.C, p. 20). Woodford argues this characterization is “clearly not true” once fiscal considerations are included: in the case of a “helicopter drop” of government debt, “the price level rises only insofar as the nominal value of net outside assets does, and the explosive growth of the latter is a mechanical consequence of the government budget deficits” – so what looks like an arbitrary self-fulfilling bubble is in fact pinned down by (and only by) the specified fiscal variables, not by unexplained shifts in expectations.

A fiscal policy regime is Ricardian if condition (3.3) – essentially, that the government’s budget automatically balances in present value – “necessarily holds, independently of what the evolution of {p_t} may be” (Section 3, p. 24). Under such a regime, “fiscal policy plays no role at all in price level determination…while the path of the money supply clearly does. Hence traditional quantity-theoretic reasoning is, in such a case, completely valid.” Woodford observes that the case “most often assumed in theoretical work in the quantity-theoretic tradition,” a regime with the supply of government bonds exogenously fixed at zero, “is necessarily of this kind,” and suspects this reflects “a tacit assumption that the policy regime should in any event be Ricardian” – an assumption the rest of the paper shows is not required for determinacy, and is not even always realistic.

Consider a regime that exogenously fixes the sequences of the interest rate on money, the interest rate on bonds, government purchases, and real tax revenue, letting the total value of government liabilities be determined residually and its composition between money and bonds be determined by households. Substituting the money-demand and Euler equations into (1.10) yields equation (4.1), which “relates the equilibrium price level at any date t to the nominal value W_t of net government liabilities at that time, and upon the anticipated value of the policy variables from that time onward”; under a mild sign condition (interest on money below interest on bonds, purchases below income), “the equation can be solved for a unique price level p_t > 0, given net government liabilities W_t > 0” (Section 4, pp. 28-29). Woodford stresses this happens “despite” the homogeneity of the money-demand equation alone: “this equilibrium condition will not exhibit the homogeneity property” of (1.8a)/(1.8b), because W_t enters as a level, not merely as a ratio to the price level, so “there may well be a unique path for the price level that satisfies this equation” even when a pure interest-rate peg is in force (Section 4, p. 28).

Households are assumed to maximize utility from consumption and from combined real balances of government money plus privately issued deposits D_t/p_t; a competitive intermediation technology lets banks fund a fraction (1-rho) of deposits by purchasing government bonds, with the remaining fraction rho absorbed in real intermediation costs, so that the interest rate on deposits equals (1-rho) times the bond rate (equation 5.1), and “the supply of money-like deposits will be perfectly elastic at this rate of interest” (Section 5, pp. 31-32). Despite unrestricted private money creation, Woodford shows condition (1.10) continues to be a necessary equilibrium condition and, under a natural fiscal-feedback rule (5.4) linking government purchases to the fiscal effects of any money-supply variation, yields a closed-form unique price level (equation 5.6): “the equilibrium path of the price level is in this case determined solely by the fiscal variables (and by the rate of interest paid on government money).” Woodford concludes plainly: “the existence of unrestricted private supply of money substitutes need pose no threat to the determinacy of the equilibrium price level” (p. 34).

“I have shown that determinacy of the price level is possible even in regimes in which the money supply is endogenous – indeed, in which it is perfectly elastic, at a given short-term interest rate – either because the central bank pegs the interest rate, or through the competitive supply of private substitutes for government-issued money. Thus avoidance of policies of these kinds is not necessary for the sake of allowing control of the general level of prices, especially if, as many have argued, such policies are desirable from the point of view of microeconomic efficiency” (Section 6, Conclusion, p. 36). He is careful to add that this is not a wholesale dismissal of quantity theorists such as Friedman, whose “insistence upon attention to the consequences of macroeconomic policies for inflation” and upon “the importance of a government commitment to a policy rule” remain valid; his target is narrower – the specific inference that avoiding monetary-aggregate control necessarily threatens price stability – and his conclusion is that “the theory of price level determination sketched here only makes it more evident that stability of expectations regarding future government policy is essential for stability of the price level,” whether or not that policy is expressed in terms of a monetary aggregate.