Monetary equilibrium

Abstract

One implication of the concept of monetary equilibrium is that the money supply should vary with money demand. In a recent paper, Bagus and Howden (Rev Austrian Econ 24:383–402, 2011) argue that this conclusion is predicated on the assumption of price stickiness. The purpose of this paper is to suggest that the foundation of monetary equilibrium is the role of money as a medium of exchange. As such, changes in the demand for money result in changes in both nominal and real spending that are welfare-reducing. This proposition is then used to examine whether a monetary policy in which the central bank varies the money supply in response to money demand can be considered optimal. In addition, the paper considers how a free banking system with competitive note issuance would vary the money supply in response to changes in money demand. In both cases, the results are consistent with the concept of monetary equilibrium. In addition, these results can be obtained even when prices are perfectly flexible if trade is decentralized (i.e. not conducted in Walrasian markets). Price stickiness is therefore not a necessary condition to suggest that the money supply should vary with money demand.

Appendix

1.1 A.1 Money demand

The model presented in this paper interprets the stochastic matching parameter, α, as a money demand shock. It will now be shown that α is isomorphic to a money demand shock in a model with money in the utility function.

Suppose that utility for an infinitely lived, representative consumer is given by:

$$\label{apputility} U = E_0 \sum\limits_{t = 0}^{\infty} \beta^t {{{c_t}^{1 - \gamma}}\over{1 - \gamma}} + \varepsilon_t {{{m_t}^{1 - \Psi}}\over{1 - \Psi}} $$

(17)

where c is consumption, m is real money balances, and ε is a stochastic preference parameter attached to real balances. The representative agent enters period t with money balances, M _t , bond balances, B _t , and receives a transfer from the central bank, T _t . During the period, the representative agent earns income, Y _t , and uses wealth and income to finance consumption, c _t , and allocate the remainder to bond and money holdings. As a result, the representative agent’s budget constraint is:

$$m_{t + 1} + b_{t + 1} + y_t = m_t + (1 + i_t) b_t + c_t + T_t$$

where i _t is the nominal interest rate and real values are expressed as lower case letters.

Maximizing Eq. 17 subject to the budget constraint yields the following first-order conditions:

$$ c_t^{-\gamma} = \lambda_t $$

$$ \lambda_t = (1 + i_t)\beta \lambda_{t + 1} $$

$$ \varepsilon_t m_t^{-\Psi} = \lambda_t - \beta \lambda_{t + 1} $$

Combining the three first-order conditions yields:

$$\label{appmd} \varepsilon_t m_t^{-\Psi} = c_t^{-\gamma} i_t $$

(18)

In equilibrium, M _t  + T _t  = M _t + 1 and B _t  = B _t + 1 = 0. Thus, from the budget constraint, y _t  = c _t . Finally, assume that money demand is unit elastic with respect to income (Ψ = γ). Then, one can re-write Eq. 18 as:

$$V_t = f(i_t, \varepsilon_t) = \bigg({{i_t}\over{\varepsilon_t}}\bigg)^{{1}\over{\Psi}}$$

From the search model, velocity is:

$$V_t = f(i_t, \alpha_t) = {{1 + \alpha_t z(q_t; \theta)}\over{z(q_t; \theta)}}$$

where velocity is a function of i through z(q; θ). Define α _t ε _t  = e, where e is some arbitrary constant. It follows that velocity in both the MIU and search framework is an increasing function of i and α.

1.2 A.2 Nominal income targeting under alternative bargaining conditions

In the discussion of nominal income targeting, it was assumed that θ = 1 (i.e. buyers determine prices in the decentralized market). Here it is demonstrated that the assumption that θ = 1 substantially simplifies the mathematics, but not alter the main results. To do so, three alternative bargaining conditions are examined. The first example considers policy under Nash bargaining. The second example considers policy in which sellers determine prices (i.e. θ = 0).²⁴ In the final example, both the buyer and the seller receive a fixed proportion of the surplus. The examples illustrate that the terms of trade in the decentralized market are of little consequence when the objective is to minimize fluctuations of trade in the DM around the steady state.

1.2.1 A.2.1 Policy under Nash bargaining

Consider Eq. 10, where z(q; θ) represents the demand for money when the terms of trade in the decentralized market are determined through Nash bargaining between buyers and sellers. In the text, it was assumed that buyers make take-it-or-leave-it offers to sellers (i.e. θ = 1). Given the functional forms in the model, this implied that z(q; θ) = c(q) = q. Here, we dispense with this assumption to demonstrate that while the assumption of θ = 1 greatly simplifies the mathematics, it does not alter the conclusion.

Log-linearizing Eq. 10, yields:

$$\xi \hat{q}_t + \eta \hat{\alpha}_t = \hat{NOM}_t - \hat{M}_t$$

where

$$\xi = {{-\gamma \theta \bigg[\theta + {{1 - \theta}\over{1 - \gamma}}\bigg] q^{1-2\gamma} - [\theta q^{-\gamma} + 1 - \theta][(1 - \gamma)\theta + (1 - \theta)]q^{1-\gamma}}\over{\bigg\{ \bigg[\theta + {{1 - \theta}\over{1 - \gamma}}\bigg] q^{1 - \gamma} \bigg\}^2}}$$

and

$$\eta = {{\alpha}\over{{{\theta q^{-\gamma} + 1 - \theta}\over{[\theta + {{1 - \theta}\over{1 - \gamma}}]q^{1 - \gamma}}} + \alpha}}$$

Plugging this log-linearized equation into the loss function yields:

$$\mathbb{W} = - \big({{\gamma \varphi}\over{2}}\big)(1/\xi)^2 E_t (\hat{M}_t + \eta \hat{\alpha}_t - \hat{NOM}_t)^2$$

It is straightforward to see that optimal policy is consistent with the rule in Eq. 13, albeit with a different definition for η.

1.2.2 A.2.2 Policy when sellers determine the terms of trade

Now consider the case in which θ = 0. In this case, from Eq. 6, it is true that:

$$z(q; \theta = 0) = u(q)c'(q)$$

Given the functional forms in the text, this implies that

$$z(q; \theta = 0) = u(q) = {{q^{1 - \gamma}}\over{1 - \gamma}}$$

Log-linearizing Eq. 10 using this definition for money demand yields,

$$\upsilon \hat{q}_t + \eta \hat{\alpha}_t = \hat{NOM}_t - \hat{M}_t$$

where

$$\upsilon = {{(\gamma - 1)(1 - \gamma) q^{\gamma - 1}}\over{(1 - \gamma)q^{\gamma - 1} + \alpha}}$$

and

$$\eta = {{\alpha}\over{(1 - \gamma)q^{\gamma - 1} + \alpha}}$$

Plugging this in to the loss function yields

$$\mathbb{W} = - \big({{\gamma \varphi}\over{2}}\big)(1/\upsilon)^2 E_t (\hat{M}_t + \eta \hat{\alpha}_t - \hat{NOM}_t)^2$$

Again, it is straightforward to see that optimal policy is consistent with the rule in Eq. 13, albeit again with a different definition for η.

1.2.3 A.2.3 Policy under a proportional solution

Suppose that buyers and sellers each receive a fixed proportion of the surplus. Denote the fraction of the surplus that goes to buyers as θ and the fraction that goes to sellers as 1 − θ. In terms of the model, this implies that

$$u(q) - \phi m = \theta [u(q) - c(q)]$$

and

$$-c(q) + \phi m = (1 - \theta) [u(q) - c(q)]$$

Combining this conditions and solving for ϕm yields a money demand equation:

$$\phi m = (1 - \theta)u(q) + \theta c(q) \equiv z(q)$$

Plugging this expression for z into Eq. 10 and log-linearizing yields:

$$\nu \hat{q}_t + \eta \hat{\alpha}_t = \hat{NOM}_t - \hat{M}_t$$

where

$$\nu = {{-\theta q + (1 - \theta)q^{1 - \gamma}}\over{{{1}\over{\theta q + {{1 - \theta}\over{1 - \gamma}} q^{1 - \gamma}}} + \alpha}}$$

and

$$\eta = {{\alpha}\over{{{1}\over{\theta q + {{1 - \theta}\over{1 - \gamma}} q^{1 - \gamma}}} + \alpha}}$$

Plugging this in to the loss function yields

$$\mathbb{W} = - \big({{\gamma \varphi}\over{2}}\big)(1/\nu)^2 E_t (\hat{M}_t + \eta \hat{\alpha}_t - \hat{NOM}_t)^2$$

Again, it is straightforward to see that optimal monetary policy is consistent with the rule in Eq. 13, albeit with a different definition of η.

1.2.4 A.2.4 A note on optimal policy under alternative terms of trade

While it is clear that the above bargaining solutions are consistent with the monetary policy rule in Eq. 13, a necessary condition for these rules to be consistent with monetary equilibrium is that η > 0. When the terms of trade are determined in the three ways described above, the functional form of the buyers utility function is of some consequence. Nonetheless, given the general functional forms used in the main text, a sufficient condition for η > 0 is that γ < 1. Whether this condition holds is therefore an empirical question. Lagos and Wright (2005) use the same functional forms as the present paper and choose the parameters of the model to fit a money demand function. They find that the best fit of the model is consistent with this condition. In fact, the highest value they obtain to fit the model is γ = 0.266 and this is not dependent on θ.