Interest Rate Policy & Money

The section introduces the central theme: the often-overlooked role of money in macroeconomic stability under interest rate policy. It challenges the prevailing New Keynesian consensus, which largely omits monetary aggregates from short-run monetary policy analysis, suggesting that efficient interest rate policies are independent of monetary aggregates (Woodford, 2003a). The paper aims to identify conditions where including real balances as an indicator for interest rate policy is beneficial, using equilibrium uniqueness, path stability, and the minimization of a welfare-based loss function as criteria. The introduction lays the groundwork for examining how a consistent specification of transaction frictions can change this widely accepted view and make real balances a crucial element of effective monetary policy. It sets the stage for a deeper exploration of how incorporating real balances can improve the uniqueness and stability of the equilibrium path.

This section details the core argument. The authors posit that transactions of goods involve costs that money, as a medium of exchange, alleviates. Consistent application of this concept (McCallum, 2001) means that the equilibrium sequences of output and inflation cannot be independently determined from real balances. While the quantitative impact of real balances is often small (Ireland, 2002), the paper argues this doesn't negate their theoretical importance. The treatment of money as a stock variable in dynamic general equilibrium models is key. The predetermined stock of money held at the beginning of a period significantly influences household behavior and becomes a relevant indicator for a stabilizing interest rate policy, especially if transaction frictions are not neglected. This highlights a crucial point: the seemingly small quantitative impact doesn’t diminish the theoretical significance of real balances in determining macroeconomic outcomes. Therefore, the paper intends to demonstrate that including money in the model leads to significantly different and more accurate predictions of macroeconomic outcomes.

The section introduces a sticky price model incorporating a shopping time specification. This specification makes the marginal utility of consumption dependent on the predetermined stock of money held at the beginning of the period. A passive interest rate policy ensures saddle path stability, while an active policy leads to explosive equilibrium paths unless the central bank adjusts the interest rate based on beginning-of-period real balances. The inclusion of real balances modifies the standard Taylor principle. Minimizing macroeconomic distortions necessitates including real balances in the interest rate feedback rule under discretionary optimization. Conversely, if end-of-period money provides transaction services, money might appear negligible for implementing the optimal plan; however, the equilibrium could become indeterminate, leading to endogenous fluctuations which are avoidable by incorporating beginning-of-period real balances into the interest rate feedback rule. This section builds the theoretical framework by introducing the core model and its key elements, demonstrating how the timing of market events and the use of real balances affect the equilibrium outcomes and the requirements for stable monetary policy.

The benchmark model introduces a crucial element: the concept of 'shopping time.' The model incorporates transactions frictions by specifying that households spend time shopping for goods, and this shopping time is reduced by holding money. Crucially, the model assumes the goods market opens before the asset market. This means beginning-of-period money holdings, the stock of money households carry into the goods market, directly affects the shopping time function and, consequently, the utility function (similar to McCallum and Nelson, 1999, and Lucas, 2000). Because real balances influence the marginal utility of consumption, aggregate demand and labor supply become dependent on these predetermined beginning-of-period real balances. This history dependence is a key feature distinguishing this model from simpler New Keynesian models. Therefore, equilibrium sequences of output and inflation are inherently history-dependent, regardless of whether monetary policy is backward- or forward-looking. Beginning-of-period real balances become a relevant endogenous state variable influencing the fundamental solution of all variables, including the nominal interest rate.

This subsection analyzes the stability properties of the model under different interest rate rules. A passive interest rate rule (where the interest rate response to inflation is less than one-to-one) ensures equilibrium uniqueness and saddle path stability. In contrast, an active interest rate rule (where the interest rate response is greater than one-to-one) results in explosive equilibrium sequences unless the central bank actively responds to changes in beginning-of-period real balances. This is a critical departure from standard New Keynesian models where active rules are often necessary for determinacy. The inclusion of beginning-of-period real balances fundamentally alters the stability conditions. The model reveals that the central bank’s reaction to changes in beginning-of-period real balances is crucial for stabilizing the economy even when using an active interest rate rule. The central bank’s response mitigates the rise in nominal interest rates, preventing macroeconomic aggregates from diverging. This emphasizes the importance of considering the dynamic interactions between real balances and other macroeconomic variables when designing interest rate policies.

The section further investigates the implications for the Taylor principle—the idea that an active interest rate rule is necessary for equilibrium uniqueness. In this benchmark model, where household behavior depends on the predetermined state variable of beginning-of-period real balances, the Taylor principle no longer applies. Passiveness, rather than activeness, guarantees stability when the central bank ignores changes in beginning-of-period real balances. However, if the central bank incorporates beginning-of-period real balances into its interest rate feedback rule, an active policy does not automatically destabilize the economy. The non-applicability of the Taylor principle in this setting is a direct consequence of the model's inclusion of real balances as a state variable which alters the dynamic behavior of the system. The dependence of consumption and inflation on predetermined beginning-of-period real balances effectively reverses the stability condition observed in standard New Keynesian models. Therefore, the model highlights the limitations of the traditional Taylor principle in situations involving explicit transaction costs and the presence of real balances as a significant state variable.

This section shifts the focus to optimal interest rate policy under discretionary optimization. The central bank's objective is to minimize a quadratic loss function that increases with the variances of output, inflation, and the nominal interest rate. This framework, similar to Woodford (2003b), is chosen to allow for analytical results and comparison with existing literature. The inclusion of the interest rate variance in the loss function is crucial; it reflects the distortion caused by transactions frictions. Changes in the interest rate (the opportunity cost of holding money) affect optimal asset holdings. Minimizing the variance of the interest rate directly addresses the additional distortion stemming from these transactions frictions. This sets up a key trade-off: the central bank needs to balance minimizing distortions from price stickiness with smoothing interest rate changes to reduce transaction friction-related distortions. The weighting of these elements in the loss function plays a critical role in determining the optimal interest rate policy.

The analysis proceeds to derive an interest rate feedback rule that implements the optimal plan under discretion. The study explicitly avoids commitment-based optimization, as this would introduce history dependence potentially obscuring the role of real balances. This section examines how the central bank’s decision-making under discretion, even though forward-looking, still influences and implicitly considers the history of real balances. The resulting optimal monetary policy doesn't ignore the impact on the endogenous state variable (real balances) in subsequent periods. Therefore, although the targeting rule is forward-looking, the implementation of the optimal plan is history-dependent due to the influence of beginning-of-period real balances on inflation and output. This history dependence stems from the fact that consumption and inflation depend on beginning-of-period money holdings. This shows that even a discretionary optimization strategy for the central bank implicitly incorporates the history of real balances into its policy decisions.

To illustrate, the section uses the Lucas (2000) shopping time function and specific parameter values to derive numerical examples of optimal interest rate feedback rules. The feedback rule takes the form of the nominal interest rate contingent on inflation and lagged real balances. The analysis shows that the money elasticity of the nominal interest rate (the coefficient on real balances in the feedback rule) is non-zero, even when the weight on interest rate variance in the loss function is zero. This is because a rise in the nominal interest rate, while reducing consumption and inflation initially, can induce a decline in future real balances. This decline in turn raises households’ demand for leisure, lowers labor supply and ultimately increases future inflation. However, if the central bank lowers the nominal interest rate in response to the decline in real balances, this effect on future prices is mitigated. Increasing the weight on interest rate variance leads to non-monotonic changes in the policy rule elasticities. This highlights how the central bank’s aversion to interest rate volatility interacts with the influence of real balances in shaping optimal policy responses.

This section introduces an alternative model specification, changing the order of market events. Instead of the goods market opening first (as in the benchmark model), the asset market opens before the goods market. This means households adjust their money and bond holdings before entering the goods market. Consequently, the end-of-period stock of money, the amount held after the goods market closes, now enters the shopping time function. This contrasts with the benchmark model using beginning-of-period real balances. This alternative timing, similar to approaches in Brock (1974) and Ljungqvist and Sargent (2000), is described as a 'cash-when-I'm-done' concept (Carlstrom and Fuerst, 2001), differing from a cash-in-advance constraint. The key difference is that the household's monetary decisions are made after receiving income and profits from the asset market, and before shopping for goods. This change in timing has significant effects on the model's dynamics and the relevance of monetary aggregates in optimal policy design.

With the asset market opening first, the model's equilibrium conditions no longer include beginning-of-period real balances. Household behavior becomes independent of the predetermined state variable. This means that if monetary policy isn't backward-looking, the model becomes entirely forward-looking. In this context, the standard Taylor principle re-emerges: an active interest rate rule (interest rate responding more than one-to-one with inflation) ensures equilibrium uniqueness (as seen in Clarida et al., 1999, and Woodford, 2001). A simple inflation-targeting interest rate rule becomes sufficient to implement the optimal plan under discretion. However, if the central bank considers transaction frictions (as opposed to ignoring them), this simple rule fails to achieve a unique equilibrium for implementing the optimal plan. Moreover, the aim to minimize transactions-friction-induced distortions—measured by the variance of the nominal interest rate—can lead to equilibrium indeterminacy. This indeterminacy opens the door for endogenous fluctuations.

The paper shows how the central bank can restore equilibrium uniqueness and eliminate endogenous fluctuations in the alternative model when transaction frictions are considered. Even though households are forward-looking, a backward-looking feedback rule that incorporates beginning-of-period real balances can be used. This strategy, while seeming contradictory to having forward-looking households, works because it leverages the endogenous state variable to ensure a stable, unique, and history-dependent equilibrium. The optimal plan under discretion, even if forward-looking for the central bank, exhibits history dependence due to household behavior. Beginning-of-period real balances become an essential element in the optimal interest rate feedback rule. This demonstrates the potential for restoring uniqueness in scenarios where the simple inflation-targeting rules fail because of the presence of significant transaction frictions. The result highlights a situation where even in a forward-looking economy, a backward-looking feedback rule is necessary for the implementation of optimal policy and to avoid endogenous fluctuations.

The conclusion emphasizes that neglecting transaction frictions leads to incomplete and potentially misleading conclusions regarding the effectiveness of interest rate policy. The paper's central finding is that because households use money to mitigate transaction costs (measured as shopping time), the equilibrium paths of consumption and inflation are inherently linked to real balances. This interdependence is crucial for understanding macroeconomic stability and designing effective monetary policy. Ignoring these frictions simplifies the model to a point where it fails to accurately capture the dynamics of the economy. The paper's analysis shows that incorporating transaction costs fundamentally alters the conditions for equilibrium uniqueness and stability, and changes the design of optimal interest rate policies.

The conclusion highlights the implications of the findings for monetary policy. The study demonstrates that when the goods market precedes the asset market, beginning-of-period real balances serve as a significant state variable, regardless of the specific interest rate policy in use. This means that a truly effective and stabilizing interest rate policy must account for changes in these beginning-of-period real balances. This is not merely a theoretical nuance; the model demonstrates how ignoring this leads to potential instability. The paper’s various models and analyses all point towards this conclusion: a robust and effective monetary policy needs to take into consideration the significant influence of real balances on the economy, especially in the presence of transaction frictions.

The conclusion suggests broader implications beyond the specific models used. The findings emphasize that the effectiveness of stabilizing interest rate policies depends critically on the accurate modeling of transaction frictions and the recognition of real balances as a state variable. The demonstrated ability to stabilize the economy by incorporating beginning-of-period real balances into the interest rate feedback rule highlights that central banks should not ignore the historical context when setting interest rates. The method of employing lagged endogenous variables, such as beginning-of-period real balances, as indicators for interest rate policy could potentially be extended to other variables and monetary policy instruments. This highlights the importance of considering the full implications of transaction costs for understanding the effectiveness of different monetary policy approaches and suggests avenues for further research into more nuanced models and applications of these findings.