A monetary equilibrium that either satisfies Friedman's rule for the optimal quantity of money or accommodates a policy of zero rate of inflation is dynamically unstable. However, without exception, all studies focus exclusively on the real effects of money and inflation on long-term balanced growth, leaving the transitory dynamics of monetary equilibrium completely unexplored. 1 For a survey of the literature on neoclassical models of money and growth, see Dornbusch and Frenkel (1973) and Wang and Yip (1992b).
By facilitating transactions, money reduces the costs of transaction services, in the spirit of the seminal work of Saving (1971). To allow for balanced growth, we assume that this portion of output used by transactions is a decreasing function of real balances. 3 In particular, Wang and Yip (1995) rely on numerical analysis to determine the property of the underlying dynamics of their model.
Most importantly, the main contributions of the paper are the complete characterization of the dynamics of monetary equilibrium in the presence of sustained economic growth.
Under Conditions U and G, there exist a unique balanced growth monetary equilibrium
Stability Analysis
As seen in Section 3 above, the dynamical system reduces in a block-recursive manner to a 2 x 2 system in terms of two transformed stationary relations, Z1 and Z2. Local stability properties can therefore be obtained by studying the linearized system of dynamics Z1 and Z2. We first reduce the system to two differential equations for Z1 and Z2, which describe the dynamics of the c/k and m/c ratios.
Denote the determinant and trace of the pre-multiplied matrix (D) on the RHS of (22) as Det(D) and Tr(D), respectively. Before our formal stability analysis, we would like to remind the reader that there is a unique BGP. However, recall that Z1 Z2 = M/(PK), where M and K cannot jump and P must be fixed in the impact from any disturbance of the rate of money expansion for any "honest government" [Auernheimer Hence, the value of Z1 Z2 must be fixed in impact.
This constrains the short-term movements of the two transformed ratios in the sense that only one of them is a free-jumping variable, while another must change according to the above constraint. Our 2-by-2 dynamical system thus exhibits the same feature as the conventional system: the saddle path case has one root with negative and one with positive real part. In the case of saddle path stability, the intersection of the Z1 Z2 = M/(PK) locus and the stable saddle represents the initial point, which has a unique transition path that converges to the BGP.
Under condition H, growth ceases and the rate of inflation is completely driven by the rate of monetary expansion. This is a setting that Cagan (1956) creates for the study of hyperinflation.16 These conditions are shown, in the following lemmas, to be useful in determining the properties of the determinant and the trace of the matrix D.
Under Condition D, Det(D) > 0
Condition D imposes an upper bound on the intertemporal elasticity of substitution (σ-1 ) that is more restrictive than the condition of limited lifetime utility (condition U).
We are now ready to state the main results concerning the stability properties of the dynamical system. To begin, we consider the Friedman rule of the optimum amount of money in that the social cost of holding money measured by the nominal interest rate is driven down to zero (i.e. i=0). The following theorem deals with the stability property of a monetary equilibrium with non-positive money growth rates.
Thus, we learn that a zero-inflation policy or a deflationary policy is not sustainable, as any small disturbance would move the economy away from the initial monetary equilibrium of balanced growth. This may explain why, in practice, central banks exercise a positive rate of monetary expansion, regardless of the welfare cost of inflation. Proof: From Lemmas 1 and 3, Det(D) > 0 and Tr(D) < 0, implying that both characteristic roots have negative real parts and therefore the dynamic equilibrium is an end.
In an episode of hyperinflation described by Cagan (1956), economic growth ceases, real returns to capital are zero, and the rates of monetary expansion and inflation are roughly the same. In contrast, our result shows that in such a situation, it is possible for the monetary equilibrium to be dynamically unstable in a perfect forecasting intertemporal optimizing framework. As a consequence, transition equilibrium outcomes are likely to be highly divergent as in many observed episodes of hyperinflation (such as the post-World War I German, post-World War II Chinese, and post-1980 Latin American experiences and Israel experiences hyperinflation). 17.
In this case the eigenvalues are -80 and 43, indicating that the steady state is a saddle point. In the money and endogenous growth literature, calibration exercises are frequently used to evaluate the growth effects of money and the welfare costs of inflation. Such quantitative analyzes are usually based on the assumption that the system is saddle-path-stable (and dynamically determined) without a formal proof.
Under Conditions U and G, the monetary equilibrium may be dynamically unstable (a
Characterization of the Saddle-Path Dynamics
An increase in the money growth rate has been shown to lead to a long-run decline in the consumption-to-capital ratio (Z1) and the real balance-to-consumption ratio (Z2). As mentioned in Section 4, the growth path of real balances (M/P) must be fixed at the impact at t=0 under the "honest government" assumption. Therefore, according to the "honest government" argument adopted in the previous section, both real balances and capital are growing at the same rate at point P.
Since the growth rate of real balances remains constant between points E and P, the inflation rate must also be unchanged. Finally, since the consumption-capital ratio (Z1) falls and the real balances-consumption ratio (Z2) rises instantaneously, the growth rate of consumption must be below the growth rates of capital and real balances at t=0. This then implies that the growth rate of capital stock must be greater than the growth rate of consumption, which in turn is greater than the growth rate of real money balances from P to E′, i.e. θm < θc < θk.
Thus, the growth rates of consumption and the capital stock must be above their new equilibrium value BGP (θ**) in the transition from P to E'. Before we leave this section, we would like to make a remark about jumps in the growth rate of consumption. To summarize, we present our main findings regarding the dynamics of the saddle path in the following proposition. Proposition 6: Under condition S, the transient dynamics of a monetary equilibrium exhibits i) Instantaneously, an increase in the nominal rate of money growth leads to a decrease in the consumption-capital ratio and an increase in the real balance-consumption ratio.
The growth rates for capital and real balances as well as the inflation rate remain unchanged, while the growth rate in consumption falls. below the initial BGP equilibrium level. ii). The growth rate of real balances (inflation rate) decreases (increases) monotonically towards its new BGP equilibrium level. While the growth rate of capital is higher than the growth rate of consumption, both rates are higher than that of the real balance.
Convex Production Technology and Elastic Labor Supply
Asymptotically, the above system will converge to the same BGP equilibrium as in the Ak model, where c, k and m grow at the same constant rate and L is constant (hence, as t 6 4, we have Z3 6 0). By completely differentiating the system and then substituting the asymptotic value for Z3, we obtain a 2H2 system of differential equations identical to (20) and (21), which can be used to analyze the stability properties. Therefore, we must have σ < 1, and therefore some previous stability properties should be repeated (since condition D is no longer valid).
It is clear that the dynamics of labor supply has no influence on the property of the dynamic system and that the dynamics of Z1 and Z2 can be characterized by the linearized system (22) with σ = α. On the one hand, in hyperinflationary episodes, the possibility of dynamic indeterminacy is strong (as before, dynamic instability may also appear). On the other hand, under a convex production technology with elastic labor, the monetary equilibrium satisfying the Friedman rule is a saddle (instead of a source).
Since this additional adjustment is a stabilizing force, it overcomes the destabilizing forces in the unstable case of Friedman's rule, leading to stability on the saddle. More importantly, Theorem 2 still holds since the monetary equilibrium can be a source, a saddle, or a sink.
Conclusions
Finally, it may be informative to relax our perfect foresight assumption by allowing for a range of expectation regimes, from the irrational adaptive expectations (slow adjustment of expectations) to a limit of short-term perfect foresight (instant adjustment of expectations). In a generalized Tobin model, Benhabib and Miyao (1981) investigate whether the dynamics depend crucially on the extent to which agents' short-term price expectations adjust. It may be interesting to investigate whether their conclusion holds in our transaction-based model rather than in the Tobin-like asset substitution framework.
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