9. Transitional Dynamics in Monetary Economies: Numerical Solutions

From Chap. 8, the transversality condition becomes, under these assumptions:

which will not be zero, since from optimality conditions, the Lagrange multiplier is equal to the marginal utility of consumption, which will generally be bounded away from zero in steady state.

Or r _ss < η < 2 + r _ss, an expression that we will use below.

Or whether we are in a monetary or in a non-monetary economy, for that matter.

Notice that \(\frac {M_{t+1}-M_{t}}{P_{t}}=\frac {M_{t+1}-M_{t}}{M_{t}}\frac { M_{t}}{P_{t}}=x_{t+1}m_{t}.\) In the steady-state analysis in the previous chapter we made a different type of transformation, leaving segniorage revenues as a function of the rate of inflation. We now prefer that the rate of money growth may explicitly appear, since we consider an exogenous time path for it, while the rate of inflation is endogenously determined.

See Mathematical Appendix (Chap. 12).

In the MATLAB program it is explained how to choose the variables that deviate from their steady-state levels.

Under this policy design, nominal interest rates are simultaneously determined with consumption and capital, while being exogenous relative to inflation and real balances. However, this could be consistent with real balances having significant explanatory power in regressions for consumption and capital using simulated data if we do not consider the nominal rate of interest as explanatory variable.

The same analysis could be conducted with the utility function in the previous section, a consumption tax and an autoregressive structure for productivity. Real balances would then enter in (9.69), which could be eliminated using (9.70). The nominal rate of interest would then appear in the transformed (9.69), but this is an exogenous variable in this policy experiment.

We could increase the dimension of the system by including the law of motion for the nominal rate of interest, as we have done in some other previous analysis. The difference is that the stability condition would then involve deviations of nominal rates around their steady state level. However, if the central bank follows a policy of maintaining constant interest rates, then there is no difference between both formulations.

Remember that seignoriage revenues SR can be written as the product of the rate of growth of money supply by the level of real balances: \(SR=\frac {M_{t+1}-M_{t}}{P_{t}}=\frac {M_{t+1}-M_{t}}{M_{t}}\frac {M_{t}}{P_{t} }=x_{t+1}m_{t}\).

Since \(\ln (1+x_{t})\simeq x_{t}\) for small x _t, it is just appropriate that we specify the money growth autoregression for \(\ln (1+x_{t})\) under the log-linear approximation and for x _t itself under the linear approximation.

A more general discussion could be made with non-zero tax rates and a non-constant productivity parameter, with the same qualitative results.

If we used M _t∕P _t as an argument in the utility function, as in previous sections, the demand for money equation would involve expectations of policy and control variables, and the analytical treatment of the model becomes more tedious.

The solution is: \(\hat {\pi }_{t}=\underset {j\rightarrow \infty }{\lim }\left ( \frac {1}{\rho _{\pi }}\right ) ^{j}E_{t}\hat {\pi }_{t+j}.-\frac {1}{\rho _{\pi } }\underset {j=0}{\overset {\infty }{\sum }}\left ( \frac {1}{\rho _{\pi }} \right ) ^{j}E_{t}s_{t+j},\) although, whenever ρ _π > 1 the limit in the first term will be zero.

For consistency with other variables, we denote by x _t+1 the rate of growth of money supply at time t. So, the value of x _t+1 is chosen at time t according to (9.134). Special cases include a deterministic rate of money growth (when \(\sigma _{x}^{2}=0),\) or even constant money growth, (if \({\sigma _{x}^{2}=\rho _{x}=0)}\).

In the Appendix to this chapter, we get

on the other hand, we have, in steady-state: \((1-\tau ^{y})\alpha A_{{\textit {ss}}}k_{{\textit {ss}}}^{\alpha -1}=\frac {1}{\beta }-(1-\delta ),\) and \(1+r_{{\textit {ss}}}=\frac {1}{\beta }.\)

Plugging both equalities into the first equation, we get

Notice that:

which is analogous to

and from the log-linear approximation to this equation, we get (9.138).

To show that result:

MATLAB program neokeyn.m computes a single realization that it presents in the form of graphics. Program nkeyprg.m presents standard statistics after simulating the model an arbitrary number of times, chosen by the user. This program calls function nkeyn.m, which it must either be placed in the same directory as the programs above, or the directory be included in the MATLAB path.

It is important to point out that the values of parameters ε and ϕ are crucial to determine whether the solution is determinate or indetermined. Remember that ε is the price elasticity of demand for intermediate goods on the part of the firm producing the final good, while ϕ measures the level of the adjustment cost of prices. The lower ε and ϕ may be, the easier will be to obtain four stable and one unstable eigenvalues, leading to an undetermined equilibrium, i.e., multiple equilibria path, all converging to the same steady state. Our MATLAB programs have been written to capture the case of determined equilibria, while in the second chapter on endogenous growth models we discussed in detail the implications of equilibrium indeterminacy and how a numerical solution can be obtained, if desired. Hence, the user must take into account the possibility that for different values of ε and ϕ we can get indeterminacy, and the provided program will not compute the right solution. The same approach used for endogenous growth models can be followed to write the program that computes the numerical solution under indeterminacy in this model.