Abstract
In this chapter we characterize optimal growth in a monetary economy, using the steady-state optimality conditions to analyze alternative designs for monetary policy. We introduce the concept of optimal steady-state rate of inflation and characterize conditions under which Friedman’s prescription for a zero nominal interest rate can be optimal. Steady-state optimality conditions are used to characterize the feasible combinations of monetary and fiscal policies. A numerical exercise is used to evaluate the welfare implications of alternative monetary policies. In another exercise, the reader will see the existence of a Laffer curve in this economy. The chapter closes with a characterization of optimal monetary policy in a dynamic Ramsey model, which takes into consideration the short- as well as the long-run effects of a policy intervention.
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Notes
- 1.
A commodity whose marginal utility at the level of zero consumption is equal to infinity must be purchased with money.
- 2.
This type of economies is not studied in this textbook. A good textbook for overlapping generations models is: Champ et al. [3].
- 3.
A discussion on alternative ways to generate a demand for money in growth models can be seen in Walsh [16].
- 4.
This choice of notation is not arbitrary. Associating a lower case letter to T t would produce an awkward notation, while the proposed use of M t is needed, because we have another money ratio, M t∕P t, for which we reserve the m t-notation.
- 5.
The private sector in the economy is made up by households and firms. The budget constraint of the private sector consolidates the exchanges between both types of agents. One part of households’ income comes form their financial investments, here represented by a portfolio of government bonds. They also receive income from the firm as a return to physical capital, that they own, as well as from their working time. Consolidating these income flows, we get (8.2). In Chap. 3 we already discussed the concept of representative agent as a way of modelling jointly the whole private sector of the economy, as well as the difference between this representative agent and a benevolent planner.
- 6.
For simplicity, we will assume along this chapter that there is not technological growth.
- 7.
We take into explicit account in this case the fact that we are dealing with nonegativity restrictions in all choice variables, which leads to the type of optimality conditions below. That allows us to discuss the possibility of zero demands for money or bonds, for instance. However, the reader must be aware that the same type of discussion could have been made in all other optimization problems in the book.
- 8.
There will always be an interior solution because of our assumption that the marginal utility of consumption becomes infinite for zero consumption.
- 9.
With population growth and the same production technology, economic growth for economy wide aggregates at the same rate than population growth could be sustained, as it is the case in the nonmonetary economies considered in previous chapters. Adding technical growth, which is not done in this chapter, would lead to a growth rate equal to the growth rate of population plus the rate of technical growth.
- 10.
This is a pure change of notation, and both expressions are used in economic modelling. The relationship can be approximated (through a Taylor series expansion) by: β = 1 − ρ, so that, for instance, β = 0.95 corresponds with ρ = 0.05.
- 11.
With the only difference that in the Cass–Koopmans chapter, the discount factor was denoted by θ, rather than ρ.
- 12.
In an economy where the government cannot use debt financing, we would have: m ss π ss = ζ ss, and the size of the lump-sum transfer would no longer be an independent policy target, being determined by the choice target for the rate of inflation.
- 13.
This policy prescription was first issued in Friedman [6].
- 14.
Condition (8.21) characterizes a maximum since, under the optimal policy,
$$\displaystyle \begin{aligned} \frac{\partial ^{2}W(\pi )}{\partial \pi ^{2}} =U_{22}(c_{{\textit{ss}}},m_{{\textit{ss}}})\left( \frac{\partial m_{{\textit{ss}}}}{\partial \pi }\right)<0 . \end{aligned}$$ - 15.
Like physical capital, for instance.
- 16.
Results would be similar if the government would choose the target level for public debt as well as for real balances, with the level of the lump-sum transfer then being endogenously determined.
- 17.
Since we have already seen that, in our model, inflation does not have any effect on consumption.
- 18.
We leave to the reader to show that the relationship between nominal and real interest rates that is obtained in this section still holds if, in addition to transfers to consumers, the government needs to finance the purchase of the single good in the economy, whatever its use may happen to be.
- 19.
Notice that
$$\displaystyle \begin{aligned} \frac{(1+n)V_{t+1}-\left( 1+i_{t}\right) V_{t}}{P_{t}}=(1+n)\frac{V_{t+1}}{ P_{t}}-\left( 1+i_{t}\right) \frac{V_{t}}{P_{t-1}}\frac{P_{t-1}}{P_{t}}=(1+n)\bar{ b}_{t+1}-\frac{1+i_{t}}{1+\pi _{t}}\bar{b}_{t}. \end{aligned}$$ - 20.
Unless agents had money illusion, it is real, and not nominal money balances, which must appear as an argument in preferences.
- 21.
We comment below on the changes to be introduced when ζ t is a transfer of some units of the consumption commodity.
- 22.
Still an alternative formulation could consider ζ t as transfers of the consumption commodity. Then D t would be defined by D t = M t − V t+1 + (1 + i t)V t, and the budget constraint for the representative consumer would be
$$\displaystyle \begin{aligned} c_{t}+\left[ k_{t+1}-\left( 1-\delta \right) k_{t}\right] +\frac{M_{t+1}}{P_{t}}\leq f\left( k_{t}\right) +\frac{D_{t}}{P_{t}}+\zeta _{t}, \end{aligned}$$and optimality conditions can be obtained without any difficulty.
- 23.
Remember from Chap. 3 that indeterminacy arises whenever the number of explosive eigenvalues of the transition matrix in the linear approximation to the equations describing the dynamics of the model is less than the number of control (i.e., decision) variables.
- 24.
Such analysis is undertaken in Chap. 9.
- 25.
The comparison is not completely fair in the sense that the θ parameter should be adjusted so that the velocity of money would remain the same in both cases, as a reflection of the fact that we are trying to match relevant characteristics of actual economies. However, this adjustment, that the reader can do as an exercise, is minor, and does not significantly change the results.
- 26.
Rates of inflation below this are not compatible with existence of a competitive equilibrium, since money would dominate physical capital and hence, the representative agent would not accumulate any of the latter, eventually leading to zero production and consumption.
- 27.
With depreciation allowances, the tax term in the budget constraint would be, \(\left ( 1-\tau _{t}^{y}\right ) [f\left ( k_{t}\right ) -\delta k_{t}]\).
- 28.
This a system of non-linear equations, whose solution may not exist, not be unique, or be unstable. In Chap. 9 we discuss numerical procedures for computing time series for the endogenous variables in economies like the one in this section.
- 29.
There is no need to impose the budget constraint for the representative agent since, as we already know, it is always satisfied, being a combination of the global constraint of resources and the government budget constraint.
- 30.
Or lump-sum taxes, if ζ t is negative.
- 31.
As shown in Sect. 3.5.2.2 when analyzing the Cass–Koopmans economy.
- 32.
We already mentioned that, with these preferences, it is not possible to compute the welfare cost of inflation relative to the optimal rate of inflation.
- 33.
Remember that there is not capital accumulation in this model economy.
- 34.
Notice that the two restrictions that follow characterize the competitive equilibrium. Indeed, we have already shown that the implementability condition summarizes (8.52)–(8.55). Together with the global constraint of resources (8.63), these conditions characterize the competitive equilibrium. It is not hard to show that if all these equations hold, so does the government budget constraint.
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Novales, A., Fernández, E., Ruiz, J. (2022). Growth in Monetary Economies: Steady-State Analysis of Monetary Policy. In: Economic Growth. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63982-5_8
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