Abstract
The effects of inflation are considered for a small open economy with overlapping generations and a cash-in-advance constraint on consumption. In an endowment economy with one good, the model recovers the adjustment mechanism underlying the monetary approach to the balance of payments, which incorporated the real balance effect in the savings function. Nevertheless, if the model has two goods that require different degrees of cash, the factor intensities of the goods also play a crucial role in determining the response of savings. In that case, the predictions of the monetary approach may be overturned; a result that is supported numerically.
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Notes
Weil (1991) considers the effects of an increase in the rate of growth of money in a closed economy with money-in-utility of Sidrauski. Money-in-utility does not correspond to the assumption underlying the real balance effect. Mansoorian (2012) discusses the closed economy implications of the real balance effect in a model with overlapping generations and CIA on consumption.
Cooley and Hansen (1991) work out the welfare implications of such asymmetric CIA constraints in a closed economy model. Mansoorian and Michelis (2010) work out some open economy implications of asymmetric CIA constraints in an infinite horizon (rather than an overlapping generations) model when both goods are traded, but one is durable.
Notice, Eq. 4 has the standard interpretation that, with logarithmic utility, total expenditures at time t (c t (s)(1 + r + ε)) as a percentage of total lifetime expenditures (a t (s) + h t (s)) is equal to the weight attached to instantaneous utility at time t (θ + π).
This assumption is convenient for our purposes, because by making the inflation rate exogenous, it abstracts completely from the off-steady state effects, arising from endogenous inflation, as emphasized by Fischer (1979). This paper is not concerned with whether inflation targeting is a superior policy. Cuche-Cueti et al. (2010) provide a comprehensive study of the advantages and disadvantages of inflation targeting versus a purely floating exchange rate system.
To obtain Eq. 7, start by aggregating Eq. 4 over cohorts to obtain \(C_{t}=\frac {\theta +\pi }{1+r+\varepsilon }\left [ A_{t}+H_{t}\right ]\), which implies \(\dot {C}_{t}=\frac {\theta +\pi }{1+r+\varepsilon }\left [\dot {A}_{t}+\dot {H}_{t}\right ]\). Next, work out Ȧ t and Ḣ t . First, consider the H t and Ḣ t equations. Note \(H_{t}=\int _{-\infty }^{t}e^{\pi (s-t)}h_{v}(s)ds\), where \(h_{t}(s)=\int _{t}^{\infty }e^{(r+\pi )(t-v)}\left [ y+\tau _{v}(s)\right ] dv\), and hence Ḣ t = (r + π)H t − (Y + Γ t ). Second, to obtain the equation for Ȧ t aggregate Eq. 3 over cohorts to obtain Ȧ t = r A t + Y + Γ t − C t − (r + ε t )M t . Finally, substitute for Ḣ t and Ȧ t into the expression for Ċ t using the results we have derived in this footnote, and use Eq. 5, together with the fact that A t = B t + M t , and hence Ȧ t = Ḃ t + Ṁ t , to obtain Eq. 7.
Notice, for saddlepath stability when r > θ the Ċ = 0 locus must be steeper than that Ḃ = 0 locus. Moreover, in the alternative case in which r > θ the Ċ = 0 will have a negative slope, and the intersection of the two curves will signify a negative steady state value for B, and, of course, a positive steady state value for C.
In the alternative case that r < θ the increase in the inflation rate would lead to an increase in consumption and a current account deficit. Most would argue that r > θ is the more reasonable assumption in an overlapping generations model; and it is the assumption made in most of the growth literature.
Non-traded goods consists mainly of services, while traded goods include all other goods. Services are traditionally more heavily financed with cash. Mansoorian and Michelis (2010) provide more details in this regard.
To derive this equation start from the no-arbitrage condition \(\dot {q}_{t}+r_{t}^{K}=rq_{t}\), multiply both sides by e r(t − v) and then integrate by parts.
To obtain Eq. 15, use the same methods as in Footnote 6 to obtain \(\dot {Z}_{t}=\frac {\theta +\pi }{1+\delta \left [ r+\varepsilon \right ] }\left [\dot {A}_{t}+\dot {H}_{t}\right ]\) and Ḣ t = (r + π) H t − (w t + Γt). Now, however, to obtain the equation for Ȧ v aggregate Eq. 11 over cohorts to obtain \(\dot{A}_{t}=rA_{t} + w_{t}+ \Gamma_{t} - [1 + \delta (r + \varepsilon)] Z_{t}\). Finally, to obtain Eq. 15, substitute for \(\dot { H}_{t}\) and Ȧ t from the expressions we have derived in this footnote into the Ż t equation, use Eq. 5, and also note that \(\dot {q}_{t}=rq_{t}-r_{t}^{K}\).
See, for example, Woodland (1982) for the properties of the GDP function, which will be used in several instances below.
Notice, with logarithmic instantaneous utility specified in Eq. 8 the demand for the non-traded good at time t is given by Z t / p~ t
Notice, this “distortion” effect involves the derivative of M̄ with respect to ε, which from the CIA constraint (16) involves the derivative of \(\frac {p_{t}}{\tilde {p}_{t}}\) with respect to ε. Now, as argued above, \(\frac {p_{t}}{\tilde {p}_{t}}\) is a measure of the distortion created by the fact that inflation implies an asymmetric tax on the traded and non-traded goods.
Notice, as at any instant a fraction π of the population dies, while at the same time a fraction π is born, the population size stays constant over time. If we normalize the population size to unity, and use the assumption that each individual supplies one unit of labour at any instant throughout his lifetime, then total labour supply at any instant will be unity. This is the assumption used in the resource constraint \(L_{t}^{T}+L_{t}^{N}=1.\)
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Acknowledgments
I would like to thank a referee and the editor for their very thoughtful comments and suggestions, which resulted in substantial improvements in this paper. I have also received very helpful comments from Leo Michelis. All the remaining errors are mine alone.
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Appendix
Appendix
In this Appendix we derive the saddlepath corresponding to the system of differential Eqs. 14, 15 and 17. Differentiating these equations totally, using (16), (18) and (19), we obtain
where bars denote steady state values, and the expressions for the coefficients α ij are as follows: α 11, α 11 = Ψ Z g p − 1, \(\alpha _{21}=-\frac {\pi (\theta +\pi )}{1+\left ( r+\varepsilon \right ) \delta }\), \(\alpha _{22}=r-\theta -\frac {\alpha \pi (\theta +\pi )}{1+r+\varepsilon }-\frac {\delta (1-\alpha )\pi (\theta +\pi )}{1+\left ( r+\varepsilon \right ) \delta }\), \(\alpha _{23}=-\frac {\pi (\theta +\pi )K}{1+\left ( r+\varepsilon \right ) \delta }\), α 32 = −Φ Z , and α 33 = r.
As B is the only predetermined variable in this system, for saddlepath stability the coefficient matrix should have one negative and two positive eigenvalues. A necessary condition for this is that the determinant Δ of the coefficient matrix for this system be negative. Let ξ denote the negative eigenvalue. Then the saddlepath of the system is given by the following equations:
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Mansoorian, A. On the Monetary Approach to the Balance of Payments. Open Econ Rev 25, 721–737 (2014). https://doi.org/10.1007/s11079-013-9293-5
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DOI: https://doi.org/10.1007/s11079-013-9293-5