APPENDIX: Derivations and Extensions
This Appendix provides details of the derivation of expressions presented in the text and extends the analysis to consider adjustment of the current account balance.
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Mr. Murphy is Assistant Professor of Economics at Boston College. This paper was written while he was a Visiting Scholar at the Research Department. The author thanks participants in seminars at the Fund, the University of Pennsylvania, and Boston College and three anonymous referees for helpful comments on an earlier draft of this paper.
See Williamson (1985) for a discussion of real exchange rate misalignment. Borensztein (1987) presents evidence that the appreciation of the U.S. dollar over the period 1980-84 is consistent with the hypothesis of a speculative bubble, although he is also unable to reject alternative explanations, Shiller (1984) describes stock prices as being determined in part by “fads.”
See Williamson (1985) for an analysis of a possible system of target zones for the exchange rates of industrial countries.
In Brock’s model, government subsidies to investment activity will drive a wedge between the market value of capital and the real exchange rate. This, though, is a one-time effect with subsequent changes in the market value of capital exactly equal to changes in the real exchange rate.
(1988) also uses an adjustment-cost framework to study capital accumulation in the open economy. His analysis, however, emphasizes the dynamics of the terms of trade rather than the adjustment of the real exchange rate.
See McKenzie (1982) for a similar adaptation of the dependent economy framework. McKenzie assumes that the traded good is the investment good and focuses on the roles of factor intensity and intertemporal substitution in determining the dynamics of the real exchange rate. The present paper follows Brock (1988) in assuming that investment requires inputs of the nontraded good and in focusing on adjustment in response to changes in fiscal policies.
See Khan and Lizondo (1987) for a related discussion of how the mix of fiscal policies accompanying a nominal devaluation is critical in determining the extent of real depreciation.
By assuming that only the traded good is consumed, the analysis leaves aside important issues concerning how the time profile of consumption is related to the domestic real interest rate. Incorporating nontraded consumption greatly complicates the analysis and obscures the role of investment in determining the real exchange rate. See Dornbusch (1983) for a model of optimal borrowing in which both traded and nontraded goods are consumed but in which there is no investment activity.
This assumption about factor intensities is sufficient to ensure that the steady-state equilibrium of the model is a unique saddlepoint. Previous work on capital accumulation in the open economy has employed nonoptimizing models in which global stability of the model required the traded sector to be relatively capital intensive (see, for example, Dornbusch (1980) and Obstfeld and Stockman (1985)). The optimizing model presented in this paper will always be “saddlepath stable” when the traded sector is capital intensive but will be globally unstable when the traded sector is labor intensive.
To highlight the role of aggregate (as opposed to sectoral) capital accumulation in determining the dynamics of the real exchange rate and the price of equity, the analysis assumes that capital and labor are costlessly reallocated across production sectors in order to equalize their respective rental rates. An alternative production setup could assume that the factors of production are sector-specific, as in Murphy (1988), or that the factors of production are reallocated gradually across production sectors, as in Mussa (1978). In the presence of aggregate capital accumulation, however, both of these alternatives become analytically intractable and would require numerical simulation for various parameter values in order to characterize the economy’s dynamics; see, for example. Murphy (1988).
An increase in P thus represents a real appreciation. The model presented in this paper focuses exclusively on the real side of the economy in order to study the interaction of relative prices and capital accumulation. To determine nominal prices and the nominal exchange rate it would be necessary to add a monetary sector to the economy. This would then raise the important issue of how to model money demand in an optimizing framework; an issue that remains controversial and that is beyond the scope of the present analysis.
As discussed in Section II. equation (3) implies that the long-run real exchange rale will be determined exclusively by the supply side of the economy. If instead capita) were sector-specific rather than mobile between sectors, then rental rates in each sector would depend on the real exchange rate and the sectoral distribution of the capital stock. As noted in Murphy (1988), though, the long-run real exchange rate would continue to be determined exclusively by the supply side provided the two sectors have the same rates of depreciation and adjustment cost functions. The reason is that in the long run rental rates will be equalized across sectors even when capital is sector specific.
The assumption that residents of the economy have perfect foresight means that they understand the structure of the economy and fully account for the effects of current and future policies on the economy in forming their expectations. While this assumption about expectations is obviously critical in determining the adjustment path for the economy, its forward-looking nature is more appealing than alternative backward-looking schemes when considering equilibrium in financial markets.
Note that the installation cost applies to the aggregate capital stock regardless of the allocation of the stock between the two sectors. Accordingly, “capital” is best viewed here as machines that can be readily shifted across production sectors and the installation cost is simply the cost of turning raw traded goods into machines.
The firm is assumed to borrow from residents at the world interest rate when its retained earnings fall short of its investment expenditures. Note that since bonds and equities are perfect substitutes in investor portfolios, the mix of external finance has no effect on the market value of the firm.
Note that if the firm were allowed to fully deduct its investment expenditures in calculating its taxable income, then changes in the tax rate would have no effect on investment incentives. Although equation (7) assumes zero deductibility, the results of this paper will continue to hold provided investment expenditures are less than fully deductible.
In the steady state, the rate of capital formation equals the rate of depreciation (x = 5) so that the capital stock is constant. Since the analysis considers adjustment in the neighborhood of the steady state, it follows that the rate of investment is always positive, implying that Q always exceeds zero.
The assumption that only the traded good is consumed leaves aside important issues concerning how changes in the real exchange rate induce intertemporal substitution effects on consumption. Allowing nontraded consumption, though, would greatly complicate the analysis and obscure the specific mechanisms highlighted here. See Dornbusch (1983) for a model of optimal borrowing in which both traded and nontraded goods are consumed but in which there is no investment activity.
The household also receives income in the form of capital gains on equity. Since the number of outstanding equity claims is assumed to be constant (recall that firms finance investment expenditure out of retained earnings or by borrowing), the capital gain exactly equals the increase in the value of equity holdings. As a result, identical capital gain terms would show up on both sides of the accumulation equation for household wealth and would therefore cancel, yielding equation (17).
The results of this paper would continue to hold if government purchases were incorporated as a separable term in the representative household’s utility function.
Since the main focus of the next section is on the dynamics of the real exchange rate and the price of equity, the adjustment of the current account is discussed in the Appendix.
The future level of lump-sum transfers to households is assumed to be the revenue variable that the government adjusts in order to ensure that the government’s budget is balanced intertemporally. Because households fully anticipate this future adjustment and because transfers are nondistortionary, the exact timing of the adjustment in transfers is irrelevant for the impact of the policy change on the economy. If instead the government adjusts a distortionary tax to ensure that its budget constraint is satisfied, then the timing of the adjustment would be critical in determining the effect of the initial tax cut on the economy.
If the increase in the price of equity is either less than or greater than the amount needed to place the economy on the stable path, then the economy will never reach the steady-state equilibrium given by the intersection of the
Continued real appreciation is more likely the longer is the time during which the tax cut is in effect. What is needed is for the capita) stock to be rising at a sufficient rate relative to the rate at which the price of equity is falling so that the change in the real exchange rate, as given by equation (29), is positive.
A change in government spending on traded goods, however, would have an effect on the price of equity, the real exchange rate, and investment if consumption included nontraded goods. The channel here would be through the change in nontraded consumption induced by the change in the future tax burden associated with the change in government spending. An increase in spending would, in this case, lower the consumption demand for nontraded goods, resulting in a real depreciation, rise in the price of equity, and capital accumulation.
In Figure 5 it is possible that the adjustment trajectory crosses into the region where the capital stock begins to rise prior to the actual reduction in spending. This will occur if the price of equity increases sufficiently relative to the real exchange rate so as to actually raise the ratio Q/P above its steady-state value. In this situation, the capital stock will begin to rise before the reduction in spending occurs.