جایگزینی موقتی، ریسک گریزی، و عملکرد اقتصادی در یک اقتصاد باز تصادفی در حال رشد

The constant elasticity utility function implies that the intertemporal elasticity of substitution is the inverse of the coefficient of relative risk aversion. With empirical evidence suggesting that this relationship may or may not hold, studies of risk and growth should decouple these two parameters. This paper provides an analytical characterization and numerical simulations of the equilibrium of a stochastically growing small open economy under general recursive preferences. We show that errors committed by using the constant elasticity utility function, even for small violations of the compatibility condition, can be substantial. Our results suggest that the constant elasticity utility function should be employed with caution.

Recently, there has been a growing interest in analyzing the effects of policy shifts and other shocks on macroeconomic performance, growth, and welfare in the context of intertemporal stochastic growth models. These studies have been conducted for both closed and open economies, and a variety of shocks have been considered. Beginning with Eaton (1981), authors such as Gertler and Grinols, 1982, Smith, 1996, Corsetti, 1997 and Grinols and Turnovsky, 1998 and Turnovsky (2000) have analyzed the effects of both monetary and fiscal shocks in stochastically growing closed economies. Parallel to this, Turnovsky, 1993 and Devereux and Smith, 1994Grinols and Turnovsky, 1994, Obstfeld, 1994a and Asea and Turnovsky, 1998, and Turnovsky and Chattopadhyay (2003) have analyzed the effects of monetary shocks, terms of trade shocks, productivity shocks, and tax changes on economic growth and welfare in small open economies. In order to obtain closed-from solutions, both the production characteristics and the preferences must necessarily be restricted, and with few exceptions the existing literature assumes that the preferences of the representative agent are represented by a constant elasticity utility function.1 While this specification of preferences is convenient, it is also restrictive in that two key parameters critical to the determination of the equilibrium growth path—the intertemporal elasticity of substitution and the coefficient of relative risk aversion—become directly linked to one another and cannot vary independently. This is a significant limitation and one that can lead to seriously misleading impressions of the effects that each parameter plays in determining the impact of risk and return on the macroeconomic equilibrium and its welfare. Conceptually, the coefficient of relative risk aversion, R say, introduced by Arrow (1965) and Pratt (1964) is a static concept, one that is well defined in the absence of any intertemporal dimension. Similarly, the intertemporal elasticity of substitution, emphasized by Hall, 1978, Hall, 1988 and Mankiw et al., 1985 and others, focuses on intertemporal preferences and is well defined in the absence of risk. A natural definition of the intertemporal elasticity of substitution is in terms of the percentage change in intertemporal consumption in response to a given percentage change in the intertemporal price. For any utility function separable both over time and states, this measure equals the elasticity of the marginal utility with respect to consumption, ϵ say; McLaughlin (1995). The standard constant elasticity utility function has the property that both parameters ϵ and R are constant, though it imposes the restriction R = 1/ϵ, with the widely employed logarithmic utility function corresponding to R = ϵ = 1. Thus it is important to realize that in imposing this constraint the constant elasticity utility function is also invoking these separability assumptions. The empirical evidence for both these parameters is quite far-ranging. Estimates for ϵ based on macro data range from near zero (say 0.1) by Hall, 1988 and Campbell and Mankiw, 1989, to near unity by Beaudry and van Wincoop (1995). Epstein and Zin (1991) provide estimates spanning the range 0.05–1, with clusters around 0.25 and 0.7. More recent estimates by Ogaki and Reinhart (1998) suggest values of around 0.4, somewhat higher than the early estimates of Hall (1988). Estimates of ϵ based on micro data introduce further sources of variation. Attanasio and Weber, 1993 and Attanasio and Weber, 1995 find that their estimate of ϵ increases from 0.3 using aggregate data, to 0.8 for cohort data, suggesting that the aggregation implicit in the macro data may cause a significant downward bias in the estimate of ϵ. Atkeson and Ogaki (1995) and Ogaki and Atkeson (1997) find evidence to suggest that the intertemporal elasticity of substitution increases with household wealth. 2 Estimates of R show even more dispersion. Epstein and Zin (1991) find values of R clustering around unity, consistent with the logarithmic utility function, while at the other extreme, issues pertaining to the “equity premium puzzle” induce authors to take R as high as 18 ( Obstfeld, 1994a) or even 30 ( Kandel and Stambaugh, 1991). However, Constantinides et al. (2002) present alternative empirical evidence to suggest that R lies most plausibly in the range 2–5, a range that appears to be gaining increasing acceptance. Again, further insight into the empirical evidence is provided by micro data where, using Pakistani village data, Ogaki and Zhang (2001) find that R decreases with wealth. Within this range of estimates one certainly cannot rule out the constraint R = 1/ϵ being approximately satisfied. For example, R = 2.5, ϵ = 0.4 provides a plausible combination of parameters for which the constant elasticity utility function is appropriate, and indeed, we shall consider this as representative of a realistic benchmark economy in simulations that we shall undertake. But, given the empirical evidence, Rϵ may plausibly range from around 5–0.1, certainly well away from 1, as the constant elasticity utility function requires. Several authors, including Epstein and Zin (1989) and Weil (1990), have represented preferences by a more general (non-separable) recursive function, which enables one to distinguish explicitly between ϵ and R. This is important for two reasons. First, conceptually, R and ϵ impinge on the economy in quite independent, and often conflicting, ways. They therefore need to be decoupled if the true effects of each are to be determined. Second, the biases introduced by imposing the compatibility condition R = 1/ϵ for the constant elasticity utility function can be quite large, even for relatively weak violations of this relationship. In this paper we apply the Epstein–Zin recursive preferences to a simple continuous time stochastic growth model of a small open economy. We shall focus on an agent having access to two assets yielding stochastic returns, and we shall identify the two assets as being domestic and foreign respectively. In this respect our analysis is related to Obstfeld (1994b) who introduces these more general preferences into a closed (one-asset) economy. But, by considering an open economy, we find that the differential effects of the two parameters, as well as their interaction, are much more complex, depending in part upon the respective risk characteristics of the two assets. The paper begins by characterizing the stochastic equilibrium of the small open economy, identifying the closed economy as a useful benchmark. We first examine the impact of risk and return analytically and characterize the bias introduced by imposing the constant elasticity utility function. We supplement our analytical results with comprehensive numerical simulations. These have the advantage of illuminating the patterns of responses as the two key parameters, R and ϵ, vary, and emphasizing the role that portfolio substitution, a key element absent from the one asset economy, plays in the risk allocation process. The structure of the equilibrium clarifies how the separation of R and ϵ is potentially important. Risk aversion impinges on the equilibrium through the portfolio allocation process and thus through the equilibrium risk that the economy is willing to sustain. It also determines the discounting for risk in determining the certainty equivalent level of income implied by the mean return on the assets. The intertemporal elasticity of substitution then determines the allocation of this certainty equivalent income between current consumption and capital accumulation (growth). The main conclusion of the numerical simulations is that the bias in using the constant elasticity utility function, even within the set of plausible parameters may be large, and further, qualitatively erroneous inferences may be drawn. The following are typical examples. First, starting from the benchmark preference parameters R = 2.5, ϵ = 0.4, the constant elasticity utility function implies that doubling R to 5 (and thus simultaneously halving ϵ to 0.2) in a closed economy will reduce the equilibrium growth rate from 1.72 to 1.12%, whereas the unrestricted utility function implies that in fact the equilibrium growth rate will be raised to 1.84%. This represents an error of 0.72%, which in a long-run growth context compounds to a serious difference in economic performance. In the open economy, the errors are comparable, though they are sensitive to the relative riskiness of the domestic and foreign assets. Second, in some cases the direction of the bias committed by the constant elasticity utility function in an open economy is reversed from what it would be in a closed economy. For example, suppose that the true preferences are R = 5, ϵ = 0.4. Our results show that increasing the risk on the domestic asset from σy = 0.04–0.05 will reduce welfare by 3.63%, whereas the constant elasticity utility function with preferences R = 2.5, ϵ = 0.4 will imply only a 1.79% reduction. In the corresponding open economy, the constant elasticity utility function will continue to understate the welfare losses as long as the foreign asset is at least as risky as the domestic. But it will mildly overstate the losses (−0.131% vs −0.117%) if the domestic asset is the riskier one. A similar situation can emerge with the effects of the mean return. Finally, in the case of an open economy, the constant elasticity utility function may wrongly predict the direction of effect, even for plausible parameters. For example, suppose the true preferences are represented by the parameters R = 1, ϵ = 0.4. Our results show that if the foreign asset is riskier, then an increase in the rate of return on the domestic asset from 8 to 8.5% will reduce the growth rate by 0.09 percentage points. In contrast, the corresponding constant elasticity utility functions, R = ϵ = 1 and R = 2.5, ϵ = 0.4, imply that the growth rate would increase by 0.06 and 0.07 percentage points, respectively. The paper is structured as follows. Section 2 sets out the analytical framework. Sections 3 and 4 derive the formal implications for the closed economy and open economy respectively. Section 5 sets out the background to the calibrations, with the numerical results for the closed and open economy being discussed in Sections 6 and 7, respectively. Some final comments are provided in Section 8.

Most intertemporal studies of risk are based on the constant elasticity utility function, which has the property that the elasticity of substitution and the coefficient of relative risk aversion are both constant, but are tightly linked to one another. With the diversity of empirical evidence suggesting that this constraint may or may not be met, it is important that studies of risk and growth decouple these two parameters, which as we have shown impinge on the equilibrium in very distinct, and in some respects, conflicting ways. Our paper has provided both an analytical characterization as well as extensive numerical simulations of the equilibrium of a stochastically growing small open economy. The general conclusion to be drawn is that errors committed by using the constant elasticity utility function, even for small violations of the compatibility condition () within the empirically plausible range of the parameter values, can be quite substantial. While one certainly cannot rule out using the constant elasticity utility function, as a practical matter, our results suggest that it should be employed with caution, recognizing that if the condition for its valid use is not met, very different implications may be drawn. The issues raised in this paper have applications to other areas. One concerns the extent to which the use of the restrictive constant elasticity utility function rather than the more general recursive preferences may yield misleading inferences with respect to the impact of fiscal policy on growth and welfare in a stochastic environment. The consequences of this for policy making may be serious and merit investigation. Another application is to relax the assumed constancy of the intertemporal elasticity of substitution and the coefficient of relative risk aversion, as suggested by the micro-based empirical literature. Rebelo (1992) indicates how this may be achieved in the simplest deterministic endogenous growth model by adding a constant subsistence consumption level to the utility function, thereby making it of the Stone–Geary form. This generates transitional dynamics and thus its introduction into a stochastic growth model is likely to be analytically intractable, precluding closed form solutions such as those derived in this paper. But it is an interesting aspect that also merits further consideration.