ریسک گریزی و رابطه عدم قطعیت سرمایه گذاری : نقش استهلاک سرمایه

Some recent contributions [Nakamura, T., 1999. Risk-aversion and the uncertainty–investment relationship: a note. Journal of Economic Behavior and Organization 38, 357–363; Saltari, E., Ticchi, D., 2005. Risk-aversion and the investment–uncertainty relationship: a comment. Journal of Economic Behavior and Organization 56, 121–125] suggest that investments react negatively to uncertainty when the risk aversion index characterizing firm’s preferences is lower than one but higher than the labour income share of output. We show that this result crucially depends on the assumption of complete capital depreciation after production. When realistic values for the capital depreciation parameter are taken into account, the investment–uncertainty relation is negative for values of the risk-aversion index larger than unity.

We assume that the technology of the competitive firms is a constant returns to scale Cobb Douglas, View the MathML sourceYt=Kt1−αLtα, where KtKt is the stock of capital at time t and LtLt is the labour employed. Notice that αα is labour’s share of output. Following Nakamura and Saltari and Ticchi, we assume that the output price, ptpt, is stochastic; more specifically, for us ptpt is a serially independent random variable with mean View the MathML sourcep¯ and variance σ2σ2. The firm’s cash flow at time t is given by View the MathML sourceπt=ptKt1−αLtα−wLt−It, Turn MathJax on where ww is the constant nominal wage.2 As in many classic papers (see, e.g., Hartman, 1972 and Abel, 1983), the firm first decides upon the capital level, then observes the realization of the price shock and finally it chooses the optimal amount of labour, which is View the MathML sourceLˆt=[(α/w)pt]ηKt, where η≡1/(1−α)η≡1/(1−α). Bear in mind that η>1η>1, since α∈(0,1)α∈(0,1). Having set View the MathML sourceLt=Lˆt, the firm’s cash flow becomes equation(1) View the MathML sourceπˆt=BtηKt−It, Turn MathJax on where Bt=BptBt=Bpt and B=(1−α)1−α(α/w)αB=(1−α)1−α(α/w)α. View the MathML sourceBtηKt represents the operating profits; in what follows we will refer to View the MathML sourceBtη as the marginal revenue product of capital. Following again Nakamura and Saltari and Ticchi, we assume that the firm’s intertemporal preferences on its cash-flows are of the constant relative risk aversion type. We deal with the firm’s intertemporal maximization problem using dynamic programming: equation(2) View the MathML sourceV(Kt,pt)=max⁡{It}πˆt1−γ1−γ+βEt[Vt+1(Kt+1,pt+1)], Turn MathJax on where V(Kt,pt)V(Kt,pt) is the maximum value function, which depends on the state variables (KtKt and ptpt), γ>0γ>0 is the relative risk aversion coefficient, and β∈(0,1)β∈(0,1) is the discount factor. The firm must take into account the capital accumulation equation: equation(3) Kt+1=(1−δ)Kt+It,Kt+1=(1−δ)Kt+It, Turn MathJax on where δ∈[0,1]δ∈[0,1] is the capital depreciation parameter. It turns out that solution for problem (2) subject to (3) implies the following investment function: equation(4) View the MathML sourceIt={[β1/γ{Et[Bt+1η+1−δ]1−γ}1/γ][Btη+1−δ]−(1−δ)}Kt. Turn MathJax on 2. The investment–uncertainty relation The effects of price uncertainty on investment can be obtained by differentiating, with respect to the variance of the shock, Eq. (4). One readily obtains equation(5) View the MathML sourceSign∂It∂σ2=Sign(1−γ)∂∂σ2Et[Bt+1η+1−δ]1−γ1−γ. Turn MathJax on The term in the big square brackets, which is the expected utility of the resources yielded in the next period by a unit of capital, is crucial in determining the sign of the uncertainy–investment relation. At this stage, Saltari and Ticchi point out that a closed form solution can be obtained only in the case δ=1δ=1, which they analyze. However, it is easy to study the general case approximating the effect of a mean-preserving spread in future prices on investment by means of second-order Taylor expansions. Therefore, we now consider the Taylor approximation of View the MathML sourceEt[Bt+1η+1−δ]1−γ/(1−γ) around View the MathML sourcep¯, which is View the MathML sourceEt[Bt+1η+1−δ]1−γ1−γ≃Et[Bηp¯η+1−δ]1−γ1−γ+[Bηp¯η+1−δ]−γηBηp¯η−1dpt+1−γ[Bηp¯η+1−δ]−(γ+1)(ηBηp¯η−1)2(dpt+1)22+[Bηp¯η+1−δ]−γη(η−1)Bηp¯η−2(dpt+1)22, Turn MathJax on where View the MathML sourcedpt+1≡(pt+1−p¯). The expression above immediately reduces to equation(6) View the MathML sourceEt[Bt+1η+1−δ]1−γ1−γ≃[Bηp¯η+1−δ]1−γ1−γ−γ[Bηp¯η+1−δ]−(γ+1)(ηBηp¯η−1)2σ22+[Bηp¯η+1−δ]−γη(η−1)Bηp¯η−2σ22. Turn MathJax on The second line in Eq. (6) highlights the risk aversion effect: an increase in the price variance exerts a negative effect on the expected utility of future resources, increasing their expected marginal utility. A simple intuition suggests that a mean-preserving spread in the future price should induce agents to invest more, to rebalance the current and the future (expected) marginal evaluations of the resources. In other words, the negative impact of the price variance on the expected utility of future resources should induce a “precautionary” increase in investment. However, the negative effect of the price variance on the next period utility does not necessarily imply an increase in investment. From (5), this happens if and only if γ>1γ>1. In fact, a larger investment, increasing the availability of capital in the future, implies larger fluctuations in future incomes for any given σ2σ2. Hence, an increase in the variance of future prices may induce a risk-averse individual to invest less, since this mitigates the increase in future income fluctuations, a desirable effect for a risk-averse agent. As is well known, with CRRA preferences the former positive effect of σ2σ2 on investment is stronger than the latter negative one when γ>1γ>1.3 The third line in Eq. (6) shows the flexibility effect: the convexity in the firm’s profit function implies that a mean-preserving spread in the output price increases the expected return of capital. However, the increase in the f uture marginal revenue product of capital, making investment more productive, favors savings (via an intertemporal substitution effect), but it also involves an income effect, which plays in favor of an increase in current “consumption”. Eq. (5), signals that the negative effect of σ2σ2 for savings is stronger than the positive one for γ>1γ>1. Notice that the flexibility effect goes against the risk aversion effect. In the very special case δ=1δ=1, Eq. (6) reduces to View the MathML sourceEt[Bt+1η]1−γ1−γ≃[Bηp¯η]1−γ1−γ+[Bηp¯η]−γηBηp¯η−2α−γ1−ασ22, Turn MathJax on and hence, from (5), Sign(∂It/∂σ2)=Sign[(1−γ)(α−γ)]Sign(∂It/∂σ2)=Sign[(1−γ)(α−γ)], as in Saltari and Ticchi (2005) and Nakamura (1999). To see what happens in the general case, notice that the variance of View the MathML sourceBt+1η, which is of the operating profit per unit of capital, can be approximated as equation(7) View the MathML sourceVar[Bt+1η]=Et[Bt+1η−Et(Bt+1η)]2=Et[Bt+12η]−{Et(Bt+1η)}2≃(ηBηp¯η−1)2σ2. Turn MathJax on Exploiting (7) we can reformulate (6) as: equation(8) View the MathML sourceEt[Bt+1η+1−δ]1−γ1−γ≃[Bηp¯η+1−δ]1−γ1−γ+[Bηp¯η+1−δ]−γ−γVar[Bt+1η]2[Bηp¯η+1−δ]+η(η−1)Bηp¯η−2σ22. Turn MathJax on By focusing on the terms in the big curly brackets, we notice that a decrease of the depreciation parameter dampens the effect of the price variance on the marginal utility of future resources. Hence, a reduction (from unity) of the depreciation parameter gives more prominence to the flexibility effect (summarized in (8) by View the MathML sourceη(η−1)Bηp¯η−2σ2/2). As we have highlighted above, the flexibility effect acts in favor of a negative investment–uncertainty relationship whenever γ>1γ>1. Therefore, a depreciation parameter lower than unity induces a negative uncertainty–investment relationship also for some γ>1γ>1. The intuition for this result is that a departure of δδ from unity reduces the normalized variance of the future firm’s resources, thereby lowering the importance of the risk aversion effect. To assess the quantitative importance of a depreciation parameter different from unity, we need to rely on some numerical exercises. The difficulty here is that convincing values for some parameters such as ββ or δδ can be easily supplied, but the choice for the moments of the marginal profit function, View the MathML sourceBt+1η, is far from obvious. To cope with this problem, we notice from (1) and (4) that the firm cash-flow is linear in capital. Hence, from (4), and (3), the average growth rate for the firm net profit can be expressed as View the MathML sourceg(σ2)=Et−1[β1/γ{Et[Bt+1η+1−δ]1−γ}1/γ[Btη+1−δ]]−1; Turn MathJax on therefore View the MathML source1+g(0)=β1/γ[Bηp¯η+1−δ]1/γ and hence, View the MathML sourceBηp¯η+1−δ=[1+g(0)]γβ. Turn MathJax on Using the result above, we substitute out View the MathML sourceBηp¯η in Eq. (6) when convenient. Grouping terms, we obtain View the MathML sourceEt[Bt+1η+1−δ]1−γ1−γ≃[Bηp¯η+1−δ]1−γ(1−γ)×1+(1−γ)ηBηp¯η−2Bηp¯η+1−δ−γη1−(1−δ)β[1+g(0)]γ+η−1σ22. Turn MathJax on The last equation allows us to conclude that equation(9) View the MathML sourceSign∂It∂σ2=Sign(1−γ)α−γ+γ(1−δ)β[1+g(0)]γ. Turn MathJax on The expression in the big square brackets of (9) can now be studied as a function relating the labour income share αα and the risk-aversion parameter γγ. To determine where this function is negative, we choose δ=0.12δ=0.12, β=0.97β=0.97, and g(0)=0.02g(0)=0.02; that is, we “calibrate” the model with a period of annual length in mind. What we obtain is that our function takes negative values in the shaded area in Fig. 1, Panel (a). Panel (b) is added for comparison: it shows the portion of the (α,γ)(α,γ) space where the investment–uncertainty relationship is negative if δ=1δ=1. Notice that in Panel (a), when α=0.7α=0.7, the investment–uncertainty relation is negative for γ∈(0,3.451)γ∈(0,3.451). Our simulations suggest that the portion of the parameter space where a mean-preserving spread of the output price induces a contraction in investment is reduced by an increase in g(0)g(0), by a increase in δδ and by a reduction in ββ. However, if we pick δ=0.15δ=0.15, β=0.95β=0.95, and g(0)=0.05g(0)=0.05, values which could be deemed on the edge of “irrealism”, when α=0.7α=0.7 the investment–uncertainty relation is negative for γ∈(0,2.465)γ∈(0,2.465).

We have shown that taking into consideration realistic values for the depreciation parameter, it is easy to obtain a negative investment–uncertainty relationship for a risk aversion parameter larger than unity. In fact, a depreciation parameter lower than unity gives more prominence to the flexibility effect because it reduces the impact of the price variance on the expected value of future income, therefore dampening the role of risk aversion. As we have already remarked, when γ>1γ>1, the flexibility effect negatively affects investment due to the negative income effect generated by an increase in the average future rate of return. We believe our result is interesting because the empirical evidence is in favor of a negative relationship between investment and uncertainty (see, e.g., Ferderer, 1993, Leahy and Whited, 1996 and Guiso and Parigi, 1999). The last paper, besides providing support for the view that irreversibility is significant in accounting for the negative relationship between uncertainty and investment, shows that such a relation is negative (although scarcely significant) also for firms characterized by low degrees of irreversibility. Caselli et al. (2003), using sectoral data, document that demand uncertainty had negative effects on investment in Europe during the 1990s. In sum, when sensible values for the capital depreciation parameter are taken into account, the investment–uncertainty relation becomes negative for realistic values of the risk aversion parameter.