The Role of Irreversibility and Fixed Costs Part 2

4.1 Baseline Parameter Values I select baseline parameter values in order to solve the model numerically. Note that the model is stylized and cannot be calibrated to confront with data. The purpose of the numerical solutions is to illustrate the working of the model mechanism, but not to match data. First, I set the corporate income tax rate θ = 0.34. I then consider parameters for preferences. I set the subjective discount rate β = 0.04 so that it is equal to the 4 percent interest rate. I choose the parameter h so that the steady-state fraction of time spent on working is equal to about 1/3. I next consider parameters for technology. I set αl = 0.64 so that the labor share is equal to 64 percent. I then choose the curvature parameter α in the profit function given in equation (6). Cooper and Haltiwanger (2006) and Hennessy and Whited (2005, 2007) estimate this parameter to be between 0.5 to 0.7. Here I set α = 0.6. Using equation (8), I can then deduce that αk = 0.216. I set the depreciate rate δ = 0.1, as standard in macroeconomics literature. As in Khan and Thomas (2003), I choose ξ = 0.003 so that fixed adjustment costs account for 1 percent of working time. I need to assign value for the remaining two parameters: λ and σ. Since there is no direct estimate for these values in the literature, I simply set σ = 0.2, as a baseline value. I also set λ = 0.5 so that half of the firms in the economy are subject to investment frictions. In summary, the parameter values are listed in Table 1. [Insert Table 1 Here.] 4.2 Welfare and Long-Run Average Capital In Section 3.4, I have analyzed the role of irreversibility and fixed adjustment costs in long-run capital accumulation in a partial equilibrium model by fixing the wage rate. I now consider the case of general equilibrium. Table 2 presents values of long-run average capital obtained, respectively, in the models with irreversible investment and fixed costs, with irreversible investment but without fixed costs, and with frictionless investment for two different volatility values σ = 0.2 and σ = 0.4. The numbers in brackets are the partial equilibrium values when 18 the wage rate is fixed at the level obtained in the model with the baseline parameter values given in Table 1. [Insert Table 2 Here.] Table 2 reveals several interesting results. First, the frictionless long-run average capital stock does not depend on the uncertainty parameter σ. This result follows immediately from Lemma 1 given my modeling of iid uncertainty. Second, introducing fixed adjustment costs in the model with irreversible investment reduces the long-run average capital stock in partial equilibrium as shown in Proposition 5. This result also holds true in general equilibrium. Third, as we discuss in Section 3.4, there is no unambiguous relation between the long-run average capital stock with frictionless investment and that with investment irreversibility in a partial equilibrium model. Table 2 shows numerically that this relation depends on the level of volatility. In particular, compared with the frictionless investment model, when σ = 0.2, introducing investment irreversibility raises the long-run average capital stock, but reduces it if fixed costs are present additionally. This result holds true in both general and partial equilibria. By contrast, when σ = 0.4, introducing irreversibility reduces the long-run average capital stock in partial equilibrium, and introducing fixed costs reduces it further. This result, however, does not hold true in general equilibrium. In general equilibrium, irreversibility raises the longrun average capital stock. In addition, the long-run average capital stock in the frictionless model is even smaller than that in the model with both irreversibility and fixed costs. This result demonstrates that the price feedback is important. Specifically, in partial equilibrium, the high level of the frictionless capital stock raises labor demand and hence causes the wage rate to rise. The high wage rate reduces marginal revenue product of capital, thereby reducing a firm’s incentive to accumulate capital. This effect may dominate so that the frictionless long-run average capital stock is actually lower than that with investment frictions. Finally, an increase in uncertainty lowers long-run average capital stock with investment frictions in partial equilibrium. This result seems intuitive because an increase in uncertainty raises the option value to wait. By contrast, because of the price feedback effect, an increase in uncertainty raises the long-run average capital stock in general equilibrium. 19 An important question is how idiosyncratic uncertainty and investment frictions influence welfare. For the frictionless model, idiosyncratic uncertainty does not affect welfare because it does not affect factor allocations and household utility. With investment frictions, capital is inefficiently allocated among firms, leading to welfare losses. I use the parameter values given in Table 1 to illustrate this point numerically. Compared to the frictionless investment model, irreversibility alone reduces the household utility from 0.5394 to 0.5387. Irreversibility and fixed costs together reduce the household utility further to 0.5371. Although an increase in uncertainty raises equilibrium aggregate capital in models with investment frictions as illustrated in Table 2, it reduces welfare. In particular, for the model with fixed costs and irreversibility, when σ is raised from 0.2 to 0.4 and other parameter values are given in Table 1, the household consumption decreases from 0.3781 to 0.3761 and working hours increases from 0.3290 to 0.3299. For the model with irreversibility and without fixed costs ξ = 0, when σ is raised from 0.2 to 0.4 and other parameter values are given in Table 1, the household consumption decreases from 0.3785 to 0.3766 and working hours increases from 0.3288 to 0.3295. These numerical examples illustrate that capital is more inefficiently allocated as idiosyncratic uncertainty increases. 4.3 Long-Run Effects of Corporate Tax Policy I now analyze the effects of an unexpected permanent corporate tax change on the economy. I consider two experiments by permanently lowering and increasing the corporate tax rate by 10 percentage points, respectively. Table 3 presents the results for the model with irreversibility and fixed costs. In particular, Columns 2-4 present the general equilibrium results, while Columns 5-6 present the partial equilibrium results by fixing the wage rate at the value obtained in the model with baseline parameter values given in Table 1. [Insert Table 3 Here.] Table 3 reveals several interesting results. First, a decrease in corporate income tax rate raises the equilibrium wage rate, output, capital stock, employment, welfare and average labor productivity. Second, a decrease in the corporate income tax rate changes the investment 20 trigger and target values. As a result, it raises the capital adjustment size and the adjustment rate. That is, changes in corporate tax have effects from both extensive and intensive margins. Figure 1 illustrates the extensive margin effect by plotting the stationary density of firms. When the corporate income tax rate is reduced, the stationary density shifts to the right. Third, the partial equilibrium impact of tax changes is much larger than the general equilibrium impact. The intuition follows from the price feedback effect as described earlier. [Insert Figure 1 Here.] I next consider the effects of fixed costs by doubling the value of the fixed costs parameter ξ. Table 4 presents the results. This table reveals that an increase in fixed costs reduces equilibrium output, capital, consumption and labor productivity. In particular, an increase in fixed costs raises the capital adjustment size, but reduces the capital adjustment rate. The extensive margin effect dominates so that it reduces aggregate capital. [Insert Table 4 Here.] Table 4 also reveals that the impact of tax changes increases with the fixed costs. But the quantitative effects are small. For example, following a 10 percentage point decrease in the corporate tax rate, the increase in output is 2.11 percent when ξ = 0.006, raising only by 0.01 percentage point of output increase in the model with ξ = 0.003. In addition, the welfare increase is extremely small, which is almost the same as in the model with ξ = 0.003. I finally compare with the models with frictionless investment and with irreversible investment, respectively. Tables 5 and 6 present the results. These tables show that irreversibility does not change the impact of tax changes on the economy in the sense that it does not affect the percentage change in equilibrium quantities. However, irreversibility is costly to the household because it reduces equilibrium consumption and labor productivity. Comparing with Table 3, I find that the impact of tax changes is larger with irreversibility and fixed costs, for both the partial and general equilibrium models (the partial equilibrium result is consistent with Proposition 7). In addition, the presence of fixed costs further reduces welfare. [Insert Tables 5-6 Here.] 21 5 Conclusion This paper presents an analytically tractable continuous-time general equilibrium model with investment irreversibility and fixed adjustment costs. In the model, there is a continuum of firms that are subject to idiosyncratic shocks to capital. I derive analytical approximate solutions that permit me to characterize the long-run stationary equilibrium explicitly up to a simple algebraic equation. I find the following main results: First, although the presence of investment frictions lowers consumer welfare, it may raise or reduce the long-run average capital stock, depending on the degree of idiosyncratic uncertainty. In addition, an increase in idiosyncratic uncertainty may raise equilibrium aggregate capital, but reduce welfare in models with investment frictions. Second, an unexpected permanent change in the corporate income tax rate affects the investment trigger and target values, and hence the size and rate of capital adjustment. Thus, tax policy has both intensive and extensive margin effects. Following this tax policy, the percentage changes in equilibrium quantities are larger when fixed adjustment costs are larger. Finally, these changes are significantly smaller in a general equilibrium model than in a partial equilibrium model. For tractability, I consider the role of irreversibility and fixed adjustment costs only. It would be interesting to consider other types of capital adjustment costs as studied in Abel and Eberly (1994) or Cooper and Haltiwanger (2006). I shall also emphasize that my model is highly stylized and cannot be used to match the data. As future research, it would be interesting to enrich the model structure and assumptions. In this case, tractability is lost and numerical methods are often needed. 22 Appendices A Frictionless Investment I write the capital accumulation equation without frictions as: dkt = −δkt dt + σkt dWt + dIt , (A.1) where It denotes investment. The firm’s problem is to solve ·Z ∞ max E −rt e 0 ¸ {((1 − θ) π (kt ; w) + θδkt ) dt − dIt } , (A.2) subject to (A.1). Substituting (A.1) into (A.2) yields: ·Z = = = = ¸ max E e {((1 − θ) π (kt ; w) + θδkt ) dt − dkt − δkt dt + σkt dWt } ¸ ·Z0 ∞ −rt e {((1 − θ) π (kt ; w) + θδkt ) dt − dkt − δkt dt} max E 0 ·Z ∞ ¸ −rt max E e {(1 − θ) (π (kt ; w) − δkt ) dt − dkt } ·Z0 ∞ ¸ −rt −rt ∞ −rt max E e (1 − θ) (π (kt ; w) − δkt ) dt − e kt |0 + kt de 0 ·Z ∞ ¸ −rt −rt k0 + max E e (1 − θ) (π (kt ; w) − δkt ) dt − re kt dt , ∞ −rt 0 ¤ £ where I have used the transversality condition: limt→∞ E e−rt kt = 0. The last equation implies that I only need to solve the first-order condition: ¡ ¢ (1 − θ) π 0 (k ∗ ; w) − δ = r. From this equation, I obtain (9). B Irreversible Investment When investment is irreversible, the value function V satisfies the Bellman equation (24) when the firm does not make investments. In addition, investment policy is characterized by a trigger value ki such that when k hits ki the firm makes investment to prevent k from falling below ki . 23 The value function V satisfies the smooth-pasting and high-contact conditions: V 0 (ki ) = 1, V 00 (ki ) = 0. In addition, V satisfies the growth condition: lim V (k) /Π (k) < ∞, k→∞ where Π (k) is given in (26). The general solution is given by: V (k) = G1 k βN + Π (k) , where G1 is a constant to be determined. Then we have G1 βN kiβN −1 + Π0 (k) = 1, G1 βN (βN − 1) kiβN −2 + Π00 (ki ) = 0. Eliminating G1 yields ¡ ¢ (1 − βN ) 1 − Π0 (ki ) = ki Π00 (ki ) . Solving yields (35). Thus, we have ³ π 0 (ki ; w) = = (1 − βN ) 1 − δθ r+δ ´ v (α) (α − βN ) (1 − θ) µ ¶ 1 − βN v (α) r +δ . α − βN r + δ 1 − θ I next derive the stationary distribution of capital. Making the transformation z = ln k, I only need to derive the stationary distribution for (zt ) . Let zi = ln ki . The stationary density f on (zi , ∞) satisfies the differential equation: µf 0 (z) = σ 2 00 f (z) , 2 where µ is given in equation (44). The general solution is given by f (z) = A1 eγz + B1 for z ∈ (zi , ∞) , 24 where A1 , B1 are constants to be determined and γ is given in equation (45). The boundary conditions are limz→∞ f (z) = 0, Z ∞ f (z) dz = 1. zi Using these boundary conditions, I can show that B1 = 0 and A1 = −γe−γzi . C Approximate Solution for the Trigger and Target I use an approximation method similar to that in Abel and Eberly (1998). I approximate J 1 (kb , kc ) and J 2 (kb , kc ) around kb = kc = ki and ξ = 0. Define k̃b = kb − ki and k̃c = kc − ki . Ignoring ξ, I apply the Taylor expansion to derive J j (kb , kc ) = J j (ki , ki ) + J1j (ki , ki ) k̃b + J2j (ki , ki ) k̃c 1 j 1 j j + J11 (ki , ki ) k̃b2 + J12 (ki , ki ) k̃b k̃c + J22 (ki , ki ) k̃b2 2 2 1 j 1 j 1 j 1 j (ki , ki ) k̃b3 + J112 (ki , ki ) k̃b2 k̃c + J122 (ki , ki ) k̃b k̃c2 + J222 (ki , ki ) k̃c3 + J111 6 2 2 6 I first derive partial derivatives for J 1 (kb , kc ). Differentiating equation (33) yields: ¡ ¢ J11 (kb , kc ) = −kcβN −1 Π00 (kb ) − (βN − 1) 1 − Π0 (kc ) kbβN −2 , and ¡ ¢ J21 (kb , kc ) = 1 − Π0 (kb ) (βN − 1) kcβN −2 + kbβN −1 Π00 (kc ) . From these equations, one can easily verify that J11 (ki , ki ) = 0, J21 (ki , ki ) = 0. Taking second-order differentiation of equation (33) yields: ¡ ¢ 1 J11 (kb , kb ) = −kcβN −1 Π000 (kb ) − (βN − 1) (βN − 2) 1 − Π0 (kc ) kbβN −3 , and 1 J12 (kb , kc ) = − (βN − 1) kcβN −2 Π00 (kb ) + (βN − 1) kbβN −2 Π00 (kc ) . 25 Evaluating at (ki , ki ) yields: ¡ ¢ 1 J11 (ki , ki ) = −kiβN −1 Π000 (ki ) − (βN − 1) (βN − 2) 1 − Π0 (ki ) kiβN −3 ¡ ¢ = kiβN −1 −Π000 (ki ) + (βN − 2) ki−1 Π00 (ki ) = α (α − 1) H (1 − θ) (βN − α) kiα+βN −4 /v (α) , and 1 J12 (ki , ki ) = 0. Similarly, we can derive that ¡ ¢ 1 J22 (kb , kc ) = 1 − Π0 (kb ) (βN − 1) (βN − 2) kcβN −3 + kbβN −1 Π000 (kc ) , 1 J22 (ki , ki ) = − (βN − 2) kiβN −2 Π00 (ki ) + kiβN −1 Π000 (ki ) · ¸ (1 − θ) H (1 − θ) H = kiβN −1 α (α − 1) (α − 2) kiα−3 − (βN − 2) α (α − 1) kiα−2 v (α) v (α) (1 − θ) H βN +α−4 ki . = α (α − 1) (α − βN ) v (α) Taking third-order differentiation of equation (33) yields: ¡ ¢ 1 J111 (kb , kc ) = −kcβN −1 Π0000 (kb ) − (βN − 1) (βN − 2) (βN − 3) 1 − Π0 (kc ) kbβN −4 , · 1 J111 (ki , ki ) ¸ −α (α − 1) (α − 2) (α − 3) (1 − θ) H α−4 −4 00 = ki + (βN − 2) (βN − 3) ki Π (ki ) v (α) (1 − θ) Hα (α − 1) = kiβN +α−5 ((βN − 2) (βN − 3) − (α − 2) (α − 3)) v (α) (1 − θ) Hα (α − 1) = kiβN +α−5 (βN − α) (α + βN − 5) , v (α) kiβN −1 1 J112 (kb , kc ) = − (βN − 1) kcβN −2 Π000 (kb ) + (βN − 1) (βN − 2) Π00 (kc ) kbβN −3 , £ ¤ 1 J112 (ki , ki ) = (βN − 1) kiβN −2 −Π000 (ki ) + (βN − 2) Π00 (ki ) ki−1 ¤ (1 − θ) H £ −α (α − 1) (α − 2) kiα−3 + (βN − 2) kiα−3 α (α − 1) = (βN − 1) kiβN −2 v (α) (1 − θ) H (βN − α) , = kiα+βN −5 (βN − 1) α (α − 1) v (α) 26 1 J122 (kb , kc ) = − (βN − 1) (βN − 2) kcβN −3 Π00 (kb ) + (βN − 1) kbβN −2 Π000 (kc ) , ¤ (1 − θ) H £ α (α − 1) (α − 2) kiα−3 − (βN − 2) α (α − 1) kiα−3 v (α) (1 − θ) H = (βN − 1) kiα+βN −5 α (α − 1) (α − βN ) , v (α) 1 J122 (ki , ki ) = (βN − 1) kiβN −2 ¡ ¢ 1 J222 (kb , kc ) = 1 − Π0 (kb ) − b (βN − 1) (βN − 2) (βN − 3) kcβN −4 + kbβN −1 Π0000 (kc ) , 1 1 J222 (ki , ki ) = −J111 (ki , ki ) . I now derive partial derivatives for J 2 (kb , kc ) . We begin with simplifying J 2 (kb , kc ) using equation (34): µ 2 J (kb , kc ) = = = = 1 βN 1 − β µ N 1 βN 1 − βN µ 1 βN µ 1 βN ¶ − 1 (kc − kb ) + Π (kc ) − Π (kb ) ¢ ¡ 0 Π (kc ) kc − Π0 (kb ) kb − ξw ¶ (1 − θ) H α δθ − 1 (kc − kb ) + (kc − kbα ) + (kc − kb ) v (α) r+δ · ¸ (1 − θ) Hα α δθ (kc − kbα ) + (kc − kb ) − ξw v (α) r+δ ¶µ ¶ µ ¶ δθ (1 − θ) H α −1 1− (kc − kb ) + 1− (kcα − kbα ) − ξw r+δ v (α) βN ¶µ ¶· ¸ δθ 1 1−α α α −1 1− kc − kb − ki (kc − kb ) − ξw. r+δ α We then derive partial derivatives: µ J12 (kb , kc ) = 1 −1 βN ¶ ¢ r + δ (1 − θ) ¡ 1−α α−1 ki kb − 1 , r+δ J12 (ki , ki ) = 0, µ J22 (kb , kc ) = ¶ ¶µ ¡ ¢ 1 δθ 1 − ki1−α kcα−1 , −1 1− βN r+δ J22 (ki , ki ) = 0, 27 ¶µ ¶ δθ 1 −1 1− = (α − 1) ki1−α kbα−2 , βN r+δ µ ¶µ ¶ 1 δθ 2 J11 (ki , ki ) = −1 1− (α − 1) ki−1 , βN r+δ µ 2 J11 (kb , kc ) 2 J12 (kb , kc ) = 0, µ 2 J22 (kb , kc ) =− ¶µ ¶ 1 δθ −1 1− (α − 1) ki1−α kcα−2 , βN r+δ 2 2 J22 (ki , ki ) = −J11 (ki , ki ) , µ ¶µ ¶ 1 δθ = −1 1− (α − 1) (α − 2) ki1−α kbα−3 , βN r+δ µ ¶µ ¶ 1 δθ 2 J111 (ki , ki ) = −1 1− (α − 1) (α − 2) ki−2 , βN r+δ 2 J111 (kb , kc ) 2 2 J112 (kb , kc ) = 0, J122 (kb , kc ) = 0, µ ¶µ ¶ 1 δθ =− −1 1− (α − 1) (α − 2) ki1−α kcα−3 , βN r+δ µ ¶µ ¶ 1 δθ 2 J222 (ki , ki ) = − −1 1− (α − 1) (α − 2) ki−2 . βN r+δ 2 J222 (kb , kc ) We finally use equations (31)-(32) and the preceding Taylor expansions to derive ³ ´ (1 − θ) H 1 (βN − α) kiβN +α−4 k̃b2 − k̃c2 0 = J 1 (kb , kc ) = α (α − 1) 2 v (α) ³ ´ 1 (1 − θ) H + α (α − 1) (βN − α) kiβN +α−5 (α + βN − 5) k̃b3 − k̃c3 6 v (α) ³ ´ (1 − θ) H 1 (βN − α) kiβN +α−5 (βN − 1) k̃b2 k̃c − k̃c2 k̃b + α (α − 1) 2 v (α) 28 and 0 = J 2 (kb , kc ) = −ξw µ ¶µ ¶ ³ ´ 1 1 δθ −1 1− (α − 1) ki−1 k̃b2 − k̃c2 + 2 βN r+δ µ ¶µ ¶ ³ ´ 1 1 δθ + −1 1− (α − 1) (α − 2) ki−2 k̃b3 − k̃c3 6 βN r+δ µ ¶ ¶µ 1 δθ = −ξw + (α − 1) −1 1− βN r+δ · ´¸ ´ 1 ³ 1 ³ 2 ×ki−1 k̃b − k̃c2 + (α − 2) ki−1 k̃b3 − k̃c3 2 3 It follows from the preceding two equations that ³ ´ ³ ´ 1 k̃b2 − k̃c2 = − ki−1 (α + βN − 5) k̃b3 − k̃c3 − ki−1 (βN − 1) k̃b2 k̃c − k̃c2 k̃b 3 µ ¶µ ¶ · ´ 1 ³ ´¸ ki−1 ³ 2 1 δθ −1 2 3 3 ξw = −1 1− (α − 1) k̃b − k̃c + (α − 2) ki k̃b − k̃c βN r+δ 2 3 (C.1) (C.2) I next substitute the right side of equation (C.1) for k̃b2 − k̃c2 in equation (C.2) and rearrange terms to derive ¶µ ¶ ´ k −2 ³ δθ 1 −1 1− (α − 1) i k̃b − k̃c ξw = βN r+δ 6 · ¸ ³ ´2 × (3 − βN ) k̃b + k̃c − 2βN k̃b k̃c . µ (C.3) Equation (C.1) also implies that ³ ´ 1 ³ ´2 1 (C.4) ki k̃b + k̃c + (α + βN − 5) k̃b + k̃c + (2βN − α + 2) k̃b k̃c = 0 3 3 ³ ´2 I will verify later that k̃b is of the order of k̃c , k̃b k̃c is of the order of k̃c2 , and k̃b + k̃c is of the order of k̃c4 . Given this fact, I ignore all terms with order higher than k̃c in equation (C.4). I then obtain that k̃b = −k̃c . Substituting k̃b = −k̃c into equation (C.3) and ignoring all terms with order higher than k̃c , I obtain equation (37). ³ ´2 I next derive k̃c . Ignoring the term k̃b + k̃c in equation (C.4), I obtain: k̃b = −3ki k̃c . 3ki + (2βN − α + 2) k̃b 29 (C.5) Given this solution, we know that k̃b k̃c is of the order k̃c2 . In addition, k̃b + k̃c = (2βN − α + 2) k̃c2 3ki + (2βN − α + 2) k̃b is of the order of k̃b2 , and ³ ´2 · k̃b + k̃c = (2βN − α + 2) 3ki + (2βN − α + 2) k̃b ¸2 k̃c4 is of the order of k̃c4 . These results verify my previous claim. D Proofs Proof of Lemma 1: See Appendix A. Proof of Lemma 2: The optimal investment policy and the trigger value ki are derived in Appendix B. From equation (35), one can immediately see that ki decreases with θ. We now show that ki < k ∗ . Using (35) and (9) and noticing that v (α) = (α − βN ) (α − βP ) r, βN βP (D.1) I only need to prove (α − βP ) (1 − βN ) δ >1+ . βN βP r Since βN < 0, the previous inequality is equivalent to δ (α − βP ) (1 − βN ) < βN βP + βN βP , r or δ α (1 − βN ) − βP < βN βP . r Using the fact that βP + βN = I can rewrite (D.2) as 2δ 2r + 1 and βN βP = − 2 , 2 σ σ µ ¶ 2δ 2δ α βP − 2 − βP < − 2 . σ σ 30 (D.2) That is βP > 2δ . σ2 ¡ ¢ This can be easily verified since v 2δ/σ 2 > 0. Q.E.D. Proof of Proposition 1: See Appendix C. Proof of Proposition 2: Proposition 1 implies that the size of capital adjustment kb − kc ≈ 2kc increases with fixed cost ξ. Let x = (1 − θ) . By (37) and Lemma 2, we can show that the sign of ∂ k̃c /∂x is the same as the sign of 2r − δ (1 − θ) (1 − α) . Since ∂ k̃c /∂θ = −∂ k̃c /∂x, we obtain the desired result. Q.E.D. Proof of Lemma 3: By Proposition 1 and the Taylor expansion, we have ! à ! à µ ¶ ki + k̃c kc ki + k̃c = ln ln = ln kb ki + k̃b ki − k̃c à ! à ! k̃c k̃c = ln 1 + − ln 1 − ki ki à !3 2k̃c 2 k̃c = + . ki 3 ki It follows from Proposition 1 that k̃c increases with ξ. Thus, the preceding equation implies that ln (kc /kb ) increases with ξ. The preceding equation also implies that to show ln (kc /kb ) increases with θ and σ 2 , we only need to show k̃c /ki increases with θ and σ 2 . By Proposition 1, it suffices to show that µ ¶ 1 δθ − 3 − 13 1− ki r+δ increases with θ and that (1 − βN ) ki decreases with σ 2 . The first result immediately follows from Lemma 2. Using Lemma 2 again, I show that ∂ [(1 − βN ) ki ] ∂σ 2 ∂βN ∂ki ki + (1 − βN ) 2 2 ∂σ · ∂σ ¸ ∂βN ∂ki = (1 − βN ) − ki ∂σ 2 ∂βN · ¸ −1 ∂βN = ki 2 − 1 < 0, ∂σ α − βN = − where the last inequality follows from the fact that βN < 0 and ∂βN /∂σ 2 > 0. Q.E.D. 31 Proof of Proposition 3: It follows from equation (40) and Lemma 3. Q.E.D. Proof of Proposition 4: The proof is similar to that of Proposition 1 in Caballero (1993). Here I outline the proof. I refer the reader to Caballero (1993) for details. Except on points zb and zc , the stationary density f (z) on (zb , zc ) ∪ (zc , ∞) satisfies the differential equation: µf 0 (z) = σ 2 00 f (z) , 2 where µ is given in equation (44). The general solution is ½ A1 eγz + B1 for z ∈ (zb , zc ) f (z) = , A2 eγz + B2 for z ∈ (zc , ∞) where A1 , A2 , B1 , and B2 are coefficients to be determined and γ is given in equation (45). The boundary conditions are f (zb ) = 0, lim f (z) = lim f (z) , z↓zc 0 z↑zc 0 lim f (z) − lim f (z) = lim f 0 (z) − lim f 0 (z) , z↑zc z↑∞ z↓zb Z ∞ f (z) dz = 1. z↓zc zb Using these boundary conditions, we can show that B2 = 0, A1 eγzb + B1 = 0, A1 eγzc + B1 = A2 eγzc , A2 γeγzc − A1 γeγzc and Z zc zb = −A1 γeγzb , Z (A1 eγz + B1 ) dz + Solving yields the desired result in Proposition 4. ∞ zc A2 eγz dz = 1. Q.E.D. Proof of Proposition 5: Note that Z Kf = ∞ ez f (z) , zb where the stationary density f is given in Proposition 4. Using Proposition 4, we can derive (47). Q.E.D. 32 Proof of Proposition 6: Using the stationary density f derived in Appendix B, we can derive the aggregate capital stock when investment is irreversible: Z Ki = ∞ zi γ zi e f (z) dz = e = γ+1 z µ ¶ σ2 1+ ki . 2δ When ξ is sufficiently small, we use Lemma 3, Propositions 3 and 5 to show that (kc − kb ) Kf = ln (kc /kb ) µ ¶ µ ¶ σ2 σ2 1+ = 1+ 2δ 2δ 2k̃c ki 2k̃c ³ ´3 . + 32 k̃kci Simplifying yields (50). By Proposition 1, k̃c increases with ξ. Thus, Kf decreases with ξ. Q.E.D. Proof of Proposition 7: Using Lemmas 1-2 and Proposition 6, I can show that ∂ ln Kf /∂θ = ∂ ln K ∗ /∂θ < 0. Using Proposition 6, I show that when ξ is sufficiently small, µ ³ ´2 ¶ ∂ ln 1 + k̃kci ∂ ln Kf ∂ ln Ki = − . ∂θ ∂θ ∂θ In the proof of Lemma 3, I show that k̃c /ki increases with θ. Thus, we have Q.E.D. 33 ∂ ln Kf ∂θ < ∂ ln Ki ∂θ < 0. 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The numbers in the brackets are values in the partial equilibrium framework by fixing the wage rate at the equilibrium value with baseline parameter values given in Table 1. Table 3. Results for the baseline model with irreversibility and fixed costs Wage w Output Y Capital K Employment L Consumption C Average productivity Y /L Duration time Adjustment size kc − kb Adjustment rate Γ θ = 0.34 0.8507 0.4369 0.5880 0.3290 0.3781 1.3279 4.5221 0.2640 0.2211 GE θ = 0.24 1.27 2.10 7.45 0.83 1.00 1.27 -4.92 2.24 5.17 θ = 0.44 -1.62 -2.58 -8.74 -0.98 -1.30 -1.61 6.07 -3.30 -5.72 PE θ = 0.24 0 8.03 13.69 8.03 4.35 0.00 -7.11 5.72 7.65 θ = 0.44 0 -9.42 -15.17 -9.42 -5.65 0.00 9.32 -7.39 -8.53 Notes: The parameter values are given in Table 1. Except for the numbers in Column 2, other numbers are in percentage. Columns 2-4 present results for the general equilibrium model. Columns 5-6 present results for the partial equilibrium model by fixing the wage rate at the equilibrium value (0.8507) in Column 2. 37 Table 4. Results for the model with irreversibility and fixed costs (ξ = 0.006) Wage w Output Y Capital K Employment L Consumption C Average productivity, Y /L Duration time Investment size kc − kb Adjustment rate Γ θ = 0.34 0.8502 0.4366 0.5870 0.3292 0.3779 1.3263 5.7220 0.3320 0.1748 GE θ = 0.24 1.28 2.11 7.47 0.83 1.00 1.27 -4.96 2.26 5.22 θ = 0.44 -1.62 -2.58 -8.76 -0.98 -1.31 -1.62 6.13 -3.32 -5.77 PE θ = 0.24 0 8.08 13.76 8.08 4.38 0.00 -7.18 5.78 7.73 θ = 0.44 0 -9.48 -15.24 -9.48 -5.69 0.00 9.44 -7.45 -8.62 Notes: Except for ξ = 0.006, other parameter values are given in Table 1. Except for the numbers in Column 2, other numbers are in percentage. Columns 2-4 present results for the general equilibrium model. Columns 5-6 present results for the partial equilibrium model by fixing the wage rate at the equilibrium value (0.8502) in Column 2. Table 5. Results for the model with frictionless investment Wage w Output Y Capital K Employment L Consumption C Average productivity, Y /L θ = 0.34 0.8519 0.4375 0.5884 0.3286 0.3786 1.3311 GE θ = 0.24 1.26 2.09 7.42 0.82 0.99 1.26 θ = 0.44 -1.60 -2.56 -8.71 -0.97 -1.29 -1.60 PE θ = 0.34 0.8507 0.4403 0.5921 0.3312 0.3811 1.3292 θ = 0.24 0 7.94 13.58 7.94 4.28 0 θ = 0.44 0 -9.32 -15.04 -9.32 -5.56 0 Notes: The parameter values are given in Table 1. Except for the numbers in Columns 2 and 5, other numbers are in percentage. Columns 2-4 present results for the general equilibrium model. Columns 5-7 present results for the partial equilibrium model by fixing the wage rate at the equilibrium value (0.8507) in the baseline model. 38 Table 6. Results for the model with irreversible investment Wage w Output Y Capital K Employment L Consumption C Average productivity, Y /L θ = 0.34 0.8516 0.4375 0.5896 0.3288 0.3785 1.3306 GE θ = 0.24 1.26 2.09 7.43 0.82 0.99 1.26 θ = 0.44 -1.60 -2.56 -8.71 -0.97 -1.29 -1.60 PE θ = 0.34 0.8507 0.4395 0.5924 0.3307 0.3803 1.3292 θ = 0.24 0 7.94 13.58 7.94 4.28 0 θ = 0.44 0 -9.32 -15.04 -9.32 -5.56 0 Notes: The parameter values are given in Table 1. Except for the numbers in Columns 2 and 5, other numbers are in percentage. Columns 2-4 present results for the general equilibrium model. Columns 5-7 present results for the partial equilibrium model by fixing the wage rate at the equilibrium value (0.8507) in the model with baseline parameter values given in Table 1. 1.6 θ=0.34 θ=0.24 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −1.2 −1 −0.8 −0.6 −0.4 −0.2 ln(k) 0 0.2 0.4 0.6 0.8 Figure 1: Stationary densities of ln(k) for θ = 0.34 and θ = 0.24. Parameter values are given in Table 1. 39