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the symmetric condition we can also obtain the aggregate production function for
…nal goods:
Yt= Kt(ZtLY;t)1 : (16)
4.2.7 Decentralized equilibrium
The decentralized equilibrium is de…ned as an in…nite sequence of allocations fCt; Kt; At; Yt; Xt; Lt; LY;t; LN;t; LZ;t; Gtg1t=0, prices fwt; rt; rA;t; rK;t; pt(j); Vt(j)g1t=0, policies f K;t; L;tg, such that at any instant of time:
a. households choose fCt; Kt; At; Ltg to maximize lifetime utility (1) taking prices and policies as given;
b. competitive …nal goods …rms choose fXt(j); LY;tg to maximize pro…t taking prices as given;
c. monopolistic intermediate …rms j 2 [0; Nt] choose fkt(j); pt(j); lZ;t(j)g to maximize pro…t taking frt; rK;t; wt; K;tg as given;
d. entrants make entry decisions taking fVt; wtg as given;
e. the …nal goods market, capital market and labor market clear;
f. the government budget constraint is balanced: K;trK;tKt+ L;twtLt= Gt.
4.3 Long-run e¤ects of capital taxation
In this section, we examine the long-run e¤ects of capital taxation on growth and entry. Before doing so, we …rst characterize the balanced-growth path (BGP) in
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this model. In the steady-state, labor allocations are stationary. Accordingly, from the resource constraint and eqs (6), (13), and (16), we can easily derive the properties of the BGP equilibrium:
K_t Kt =
Z_t Zt =
Y_t Yt =
C_t Ct =
G_t Gt = w_t
wt; _Nt= 0: (17) In the following analysis we denote x;t as the growth rate of any generic variables x and denote as the balanced growth rate of all growing variables.
This section focuses on the steady-state e¤ects of capital taxation. For this purpose, we de…ne the transformed variables ! = w=K, c = C=K, z = Z=K, y = Y =K, and drop the time subscript to denote the stationary levels of (trans-formed) variables in the steady state. By using the property of eq. (17), the BGP equilibrium of the macro economy can be described by the following set of equations:
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The detailed derivation of eq. (18) is presented in Appendix A. It should be noted that in this model the labor income tax L is treated as an endogenous variable because it will endogenously adjust in order to balance the government constraint as the capital income tax changes.
We are now ready to solve the balanced growth rate. By inserting eq. (18c) into eq. (18h) we can obtain LNY = ' . By inserting eq. (18c) into eq. (18i) we can obtain lZ = LNY = ' . Combining these two expressions and putting into eq. (18g) yields the closed-form balanced growth rate:
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= 1 '
: (19)
In eq. (19) we have assumed that > 'to ensure that both lZ and are positive.
It directly follows from eq. (19) that:
@
@ K
= 0: (20)
Before explaining the result of eq. (20), it is useful to …rst discuss the long-run e¤ects of the capital income tax on labor allocations and entry. By inserting
LY
N = ' and lZ = ' into (18m), we can obtain the relation L = LY where 1 + ' > 0is a composite parameter independent of K. With L = LY, we can directly infer from (18j) that L is a decreasing function of K:
L = 1 2 K
(1 ) : (21)
eq. (21) states that an increase in the capital tax is coupled with a decrease in the labor tax. This result is termed as “tax shifting” in the literature.
Next, we put eqs (18c)-(18e), (18k), and (18l) into eq. (18b) to eliminate !, c, y, z and p, and then insert eqs (18a) and (18f) to eliminate r and rK; accordingly, we attain the following expression:
L = LY = (1 L)(1 )
(1 ) 2(1 K)=( + ) : (22)
eq. (22) is a closed form solution of equilibrium labor force given eqs (19) and (21). Di¤erentiating L with respect to K yields:
‧
the situation under which the government has another policy instrument such as a lump-sum tax (or transfer). The lump-sum tax adjusts to balance the govern-ment constraint as the capital tax increases, so that the labor income tax is left unchanged. In this case, eq. (21) is absent, and thus in eq. (22) L is treated as an exogenous variable. As such, we can easily see from eq. (22) that an increase in the capital tax results in less labor supply. The intuition can be explained as follows. A unilateral increase in the capital tax depresses the intermediate-goods sector. Speci…cally, it increases the production costs of intermediate …rms, and in turn results in less intermediate goods for …nal goods production, leading to a lower marginal product of …nal-goods labor. On the other hand, the pro…ts of intermediate …rms decreases, which also depresses the returns of in-house R&D labor. Both e¤ects point to a lower labor supply because the returns on both labor decrease. Now we turn to the case of tax shifting. In this case, to hold the ratio of government spending to GDP constant, an increase in the capital tax is accompanied with a decrease in the labor tax. The decrease of the labor tax boosts the households’labor supply. Although the abovementioned negative e¤ect is still present, the positive e¤ect triggered by the lower labor tax is stronger. As a consequence, the total labor supply increases in response to the rise of the capital tax under the case of tax shifting.
Finally, from the expressions L = LY, LNY = ' , and LZ = N lZ = N ( ' ), we can also derive the e¤ects of the capital tax on …nal-goods labor, in-house R&D labor, and the number of intermediate …rms:
@LY
@ K > 0; @LZ
@ K > 0; @N
@ K > 0; @lZ
@ K = 0:
The following proposition highlights our …ndings in this section:
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Proposition 5 In the long run, an increase in the capital income tax has a positive e¤ect on total labor force, …nal-goods labor, aggregate in-house R&D labor, and the number of intermediate …rms, while it has a neutral e¤ect on in-house R&D labor per …rm and the growth rate.
We have explained above that the rise of the capital income tax boosts to-tal labor force by reducing the labor income tax. However, this increase of the labor force has no e¤ect on long-run growth. This is the distinct feature of the second-generation R&D-based growth model, in which the scale e¤ect (of labor) is removed. For example, Peretto (1998) demonstrates that the steady-state produc-tivity growth does not depend on population size. Changes in population size have only transitory e¤ects on economic growth. Likewise, in our model, the long-run growth rate is independent of the aggregate labor force. More important insights into the e¤ects of capital taxation can be obtained by carrying out a quantitative analysis of the e¤ects along the transition, which we present in the next section.
4.4 Quantitative analysis
In this section, we provide a quantitative analysis to explore the e¤ects of capital taxation along the transition.7 We generalize the utility function as follows:
U = Z 1
0
e t ln Ct+ (1 Lt)1
1 dt; (24)
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are within the plausible ranges used in the literature. First, in line with Andolfatto et. al. (2008) and Acemoglu and Akcigit (2012), the capital income share and the discount rate are set to the values 0.4 and 0.05, respectively. Second, the initial capital tax rate K is set to 0.3 based on the average US e¤ective tax rate estimated by Carey and Tchilingurian (2000). As for the government size (the ratio of government spending to output), data of US exhibits around 20 percent (Gali, 1994), and has slightly increased in recent years. We set to be 0.22, which is the average level during 2001-2013. Third, the parameter for leisure preference is chosen as 0.9135 such that total hours worked is around one third of time endowment. Moreover, we set = 1:67, implying a Frisch elasticity of 1.2; see Chetty et al. (2011). For the in-house R&D productivities, we choose ' = 8:94 to target the empirical level of the output growth rate in the US, which is around 2%. As for the R&D spillovers, we choose = 0:2052 such that the ratio of R&D to GDP is 1.97%; see King (2004). Lastly, we assume that an entrant incurs 0.5 units of labor as a setup cost, indicating the value = 2. Table 4.1 reports our calibrated parameter values.
Table 4.1: Calibrated parameter values
K '
0.4 0.05 0.3 0.22 0.9135 1.67 8.94 0.2052 2
We conduct a policy experiment of a small increase in the capital income tax rate from its initial value 30% to 31%. Figures 4.1-4.8 depict the e¤ects on the growth rate and important variables along the transition path. As shown in Figure 4.1, in the short run where the number of intermediate …rms is …xed, raising the capital tax has a negative impact on economic growth.
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Figure 4.1: Transition path of the output growth rate
Figure 4.2: Transition path of total labor force
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wage. The decreased wage has an immediate bene…cial e¤ect on three segments that use labor as inputs: the …nal-goods sector, the house R&D sector of in-termediate …rms, and potential entrants. However, for the …nal-goods sector, the higher capital tax decreases the supply of intermediate goods; for the intermedi-ate …rms, the higher capital tax increases their production costs. Only the entry labor can enjoy the pure bene…t of the lower wage without being directly (and negatively) a¤ected by the higher capital tax. Therefore, in the …nal-goods sector and in-house R&D sector, the marginal product of labor is temporally less than marginal product of labor used for entry. This subsequently causes labor to ‡ow out from these two sectors to the entrants as a sudden response. This is what we see in Figures 4.3, 4.5 and 4.6.
Figure 4.3: Transition path of aggregate in-house R&D labor
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Figure 4.4: Transition path of in-house R&D labor per …rm
Figure 4.5: Transition path of …nal-goods labor
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‡ows out. Furthermore, in the short-run, the number of intermediate …rms is …xed.
This indicates that the in-house R&D labor per …rm also decreases (Figure 4.4), which then is associated with a lower growth rate. Thus we see in Figure 4.1 that the growth rate immediately jumps down as the policy shock occurs.
Figure 4.6: Transition path of labor used for entry
Figrue 4.7: Transition path of …rm number
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Along the transition process, new intermediate …rms start to enter the market.
Thus the number of intermediate …rms rises (Figure 4.7). The pro…ts of intermedi-ate …rms declines, which reduces the gap between the value of intermediintermedi-ate …rms and entry cost, and therefore slows down the speed of entry. Moreover, because the bene…t of entry declines, labor gradually ‡ows back to the …nal-goods sector and in-house R&D sector. In particular, although during the transition process both aggregate in-house R&D labor and …rm number are increasing, it turns out that the former exhibits a faster rate of growth. Therefore, the …rm size gradually increases, leading the growth rate eventually to return to its original value. Our results with regard to the diverse growth e¤ects of capital taxation in the short run and in the long run may provide a possible explanation for the mixed empirical observations between capital taxation and economic growth.
Figure 4.8: Transition path of the investment rate