Policy implications of R&D externalities

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Figure 3.1: The level of welfare and the rate of capital income tax

a consequence, there is less labor devoted to the production in the R&D sector, which then results in fewer equilibrium varieties for the …nal-good production, and ultimately depresses the level of consumption and welfare. In summary, to achieve the social optimum, it is necessary to balance both distortions in capital and labor market. Accordingly, an extreme case of the zero capital tax is unlikely to be optimal.

3.4.3 Policy implications of R&D externalities

This subsection investigates how the optimal capital tax responds to relevant para-meters, in particular those related to the features of innovation. More importantly, we shed some light on the roles of R&D externalities in the design of optimal tax

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our robustness analysis.

Figure 3.2: The optimal capital tax rate and the stepping on toes e¤ect

First, the optimal capital tax is increasing in (the stepping on toes e¤ect) and (the standing on shoulders e¤ect). With su¢ ciently small values of and , the optimal capital income tax can be negative (see Figures 3.2 and 3.3). The underlying intuition behind the result can be explained as follows. Notice that a higher implies that the negative duplication externality is small, and a higher means that the positive spillover e¤ect of R&D is relatively strong. Both cases indicate a similar circumstance in which the innovation process is more productive, and in which underinvestment in R&D is more likely. Under such a situation, the welfare cost of depressing innovation by raising the labor income tax is larger.

Therefore, the government should increase the capital tax while reducing the labor tax.

Second, the optimal capital income tax and the substitution parameter dis-play an inverse-U shaped relationship (see …gure 3.4). A lower is associated with a higher monopolistic markup , regardless of whether the adoption constraint is binding or not. The markup mainly a¤ects the optimal capital tax in two oppo-site ways. The …rst e¤ect (the monopoly e¤ect) is that, when is large (when

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Figure 3.3: The optimal capital tax rate and the standing on shoulders e¤ect is small), the degree of the intermediate …rms’ monopoly power is strong. To correct this distortion, the government tends to subsidize capital to o¤set the gaps between price and marginal costs; see Judd (1997, 2002). The second e¤ect (the R&D e¤ect) is that, a large implies that the pro…ts of intermediate …rms are high, so will be the value of a successful innovation. This means that the R&D sector is crucial, and the welfare cost of slowing down innovation by raising the labor income tax is bigger. Thus, the government tends to tax capital income instead of taxing labor income. It is illustrated in Figure 3.2 that, with an initially very large (a very small ), the monopoly e¤ect dominants, such that the optimal capital tax is negative. As becomes smaller (i.e., as goes up), both e¤ects decline.

However the monopoly e¤ect diminishes more rapidly than the R&D e¤ect. The incentives to subsidize capital falls sharply, and thus the optimal capital income tax begins to increase with a rise in . Finally, when is very small (a su¢ ciently high value of ), there are few rents ‡owing to the R&D sector, rendering the R&D

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Figure 3.4: The optimal capital tax rate and the substitution parameter

Third, the optimal capital tax increases in response to a rise in the size of innovation cluster (creative destruction). The intuition is as follows. Given our baseline parameterization, the markup is not limited by the adoption constraint.

In this case, simply functions as a negative R&D externality, like the stepping on toes e¤ect does. A higher means that the negative externality is larger, thereby decreasing the importance of the R&D sector. Therefore, a higher makes taxing labor more favorable than taxing capital (see …gure 3.5).

Figure 3.5: The optimal capital tax rate and creative destruction

Finally, the optimal capital tax is increasing in the government spending ratio

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(see …gure 3.6). This result is in consistence with the Aghion et al. (2013) …nding.

When the need for public expenditure is su¢ ciently small, the government can collect labor tax revenues to …nance the government spending and also to subsidize capital. Note that in this case the monopoly e¤ect dominants the R&D e¤ect so that the optimal capital tax is negative. As the size of government expenditure increases, it is not promising to count solely on raising the labor tax, because the distortion to the R&D sector would be su¢ ciently strong. Moreover, as we have discussed in Section 3.3, when the wasteful government increases, the government has an incentive to restore consumption by raising the capital tax. These e¤ects turn the optimal capital income tax rate to gradually become positive.

Figure 3.6: The optimal capital tax and the government size

3.5 Conclusion

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the result of a positive optimal capital income tax is robust with respect to varying the degrees of various types of R&D externalities.

Some extensions for future research are worth noting. First, to re‡ect the em-pirical reality, it would be useful to consider more complex optimal tax structures.

Second, since R&D investment usually has liquidity problems (see, e.g., Lach, 2002), it would be relevant to introduce credit constraint on R&D investment into our model. Third, our model has assumed in…nitely-lived agents. In the vein of optimal capital taxation, however, an important issue concerns the intergenera-tional consideration. Thus, it would be important to analyze the implications of innovation in a model with …nitely-lived individuals and bequests. These future directions would generate new insights to the debate on the Chamley-Judd result.

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Appendix 3.A. Deriving the steady-state growth rate

To solve for the steady-state growth rate of the economy, from eqs (14) and (15) we have:

A__t

A_t = &

1 + L_A;t

A¹_t : (A1)

where gA;t = _A_t=A_t:Let gZ denote gZ;t = ^Z_Z^_ the growth rate of any generic variable Z, and drop the time subscript to denote for any variables in the steady state. The steady-state growth rate of varieties is given by:

gA= &

1 + L_A

A¹ : (A2)

Moreover, The R&D labor share is st = L_A;t=(N_tl_t). By doing so, eq. (A2) can alternatively written as:

g_A= &

1 + (sN l)

A¹ : (A3)

By taking logarithms of eq. (A3) and di¤erentiating the resulting equation with respect to time, we have the following steady-state expression:

g_A=

1 n: (A4)

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Taking logarithms of eq. (A5) and di¤erentiating the resulting equation with respect to time, we can infer the following result:

g_Y = (¹ )

1 g_A+ n: (A6)

Inserting eq. (A4) into eq. (A6) yields:

g_Y = n; (A7)

where 1 + ⁽

1 )

1 1 is a composite parameter.

We now turn to solve the steady-state R&D labor share. In the long run substituting _At = gAAt and di¤erentiating the resulting equation with respect to time give rist to:

P__A=P_A= g_Y g_A (A8)

From eqs (13), (17), (18a), in the steady state we have:

x = 1 Y

Subsituting eqs (A8), (A9), and (A10) into eq. (A11) yields the result:

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r =

1 Y =A

s 1 s

(1 )Y =A (1+ )gA

+ g_Y (1 + )g_A (A12)

Based on eq. (A12), we have the staionary R&D labor share s as follows:

s =

1

1 (1 + )g_A

r gY + (1 + ¹₁ )(1 + )gA

(A13)

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This appendix solves the dynamic system of the model under tax shifting from labor income taxes to capital income taxes. The set of equations under the model is expressed by:

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The above 16 equations determine 16 unknown fc^t; l_t; A_t; K_t; L_Y;t; x_t; r_K;t; _x;t; r_t; G_t; _L;t; Y_t; q_t; L_A;t; P_A;t; w_tg, where q^t is the Hamiltonian multiplier, Ct= N_tc_t, K_t N_tk_t= A_tx_t;and st= L_A;t=N_tl_t. Based on Kt= N_tk_t= A_tx_t, eqs (B1), (B2), (B5), and (B12), we can obtain:

= 1

c_t(1 _L;t)(1 ) Y_t

L_Y;t(1 l_t): (B17a)

From eqs (B5), (B7), and (B11), we have:

L;t = ^K

1 (L_Y;t

N_tl_t): (B17b)

Moreover, to solve the balanced growth rate, we de…ne the following transformed variables:

^k_t Kt

N_t ; ^c_t Ct

N_t ; ^y_t Yt

N_t ; ^a_t At

N_t^{=(1 )}, lY;t (1 s_t)l_t; s_t L_A;t=N_tl_t: (B18)

Based on eqs (B16), (17), (18a), and the above de…nitions, we can obtain:

(1 l_t) = 1

^

c_t(1 ^K

1

l_Y;t

l_t )(1 )^a¹⁼_t (^k_t=l_Y;t) : (B19a) From eq. (B19a), we can infer the following expression:

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From eqs (B10), (B12), (B13), and (B18), we can directly infer:

g^k;t

d^k_t=dt

^k_t = (1 )(^a_t)¹⁼ (l_Y;t

k^_t )¹ ^c_t

k^_t g_Y: (B22) According to eqs (B14) and (B18), we can futher obtain:

g_a;t^ d^a_t=dt

In what follows, to simplify the notation we suppress those arguments of the laobr supply function. From eq. (B18), taking logarithms of eqs (B19a) and (B12) and di¤erentiating the resulting equations with respect to time, we have:

g_y;t^ = (1= )g_^a;t+ g_^k;t+ (1 )( _l_Y=l_Y); (B24)

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Taking logarithms of eq. (B15) di¤erentiating the resulting equation with respect to time, we obtain:

P__A;t

P_A;t = (g_y;t^ + g_Y) n + (1 ) _l_t

1 l_Y;t [1 + (1 ) l_Y;t

l_t l_Y;t]( _l_Y;t=l_Y;t): (B26) Combinning eqs (B9), (B15), (B18), (B21), (B24), (B25), and (B26) together, we obtain:

Inserting eq. (B18) into eq. (B17b) yields:

L;t= ^K

1

l_Y;t

l_t : (B28)

Based on eqs (B21), (B22), (B23),(B27), and (B28), the dynamic system can

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Linearizing eqs (B29a), (B29b), (B29c), and (B29d) around the steady-state equilibrium yields:

Due to the complicated calculations, we do not list the analytical results for bij; where i 2 f1; 2; 3; 4; 5g and j 2 f1; 2; 3; 4; 5g:

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Due to the complexity calculations of the four characteristic roots, we do not try to prove the saddle-point stability analytically. Instead, we show that the dynamic system exists two positive and two negative characteristic roots via a numerical simulation. For expository convenience, in what follows let `1and `2be the negative root as well as `3 and `4be the positive roots. The general general solution is given by:

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4^j =

b₁₂ b₁₃ b₁₄ b₂₂ `<sub>j</sub> b<sub>23</sub> b<sub>24</sub> b<sub>32</sub> b<sub>33</sub> `_j b₃₄

; j 2 f1; 2; 3; 4g; (B31b)

h_2j =

`<sub>j</sub> b<sub>11</sub> b<sub>13</sub> b<sub>14</sub> b<sub>21</sub> b<sub>23</sub> b<sub>24</sub> b<sub>31</sub> b<sub>33</sub> `_j b₃₄

=4^j ; j 2 f1; 2; 3; 4g; (B31c)

h_3j =

b₁₂ b₁₁ b₁₄ b₂₂ `_j b₂₁ b₂₄ b32 b31 b34

=4^j ; j 2 f1; 2; 3; 4g; (B31d)

h4j =

b₁₂ b₁₃ `<sub>j</sub> b<sub>11</sub> b<sub>22</sub> `_j b₂₃ b₂₁

b₃₂ b₃₃ `_j b₃₁

=4^j ; j 2 f1; 2; 3; 4g: (B31e)

The government changes the capital tax rate K from K0to K1 at t=0, based on eqs (B31a)-(B31e), we employ the following equations to discribe the dynamic adjustment of ^k_t, ^a_t, ^c_t and lY;t:

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where 0 and 0⁺ denote the instant before and after the policy implementation, respectively. The values for D1; D₂; D₃ and D4 are determined by:

) remain intact at the instant of policy implementation since Kt, At, and Nt are

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Inserting eqs (B33c), (B34a), and (B34b) into eqs (B32a)-(B32d) yields:

k^_t =

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Appendix 3.C. Proof of comparative statics

From eqs (B29a)-(B29d), we have:

d^k_t=dt

Y;t = 0;we then have the following steady-state results:

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Inserting eq. (C2a) into eq. (C1h) yields:

s =

1

1 (1+ )g_A + 1 + ¹₁ (1+ )g_A

: (C2b)

From eqs (B3) and (C1g), we can obtain

r g_Y = : (C3)

Equation eq. (C1g) can be rearranged as:

^

y=^k = (^a)¹⁼ (l_Y

k^)¹ = ( + + g_Y)

(1 _K) : (C4a)

Substituting eq. (C4a) into eq. (C1e) gives rise to:

^ To ensure the steady-state consumption-output ratio ^c=^y is positive, we impose the restriction (1 ) ( + g_Y) _{( + +g}^{(1 K)}

Y) > 0 for all values of the time preference rate . As a consequence, lim _!0 ^c=^y>0 implies:

(1 ) (1 _K) Based on eq. (B28) and lY (1 s)l , we can infer the following expression:

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Inserting eqs (C5a) and (C7a) into eq. (C8) yields:

l =

Combinning eqs (C2b), (C6), and (C9b) together, we can derive

^

a = [ &

(1 + )g_A]^{1=(1 )}(sl) ^{=(1 )}; (C10a) where

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According to eqs (C4a), (C5a), and (C11b) yields:

^k = (1 _K)

( + + g_Y)y;^ (C12a)

^

c = [(1 ) ( + g_Y) (1 _K)

( + + g_Y)]^y; (C12b) Inserting eq. (C11a) into (C12a) and (C12b), we can derive the following compar-ative statics:

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CHAPTER 4

SHORT-RUN AND LONG-RUN EFFECTS OF CAPITAL TAXATION ON ECONOMIC GROWTH IN A R&D-BASED

MODEL WITH ENDOGENOUS MARKET STRUCTURE

在文檔中 R&D經濟成長模型：資本稅之探討 - 政大學術集成 (頁 73-95)