洪福聲陳建印

Investment of public inputs can stimulate economic growth (Eberts, 1986; Garcia-Mila and McGuire, 1988; Aschauer, 1989; Barro, 1990; Kocherlakota and Yi, 1992)

Public inputs: Rival vs. Non-rival

Excludable vs. Non-excludable

Pure public input:Barro (1990):public input (flow) into

producti- on function.

Futagami, Morita, and Shibata (1993): public input (stock) into production function.

The size of user fee of revenue is increasing

for the governments in many countries (Borge, 1995; Dewees, 2002; Blomquist and Christiansen, 2005).

Public input (subject to congestion ):Barro and Sala-I-Martin (1992): income tax

rate is superior to lump-sum taxation. Glomm and Ravikumar (1994): optimal

income tax rate (second best) is independent of the degree of congestion.

Public Input (Excludability)Borge (1995) 、 Dewees(2002) 、 Blomquist

and Christiansen(2005) : In many countries there has been a recent trend towards more user fees.

Ott (2001): an increase in the degrees of congestion leads to an increase in the income tax rate, a decrease the user fee, and a decrease in the consumption tax rate.

Ott and Turnovsky(2006): the government provided two public inputs: excludable and non-excludable.

The key feature that distinguishes our study from

Ott (2001) and Ott and Turnovsky (2006) is the

incentive effect of the user fee.

Individual:

Individual uses a technology given as

(1)

and are the actual levels of excludable public inputs and non-excludable public inputs, respectively.

i

yi = k®i E

¯i G

°i ;®+¯ +° = 1;0< ®;¯;° · 1

Ei Gi

Excludable public inputs:

(2a)

:aggregate supply of excludable public inputs :aggregate capital stock :usage of excludable public inputs chosen by

the individual :aggregate demand of excludable public inputs.

E i = E[(ki

K)Áe(

edi

Ed)1¡ Áe]²e; 0< ²e;Áe · 1

E

K

edi

Ed

Non-Excludable public inputs:

(2b)

:aggregate supply of excludable public inputs

:usage of excludable public inputs chosen by the individual

:aggregate demand of excludable public inputs.

Gi = G[(ki

K)Ág(

gdi

Gd)1¡ Ág]²g; 0< ²g;Ág · 1

G

gdi

Gd

The utility function of a representative indivi- dual is given as

(3)

budget constraint is given as

(4)

: user fee

: tax rate

U =Z 1

0

c1¡ ¾i ¡ 11¡ ¾

e¡ ½tdt

_ki = (1¡ ¿)yi ¡ qedi ¡ ci

q

¿

Government :

government budget constraint is given by

(5)

where is total output of the economy.

assume that

(6a)

and

(6b)

E +G = ¿Y +qE

Y (= nyi)

E = µeY

G = µgY ; µe; µg 2 (0;1)

the social planner will realize that

The production function faced by the social planner become

(7)

The resource constraint given as

(8)

ki = k;yi = y;edi = ed = Ed=n;gd

i = gd = Gd=n;K = nki

y= k®E ¯G°(1=n)¯ ²e+°²g

_k = (1¡ µe ¡ µg)y ¡ c

The optimality conditions

(9a)

(9b)

growth rate

(10)

c¡ ¾= ¸C

¸C(1¡ µe ¡ µg)@y@k

= ¡ _̧C +½̧C

_cc

=1¾[(1¡ µe ¡ µg)

@y@k

¡ ½]

Substitute eqs. (6a) and (6b) into eq. (7)

(11)

where

(12)

Growth rate

(13)

y = µ¯®e µ

°®g ¹nk

¹n = n1®[̄ (1¡ ²e)+°(1¡ ²g)]

@y@k

= µ¯®e µ

°®g ¹n

_cc

=1¾[(1¡ µe ¡ µg)µ

¯®e µ

°®g ¹n ¡ ½]

Proposition 1. Under a centralized economy, the optimal expenditure share of excludable public inputs is given as

(14a)(14b)

The growth rate of the economy under the centralized economy

(15)

µ¤e = ¯

µ¤g = °

Фc =

_cc

=1¾[(1¡ ¯ ¡ °)¯

¯®°

°®¹n ¡ ½]

The optimality conditions

(16a)

(16b)

or if (16c)

(16d)

c¡ ¾= ¸D

(1¡ ¿)@yi

@edi= q

(1¡ ¿)@yi

@gdi

= 0 gdi =

Gn

(1¡ ¿)@yi

@gdi

> 0

¸D(1¡ ¿)@yi

@ki= ¡ _̧

D +½̧D

growth rate of consumption

(17)

The production function faced by the individual

(18)

_cc

=1¾[(1¡ ¿)

@y@k

¡ ½]

y = k®+Áe¯+Ág°f E [(1=K )Áe(ed=E d)1¡ Áe]²eg¯

fG[(1=K )Ág(gd=Gd)1¡ Ág]²gg°

A comparison between eq. (18) and eq. (7) reveals the following two lemmas.

Lemma 1.

(i)

(ii)

(iii)

MPK f aced by an individual > MPK social planner

MPK f aced by an individual "²e;²g;Áe;Ág " )MPK social planner "

²e;²g;Áe;Ág " )

gapof MPK f aced by an individual ¡ M PK social planner "

(19)

(20)

@y@ksocial planner

= ®yk

@y@kdecentralized economy

= (®+Áe²e¯ +Ág²g°)yk

Lemma 2.

(i)

(ii)

MPeddecentralizedeconomy < MPed

social planner

²e " or Áe # ) MPeddecentralizedeconomics "

Lemma 2.

(21)

(22)

@y@edsocial planner

= ¯yed

@y@eddecentralized economy

= (1¡ Áe)²e¯yed

Marginal productivity of capital in the decentralized economy as

(23)The growth rate of the economy under the

decentralized economy

(24)

@y@k

= (®+¯Áe²e +°Ág²g)¯¯®°

°®¹n

ФD =

_cc

=1¾f (1¡ ¿)(®+¯Áe²e+°Ág²g)¯

¯®°

°®¹n ¡ ½g

(25a)

(25b)

¿¤f irst =

¯²eÁe +°²gÁg

1¡ ¯(1¡ ²eÁe) ¡ °(1¡ ²gÁg)

q¤f irst =

®(1¡ Áe)²e1¡ ¯(1¡ ²eÁe) ¡ °(1¡ ²gÁg)

the government expenditure share

(26a)

the government revenue share

(26b)

E +GY

= µ¤e +µ¤

g = ¯ +°

¿¤f irstY +q¤

f irstE

Y= ¿¤

f irst +q¤f irstµ

¤E

= ¯ +° ¡(1¡ ¯ ¡ °)[̄ (1¡ ²e) +°(1¡ ²gÁg)]

1¡ ¯(1¡ Áe²e) ¡ °(1¡ Ág²g)| {z }Lump¡ sum tax rate

Corollary 1.

(i)

(ii)

²e ")¿¤f irst; q¤

f irst "

Lump¡ sumtax rate#

Áe ")¿¤f irst "

q¤f irst; Lump¡ sumtax rate#

Corollary 1.

(iii)

(iv)

²g ")¿¤f irst "

q¤f irst; Lump¡ sumtax rate#

Ág ")¿¤f irst "

q¤f irst; Lump¡ sumtax rate#

Comparison with Ott and Turnovsky (2006) If , then our model reduces to Ott

and Turnovsky. If , then the individual firm still over-

estimate the MPK. Hence, an increase in or leads to an increase in the tax rate.

However, such a change will not affect the MPe if . Eq. (16b) implies that the user fee should

decrease.

(1¡ ¿)

@yi

@edi= q

The fraction of government revenue allocated to excludable and non-excludable public inputs.

E = vE (¿Y +qE)= ºE [¿ +(1¡ ¿)(1¡ Áe)²e¯]Y

G = (1¡ ºE )(¿Y +qE)= (1¡ ºE )[¿ +(1¡ ¿)(1¡ Áe)²e¯]Y

Using these two definitions as well as market equilibrium conditions, eq. (18) implies that

(27)

Substituting eq. (27) in to eq.(20b)

(20c)

yk

= º¯®E (1¡ ºE )

°®[¿ +(1¡ ¿)(1¡ Áe)²e¯]

¯ +°® ¹n

@y@k

= (®+¯Áe²e+°Ág²g)º¯®E (1¡ ºE )

°®

[¿ +(1¡ ¿)(1¡ Áe)²e¯]¯ +°

® ¹n

Substitute eq.(20c) into eq.(17), the growth rate is given as

(28)

_cc

=1¾[(1¡ ¿)| {z }

A

[®+¯Áe²e +°Ág²g]º¯®E (1¡ ºE )

°®

[¿ +(1¡ ¿)(1¡ Áe)²e¯]¯ +°

®

| {z }B

¹n ¡ ½]

Proposition 3. (Second-best optimal policies) The second-best optimal fractions of govern- ment revenue that are allocated to excludable and non-excludable public inputs are given as

(29a)

(29b)

º¤E =

¯¯ +°

1¡ º¤E =

°¯ +°

The second-best optimal tax rate and user fee are given as

(29c)

(29d)

¿¤second =

¯ +° ¡ (1¡ Áe)¯²e1¡ (1¡ Áe)¯²e

q¤second =

®(1¡ Áe)²e1¡ (1¡ Áe)¯²e

Under the second-best optimum

(30a)

and

(30b)

E=Y = º¤E [¿¤

second +(1¡ ¿¤second)(1¡ Áe)²e¯]= ¯

G=Y = (1¡ º¤E )[¿¤

second +(1¡ ¿¤second)(1¡ Áe)²e¯]= °

Corollary 2. The optimal tax rate under the second-best optimal is less than the output elasticity of public inputs, if the incentive effect of the user fee for the excludable public inputs is present.

¿¤second < ¯ +° if Áe < 1:

The second-best optimal tax rate is equal to the output elasticity of public inputs (Barro, 1990).

It is, however, generally felt that the optimal tax rate is not equal to the output elasticity of public inputs.

(Chen, 2003; Hung, 2005; Ho and Wang, 2005) With congestion, this optimal tax rate is

independent of the degree of congestion (Glomm and Ravikumar, 1994).

With the presence of the incentive effect of user fees, the optimal tax rate is less than the output elasticity of public inputs.

Corollary 3. (with the incentive effect) Eq. (28)

(i)

(ii)

²e ")¿¤second #

q¤second "

Áe ")¿¤second "

q¤second #

Corollary 3.

(iii) ²g;Ág ")¿¤second !

q¤second !

Total revenue generated by the user fee is sufficient to finance excludable public inputs if

Centralized economy

Decentralized economy

q¤f irst ¸ 1 if ²e ¸

1¡ ¯ ¡ °(1¡ ²gÁg)®(1¡ Áe) ¡ ¯Áe

´ ²ue > 1cannot hold

q¸ 1:

q¤second ¸ 1 if ²e ¸

1(1¡ Áe)(®+¯)

> 1cannot hold

Proposition 4. The revenue generated by the user fee alone is not sufficient to finance excludable public inputs under the first- and second-best optima.

 

the optimal expenditure shares of non-exclud able and excludable public inputs are decided by their corresponding productive elasticity, regardless of the first- or second-best optimum.

an increase in the overall degree of congestion for excludable public inputs, on the one hand, always leads to an increase in the user fee and, on the other hand, leads to an increase in the first-best tax rate but a decrease in the second-best tax rate.