M. WATANABE

290

From Equation (9), (10), (12), (13), and (15), is

given by: t

G

111

11

1

tt

A

Gu

tt

uK

(16)

Tax rate t

is an endogenous variable delivered by a

balanced budget and this 4assumption is the same as in

Fanti and Gori [4]. Equation (15) shows that the tax rate

is an 5increasing function of the unemployment rate and

the ratio of public capital to tax revenue. The intuition is

described as follows. If unemployment rate increases,

then minimum wage increases and interest rate decreases

because labor force becomes relatively scarce to capital.

On the other hand, unemployment benefit t also in-

creases with an increase ,mt because unemployment

benefit is fraction of minimum wage, and the total unem-

ployment benefit tt

increases. We assume the ratio of

total unemployment benefit to tax revenue is constant

and balanced budget. From Equation (12) and (14), the

relationship between total expenditure of unemployment

benefit and tax revenue is denoted as tttt .

To satisfy balanced budget when total expenditure of

unemployment benefit, tt

, increases and interest rate,

t, decre ases with an incr ease in unemployment rate, t

b

1

w

ub

ub

ub rK

r

should increase. Hence tax rate is an increasing function

of unemployment rate.

From Equation (16), public investment is also an in-

creasing function of unemployment rate; the reason for

this is described as follows. When the unemployment

rate increases, there are two effects to public investment.

First, an increase in the unemployment rate increases t

from Equation (15), and this enlarges public investment

because the tax revenue increases. Second, an increase in

the unemployment rate decreases t because labor force

becomes relatively scarce to capital, and this decrease

public investment because the tax revenue decreases.

Comparing the two effects, the first effect dominates the

second effect.

r

3. Equilibrium

In a basic overlapping generations model, the capital

stock in period is equal to the savings in period t.

The relationship between ca pi tal and savi ngs is:

1t

11e

ttt

u

tt

us us

(17)

where 1t

is the capital in period , is

the total savings of employees, and is the total

savings of unemployment. The dynamics of this econ-

omy is shown as follows:

1tu

tt

us

1e

tt

us

11 1

11

1

tttt

t

KZuuu

K

(18)

1

1

11

ZA

where 1tt

K

is the growth rate in the economy. To

satisfy sustained growth, a large A is assumed. From

Equation (11), the unemployment rate, t, is constant

and an increasing function of the constant mark-up rate

u

. The derivative of growth rate with respect to

gives:

ddd0

ddd

ttt

t

ggu

u

(19)

1ttt

KK

Therefore the following proposition is established.

Proposition 1

An increase in the constant mark-up rate increases the

minimum wage and promotes economic growth.

The total unemployment income tt

increases when ub

increases. Therefore the savings of unemployment

increases. On the other hand, an increase in

has two

effects on the total income of employees tmt

.

First, an increase in

1

,

uw

directly decreases the total in-

come of employees because unemployment increases.

Second, an increase in

increases ,mt

wbecause labor

force becomes scarce and labor productivity is promoted.

Comparing the two effects, the second effect dominates

the first effect. Because the total income increases with

an increase in

, the savings this economy also in-

crease. Therefore economic growth is promoted with an

increase in

in

.

Finally, we focus on the effect of the ratio of public

investment tax revenue on growth rate. Public investment

increases with an increase in the constant ratio of public

capital to tax revenue, from Equation (16). The deriva-

tive of growth rate with respect to

gives:

d0

dt

g

(20)

Therefore the following proposition is established.

Proposition 2

An increase in the ratio of public investment to tax

revenue promotes economic growth.

An increase in the ratio of public investment to tax

revenue,

, promotes public investment, and this in-

creases the minimum wage and unemployment benefit,

from Equation (5). Therefore both the total wage income

and total unemployment benefit increase with an increase

in the ratio of public investment. This means that the

4Fanti and Gori [6] also consider the exogenous replacement rates in

extension.

5From equation (15), the derivative of tax rate with respect to

unemployment rate gives:

2

2

11

d0

d11

t

t

tt

uu

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