# Fiscal Policy by dfsiopmhy6

VIEWS: 3 PAGES: 10

• pg 1
```									Fiscal Policy

- Sources and e¤ects of Government budget de…cits
- Optimal taxation

1     The Government budget constraint

Present value of purchases           Initial wealth plus
of goods and services                present value of tax receipts
Rt
Let R(t) = =0 r( )d
where r( ) is the real interest rate. Then, a unit of output discounted
back to 0 is e R(t) .

1.1    The constraint
Z   1                                       Z   1
R(t)                                        R(t)
e          G(t)dt       D(0) +              e          T (t)dt   (1)
t=0                                       t=0
Where G(t) is real purchase, D(0) is debt, and T (t) real taxes.
In addition, the present value of debt cannot be positive:

R(t)
lim e          D(s)       0
s!1

If D(t) = D (Constant) and r(t) > 0, then there is no need to
repay.
If growth of D(t) is less than the interest rate, then there is no
need to repay.

1
1.2    De…nition of de…cit

Rate of change of debt = Government De…cit

_
D(t) = [G(t)     T (t)] + r(t)D(t)                        (2)

We can think of G(t) as the purchase and of T (t) as the revenue.
The term [G(t) T (t)] is considered the "Primary
De…cit" and r(t) is the interest on debt.

R1            R(t)
From the budget constraint (1) )        t=0
e          [T (t) D (0)
G(t)] dt
(3)
The budget equation involves the entire path of purchases and taxes.

2     Measurement issues on de…cits
2.1    In‡ation vs De…cit
Let B(t) be the nominal debt, then

_
B(t) = P (t) [G(t)   T (t)] + i(t)P (t)D(t)                   (4)

P (t) is the price level and i(t) is the nominal interest rate.
But i(t) = r(t) + (t) then (4) becomes                      h          i
_                                                            _
B(t) = P (t) [G(t) T (t)]+[r(t) + (t)] P (t)D(t) = P (t) D(t) + (t)D(t)
so
_
B(t)
_
= D(t) + (t)D(t)                              (5)
P (t)

Consequently, a rise in in‡   ation rises the de…cit even when
it is de‡ated by price level. Why? Because higher interest payment
are needed, but this is to compensate the fact that debt is loosing value
due to in‡ation.

2.2    Sale of asset
Reduces de…cit without a¤ecting the budget constraint.

2
2.3         Unfunded liability
It leaves de de…cit unchanged but a¤ects the constraint. Example: social
and health insurance.
) Problems because of lack of relationship between the con-
straint and the de…cit.

2.4         Ponzi Games
s
If there is an in…nity of agents the present value of the private sector’
total spending may be less than the value of its total after tax income.
Example: Overlaping-generations model. At any time there is an agent
who saved and not yet dissaved, then the government can issue debt and
roll it forever.

3       The Ricardian Equivalence
3.1         Issue: Are bond and taxes equivalent?

With no uncertainty and no market imperfections.

s
Household’ Budget Constraint

Z       1                                               Z       1
R(t)                                                   R(t)
e           C(t)dt     K(0) + D(0) +                    e          [W (t)          T (t)] dt          (6)
t=0                                                      t=0

Where C(t) is consumption, K(0) is capital, D(0) represents the
government bonds, and W (t) labour income.

From (6) we have:

Z   1                                                 Z    1                                  Z    1
R(t)                                                    R(t)                                    R(t)
e            C(t)dt         K(0) + D(0) +               e           W (t)dt                     e          T (t)dt
t=0                                                      t=0                                     t=0
(7)

But the Government satis…es (1) in equality
Z 1                         Z 1
R(t)                                                            R(t)
e      G(t)dt = D(0) +       e                                           T (t)dt                     (8)
t=0                                         t=0

So, substituting (8) in (7)

Z   1                                     Z   1                                 Z     1
R(t)                                  R(t)                                    R(t)
e              C(t)dt    K(0) +           e          W (t)dt                      e          G(t)dt (9)
t=0                                       t=0                                    t=0

3
No reference to …nancing, i.e. bonds or taxes.
s
The path of houses does not enter into household’ budget con-
t
straint or preferences, so it doesn’ a¤ect consumption. Only govern-
ment purchases, not taxes, a¤ect investment and capital accumulation
since I = Y C G.

Then, only G, not the division of …nancing of those pur-
chases between taxes and bonds a¤ect the economy.

3.2      Does de Ricardian equivalence hold?
t?
When it doesn’
How Much?

(1) Turn over in the population.- "Some of the future tax burden as-
sociated to bond issue is borne by individuals not yet alive when
bond is issued" (Overlap Generations Model).

(1) Individuals linked by bequests (Barro 1974)

(2) Lifetimes are long enough. Example: Most of the bonds issued for
IIWW debt borne by agents living/working during war.

4      Tax-Smoothing
How optimal taxes look like?
What determines the de…cit?
Governments need revenue but decide to minimize distortions in-
duced by taxation. If distortions rise more than linearly with the raised
revenue then there is a reason for tax smoothing (over time).

4
4.1          s
Barro’ Model 1979

- A discrete economy
- Yt , Gt , rt exogenously given.
- Initial debt D, given.
- Government chooses T as to maximize distortion costs created by
taxes only.
Tt
Ct = Yt f       ,     f (0) = 0, f 0 ( ) > 0, f 00 ( ) > 0.
Yy

4.1.1             s
Government’ Problem

X
1
1          Tt
M in                        Yt f                 (10)
T0 ;T1 ;:::
t=0   (1 + r)t      Yt
Subject to
X
1
1                      X
1
1
Tt = D0 +                           Gt
t=0   (1 + r)t                    t=0   (1 + r)t
4.2     Solving (10) by perturbation

Let T 0 be such that
8                                         9
>
>             f or
T                  t0 < t >
>
<                                         =
T    T    f or                 t0 = t
T0 =
> (1 + r) T + T f or t0
>                                    = t + 1>
>
:                                           ;
T      f or t0              >t+1
No e¤ect of present value of revenues. So, no e¤ect on budget con-
straint. So, no e¤ect on optimality.

4.2.1   Bene…t of Marginal Change

1          Tt 1              1         Tt
MB =           Yt f 0           T =           f0        T
(1 + r)t        Y t Yt         (1 + r)t     Yt
1                Tt+1 (1 + r)              1        Tt+1
MC =               Yt+1 f 0                  T =             f0           T
(1 + r)t+1            Yt+1 Yt+1            (1 + r)t+1    Yt+1
Tt           Tt+1
) f0       = f0             )
Yt           Yt+1
Tt   Tt+1
=
Yt   Yt+1

5
The tax rate must be constant.

4.3     Tax Smoothing under uncertainty
s
Uncertainty about G. Now, the government’ problem is:
X
1
1              Tt
M in E0                         t
Yt f
T0 ;T1 ;:::
t=0   (1 + r)             Yt
Subject to
X
1
1                          X
1
1
t
Tt = D0 +          G
t t
t=0    (1 + r)                t=0 (1 + r)
From T to T 0 : If T optimal, T 0 ( small deviation from T ) should also
be such.
8                                     9
>
>           T         f or t0 < t >   >
<                                     =
0          T        T     f or t0 = t
T =
> (1 + r) T + T f or t0 = t + 1 >
>                                     >
:                                     ;
T         f or t0 > t + 1
T ! T 0 should not a¤ect expected present value of distortion.
Tt                Tt+1
M B = Et (M C) ) f 0             = Et f 0
Yt                Yt+1
Tt+1          Tt+1
If f ( ) is quadratic ) f 0 ( ) is linear. Thus Et f 0         = f 0 Et
Yt+1          Yt+1
Tt             Tt+1
and f 0        = f 0 Et
Yt             Yt+1
There cannot be predictable changes in the tax rate. Tax rate
follows a random walk.

4.3.1   Implications for de…cits.
If government purchases, as a share of output, are random walk, there
will be no de…cit. The tax rate will follow a random walk. De…cits are
G
due to expected changes in .
Y
4.3.2 Wars and recessions.
These are predictable so induce de…cits. Governments run de…cit in war
times and depressions. Data: con…rmed.
4.3.3   Extensions
Including capital accumulation allows to tax capital. Ex–    ante zero cap-
ital taxation is optimal, but ex-post it is not as it has no distortions. So
there is a time consistency issue.

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4.4    Political - Economy theories of budget de…cits.

Tax-smoothing provides an answer to why governments react to tempo-
rary expenditures with de…cits. But what about the systematic tendency
to large de…cit?

Is there a bias in …scal policy toward de…cits?

The "government" is not benevolent but composed of individuals
maximizing their objectives.
Ine¢ ciencies produced by political process with voters and candi-
dates.

(1) Strategic debt accumulation. Debt accumulation may restrict the
spending of future policy markets.

(2) Delayed Stabilization. When no single group controls policy, …scal
reform is delayed because each group delays agreement because
hope other group will bear the burden.

4.5    The cost of de…cit

What are the costs of large de…cits? Poorly understood!

a) Sustainable de…cits

i) A departure from tax-smoothing. If taxes are high might have
an e¤ect. E.g. UK during the war.
ii) When Ricardian equivalence fails.
s
De…cit increase consumption and lower’ economy total
future wealth. But individuals o¤sets the lack of "sav-
ings" by the government.
Depends on why the Ricardian equivalence fails. If liq-
uidity constraints present consumption may have a large
marginal utility. So bene…cial...
Distribution e¤ects
From present to future generations.
From workers to capitalists... (as reduce K, reduce
wages, rise r).

b) Unsustainable de…cits

7
Ever raising ratio of debt to GDP. This is unsustainable: it cannot
continue inde…nitely.
The government behaves in a way that violates the budget constraint.
How can this stop? An "external" shock!!
The outcome is like to be a "crisis", perhaps a default on the debt.
Problem? A large crisis imply a fall in the capital account surplus: For-
s
Depreciation of exchange rate. Price of foreign goods", very expensive.
) Welfare reduction

Direct (goods")

Indirect (capital markets, investment)

5      Debt Crises
Situations in which investors are not willing to buy debt at any interest
rate.

5.1      Model
Debt in one period: D.

Real interest rate R   1.

Potential tax revenue the following period T (random).

If T > RD government pay its debts orders.

If T < RD, it defaults.

Investors are risk neutral.

5.2      Equations:
Let      be probability of default.
Let R be the risk free payo¤s.
R R
(1    )R = R ) =
R

8
pi
1

R1                             R

on the other hand, the government defaults only if T < RD:So   =
F (RD). F ( ) is the distribution function of T .

pi
1
pi=F(RD)

T1 R1A             B            T2 R

with T.- minimum of T ; T .- Maximum of T .1
¯
T                 T
So, if R < ¯ ) = 0. If R >        ) =1
D                  D
If T is bell shaped, is as shown.

1
In the graphs, we have the next representation:
pi =
R1= R
T
T1= ¯
D
T
T2=
D

9
5.3   Implications:
(1) Two equilibria in t default B is likely to be unstable. If investors
believe in > B , their R needed to hold debt is higher than RB .
Their estimate ".

(2) Di¤erences in fundamentals are not needed for large di¤erences in
fundamentals. Why? Multiplicity of equilibria.

(3) Default may be unexpected.

pi
1
pi=F(RD)

R

)     default very small or 1.

(4) Default depend on fundamentals. ex. R ", Increase R at any stable
equilibrium.

pi
1
pi=F(RD)

T1 R1A A'       B              T2 R

Stable equilibrium A ! A0 increasing R.

10

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