Optimal Govnt Policy

8/10/2019 Optimal Govnt Policy

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1 Determination of optimal government policy

Timing: The government chooses y Y first and then the household (HH) chooses x X(y). We

assume that the government is rational and has no commitment problem.

Let the HHs preferences be given by U(x).

When the government sets its policy, it has to take into account how the HH responds to it, in particular

it needs to know how the HHs equilibrium choice x varies with the governments choice y .

Knowing how x varies with y , the government chooses the y that maximizes the HHs utility. In some

special cases, there might be only one feasible y value. Then, the government would simply pick that y

to be the equilibrium government policy.

Thus, Step 1in finding the equilibrium government policy is to obtain the HHs choice x as a function

of governments choice y . In particular, let x(y) be the equilibrium choice of the HH for an arbitrary

y. x(y) solves

x(y) = arg max

x

X(y)

U(x).

Step 2: Given x(y), the governments problem is

maxyY

U(x(y)).

1.1 Example

Consider a two-period economy.

The HHs preferences is given byu(c1) + u(c2) (whereu(c) = log(c)).

Assume that the government uses consumption tax (1c

, 2c

) to raises revenue to finance (G1, G2).

The HHs budget constraint is

c1(1 + 1c ) +

c21 + r

(1 + 2c ) = e1+ e21 + r

.

Step 1: Find the HHs consumption allocation for some arbitrary (1c , 2c )

maxc1,c2

u(c1) + u(c2)

s.t.

c1(1 + 1c ) +

c21 + r

(1 + 2c ) = e1+ e21 + r

.

The HHs optimal choice should satisfy

u(c1) = (1 + r)1 + 1c1 + 2c

u(c2)

and the lifetime budget constraint.

One can see that the optimal (c1, c2) will be functions of (1c ,

2c ).Denote the consumers optimal choice

(c1(1c ,

2c ), c

2(1c ,

2c )). These functions describe the equilibrium consumption choice of the HH as a

function of (1c , 2c ).

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The consumers problem becomes

max1c,2

c

log

1

1 + 1c

+ log

1

1 + 2c

+a constant that does not depend on s

subject to

1

c

1 + 1c+

2

c

1 + 2c= (1 + ) P V G

P V e.

Subtract both sides of the budget constraint from 1 + to obtain

1

1 + 1c+

1 + 2c= (1 + )

1

P V G

P V e

.

Lettingx = 11+1c

andy = 11+2c

, we can rewrite the problem as

maxx,y

log(x) + log(y) +a constant

subject to

x + y = (1 + )

1

P V G

P V e

.

Setting up the Lagrangian, we obtain 1/x= and /y= , which implies thatx = y, thus1c =2c =

. Inserting this to the governments budget constraint, we obtain

= P V G

P V e P V G.

We found that it is optimal to have the same consumption tax rate in both periods. Why is this

optimal? By imposing the same tax rate, the government avoids creating distortions in consumption-

savings decision of the consumer. If the taxes were distortionary, i.e. if the consumer could avoid taxes

by changing her consumption allocation, she would strategically change her allocation to avoid taxes,

which would result in a less efficient allocation. Since the consumer cannot avoid taxes by changing herconsumption choices when the consumption tax in both periods are the same, she chooses the efficient

allocation.

Note that any feasible tax choice of the government has to satisfy PV govnt budget constraint

c1(1c ,

2c )

1c +

c2(1c ,

2c )

1 + r 2c =G1+

G21 + r

.

Note also that any feasible choice (c1, c2) (not necessarily the optimal choice) of the consumer has to

satisfy

c1(1 + 1c ) +

c21 + r

(1 + 2c ) = e1+ e21 + r

.

When consumer makes her choice, she faces this budget line. The optimal choice of the consumer on

the other hand has to satisfy both this budget constraint and the governments budget constraint above

(since the govnt bases its policies on the consumers optimal choice). That is

c1(1c ,

2c )(1 +

1c ) +

c2(1c ,

2c )

1 + r (1 + 2c ) = e1+

e21 + r

and

c1(1c ,

2c )

1c +

c2(1c ,

2c )

1 + r 2c =G1+

G21 + r

.

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Substracting the second from the first constraint, we obtain

c1(1c ,

2c ) +

c2(1c ,

2c )

1 + r =e1+

e21 + r

G1 G21 + r

.

The last expression is called the resource constraint for the economy, which states that the present

value of consumers consumption and government spending has to be equal to present value of income.Important point to notice is that the resource constraint holds at the optimal consumption choice of

the consumer, not necessarily in any other point of the consumers budget constraint (unless in special

cases in which the resource constraint and consumers budget constraint are identical, which happens

with lump-sum taxes for example).

The analysis above show how to obtain equilibrium allocation. In particular, the equilibrium allocations

(c1(1c ,

2c ), c

2(1c ,

2c )) and tax policy (

1c ,

2c ) have to satisfy

1. The consumers budget constraint:

c1(1 + 1c ) +

c21 + r

(1 + 2c ) = e1+ e21 + r

.

2. The economys resource constraint:

c1+ c21 + r

=e1+ e21 + r

G1 G21 + r

.

3. The consumers optimality (tangency) condition:

u(c1) = (1 + r)1 + 1c1 + 2c

u(c2).

First best allocation: We know that the equilibrium consumption allocation under any feasible

government policy has to satisfy the economys resource constraint. Thus, if we maximize the consumers

utility subject to economys resource constraint, we obtain the first best allocation, which gives themaximum possible utility that can be achieved by the best government policy. Note that not all

government policies can achieve this first best. With lump-sum taxes, we know that the consumers

budget constraint becomes exactly the same as the economys resource constraint. Thus, the first best

allocation is achieved when the government uses lump-sum taxes. Figure 1 illustrates the first best

allocation.

Examples:

1. 1c =2c =: In this case, the consumers budget constraint is

c1(1 + ) + c21 + r

(1 + ) = P V e

Rewriting, we obtain

c2 = (1 + r)c1+ (1 + r)P V e

1 + .

Note that this budget constraint has the same slope as the economys resource constraint. Since

the consumers optimal consumption choice has to be on both the consumers budget constraint

and the economys resource constraint, the only possible case is that the two constraints overlap as

in Figure 2. Thus, when the consumption tax is the same in both periods, the first best allocation

is achieved.

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2. 1c = 0 and 2c >0: The consumers budget constraint is given by

c2 = 1 + r

1 + 2cc1+

1 + r

1 + 2cP V e .

Note that this constraint has a flatter slope than the economys resource constraint. The equilib-

rium consumption allocation is given by point Ein figure 3. Figure 3 also illustrates the first bestallocation (denoted by F B). It can be seen that the consumption allocation with this particular

tax system generates a lower utility than the first best allocation.

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Redistributive tax policy example

Question: Consider a static (one-period) economy with two types of households, who differ in their wage

rates. i represents the household type (i {1, 2}). There are equal number of each type. Household i

earns wage rate wi. Assume thatw2 > w1. The preference of each household is given by Ui = log(ci n2i ),

whereci is the consumption and ni is the labor supply of household i. The government wants to implementa redistribution policy. For this purpose it taxes labor income of both households at rate l and gives

back a lump-sum transfer T to both households. Thus, the consumption of household i is given by ci =

(1l)wini+ T. There is no government expenditure and the government budget balances, i.e. the total

labor income tax collected is equal to the total lump-sum transfer distributed. The government cares both

types of households equally, that is, it aims to maximize the sum of the utility of both types of households.

a. Show that the optimal l (and thus T) is positive.

b. Show that the optimal policy in part (a) redistributes income from the high earner (i = 2) to the low

earner (i= 1).

c. Find the first best allocation that the government can possibly achieve.

d. Is the first best allocation redistributive relative to the laissez-faire (the equilibrium allocation without

government interntion, i.e. with l= 0 and T= 0)?

e. Can the policy in part (a) achieve the first best allocation? Explaing why.

TRY TO SOLVE THIS QUESTION WITHOUT LOOKING AT THE ANSWER. UNDER-

STANDING THE SOLUTION IS NEVER ENOUGH.

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Answer:

a. Step 1: Solve the problem of each household for a given policy (l, T). Householdis problem is given by

Ui(l, T) = maxci,ni

log(ci n2i )

s.t.

ci = (1 l)wini+ T.

We will denote the optimal consumption and leisure choices of household i by ci (l, T) and n

i (l, T)

respectively. The solution to the problem above gives

ni (l, T) =(1 l)wi

2

and

ci (l, T) =(1 l)2w2i

2 + T.

Step 2: Givenci (l, T) andn

i (l, T), the govenment maximizes the sum of the utilities of both house-

holds subject to the government budget constraint. Note that the government bases its decisions takinginto account how the households responds to its tax policy. Thus, the governments budget is given by

2T=w1n

1(l, T)l+ w2n

2(l, T)l.

The left hand side of the expression above is the total transfers distributed and the right hand side is

the total labor income tax collected.

Givenci (l, T) and n

i (l, T), the govenments problem then is

maxl,T

log

c1(l, T)(n

1(l, T))2

+ log

c2(l, T)(n

2(l, T))2

s.t.

2T =w1n

1(l, T)l+ w2n

2(l, T)l.

Plugging the expressions for ci (l, T) and n

i (l, T), householdis utility function becomes

Ui(l, T) = log

ci (l, T)(n

i (l, T))2

= log

(1 l)2w2i

4 + T

,

and the government budget constraint becomes

T = l(1 l)

4

w21+ w

22

.

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Then, the problem of the government becomes

maxl,T

log

(1 l)

2w214

+ T

+ log

(1 l)

2w224

+ T

s.t.

T = l(1 l)4

w21+ w22

.

Note that we can substitute T from the budget constraint into the objective function of the government.

Then, the governments objective function becomes

log

(1 l)2w21

4 + T

+ log

(1 l)2w22

4 + T

= log

(1 l)

2w214

+l(1 l)

4

w21+ w

22

+ log

(1 l)

2w224

+l(1 l)

4

w21+ w

22

.

Denote this objective function by W(l). Using properties of log utility, W(l) can be simplified to

W(l) = 2log(1 l) + log(w21+ lw

22) + log(w

22+ lw

21)2 log(4).

Typically, we take derivative of objective function with respect to l and set it to zero in order to find the

optimall. If you try to do this, you will see that the expression turns out to be complex. Instead, what we

could do is to look at how the derivative of the objective function behaves. We will use the following three

properties of the derivative:

1. The derivative of the objective function at l = 0 is positive, i.e ddl

W(0)> 0 whenever w1 =w2.

2. ddl

W(l) is decreasing in l, i.e. the second derivative of the objective function with respect to l is

negative for all values ofl.

3. Finally, ddl

W(l) is minus infinity atl= 1.

These properties of the derivative suggest that ddl

W(l) and W(l) are as in the following figure. That is,

there is a unique 0 < l < 1, such that ddl

W(l) = 0 and that the objective function is increasing until

l

and then it is decreasing. Thus, the maximum is achieved at l =l. Thus, the optimal policy is to have a

positivel.

To prove the three properties above, take the derivative of the objective function to obtain

d

dlW(l) =

2

1 l+

w22w21+ lw

22

+ w21

w22+ lw21

.

It is easy to show that properties 2 and 3 above hold. To prove property 1, evaluate ddl

W(l) at l = 0,

d

dlW(0) =2 +

w22w21

+w21w22

=w41 2w

21w

22+ w

42

w21w22

=

w21 w

22

2w21w

22

.

Note that ddl

W(0)> 0 wheneverw1 =w2. Thus, the three properties above imply that it is optimal for the

government to impose a labor income tax and give a lump-sum transfer back to both agents.

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b. To illustrate why l> 0 is a redistributive policy, first note that w2 > w1, which implies that n

2(l, T)>

n1(l, T) and thusw2n

2(l, T)> w1n

1(l, T). We will show that the low earner receives a transfer more

than the tax she pays, while the high earner receives a transfer less than the tax she pays. Thus, this

system redistributes income from the high earner to the low earner. To see this, note that the govenment

budget constraint is

2T=w1n

1(l, T)l+ w2n

2(l, T)l.

Together with the fact that w2n2(l, T) > w1n

1(l, T), the government budget constraint implies

that lw2n

2(l, T) > T > lw1n

1(l, T), which in turn implies that T lw1n

1(l, T) > 0 and T

w2n

2(l, T)< 0.

c. The first best allocation is obtained by maximizing the sum of utilities of both housholds subject to the

economy-wide resource constraint. Although, the economy-wide resource constraint is independent of

which tax policy the government uses, it would be convenient to derive it from the budget constraint

of the consumers and the government in part (a). For the consumers, we havec1 = (1l)w1n1+ T

and c2 = (1l)w2n2+T, and for the government we have 2T = w1n1l+ w2n2l. Adding these we

obtain the economys resouce constraint, which is

c1+ c2 = w1n1+ w2n2.

(Small digression: Note that the resource constraint for the economy basically states that total expen-

ditures on goods should be equal to total income. Here, we do not have government expenditures and

investment expenditures. Thus, government expenditures and investment expenditures do not appear

on the expenditure side.)

The first best solution is given by the solution to the following problem

maxc1,c2,n1,n2

log(c1 n21) + log(c2 n

22)

s.t.c1+ c2 = w1n1+ w2n2.

Setting up a Lagrangian, we obtain the following FOCs:

1

c1 n21=,

1

c2 n22=,

2n1c1 n21

=w1,

2n2c2 n22

=w2.

These FOCs imply that c1 n21 = c2 n

22, n1 =

w12

, and n2 = w22 . Using these expressions and the

resource constraint for the economy, we obtain optimal consumptions arec1 = 3w2

1+w2

2

8 andc2 = 3w2

2+w2

1

8 .

d. Note that the equilibrium consumption allocations in part (a) in laissez-faire are c1 = w2

1

2 and c2 = w2

2

2 .

Since 3w2

1+w2

2

8 > w

2

1

2 and 3w2

2+w2

1

8 < w

2

2

2 , the first best allocation gives more consumption to the poor

household relative to what she would consume in laissez-faire and less consumption to the rich household

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relative to what she would consumer in laissez-faire. Thus, the first best allocation is redistributive

relative to laissez-faire. This suggests that the government wants to redistribute income from rich to

poor. However, whether it can achieve the first best allocation or not depends on the type of the

policy that it uses to do redistribution. In general, since all policies in real life are distortionary, the

government cannot implement the first best allocation. See part (e) below as an example.

e. The policy in part (a) cannot achieve the first best allocation because ni = wi2 under the first best

allocation, butni = (1l)wi2 (with l > 0) under the optimal policy in part (a). This is because of

the fact the labor income tax distorts labor supply decision of households and it leads to a lower labor

supply than what can be achieved under the first best.

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Optimal Govnt Policy