程序代写代做代考 finance UCLA

UCLA
Professor Lee Ohanian

Notes on Taxes and Government Spending, and the Impact on
Economic Activity

We will analyze two types of government spending: purchases of goods and
services, which range from national defense expenditures to parks, to roads and
bridges, and transfer payments, which is the act of the government transfering
income from one person to another. Social security and medicare are examples of
transfer payments. Most of the federal government budget is transfer payments.

To pay for government purchases or transfer payments, the government needs
revenue. We will analyze different types of taxes below.

The impact of government spending and taxation on the economy depends
on the type of spending, and how it is financed.

First, let’s look at the simplest case, which is the case of government spending
is national defense expenditures, and they are financed with lump sum taxes.

A lump sum tax is a tax that is levied on each individual (or household), and
the tax liability that the individual (or household) incurs doesnt depend on the
individual’s characteristics. That is, the amount of the tax that must be paid
doesnt depend on income, or the level of consumption, or wealth, or any other
possible attribute of the taxpayer. Everybody owes the same tax. Lump sum
taxes are sometimes called “head taxes”, because it is a tax that is identically
levied on each tax payer.

When we talk about national defense spending, we will assume that this is
output purchased by the government that doesnt impact the household’s utility,
nor does it impact the production technology. The only affect it has is that it
takes away resources from the private sector. Note that we will call this national
defense, but it could be other government purchases that dont affect marginal
utility or the production technology.

Example 1 – National Defense Spending with Lump Sum Taxes

There is one type of household. To make this as simple as possible, we do
not consider capital in the production function.

The preferences are

maxu(c)− v(h) (1)
(2)

(3)

1

The consumer’s budget constraint is

wh = c+ T (4)

Government spending is exogenous. The government’s budget constraint is:

G = T (5)

The economy’s resource constraint is

Y = C +G (6)

The firm’s profit maximization problem is

maxAh− wh (7)

The first order condition for labor and consumption are the sames ones we
have worked with before and equate the marginal cost of working to the marginal
benefit of working:

v′(h) = u′(c)w (8)

The firm’s profit maximization first order condition equates the wage rate
to the marginal product of labor:

F ′(h) = A = w (9)

(10)

(11)

This means we can combine those two equations to get:

v′(h) = u′(c) ∗ F ′(h) = u′(c) ∗A (12)

This last equation, along with the fact that Y=Ah = c+G gives us two
equations in the two unknowns, h and c.

2

We can see how changes in government spending impact the economy if we
specify the utility funciton. .Suppose that the utility function is ln(c)− h

2

2
.

Then we have for the first order condition for labor:

h =
A

wh− T
=

A

Ah− T
(13)

(14)

Now let’s vary taxes (which is the same thing as varying government spend-
ing) to see what happens to hours worked, consumption, and output. Note that
as T gets large, the marginal utility of consumption rises, all other things equal.
This will lead the household to increase the amount of hours that they work
compared to T that is small.

Why do taxes lead to more work? This is because taxes are lump sum. The
tax doesnt change the incentive to work, as a labor income tax would change
that incentive (and we will see how that works a bit later).

Specifically, if government is going to take a lot of output, then there will
be little left for private consumption, and that means that the marginal utility
of consumption rises. This higher marginal utility motivates the household to
work more. Economists call this effect the income effect. That is, taxes reduce
your income, and this leads the household to work more. In the next section,
we will introduce the substitution effect of taxes. This substitution effect doesnt
appear here because taxes dont change the return to working.

Now we will show that this economy is Pareto optimal. To see this, we solve
the social optimum program, which maximizes utility subject to the constraints,
which in this case is the resource constraint, which divides output between
the consumer and the government.(Recall that government will be taking an
exogenous amount of output). Moreover, recall that with the social optimum,
there is no firm maximization problem, no household maximization problem.

maxu(c)− v(h) (15)
(16)

(17)

subject to the resource constaint

Ah = c+G (18)

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The first order conditions for consumption and labor are exactly the same as
above:

v′(h) = u′(c) ∗Ah (19)

and we have the resource constraint, Y = c +G.

We have two equations in the two unknowns c and h, and the equations
are exactly the same as above. Therefore the values of h and c that solve the
market equilibrium also solve the social optimum problem, which means that
the market economy is effi cient – it is Pareto optimal.

Lump Sum Taxes over Time: Ricardian Equivalence

You probably have read about the size of the national debt, and how it
continues to grow. Presently, the debt held by the public is about 80 percent
of one year of GDP. An important issue is how to pay off debt over time. We
will now discuss this issue in economies with lump sum taxes, and we will see a
principle emerge that is called Ricardian Equivalence.

Ricardian equivalence means that the timing of taxes does not affect eco-
nomic performance. Specifically, an economy that issues taxes today to pay for
spending, compared to the same economy that issues debt to pay for spend-
ing, and then pays off the debt with future taxes, will have the exact same
allocations. ’

Consider an economy with two periods, periods 1 and 2. There is government
spending in period 1, but not in period 2. The spending is G1.The population
size is normalized to 1. We will see that the amount of employment, output,
and consumption will be the same whether we pay for government today with
taxes, or whether we pay for government with debt.

Case 1 – Pay for government spending with taxes today

The household’s problem is:

max ln(c1)− φh1 + ln(c2)− φh2,

where the coeffi cient φ is the number that governs the disutility of working.
The budget constraint in the first period sets income equal to consumption,

and the tax payment (τ) :

w1l1 = c1 + τ1

The government budget constraint each period is:

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G1 = τ1, G2 = 0

There is a competitive firm that hires labor to maximize profits at both
dates, where the term “i” denotes periods 1 and 2:

maxAli − wihi
The budget constraint in the second period is

w2l2 = c2 (20)

Form the Lagrangian:

L = max{ln(c1)− φh1 + ln(c2)− φh2 (21)
+λ1[w1h1 − c1 − τ1] + λ2[w2h2 − c2]} (22)

Differentiating with respect to c1, h1, c2, and h2 and setting the derivatives
to zero give us the first order conditions.

The first order conditions for consumption are:

λ1 =
1

c1
, λ2 =

1

c2
(23)

The first order conditions for labor are:

φ = λ1w1, φ = λ2w2 (24)

Note that the wage rate will be equal to the marginal product of labor, which
is A. Let’s also substitute out for λ1 and λ2.

φ =
w1
c1
=
A

c1
, φ =

w2
c2
=
A

c2

We also have the budget constraints for the two periods. Using the fact that
c1 = w1h1 − τ , and c2 = w2h2, let’s substitute in for c1 and c2 to get:

φ =
A

(Ah1 − τ1)
, φ =

A

(Ah2)
(25)

We get the solutions for h1 and h2 as:

h1 =
1

A

(
A

φ
+ τ

)
, h2 =

1

φ
(26)

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Suppose that A = 1, and φ = 2, and that τ = G1 = .1. Plugging these values
in, we get:

h1 = .6, h2 = .5 (27)

Case 1 – Pay for government spending with debt today, pay the
debt tomorrow

The household’s problem is:

max ln(c1)− φh1 + ln(c2)− φh2,

Now, the government issues a bond, or an “IOU”, in which the household
gives the government resources, and the government promises to pay the debt
back in the second period.

The first period budget constraint is (note that b1 is the amount of resources
the household loans to the government):

w1l1 = c1 + b1

The government budget constraint each period is:

G1 = b1, (1 + r)b1 = τ2 (28)

Note that the government still spends the exact same amout in period 1. In
period 2, it pays the debt back by issuing taxes.

There is a competitive firm that hires labor to maximize profits at both
dates, where the term “i” denotes periods 1 and 2:

maxAli − wihi
The budget constraint in the second period now includes the government

repaying debt, with interest, which is (1 + r)b1. However, the government must
also issue taxes today to pay for that debt, which we denote as τ2

(1 + r)b1 + w2l2 = c2 + τ2 (29)

L = max{ln(c1)− φh1 + ln(c2)− φh2 (30)
+λ1[w1h1 − c1 − b1] + λ2[(1 + r)b1 + w2h2 − c2 − τ2] (31)

Differentiating with respect to c1, h1, c2, h2 and b1,and setting the derivatives
to zero give us the first order conditions. Note that the wage rate will be equal to
the marginal product of labor, which is A.We get the same first order conditions

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as before for the equations that govern the marginal cost of working and the
marginal benefit of working:

φ =
w1
c1
=
A

c1
, φ =

w2
c2
=
A

c2

We will also have the household choosing how much to loan the government.
That is, differentiate L with respect to b1 and set to zero. We get:

λ1 = λ2(1 + r) (32)

or

1

c1
=
1

c2
(1 + r) (33)

Now, let’s see how much the household works. We will proceed as before,
by taking the equations

φ =
A

c1
, φ =

A

c2
(34)

and use the budget constraints to substitute out for c1 and c2 :

φ =
A

(Ah1 − b1)
, φ =

A

((1 + r1)b1 +Ah2 − τ2)
(35)

Now, recall that b1 = G1. We substitute out for b1 and get:

φ =
A

(Ah1 −G1)
, φ =

A

((1 + r1)G1 +Ah2 − τ2)
(36)

First, let’s work with the equation for h1. We can write that equation as

h1 =
1

A

(
A

φ
+G1

)
(37)

Note that this is the exact same equation as before, recognizing that τ1 = G1.
Plugging in our values of A = 1, φ = 2, and G1 = .1, we find that h1 = 0.6,
just as in case 1.

Let’s compare the equilibrium conditions at date 0

ult = uctwt

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uct = βuct+1(rt+1 + 1− δ)

F (Kt, Lt) = Ct + It +Gt

Gt = b1

FLlt + FKkt + (1− δ)kt − ct − kt+1 − b1 = 0

ct = Ct, lt = Lt, kt+1 = Kt+1, b1 = B1

Note that every equation is the same, because b1 = τ0 from the above
economy.

What about the equations at date 1? All are identical. To see this, con-
sider the household’s constraint, and substitute out for τ1 using the government
budget constraint:

rtkt + wtlt + (1− δ)kt +Rtb1 − ct − kt+1 −G1 −Rtb1
This is identical to the household’s constraint at date 1 if government debt

was zero as above.

Since all the equilibrium conditions between the two economies are the same,
then the allocations that satisfy those equilibrium conditions are also the same.
This equivalence could also be established if the debt was re-paid at any other
date(s). Ricardian equivalence requires some special assumptions: lump sum
taxes, infinitely lived households, no borrowing constraints.

Example 2 – Transfer Payments (or Government Spending that
substitutes for private spending) Financed with Labor Income Taxes
Now we will work with labor income taxes, in which your labor income will

be taxed at the tax rate, τ , 0 ≤ τ ≤ 1.

This tax rate on labor income will change the return to working by reducing
the after-tax pay that you receive. We will see that this will impact how much
work the household does. Before doing that, we will discuss the income and
substitution effects of taxes on labor supply. Suppose we have the utility
function u(c)− v(h). Suppose also that your wage is taxed at rate τ .

We have

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maxu(c)− v(h) (38)
(39)

subject to

(1− τ)wh+ T = c (40)

(41)

Note that the budget constraint includes two sources of resources. The first
is after-tax labor income, which is (1−τ)wh. The second is the transfer payment
from the government. Specifically, the government collects trax revenue, which
is τwh, and then gives it back to you. However, the transfer is viewed as
exogenous by the household. Thus, we call the exogenous transfer “T”.

The government budget constraint is given by

T = τwh (42)

Since there are no government expenditures, the resource constraint is given
by:

Y = C (43)

Now, we can see how taxes impact the decision to work. The usual condi-
tion that equates the marginal cost of work with the marginal benefit of work
changes, because you dont keep the whole wage, you just keep the after-tax
wage. So the equation is given by:

v′(h) = u′(c)(1− τ)w (44)

rather than

v′(h) = u′(c)w (45)

which is what we have had in this course prior to studying labor income
taxes.

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Aside – Note that we will get the same result if the government uses the tax
revenue to purchases consumption goods for households that households value,
such as school lunch programs or other government-provided food, government-
provided healthcare, government-provided recreation facilities, etc. In this case,
we modify the consumers utility function as U(c + G) − v(h), where c is pri-
vate consumer spending and G is spending for consumers done by the govern-
ment.Now, substitute the household budget constraint into the utility function
to get U((1−τ)wh+G). Next, recall that G = τwh. This means that the utility
function, with these substitutions, becomes: U((1− τ)wh+ τwh) = U(wh).

We can now talk about income effects of taxes and substitution effects of
taxes. We can see this in the consumers first order condition. First note that
taxes reduce the incentive to work by directly reducing your take home pay. The
higher the tax, the lower is the take home pay: (1−τ)w, which is a disincetive to
work. However, as we saw in the example above, taxes can reduce the resources
available to the private section, which raises the marginal utility of consumption.
This factor, tends to raise the incentive to work. We call these two effects the
substitution and income effects of taxes, respectively.

In this particular case of a labor income tax, and the tax revenue is used
only as a transfer payment, there is only a substitution effect of taxes. To see
this, substitute the budget constraint into the first order condition. Recall the
budget constraint is (1− τ)wh+ T = c, and therefore the first order condition
becomes:

(46)

v′(h) = u′((t− τ)wh+ T ) ∗ (1− τ)w (47)
Next, note that the transfer payment, T, is equal to the tax revenue collected,

τwh. This means we have:

v′(h) = u′((t− τ)wh+ τwh) ∗ (1− τ)w (48)

Combining the terms −τwh and τwh, we get

(49)

v′(h) = u′(wh) ∗ (1− τ)w (50)
In this case when the government is not taking any resources, so the marginal

utility of consumption is not impacted by any government decisions, because the
tax revenue is transfered back to households, the tax decision doesnt impact the

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consumer’s income or marginal utility, as the household consumes all of output.
The income effect is therefore zero in this case. We therefore only have the
substitution effect.

Next, note that the size of the substitution effect is determined the marginal
cost of working, v′(h).

To see this, consider the function form for v, h
α

α
, where α > 1. This functional

form is a standard one that is used in the analysis of taxation. We can now
see how the value of the coeffi cient α is involved with how much taxes influence
hours worked. To see this, note that in this case, v′(h) is hα−1.

The income effect means that taxes reduce the household income, and we see
this when we substitute out for consumption in the marginal utility of consump-
tion. This means that all other things equal, taxes increase the marginal utility
of consumption, which motivates you to work harder. The substitution effect
means that the return to working is impacted by taxes, as you keep (1 − τ)w,
rather than keeping w. To isolate the impact of the substitution effect, we will
assume that all tax revenue is redistributed back to the household, as discussed
above. Since we will be holding consumption constant, we will indicate that
with an overbar:

hα−1 = u′(wh) ∗ (1− τ)w (51)

Next, isolate the term h on the left hand side by exponentiating by 1
α−1

h =
(
u′(wh) ∗ (1− τ)w

) 1
α−1

(52)

Now, let differentiate h with respect to (1−τ)w, which allows us to caclulate
the impact of a change in the after-tax wage on hours.

dh

d((1− τ)w)
=

1

α− 1
(u′(wh) ∗ (1− τ)w)

a
α−1u′(wh) (53)

This seems complicated, but let’s use that expression to calculate the elas-
ticity of hours with respect to after-tax wages, which is much simpler, and we
do this using the normal formula to construct an elasticity. Recall that the
elasticity of the variable y with respect to x is given by dy

dx
x
y

dh

d((1− τ)w)
((1− τ)w)

h
=

1

α− 1
(54)

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So the elasticity of hours worked with respect to after-tax income, is just
determined by the coeffi cient α..Thus, high values of α mean that the elasticity
is low, which means that changes in after-tax wages will have a small effect on
hours worked. A reasonable range for this coeffi cient is around 3.. This means
that a 1 percent change in the after-tax wage rate will result in around a 1/2
percent change in hours worked.

The paper by Ohanian, Raffo, and Rogerson looks at historical evidence on
how taxes changed across countries and across time, and uses a model similar
to this one to evaluate how much of the change in hours worked in different
countries is due to taxes. We will discuss this paper in class.

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