lecture note for elements of economic analysis iv1

8/19/2019 Lecture Note for Elements of Economic Analysis IV1

1/140

Lecture Note for

Elements of Economic Analysis IV1

Seung Mo Choi

([email protected])

November 27, 2006


2/140

Contents

1 Labor Market Policy 11.1 Unemployment Insurance 1

1.2 Minimum Wage, Labor Union and Firing Restriction 5

2 Fiscal Policy (I) 18

2.1 Lump-Sum Tax 18

2.2 Distortionary Tax: Capital Income Tax 22

3 Fiscal Policy (II) 32

3.1 Intertemporal Budget Constraint, Government Debt and

Lump-Sum Tax (“Ricardian Equivalence”) 32

3.2 Government Debt and Distortionary Tax (“Ramsey Problem”) 40

3.3 Social Security 45

4 Monetary Policy (I) 55

4.1 Seigniorage and Ination 55

4.2 Price Level, Output and (Un)Employment (“Monetary

Neutrality”) 63

5 Monetary Policy (II) 75

5.1 Ination, Unemployment and Optimal Monetary Policy

(“Expectations-Augmented Phillips Curve”) 75

5.2 Interest Rate and Optimal Monetary Policy (“FriedmanRule”) 79

6 International Trade 89

6.1 Exchange Rate (“Purchasing-Power Parity”) 89

6 2 Gains from International Trade (“Comparative Advantage”) 95


3/140

ii Contents

7.1 Fundamental Equation of Asset Pricing 105

7.2 Contingent Claims (“Lucas-Tree Model”) 112

8 Appendix: Tables and Figures 125


4/140

Chapter 1

Labor Market Policy

This chapter discusses labor market policies. The effects of (i) unemploymentinsurance, (ii) minimum wage, (iii) labor union, and (iv) ring restriction are

discussed.

1.1 Unemployment Insurance

Question: Does a More Generous Unemployment Insurance Increases the

Long-Term Unemployment? (Reference: Chapter 10 of Barro. Nan Li's

ECON 203 Material.)

The unemployment insurance provides unemployed workers with income

while they are searching for new jobs. Before we start any theoretical analysis, it

is useful to understand related concepts clearly.

(1) Unemployment (U ): The number of people who are looking for work but have no job. (On the other hand, employment (L) is the number of people

who have jobs.)

(2) Labor force: The sum of unemployment and employment. (Hence, peo-

ple who neither have a job nor are looking for one (e.g., students) are classied

as outside of the labor force.)

(3) Unemployment rate (u): Unemployment divided by labor force (i.e.,U=(U + L)).

- The unemployment rate is usually countercyclical . When the economy is

good (bad), the unemployment rate is low (high).

- A “stable” level of unemployment rate is often called the natural unem-

ployment rate.

Some datasets regarding the unemployment can be downloaded at the Bu-

reau of Labor Statistics website (http://www bls gov/s/availability htm) Also


5/140

2 Chapter 1 Labor Market Policy

much shorter in the U.S. relative to many European countries.

Higher rate of (long-time) unemployment in Europe may be because Euro-

pean countries, in general, provide more generous unemployment insurances.

This theory will become clearer after we introduce the following model.

1.1.1 Model Description and Solution

The generosity of unemployment compensation may means either more money

(“amount”) or a longer period (“duration”), or perhaps both. In our model, as-

sume that the unemployment compensation is paid forever as long as a worker

stay unemployed. This enables us to focus on the amount rather than the duration.

(We will think about the duration in Exercises.)

Consider an innitely-lived worker, who is initially unemployed at the current

period (say, period 0). She meets one employer at each period of job search. The

employer may offer a high wage wh or a low wage wl. The probability that ahigh wage is offered is p. Looking at the wage offer, the worker may accept or reject : If she accepts the offer, she receives the specied wage at each period,

starting from this period. To make the problem simple, assume there is no risk of

voluntary or involuntary job loss. If she rejects, she receives the unemployment

benet b (where b < wl < wh) at this period, and continues to search at the next

period. The environment in the next period (say, period 1) is exactly the same asthe environment in this period (say, period 0) if she rejects the offer. That is, she

again meets an employer who offers wh or wl in the next period. Our interest isto see what happens to this unemployed worker's decision when b increases (i.e.,the unemployment insurance becomes more generous).

Before we start any analysis, let us think about what this worker gets if she

accepts the wage offer. If a high wage wh is offered and accepted by this worker,

she gets wh dollars at each period. In terms of utility level, this is u(wh), so thediscounted value of the utility levels is

u(wh) + u(wh) + 2u(wh) + ::: =

1Xt=0

tu(wh) = u(wh)

1 ; (1.1)


6/140

1.1 Unemployment Insurance 3

units of discounted value of utility.

Now let's think about the decision of this worker in detail. First, suppose a

high wage (wh) is offered. The unemployed worker has two options: “accept”or “reject.” But it is clear that she will accept the offer because it is the best

possible offer. If she rejects the offer, she will only receive b (unemploymentcompensation) in this period, and she will have to draw another wage offer in the

future which can never be higher than wh. So it is not a reasonable solution for her to reject a high wage offer.

Summary: In this model, if a high wage is offered, the worker will denitely

accept it.

But things are somewhat different if a low wage (wl) is offered. If the worker

rejects, she will only receive b at this period, but instead, she will have a chanceto draw a high wage offer (wh) in the next period. That is: (i) Cost of rejecting:Receives onlyb which is smaller than wl. (ii) Benet of rejecting: Has a chance towin a higher wage (wh) in the future. Hence, to determine the worker's decision,we have to compare the discounted utility values of “accept” and “reject”.

First, let us see her discounted utility level when she accepts the offer, de-

noted by V accept. From (1.2), it is clear that

V accept = u(wl)1 : (1.3)

Second, consider her discounted utility level when she rejects the offer, de-

noted by V reject. Assume the worker is “consistent” in her decision. That is, if she chooses to reject a low-wage offer at this period, then she will continue to

do so when a low wage is offered in later periods. (This assumption is relevant

because the environment tomorrow is exactly the same as today's if she rejects.)

If she rejects, she gets b at this period, so we can write

V reject = u(b) + (discounted expected utility of the next period): (1.4)Then, at the next period, a high wage is offered with probability p, which willgive (1.1) in terms of utility level, and a low wage is offered with probability


7/140


We can solve this equation for V reject. That is,

[1 (1 p)]V reject = u(b) + p u(wh)1 ;

or equivalently,

V reject = u(b) + pu(wh)=(1 )

1 (1 p) : (1.5)

1.1.2 Discussion

The bottom line is that a worker with a low-wage offer will reject it if

V reject > V accept;

or equivalently, from (1.3) and (1.5),

u(b) + pu(wh)=(1 )1 (1 p) >

u(wl)

1 : (1.6)

Whether this inequality holds depends on the values of parameters (e.g., , p,wh, and wl). But it is clear that this inequality is more likely to hold as b in-creases. That is, suppose (1.6) does not hold when b = 0 so that an unemployed

worker accepts the low-wage offer. As you increase b, the left-hand side of (1.6)increases, so it may be the case that (1.6) now holds if b is big enough. Therefore,a more generous unemployment benet is expected to delay the acceptance of a

job offer.

Summary: In this model, if a low wage is offered, the worker's decision

depends on parameter values. But the worker is more likely to reject the offer as

the unemployment benet becomes more generous.

In many European economies, it takes longer for an unemployed worker toaccept a new job relative to the U.S. Perhaps one of the reasons is because the

European system of unemployment insurance is more generous. You have to un-

derstand that the introduction of this unemployment insurance has two different

effects, positive and negative: (i) A positive effect is that an unemployed worker

b d hil h i l d (ii) A i ff i h hi i


8/140


optimal insurance system by considering these two effects together, which is one

of major challenges in macroeconomics.

The model introduced in this section is simple. (In fact, our conclusion is

so clear that we do not need such a mathematical model.) But this becomes a

nice tool to analyze more complicated problems. You will see some examples in

Exercises.

True or False? —

1. We should not offer the unemployment insurance because it will only

distort the worker's decision.

2. The United States pays the unemployment benets for about six months

after a worker gets red. As in many European countries, the United States

should lengthen the duration of the benets, for example, to two years.

1.2 Minimum Wage, Labor Union and Firing Restriction

Question: How Do the Minimum Wage, Labor Union, and Firing Restric-

tion Affect the Economy? (Reference: Chapter 10 of Barro.)

This section briey discusses three other labor market policies.

1.2.1 Minimum Wage

Think about the labor market for low-skilled workers. Just as any prices, their

wage rates are determined in the market, based on a demand-supply story. A

graphical analysis makes it clear that if the minimum wage is higher than the

equilibrium wage, a smaller number of workers will be employed because rms

need to pay more for each worker hired. [Draw a graph.] Hence, the mini-

mum wage policy may increase the wage rate of the employed workers who are

low-skilled, but it leaves more workers unemployed. A positive effect is that

some employed workers can be protected, but a negative effect is that it may

decrease the employment of low-skilled workers. The size of this negative effect

depends on the shapes of demand and supply curves. [Show elastic/inelastic


9/140


1.2.2 Labor Union

The labor union tries to protect union members. In some industries, unions are

strong enough to raise the wage rates (for example, by threatening strikes). Here

we will consider a simple model regarding the consequences of union activities.

The environment here is similar to the model about the minimum wage, in the

sense that the wage rate is set articially high relative to the equilibrium wage

rate. One difference is that labor unions are strong only in some industries, so the

wage rates are articially set in those industries, but are determined in the market

equilibrium in others. We will see that workers in not unionized industries may

have some side-effects.

Suppose there are two industries. Each industry has 5 rms. All 10 rm are

identical and have the following production function:

Y = 2p

L; (1.7)

where Y is the output (in terms of dollars) and L is the labor input (in terms of workers). There are 1,000 workers, inelastically supplied. (This means that all of

them are employed and that the wage rate is determined so that all are employed.

This assumption makes the problem simple: Without it, we have to consider the

choice of workers on work and leisure and to determine which workers work

for how many hours, etc.) Finally, let us denote the wage per worker (in terms

of dollars) by w . We will rst think about how the equilibrium wage rates aredetermined without unions. Then we introduce unions to see how the wage rates

are determined.

1. No Union: Assume there is no labor union. Each rm maximizes its prot

(output minus cost),

2

p L wL;

where w is treated as given. The rst-order condition implies

1p L w = 0;

or equivalently


10/140


1,000 workers in this economy, a market clearing (“quantity demanded equals

quantity supplied”) implies

10

w2 = 1000;

or equivalently,

w = 0:1:

Under this wage rate of 0.1 dollars per worker, each rm hires 100 worker (i.e.,

L = 100) from (1.8) and produces $20 (i.e., Y = 20) from (1.7). Since thereare 10 rms each of which produces $20, the GDP is $200.

2. Unionization: Assume there is a labor union in one industry, and it forces

the industry (i.e., 5 rms) to raise the wage rate from $0:1 to $0:2. There areno unions in the other industry. From (1.8), if the wage is given by w1 = 0:2,the labor demand of each rm in this unionized industry becomes 25. So this

industry hires 255=125 workers. Each rm produces $10 from (1.7), so thisindustry produces $50.

Then, the wage rate for the other (not unionized) industry is determined so

that the remaining 1000125=875 workers are all hired in an equilibrium. Thelabor demand for each rm is (1.8), and there are 5 rms in this industry, so themarket-clearing condition implies

5

w22= 875;

or equivalently, w22 = 1175

. Solving this equation, we have the equilibrium

wage of 0:08. Each rm in this industry hires 875=5=175 workers, so (1.7)

implies that each rm produces $26.5 in value. Hence, the industry produces$26.55=$132.5.

Therefore, the entire economy (i.e., two industries combined) produces $50+$132.5=$182.5,which becomes the new GDP. This level is lower than the previous GDP level of

$200.

3 C i h i i i i l h l b i i i


11/140


decrease the GDP level because the rms under unionized industries, forced to

pay higher wage rates, are not able to produce at optimal levels.

In sum, labor unions help the workers to get what they deserve, but their

activities may also have some side effects. We are not saying that union activities

should be discouraged. But we should remember that as always, there are positive

and negative effects of their activities.

1.2.3 Firing Restriction

To protect the employed workers, many countries enact restrictions so that ring

a worker is costly or difcult. Obviously, a positive effect is that the employed

workers are protected with more secure jobs, but a negative effect is that the

rm's decision would be distorted: Because it is costly or difcult to re workers

later, rms will hire less workers from the beginning . Let's consider a model tomake this intuition clear.

For our purposes, we move our attention from how equilibrium wage rates

are determined. Rather, we introduce a partial equilibrium model: There are a

lot of rms in the economy, and the wage is set to be w = 1 in the labor market.We consider a rm, which is small in the sense that it cannot affect the wage level

of the market. (So in this model, we treat the wage rate as given.) There are two

periods in this economy, but the production functions are different:

Y 1 = 4p

L1 at period 1;

Y 2 = 2p

L2 at period 2;

where Y 1 and Y 2 are outputs (in dollars) at periods 1 and 2 and L1 and L2 arelabor inputs (in number of workers) at period 1 and 2. Why should we assume

different production functions for different periods? The idea is that we want to

introduce some “uctuations” so that there are booms and recessions. So the rm

makes a lot during booms (or at period 1) but less during recessions (or at period

2).

1. No Firing Restriction: If there is no ring restriction, and if the rm


12/140


The rst-order conditions imply that

L1 = 4; L2 = 1:

So four workers are employed at period 1, but only one worker is employed at

period 2, so three workers are red at the end of period 1.

2. Firing Restriction: Now introduce a ring restriction that no one can be

red (i.e., L1 = L2) between the two periods. The rm maximizes

(4p

L 1 L | {z } prot at period 1

) + 1

1 + r(2p

L 1 L | {z } prot at period 2

):

To make the algebra simple, let's assume r = 0. (r is close to 0 anyway.) The

rst-order condition is3p L 2 = 0;

or equivalently,

L = 9

4:

3. Conclusion: Workers are red when the business is bad, but everyonewants to avoid the risk of getting red. The ring restrictions help them to have

more secure jobs. But under these restrictions, recognizing the costs of ring,

rms will hire less workers from the beginning. That is, the government has made

the labor market less exible, the benet is that the employed become happier,

but the cost is that we may have more unemployed workers. In the previous

example, the rm without any ring restrictions hired 4 workers and 1 worker at

periods 1 and 2. In total, it hired 5 workers cumulatively. If ring is prohibited,

the rm only hires 9/4 workers at each period. So, cumulatively it only hired 9/2

workers, which is less than 5. Again, this economic policy of ring restrictions

have good and bad effects at the same time.

1.2.4 Final Comments

ll i h k k h i All li i i i i id d


13/140


acceptance, which may further increase the unemployment. By the minimum

wage policy, low-skilled workers can get more money for their lives, but it might

discourage rms from hiring low-skilled workers. Labor unions help the hired

workers to have what they deserve, but they may decrease the employment (and

hence output) and may also affect the wage rates of the rms in other industries.

Firing restrictions make the jobs more secure, but rms may hire less workersfrom the beginning because ring is costly. All economic policies have costs and

benets – No exceptions!

True or False? —

1. Labor unions help their member workers to get what they deserve. Hence,

the government needs to subsidize any activities of labor unions.2. The United States labor market is too exible in the sense that the rms

can re workers easily compared to European countries. If the United States

government makes ring more costly (for example, by collecting “ring taxes”

from rms), the rms may be hurt, but denitely the workers will always benet.

3. Assume that there are two industries which produce goods by hiring work-

ers, where all workers are identical and may work in either industry. Further,

in the market equilibrium, the wage per worker is $1 (of course, the same inboth industries). If the labor union in one industry articially increases the wage

per worker to $1.5 in that industry only, then the other industry will hire more

workers with a lower level of equilibrium wage per worker.


14/140


Exercises

1. (Unemployment Insurance) Consider a worker who maximizes the ex-

pected discounted value of consumptions. (This is equivalent to assumingu(C ) =

C . At period 0, she maximizes E P1t=0 tC t for 0 < < 1.) While unemployed, a worker meets one employer every period, who offers either a high wagewh (with probability p) or a low wage wl (with probability 1 p). Looking atthe wage offer, the worker may accept or reject it. If she accepts, she receives

the specied wage at each period, starting from this period. There is no risk of

getting red, and it is not allowed to search for a new job once employed.

The unemployment benet is provided only for one period . To be specic, if

a worker starts a job search (say, at period 0) and rejects the rst wage offer, theunemployment benet b (where b < wl < wh) is paid immediately. But if sheremains unemployed and rejects other offers in later periods (at periods 1, 2, ...),

no benets are provided.

(a) Will an unemployed worker accept a high-wage offer? Why or why not?

(b) Consider an unemployed worker who previously received the unemploy-

ment benet. Show that the “value” (i.e., expected discounted value of consump-

tions) of this worker when a low wage is offered and is rejected by her is

V reject = pwh=(1 )

1 (1 p) :

(The worker is consistent with her decision. That is, if she rejects a low wage

offer once, then she does so under the same situation in the future.) [Hint:

P1t=0

t = 1=(1 ) for 0 <


15/140


(f) Summarize your ndings in the previous two questions in plain words.

(g) Suppose the government decreases the unemployment benet tob = 0:1.Will this worker (at the initial period of her job search) accept or reject a low-

wage offer? What is the implication of this result?

(h) (OPTIONAL) Continue to assume b = 0:1. Suppose the government nowdecides to increase the duration of unemployment benet from one period to two

periods. Consider an unemployed worker at the initial period (period 0) of the job

search. Will she accept a low-wage offer? What about in the next period (period

1)? What about future periods (periods 2, 3, ...)? Summarize what happens when

the government increases the duration from one period to two periods.

2. (Unemployment Insurance) In our discussions on economic effects of

unemployment benets, we considered the following set-up (except that u(C ) =C is assumed here):

Consider a worker who maximizes the expected discounted value of

consumptions. (This is equivalent to assuming u(C ) = C .) For example,at period 0, she maximizes

E " 1

Xt=0 tC t# ; 0 <


16/140


next period. An employed worker with wh (high wage), on the other hand, keepsthe job forever without being red.

(a) Suppose that an unemployed worker always rejects the offer if the offered

wage is wl. Let V 1 denote the “value” (i.e., expected discounted “value” of con-sumptions), at the beginning of each period (i.e., before wage or compensation is

paid), for an unemployed worker who has just rejected the wage offer. Show that

V 1 = b + pwh=(1 )

1 (1 p) :

[You have to clearly explain why this is true. Hint:P1

t=0 t = 1=(1 ) for

0 < V 2 or V 1 < V 2? As b increases, is she more likely to accept one? As q increases, is shemore likely to accept one?

(d) Why does an increase in a ring risk for low-wage jobs (q ) affect thedecision of an unemployed worker in accepting a low-wage offer in the way you

describe in the above question? Provide clear intuitions.

3. (Unemployment Insurance) In our discussions on economic effects of

unemployment benets, we considered the following set-up (except that u(C ) =C is assumed here):

Consider a worker who maximizes the expected discounted value of

( )


17/140


While unemployed, a worker meets one employer every period, who offers

either a high wage wh (with probability p) or a low wage wl (with probability 1 p). Looking at the wage offer, the worker may accept or reject.If she accepts, she receives the specied wage at each period, starting from

this period. There is no risk of getting red, and it is not allowed to search

for a new job once employed. If she rejects, she receives the unemploy-

ment benet of b (which is lower than wl) in this period and continues tomeet an employer in the next. The unemployment benet is paid forever,

as long as she is unemployed.

Let us make the following changes:

(1) An unemployed worker meets two employers each period. (But of course,

the worker can accept only one if she chooses to accept.) Each employer still

offers wh and wl with probabilities p and 1 p, respectively. (If both employersoffer wh, then the worker will accept either one. If one employer offers wh andthe other wl, the worker will accept wh. If both employer offers wl , then theworker will either accept one of them or reject both.)

(2) An employed worker with wl (low wage) can also search for a job. (Noticethat a worker with wh does not need to search because she already has the bestoffer.) In particular, she meets only one employer at the beginning of every

period (i.e., before she works and receives the wage). An employer offers whand wl with probabilities p and 1 p, respectively. If wh is offered, then she willdenitely accept a new job immediately, receiving wh from the current period.Otherwise assume she stays at the current position.

(a) Suppose that an unemployed worker always accepts one offer if both

of offered wages are wl. Let V 1 denote the “value” (i.e., expected discounted“value” of consumptions), at the beginning of each period (i.e., before wage or

compensation is paid), for an unemployed worker who has just accepted one of the offers where both offers are wl. Show that

V 1 = wl + pwh=(1 )

1 (1 p) :

[Y h t l l l i h thi i t Hi tP1 t 1=(1 )


18/140


(i.e., before wage or compensation is paid), for an unemployed worker who has

just rejected both offers where both offers are wl. Show that

V 2 = b + (1 (1 p)2)wh=(1 )

1 (1 p)2 :

(c) When does a worker with two wl offers accept either offer: V 1 > V 2 or V 1 < V 2? As the unemployment benet (b) increases, is an unemployed worker more likely to stay longer as unemployed?

(d) Assume p = 0. Do we have V 1 > V 2 or V 1 < V 2? Explain this resultin plain words. What do you think will happen as p increases (other things beingequal)? In light of your answers, explain “good” and “bad” effects to herself

when a consumer chooses to accept one of offers, relative to when she rejects

both offers, under the condition that she has two wl offers.

4. (Minimum Wage) Consider a rm that hires both “skilled” and “unskilled”

workers. The production function is given by

Y (LS ; LU ) = 2LS + LU + 2p

LS LU ;

where Y is the output in dollars, and LS and LU are the numbers of skilledand unskilled workers hired. There are 75 skilled and 75 unskilled workers who

want to be hired regardless of wages by this rm. (Hence, labor is “inelastically”

supplied for this rm.) Denote the wages by wS for the skilled and wU for theunskilled.

(a) Set up the prot-maximization problem of the rm, given the wages (wS and wU ). Obtain the rst-order conditions.

(b) The market-clearing condition implies that equilibrium wages are de-

termined so that all workers who want to be hired are hired. What are the

equilibrium wages for skilled and unskilled workers, respectively?

(c) Suppose the government sets the minimum wage to 2.5. If this is higher

than equilibrium wage, the market clearing is broken and the rm may not hire

all workers who want to be hired How many unskilled workers are now hired?


19/140


(e) Compare the outputs in terms of dollars in two cases (no minimum wage

vs. minimum wage). By introducing the minimum wage of 2.5, has the output

increased or decreased? Why?

5. (Immigration of The Unskilled) Consider an economy with “skilled” and

“unskilled” workers. Each individual rm has an identical production function:

Y (LS ; LU ) = (LS )2=3(LU )

1=3;

where Y is the output in units of consumption goods, and LS and LU are thenumbers of skilled and unskilled workers hired by an individual rm, respec-

tively. (LS has a higher power, 2/3, which reects that those workers are skilled.)Denote wage rates by wS for a skilled worker and by wU for an unskilled.

(a) An individual rm chooses how many skilled and unskilled workers tohire (LS and LU ), taking wage rates (wS and wU ) as given. Set up a prox-maximization problem of an individual rm. Obtain the rst-order conditions.

(b) (Since the production function is constant-returns-to-scale, the number

of rms existing in this economy does not matter to obtain the total output of

the economy.) Assume there is (only) one “representative” rm in this economy

with the above production function. Labor is “inelastically” supplied for this

rm, and there are 125 skilled and 125 unskilled workers who want to be hired.A market-clearing condition implies that equilibrium wage rates are determined

so that all workers (i.e., 125 workers in each labor market) are hired. (That is,

LS = 125 and LU = 125.) What are the equilibrium wage rates for skilledand unskilled workers, respectively? What is the GDP (i.e., total output) of this

economy? GDP per capita (i.e., GDP divided by 250)?

(c) Suppose there are 91 unskilled immigrants. So now there are 125+ 91 =

216 unskilled workers in this economy. Obtain equilibrium wage rates for skilledand unskilled workers, GDP, and GDP per capita. (Hint: 216 = 63: Feel free touse 25=108 0:231 and 150=341 0:440.)

(d) Describe your ndings in plain words so that your friend (whose major

is not economics) can understand. Your solution should contain explanation on


20/140


- What happens to GDP per capita and why?


21/140

Chapter 2

Fiscal Policy (I)

This chapter addresses several issues in taxation. In particular, we discuss how

some types of taxes - lump-sum tax and capital income tax - affect the economy.

2.1 Lump-Sum Tax

Question: How Do Lump-Sum Taxes Affect an Economy? (Reference:

Section 12.1 of DLS.)

How taxes affect production, consumption, investment, capital stock or gov-ernment revenue is an important question in macroeconomics. We start our

discussion with lump-sum tax. This type of tax simply collects a xed amount

of money from a taxpayer, regardless of any individual characteristics such as

income or wealth. It does not take 10% of your income (like income tax) or 5%

of your wealth (like property tax). It takes, for example, $1,000 from you, no

matter who you are, how much you earn, or how rich you are.

Of course, this type of tax is somewhat unrealistic because no governmentin reality collects the same amount of money from everyone. But it is simple to

analyze, so it often becomes a starting point of an economic analysis on taxa-

tion. But more importantly, there is another reason why this type of taxation is

important, which will be revealed after

2.1.1 Model Description

Our goal is to compare two economies: one without any taxation, and the other

with $G of lump-sum tax per worker. There are no other taxes. We study how productions, consumptions, investments, capital stocks and government revenues

come to be different between these two economies. To do this, we use a general

equilibrium setup, which is familiar to you from earlier courses.

In our ctional economy, there is only one type of consumption good. (For


22/140

2.1 Lump-Sum Tax 19

same time. She has the following preferences:

1Xt=0

tU (C t); (2.1)

where C t is consumption at time t, U is a utility function (that is increasing,concave, and twice-differentiable), and 0 < < 1 is a time discount factor.

Notice that (2.1) is an extended version of a two-period setup, U (C 0) + U (C 1),so that the representative consumer can live forever.

Now consider a production function. The representative consumer has an

access the following technology:

Y t = (K t); 0 <


23/140

20 Chapter 2 Fiscal Policy (I)

then K 1 is determined, then the consumer again decides I 1, and so on.

This completes the model description. At each period, the representative

consumer decides how much to consume (C t) and how much to invest (I t), giventhe capital stock (K t) and the lump-sum tax (G). That is, the representativeconsumer maximizes (2.1) subject to all other constraints, (2.2), (2.3) and (2.4).

We have one function to maximize, and three constraints binding it.

2.1.2 Solution and Discussion

An easy way to solve this problem is to eliminate the constraints as many as we

can. (2.2) and (2.3) yields

C t + I t + G = (K t);

(which eliminates Y t,) and again with (2.4), it becomes

C t + K t+1 (1 )K t + G = (K t);(which eliminates I t,) or equivalently,

C t = (1 )K t + (K t) K t+1 G: (2.5)We can even make this problem simpler by eliminating C t and transforming itto an unconstrained problem. That is, inserting (2.5) into (2.1), our problem

becomes

maxfK t+1g1t=0

1Xt=0

tU ((1 )K t + (K t) K t+1 G): (2.6)

Now all we have to do is to obtain the rst-order conditions.

We must be careful in doing this because there are both K t and K t+1 in one parenthesis. That is, (2.6) is extended to

maxK 1;K 2;:::

fU ((1 )K 0 + (K 0) K 1 G)::: + tU ((1 )K t + (K t) K t+1 G)


24/140

2.1 Lump-Sum Tax 21

K t+1. The rst-order condition is

tU 0(C t)[1] + t+1U 0(C t+1)[1 + (K t+1)1] = 0; for all t = 0; 1;:::;or equivalently,

U

0

(C t) = U

0

(C t+1)[1 + (K t+1)1

]; for all t = 0; 1;::: (2.7)Inserting t = 0; 1;::: provides all the rst order conditions. This equation deter-mines how fK t+1g evolves over time.

It is not easy and beyond our scope to investigate (2.7) in detail. But the

analysis is simple if we look at the steady state only. It turns out that in the long

run, we have a steady state in which the capital stock stays at a constant level

K SS forever. (Proving this argument is difcult. At this stage, simply assume it.)

From (2.5), we know that the consumption also stays at a constant level C SS . So,in a steady state, (2.7) becomes

U 0(C SS ) = U 0(C SS )[1 + (K SS )1];

or,

K SS = 1= + 1

1

1

: (2.8)

This is the long run level of capital and we are now ready to examine how the

government spending G affects this level.

1. Capital Stock (K SS ): As you can see from (2.8), G does not affect the steady-state level of capital . (We will see how other types of taxes might affect

this level later in this chapter.)

2. Investment (I SS ): From (2.4), we can easily see that

K SS = (1 )K SS + I SS ;or equivalently,

I SS = K SS ;


25/140


and we know this is not affected by G because K SS is not affected by it. Nowthink about this: The production is not affected by this lump-sum tax. But

the government is spending something. Then how can the economy nance

this? Output and investment stay at the same levels, and the only thing left is

consumption. Yes, the consumer decreases consumption to nance government

spending!4. Consumption (C SS ): The consumption becomes from (2.5),

C SS = K SS + (K SS ) G:This means if we assume


26/140


fair to force each of them to pay $100 for taxes. If someone earns more dollars,

we think he/she should pay more dollars. (Furthermore, in many countries, those

people pay a higher fraction (e.g., 40%) of their incomes than middle-income

workers do (e.g., 20%). Here, to make the analysis simple, we will assume that

everyone pays the same fraction of their incomes.) In this section, we analyze

the effects of (capital) income tax. If you have your money in bank accountsor hold stocks in stock markets, you should pay some part of your income from

those sources to the government. At this point, you may guess that because your

capital income is now taxed, you want to hold less stock of capital. In that sense,

this type of tax is different from lump-sum tax. We now conrm this conjecture

in a clear way.


The model is the same as before, but a new assumption here is that the govern-

ment will take a fraction (say, 10%) of your capital income (and not a xedamount of $G) as a tax. For convenience, we rewrite expressions (2.1), (2.2) and(2.4). The preferences of a representative consumer are

1

Xt=0 tU (C t); (2.10)

and private production and capital accumulation follows

Y t = (K t); 0 <


27/140


or equivalently,

C t + I t = (1 )Y t; (2.13)which becomes a new resource constraint. This completes a model description.

2.2.2 Solution and Discussion

How to solve this model is similar to our previous setup. (2.11) and (2.13) yield

C t + I t = (1 )K t; (2.14)and (2.12) implies

C t + K t+1 (1 )K t = (1 )K t;or equivalently,

C t = (1 )K t + (1 )K t K t+1: (2.15)We can make the problem simpler by inserting (2.15) into (2.10): Given K 0, theconsumer chooses fK t+1g for all t = 0; 1; 2;::: to maximize

1

Xt=0 tU ((1 )K t + (1 )K t K t+1):

The rst-order condition with respect to K t+1 is

tU 0(C t)[1] + t+1U 0(C t+1)[1 + (1 )(K t+1)1] = 0;or equivalently,

U 0(C t) = U 0(C t+1)[1 + (1 )(K t+1)1]: (2.16)

This equation determines how fK t+1g (and other variables) evolves over time.As before, let us focus on a steady state. In a steady state, the capital stock

stays at a constant level K SS , and from (2.15), it turns out that the consumptionalso stays at a constant level C ss. In this steady state, (2.16) implies

U 0(C ) U 0(C )[1 + (1 )(K )1]


28/140


On the left-hand side of this equation, I wrote K SS as a function of to empha-size that K SS may depend on .

1. Capital Stock (K SS ): As increases (for example, from 0% to 10%),the nominator in the parenthesis of (2.17), (1 ), decreases. So the entire

part in parenthesis, 1=+1(1 )

, increases (because from since 0 < < 1, i.e.,

1= 1 > 0, we have 1= + 1 > 0). Hence, as 11


29/140


Hence, RSS ( ) may increase or decrease as increases. A clear way to seewhether it increases or decreases is to differentiate (2.19) with respect to andsee the sign of it. A differentiation yields

R0SS ( ) = (1 ) 1

1

(1 ) 11

(constant) 1

=

(1 ) 1

(1 ) 11 (constant) 1

=

1 1

1 (1 ) 11 (constant) 1

where “(constant)” is positive (since 0 < < 1, i.e., 1= 1 > 0) and (1 )

1

1 is also positive (since 0 < < 1). Then, it is clear that R 0SS ( ) is

positive if

1 11 > 0;

or equivalently,

1

. So as the government increases the

tax rate between 0 and 1 , its revenue increases. But if it increases it from1 to a higher level, (i.e., if the tax rate becomes too high,) the governmentrevenue actually decreases. [Draw a graph.] The interpretation is as follows.

The government may get more revenue if it increases the tax rate because the

consumer will pay a higher share of her income. But this tax will distort the

decision of consumer, who will lower the private production, and it may be the

case that this decrease in production is so big that the government revenue (which

is a fraction of the private production) actually decreases. The “curve” that showshow the government revenue reacts as the government increases the tax rate is

called a Laffer curve. Needless to say, too high level of taxation (for example, a

tax rate higher than 1 in this model) should be avoided because it will distortthe economy while the government will suffer from a loss of revenue at the same


30/140


or equivalently,

C SS ( ) = (1 )K SS ( ) K SS ( ):Plugging (2.17),

C SS ( ) = (1 )1= + 1(1 )

1 1= + 1(1 )

1

1

= (1 )1+ 1

1= + 1

1

(1 ) 11

1= + 1

11

= (1 ) 11 h

(constant) 1 (constant) 11

iIf you increase , the rst term (1 )

1

1

denitely decreases. But it is not easyto determine the sign of the second term,

h(constant)

1 (constant) 11i. It

depends on parameter values. So, depending on the sign of the second term,

C SS ( ) can either increase or decrease. Then why might consumption increaseor decrease? The answer lies in (2.21), or more fundamentally, (2.14). (2.14) can

be written in terms of steady-state variables,

C SS ( ) + I SS ( ) = (1

)K SS ( ):

The key point is that an increase in has two different effects. First, it decreasesthe right-hand side, (1 )K SS ( ) (since both (1 ) and K SS ( ) decreases),which means now the consumer has less after-tax income, which will perhaps

decrease her consumption level. Second, on the other hand, it also decreases

the investment, I SS ( ), so the consumer want to consume rather than invest,which will increase her consumption level. Which one dominates is an empirical

question.In summary, capital income tax is distortionary. Unlike lump-sum tax, it

affects all macroeconomic variables: capital stock, investment, production and

consumption. [Draw gures. vs. variables: (i) vs. K SS , (ii) vs. I SS , (iii) vs. Y SS , (iv) vs. C SS .]


31/140


government wants to tax the capital income, but it does not want to discourage

the investment. So it decides not to tax the portion of capital income that is used

for investment. The budget constraint in this case will be

C t + I t + 1(Y t I t) = Y t:

Examine how this type of tax affects macroeconomic variables for your exercise.To take another example, suppose the government has decided to tax the

consumption only. You still have a full amount of capital income, but if you

want to spend $1 for your consumption, you should pay $ 2 to the government.The budget constraint in this case becomes

C t + I t + 2C t = Y t:

You can similarly analyze how this tax will affect the economy (or you can showwhy this budget constraint is, in fact, the same as the previous one).

True or False? —

1. Increasing the tax rate for capital income will serve for social justice in

a sense that richer people pay more taxes in terms of dollars, without any costs

that the economy should bear.

2. Suppose the government has nanced its spending solely by collecting

capital income taxes for a long time, and that the government revenue at each

period has been $100. If the government stops collecting capital income taxes

and starts collecting lump-sum taxes of the same amount ($100), then the invest-

ment, capital stock, and output in a steady state will increase.

3. Suppose the government nances all its spending by capital income taxes.

That is, the consumers need to pay a at rate of their capital income to the

government. As the government increases , the government revenue in a steady state also increases because consumers need to pay more taxes.


32/140


Exercises

1. (Consumption Tax) Suppose the government nances its spending solely

by the consumption tax with a xed rate of . That is, if you consume $1, youshould pay $ to the government. No tax is levied on the investment. Your job isto consider the effect of increasing this tax ratio (for example, from 10% to 20%)

on macroeconomic variables in a steady state.

Let us use the model described in the classes. The preferences of the repre-

sentative consumer are to maximize

1

Xt=0 tU (C t);

where C t is consumption at time t. Production (Y t) and capital accumulation (K t)follows

Y t = K t; 0 <


33/140


2. (Capital Holding Tax) Suppose the government nances its spending

solely by this “capital holding tax” with a xed rate of . That is, if you hold$1 value of capital stock, you should pay $ to the government at each period.The representative consumer maximizes

1Xt=0

t

U (C t);

where C t is consumption (in terms of dollars) at time t. Production (Y t) and stock of capital (K t) (both in terms of dollars) follow

Y t = (K t); 0 <


34/140


Laffer curve should look like yours.]

3. (Labor Income Tax) Consider a representative consumer with a utility

function over consumption C (in terms of dollars) and hours worked L (in termsof hours):

U (C; L) = 2p C L:The wage rate per hour is w, so the labor income is

Lw

in terms of dollars. However, the government collects a ratio of this labor income, so the consumer only gets

(1 )Lw:

(a) Set up the problem of this consumer. (This is a one-period problem. The

consumer chooses C and L and takes w as given.)

(b) Derive the rst-order condition(s). How many hours does this consumer

work? If the government increases the tax rate , does this consumer work moreor less?

(c) Obtain the government revenue. Does the government revenue increase

as the government raise the tax rate? Explain.


35/140

Chapter 3

Fiscal Policy (II)

3.1 Intertemporal Budget Constraint, Government Debt andLump-Sum Tax (“Ricardian Equivalence”)

Question: In an Economy with Lump-Sum Taxes only, How Does an Intro-

duction of Government Debt Affect the Consumption Level? (References:

Sections 14.1 and 14.2 of DLS.)

There are three main sources to nance government spendings:

(1) Collect taxes. We analyzed how taxes affect an economy in the last

chapter.

(2) Issue government debts. (Borrow at a nancial market.) How govern-

ment debt affects an economy is of our main interests in this chapter.

(3) Print money. This will be discussed in a later chapter.

Two main results derived in the last chapter are that lump-sum taxes are not

distortionary and that capital income taxes (and most of other taxes in reality)are. In this section, we model an economy with some combination of lump-sum

taxes and government debts. (We replace lump-sum taxes by distortionary taxes

in the next section.) The goal is to see how the size of government debt affects

our ctional economy. When the government borrows instead of taxing, will

consumers be happy because of lower taxes?


Into our model we introduce a perfect nancial market available both to a rep-

resentative consumer and to the government. This nancial market is perfect in

a sense that anyone can borrow or lend, (i) up to the amount he/she wishes, and

(ii) at a xed interest rate r. So you are free to borrow $100 in this period, butof course you should pay $100(1 + r) back in the next period You can lend (or


36/140

3.1 Intertemporal Budget Constraint, Government Debt and Lump-Sum Tax (“Ricardian Equivalence”)33

economy are not affected by these taxes anyway. This allows us to forget about a

production procedure and to simply assume that a xed amount of consumption

goods are given. That is, our economy in this model is an endowment economy,

in which the representative consumer simply gets her income from the heaven

without working (i.e., providing labor) or accumulating capital. Thanks to this

assumption, we are able to concentrate on the role of perfect nancial market. Now we describe the model in detail. Assume there are only two periods, 0

and 1. The representative consumer is endowed with Y 0 and Y 1 units of consump-tion goods at periods 0 and 1, respectively. These values are known in advance

to her. By borrowing and lending, she maximizes her discounted utility:

U (C 0) + U (C 1); 0 <


37/140

34 Chapter 3 Fiscal Policy (II)

then the problem becomes to maximize (3.1) subject to

C 0 = Y 0 + Y 1 C 1

1 + r ;

or equivalently,

C 0 + C 11 + r = Y 0 + Y 11 + r : (3.4)

This constraint is worth discussing in detail. The left-hand side of this equa-

tion, C 0 + C 11+r

, is a present value of consumptions at periods 0 and 1. Similarly,

the right-hand side, Y 0+ Y 11+r

, is a present value of endowments at periods 0 and 1.

So (3.4) implies that the present value of consumptions equals the present value

of endowments (i.e., incomes). This equation is often called an intertemporal

budget constraint. This constraint shows the role of a nancial market. Withoutthis market, the consumer's decision should satisfy

C 0 = Y 0;

C 1 = Y 1:

That is, she should consume what she is endowed with at each period because

there is no way to borrow or lend (or store). Now, the consumer's problem

becomes somewhat “exible” in a sense that these two budget constraints, sayingthat consumption equals endowment at each period, are replaced by only one in-

tertemporal budget constraint, (3.4), saying that a present value of consumptions

equals a present value of endowments. Of course, this difference is made by an

introduction of the nancial market.

The representative consumer maximizes (3.1) with respect to (3.4). The prob-

lem becomes an unconstrained problem as C 1 is eliminated. That is,

maxC 0

U (C 0) + U ((1 + r)Y 0 + Y 1 (1 + r)C 0):

The rst-order condition is

U 0(C 0) + U 0(C 1)[(1 + r)] = 0;


38/140


the same as a discount factor in the nancial market) in this type of setup. Then,

this result reduces to

U 0(C 0) = U 0(C 1);

hence (since U is strictly concave),

C 0 = C 1:

This means that the consumer wants to “smooth” the consumption over time

although endowments may be different between periods. This consumption

smoothing turns out to be important in many types of economic analyses.

2. Government Spendings Financed Solely by Taxes: Now we have a

public sector, so the government needs to spend G0 and G1 at periods 0 and

1. (The government does not produce anything from these.) G0 and G1 are pre-determined constants. The government collects lump-sum taxes to nance

them: The consumer are forced to pay T 0 and T 1 as lump-sum taxes to thegovernment, where T 0 and T 1 are chosen by the government, so the consumer's

budget constraint (3.4) should be replaced by

C 0 + C 1

1 + r

= Y 0

T 0 +

Y 1 T 11 + r

: (3.5)

Notice that (Y 0 T 0) and (Y 1 T 1) are after-tax endowments. Since the gov-ernment spending is nanced solely by the tax at each period, we should have

G0 = T 0;

G1 = T 1;

so (3.5) is replaced by

C 0 + C 11 + r

= Y 0 G0 + Y 1 G11 + r

: (3.6)

The consumer maximizes (3.1) with respect to (3.5). This is almost identical to

the problem of “no government” case and we can similarly get the consumption


39/140


government debt.) Then, at period 0, the government constraint becomes

G0 = T 0 + B0; (3.7)

which means that the government nances its spending (G0) by collecting taxes(T 0) and by issuing debt (B0). At period 1, it is

G1 + (1 + r)B0 = T 1; (3.8)

which means the government spends (G1) and pays back ((1 + r)B0) by collect-ing taxes (T 1). So as long as these two constraints are satised, the governmentis free to choose T 0, T 1 and B0.

But notice that we can eliminate one constraint. B0 is automatically deter-mined by (3.7) once T 0 is chosen. Eliminating B0, (3.7) and (3.8) reduce to

G0 = T 0 + T 1 G1

1 + r ; (3.9)

or equivalently,

G0 + G11 + r

= T 0 + T 11 + r

: (3.10)

The government had to satisfy two conditions, G0 = T 0 and G1 = T 1, in the previous setup, so government spending should equal tax revenue at each period.

But now, (3.10), saying that the present value of government spendings equals

the present value of taxes collected, is enough.

Our goal is to see how this new environment affects the consumer's problem.

The consumer still maximizes the same objective function, (3.1), subject to one

constraint, (3.5). The levels of T 0 and T 1 are selected by the government, which

satisfy (3.10). But wait. Consumer's constraint (3.5) can be written as

C 0 + C 11 + r

= Y 0 + Y 11 + r

T 0 + T 11 + r

; (3.11)

which is rewritten using (3.10) as,


40/140


and this is the same as (3.6)! This implies that the consumer's problem is ex-

actly the same whether the government borrows or not. This result is called the

Ricardian equivalence (also known as the Barro-Ricardo equivalence).

3.1.2 Discussion

1. Implication: Finally, the consumer's problem when the government is al-

lowed to borrow reduces to maximizing (3.1) with respect to (3.12). ((3.12) is

the same as (3.6).) There are no taxes (T 0 and T 1) or debts (B0) in this problem.Only G0 and G1 are there. And they are in the form of G0+

G11+r

. This means that

only the present value of government spendings is important in the consumer's

decision on consumption. How much to be taxed today or tomorrow does not

matter at all as long as the present value of government spendings remains con-

stant.

What should we learn from this result? In consumer's budget constraint

(3.11), what is important to the consumer is the present value of taxes: If the

government decreases T 0 (today's tax) by increasing debt, then T 11+r

(discounted

value of tomorrow's tax) is increased by the same amount. A decrease in tax

payment today made possible by an increase in government debt offsets by an

increase in tax payment tomorrow. So the consumer has no reason to feel happy

in the news of tax cuts nanced by debts.

To understand the importance of this implication, it is useful to review some

history of macroeconomics. The idea of Ricardian equivalence originally came

from David Ricardo in the 19th century. But for a long time, economists and

policy makers had not paid so much attention to this argument. After the in-

uence of John Keynes in 1930s, many Keynesian economists thought that the

government could “stimulate” the economy by spending more money. Suppose

the government has borrowed one million dollars from Tom. Tom is not hurt at

all because he still has a claim to one million dollars. But the government nowhas new one million dollars out of nowhere, so consumers do not have to pay one

million dollar taxes today, which makes them one million dollars richer. Hence,

consumers will spend more money, which will increase the consumption of an

economy The Ricardian equivalence tells you why this idea is not effective


41/140


cludes lump-sum taxation. So in reality (full of distortionary taxes), we do not

really expect that this result would perfectly hold. That is, a change in govern-

ment debt will affect the tax rate, which will change output and eventually (and

indirectly) consumption. But an important implication of the model can never be

underestimated. Yes, the decision of government regarding how much to borrow

will perhaps affect the consumer's decision to some extent in reality, but she isnot foolish enough to take those tax cuts as “additional incomes”.

3. Extension of the Model: Our model here is a two-period model. It is not

so difcult to extend our current model to an innite-period model. At time t, thegovernment constraint is

Gt + (1 + r)Bt1 = T t + Bt; for all t = 0; 1; 2;:::;

where B1 = 0 is given (i.e., there is no government debt that should be paid at period 0). Recall that Bt is a one-period government debt. Since this implies

B0 = G0 T 0if t = 0 and

G1 + (1 + r)B0 = T 1 + B1

if t = 1, we can eliminate B0

to have

G0 + 1

1 + rG1 = T 0 +

1

1 + rT 1 +

1

1 + rB1:

We can continue to do this algebra to eliminate B1, B2, ..., BS 1 to have

S

Xt=0

1

1 + r

tGt =

S

Xt=0

1

1 + r

tT t +

1

1 + r

S BS : (3.13)

Now suppose we are looking at a very long time horizon. That is, suppose S is very very large (S ! 1). In this case, we usually assume that

limS!1

1

1 + r

S BS = 0; (3.14)


42/140


do not want the government to accumulate the debt forever to an extremely huge

amount. No one can nance its spending by borrowing more and more forever.

If we take (3.14), we can simply write (3.13) as

1

Xt=0 1

1 + rt

Gt =1

Xt=0 1

1 + rt

T t: (3.15)

This, again, means that the present value of government spendings should be

equal to the present value of tax revenues. The representative consumer's prefer-

ences are to maximize

1Xt=0

tU (C t);

subject to the budget constraint

1Xt=0

1

1 + r

tC t =

1Xt=0

1

1 + r

t[Y t T t]; (3.16)

implying that the present value of consumptions equals the present value of after-

tax endowments. ((3.16) can be obtained similarly just as we have derived the

government's constraint.) But wait. (3.16) can be written using (3.15) as

1Xt=0

1

1 + r

tC t =

1Xt=0

1

1 + r

t[Y t Gt]:

Notice that neither fT tg nor fBtg appears hear. Only the present value of govern-ment spendings is important for the consumer. She does not even look at whether

government spendings are nanced by taxes or government debts, and she will

not be fooled by tax cuts made possible by issuing more government debts.

True or False? —

1. Suppose the Ricardian equivalence holds in an economy. If the government

wants to increase the private consumption this year then it is a desirable scal


43/140


Problem”)

Question: Can the Government use Government Bonds to minimize the

losses caused by Distortionary Taxes? (Reference: Barro, Robert J. (1979),

“On the Determination of the Public Debt,” Journal of Political Economy,

87(5), 940-971.)

Higher rates of distortionary taxes are costly. But the government sometimes

should spend a lot for many reasons – for example, to nance a war or to recover

from a disaster. If the government cannot borrow at all, then the tax rate should

be high for those periods, which will increase the deadweight losses (i.e., costs

arising from taxation) for those periods. Can we do something if the government

debts are introduced? In other words, while we know distortionary taxes are

costly, can we use the government debts to minimize these deadweight losses?

3.2.1 Model Description and Solution

Let us go back to the endowment economy in the previous section. Suppose

the endowments (Y 0, Y 1, Y 2, ...) are pre-determined prior to period 0, and thegovernment is free to choose the amount of taxes to be collected at each period.

That is, the government can determine T 0, T 1, T 2, etc. We say that T 0=Y 0, T 1=Y 1,T 2=Y 2... are the overall tax rates for given periods. Higher tax rates will be more

costly, so let a function,

f (T t=Y t);

denote the fraction of income that the economy loses by deadweight losses. f isassumed to be increasing (f 0 > 0) and convex (f 00 > 0). [Draw a gure.]

The convexity of this loss function can be justied by our results in the last

chapter. That is, (2.17) and (2.18) imply that

Y SS ( ) = (1 ) 1 (constant):Here, the deadweight losses are, by denition, Y SS (0) Y SS ( ). This is thedifference between two outputs, with and without capital income taxes. As you

increase , you will easily see that the deadweight losses are increasing and


44/140


To summarize, the environment of the model is as follows. Suppose at time t, thegovernment collects taxes of T t units of consumption goods (or dollars). Then,the consumer, whose original endowment is Y t, loses T t because this amountis transferred to the government. At the same time, she also loses Y tf (T t=Y t)

because of distortions caused by taxation. These losses simply disappear from

this economy.Assume that the government minimizes the present value of these deadweight

losses:

1Xt=0

1

1 + r

tY tf

T tY t

: (3.17)

And the constraint is that the present value of government spendings should be

smaller than or equal to the present value of tax revenues. That is,1Xt=0

1

1 + r

tGt

1Xt=0

1

1 + r

tT t: (3.18)

(This constraint is adopted from (3.15).) So we have a constrained maximization

problem of choosing fT tg to minimize (3.17) with one constraint (3.18).The Lagrangian function is

L =1Xt=0

1

1 + r

tY tf

T tY t

" 1Xt=0

1

1 + r

tT t

1Xt=0

1

1 + r

tGt

#;

where is a Lagrangian multiplier, so the rst-order condition with respect to T tis

11 + rt

f 0T tY t 11 + rt

= 0;

or equivalently,

f 0

T t

=


45/140


holds with equality, i.e.,

1Xt=0

1

1 + r

tGt =

1Xt=0

1

1 + r

tT t : (3.19)

(If (3.18) holds with inequality, then taxes collected are too much relative to

required spendings. Why collect unnecessary taxes?) It also requires that (ii) theoverall tax rate ( T t =Y t ) is constant over time. That is, regardless of the sequenceof government spendings fGtg, the government should keep the overall tax ratesas constant. Recall that the consumer loves consumption smoothing. Similarly,

the government loves tax smoothing.

3.2.2 Discussion

To understand the implication of this model, let us consider several examples.

Example 1. Suppose that both the endowment and government spending are

constant over time, i.e., Y t = Y and Gt = 1 < Y for all t = 0; 1; 2;::: [Drawgures for fY tg and fGtg.] Since T t =Y t should be constant, it is clear that T t isalso constant, i.e., T t = T . From (3.19), we have

T t = T = 1;

so the government spending in each period is always fully nanced by taxationat the same period. [Draw a gure for fT tg.] Hence, there is no government debtat any periods, i.e., Bt = 0 for all t = 0; 1; 2;::: [Draw a gure for fBtg.]

Example 2. Suppose that the endowment is constant over time, i.e., Y t = Y ,and the government needs to nance an immediate temporary spending. Let's

say G0 = 1 and G1 = G2 = ::: = 0. [Draw gures for fY tg and fGtg.] SinceT t =Y t should be constant, it is clear that T

t is also constant, i.e., T

t = T . From

(3.19), we have

1 = T 1Xt=0

1

1 + r

t:

UsingP1

t 0 t = 1=(1 ) for 0 < < 1, we can rewrite this as


46/140


or equivalently,

T = r

1 + r

So in order to nance a temporary spending, the government collects this amount

of constant tax at each period.

Then, we can obtain a sequence of the optimal government debts. At period

0, it is

B0 = G0 T 0 = 1 r1 + r

= 1

1 + r:

This means the following. The government nances its temporary spending, $1,

by taxing $r=(1 + r) and borrowing $1=(1 + r) at period 0. Then what willhappen at period 1? We have

B1 = G1 + (1 + r)B0 T 1= 0 + 1 r

1 + r

= 1

1 + r

and we can show that B0 = B1 = B2 = ::: = 1=(1 + r). So the governmentwill continue to have a debt amount of $1=(1 + r) forever at all periods. (Thatis, the government rolls over forever.) [Draw gures for fT tg and fBtg.] (Youcan easily extend the result to consider a more general case: G0 = G1 = ::: =Gn = G and Gn+1 = Gn+2 = ::: = g where G > g .)

Example 3. Continue to assume that the endowment is constant over time,

i.e., Y t = Y

, but at this time, the government should nance a future spending.

Let's say the government does not spend anything except for period 2. SoG2 = 1and Gt = 0 for t = 0; 1; 3; 4;::: [Draw gures for fY tg and fGtg.] Since T t =Y tshould be constant, it is clear that T t is also constant, i.e., T

t = T . From (3.19),

we have


47/140


or equivalently,

T =

1

1 + r

2=

1 + r

r

=

r

(1 + r)3

Hence, we have

B0 = G0 T 0 = 0 r(1 + r)3

= r(1 + r)3

;

B1 = G1 + (1 + r)B0 T 1 = 0 r(1 + r)2

r(1 + r)3

= r(r + 2)(1 + r)3

B2 = G2 + (1 + r)B1

T 2 = 1

r(r + 2)

(1 + r)2

r

(1 + r)3

= 1

(1 + r)3

B3 = G3 + (1 + r)B2 T 3 = 0 + 1(1 + r)2

r(1 + r)3

= 1

(1 + r)3

and it iseasy to seeB4 = B5 = ::: = 1=(1+r)3. So the government accumulates

assets up to period 2 by collecting taxes that are constant over time. At period 2,

the government debt jumps up to a positive level (because of the spending), and

then it stays there forever. (You can easily extend the result to consider a more

general case: Gn = G and Gt = g for all t = 0; 1;:::;n 1; n + 1; n + 2:::;where G > g .)

The Ramsey problem asks how to design a tax system to make the consumer

the happiest, for a given schedule of government spendings. In this section, we

have seen a nice example of this Ramsey problem: the optimal taxation problem

across different periods. There is another interesting example of this problem,

that asks how to tax different goods. According to the result, goods with higher

demand elasticities should be taxed less, and this result is often called the Ram-

sey rule.

T F l ?


48/140


2. Just as any economic agents, the government needs to pay interests for

borrowing. Hence, the government should not issue government bonds (i.e.,

should not borrow in the nancial market and should nance its spending solely

by collecting taxes).

3.3 Social SecurityQuestion: (i) When Do (and Don't) the Consumers Prefer a Pay-as-you-go

Pension System? (Reference: Section 12.3 of DLS.) (ii) Does the Redistribution

Policy Affect the Output of an Economy? (Reference: Section 13.4 of DLS.)

3.3.1 Model 1: Pension

The pension system is administrated by the government in many countries. Broadly

speaking, there are two pension schemes: (i) In a funded pension scheme, thegovernment taxes young workers and saves tax revenues, and later when they

retire, it transfers the money back to them. So this scheme is a “forced” savings

system. (ii) The U.S. is roughly on the other system, which is the unfunded or

pay-as-you-go pension scheme. Here, payments to retirees are collected from

current young workers.

A brief discussion on these two systems will help you to understand the

debate around the social security. Suppose the consumers live for two periodsonly. Generation-t consumers are born in period t and dies in period t + 1, andthey work when they are young (in their rst period, which is t) to obtain income.When they are old (in their second period, which is t + 1), they are not able towork, so their income become zero. (Assume generation-(t 1) consumersalways exist. That is, there is nothing like the “rst-ever” generation like Adam

and Eve.)

The population of generation-t consumers, denoted by N t, evolves as

N t+1 = (1 + n)N t; 0 < n


49/140


50/140


for i = A, B, where Y i is the output (thus the pre-tax income) and wi is her wage rate (or labor productivity). Assume wA > wB so that agent A is more

productive.

For redistribution purposes, the government collects a labor income tax from

agent A with a constant rate , and gives this to agent B as a lump-sum transfer.

So two agents face budget constraints of the form:

C A = (1 )wALA; (3.22)C B = wBLB + F; (3.23)

where is the at tax rate and F is the lump-sum transfer. Of course, thegovernment should balance revenue and expenditure, so

wALA = F:

Agent A maximizes (3.21) subject to (3.22). That is, she maximizes with

respect to LA,

2p

(1 )wALA LA:The rst-order condition is

(1

)wAp (1 )wALA 1 = 0;

or equivalently, p (1 )wA =

p LA;

or equivalently,

LA( ) = (1 )wA: (3.24)This result implies that as the government increases the tax rate , agent A willdecrease her labor supply in this model.

In fact, the result really depends on the utility function. For other types of

utility functions, the hours worked may even increase as the tax rate increases.

(S S i 13 4 f S ) ll f i i h d


51/140


want to work less. If the former effect dominates, an increase in will increasethe hours worked. If the latter effect does, it will decrease the hours worked. In

this model, the latter dominates.

Similarly, agent B maximizes (3.21) subject to (3.23), or equivalently, she

maximizes with respect to LB,

2p wBLB + F LB:The rst-order condition is

wBp wBLB + F

1 = 0;

or equivalently,

(wB)2 = wBLB + F;

or equivalently,

LB = (wB)

2 F wB

: (3.25)

This implies that agent B decreases her hours worked as the transfer F increases.This is because the transfer has only an income effect : Now you have more

money, so you want to consume more leisure. Notice that the lump-sum transfer

does not alter any price for agent B, so there is no substitution effect.

So both agents A and B decreases hours worked as the government strength-

ens the redistribution policy. A few remarks on this result:

(1) This result is based on a specic example of utility function. A more

general analysis can be found in Section 13.4 of DLS. Still, for a broad family

of utility functions, the section shows that the income of an economy (i.e., A's

income plus B's income) decreases as the government strengthens the redistribu-

tion system.

(2) Notice that (3.25) implies that the agent responds to lump-sum transfers.

Similarly we can show that the agent responds to lump-sum taxes. While we

argued that lump-sum taxes are not distortionary when the model has only the

i l f l i i h i h h l d ff


52/140


True or False? —

1. Given the current levels of population growth and interest rate in the

United States, an average worker will prefer being engaged in the pay-as-you-go

pension system rather than using his/her own savings accounts or other invest-

ment methods.

2. The redistribution policy will serve for our social justice in the sense that we can help old or poor people while it will not affect the GDP level.

3. Consider a consumer who has a utility function over consumption C (indollars) and labor input L (in hours worked), where she prefers leisure to work.(For example, you may want to assume U (C; L) = 2

p C L.) Also, assume this

consumer gets a xed amount of wage for each hour worked. (So if w denotesa wage rate, she earns wL dollars.) If the government introduces a lump-sum

tax so that a xed amount of tax is collected from this consumer, then the incomeeffect implies that this consumer will increase the number of hours worked, while

the substitution effect implies that this consumer will decrease it, so the total

effect is uncertain.

4. In a paper published in 2004, Edward Prescott (2004 Nobel-prize winner)

argues that the total number of hours worked of an economy (e.g., the U.S. or a

European country) decreases by 0.8% as the wage rate decreases by 1%. This

implies that the substitution effect dominates the income effect. (Assume that leisure is normal.)


53/140


Exercises

1. (Optimal Taxation) Consider an endowment economy in which the en-

dowment (i.e., income) at period t = 0; 1; 2;::: is given by Y t. The govern-ment should spend Gt at period t = 0; 1; 2;::: The values for Y t and Gt for

all t = 0; 1; 2;::: are pre-determined. As discussed in the class, suppose thegovernment wants to minimize the present value of deadweight losses subject

to the (intertemporal) budget constraint. The optimal tax revenue at each period

t = 0; 1; 2;:::, denoted by T t , should satisfy the following two conditions:

(i)

1Xt=0

1

1 + r

tGt =

1Xt=0

1

1 + r

tT t ;

i.e., the (intertemporal) budget constraint of the government is satised, and,

(ii) T t

Y tis constant;

i.e., the overall tax rate is constant over time for all t = 0; 1; 2;::: Assume Y t =Y , where Y is constant and greater than 2. (So the endowment is always thesame.) Also, assume there is no government debt that should be paid at time 0.

Suppose G0 = 2 and G1 = G2 = ::: = 1. That is, the government faces atemporary increase in government spending at time 0 in which the government

spending is $2, but after time 0, the spending stays at $1 forever.

(a) How much should the government collect as taxes at each period under

the optimal taxation? (Hint: If 0 < Gt for


54/140


(a') Answer the same question above. (Use the same hint. That is all you

need for the algebra.)

(b') Answer the same question above.

[End of the question. There is no (c').]

2. (Ramsey Problem with Labor Income Tax) The goal of this question is toderive an optimal scal policy when government spendings are nanced solely

by labor income taxes and government debts. There are only two periods, 0 and

1, (and period 1 is the end of the world). Consider a representative consumer

with a discounted utility function over consumptions and hours worked:

U (C 0; L0) + U (C 1; L1);

where C 0 and C 1 are consumptions (in units of consumption goods) at periods 0and 1, and L0 and L1 are hours worked (in hours) at periods 0 and 1. In particular,assume a specic form of U so that

U (C t; Lt) = 2p

C t Lt;for t = 0 and 1. That is, the consumer maximizes

2p C 0 L0 + (2p C 1 L1): (3.26)This representative consumer's output (which becomes the output of this econ-

omy) follows at each period,

Y t = Lt;

for t = 0 and 1, where Y t is the output in units of consumption goods. However,income tax rates are 0 and 1 at periods 0 and 1 (which may be different between

these two periods), so the government collects a fraction 0 or 1 of these outputs.Hence the consumer only gets

(1 t)Lt;as disposable income at period t 0 and 1 Throughout the question the


55/140


note these optimal hours worked by L0( 0) and L1( 1), respectively.) Does the

worker work more or less as the government increase the tax rate in either period?

(b) Obtain the government revenue at each period. Draw a period-0 Laffer

curve (that relates period-0 tax rate 0 and period-0 tax revenue). Draw a period-1 Laffer curve. Suppose the government needs to spend $0.25 (=$1

4) and $0

(nothing) at periods 0 and 1, respectively. The government cannot borrow, sogovernment spendings should be nanced solely by labor income taxes. What

should be the tax rates at periods 0 and 1?

(c) Continue to assume that the government needs to spend $0.25 (=$14

) and

$0 (nothing) at periods 0 and 1, respectively. But now the government has an

access to the perfect nancial market, which offers a xed interest rate, r. Thegovernment debt at period 0, denoted by B0 (in dollars), should be paid back at

period 1 with interests. Period 1 is the last period, so it is not allowed to borrow.That is, the government should choose 0, 1 and B0 to satisfy

1=4 = 0L0( 0) + B0;

0 + B0(1 + r) = 1L1( 1):

The consumer does not borrow or lend, and hence she acts as you described in

question (a). The (benevolent) government wants to choose a scal policy ( 0, 1

and B0) that maximizes the discounted utility of consumer, which is expression(3.26). (Of course, the government knows that consumer will act as you describedin question (a).) Set up the problem of the government. Go as far as you can to

“characterize” the optimal scal policy. (Your characterization should be enough

to answer the questions in (d). Feel free to assume = 1=(1 + r).)

(d) Describe your ndings in plain words so that your friend (whose major

is not economics) can understand. Your solution should contain explanation on

all of the following questions:- To make the consumer the happiest, how should the government design the

tax system? Are the tax rates the same across the periods?

- Should the government borrow?

I h hi di d i l h i i i h di ?


56/140


same steps, but with a slightly different utility function. There are two agents,

A and B, who have a common utility function over consumption C i (in terms of dollars) and labor input Li (in terms of hours worked) for i = A; B:

U (C i; Li) = log(C i) Li:Suppose each agent has a technology transforming labor effort into the produc-

tion:

Y i = wiLi;

for i = A, B, where Y i is the output (thus pre-tax income) and wi is her wage rate(or labor productivity). Assume wA > wB so that agent A is more productive.

For redistribution purposes, the government collects the labor income tax

from agent A (with a constant rate ) and gives this to agent B as a lump-sumtransfer. So two agents face the budget constraints of the form:

C A = (1 )wALA;C B = wBLB + F;

where is the at tax rate and F is the lump-sum transfer. Of course, thegovernment should balance revenue and spending, so

wALA = F:

(a) How does agent A's hours worked (LA) react to the tax rate ? Explainwhy you have such a result using the following two concepts: income effect and

substitution effect.

(b) How does agent B's hours worked (LB) react to the lump-sum transfer

F ? Explain why you have such a result using the following two concepts: incomeeffect and substitution effect.

(c) Suppose the government has just introduced this redistribution policy. So

agent A now faces the tax rate (which used to be 0) and agent B now obtains

lecture note for elements of economic analysis iv1

Documents