# reconciling macroeconomic and microeconomic approaches to lump

Management

Oct 28, 2013 (4 years and 6 months ago)

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Johnnie B. Linn III

Concord College

Athens, WV

Reconciling Macroeconomic
And Microeconomic
Approaches To Lump Sum And
Proportional Taxes That
Collect The Same Revenue:
Approach

The Problem

Keynesian equilibrium

identical impact,
leaving the economy
with identical levels of
employment and
income

Output

A

T

The Problem, Continued

Microeconomic theory
of labor supply

a proportional tax
reduces the
opportunity cost of
leisure (substitution
effect)

a lump sum tax does
not (no substitution
effect)

Labor

Leisure

C

B

S

S’

How do we Reconcile These?

Key: A change in worker productivity and
a change in the work week

Why? To match output before and after a
tax change

How? Change the capital/labor ratio

From Lump Sum to Proportional
Tax

The amount of labor
offered per worker
decreases.

To maintain the same
level of output, employers
must increase the ratio of
capital to labor.

This will raise the
productivity of labor and
raise its after
-
tax wage.

From Proportional to Lump Sum
Tax

The households offer to
work a greater number of
hours per week

Too much output is
produced compared to
the increased amount of
aggregate demand.

Employers therefore
withdraw some capital
from the production
process, reducing the
productivity of labor.

How Can the Macro
-
Micro
Reconciliation Best be Taught?

Graphical Analysis

Keynesian Cross

Aggregate Labor Demand Function

Household Labor Supply Function

Aggregate Demand Schedule

Aggregate Labor Demand

Individual Labor Supply

Pros and Cons

Graphical Analysis

Well known

Good for partial equilibrium analysis

Difficult to integrate macro and micro levels

Spreadsheets not yet familiar to all

Less visual appeal

Easier to integrate macro and micro levels

Our Assumptions

Fixed technology (Cobb
-
Douglas labor
-
capital production function)

Labor’s and capital’s shares of total iIncome
are invariant (75% Labor, 25% Capital)

For a given level of productivity, labor and
capital are hired and laid off in the same
proportion.

Workweek is the same for all employed
households.

Assumptions, Continued

All households, whether employed or not,
own equal shares of capital.

Capital is expressed in labor
-
equivalent units.

Each household owns 40 shares of capital.

Arguments of household’s utility function
are leisure and income.

Marginal propensity to consume is imputed at
macro level.

Marginal propensity to consume is constant.

Assumptions, Continued

No income effect in labor supply.

Proportional tax is levied on all income,
earned and unearned.

The Graphical Analysis

1. Full Employment: The
Keynesian Cross and the “Sailboat”

A

B

C

D

E

G

F

OUTPUT

LABOR

The Labor Utilization Curve (EB)

Slope of EB is the
productivity of labor.

Productivity changes
only if there is a change
in the capital
-
labor ratio.

Point B is 100%
employment of labor.

B

C

D

E

G

F

LABOR

The Household Income Function
(DCB)

CE and BF are
always in fixed
proportions, so DCB
is always a straight
line

Slope of DCB is the
wage.

Utility Function is
Tangent to CB at B.

B

C

D

E

G

F

LABOR

2. Equilibrium at Less than Full
Employment, No Taxes

A

B

C

D

E

H

J

K

L

M

N

P

OUTPUT

LABOR

HH. INCOME

The Aggregate Labor Demand
Function (LKJ)

Reduction in
aggregate demand
from A to H results in
reduction of labor
demand from B to J.

Demand for capital
(EL) is reduced in
same proportion as
demand for labor
(EJ).

A

B

C

D

E

H

J

K

L

OUTPUT

LABOR

The Household Labor Supply
Function

Earnings of capital
(EK) is divided
equally among all
households (NM).

Unemployed
Households are at
point M, employed
households are at
point P.

B

C

E

J

K

M

N

P

LABOR

HH. INCOME

3. Proportional and Lump Sum
Taxes, Full Employment Case

A

B’

D

R

S’

S

OUTPUT

LABOR

B

T

A

Aggregate

output

This

column

is

set

up

in

such

a

way

that

the

scale

is

set

by

extrapolation

from

the

first

two

entries
.

B

Productivity

Ratio

of

output

to

labor

hourly

input
.

C

Labor

hours

demanded

Output

/

productivity

D

Workweek

Exogenous
.

E

Labor

demanded

Labor

hours

demanded

/

workweek
.

F

Labor

force

Exogenous
.

G

Unemployment

rate

(labor

force

labor

demanded)

/

labor

force

H

Wage

Productivity

×

labor

share

of

income

I

Labor

supply

in

hours/week

40

×

(after

tax

wage

/

7
.
5
)
.

J

Capital

demanded

Derived

from

Cobb
-
Douglas

function

(see

text)

K

Capital

base

Exogenous
.

L

Earnings

per

share

(
1
/
40
th

of

capital’s

share

of

income)

/

labor

force
.

M

Excess

capital

supplied

Capital

base

less

capital

demanded
.

N

45
-
degree

line

Duplicate

of

column

A
.

O

Tax

rate

Exogenous
.

P

Fixed

(lump

sum)

tax

Exogenous
.

Q

Total

tax

Fixed

tax

+

(tax

base

×

tax

rate)
.

The

base

of

the

proportional

tax

is

all

income,

earned

and

unearned
.

R

Disposable

income

Output

total

tax

S

Marginal

propensity

to

consume

Exogenous

T

Consumption

demand

(
100

×

labor

force)

+

marginal

propensity

to

consume

×

disposable

income

U

Saving

Disposable

income

consumption

demand

V

Investment

Exogenous

W

Government

purchases

of

goods

and

services

Exogenous

X

Government

budget

Total

tax

government

purchases

of

goods

and

services

Y

Aggregate

demand

Consumption

+

investment

+

government

purchases

of

goods

and

services

Z

Unintended inventory investment

Output

aggregate

demand

Lump
-
Sum Tax

Lump
-
Sum Tax:

Full employment output is \$1200

Lump
-
sum tax of \$100 is 8.33% of GDP

Workweek is 40 hours

There are 3 households each earning
\$400 before taxes.

Unearned income for each household: \$100.

Earned income is 40 hours @ \$7.50 or \$300.

Lump
-
Sum to Proportional Tax:

Lump
-
Sum to Proportional Tax:

Equilibrium output is \$1200.

Proportional Tax is 8.33% or \$100

The household labor market is not in
equilibrium

The after
-
tax wage is \$6.88.

Households offer to work 36.67 hours, or
8.33% less than before.

Employers still want a 40
-
hour workweek.

Lump
-
Sum to Proportional Tax:

Employer Reaction

A 10% boost in output is needed to meet
demand.

Part of this can be met by raising worker
productivity.

Increased productivity means higher after
-
tax
wage.

Higher after
-
tax wage will boost hours offered
per week and meet remainder of output
target.

Lump
-
Sum to Proportional Tax:

Employer Reaction (Continued)

Try a 5% increase in productivity.

Productivity will be raised from 10.0 to 10.5 (part of
this can be attained immediately because some
surplus capital is initially available).

An eventual increase in the capital base from 120 to
139 will be needed.

Workers will be constrained to working more hours
than they would like until the capital base is built up.

Split the Difference on the Workweek.

Midpoint of spread is 38.33 hours.

Proportional Tax: First Iteration

Result of First Iteration

The 5% productivity boost has overshot its target
slightly.

The before
-
tax wage is \$7.88.

The after
-
tax wage is \$7.22.

The workweek desired by employers is 38.33 hours.

The workweek offered by labor is 38.50 hours, a .17
hour surplus.

The next iteration should be a small negative
change in productivity and a small increase in
the workweek.

Proportional Tax: Solution

Final Figures

Productivity is 10.44

Before
-
tax wage is \$7.83.

After
-
tax wage is \$7.18.

Workweek is 38.3 hours.

Capital base is 137.

Proportional Tax to Lump
-
Sum Tax

Labor desires a workweek that is too long.

Employers will reduce the amount of
capital applied to labor.

Lower productivity will reduce drag of too
much output.

Lower wage will shorten workweek and
eliminate remainder of drag on output.

Change can be rapid because there is surplus
capital.

Reprise

Graphical Approach:

Compact.

Individual components can be studied in
isolation.

Process is less transparent.