Johnnie B. Linn III
Concord College
Athens, WV
Reconciling Macroeconomic
And Microeconomic
Approaches To Lump Sum And
Proportional Taxes That
Collect The Same Revenue:
An Interactive Spreadsheet
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
•
Spreadsheet Analysis
–
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
•
Spreadsheet Analysis
–
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
The Spreadsheet Approach
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
Spreadsheet Approach:
Lump

Sum Tax
Lump

Sum Tax:
Spreadsheet Approach (Continued)
•
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:
Spreadsheet Approach
Lump

Sum to Proportional Tax:
Spreadsheet Approach (Continued)
•
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.
•
Spreadsheet Approach:
–
Less compact.
–
Process is more transparent.
We now return you to your
regular programming.
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