Johnnie B. Linn III

Concord CollegeAthens, 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

SS’

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

FOUTPUT 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

FLABOR

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

FLABOR

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

LOUTPUT 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

KM

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 (1/40th 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

PRODUCTIVITY

LABOR HOURS

WORK WEEK LABOR

LABOR FORCE

UNEMP. RATE WAGE

LABOR SUPPLY HRS/WK CAPITAL

CAPITAL BASE

EARNINGS PER SHARE

CAPITAL EXCESS SUPPLY OUTPUT

TAX RATE FIX.TAX

10 0.00 40 0.00 3 100.0% 7.5 40 0 120 0.00 120 0 0.00% 10010 10.00 40 0.25 3 91.7% 7.5 40 10 120 0.21 110 100 0.00% 10010 20.00 40 0.50 3 83.3% 7.5 40 20 120 0.42 100 200 0.00% 10010 30.00 40 0.75 3 75.0% 7.5 40 30 120 0.63 90 300 0.00% 10010 40.00 40 1.00 3 66.7% 7.5 40 40 120 0.83 80 400 0.00% 10010 50.00 40 1.25 3 58.3% 7.5 40 50 120 1.04 70 500 0.00% 10010 60.00 40 1.50 3 50.0% 7.5 40 60 120 1.25 60 600 0.00% 10010 70.00 40 1.75 3 41.7% 7.5 40 70 120 1.46 50 700 0.00% 10010 80.00 40 2.00 3 33.3% 7.5 40 80 120 1.67 40 800 0.00% 10010 90.00 40 2.25 3 25.0% 7.5 40 90 120 1.88 30 900 0.00% 10010 100.00 40 2.50 3 16.7% 7.5 40 100 120 2.08 20 1000 0.00% 10010 110.00 40 2.75 3 8.3% 7.5 40 110 120 2.29 10 1100 0.00% 10010 120.00 40 3.00 3 0.0% 7.5 40 120 120 2.50 0 1200 0.00% 10010 130.00 40 3.25 3 -8.3% 7.5 40 130 120 2.71 -10 1300 0.00% 100

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

PRODUCTIVITY

LABOR HOURS

WORK WEEK LABOR

LABOR FORCE

UNEMP. RATE WAGE

LABOR SUPPLY HRS/WK CAPITAL

CAPITAL BASE

EARNINGS PER SHARE

CAPITAL EXCESS SUPPLY OUTPUT

TAX RATE FIX.TAX

10 0.00 40 0.00 3 100.0% 7.5 36.666667 0 120 0.00 120 0 8.33% 010 10.00 40 0.25 3 91.7% 7.5 36.666667 10 120 0.21 110 100 8.33% 010 20.00 40 0.50 3 83.3% 7.5 36.666667 20 120 0.42 100 200 8.33% 010 30.00 40 0.75 3 75.0% 7.5 36.666667 30 120 0.63 90 300 8.33% 010 40.00 40 1.00 3 66.7% 7.5 36.666667 40 120 0.83 80 400 8.33% 010 50.00 40 1.25 3 58.3% 7.5 36.666667 50 120 1.04 70 500 8.33% 010 60.00 40 1.50 3 50.0% 7.5 36.666667 60 120 1.25 60 600 8.33% 010 70.00 40 1.75 3 41.7% 7.5 36.666667 70 120 1.46 50 700 8.33% 010 80.00 40 2.00 3 33.3% 7.5 36.666667 80 120 1.67 40 800 8.33% 010 90.00 40 2.25 3 25.0% 7.5 36.666667 90 120 1.88 30 900 8.33% 010 100.00 40 2.50 3 16.7% 7.5 36.666667 100 120 2.08 20 1000 8.33% 010 110.00 40 2.75 3 8.3% 7.5 36.666667 110 120 2.29 10 1100 8.33% 010 120.00 40 3.00 3 0.0% 7.5 36.666667 120 120 2.50 0 1200 8.33% 010 130.00 40 3.25 3 -8.3% 7.5 36.666667 130 120 2.71 -10 1300 8.33% 0

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

PRODUCTIVITY

LABOR HOURS

WORK WEEK LABOR

LABOR FORCE

UNEMP. RATE WAGE

LABOR SUPPLY HRS/WK CAPITAL

CAPITAL BASE

EARNINGS PER SHARE

CAPITAL EXCESS SUPPLY OUTPUT

TAX RATE FIX.TAX

10.5 0.00 38.33 0.00 3 100.0% 7.875 38.5 0 120 0.00 120 0 8.33% 010.5 9.52 38.33 0.25 3 91.7% 7.875 38.5 12 120 0.21 108 100 8.33% 010.5 19.05 38.33 0.50 3 83.4% 7.875 38.5 23 120 0.42 97 200 8.33% 010.5 28.57 38.33 0.75 3 75.2% 7.875 38.5 35 120 0.63 85 300 8.33% 010.5 38.10 38.33 0.99 3 66.9% 7.875 38.5 46 120 0.83 74 400 8.33% 010.5 47.62 38.33 1.24 3 58.6% 7.875 38.5 58 120 1.04 62 500 8.33% 010.5 57.14 38.33 1.49 3 50.3% 7.875 38.5 69 120 1.25 51 600 8.33% 010.5 66.67 38.33 1.74 3 42.0% 7.875 38.5 81 120 1.46 39 700 8.33% 010.5 76.19 38.33 1.99 3 33.7% 7.875 38.5 93 120 1.67 27 800 8.33% 010.5 85.71 38.33 2.24 3 25.5% 7.875 38.5 104 120 1.88 16 900 8.33% 010.5 95.24 38.33 2.48 3 17.2% 7.875 38.5 116 120 2.08 4 1000 8.33% 010.5 104.76 38.33 2.73 3 8.9% 7.875 38.5 127 120 2.29 -7 1100 8.33% 010.5 114.29 38.33 2.98 3 0.6% 7.875 38.5 139 120 2.50 -19 1200 8.33% 010.5 123.81 38.33 3.23 3 -7.7% 7.875 38.5 150 120 2.71 -30 1300 8.33% 0

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

PRODUCTIVITY

LABOR HOURS

WORK WEEK LABOR

LABOR FORCE

UNEMP. RATE WAGE

LABOR SUPPLY HRS/WK CAPITAL

CAPITAL BASE

EARNINGS PER SHARE

CAPITAL EXCESS SUPPLY OUTPUT

TAX RATE FIX.TAX

10.444122 0.00 38.3 0.00 3 100.0% 7.83309 38.295113 0 137 0.00 137 0 8.33% 010.444122 9.57 38.3 0.25 3 91.7% 7.83309 38.295113 11 137 0.21 126 100 8.33% 010.444122 19.15 38.3 0.50 3 83.3% 7.83309 38.295113 23 137 0.42 114 200 8.33% 010.444122 28.72 38.3 0.75 3 75.0% 7.83309 38.295113 34 137 0.63 103 300 8.33% 010.444122 38.30 38.3 1.00 3 66.7% 7.83309 38.295113 46 137 0.83 91 400 8.33% 010.444122 47.87 38.3 1.25 3 58.3% 7.83309 38.295113 57 137 1.04 80 500 8.33% 010.444122 57.45 38.3 1.50 3 50.0% 7.83309 38.295113 68 137 1.25 69 600 8.33% 010.444122 67.02 38.3 1.75 3 41.7% 7.83309 38.295113 80 137 1.46 57 700 8.33% 010.444122 76.60 38.3 2.00 3 33.3% 7.83309 38.295113 91 137 1.67 46 800 8.33% 010.444122 86.17 38.3 2.25 3 25.0% 7.83309 38.295113 103 137 1.88 34 900 8.33% 010.444122 95.75 38.3 2.50 3 16.7% 7.83309 38.295113 114 137 2.08 23 1000 8.33% 010.444122 105.32 38.3 2.75 3 8.3% 7.83309 38.295113 125 137 2.29 12 1100 8.33% 010.444122 114.90 38.3 3.00 3 0.0% 7.83309 38.295113 137 137 2.50 0 1200 8.33% 010.444122 124.47 38.3 3.25 3 -8.3% 7.83309 38.295113 148 137 2.71 -11 1300 8.33% 0

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.