Marginal Propensity to Consume

Zheli He

Penn Wharton Budget Model

July 2018
Introduction

Questions:

What is the marginal propensity to consume (MPC) out of permanent

income and income shocks? How does it differ across gender, race,

education and age groups?

Why is it important:

Implications for predicting household responses to tax reforms and

other redistributive policies

What has been done:

In the literature, a common measure of permanent income is the

average annual income over some time period

What is new:

Use a large household dataset which includes measures of both total

consumption expenditure and income

Take into serious consideration the changing individual characteristics

when calculating permanent income

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 1 / 23
This Project

Provide a regression framework for analyzing the effects of individual

attributes on total family after-tax income

I the Consumer Expenditure Survey

Construct transition path of these attributes over the life cycle

I the Panel Study of Income Dynamics

Use the estimated regression coefficients and the projected

demographic profiles to predict family permanent income

Calculate the MPC by regressing consumption expenditure on

permanent income and income shocks

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 2 / 23
Preview of Findings

On average, households spend 22.37 cents out of each dollar of

income shocks

$1 increase in permanent income drives up current consumption

expenditure by 2.06 cents

The MPC out of income shocks is not statistically different for people

at different permanent income levels

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 3 / 23
Related Literature

Predictions about the level of consumption in relation to the level of

the permanent component of income

I Friedman (1957), Mayer (1966), Evans (1969), Mayer (1972), Dynan et

al. (2004), Bozio et al. (2013), Alan et al. (2015) and Straub (2018)

Predictions about changes in consumption in response to predictable

or unpredictable, transitory or permanent, income changes

I Shapiro and Slemrod (2009), Parker (1999), Souleles (2002), Johnson

et al. (2006), Huntley and Michelangeli (2014), Banks et al. (1998),

Hurd and Rohwedder (2006), Aguiar and Hurst (2007), Blundell et al.

(2008), Jappelli and Pistaferri (2008)

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 4 / 23
Consumption Function

The life cycle and the permanent income models have constituted the

main analytical tools to the study of consumption behaviour, both at

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Total consumption is much smoother than disposable income, and the

consumption profile is flat after accounting for changes in family size

and composition I use the following equivalence scale: assign weight 1 to the male adult,

0.8 to the female adult, and 0.4 to a kid

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 5 / 23
Consumer Expenditure Survey

Data on expenditures, income, assets, and demographic characteristics

of consumers in the United States

The Quarterly Interview Survey is designed to collect data on large

and recurring expenditures

Use the subsample of households that have completed all four

quarterly interviews

I Their annual consumption expenditure chronologically corresponds to

the reported income, which is over the last twelve months

Aggregate expenditures from quarterly to annual frequency using data

between 1990 and 2016

Convert all values into real 2016 dollars based on chained CPI

Imputation of Chained CPI Before 1999

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 6 / 23
Regression Framework

Apply weighted least squares, minimizing the sum of squared residuals

weighted by the inverse probabilities of selection:

ln yi = ↵0 + ↵1agei + ↵2age2

i + 0xi + 1xi agei + 2xi age2i + zi + ⌘i

yi is total family after-tax income in 2016 dollars

x includes marital status, number of children, gender, employment

status, employment status*I(employer pension contributions),

employment status*I(self-employment), full-time equivalent of weeks

worked, educational attainment and race

zi is a set of birth-year dummies to control for cohort effects

⌘i is an income shock

Use robust estimator of variance, which gives

heteroskedasticity-consistent standard error

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 7 / 23
Top-coding

The CE protects the respondents’ identity by changing sensitive data

with topcoding

I Topcoding refers to the replacement of data if the value of the original

data exceeds prescribed critical values

Each observation that falls outside the critical value is replaced with a

topcoded value that represents the mean of the subset of all outlying

observations

I Do not correct the top-coding in household expenditures, income and

assets

I Bosworth, et al. (1991): the top coding has only a minimal effect on

the surveys conducted after 1981

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 8 / 23
Income Imputation

Starting with the publication of the 2004 data, the CE includes income

data that have been produced using multiple imputation

I fill in blanks due to nonresponse, i.e., the respondent does not know or

refuses to provide a value for a source of income

I The process preserves the mean of each source of income, and also

yields variance estimates that take into account the uncertainty built

into the data from the fact that some observations are imputed, rather

than reported.

Use reported values if available. Otherwise, use imputed values and

apply the method of repeated-imputation inference

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 9 / 23
Repeated-Imputation Inference

The proper estimation uses all five imputations for income by estimating

the regression model once with each imputation

The point estimates, b ̄ =

Pm

i=1 bi

m , where m equals the number of

imputations

The variance of the point estimate, Tm = U ̄m + (1 + m1)Bm

I U ̄m =

Pm

1=1 Ui

m and Ui is the variance of the estimated coefficient for

imputation i

I Bm =

Pm

i=1(bib ̄)

2

m1 , which takes into account the uncertainty involving

the point estimate Regression Results

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 10 / 23
Panel Study of Income Dynamics

Originally designed to study the dynamics of income and poverty

I an over-sample of 1,872 low income families from the Survey of

Economic Opportunity (the “SEO sample”)

I a nationally representative sample of 2,930 families designed by the

Survey Research Center at the University of Michigan (the “SRC

sample”)

“Sample persons” include all persons living in the PSID families in

1968 plus anyone subsequently born to or adopted by a sample person

I All sample members are followed even when leaving to establish

separate family units (FUs)

The most common example of “non-sample persons” are those who

after 1968 marry sample persons.

I Information on non-sample persons is collected while they are living in

the same family unit as a sample person

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 11 / 23
Transition Matrix of Attributes

Use the PSID to construct transition matrix of all attributes for each

gender, race, age and education group:

Pg

ij = Pr(Xg

t+1 = j|Xg

t = i)

= ng

P

ij

j ng

ij

P

j Pg

ij = 1 and Pgij  0. ngij is the weighted sum of the number of

observations where individuals in group g transition from state i at

age t to state j at age t + 1 in attribute X

Impose the restrictions that Educt  Educt+1  Educt + 1. In

addition, if one’s education increases from high school graduates to

some college in year t, then there can be no more increase in the next

2 years

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 12 / 23
Assumptions I

Include an observation if and only if the individual has a positive

sample weight in both last year and the current year, and there is no

missing information about their age, gender, race and education as

well as the variable in question

If there is not enough information to calculate the transition

probability of a variable for a given group at a given age, then assume

that the next state is the same as the initial state with probability 1

Transition probabilities after age 65 are the same as those at age 65

due to a lack of data

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 13 / 23
Assumptions II

Split the PSID sample into three time periods: 1990-1997, 1997-2005

and 2005-2015 to allow transition matrix to vary over time

I Suppose an individual was 20 in 1996, then apply the 1990-1997

transition matrix to infer the probabilities of their attributes at age 21,

and apply the 1997-2005 transition matrix for predications at age 22

In order to classify an individual as a self-employed or wage worker

who claims to be both in the PSID, compare their labor and asset

income from business with their wages and salaries

I If the former is larger, identify the individaul as self-employed

I If this information is unavailable, assume an individual is a wage worker

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 14 / 23
One-Year Transition Matrix

PSID was conducted annually until 1996, and biennially since 1997

Impute 1-year transition matrix by calculating the principal square root

of 2-year transition matrix

I The unique square root for which every eigenvalue has nonnegative real

part

Underlying assumption: transition probabilities are the same from t to

t + 1 and from t + 1 to t + 2 between 1997 and 2015

If [V, D] = eig(A), then the squareroots have the general form

Y = V ⇤ S/V, where D = S ⇤ S, and the unique principal square root

is chosen

Square Root of a Matrix

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 15 / 23
Transition Path of Attributes

Assume sequence of all attributes follows a Markov chain, then the

conditional probability is:

Pr(Xi,t+1 = xi,t+1|Xi,0 = xi,0, Xi,1 = xi,1, ..., Xi,t = xi,t)

= Pr(Xi,t+1 = xi,t+1|Xi,t = xi,t)

= Pg

xi,txi,t+1

The probability of each realization of the random variable Xi,t for

individual i 2 g is:

Pr(Xi,t+k = xi,t+k ) = X

{xi,t+k}

..." X

{xi,2}

(

X

{xi,1}

Pg

xi,0xi,1Pgxi,1xi,2 )Pgxi,2xi,3

#

...Pg

xi,t+k1xi,1+k

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 16 / 23
Permanent Income

Calculate PIi = Ai,t + PTt

k=0

 

1

1+r

!k

Et[Yi,t+k ], where

Et[Yi,t+k ] = i,t+k

X

l

Pr(Yi,t+k = yi,l) ⇤ yi,l

Pr(Yi,t+k = yi,l) ⇤ yi,l = Pr(Xi,t+k = xi,t+k )⇤

[ ˆ↵0 + ˆ↵1agei,t+k + ˆ↵2age2

i,t+k + ˆ0xi ,t+k + ˆ1xi ,t+k agei,t+k + ˆ2xi ,t+k age2i,t+k + ˆzi]

Ai,t is a CU’s current net worth

Assume that r = 0.04; t corresponds to the age of the individual when

interviewed; T = 94, the oldest age in the CE

Mortality rates, i,t, is a function of education levels and martial

status along with other demographic variables (gender, race, age)

Use the maximum of the head’s and the spouse’s predicted permanent

income as the family permanent income Life-Cycle Profiles of Permanent Income

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 17 / 23
Average Propensity to Consume

Compute the APC by dividing consumption expenditure by permanent

income

more variation across households as they age

increase monotonically with age, rising from 2% at age 20 to over

10% at age 90

adjustment using the equivalence scale increases its curvature

similar results after talking into account the economies of scale of joint

living

not a lot of variation across race nor education groups

Life-Cycle Profiles of APC

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 18 / 23
Average Propensity to Consume

Table 1: Summary Statistics for the APC

median standard deviation

APC .0380666 .0825273

APC per adult equivalent .0202603 .0853685

APC joint living .0248291 .0819932

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 19 / 23
Marginal Propensity to Consume

Compute the MPC out of income shocks, ✓1, and out of permanent

income, ✓2:

ci = ✓0 + ✓1⌘ˆi + ✓2PIi

+↵1agei + ↵2age2

i + 0xi + 1xi agei + 2xi age2i + zi + ✏i

Measure income shocks as:

⌘ˆi = ln yi  ( ˆ↵0 + ˆ↵1agei + ˆ↵2age2

i + ˆ

0xi + ˆ1xi agei + ˆ2xi age2

i + ˆzi)

Table 2: Marginal Propensity to Consume

obs = 109,702

R-squared = 0.4635

ci Coef. Bootstrapped Std. Err. t P>|t|

⌘ˆi 0.2237425 0.0048216 46.40 0.000

PIi 0.0205654 0.0010507 19.57 0.000

Households spend 22.4 cents out of each dollar of income shocks

If permanent income goes up by $1, then consumption expenditure goes up

by 2.1 cents

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 20 / 23
Life-Cycle Profiles of MPC          

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Before 55, younger households spend more out of each dollar of

income shocks; after 55, older households do

Before 55, older households spend more out of each dollar increase in

permanent income; after 55, younger households do

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 21 / 23
Life-Cycle Profiles of MPC

To test this hypothesis, examine MPC calculated based on

consumption expenditure per adult equivalent         

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The shape of MPC out of income shocks does not change much

MPC out of permanent income monotonically increases with age after

change in family size is taken into account

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 22 / 23
MPC of Income Shocks at Different PI Levels

Table 3: MPC of Income Shocks at Different PI Levels

obs = 109,702

R2 = 0.4453

ci Coef. Bootstrapped Std. Err. t P>|t|

⌘ˆi ⇤ PIi 1.24e-08 1.20e-08 -1.03 0.303

⌘ˆi 0.3871338 0.0615386 6.29 0.000

⌘ˆi ⇤ age -0.0054707 0.0023142 -2.36 0.018

⌘ˆi ⇤ age2 0.0000491 0.0000243 2.02 0.043

PIi 0.0178924 0.0055083 3.25 0.001

PIi ⇤ age 0.0003511 0.0002047 1.71 0.086

PIi ⇤ age2 -4.81e-06 1.88e-06 -2.56 0.010

MPC out of income shocks is not statistically different for people at

different permanent income levels

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 23 / 23
Appendix: Imputation of Chained CPI

Use the series of chained CPI of all items in U.S. city average for all

urban consumers from the Bureau of Labor Statistics, which is

available from 1999 to 2016

To impute chained CPI for the missing years:

I First, compute the difference in average annual growth rate of chained

and unchained CPI between 1999 and 2016

I Second, use that difference to adjust the annual growth rate of

unchained CPI between 1990 and 1999

Use the same deflator for income and expenditures to keep them

consistent

back

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 1 / 11
Appendix: Regression Results

Table A1: Mean and Variance of the Estimated Coefficients

Y Coef. Robust Std. Err. Y Coef. Robust Std. Err.

marital 0.3271049 0.0566488 kid3Xage 0.000208 0.005864

maritalXage 0.0153051 0.002264 kid3Xage2 0.0001387 0 .0000642

maritalXage2 -0.0001664 0.0000209 kid4 -0.098393 0.2316845

kid1 -0.1385342 0.0661254 kid4Xage -0.005011 0.0106499

kid1Xage 0.0024829 0.0026706 kid4Xage2 0.0002041 0.0001195

kid1Xage2 0.0000598 0.0000257 age 0.0054983 0.0036854

kid2 -0.2628895 0.0877544 age2 0.000048 0.000032

kid2Xage 0.005837 0.0038422 sex 0.0568098 0.0525424

kid2Xage2 0.0000717 0.0000412 sexXage 0.0052125 0.0020828

kid3 -0.1531035 0.1301121 sexXage2 -0.0000892 0.0000193

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 2 / 11
Appendix: Regression Results

Table A1: Mean and Variance of the Estimated Coefficients

Y Coef. Robust Std. Err. Y Coef. Robust Std. Err.

wk -0.4799289 0.108972 wksemp2Xage -0.0130549 0.0041925

wkXage 0.0255648 0.0044286 wksemp2Xage2 0.0000715 0.000043

wkXage2 -0.0002041 0.0000424 wksemp3 0.7996495 0.1193799

wkXemplcont 0.1028728 0.0822613 wksemp3Xage -0.0077052 0.0052985

wkXemplcontXage 0.0032771 0.0037042 wksemp3Xage2 0.0000136 0.000056

wkXemplcontXage2 -0.0000367 0.0000401 educ2 0.0126893 0.0717158

wkXself_emp 0.4755819 0.1537953 educ2Xage 0.0050793 0.0028126

wkXself_empXage -0.0250893 0.0062541 educ2Xage2 -0.000038 0.0000255

wkXself_empXage2 0.0002509 0.0000605 educ3 -0.2245933 0.0776458

wksemp2 0.7360999 0.0963762 educ3Xage 0.0203403 0.0030635

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 3 / 11
Appendix: Regression Results

Table A1: Mean and Variance of the Estimated Coefficients

Y Coef. Robust Std. Err. Y Coef. Robust Std. Err.

educ3Xage2 -0.0001744 0.0000282 race2Xage -0.0043084 0.003571

educ4 -0.0304018 0.0876632 race2Xage2 0.0000282 0.0000334

educ4Xage 0.0242336 0.0034926 race3 -0.0147728 0.229137

educ4Xage2 -0.0002314 0.0000326 race3Xage -0.0065156 0.0095975

educ5 0.0633747 0.1127781 race3Xage2 0.0000704 0.0000927

educ5Xage 0.0244482 0.0044432 race4 0.0593875 0.1540429

educ5Xage2 -0.0002159 0.0000413 race4Xage 0.0017018 0.0063389

race2 -0.1273645 0.0893931 race4Xage2 -0.0000672 0.0000614

back

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 4 / 11
Appendix: Overview of Multiple Imputation

The method used to derive the multiple imputations is

regression-based I A regression is run to provide coefficients for use in estimating values

for missing data points

I The coefficients are then “shocked” by adding random noise to each,

and missing values are estimated using the shocked coefficients

I To each of these estimated values, additional random noise is added, to

ensure that consumer units (or members) with identical characteristics

(e.g., urban service worker aged 25 to 34) will not receive identical

estimates for their income

I The resulting values are used to fill in invalid blanks where they occur,

while reported values are retained

I This process is then repeated four times

For the small number of cases in which the respondent does not report

receipt of any source of income, receipt of each source is imputed

using logistic regression. I The income value is treated as a missing data point, and is imputed

using the method described above

back

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 5 / 11
Appendix: Square Root of a Matrix

More generally, an n×n matrix with n distinct nonzero eigenvalues has

2n square roots

I A = VDV 1, where V is the matrix whose columns are eigenvectors of

A; D is the diagonal matrix whose diagonal elements are the

corresponding n eigenvalues i

I The square roots of A are given by A = VD 1

2 V, where D 12 is any square

root matrix of D, which, for distinct eigenvalues, must be diagonal with

diagonal elements equal to square roots of the diagonal elements of D

I Since there are two possible choices for a square root of each diagonal

element of D, there are 2n choices for the matrix D 1

2

M is called positive semidefinite (or sometimes nonnegative definite) if

x ⇤ Mx  0 for all x in Rn

I A positive-semidefinite matrix has precisely one positive-semidefinite

square root, which can be called its principal square root

back

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 6 / 11
Appendix: Life-Cycle Profiles of Permanent Income 10 12 14 16 Constant 2016 Dollars

20 40 60 80 100

Age

Raw Smoothed

Predicted based on individual characteristics and adjusted for mortality rate; 4% discount rate assumed.

Permanent Income

11 12 13 14

Constant 2016 Dollars

20 40 60 80 100

Age

Excluded Included

Predicted based on individual characteristics and adjusted for mortality rate; 4% discount rate assumed.

Permanent Income Including and Excluding Initial Net Worth

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 7 / 11
Appendix: Life-Cycle Profiles of Permanent Income       &RQVWDQW'ROODUV

    

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Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 8 / 11
Appendix: Life-Cycle Profiles of APC

0 .1 .2 .3 .4 20 40 60 80 100

Age

Raw Smoothed

Raw predicted values at the top and bottom 5% are trimmed.

Average Propensity to Consume

         

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Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 9 / 11
Appendix: Life-Cycle Profiles of APC .02 .04 .06 .08 .1 20 40 60 80 100

Age

Predicted based on individual characteristics and adjusted for mortality rate; 4% discount rate assumed.

Average Propensity to Consume: Adult Equivalent

.02 .04 .06 .08 .1 20 40 60 80 100

Age

Predicted based on individual characteristics and adjusted for mortality rate; 4% discount rate assumed.

Average Propensity to Consume: Joint Living

Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 10 / 11
Appendix: Life-Cycle Profiles of APC

        

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Zheli He (Penn Wharton Budget Model) Marginal Propensity to Consume July 2018 11 / 11