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The Personal Discount Rate: Evidence From Military Downsizing Programs

March 2001
American Economic Review 91(1):33-53

DOI:10.1257/aer.91.1.33

Source
RePEc

Authors:

John T. Warner

Clemson University

Saul Pleeter

American University Washington D.C.

The military drawdown program of the early 1990s provides an opportunity to obtain estimates of personal discount rates based on large numbers of people making real choices involving large sums. The program offered over 65,000 separatees the choice between an annuity and a lump-sum payment. Despite break-even discount rates exceeding 17 percent, most of the separatees selected the lump sum--saving taxpayers $1.7 billion in separation costs. Estimates of discount rates range from 0 to over 30 percent and vary with education, age, race, sex, number of dependents, ability test score, and the size of payment.

NUMBER ELIGIBLE FOR SEPARATION PAYMENTS, NUMBER SEPARATING, AND NUMBER CHOOSING SSB

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SIMPLE PROBIT AND LINEAR PROBABILITY MODELS OF SEPARATION PAYMENT CHOICE AND IMPLIED MEAN PERSONAL DISCOUNT RATE

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PROBIT EQUATIONS FOR PROBABILITY OF CHOOSING SSB AND IMPLIED PDR EQUATIONS, OFFICERS

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PROBIT EQUATIONS FOR PROBABILITY OF CHOOSING SSB AND IMPLIED PDR EQUATIONS, ENLISTED PERSONNEL

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MEAN NOMINAL DISCOUNT RATES

…

Figures - uploaded by Saul Pleeter

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Content uploaded by Saul Pleeter

Content may be subject to copyright.

The Personal Discount Rate: Evidence from Military

Downsizing Programs

By JOHN T. WARNER AND SAUL PLEETER*

The military drawdown program of the early 1990’s provides an opportunity to

obtain estimates of personal discount rates based on large numbers of people

making real choices involving large sums. The program offered over 65,000

separatees the choice between an annuity and a lump-sum payment. Despite

break-even discount rates exceeding 17 percent, most of the separatees selected the

lump sum—saving taxpayers $1.7 billion in separation costs. Estimates of discount

rates range from 0 to over 30 percent and vary with education, age, race, sex,

number of dependents, ability test score, and the size of payment. (JEL D91)

The rate at which individuals trade current for

future dollars, or personal discount rate, is a

provocative subject with important implications

for many aspects of economic behavior and

public policy.

In this paper we report the results of a natural

experiment that provides a unique opportunity

to measure personal discount rates that resulted

from the U.S. military drawdown program be-

ginning in January 1992. The U.S. Department

of Defense (DoD) began offering two separa-

tion beneﬁt packages to mid-career military per-

sonnel in selected occupations that consisted of

a choice between a lump-sum separation beneﬁt

or an annuity.

We observe the separation payment choices

of 11,000 ofﬁcers and 55,000 enlisted personnel

who faced before-tax break-even discount rates

(the rate which equated the present value of the

annuity with the value of the lump-sum pay-

ment) of between 17.5 and 19.8 percent. Based

on conventional interest rates, economists in

DoD predicted, prior to implementation of the

program, that about half of the enlisted person-

nel, but virtually no ofﬁcers, would take the

lump sum rather than the annuity. In fact, over

half of the ofﬁcers and over 90 percent of the

enlisted personnel took the lump-sum payment,

implying that the vast majority of personnel had

discount rates of at least 18 percent. Since gov-

ernment borrowing rates are far below these

personal discount rates, we estimate that offer-

ing lump-sum payments saved taxpayers $1.7

billion in severance costs.

We believe that the evidence from our analysis

is more compelling than some of the previous

evidence about personal discount rates for two

reasons. First, the choices are real, not hypotheti-

cal, and involve decisions over quite large sums.

The typical ofﬁcer faced a choice over an annuity

versus a lump sum worth close to $50,000. The

typical enlisted choice involved a lump sum of

approximately $25,000. These are generally much

larger sums than those involved in previous non-

experimental studies. Second, unlike many past

studies, which have been based on experiments

with college students or analyses of low-income

populations, our data are representative of differ-

ent segments of the U.S. population. Whereas 24.5

percent of individuals aged 25–35 in the March

1992 and March 1993 Current Population Sur-

veys have a college education or better, virtually

all of the ofﬁcers in our data do. Ofﬁcers comprise

about 20 percent of our sample. The rest of our

* Warner: Department of Economics, 222 Sirrine Hall,

Clemson University, Clemson, SC 29634; Pleeter: Ofﬁce of

the Undersecretary of Defense for Personnel and Readiness,

Compensation Directorate, Room 2B279, U.S. Department

of Defense, Pentagon, Washington, DC 20301, and Amer-

ican University. We thank Dan Benjamin, Gene Devine,

Bill Dougan, Matt Goldberg, Stan Horowitz, Richard Ip-

polito, Matt Lindsay, Bruce Lintner, Walter Oi, Sherwin

Rosen, Curtis Simon, Robert Tamura, Chris Thornberg,

Myles Wallace, an anonymous referee, and seminar partic-

ipants at American University for comments on previous

drafts. Special thanks are due Terry Cholar of the Defense

Manpower Data Center for assembling the data set used for

this study. We are responsible for any remaining errors.

sample is comprised of enlisted personnel, among

whom the modal educational level is a high-

school degree, which is also the modal degree of

individuals aged 25–35 in the March 1992 and

March 1993 Current Population Surveys (37 per-

cent). In terms of earnings, military ofﬁcers have

about the same taxable earnings as male civilians

aged 25–35 with a college degree and the enlisted

personnel have about the same earnings as civilian

male high-school graduates in the 25–35 age

range. In terms of educational achievement and

earnings, our data are representative of different

segments of the U.S. population.

After describing the drawdown programs in

Section I, Section II reviews the relevant liter-

ature in more detail. Then Section III presents

our analysis of the drawdown data. Section IV

concludes the paper.

I. Drawdown Programs

In light of the collapse of the Soviet Union and

lesser need for a large standing military force, the

1991 Defense Authorization Act directed DoD to

reduce active duty strength by 400,000 by FY

1995, a 25-percent reduction. To maintain a bal-

anced force, reductions would have to come from

every experience level, including career personnel

not yet vested in the military retirement system,

which “cliff-vests” at 20 years of service. Con-

gress also directed that involuntary separations of

career members be minimized.

To assist DoD in attaining this voluntary re-

duction, two temporary ﬁnancial incentive pro-

grams were developed: the Voluntary

Separation Incentive (VSI) and the Selective

Separation Beneﬁt (SSB). The VSI program

provided an annuity to the separating member

equal to 2.5 percent of annual basic pay multi-

plied by the member’s years of service (YOS).

Payments were to be received for a period equal

to twice the member’s YOS. The VSI formula is

similar to that for determining military retire-

ment beneﬁts. But, unlike military retirement

beneﬁts, the VSI annuity was not indexed for

inﬂation. The SSB program provided a lump-

sum payment of 15 percent of annual basic pay

multiplied by the member’s YOS.

Although VSI payments were not assignable

(transferable), the payments continue to desig-

nated beneﬁciaries in the event of the member’s

death. Both the annuity and the lump-sum pay-

ment are taxable in the year received. As we

show below, the lump-sum payment is likely to

be taxed at a higher marginal rate than the

annuity, or the same rate at best.

Importantly, DoD went to great lengths to in-

form personnel about the programs. The DoD

Compensation Directorate prepared a pamphlet

explaining the program and distributed it to all

affected personnel. Furthermore, articles in the

Service newspapers and in the general media ex-

plained the characteristics of each program. Indi-

vidual counseling was also made available to

individuals needing further clariﬁcation. The pam-

phlets contained comparisons between the lump-

sum SSB and the present value of the annuity for

selected grades and YOS. Discounting was done

using a nominal 7-percent interest rate, the rate

being paid for money market funds in 1991. At

this discount rate, the annuity compared quite fa-

vorably with the lump sum, as Table 1 illustrates.

The table provides illustrative examples of the

lump-sum amounts offered ofﬁcers and enlisted

personnel in some of the larger rank/YOS group-

ings affected by the program. Annuity amounts

and present values using DoD’s assumed 7-per-

cent discount rate are also shown. At this discount

rate, the present value of the annuity was in many

cases more than double the value of the lump-sum

payment.

Table 1 also shows the present value of the

annuity at discount rates of 10, 20, and 30

percent, respectively. At discount rates of 20 or

30 percent, the lump-sum amount exceeds the

present value of the annuity. In these compari-

sons, which do not account for the tax conse-

quences of the choices, break-even discount

rates range from 17.5 percent for those with 7

years of service to 19.8 percent for those with

15 years of service. Apparently, personnel did

not ﬁnd DoD’s information pamphlet, with its

comparison of the two separation choices, very

convincing. Among the ofﬁcers with less than

10 years of service, more than half took the

lump sum. Among the E-5 enlisted personnel

with less than 10 years, over 90 percent did so.

Almost 75 percent of E-7 enlisted personnel

with 15 years of service took the lump sum.

Even among the more senior ofﬁcers, 30 per-

cent or more took the lump sum. Overall, about

half of the ofﬁcers chose the lump sum while

over 90 percent of the enlisted personnel did so

(see Table 2).

34 THE AMERICAN ECONOMIC REVIEW MARCH 2001

The evolution of the drawdown programs is

interesting. In July of 1991, DoD submitted a bill

to Congress requesting authorization of the VSI.

DoD’s proposal would have allowed separatees

who did not want to receive their separation ben-

eﬁts in annuity form to sell their annuities in the

private market. Congress balked at this idea, and

an impasse prevailed between DoD and the Con-

gress for several months. At the last minute, the

Congress included a provision in the FY 1992

Defense Authorization Act allowing separatees to

take a lump-sum payment of 1.5 times the pay-

ment for involuntary separation in lieu of the

annuity proposed by DoD.

(It is fortunate that the

Congress inserted the SSB option, because it set

up the natural experiment that we analyze!) Per-

sonnel who took the lump sum were also to re-

ceive all the other separation beneﬁts available to

involuntary separatees, such as medical beneﬁts,

The lump-sum alternative was devised by Fredrick

Pang, a retired Air Force Colonel who at the time was the

staff director of the Senate Armed Services Committee.

Colonel Pang was familiar with previous military man-

power research suggesting high personal discount rates

among military personnel.

ABLE 2—NUMBER ELIGIBLE FOR SEPARATION PAYMENTS,NUMBER SEPARATING, AND NUMBER CHOOSING SSB

Army Navy Air Force Total

Ofﬁcers

Eligible 26,159 9,777 23,272 59,208

Separated 6,447 645 4,120 11,212

(24.7) (6.6) (17.7) (18.9)

Lump sum 3,459 401 1,854 5,714

(53.7) (62.2) (45.0) (51.0)

Enlisted

Eligible 76,118 38,794 120,040 234,952

Separated 24,213 9,235 21,823 55,271

(31.8) (23.8) (18.2) (23.5)

Lump sum 22,994 8,080 19,802 50,876

(95.0) (87.5) (90.7) (92.1)

TABLE 1—VSI AND SSB BENEFITS,SELECTED EXAMPLES

Lump-

sum

amount

Annuity

amount

Present value of annuity

Break-even

discount

rate

Percent

lump

sum

Percent

separating7 percent 10 percent 20 percent 30 percent

Ofﬁcer

O-3 with 7

YOS $34,709 $5,785 $54,129 $46,875 $32,002 $24,430 0.175 70.7 35.5

O-3 with 9

YOS $46,219 $7,703 $82,908 $69,497 $44,485 $33,085 0.189 52.1 47.8

O-4 with 12

YOS $72,006 $12,001 $147,276 $118,005 $71,106 $51,904 0.196 36.2 8.0

O-4 with 15

YOS $94,114 $15,686 $208,274 $162,645 $93,722 $67,950 0.198 29.8 6.8

Enlisted

E-5 with 7

YOS $16,655 $2,776 $25,973 $22,492 $15,356 $11,722 0.175 95.1 6.3

E-5 with 9

YOS $22,283 $3,714 $39,972 $33,506 $21,447 $15,951 0.189 94.8 28.1

E-6 with 12

YOS $35,549 $5,925 $72,710 $58,259 $35,105 $25,625 0.196 88.1 13.2

E-7 with 15

YOS $51,216 $8,536 $113,342 $88,510 $51,003 $36,978 0.198 74.3 8.0

35VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

moving and furniture storage expenses, etc. The

act also adopted DoD’s VSI proposal along with

DoD’s provisos relating to other beneﬁts available

to VSI recipients.

Unfortunately for the VSI recipients, all

other aspects of the choice favored the SSB.

VSI required an afﬁliation with a reserve

component for the life of the payment

whereas SSB required only a three-year com-

mitment in the Ready Reserve.

Transition

beneﬁts associated with SSB included ex-

tended commissary and exchange privileges,

60- to 120-day extension of medical cover-

age, shipment of household goods, and pos-

sible extension of military housing. VSI had

none of these transition beneﬁts. In terms of

possible civil service employment, SSB al-

lowed the member to count military service

toward federal civil service retirement

whereas VSI would not allow military service

to be counted toward federal civil service

retirement. Because of these additional con-

siderations, inferences about personal dis-

count rates from the FY 1992 data alone

would be clouded. Even if a person had a low

discount rate, he or she might be induced to

select the SSB because of the higher value of

the other beneﬁts associated with that choice.

Recognizing in retrospect the disadvantages

that DoD’s plan placed on VSI recipients, the

FY 1993 Defense Authorization Act autho-

rized VSI recipients the same separation

beneﬁt package as that provided to SSB re-

cipients. Therefore, for those separating after

the start of FY 1993, any choice of annuity/

lump-sum programs would be based upon the

ﬁnancial characteristics of the two choices.

II. Past Studies of the Personal Discount Rate

Over the past two decades, economists and

psychologists have devoted considerable effort

to the measurement of the personal discount rate

(D).

Experimental studies (see Richard H.

Thaler, 1981; Matthew Black, 1984; Uri Ben-

zion et al., 1989) have confronted subjects (of-

ten college students) with hypothetical

intertemporal choices and estimated D from the

choices. Nonexperimental studies (Harry J. Gil-

man, 1976; Jerry A. Hausman, 1979; Dermot

Gately, 1980; Steven Cylke et al., 1982; Henry

Ruderman et al., 1986) have attempted to infer

the personal discount rate from actual rather

than experimental choices.

These studies offer three general ﬁndings.

First, individuals do not discount all future val-

ues at the same rate. Experimental studies by

Thaler (1981) and Benzion et al. (1989) have

found D to be higher for hypothetical choices

involving relatively small sums. This result

seems to be corroborated in inferential studies.

Hausman (1979) inferred personal discount

rates by comparing appliance purchases—a ref-

erence model compared with more energy-efﬁ-

cient models. A “break-even” discount rate was

calculated based on capital cost and energy-

saving differences. Hausman estimated an aver-

age personal discount rate of about 25 percent.

Gately (1980) studied purchases of refrigerators

and found even larger discount rates. Studying

purchases of various appliances, Ruderman et

al. (1986) estimated discount rates ranging from

17 to 243 percent depending upon type of ap-

pliance. Ignorance, illiquidity, and shortened

“payback” periods were offered as explanations

for such high rates.

Cylke et al. (1982) examined the differen-

tial impacts of lump-sum and installment bo-

nuses upon the propensity of Navy enlisted

personnel to reenlist and inferred discount

rates of 15 to 18 percent from the stronger

reenlistment effect of lump-sum bonuses. Gil-

man (1976) inferred personal discount rates

from the propensity of employees of four

nonproﬁt organizations to participate in their

Individual Ready Reserve members are not required to

participate in monthly drills, but may be activated during an

emergency.

Note that many FY 1993 separatees may have de

cided to leave in FY 1992. Although separatees were

asked to make a separation beneﬁt choice at the time they

announced their separation decision, DoD permitted

switching between VSI and SSB up to the date of actual

separation.

In the standard neoclassical model of intertemporal

choice, D ⫽ MRTP ⫺ 1, where MRTP is the individual’s

marginal rate of time preference at the utility-maximizing

mix of current and future consumption. D is distinct from

the pure rate of time preference, which equals MRTP ⫺ 1

measured at equal amounts of current and future consump-

tion along the individual’s intertemporal budget constraint

(Emily C. Lawrance, 1991).

36 THE AMERICAN ECONOMIC REVIEW MARCH 2001

organization’s retirement plan. He estimated

personal discount rates ranging between 8.5

and 16.2 percent for one model speciﬁcation

and between 1.3 and 19.6 percent for an-

other.

Black (1984) estimated discount rates

from survey questions about alternative re-

tirement systems for the military. In this sur-

vey, military personnel were asked a series of

questions regarding preferences for alterna-

tive hypothetical military retirement plans.

He estimated an average discount rate of 10.3

percent for ofﬁcers and 12.5 percent for en-

listed personnel. The generally lower esti-

mates of D obtained in the latter three studies

than in the appliance purchase studies may be

due to the fact that the individuals in the latter

studies were making choices over signiﬁ-

cantly larger sums.

A second ﬁnding from earlier studies is that

D varies with the time delay of the reward or

penalty. Individuals appear to discount future

amounts hyperbolically, applying higher dis-

count rates to amounts with a short delay than to

amounts to be received farther into the future

(see, e.g., Thaler, 1981; Benzion et al., 1989;

and the literature review by George Ainslie and

Nick Haslam, 1992).

Third, there is evidence that D varies with in-

come and other personal characteristics. Both Gil-

man and Black found that personal discount rates

decline with income, education, and age. They

also found that blacks have higher discount rates

than whites.

However, results with respect to

gender and marital status were mixed.

Why personal discount rates might be related

to personal attributes is not entirely clear. In the

standard neoclassical model with perfect capital

markets, utility-maximizing agents borrow or

lend until D equals the market rate of interest

for borrowing or lending (R). If different indi-

viduals face different borrowing or lending

rates, D will be correlated with personal at-

tributes to the extent that such attributes affect

R. For example, more-educated, higher-income

individuals may be able to borrow at lower rates

than less-educated, lower-income individuals.

And to the extent that blacks and other minor-

ities face discrimination in credit markets, they

will face higher borrowing rates and therefore

exhibit higher personal discount rates. D might

also decline with age if lenders perceive older

individuals to be better credit risks than younger

ones.

When capital markets are imperfect, per-

sonal attributes could affect personal discount

rates through their inﬂuence on intertemporal

preferences. One such imperfection is the ex-

istence of borrowing constraints. In the pres-

ence of borrowing constraints, individuals

will be unable to expand current consumption

(at the expense of future consumption) to the

point where D equals R. D will exceed R at

the observed mix of current and future con-

sumption.

In such a case, D could be related

to personal attributes such as education be-

cause these factors have directly shaped pref-

erences for current versus future consumption

or because they are related to preferences

through previous choices. For example, more

education might reduce D directly by helping

people better understand intertemporal choice

and delay immediate gratiﬁcation for future

rewards. But education might be negatively

associated with D simply because more pa-

tient individuals acquired more education in

the past.

Identifying why D varies with personal at-

tributes is beyond the scope of this paper. Our

goal is more modest—estimate reduced-form

relationships between D and various personal

attributes and other factors with data from the

drawdown experiment. We now turn to our

model and the empirical analysis.

Two of the organizations were the Center for Naval

Analyses and the University of Rochester. The other two

could not be disclosed for conﬁdentiality reasons.

Lawrance (1991) estimated pure rates of time prefer

ence from Panel Survey of Income Dynamics data on food

consumption. She found the pure rate of time preference to

vary with education, age, and race. Estimates ranged from

12 percent for college-educated whites in the top 5 percent

of the income distribution to 19 percent for nonwhites

without a college education in the bottom ﬁfth of the income

distribution.

Stephen P. Zeldes (1989) ﬁnds evidence that those

with low wealth are more likely to be liquidity con-

strained than those with more wealth. If those with low

wealth are more likely to be liquidity constrained, a

ﬁnding that they have higher personal discount rates than

more wealthy individuals could reﬂect differences in

preferences for current versus future consumption as well

as differences in the market interest rates the two groups

face.

37VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

III. Analysis of the Drawdown Data

A. Probit Model of the Separation Payment

Choice

Our statistical procedure was suggested in

the discount-rate studies by Gilman and

Black, although they did not or could not fully

implement the procedure. Orley Ashenfelter

(1983) employed the same procedure to study

program participation and earnings in the

Negative Income Tax experiments of the

1970’s. Let D* denote the break-even dis-

count rate, i.e., the rate that equates the after-

tax value of the SSB lump sum with the

after-tax present value of the VSI annuity.

Individuals choose the lump sum if D ⬎ D*

and the annuity otherwise. The break-even

rate D* is the discount rate such that

(1a) 共1 ⫺ t

兲SSB

⫽⌺

t ⫽ 0

2YOS⫺ 1

共1 ⫺ t

兲A共1 ⫹ D*兲

⫺t

where SSB is the lump-sum payment, t

is the

tax rate applied to the SSB, A is the annuity

amount, and t

is the tax rate applied to the

annuity amounts. But since SSB ⫽ 0.15 ⫻

YOS ⫻ ﬁnal basic pay and A ⫽ 0.025 ⫻

YOS ⫻ ﬁnal basic pay, the break-even discount

rate solves

(1b) 6

共1 ⫺ t

兲

共1 ⫺ t

兲

⫽ ⌺

t ⫽ 0

2YOS⫺ 1

共1 ⫹ D*兲

⫺t

D* was calculated for each individual in

the sample. To do so, we estimated the mar-

ginal tax liability associated with each choice

using the tax tables in effect at the time of

separation and assuming that individuals take

the standard deduction based on their family

size. The tax rates in equations (1a) and (1b)

are ratios of the respective tax liabilities to

separation payment. Calculations were made

assuming that the same marginal tax bracket

would be maintained after separation. Earn-

ings were imputed to the spouses of married

personnel based on results of a 1992 survey of

earnings of military spouses. The average

marginal tax rate on annuity income was cal-

culated to be 16.8 percent while the average

tax rate on the lump sum was calculated to be

19.9 percent.

Because the payout period rises with YOS,

D* rises with the military member’s years of

service. It also varies with the marginal tax rates

to which the two payments would be subject.

D* increases with the difference between the

two tax rates, as is the case with individuals

with greater income and for single personnel.

The sample average value of D* is 19.8 percent.

Individuals choose the SSB if D ⬎ D* and

VSI otherwise. To model the VSI/SSB choice

formally, let D be a linear function of the ob-

served characteristics of the individual (X) and

random error (␧)

(2) D ⫽ X

␤

⫹␧

where

␤

is a parameter vector. We assume

that ␧ ⬃ N(0,

␴

). Of course, D cannot be

directly observed. But it can be estimated as

follows. Since individuals select the lump-

sum SSB payment if D ⬎ D*, they choose

SSB if X

␤

⫹ ␧ ⬎ D*orX

␤

⫺ D* ⬎⫺␧.

The probability of choosing the lump-sum

payment is given by:

(3) P共SSB兲 ⫽ P共X

␤

⫺ D* ⬎ ⫺␧兲

⫽ P

冉

␤

␴

⫺

␴

D* ⬎ ⫺

␧

␴

冊

⫽ P共X

␦

⫺

␣

D* ⬎

␰

兲

⫽ ⌽共X

␦

⫺

␣

D*兲

where

␦

⫽

␤

␴

␣

⫽ 1/

␴

␰

⫽⫺␧/

␴

, and ⌽⵺

denotes standard normal distribution function.

Since

␰

is standard normal, equation (3) is an

ordinary probit model. Unlike usual probit mod-

els, in which parameters of equations underly-

ing the choice [e.g., equation (2)] can only be

estimated up to a scale factor, the underlying

Most enlisted personnel were in the 15-percent federal

tax bracket and would stay there whether they took the

annuity or the lump sum. The more highly paid ofﬁcers

were in higher marginal brackets and for them the marginal

tax rate on the lump sum was higher.

38 THE AMERICAN ECONOMIC REVIEW MARCH 2001

parameters

␤

and

␴

are identiﬁed from esti-

mates of

␦

and

␣

A disadvantage of the linear formulation is

that it permits individuals to have negative dis-

count rates. A lognormal model would restrict

D to be strictly positive. Let D ⫽

exp{X

␤

}*exp(␧) where ␧ ⬃ N(0,

␴

). The

expected discount rate is E(D) ⫽ exp{X

␤

⫹

0.5

␴

} and the standard deviation is

E(D)(exp{

␴

} ⫺ 1)

1/2

. As before, individuals

choose the lump sum if D ⬎ D*, or in this case

␤

⫹ ␧ ⬎ ln D*. In addition to the imposition

of strictly positive discount rates, the lognormal

model differs from the linear model in that the

effect on E(D) of changes in X is not constant

but depends on the level of X.

B. Regressors

Section II discussed reasons why personal

discount rates might be associated with personal

attributes such as income, education, age, sex,

race, and marital status. Our empirical models

include most of the same variables included in

past studies. One is the military member’s wage

or income (WAGE). In the analysis below,

WAGE is the individual’s total taxable military

pay plus allowances in the calendar year pre-

ceding separation. WAGE should be negatively

related to the likelihood of choosing the SSB

and to D for reasons discussed earlier. How-

ever, because military human capital may not be

very transferable to the private sector (Matthew

S. Goldberg and Warner, 1987), the military

wage may not be a very good indicator of an

individual’s permanent private-sector earnings

capacity.

Permanent earnings capacity is probably bet-

ter measured by education, which is also ex-

pected to be negatively related to the likelihood

of choosing the SSB and to D. The ofﬁcer

models presented below include dummies for

graduate education and for college degree. The

enlisted models include dummies for some col-

lege education (or better) and for high-school

degree. Age is included in the model and, due to

the limited age range of the individuals in our

data (about 25 to 35), is expected to be nega-

tively related to the likelihood of choosing the

SSB.

D may be related to mental test score as well

as education. If individuals who score better on

mental tests have higher earning capacity, D

will be negatively related to measures of mental

ability through the effect of test scores on earn-

ings (see, e.g., Derek A. Neal and William R.

Johnson, 1996). In addition, individuals scoring

better on mental tests may be better able to

process information and understand intertempo-

ral choices. To control for test-score effects in

the enlisted models we included three standard

measures based on entry test score—mental

group I [score of 94 or above on the Armed

Forces Qualiﬁcation Test (AFQT)], mental

group II (65–93 on the AFQT), and mental

group IIIA (50–64 on the AFQT). The omitted

group consists of individuals scoring below 50

on the AFQT. Unfortunately, no test-score mea-

sure is available for ofﬁcers.

Other demographic controls include dum-

mies for race (black, white) and sex (female)

and the number of dependents. To the extent

that borrowing opportunities vary by race, or

that race is a proxy for permanent (as opposed

to current) income, the VSI/SSB choice, and

thus the personal discount rate, will vary by

race. We have no priors about the effect of

gender, and previous studies have yielded

mixed results about the relationship between

gender and D.

Although there has been little work on the

relationship between family size and D,a

model of human-capital investment decisions,

fertility, and economic growth by Gary S.

Becker et al. (1990) relies on the assumption of

a positive relationship. In their analysis, parents

who discount the future more highly have more

children and invest less in each one. However,

there is no existing evidence about the relation-

ship between family size and discount rates, so

our estimates are the ﬁrst.

D might vary with one’s military occupation.

One reason is that individuals who discount the

future more highly may be attracted to the

Gilman estimated equation (2) excluding D* and had

to assume a value of

␴

in order to estimate D. He assumed

it to be 0.06. Because D* was correlated with included

regressors, Gilman’s estimates were biased due to the ex-

clusion of D*. In Black’s survey data there was no variation

in D*, so it could not be included as a regressor. He inferred

values of

␴

from (seemingly arbitrary) subsidiary analyses

and found it to be about 0.04 to 0.06. An improvement of

our study is that we can actually estimate

␴

39VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

military skills that offer more up-front pay and

beneﬁts at entry. Such skills also tend to be the

more dangerous (e.g., combat arms), so occu-

pational differences in D might also reﬂect dif-

ferences in attitudes towards risk. To the extent

that military occupation is a surrogate for future

civilian employment opportunities, and, there-

fore, permanent income, we would expect D to

vary with military occupation.

According to several studies cited in Section

II, people discount future beneﬁts hyperboli-

cally. Hyperbolic discounting means that the

discount rate applied to each future amount in

equation (1a) or (1b) will be dependent upon

time and furthermore that the near-term

amounts will be discounted more heavily, and

far-term amounts less heavily, than would be

implied by discounting at a constant rate. It

would be impossible to estimate a whole time

proﬁle of discount rates. But note that because

of the construction of the VSI/SSB program,

people with more years of service who select

the annuity will receive that annuity over a

longer time span and will thus have to wait

longer to receive the “average” dollar of annuity

payment. If hyperbolic discounting is present,

then holding constant age and D*, YOS will

affect D to the extent that it measures the aver-

age delay in receiving annuity payments. If it is

present, D should fall with YOS.

It should be pointed out that the VSI was not

indexed to inﬂation, as is the standard military

retirement annuity. Therefore the discount rates

that we estimate are nominal rates and not real

ones. Even those individuals with very low real

discount rates could have been induced to select

the SSB if they expected high future inﬂation.

Although we cannot observe the inﬂationary

expectations of the individuals in our data, we

doubt that expectations of high or accelerating

future inﬂation were inﬂuencing their choices.

The inﬂation rate averaged about 3 percent dur-

ing 1991–1992 and inﬂation over the period

1988–1992 averaged 4.2 percent. It is likely

that individuals were anticipating inﬂation rates

in the range of 3–4 percent when they were

making their payment choice decisions.

C. Sample Selection Issues

Since the sample of individuals who made

separation payment choices is observed only for

those who actually separated, an analysis that

ignores the separation process may suffer from

the problem of sample selection bias. A priori, it

is not obvious which way the sample selection

bias will go. If the drawdown program had been

strictly voluntary, the selection bias would most

likely be positive. In an environment of volun-

tary retention decisions on the part of service

members, each individual would make a stay-

leave decision based on the expected present

value of future income from a stay decision with

the present value of a leave decision (Thomas

Daula and Robert Mofﬁtt, 1995). Because cur-

rent period pay tends to be lower in the military

than the private sector, and because military

retired pay is only vested after 20 years of

service, military compensation is more heavily

deferred than private-sector compensation. The

system is therefore more attractive to individu-

als with low discount rates. Consequently,

stayers will tend to be individuals with lower

discount rates while leavers will tend to have

higher discount rates. Discount rates estimated

from a sample of leavers would, in an environ-

ment of fully voluntary retention decisions,

overstate the mean discount rate in the whole

military population. That is, the selection bias

would be positive.

On the other hand, retention decisions in the

environment of the military drawdown were not

strictly voluntary. Personnel in the Army and

Air Force who were in skills selected for sepa-

ration payments were told that if separation

rates in those skills were not high enough with

the voluntary payments, they would be subject

to involuntary separation. Navy personnel were

not so threatened.

Furthermore, at the start of

the drawdown, the services imposed up-or-out

Some recent analyses of inﬂation suggests that inﬂa

tion is integrated of order 1 (I(1)) and requires ﬁrst-differ-

encing to be stationary (see, e.g., Myles S. Wallace and

Warner, 1993). Since inﬂation is I(1), if individuals use the

time-series properties of past inﬂation to forecast future

inﬂation, the previous period’s inﬂation rate will eventually

dominate the long-run forecast.

Recall that the SSB payment was set at 1.5 times the

involuntary separation payment. The regret for a person

with 10 years of service who did not separate and was

subsequently involuntarily separated was half a year’s basic

pay (if the person opted for the lump sum).

40 THE AMERICAN ECONOMIC REVIEW MARCH 2001

rules on nonretirement-eligible personnel which

previously did not exist. The Army, for in-

stance, imposed mandatory separation rules on

enlisted personnel in the rank of E-4 who had

more than 8 years of service and E-5’s with

more than 13 years of service. Previously, such

personnel were mandatorily separated only

upon completion of 20 years of service. As

another example, although Captains usually get

two chances at promotion to Major before fac-

ing involuntary separation, Army and Air Force

Captains who had failed their ﬁrst chance at

promotion to Major were told that they had to

leave. In such an environment, where mid-

career personnel are confronted with the sudden

imposition of tighter standards for continuation,

cohorts of separatees are likely to be comprised

of numerous individuals with low discount rates

as well as individuals with high rates.

The problem of selection bias is handled with

a bivariate probit model with partial observabil-

ity (William H. Greene, 1997 p. 912). Let S* ⫽

␥

⫹ u be an index function for the net value

of separation, where Z is a vector of regressors,

u is a standard normal random error, and where

S ⫽ 1 if separation occurs and 0 if it does not.

Estimates of the separation payment choice

equation are biased if the residuals in the sepa-

ration equation (u) and the separation payment

choice equation (␧/

␴

) are correlated. Consistent

maximum-likelihood estimates may be obtained

by estimating the separation and payment

choice equations jointly, allowing for correla-

tion between u and ␧/

␴

D. Summary Statistics

Data for our analysis were provided by the

Defense Manpower Data Center (DMDC). At

the start of the drawdown, DoD instructed the

services to provide DMDC with a report on

each service member’s eligibility on a quarterly

basis along with information about each mem-

ber’s actual separation under the program. Our

data set is a match of (1) the service reports of

eligibility, (2) DMDC’s master ﬁle records con-

taining information about race, sex, education,

rank, years of service, etc., and (3) Joint Uni-

form Military Pay System (JUMPS) records

containing information about each service

member’s military compensation and the sepa-

ration payment actually received. The Air Force

and the Navy complied with reporting require-

ments. Unfortunately, the Army and the Marine

Corps only reported actual separations under

the program and failed to report each member’s

eligibility. We were able to construct a variable

for eligibility of Army personnel based on cri-

teria for eligibility that the Army published and

distributed periodically during the drawdown.

Unfortunately, we could not repeat this proce-

dure for the Marine Corps, which is therefore

deleted from the analysis. But because it is a

small service and had few separatees receiving

separation payments, we do not lose much in-

formation by ignoring the Marine Corps.

Approximately 59,000 ofﬁcers and 235,000

enlisted members in the Army, Navy, and Air

Force were eligible to participate in the pro-

gram in ﬁscal years 1992 and 1993. Table

2 shows the base number eligible in each

service, the number (and percent) separating,

and the number (and percent) selecting the

lump-sum option. Overall, 18.9 percent of

eligible ofﬁcers separated under the program

and 23.5 percent of eligible enlisted personnel

separated. Of those separating, about half of

the ofﬁcers chose the lump sum while over 90

percent of the enlisted personnel did so.

Means of the variables used in the empirical

analysis are provided in Appendix Table A1.

E. Point Estimates of Personal Discount

Rates

Estimates of a single discount rate may be

obtained by estimating a simple probit model

for the sample of separatees with D* as the only

regressor. Three simple probit models are re-

ported in Panel A of Table 3, a model that pools

the ofﬁcer and enlisted separatees together and

models for each group separately. Tax adjust-

ment of break-even discount rates introduces

more variation into D* than exists in the non-

tax-adjusted values, so it is of interest to know

how sensitive the estimated coefﬁcients on D*

(and consequently estimates of

␴

) are to tax

adjustment. Panel A of Table 3 reports models

with tax-unadjusted and tax-adjusted break-

even rates, respectively. The implied personal

discount rate is the product of the intercept and

the inverse of the coefﬁcient on D*.

For interest, Panel B of Table 3 provides

linear probability model estimates of the

41VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

relationship between separation payment choice

and D*. The linear probability model is the

appropriate functional form when personal dis-

count rates are distributed uniformly. The linear

probability model estimates of discount rates in

Panel B are larger than the comparable Panel A

probit estimates. The disparities in the discount-

rate estimates are larger for enlisted personnel

than for ofﬁcers. This outcome was to be ex-

pected because estimates based on a uniform

distribution will not deviate as much from esti-

mates based on a normal distribution when the

sample mean choice rate is 0.51 (ofﬁcer case)

than when the mean choice rate is 0.921 (en-

listed case). Identiﬁcation of the true underlying

distribution would require observation of D,

something we cannot do, but which Ashenfelter

(1983) could for the analogous variable. Al-

though it cannot be tested, the assumption of

normality is inherently more plausible.

We now focus on Panel A of Table 3. In all

models the probability of selecting the lump-

sum payment is negatively related to the break-

even discount rate. Coefﬁcients on D* using the

tax-adjusted values of D* are uniformly

smaller, and estimates of

␴

consequently larger.

In the enlisted case, the tax-adjusted estimate of

␴

(0.039) is almost double the unadjusted esti-

mate (0.022). In the ofﬁcer case, the tax-

adjusted estimate is over three times larger

(0.087 versus 0.025). Tax-unadjusted and tax-

adjusted estimates of the personal discount rate

are similar, 0.191 compared with 0.201 in the

ofﬁcer case and 0.224 compared with 0.256 in

the enlisted case. That the discount rate would

be this high was to be expected given that the

break-even discount rates exceeded 17 percent

and over half of the separatees still selected the

lump sum. Recall that these are nominal rates.

Based on an expected inﬂation rate of 3–4 per-

TABLE 3—SIMPLE PROBIT AND LINEAR PROBABILITY MODELS OF SEPARATION PAYMENT CHOICE AND IMPLIED MEAN

PERSONAL DISCOUNT RATE

Variable

Pooled Ofﬁcer Enlisted

Parameter

estimate

Standard

error

Parameter

estimate

Standard

error

Parameter

estimate

Standard

error

Panel A: Probit Models

1. Without tax adjustment

Intercept 5.320 0.186 7.553 0.308 10.240 0.304

D* ⫺22.222 0.966 ⫺39.530 1.614 ⫺45.638 1.565

Sigma 0.045 0.025 0.022

Log-likelihood ⫺27,688.84 ⫺7,460.74 ⫺14,844.03

Sample size 66,483 11,212 55,271

Mean discount rate 0.239 0.191 0.224

2. With tax adjustment

Intercept 4.549 0.104 2.307 0.212 6.579 0.140

D* ⫺17.531 0.516 ⫺11.463 1.064 ⫺25.681 0.689

Sigma 0.057 0.087 0.039

Log-likelihood ⫺27,372.87 ⫺7,711.03 ⫺14,599.53

Sample size 66,483 11,212 55,271

Mean discount rate 0.260 0.201 0.256

Panel B: Linear Probability Models

1. Without tax adjustment

Intercept 1.777 0.040 3.408 0.115 1.948 0.035

D* ⫺4.823 0.209 ⫺15.225 0.603 ⫺5.340 0.182

0.008 0.054 0.015

Sample size 66,483 11,212 55,271

Mean discount rate 0.368 0.224 0.365

2. With tax adjustment

Intercept 1.631 0.023 1.411 0.212 1.644 0.019

D* ⫺3.922 0.114 ⫺4.529 1.064 ⫺3.640 0.094

0.017 0.010 0.027

Sample size 66,483 11,212 55,271

Mean discount rate 0.416 0.312 0.452

42 THE AMERICAN ECONOMIC REVIEW MARCH 2001

cent, the real rates implied by these estimates

range from about 16 percent to 23 percent.

Estimates of the unobserved heterogeneity in

personal discount rates are clearly sensitive to

tax adjustment of the break-even discount rate.

Which estimates are to be preferred? In the

enlisted case the log-likelihood function value

for the tax-adjusted model is higher than the

log-likelihood value for the tax-unadjusted

model. Unfortunately, in the ofﬁcer case the

reverse is true. Marginal tax rates on the lump-

sum payments deviated more from the marginal

tax rates on the annuities in the case of ofﬁcers

than in the case of enlisted personnel. It is

therefore possible that more measurement error

exists in the tax-adjusted estimates for ofﬁcers

than in the enlisted estimates and that the mea-

surement error has the effect of biasing down-

ward the estimated coefﬁcient on D* and

overstating the unobserved heterogeneity. On

the other hand, tax adjustment seems necessary

because the tax-unadjusted estimates of

␴

are

implausibly low in light of the high mean rates.

There is another reason why the estimates of

␴

reported in Table 3 are too low. D* is corre-

lated with several other observable variables

that we ﬁnd below to affect the personal dis-

count rate (e.g., age and the size of the lump-

sum payment). Inclusion of other variables

reduced the coefﬁcient on D* and increased the

estimates of

␴

. We also found in the enlisted

case that estimates of

␴

were very close when

we included other variables and that the log-

likelihood function values were always higher

for equivalently speciﬁed models with tax ad-

justment. In the ofﬁcer case, estimates of

␴

rose

in both tax-adjusted and tax-unadjusted models,

but did not tend to converge as in the enlisted

case. However, in the case of simple probit

models of ofﬁcer separation beneﬁt selection,

differences in the log-likelihood function value

were virtually eliminated by inclusion of other

variables. Furthermore, in the bivariate probit

analysis of the ofﬁcer data, the models with tax

adjustment had higher log-likelihood values

than models without tax adjustment. The tax-

adjusted estimates seem more plausible than

estimates without adjustment.

F. Bivariate Probit Analysis

1. Separation Equation Estimates.—The spec-

iﬁcation of the Z vector in the separation equation

requires some discussion because identiﬁcation of

the model requires some variables in the separa-

tion equation that do not also appear in the pay-

ment choice equation. As implied by previous

discussion, differences in service separation poli-

cies generated considerable variation in separation

rates across the various service-rank-YOS groups.

It is these policy-induced differences in separation

rates that help identify the model. We capture

these policy effects through a set of service-rank-

YOS interactions in the separation equation. Thus,

the ofﬁcer separation equation includes 23 inter-

action variables between service (Army, Navy,

Air Force), rank (O-3 and O-4), and YOS group-

ings (7–8, 9–10, 11–12, and 13–15).

The en-

listed model contains 43 interactions between

service, rank (E-4, E-5, E-6, and E-7), and YOS

groupings.

Another policy-related separation variable is

whether the individual was at the end of a

contracted term of service (ATETS). As part of

the drawdown, individuals who had not yet

completed an enlistment were permitted to vol-

untarily separate with a payment. But the con-

verse was not true—the services could not force

individuals to leave prior to completion of the

contracted enlistment. However, the services

could refuse to retain individuals who had com-

pleted their contracted service. Thus, those still

under an enlistment contract were afforded pro-

tection against an undesired separation not

available to those whose contracts had expired.

Again, we allow for cross-service differences

by interacting ATETS with service. Along with

the rank-service-YOS interactions, ATETS is

an important identiﬁer of the model because it

clearly belongs in the separation equation, and

there is no theoretical reason to believe that it

should have a direct effect on the separation

payment choice.

Other variables in the separation equations

included dummies for race, sex, education level,

military occupation, geographic region, the

number of dependents, and the military wage.

The enlisted separation equation also included

dummies for mental group. In the case of the

One cell was empty. The Air Force had no O-4s in

YOS 7–8.

In fact, ATETS has virtually 0 coefﬁcients and t-

statistics when included in the separation payment choice

equations.

43VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

ofﬁcer model, we fully interacted these vari-

ables with dummies for service. To the extent

that there exist cross-service differences in the

effects of these variables upon separation, fully

interacting these variables with service is a bet-

ter speciﬁcation of the separation equation. Be-

cause of computer limitations, in the case of the

enlisted model we were only able to interact

service with the military wage. The effects of

other variables were thus constrained to be the

same across services. Experiments with a sub-

sample of the data indicated that the lack of

interaction of these other variables and service

had a negligible effect on the estimated separa-

tion payment choice equation and the estimated

correlation of the residuals in the two equations.

Separation equations are reported in Appendix

B, Tables B1 and B2. The coefﬁcient on a partic-

ular service-rank-YOS combination forms the in-

tercept for that combination. Differences across

services in the application of drawdown policies

are apparent in these intercept estimates. In the

Army, for example, Captains with seven or eight

years of service had a lower probability of sepa-

ration than Captains with more experience. Since

promotion from O-3 to O-4 occurs in the Army at

around the 10- or 11-year mark, the O-3 separa-

tions in the higher YOS groups no doubt con-

tained many who had been previously passed over

for promotion. Among enlisted personnel, there

exist quite different patterns in the probability of

separation in the different service-rank-YOS

groups. Among the lower-ranking groups (E-4–

E-5) the probability of separation is either very ﬂat

or increasing as YOS increases; in the higher-

ranking groups, the probability of separation de-

clines as YOS increases.

Being at the end of a contracted period of

service had a signiﬁcant positive effect on the

separation of Army ofﬁcers, but not of Navy or

Air Force ofﬁcers. The lack of a signiﬁcant

effect was not surprising in the Navy case be-

cause the Navy was not forced to downsize to

the same extent as the Army. The lack of a

signiﬁcant ATETS effect is somewhat puzzling

in the case of Air Force ofﬁcers. In the enlisted

case, the ATETS effects are strongly supportive

of a priori expectations. The probability of sep-

aration is higher for Army and Air Force per-

sonnel whose enlistment contracts expired in

FY 1992–1993 than for personnel whose con-

tracts did not expire in that period. Among

Navy personnel, the estimated ATETS effect is

insigniﬁcant.

Among Army ofﬁcers and Army enlisted per-

sonnel the probability of separation is estimated

to rise with the individual’s military wage. But

in the Navy it is estimated to decline. Among

both Air Force ofﬁcers and enlisted personnel,

the probability of separation was independent of

the military wage. That the probability of sep-

aration actually rises with wage in the Army but

declines in the Navy is consistent with the fact

that separation decisions were voluntary in the

Navy but not in the Army. In the latter service,

the more highly paid may have felt the possi-

bility of future involuntary separation more

acutely.

Demographic differences in the probability

of separation were also apparent. Males have

signiﬁcantly lower probabilities of separation in

all cases. Those with more education generally

have signiﬁcantly lower probabilities of separa-

tion. Having more dependents does not affect

the probability of separation. Among both of-

ﬁcers and enlisted personnel there exist signif-

icant occupational differences in the probability

of separation. In the ofﬁcer equations, the omit-

ted occupation group is tactical operations ofﬁ-

cer. The omitted enlisted group is combat arms.

For each category of personnel, the separation

probability is generally highest in the omitted

occupation group. This result is not surprising

because these are the military-speciﬁc skills that

were most in need of a reduction in force and in

which the threat of involuntary separation in the

event of insufﬁcient voluntary separations was

highest.

2. Ofﬁcer Separation Payment Choice Equa-

tion Estimates.—Table 4 contains probit esti-

mates of the linear and loglinear separation

payment choice equations for ofﬁcers. The neg-

ative of the coefﬁcient on the break-even dis-

count rate D* (or ln D*) is equal to the inverse

of the standard deviation of the unobservable

factors in the equation for D (or ln D). Table

4 reports the estimate of

␴

and its standard

error.

In the linear model, the coefﬁcient on

D* is negative and highly signiﬁcant—the

higher the break-even discount rate the lower

Since

␴

ˆ ⫽ 1/

␣

ˆ, V(

␴

ˆ ) ⫽ (⭸

␴

ˆ /⭸

␣

ˆ )

␣

ˆ ) ⫽

␴

␣

ˆ ).

44 THE AMERICAN ECONOMIC REVIEW MARCH 2001

the probability of selecting the lump-sum pay-

ment. In the linear model the estimated value of

␴

is 0.1697, about double the tax-adjusted esti-

mate reported in Table 3.

The coefﬁcient on ln

D* is also highly signiﬁcant in the loglinear

models. Both estimates indicate that as the

break-even discount rate increased, fewer per-

sonnel chose the lump sum.

The third column under each model in Table

4 shows the effect of each variable on the prob-

ability of selecting the SSB.

The fourth col-

umn shows the estimates of discount-rate

equation parameters (linear model) or the mar-

ginal effect of each regressor on D (loglinear

model). The latter is calculated at the sample

mean value of exp{X

␤

⫹ 0.5

␴

}, which is

each individual’s predicted discount rate in the

The log-likelihood function for this model,

⫺32,774.4, was higher than for a comparable model without

tax adjustment, ⫺32,783.4.

Recall that ⭸P/⭸X

⫽

␤

␾

(z) where

␾

(z) denotes the

standard normal-density function evaluated at z ⫽ X

␦

⫺

␣

D*. Since, for ofﬁcers, the mean SSB choice rate is very

close to 0.5, z is approximately 0 at the sample means and

␾

(0) is 0.4.

TABLE 4—PROBIT EQUATIONS FOR PROBABILITY OF CHOOSING SSB AND IMPLIED PDR EQUATIONS,OFFICERS

Variable

Linear model Loglinear model

Parameter

estimate

Standard

error ⌬P

␤

Parameter

estimate

Standard

error ⌬P ⭸D/⭸X

Intercept 2.3647 0.4109

0.401 ⫺0.7205 0.7311

Male 0.0468 0.0394 0.019 0.008 0.0468 0.0394 0.020 0.009

Black 0.3731 0.0738

0.149 0.063 0.3733 0.0704

0.150 0.069

White ⫺0.0987 0.0636 ⫺0.039 ⫺0.017 ⫺0.0987 0.0636 ⫺0.039 ⫺0.018

Number of dependents 0.1089 0.0123

0.044 0.018 0.1088 0.0122

0.044 0.020

Graduate education ⫺0.4397 0.1330

⫺0.176 ⫺0.075 ⫺0.4391 0.1330

⫺0.173 ⫺0.080

College education ⫺0.1702 0.1287 ⫺0.068 ⫺0.029 ⫺0.1696 0.1286 ⫺0.067 ⫺0.031

Wage ($10K) ⫺0.0057 0.0258 ⫺0.002 ⫺0.001 ⫺0.0057 0.0258 ⫺0.002 ⫺0.001

After-tax lump sum

($10K) ⫺0.3253 0.1019

⫺0.130 ⫺0.055 ⫺0.3280 0.1020

⫺0.127 ⫺0.059

Fiscal year 1992 0.4286 0.0283

0.171 0.073 0.4281 0.0283

0.173 0.080

Age ⫺0.0190 0.0055

⫺0.008 ⫺0.003 ⫺0.0190 0.0055

⫺0.008 ⫺0.004

Years of service 0.0180 0.0456 0.007 0.003 0.0197 0.0456 0.006 0.003

South 0.0422 0.0334 0.017 0.007 0.0421 0.0334 0.017 0.008

West 0.0476 0.0372 0.019 0.008 0.0476 0.0372 0.019 0.009

Midwest ⫺0.0479 0.0437 ⫺0.019 ⫺0.008 ⫺0.0480 0.0437 ⫺0.019 ⫺0.009

Army 0.0773 0.0299

0.031 0.013 0.0778 0.0299

0.031 0.014

Navy 0.2889 0.0733

0.116 0.049 0.2875 0.0733

0.119 0.055

Intelligence 0.1623 0.0501

0.065 0.028 0.1624 0.0500

0.065 0.030

Engineering ⫺0.2036 0.0368

⫺0.081 ⫺0.035 ⫺0.2034 0.0368

⫺0.082 ⫺0.038

Scientist or professional ⫺0.1114 0.0693

⫺0.045 ⫺0.019 ⫺0.1115 0.0693 ⫺0.044 ⫺0.020

Health 0.0453 0.0875 0.018 0.008 0.0455 0.0875 0.019 0.009

Administration ⫺0.0907 0.0457

⫺0.036 ⫺0.015 ⫺0.0906 0.0457

⫺0.037 ⫺0.017

Support ⫺0.0934 0.0414

⫺0.037 ⫺0.016 ⫺0.0934 0.0414

⫺0.038 ⫺0.017

Other ⫺0.2132 0.1498 ⫺0.085 ⫺0.036 ⫺0.2130 0.1498 ⫺0.084 ⫺0.039

D* ⫺5.8930 1.9223

(Sigma) (0.1697) (0.0554

)

ln D* ⫺1.1770 0.3769

(Sigma) (0.8496) (0.2720

)

Rho 0.1449 0.0749

0.1456 0.0748

Bivariate probit sample

size 59,208 59,208

Bivariate probit log-

likelihood ⫺32,774.40 ⫺32,774.21

Signiﬁcant at the 0.01 level.

Signiﬁcant at the 0.05 level.

Signiﬁcant at the 0.10 level.

45VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

loglinear model. For ofﬁcers, the estimated ef-

fect of each regressor on the personal discount

rate is quite similar between the linear and

loglinear models.

Notice that the estimated correlation in the

residuals between the separation and payment

choice equations is about 0.145 in both models

and is signiﬁcant at the 0.05 level. The positive

correlation suggests that those who, for unob-

servable reasons, were more likely to separate

were also more likely to choose the lump sum.

D will be overestimated when the separation

process is ignored.

Many of the demographic variables in-

cluded in the separation payment choice

equations are signiﬁcant at conventional lev-

els of signiﬁcance and have the expected

signs. Compared with other nonwhites, blacks

are more likely to take the SSB; the white

coefﬁcient is negative but not signiﬁcant. The

implied racial difference in D is large—

blacks are estimated to have over a 0.063

higher discount rate than other nonwhites.

The propensity to select the SSB falls with

education level, with the difference most pro-

nounced for ofﬁcers with graduate educations,

who are estimated to have a 0.075 lower dis-

count rate than those ofﬁcers without a college

degree. Ofﬁcers possessing a college degree

have about a 0.03 lower rate. Army and Navy

ofﬁcers have a higher propensity to select the

SSB compared with Air Force ofﬁcers and

somewhat higher implied discount rates.

The propensity to select the SSB rises with

the number of dependents, but declines with

age. Each dependent adds almost 0.02 to the

discount rate; each 10 years of age reduces D

by about 0.03. That D increases with the

number of dependents lends credence to the

assumption by Becker et al. (1990) of a pos-

itive relationship.

There were signiﬁcant occupational differ-

ences in the propensity to select the SSB, with

tactical operations ofﬁcers having a higher pro-

pensity to take the lump sum than individuals in

most other occupation groups. These occupa-

tional differences may reﬂect differences in the

transferability of human capital acquired in the

military to the private sector. There were no

apparent geographic or gender differences in

ofﬁcers’ propensity to select the lump sum.

The two most signiﬁcant determinants of D

in the ofﬁcer analysis are the dummy for FY

1992 and the value of the after-tax lump-sum

payment. The FY 1992 dummy is positive, a

result consistent with the fact that in FY 1992

other aspects of the separation choice were less

generous for annuity recipients. When the dis-

advantages associated with the annuity choice

were eliminated at the beginning of FY 1993, a

signiﬁcantly higher fraction of ofﬁcers began

choosing the annuity. The disadvantages asso-

ciated with the VSI separation package added

the equivalent of about 0.07 to the ofﬁcer dis-

count rate.

The probability of choosing the lump sum

declines sharply with the size of the after-tax

lump-sum amount. This result implies that in-

dividuals do in fact discount larger sums at a

lower rate than smaller sums, with D estimated

to decline by over 0.05 for each $10,000 in-

crease in the lump-sum amount. This result is

consistent with ﬁndings of experimental studies.

However, controlling for the size of the lump-

sum payment, YOS has no effect on the sepa-

ration payment choice. A negative coefﬁcient

on YOS would have been consistent with hy-

perbolic discounting; however we ﬁnd no evi-

dence of it here.

3. Enlisted Separation Payment Choice

Equation Estimates.—Table 5 contains the en-

listed estimates. As was the case in the ofﬁcer

analysis, the estimated coefﬁcients on D* (in the

linear model) and ln D* (in the loglinear model)

In fact, the essential difference between the bivari

ate probit estimates found in Tables 3 and 4 and simple

probit estimates that ignore the selection process is in the

intercept estimate of the separation payment choice equa-

tion. Consider the ofﬁcer models. Simple probit estimates

of the linear- and loglinear-model intercepts are 2.5509

and ⫺0.4834 compared with bivariate probit estimates of

2.3647 and ⫺0.7205, respectively. In the enlisted mod-

els,the simpleprobit estimatesof the linear- and loglinear-

model intercepts are 3.9566 and 0.3570, compared with

the bivariate probit estimates of 3.9732 and ⫺0.0002.

Other estimates are similar in sign, magnitude, and sig-

niﬁcance in the simple and bivariate probit models. The

simple probit models give larger estimates of personal

discount rates than the selectivity corrected estimates

(about 0.03 higher in both the ofﬁcer and enlisted cases).

But estimates of how discount rates vary with personal

characteristics are quite similar between the two proce-

dures. Simple probit estimates are not shown to save

space, but are available upon request.

46 THE AMERICAN ECONOMIC REVIEW MARCH 2001

are negative and highly signiﬁcant. The implied

estimate of

␴

in the linear model is 0.1284 and is

indicative of less unobservable heterogeneity in

personal discount rates than existed in the ofﬁcer

data. The estimated correlation in the residuals of

the separation and payment choice equations is

also lower, only 0.049.

Although enlisted personnel had a much

higher average propensity to select the lump

sum, the propensity to select the lump sum

varies considerably with personal attributes and

in the same direction as in the ofﬁcer analysis.

Again, blacks are estimated to be signiﬁcantly

more likely to take the lump sum than other

nonwhites while whites are signiﬁcantly less

likely. Those with more education are again

found to be less likely to take the lump sum and

to have lower discount rates, as are older per-

sonnel. (The age coefﬁcient is almost the same

as in the linear ofﬁcer model.) Although the

TABLE 5—PROBIT EQUATIONS FOR PROBABILITY OF CHOOSING SSB AND IMPLIED PDR EQUATIONS,ENLISTED PERSONNEL

Variable

Linear model Loglinear model

Parameter

estimate

Standard

error ⌬P

␤

Parameter

estimate

Standard

error ⌬P ⭸D/⭸X

Intercept 3.9732 0.3707

0.510 ⫺0.0002 0.8938

Male 0.0750 0.0254

0.012 0.010 0.0749 0.0254

0.012 0.009

Black 0.2712 0.0390

0.044 0.035 0.2711 0.0390

0.043 0.034

White ⫺0.0639 0.0367

⫺0.010 ⫺0.008 ⫺0.0647 0.0367

⫺0.012 ⫺0.009

Number of dependents 0.0576 0.0081

0.009 0.007 0.0587 0.0079

0.009 0.007

Some college ⫺0.3766 0.0442

⫺0.061 ⫺0.048 ⫺0.3769 0.0442

⫺0.062 ⫺0.049

High-school graduate ⫺0.1171 0.0307

⫺0.019 ⫺0.015 ⫺0.1174 0.0307

⫺0.020 ⫺0.015

Mental group I ⫺0.1217 0.0552

⫺0.020 ⫺0.016 ⫺0.1220 0.0552

⫺0.020 ⫺0.016

Mental group II ⫺0.0462 0.0216

⫺0.008 ⫺0.006 ⫺0.0464 0.0216

⫺0.008 ⫺0.006

Mental group IIIA 0.0190 0.0221 0.003 0.002 0.0168 0.0221 0.003 0.002

Wage ($10K) 0.0058 0.0236 0.001 0.001 0.0040 0.0234 0.000 0.000

After-tax lump sum

($10K) ⫺0.4628 0.0683

⫺0.075 ⫺0.059 ⫺0.4674 0.0681

⫺0.075 ⫺0.058

Fiscal year 1992 0.2385 0.0200

0.039 0.031 0.2384 0.0201

0.039 0.031

Age ⫺0.0199 0.0030

⫺0.003 ⫺0.003 ⫺0.0198 0.0030

⫺0.003 ⫺0.003

Years of service 0.0271 0.0191 0.004 0.003 0.0271 0.0191 0.004 0.003

South 0.1261 0.0215

0.021 0.016 0.1261 0.0215

0.021 0.016

West 0.1042 0.0226

0.017 0.013 0.1044 0.0226

0.017 0.013

Midwest 0.0789 0.0326

0.013 0.010 0.0803 0.0326

0.013 0.010

Army 0.2383 0.0284

0.039 0.031 0.2363 0.0284

0.039 0.030

Navy ⫺0.0012 0.0267 0.000 0.000 ⫺0.0036 0.0266 0.000 0.000

Electronics ⫺0.1399 0.0350

⫺0.023 ⫺0.018 ⫺0.1399 0.0350

⫺0.023 ⫺0.018

Communication ⫺0.0805 0.0392

⫺0.013 ⫺0.010 ⫺0.0804 0.0392

⫺0.014 ⫺0.011

Medical ⫺0.1374 0.0576

⫺0.022 ⫺0.018 ⫺0.1375 0.0576

⫺0.023 ⫺0.018

Other technical ⫺0.1899 0.0540

⫺0.031 ⫺0.024 ⫺0.1897 0.0540

⫺0.031 ⫺0.025

Administration ⫺0.1900 0.0308

⫺0.031 ⫺0.024 ⫺0.1902 0.0308

⫺0.032 ⫺0.025

Electrical/mechanical

equipment repair ⫺0.1134 0.0299

⫺0.018 ⫺0.015 ⫺0.1132 0.0299

⫺0.019 ⫺0.015

Craftsman ⫺0.1555 0.0439

⫺0.025 ⫺0.020 ⫺0.1556 0.0439

⫺0.026 ⫺0.020

Supply ⫺0.1067 0.0369

⫺0.017 ⫺0.014 ⫺0.1068 0.0369

⫺0.018 ⫺0.014

D* ⫺7.7853 2.4133

(Sigma) (0.1284) (0.0397

)

ln D* ⫺1.5082 0.4703

(Sigma) (0.6630) (0.2067

)

Rho 0.0485 0.0251

0.0486 0.0251

Bivariate probit

sample size 234,952 234,952

Bivariate probit log-

likelihood ⫺115,039.7 ⫺115,039.6

Signiﬁcant at the 0.01 level.

Signiﬁcant at the 0.05 level.

Signiﬁcant at the 0.10 level.

47VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

effect of dependents is estimated to be smaller

for enlisted personnel than ofﬁcers, those with

more dependents are still estimated to be more

likely to take the SSB and to have higher dis-

count rates. Although there was no gender dif-

ference among ofﬁcers, male enlisted personnel

are more likely to take the lump sum, and have

higher discount rates, than females.

The test-score effect is the most interesting

new result in the enlisted analysis. Individuals

in the top two mental groups are more likely

than others to select the annuity and to have a

lower discount rate. Higher test scores may

reﬂect better capacity to understand or pro-

cess the information about intertemporal

choices. Combat arms personnel were more

likely to take the SSB and have higher dis-

count rates. Potential reasons for such a result

were discussed above. As was the case in the

ofﬁcer models, the military wage was insig-

niﬁcant in both enlisted models. Signiﬁcant

regional differences exist in the propensity of

enlisted personnel to select the SSB, whereas

none were found for ofﬁcers.

As in the ofﬁcer results, enlisted personnel

separating in FY 1992 were more likely to take

the lump sum. And, the size of the lump-sum

payment is again negative and highly signiﬁ-

cant. Furthermore, YOS has no inﬂuence on the

separation payment choice once the size of the

payment, age, and D* are controlled for.

4. Estimates of Personal Discount Rates

from Bivariate Probit Models.—The models re-

ported in Tables 4 and 5, respectively, were

used to calculate personal discount rates for the

whole force, stayers, and leavers separately, and

the whole force at each YOS. Means were ob-

tained by using the estimated models to predict

each individual’s discount rate and then averag-

ing the predictions. Table 6 provides the esti-

mated mean rates. The FY 1992 dummy was set

to 0 in these calculations to remove the effects

associated with other aspects of the SSB/VSI

choice. The predicted rates are nominal rates.

Real discount rates would be 3–4 percentage

points below the numbers in Table 6.

Several patterns emerge from Table 6. As to

be expected, estimates derived from the loglin-

ear model are higher than from the linear model

and are strictly positive. The mean nominal

discount rate for all ofﬁcers estimated from the

linear model is 0.104 while the mean estimate

from the loglinear model is 0.187. The impor-

tance of controlling for sample selection is ev-

ident in the difference between the estimated

mean rates for stayers and leavers. Among of-

ﬁcers, the estimated discount-rate difference be-

tween stayers and leavers is 0.030 in the linear

model and 0.028 in the loglinear model. Among

enlisted personnel, the estimated difference is

about 0.019 for the linear model and 0.047 for

the loglinear model.

Third, D declines with YOS. Although YOS

itself does not affect D, the decline reﬂects the

combined effect of an increase in the size of the

lump-sum payment and higher age. Among of-

ﬁcers, the linear model predicts a discount rate

of almost 0 by YOS 15, with the implication

that a signiﬁcant number of more senior ofﬁcers

have negative discount rates. By imposing the

restriction of strictly positive discount rates, the

loglinear model avoids this unfortunate predic-

tion. However, both models ﬁt the data equally

well; we cannot distinguish between them on

empirical grounds.

The very high discount rates estimated for

enlisted personnel might seem to be a puzzle.

But remember that enlisted personnel received

lump-sum payments that were only about half

of the average payment to ofﬁcers (Table

1). Furthermore, enlisted personnel have lower

education levels than ofﬁcers and have other

characteristics that would make them more

prone to select the lump sum. To see how much

the difference in personal attributes and the

lump-sum payment difference made to the esti-

mated difference in discount rates, we used the

linear enlisted model to predict the mean en-

TABLE 6—MEAN NOMINAL DISCOUNT RATES

Ofﬁcers Enlisted personnel

Linear

model

Loglinear

model

Linear

model

Loglinear

model

All 0.104 0.187 0.354 0.536

Stayers 0.099 0.182 0.350 0.525

Leavers 0.129 0.210 0.369 0.572

All in YOS:

7 0.205 0.291 0.410 0.714

9 0.159 0.232 0.381 0.607

11 0.111 0.180 0.353 0.527

13 0.046 0.132 0.327 0.459

15 0 0.099 0.294 0.389

48 THE AMERICAN ECONOMIC REVIEW MARCH 2001

listed discount rate when enlisted personnel are

given the same lump-sum payment as received

by ofﬁcers, the same average number of depen-

dents, and the same distributions of sex, race,

and education. Giving enlisted personnel the

same average number of dependents and the

same distribution of personal attributes reduces

the mean discount rate from 0.354 to 0.288.

Giving them the same lump-sum payments as

ofﬁcers reduces their mean discount rate to

0.22. Giving them the same lump-sum payment

and the same demographic characteristics re-

duces their mean rate to 0.173. This mean rate is

to be compared with the mean rate of 0.104 for

all ofﬁcers from the linear ofﬁcer model. Thus,

over half of the estimated mean discount-rate

difference between ofﬁcers and enlisted person-

nel is attributable to observable demographic

differences and lump-sum payment differences

between the two groups. Furthermore, although

we do not observe the mental ability of ofﬁcers,

they would no doubt score better on mental tests

than enlisted personnel, and we know from the

enlisted results that brighter individuals have

lower discount rates.

G. Markets and Policy

We ﬁnd evidence of high personal discount

rates in our sample of military separatees. Read-

ers might remain skeptical that the rates we

estimate are representative of the general pop-

ulation. But other anecdotal evidence points to

high discount rates in subsets of the general

populace. Lawrence M. Ausubel (1991) ﬁnds

that over three-quarters of credit card holders do

not pay their balances monthly but pay interest

at rates often exceeding 18 percent on outstand-

ing balances averaging over $1,000. Another

piece of anecdotal evidence is the development

of a rapidly growing market in the United States

in which ﬁrms buy annuity streams from recip-

ients of divorce and personal-injury settlements,

and other forms of delayed payment. The Wall

Street Journal reports that between 1995 and

1998, J. G. Wentworth and Company, the larg-

est ﬁrm in this market, acquired the rights from

over 7,000 people to over $250 million in de-

ferred payments.

The average discount rate on

these purchases was 21 percent. Individuals

selling their deferred streams to Wentworth and

other such ﬁrms tended to be persons with

claims to relatively small annual payments and

small incomes from other sources.

Clearly, the drawdown program’s lump-

sum alternative to annuity payments was wel-

fare enhancing and saved the federal

government (and taxpayers) money. The fact

that individuals chose the lump-sum alterna-

tive in spite of DoD’s attempts to discourage

them from doing so indicates that they viewed

themselves as better off with that choice. The

lump-sum alternative saved the federal gov-

ernment a substantial amount. Using the

7-percent discount rate on government bonds

prevailing at the time of the program, we

calculate that if only the annuity alternative

had been available, the present value of the

annuity payments would have been $4.2 bil-

lion. The present value of the actual annuity

payments plus lump-sum payout was $2.5

billion. The lump-sum alternative thus saved

the federal government $1.7 billion.

IV. Conclusions

The size of the personal discount rate has been

the subject of numerous studies. But past studies

have not been entirely convincing because esti-

mates have for the most part been based on hy-

pothetical choices or choices involving relatively

small sums, or were inferred from other behavior.

What makes this study interesting, and we think

more convincing, are the data: the military draw-

down with its accompanying separation programs

has provided a large-scale natural experiment in-

volving large numbers of individuals making

choices over substantial sums of money, individ-

uals whose earnings, education levels, and other

attributes are representative of the wide spectrum

of American society. We ﬁnd signiﬁcant demo-

graphic variation in discount rates in directions

suggested by theory and by recent experimental

work. We ﬁnd relatively high discount rates

among military personnel, especially the enlisted

The Wall Street Journal (February 25, 1998).

Another ﬁrm buying such income streams, the Peach

tree Settlement Company of Atlanta, GA, advertises almost

nightly on the Prevue cable channel. It explicitly tells view-

ers to convert their small claims into one large lump-sum

payment.

49VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

personnel. But we also ﬁnd discount rates to vary

inversely with the size of the lump-sum payment

such that individuals place low discount rates on

large sums. Our estimates are similar to discount

rates in the rapidly developing market for pur-

chases of deferred income streams.

Finally, it is useful to mention that, like the

military, large corporations downsizing in the

early 1990’s also offered separatees the opportu-

nity to take lump-sum buyouts of their accumu-

lated pension beneﬁts. While systematic data on

the terms of the buyouts and the implied break-

even discount rates are not available, one anec-

dotal report indicates that a high percentage of

private-sector separatees is selecting lump-sum

buyouts in lieu of deferred annuities.

When they

become available, analysis of private-sector data

will be another interesting avenue of future re-

search on personal discount rates.

See The Wall Street Journal (July 31, 1995).

APPENDIX A

TABLE A1—SAMPLE MEANS

Variable Ofﬁcer Enlisted

Male 0.855 0.887

White 0.863 0.656

Black 0.105 0.295

Wage $43,901 $21,563

After-tax lump sum $42,091 $23,507

Graduate education 0.417

College degree 0.573

Some college 0.094

High-school graduate 0.844

Dependents 3.04 3.12

Age 33.9 30.9

YOS 10.9 11.0

ATETS 0.952 0.417

Mental group I 0.024

Mental group II 0.296

Mental group IIIA 0.238

Tactical operations 0.447

Intelligence 0.067

Engineering 0.169

Science 0.065

Health 0.029

Administration 0.099

Support 0.110

Other 0.014

Combat arms 0.140

Electronics 0.126

Communication 0.071

Medical 0.023

Other technical 0.032

Administration 0.219

Electrical/mechanical equipment repair 0.266

Craftsman 0.044

Supply 0.079

50 THE AMERICAN ECONOMIC REVIEW MARCH 2001

APPENDIX B

TABLE B1—PROBIT EQUATION FOR PROBABILITY OF OFFICER SEPARATION (BASED ON LINEAR MODEL)

Variable

Army Navy Air Force

Parameter

estimate

Standard

error

Parameter

estimate

Standard

error

Parameter

estimate

Standard

error

Black 0.040 0.048 0.050 0.174 ⫺0.150 0.058

White ⫺0.115 0.044

0.126 0.148 ⫺0.119 0.058

Male ⫺0.274 0.030

⫺0.126 0.097 ⫺0.105 0.032

College graduate 0.139 0.097 ⫺0.282 0.094

⫺0.046 0.227

Graduate education ⫺0.040 0.099 ⫺0.405 0.099

⫺0.520 0.227

Dependents 0.006 0.006 ⫺0.035 0.018

⫺0.011 0.007

Wage 0.105 0.0181

⫺0.186 0.049

⫺0.048 0.020

O-3, YOS 7–8 ⫺1.954 0.314

⫺0.143 0.328 ⫺0.049 0.250

O-3, YOS 9–10 ⫺1.683 0.315

⫺0.396 0.336 ⫺0.188 0.252

O-3, YOS 11–12 ⫺1.740 0.317

⫺0.637 0.354 ⫺0.089 0.254

O-3, YOS 13–15 ⫺1.813 0.318

⫺0.696 0.356

⫺0.244 0.256

O-4, YOS 7–8 ⫺2.021 0.413

⫺0.989 0.514

O-4, YOS 9–10 ⫺2.349 0.354

⫺1.297 0.427

⫺0.935 0.387

O-4, YOS 11–12 ⫺2.540 0.322

⫺0.945 0.357

⫺0.786 0.263

O-4, YOS 13–15 ⫺2.360 0.319

⫺0.972 0.369

⫺0.815 0.260

Intelligence ⫺0.083 0.033

⫺0.043 0.141 0.227 0.045

Engineering 0.008 0.029 ⫺0.526 0.096

0.152 0.029

Science ⫺0.467 0.051

⫺0.330 0.129

0.100 0.043

Health ⫺0.405 0.046

0.370 0.010

Administration ⫺0.088 0.033

⫺0.216 0.104

0.255 0.038

Support ⫺0.008 0.028 ⫺0.225 0.084

0.159 0.038

Other ⫺0.358 0.068

⫺0.558 0.151

⫺0.445 0.465

South 0.002 0.021 0.322 0.061

0.040 0.033

Midwest ⫺0.062 0.032

0.561 0.144

0.012 0.036

West ⫺0.011 0.027 0.028 0.063 0.058 0.032

At end of term of service 1.106 0.289

0.171 0.189 ⫺0.032 0.033

Signiﬁcant at the 0.01 level.

Signiﬁcant at the 0.05 level.

Signiﬁcant at the 0.10 level.

ABLE B2—PROBIT EQUATION FOR PROBABILITY OF ENLISTED SEPARATION (BASED ON LINEAR MODEL)

Variable

Army Navy Air Force

Parameter

estimate

Standard

error

Parameter

estimate

Standard

error

Parameter

estimate

Standard

error

Black 0.157 0.014

0.157 0.014

White 0.358 0.013

0.358 0.013

Male ⫺0.057 0.010

⫺0.057 0.010

Some college education ⫺0.178 0.017

⫺0.178 0.017

High-school graduate ⫺0.067 0.012

⫺0.067 0.012

Mental group I 0.052 0.022

0.052 0.022

Mental group II 0.053 0.008

0.053 0.008

Mental group IIIA 0.053 0.008

0.053 0.008

Dependents 0.024 0.002

0.024 0.002

Wage 0.478 0.013

⫺0.219 0.023

0.021 0.013

E-4, YOS 7–8 0.595 0.200

⫺1.524 0.036

E-4, YOS 9–10 ⫺0.060 0.035 ⫺0.273 0.161 ⫺0.089 0.039

E-4, YOS 11–12 0.119 0.074 ⫺0.297 0.325 ⫺0.078 0.104

E-4, YOS 13–15 ⫺0.159 0.136 ⫺0.306 0.423 ⫺0.188 0.163

E-5, YOS 7–8 ⫺3.073 0.035

0.584 0.059

⫺1.641 0.036

E-5, YOS 9–10 ⫺1.651 0.030

⫺0.428 0.055

⫺0.984 0.036

51VOL. 91 NO. 1 WARNER AND PLEETER: PERSONAL DISCOUNT RATES AND THE MILITARY

TABLE B2—Continued

Variable

Army Navy Air Force

Parameter

estimate

Standard

error

Parameter

estimate

Standard

error

Parameter

estimate

Standard

error

E-5, YOS 11–12 ⫺0.989 0.031

⫺0.581 0.057

⫺1.005 0.038

E-5, YOS 13–15 ⫺0.970 0.032

⫺0.544 0.060

⫺1.093 0.038

E-6, YOS 7–8 ⫺3.651 0.068

0.597 0.069

⫺2.036 0.210

E-6, YOS 9–10 ⫺2.526 0.035

⫺0.420 0.060

⫺1.895 0.063

E-6, YOS 11–12 ⫺2.335 0.034

⫺0.603 0.061

⫺2.055 0.047

E-6, YOS 13–15 ⫺2.319 0.033

⫺0.671 0.062

⫺2.108 0.042

E-7, YOS 7–8 ⫺2.522 0.426

E-7, YOS 9–10 ⫺2.851 0.131

E-7, YOS 11–12 ⫺2.822 0.064

0.791 0.146

⫺2.222 0.110

E-7, YOS 13–15 ⫺2.385 0.038

1.351 0.108

⫺2.385 0.057

Electronics ⫺0.122 0.013

⫺0.122 0.013

Communication ⫺0.065 0.015

⫺0.065 0.015

Medical ⫺0.203 0.022

⫺0.203 0.022

Other technical ⫺0.157 0.020

⫺0.157 0.020

Administration 0.002 0.011 0.002 0.011 0.002 0.011

Electrical/mechanical equipment

repair

⫺0.107 0.109

Craftsman 0.103 0.017

0.103 0.017

Supply 0.141 0.014

0.141 0.014

South 0.040 0.008

0.040 0.008

Midwest 0.127 0.026

0.127 0.026

West 0.082 0.017

0.082 0.017

At end of term of service 0.280 0.012

⫺0.050 0.015 0.184 0.009

Signiﬁcant at the 0.01 level.

Signiﬁcant at the 0.05 level.

Signiﬁcant at the 0.10 level.

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