Entrepreneurship Does Pay

December 2013

Abstract

Existing evidence suggests that returns to entrepreneurship are low relative to

the returns to wage work. These findings have been associated with non-pecuniary

benefits, and more generally with heterogeneity in preferences, rationality or be-

liefs. In this paper I challenge this view. I extend the data and show that the

differential in earnings is in fact U-shaped, with entrepreneurs earning more than

wage workers both early and late in their tenure within a firm or business. I argue

that the difference in the earnings profile can be rationalized in the context of a

life-cycle model of occupational choice where agents are fully informed, face no un-

certainty, are fully rational, and ex-ante identical. I show that the lifetime returns

to entrepreneurship can in fact be identical to those of wage work.

1

1 Introduction

One surprising result in the literature on entrepreneurship is that median entrepreneurs

fare worse than wage-workers. In fact, the median entrepreneur is “poor” relative to his

counterparts in the labor force. Hamilton (2000), for example, focuses on entrepreneurs

and workers who have between 10 and 35 years of labor market experience. Using the 1984

Survey of and Program Participation he shows that for all those individuals, the predicted

median wage worker’s hourly wage is higher than that of the median entrepreneur.1 Other

studies have looked at a subset of entrepreneurs and have reached similar conclusions: the

payoff to becoming an entrepreneur is low relative to wage work.2 This is even more of a

surprise given the fact that entrepreneurs tend to be more educated than wage workers.

The disparity in earnings is such a puzzle that so far models trying to explain it

have relied mainly on heterogeneity between entrepreneurs and workers, for example in

their preferences, rationality or beliefs.3 While some of these explanations have been

considered sensible, a re-examination of the data calls them into question.

In this paper, I start by revisiting the empirical evidence and show that the median

earnings differential between entrepreneurs and wage workers is in fact U-shaped as a

function of time over their tenures within a firm or business. Entrepreneurs earn more

than workers both early and late in their tenures. Previous empirical evidence focused

mainly on the middle period of individuals’ lives, leading to the incorrect finding that

entrepreneurs earn less than wage workers. Thus, the puzzle needs to be redefined: the

puzzle is not that entrepreneurs earn less than wage workers, but rather that the earnings

differential is U-shaped. As in Hamilton (2000), the empirical evidence is on wages and

does not take into account returns on equity invested in the company (in the form of

1In 1989, entrepreneurs made up 9% of the labor force and although they held 39% of its wealth,the distribution of wealth among entrepreneurs was heavily skewed. Thus, the conventional view of ahigh-tech entrepreneur who may struggle early on to make it big after a few years is not representative(see Gentry and Hubbard, 2004).

2For example, Hall and Woodward (2010) study entrepreneurs backed by venture capital for exampleand look at exit value and also find that the return to entrepreneurship is low.

3See Section 1.1.

2

dividends for example). For entrepreneurs, this can be a source of income. What I show

in this paper, is that even without taking such payments into account, entrepreneurs do

not necessarily fare less well than wage workers. Indeed, one direct implication of the

U-shape differential in earnings is that the evidence is not necessarily in favor of workers:

since future earnings are discounted, the fact that entrepreneurs earn less than wage

workers in the middle of their lives can be compensated by higher earnings earlier on.

I then formulate a theory to account for these facts. Contrary to previous work, I

do not rely on ex-ante heterogeneity between workers and entrepreneurs, irrationality

or uncertainty of any kind. In the model, all agents are identical at the start of their

lives and choose to either become an entrepreneur or to become a worker that will work

for an entrepreneur.4 What happens to agents after their choice depends, however, on

which occupation they selected. Ex-post, agents face different opportunities. For workers,

consumption decisions do not affect their labor income: they will, in each period, earn

the same wage. This is not true of entrepreneurs. Indeed, an entrepreneur faces a

tradeoff between paying himself and consuming on the one hand, and re-investing the

firm’s output back into the firm on the other. Thus, how much an entrepreneur decides

to pay himself at time t affects the firm’s output at time t+ 1, and therefore also affects

his future labor income. In other words, entrepreneurs have control over the production

process and their labor income stream while workers do not. So, in this model, workers

and entrepreneurs are fundamentally different in a single dimension: the former group

is committed to a wage while the latter has the ability to determine it.5 This does not

result from any ex-ante difference. It results from the essence of what it means to be an

entrepreneur versus a worker.

The shape of the difference in labor income between entrepreneurs and workers is

driven by their rate of time preferences. I do not restrict the discounting to be exponential;

4At this time, agents compare their utility from being wage workers with their utility from becomingan entrepreneur. Because agents are identical, this implies that the values of becoming an entrepreneuror a worker are the same, so they are indifferent between occupations. Selection into either occupationis only pinned down by a labor market equilibrium condition.

5In each period, an entrepreneur produces, decides how much to pay himself and how much to investfor next period’s production. Workers, on the other hand simply consume their wage.

3

I instead use a more general class of discounting: quasi-geometric discounting.

With quasi-geometric preferences, agents can discount more between the present and

the next period, compared with how much agents discount between any two other con-

secutive dates. Essentially, if an exponential discounter discounts periods 1, δ, δ2, δ3, ...

, the quasi-geometric discounter will weight future periods using the discount rates 1, βδ,

βδ2, βδ3, ... So, a quasi-geometric discounter will discount tomorrow with a weight βδ

relative to today. However, the factor by which he discounts periods 2 and 3 relative to

period 1 is simply δ. I assume that agents are “sophisticated” (as opposed to “naive”) in

that they know themselves perfectly and are aware of the present bias in their preferences.

I obtain a closed form solution for an entrepreneur’s decisions, both for the case wherein

agents have a commitment technology and make all their decisions at time zero, as well

as the case wherein they do not. In the no-commitment case agents cannot commit to

their time-0 decisions, and one may think of agents having different “selves:” the decision

an agent makes is one that solves the subgame perfect equilibrium.6

Quasi-geometric discounting affects entrepreneurs and workers asymmetrically. A

paid employee is committed to a constant wage. So while his preferences would affect

other investment decisions, they do not affect his labor income. This is not the situation

of an entrepreneur since he controls his means of production and therefore his labor

income. The intuition behind the U-shape pattern is the following:

• A relatively high discount rate pushes the entrepreneur to shift production towards

the present, reinvest less and consume early. This explains why entrepreneurs’ labor

income is higher than that of workers in the early periods.

• By doing this, entrepreneurs consume their means of production and impoverish

their future selves. So, while an entrepreneur enjoys higher income early in his life,

in the middle run, his situation will deteriorate relative to workers.

• Towards the end of his life, the entrepreneur will see his relative situation improve.

6See the Additional Literature part in this section for more on quasi-geometric preferences.

4

Indeed, in the last period of his life, the entrepreneur will liquidate his firm. The

agents’ preference for consumption smoothing will make them spread this higher

consumption over a few periods preceding the last one. If the liquidation value is

high enough, entrepreneurs will earn more than wage workers.

So, while the non-age adjusted average median wage of entrepreneurs is less than that of

wage workers, it is precisely because they are able to shift their wages towards the early

periods that the decision to become an entrepreneur can be rationalized.

Quantitatively, for the difference to be U-shaped, assuming exponential discounting,

the yearly discount rate δ would have to be lower than 0.65. Indeed, for agents to be

indifferent between a path of labor income that is constant (equal to the equilibrium

wage) or relatively high early on and then very low for many periods, agents would have

to be very impatient. While a yearly discount rate δ of 0.65 seems implausible if agents

discount periods in an exponential way, it is much more reasonable if we allow agents to

have quasi-geometric discounting, as discussed in Section 5.

This paper is, to the best of my knowledge, the first one to empirically analyze the

entire life-cycle of workers and entrepreneurs, and to provide a parsimonious model that

is also consistent with the empirical data. The rest of the paper is organized as follows.

In Section 2, I present the data using the Survey of Income and Program Participation

for the year 2001, extending the empirical evidence to include the entire life-cyle profile

of individuals. The model is in Section 3. In Section 5, I present the numerical results

and estimates of the model. In Section 6, I discuss other possible explanations. Section

7 concludes.

1.1 Related Literature

The returns to entrepreneurship puzzle is one that goes beyond the simple comparison

of wage earnings. Entrepreneurs’ earnings are more volatile than those of wage workers

5

they experience a significant increase in income risk.7,8

I am not the first to call into question the “low” returns to entrepreneurship. Very

recently, Manso (2013) shows that if one takes into account the option value of self-

employment, the returns to entrepreneurship are more attractive than what previous

studies have shown. Levine and Rubinstein (2013) show that there is heterogeneity among

entrepreneurs and that those who are incorporated can earn more per hour than their

salaried counterparts. Other studies that use consumption data show that entrepreneurs

have consumption levels that are at par or even above those of wage workers.9 Finally,

there is a large question as to the quality of self-reported income data, particularly for

the self-employed, as documented in Astebro and Chen (2013).10

A related puzzle in the entrepreneurship literature is the “private equity premium

puzzle:” Moskowitz and Vissing-Jorgensen (2002) use the Survey of Consumer Finances

from 1989 to 1998 and show that entrepreneurs’ returns in their private business are

similar to returns of public businesses. Given Hamilton’s (2000) findings and the fact that

entrepreneurs concentrate more than 70% of their wealth in their business, the decision

to become an entrepreneur is even more puzzling.11 Moskowitz and Vissing-Jorgensen

(2000) suggest that non-pecuniary benefits and heterogeneity between entrepreneurs and

workers, for example in their rationality, preferences or beliefs account for this.12

Regarding quasi-geometric preferences, there is vast experimental evidence that peo-

ple exhibit discounting may not be not exponential. Indeed, in laboratory and field

experiments, agents tend to discount the current and next period differently than any

7See Evans and Leighton (1989), Hamilton (2000), Hyytinen and Rouvinen (2013) for example.8See Parker (2009) for an excellent review on the topic of entrepreneurship.9See for example, Hurst et al (2013), Sarada (2013) and earlier work by Pissarides and Weber (1989).

10I return to this issue in Section 6.11It has been established that entrepreneurs tend to hold poorly-diversified portfolios. See Gentry and

Hubbard (2004), Moskowitz and Vissing-Jorgensen (2002) and Heaton and Lucas (2000).12Hopenhayn and Vereshchagina (2009) develop a partial equilibrium model of “endogenous risk tak-

ing that explains why self-financed entrepreneurs may want to invest in risky projects offering no riskpremium.” Their finding is explained by the fact that entrepreneurs have an “outside” option (wagework) and so can take on riskier projects. Kihlstrom and Laffont (1979) say that entrepreneurs have alower degree of risk aversion and so do not require a risk-premium. For explanations relying on optimismsee Landier and Thesmar (2009) and Puri and Robinson (2005).

6

two other consecutive future periods. This matches many field and lab experiments that

find that discount rates are larger in the short run than in the long run.13 There has

also been non-experimental evidence that supports quasi-geometric discounting, for ex-

ample in savings/borrowing decisions.14 Nonetheless, the assumption of quasi-geometric

discounting is not uncontroversial.15

2 Empirical Evidence

The data presented here is from the first wave of the 2001 Survey of Income and Program

Participation.16 Following Hamilton (2000), women were dropped from the analysis to

avoid labor force participation issues. An individual is called an “entrepreneur” if he

is self-employed, has had to invest in his business, and if their business is non-casual

(the business generates at least $2,500 a year). Both incorporated and non-incorporated

businesses are in the sample. People who are both entrepreneurs and work for a wage

at another job are dropped from the sample.17 Finally, anyone in the farming sector as

well as doctors and lawyers were dropped from the analysis (the former because much of

their earnings may come from subsidies and the latter because as they progress they are

likely to become partners and change from paid-employment to self-employment).

The final sample consists of 21,733 individuals among which 2,261, or roughly 10.4%,

qualify as entrepreneurs, a percentage consistent with previous research.18 Figures 1(a)

and 1(b) show the distribution of tenure in a firm (for the workers) and firm age (for the

entrepreneurs). They are, as expected, heavily skewed to the right, with high turnover

13See Lowenstein and Thaler (1989) for an example of laboratory evidence. See Anslie (1992) for fieldevidence. There is also neurological evidence of this: see Berns, Laibson and Lowenstein (2007). For areview see Frederick, Lowenstein and O’Donoghue (2002).

14See Harris and Laibson (2001)15See Kocherlakota (2001) for example.16The same analysis was conducted using the 1996 and 2005 data sets with similar qualitative results.17This has very little bearing on the results: similar results were found by assigning people to the

occupation that generated the higher earnings or to the occupation in which they worked the mosthours.

18See Evans and Leighton (1989) as an example.

7

0.05

.1.15

.2.25

Density

0 5 10 15 20 25 30 35 40 45Tenure

(a) Density of Workers by Tenure

0.02

.04

.06

.08

.1Density

0 5 10 15 20 25 30 35 40 45Tenure

(b) Density of Entrepreneurs by Tenure

Figure 1: Density of the number of observations by tenure for workers and entrepreneurs.

both in wage work and in self-employment.19

Raw Data

In the SIPP, individuals who held a job were asked were asked how long they had been

in that same job, and individuals who owned a business were asked when the business

was created. Those are the values that correspond to tenure in these graphs. People

who were workers were asked if they had an hourly wage. If so, this number was used.

19While the data contains information on occupational tenure for workers, there is no equivalent tooccupational tenure for an entrepreneur. Running the analysis using occupational or job tenure forworkers yields the same qualitative results.

8

10

15

20

25

30

35

0 10 20 30 40 50Tenure

Worker Entrepreneur

Worker (fitted) Entrepreneur (fitted)

(a) Hourly wages of entrepreneurs and workers bytenure with smoothed fitted lines.

-20

-10

010

20

0 10 20 30 40 50Tenure

Wage Difference (fitted) Wage Difference (data)

Zero

(b) Difference in absolute hourly wages of en-trepreneurs and workers by tenure with smoothed fit-ted lines.

-20

020

40

60

0 10 20 30 40 50Tenure

Percentage Difference (fitted) Zero

(c) Smoothed percentage difference in hourly wagesof entrepreneurs and workers by tenure.

Figure 2: Difference in hourly wages of entrepreneurs and workers by tenure.

9

For those not paid by the hour and for entrepreneurs, monthly income received from the

business or employer was divided by the number of hours worked. The model in this

paper is silent on whether the variable to compare should be hourly earnings or total

working income. The data work was done on both measures20 with similar results.21

Figure 2(a) shows median hourly wages by tenure.22 Both self-employed and salaried

workers see their wages increase with tenure. However, as is clear, for a majority of the

first 10 years of tenure, entrepreneurs earn more than their counterparts in the wage

world.23 Further, the initial wage differential is almost 20% higher for entrepreneurs.

Figure 2(b) shows the difference in wages between workers and entrepreneurs as a func-

tion of tenure, and Figure 2(c) shows the percentage difference in hourly wages between

entrepreneurs and workers as a function of tenure. As is clear from all those graphs,

entrepreneurs’ hourly wages are higher for entrepreneurs relative to workers both early

and late in their tenures within a firm. The opposite is true for levels of tenure between

9 and 39 years.24

To remain consistent with prior literature, in the paper I will use hourly earnings.

The evidence is even more striking if we look at the graphs on total monthly income in

Appendix B: the average income differential is close to 35% in favor of entrepreneurs for

the first 10 years.

Hamilton (2000) showed that the tenure-dependent earnings profile of entrepreneurs

is below that of worker’s wage profile at any level of tenure. This can be seen in Figures

20On average entrepreneurs work more hours per week than workers so the results could have beendifferent if the plot were of income against tenure.

21See Appendix B for the “income” graphs.22This paper focuses on median earnings, as does Hamilton (2000). As Rosen (1981) points out in his

superstar theory, the mean is likely to be greatly influenced by only a handful of individuals. In the data,those superstar earners are more likely to be in the “high tech” business. The average entrepreneur isnot in the high tech business and so it would be incorrect to use mean earnings since that entrepreneurhas no hope to be a “top earner.” In other words, to use the mean, one would have to have rich enoughdata that is industry and occupation specific.

23Further in this same section, a regression controlling for various observables shows that this is truefor roughly the first 12 years.

24All regressions and statistical test were performed on the data using the weights provided by theSIPP data set.

10

2 and 3 of his manuscript. The driving force that explains the difference between our

findings is that the regressions plotted in Figures 2 and 3 in Hamilton (2000) are drawn for

individuals with precisely 10 years of labor market experience who would be just starting

a new business or a new job. In Figure 2 of the present manuscript I use all possible

values of labor market experience. While entrepreneurs who have been in their current job

between 10 and 30 years do in fact earn less than their wage worker counterparts (which

is consistent with the Hamilton (2000)), individuals who are just starting a business earn

more than individuals who are just starting a new job. Restricting the analysis to those

who have exactly 10 year of prior experience masks this important fact. In the last part

of this section, I use the 1984 SIPP panel (the same panel as Hamilton (2000)) and

present further support for the new findings in this paper, showing that the evidence is

not restricted to the 2001 panel.25

The results of a series of weighted median tests (one for each level of tenure) shows

that the interpretation of the previous graphs rest on statistically solid ground.26 So,

the smoothed lines in Figure 2 reflect statistically significant differences between en-

trepreneurs and workers even when looking at the data for each level of tenure indepen-

dently.27

As can be seen from Figure 1, the number of individuals with very high levels of

tenure is rather low for tenure levels beyond 30 years and so one may question an anal-

ysis that use these observations. It is important to note that the main contribution of

the present work is to show that the returns to entrepreneurship can in fact be equal

to those of wage work. The driving force behind this result is not the final “upswing”

of entrepreneurial earnings relative to wage workers, but instead the initial higher earn-

ings. Removing observations of individuals with more than 30 years of tenure does not

change our conclusions: because individuals discount the future, individuals’ earnings 30

25Or for that matter the 1996 or 2005 panels that also show a U-Shaped earnings differential.26The null is that for each level of tenure t, the median of the wages of entrepreneurs equals the median

of the wages of wage workers.27For eight of the first ten years, as well as for the last five, the median wage of entrepreneurs is higher

than that of the wage workers. For all but four of the remaining years of tenure, the converse is true.

11

or more years into the future do not represent a large part of the present discounted value

of lifetime earnings. Thus, what matters for the analysis, and for the conclusion that the

returns to these two occupations can be the same is that entrepreneurs earn more than

wage workers at some point during their tenure.

Quantile Regression

Figure 2 showed hourly wages of entrepreneurs and wage workers by tenure without

controlling for observables. Here we control for observables: Table 2 and Figure 3 describe

the predicted hourly wages for entrepreneurs and workers using a quantile regression at

the median of the following equation: 28

Wit = β0Hit + β1f (Tenureit) + εit,

where Hit is a vector of observed individual characteristics, f(x) = x + x2 + x3, Tenure

is the number of years in a specific job or business, and εit is a random error term. The

individual characteristics used are age, age squared, as well as dummies on whether an

individual received a college degree or higher, race and marital status.29 I also included

health limitations, whether an individual has already retired as well as presence of children

or not, the qualitative results are largely unchanged. Adding potential labor market

experience, which is the experience implied by subtracting years of education from age

as in Hamilton’s work are not significant in the regressions and are not included in these

results.

These results confirm that if we take into account the early and late periods of in-

dividuals’ lives, the differential in earnings is U-shaped, even controlling for observables

such as the ones in the regression described above.30

28I used mean values (by occupation) for all the explanatory variables, so that these graphs representthe predicted wage path of an average individual who is about to start a business and the predicted wagepath of that same average individual if he were instead to start a new job as a hired employee.

29See Table 1 for a full list, description and summary statistics.30It is worth pointing out that while quantitative features such as the levels of the curves as well as

where they intersect exactly obviously depends on the the specification of the regression, in almost all

12

Variable Description Worker Entrepreneur

AgeStart Age when tenure is zero 33.2 40.5Tenure Years in current job or business 8.5 13.5CollegePlus Has at least a college degree .25 .37Married Married, spouse present .58 .73White Race is white .84 .90

Table 1: Summary Statistics

1015

2025

Pred

icte

d H

ourly

Wag

e

0 10 20 30 40 50 60Tenure

Entrepreneurs Workers

Figure 3: Quantile regression at the median. Fitted Lines: Predicted Hourly Wages by Tenurefor Entrepreneurs and Workers.

13

Wage Workers (N=20,154) Entrepreneurs (N=2,280)R2 = 19% R2 = 4.5%

Control Coefficient Coefficient

Tenure .469∗∗∗ .819∗∗∗

Tenure2 -.011∗∗∗ -.044∗∗∗

100 x Tenure3 .016∗∗∗ .069 ∗∗∗

Age .565∗∗∗ .023∗∗∗

100 x Age2 .633∗∗∗ -0.057∗∗∗

CollegePlus 8.49∗∗∗ 11.34∗∗∗

Married 1.96∗∗∗ 2.15 ∗∗∗

White 1.02∗∗∗ 2.79 ∗∗∗

Constant -4.53∗∗∗ 4.18∗∗∗

Table 2: Quantile Regression at the Median

Panel data: evidence on starting wages

The 1984 panel asks an additional question relative to more recent panels. They asks all

responders to report the wage they had when they first started their current job/business.

The 1984 data corroborates the 2001 data in three ways.

1. We now have we have a nationally representative population and all individuals’

starting wages (as opposed to having the starting wages of only those individu-

als who have actually just started their job/business). Figure 4(a) represents the

probability distribution function of the starting wages of workers an entrepreneurs.

Dollars are normalized to 1984 dollars (using the CPI). The vertical lines represent

the medians of those distribution. Here again, starting wages of entrepreneurs are

indeed higher than those of wage workers, and the difference in medians is about

20%, close to what the data from 2001 suggest is the difference median starting

wages.

2. Figure 4(b) summarizes the median responses of starting wages for each level of

tenure.31 This shows that for most years, the median entrepreneur earned more

than the median worker. For example, one can see that entrepreneurs who had 30

specifications, the U-shaped pattern is preserved.31These are nominal amounts.

14

years of tenure in 1984 had a median starting wage of about $3, while the median

starting wage for workers who had 30 years of tenure in 1984 was a little less than

$2. A weighted test of medians examines whether the distribution of hourly wages

for entrepreneurs and workers come from distributions with the same median. The

conclusion from this test that in about 80% of the tenure levels, when the median

entrepreneur started his business, he earned more than the starting median wage

worker. This is evidence that the finding that entrepreneurs earn more than wage

workers early in their tenure is robust across years.

3. Finally, the 1984 data allows us to address what is perhaps the biggest limitation

of the later data. The 2001 cross-section data do not tell us how well individuals

who are currently “at the bottom of the U” (the mid-ternured individuals) fared

when they were at tenure 0. One possible explanation of the U-shape is that

successful entrepreneurs exit and sell their businesses so that as time goes by, the

median entrepreneurs’ earnings will fall below those of wage workers because only

the “bad fruit” will remain in the data. The 1984 data show that this hypothesis

can be rejected. In fact, more than two thirds of entrepreneurs make more than the

median wage workers who started their job at the same time that they started their

business. Figure 4(c) shows the probability distribution function of starting wages

of individuals who in 1984 had between 10 and 35 years of tenure (also normalized to

1984 dollars). In other words, even those entrepreneurs who in 1984 were earning

less than the median wage worker of the corresponding tenure had higher wages

than the median wage worker when they just started their business.

SIPP Verus Other Data Sets32

The SIPP data set is unique in several aspects. It asks individuals if they own a business,

if they are active in that business and whether they have had to make capital investments

in that business. Only those individuals who respond affirmatively to these questions may

32There are also non-US data that allow for the study of entrepreneurs. See Woodruff and Fairlie(2005).

15

0.0

2.0

4.0

6.0

8

0 10 20 30 40Real Starting Wage

Density: Entrepreneurs Density: Workers

(a) Distribution of median hourly wages for the en-tire sample.

02

46

8H

ourly W

age

0 10 20 30 40 50Tenure

Workers Entrepreneurs

(b) Median starting hourly wages by current tenure.

0.0

2.0

4.0

6.0

8

0 10 20 30 40Real Starting Wage

Density: Entrepreneurs Density: Workers

(c) Distribution of median starting hourly wages forthose who in 1984 had betwen 10 and 35 years oftenure.

Figure 4: Differences in starting wages of entrepreneurs and workers, 1984 SIPP.

be counted as entrepreneurs.33 In the CPS, CEX, PSID or even the SCF, the question

defining employment type is whether someone is “self-employed”, which captures many

more individuals.34

Another major difference is that the SIPP asks individuals about their tenure within

their firm. The PSID and SCF also do, but the sample size of a few thousand with fewer

33Along with the other criteria mentioned in Section 2.34For example, a baby-sitter might be counted as self-employed in other data sets, but would not be

considered an entrepreneur in the SIPP. The inclusion of them in data sets changes the composition ofthe self-employed and may lower the average and median returns to self-employment relative to wagework.

16

than 300 self-employed individuals makes an analysis using those data sets quite difficult

for this particular question.35

While the SIPP isn’t specifically tailored to entrepreneurs and therefore doesn’t in-

clude information on important aspects of entrepreneurship (for example on funding),

its sample size and representativeness make it particularly interesting for comparisons

between occupations.

3 Model

Environment

The environment is a dynamic model of overlapping generations with no population

growth. Agents are identical, live for (T+1) periods and have commitment problems in

the absence of a commitment technology. At the start of their lives, agents make an

irreversible decision between becoming entrepreneurs (or managers) who hire workers,

or becoming workers who work for the entrepreneurs. This strong assumption could

be replaced by a weaker one that would introduce a cost of switching occupation that

would increase with time. For example, one could assume that entrepreneurs loose their

traditional labor market skills over time, or find it harder to readapt to wage work after

experiencing “life as an entrepreneur.”36 At the beginning of each period, workers receive

a wage w (per efficiency unit of skill); entrepreneurs receive the production of each worker

and choose how much to pay themselves (and consume) and how much to invest in next

period’s production. Agents have no initial wealth and there are no capital markets.37

35With the SCF, in 5 of the first 8 years of tenure, the median self-employed earns more than themedian worker.

36Bruce and Schuetze (2004) show that returning to wage work after a self-employment spell canactually decrease average hourly earnings. Another possibility is that entrepreneurs delay exiting abusiness because of the emotional cost that failure represents as in Shepherd, Wiklund and Haynie(2009).

37Allowing for hyperbolic discounting complicates the theoretical problem greatly. The assumption ofno capital markets aids in finding the closed form solution to the problem. In the exponential discountingcase (β = 1), allowing for a one-period bond market where the interest rate is endogenously determinedwould make consumption equal for all agents of identical tenure (regardless of occupation). However,the portion of consumption coming from savings versus income would differ across occupations. The

17

Therefore, labor income and consumption are identical.

Preferences

Using the idea introduced by Strotz (1956), Phelps and Pollack (1968), and Laibson

(1997), agents rank streams of consumption according to the following quasi-geometric

preferences:

u(c0) + β

T∑t=1

δtu(ct)

The parameter β represents the commitment level of an agent. If β = 1 we are in the

case of standard exponential preferences and there are no commitment issues: agents do

not deviate from their time-zero decisions. If agents do not have access to a commitment

technology, as β decreases, agents face commitment problems and deviate from time-zero

decisions: the agents are time-inconsistent. Agents are sophisticated and realize that

they will have this commitment issue and make their decisions based on that knowledge.

There are two cases, which I will address in turn: the case where an entrepreneur has

access to a commitment technology and makes all decisions at time 0, and the case where

an entrepreneur does not have such a technology, makes his decisions at each period, and

is time-inconsistent.38

The utility function u(c) is continuously differentiable, increasing and concave.

The Production Process

Output is increasing and proportional to entrepreneurial ability for any level of labor

hired by the firm. Labor is the only factor of production. More precisely:

outputt = f(lt),

differential in labor income would still be U-Shaped for very low discount rates, as is the case withhyperbolic discounting - See Section 5.

38After the time-zero occupational decision, agents make sequential decisions. Agents solve the sub-game perfect equilibrium being aware that they are quasi-geometric discounters.

18

where f(l) is increasing in l.

Decision Variables

Entrepreneurs

Entrepreneurs take w as given and at each date decide how much to pay themselves, how

much labor to hire in the next period. An important feature is that entrepreneurs have

to decide how much to spend on labor one period in advance. If lt+1 is the amount of

labor in the firm at date t + 1, the entrepreneur has to keep wlt+1 aside at date t. In

other words, investment lt equals wlt+1.

A agents who has access to a commitment technology solves the following maximiza-

tion problem at time 0:

maxct,lt+1

u(c0) + βT∑j=1

δju(cj)

subject to:

ct + wlt+1 ≤ f (lt) ∀t

On the other hand, an entrepreneur who doesn’t have access to a commitment tech-

nology solves a maximization problem at every date and deviates from time-zero choices.

At time t, such an entrepreneur would face the following maximization problem:

maxct,lt+1

u(ct) + βT∑

j=t+1

δju(cj)

subject to:

ct + wlt+1 ≤ f (lt)

So, consumption plus spending on next period’s wage bill is financed with output.

In both cases an exception is period 0, where workers are paid contemporaneously. The

19

budget constraint at date 0 is:

c0 + wl0 + wl1 ≤ f(l0)

Given their choice of lt+1, the wage (and consumption) of an entrepreneur at period

t is:

wet = f(lt)− wlt+1

Workers

Workers supply labor inelastically. Because workers don’t face the same decision-problem

as the entrepreneur, i.e., do not have to choose their wages, the workers’ maximization

problem is independent of whether there is commitment or not. They solve:

maxct

u(ct) + βT∑t+1

δt+1u(ct+1)

subject to:

ct ≤ w

and clearly consume their entire wage each period.

Figure 5 shows the interaction between different generations of individuals over their

lives. In the last period entrepreneurs do not have a wage bill for the “next period:” they

liquidate their firm and consume the production entirely.

In a world without a commitment technology an entrepreneur’s labor income path will

have higher variance than in a world with a commitment technology. This is easier to

explain if we imagine a partial equilibrium setting where the wage is fixed. An agent who

cannot commit will, at each period, shift his labor income towards the present and will

deplete his wealth much faster than what that same agent would do if he could commit

to his time-zero decisions. This is because in the model with commitment, the present

20

ProduceTransfer

Consum

e

ReceiveWage

Consum

e

PayProduce

TransferConsum

e

ReceiveWage

Consum

e

ProduceConsum

e

ReceiveWage

Consum

e

Timeline

!w

l1

wl1

wl0

Perio

d0

Perio

d0

<t<

TPerio

dT

wlt+

1

ww

ww

ww

f(l0 )

f(lT

)f(l0 )!

wl0

f(lt )

f(lt )!

wlt+

1f(lT

)

Figure 5: Timeline.

21

bias exists only in the first period and there will be a less drastic decrease in investment

because of it. So, the difference between the two models will be more apparent in the

later part of an agent’s life than the earlier part, and for a given pair (β, δ), an agent

will end his life with a higher income under the commitment case than under the non

commitment case.

Market Clearing Conditions and Selection into Occupations

The total demand for workers in a given period is the sum of the demands from all the

entrepreneurs. Let e be the proportion of agents of a particular generation who choose to

become entrepreneurs. As all agents are identical ex-ante, they are indifferent between

which occupation to select into. The proportion e of agents becoming entrepreneurs is

pinned down by imposing that the labor market clears:39

eT∑t=0

(lt) = (T + 1) (1− e) ,

The Equilibrium

The equilibrium is characterized by a wage w, a stream of consumption for entrepreneurs

{ct}T0 , a stream of consumption for workers {cwt }T0 , and a proportion of entrepreneurs e

such that:

1. Agents’ present discounted value of utility are equal, so that they are indifferent

between becoming entrepreneurs or workers.

2. All decisions are optimal.

3. The labor market clears.

39More generally, if there is a distribution Φ of ability in the population, and z is ability of the marginalmanager (the one who is indifferent between becoming a manager or remaining a worker as in Jovanovic(1994)), we instead have:

Total Demand of workers=∑Tt=0

∫∞zltdΦ(x) = (T + 1)Φ(z).

22

4 Theoretical Results

Heretofore, the production function is f(x) = xα, and the utility is u(x) = log(x).

Theorems 4.1 and 4.2 show what levels of labor an entrepreneur will hire at each period

when he doesn’t and does have access to a commitment technology, respectively. The

expressions for labor investment in the two cases are very similar: the difference being

that in the case with a commitment technology, consecutive periods are discounted by a

factor of δ and so β only appears in the expression between period 0 and period 1.

Appendix A contains the proofs for Theorems 4.1 and 4.2.

Let’s define Sτ :

Sτ =τ∑i=1

αiδi

Theorem 4.1. If σ = 1, for a given wage w and parameters α, β and δ, at date t > 0,

entrepreneurs without a commitment technology will choose a level lt equal to:

lt =

(1

w

)∑t−11 αi [(

l0w

)α− l0

]αt−1

Πt−1j=0

[αβδ (1 + ST−t+j)

(1 + βST−t+j+1)

]αjl0 =w

(wα

) 1α−1

Theorem 4.2. If σ = 1, for a given wage w and parameters α, β and δ, at date t > 0,

entrepreneurs with a commitment technology will choose a level lt equal to:

lt =

(1

w

)∑t−11 αi [(

l0w

)α− l0

]αt−1

Πt−2j=0

[αδ (1 + ST−t+j)

(1 + ST−t+j+1)

]αj [αβδ (1 + ST−1)

1 + βST

]αt−1

l0 =w(wα

) 1α−1

Given the above expressions, we can compare the equilibrium investment levels and

wages under both commitment and non-commitment. This leads to the next two theo-

rems.

23

Theorem 4.3. For a given wage w, investment under commitment is higher than without

commitment.

Proof. Given the expressions from Theorems 4.1 and 4.2, comparing investment amounts

boils down to comparing Πt−2j=0

[αβδ(1+ST−t+j)(1+βST−t+j+1)

]αjand Πt−2

j=0

[αδ(1+ST−t+j)(1+ST−t+j+1)

]αj, which is equiv-

alent to comparing β1+βST−t+j+1

and 11+ST−t+j+1

. Clearly, the former is smaller than the

latter and we conclude that for a given wage w, investment under commitment is higher

than without commitment.

Theorem 4.4. The equilibrium wage under commitment is higher than without commit-

ment.

Proof. With commitment, assuming the same wage and discounts factors, we see that

the amount of labor that is hired in each period is higher than without commitment.

Given that l0 and l1 are the same whether there’s commitment or not, we can see that

the utility under commitment is higher than without commitment (for a same w).40

So, it must be that the equilibrium wage under commitment is higher than without

commitment because of the equilibrium condition that agents are indifferent between the

two occupations. Start with a world without commitment with a given β and δ. Workers

and entrepreneurs have the same PDV of utility. Now take that equilibrium wage, β and

δ and imagine that we are in a world with commitment. Now entrepreneurs have higher

PDV of utility than workers. So for that β and δ it must be that the real equilibrium

wage in this economy is higher so that the PDVs are equal.

So, in a world with a commitment technology, investment, wages and utilities are

higher. In the context of this model, a commitment technology would be one that forces

entrepreneurs to make labor hiring and firing decisions less frequently.41

Theorem 4.5. The equilibrium exists and is unique.

40((x/w)α − x) is increasing in x.41As an example, stricter labor laws could be such a technology.

24

Proof. A worker’s present discounted value of utility is:

[1 + βδ

1− δT1− δ

]log(w),

which is increasing in w and the range is R+.

A entrepreneur’s PDV of utility is clearly a decreasing function of w and for a wage of

zero, is infinity.

Therefore, the two PDVs always intersect, and intersect exactly once. Thus, the

equilibrium wage always exists and is unique.

5 Numerical Results

All simulations were done with α = .842.

Intuitively, the higher the degree of impatience, the less entrepreneurs are going to

delay gratification. Figures 6 and 7 present the combination of β and δ for which an

entrepreneur earns more than a worker in the first period (figures 6(a) and 7(a)), more

than a worker in the last period (figures 6(b) and 7(b)), and the intersection of the two

corresponding to the U-shaped differential (figures 6(c) and 7(c)), for the cases without

and with a commitment technology, respectively. The more opaque regions correspond

to higher percentage differences.

Roughly, the lower the product βδ, the more an entrepreneur transfers his labor

earnings towards the initial period(s). Naturally, the opposite is true if agents are patient.

Similar reasoning applies to the last period: if an agent is patient, he is willing to wait

until later periods to consumption. An extremely impatient entrepreneur will push all

consumption towards the early periods and end with life “poor” relative to wage workers,

whereas an extremely patient entrepreneur will start with a consumption lower than that

42See Atkeson and Kehoe (2005), Atkeson, Khan and Ohanian (1996), Bahk and Gort (1993). Ro-bustness checks for α ∈ [.6, .8] show no change in the qualitative results.

25

(a) (b)

(c)

Figure 6: No Commitment: Combination of β and δ for which the differential in earnings is(a) positive in the first period, (b) positive in the last period, (c) U-shaped over the life-cycle.

26

of wage workers, but will end his life consuming more than his wage worker counterparts.

To observe a U-shape, entrepreneurs and workers have to have “intermediate” discount

rates.

Without a commitment technology, we can see that the combination of β and δ for

which the differential is U-shaped is much wider than for the case without a commitment

technology. The intuition is that for a given level of income at time t, an individual with

a commitment technology will invest more of it in his business than an individual without

a commitment technology. Therefore, he will have a higher labor income stream in the

future and in some circumstances may be able to earn more than a wage worker towards

the end of his life while an entrepreneur without this commitment technology will not.

What is clear from Figures 6(a) through 7(c), is that if we assume standard expo-

nential discounting (β = 1), δ > .9 and standard assumptions we cannot explain the

data: with standard exponential discounting the model predicts that in the early periods

entrepreneurs earn less than wage workers, and then at the end earns more so that the

differential in earnings is monotonic and increasing. This is true whether or not there

is a commitment technology. Interestingly, not only is the new evidence not explained

if we make standard assumptions, but these standard explanations cannot explain the

“old” evidence, leading to the 10-year puzzle of why someone would choose to become an

entrepreneur.

While allowing for quasi-geometric discounting can only produce better results (since

it has an additional degree of freedom), it is not obvious that the resulting discount rates

would be reasonable. However, the range of β and δ that can explain the U-shape pattern

is consistent with prior literature. For example, Laibson, Repetto and Tobacman (2007)

find a pair (β, δ) = (.7, .95), and Fang and Yang (2009) find a product βδ = .86.

27

(a) (b)

(c)

Figure 7: With Commitment: Combination of β and δ for which the differential in earnings is(a) positive in the first period, (b) positive in the last period, (c) U-shaped over the life-cycle.

28

6 Discussion

In this paper, I argue that the puzzle concerning entrepreneurial choice needs redefining.

Namely, the puzzle isn’t that entrepreneurs’ wages are lower than those of wage workers,

but rather that the differential in earnings is U-shaped. While the model I use in this

paper rests on some strong simplifying assumptions, the goal was to provide an expla-

nation that did not rest on heterogeneity of agents. Below I list the main explanations

in the literature that were consistent with the prior conception of the puzzle. However,

while a combination of these explanations could be consistent with the data, none taken

alone can account for the non-linearity of the earnings differential.

The Better Entrepreneurs Sell The Business Early

One possibly pealing explanation could be that the initial higher relative earnings are

driven by successful entrepreneurs who then sell their business early. Once they sell their

businesses they are no longer part of the self-employed category, and so as time goes

by the pool of entrepreneurs is of lower quality and wage workers start to earn more

relative to them. There are two problems with such an explanation. First, the evidence

from the 1984 panel shows that most those entrepreneurs who are “at the bottom of the

U” did earn more than the median wage worker when they first started their business.

Second, the evidence presented in this paper focuses on the median. Very successful

entrepreneurs are few and far between, and so their behavior cannot explain empirical

facts that concentrate on the median.

Entrepreneurs Need to Invest to Grow their Business and so Postpone their Earnings

One can imagine that perhaps the high relative earnings could be driven by entrepreneurs

who under-invest in their business (and hence over-consume). Those entrepreneurs would

then be weeded out because their businesses would be more likely to fail. The remaining

entrepreneurs would then be those who are re-investing efficient amounts back into their

businesses and because of that they would be earning relatively less than wage workers.

Over time their businesses would grow and they would then earn more than the wage

29

workers of equivalent tenure. While this could deliver a U-shaped earnings profile, this

would imply that these entrepreneurs are highly irrational, or highly discount the present.

Indeed, one cannot rationalize this behavior as it would mean that these individuals are

prepared to earn less than their wage worker counterparts for almost four decades in

order to have a higher payoff only very late in their lives.

Precautionary Saving and Bequest Motives

The presence of these two motives could explain why entrepreneurs may earn less relative

to wage workers on paper since entrepreneurs can “hide” their wages by keeping them in

the firm. One reason for doing this is that entrepreneurs may want to save within their

firm in order to have a healthier retirement or in order to pass the firm down to potential

heirs. If this is the case, we would indeed observe that entrepreneurs earn less than wage

workers, but this does not imply that the earning differential is U-shaped. Further, if an

entrepreneur has a precautionary savings motive or a bequest motive, then he would be

better off retrieving a higher amount from his firm and investing it in equity that is less

risky. Indeed, the returns to entrepreneurship do not compensate for the high amount of

risk therein.43

Non-pecuniary benefits

There is evidence that entrepreneurs enjoy non-pecuniary benefits such as not having

a boss. Given the data, this would mean that on average entrepreneurs are willing to

give up 15% of hourly wages each year relative to wage workers.44 While these may

be reasonable amounts, there is no straightforward reasons why this would depend on

occupational tenure and therefore cannot explain the U-shape pattern.

Preference for Skewness

Age-unadjusted data show that the distribution of earnings for self-employed people is

heavily skewed to the left. If entrepreneurs have a preference for skewness, they do

not necessarily demand higher earnings to compensate for the volatility of their earnings.

43See Moskowitz and Vissing-Joergensen (2002).44The difference in earnings is actually likely to be understated given that paid employees benefit from

non-wage compensation such as paid vacation or health insurance.

30

While this could account for the years 10-35, it is difficult to see this driving the result that

both early and late in their lives entrepreneurs earn more relative to workers. Further, as

Moskowitz and Vissing-Jorgensen (2002) point out, skewness can be obtained more easily

through options markets and other strategies. Thus, this isn’t a strong explanation for

the years for which entrepreneurs do earn less than wage workers.

Optimism

An existing body of work that shows that entrepreneurs are highly optimistic when

asked about the survival of their firm. This may explain why entrepreneurs persist in a

business longer than they perhaps should, but cannot explain any non-monotonicity in

the differential in earnings.

Mismeasurement

Recent work has explored the possibility that the gap in the reported earnings of self

and paid employed individuals is due to mismeasurement issues. For example, Hurst et

al. (2013) show that the self-employed a likely to be underreporting their income. If the

self-employed become better and better at doing so, then one would observe that the gap

between the self and paid employed would widen. There is no reason, however, that the

self employed would start off with lower wages that their paid employed counterparts. In

other words, while we would observe the “first half” of the U-shape, there is no reason

that the initial difference would be positive.

7 Conclusion

In this paper I redefine the puzzle regarding the choice to become an entrepreneur. I

have:

1. Extended the existing empirical evidence on the median wage differential between

entrepreneurs and workers. I have paid particular attention to tenure and shown

31

that the wage differential is in fact U-shaped as a function of tenure. This new evi-

dence contrasts with previous work that has focused predominantly on the bottom

section of the U-shape, i.e., the part where entrepreneurs’ wages are dominated by

those of wage workers. With this new evidence, we see that the lifetime returns to

entrepreneurship depend on how agents discount time. Thus, the evidence is not

necessarily in favor of workers.

2. Presented a life-cycle model to show that while age-nonadjusted labor income may

be on average lower for entrepreneurs, the present discounted value of utility can

be equalized between the two occupations.

3. Calibrated the model and shown that hyperbolic discounting is enough to rationalize

the data, and that there is no need for any ex-ante heterogeneity assumption to

explain the U-shape pattern in the data.

In the data section, using the 2001 panel year of the SIPP data set, I control for

observables via a quantile regression at the median and show that the differential in wages

is U-shaped. With a test of medians, I further check that taking each year independently,

the first ten years and the last six show the median entrepreneur earning more than the

median worker, while the reverse is true for the years in between. These findings are

confirmed using additional information from the 1984 panel.

In this model, agents are fully rational, face no uncertainty, are fully informed and

are identical before making an occupational decision. There is, however, a non-standard

component in the way agents discount. I allow for an extra degree of freedom by allowing

for agents to be quasi-geometric discounters. That is, they discount more between the

present period and the next than between any two other consecutive periods.

Naively averaging the data is likely to over-focus on middle-aged entrepreneurs and

workers, and produce the erroneous conclusion that entrepreneurs earn less than wage

workers (or more generally that they do not make rational decisions). Given the very

different patterns in earnings early and late in agents’ lives that I have shown in this

32

paper, it is crucial that length of tenure is taken into account, and that earnings are

discounted appropriately.

33

References

[1] Atkeson, Andrew and Kehoe, Patrick J., 2005 Modeling and Measuring Organization

Capital, Federal Reserve Bank of Minneapolis, Research Department Staff Report

291.

[2] Atkeson, A., Khan, A., Ohanian, L., 1996, Are Data on Industry Evolution and

Gross Job Turnover Relevant for Macroeconomics?, Carnegie-Rochester Conf. Ser.

Public Pol 44: 216-50.

[3] Astebro, Thomas and Jing Chen, forthcoming, The Entrepreneurial Earnings Puzzle:

Mismeasurement or Real?, Journal of Business Venturing.

[4] Bahk, Byong-Hyong and Gort, Michael, 1993, Decomposing Learning by Doing in

New Plants, The Journal of Political Economy, 101(4): 561-583.

[5] Bruce, D., Schuetze, H.J., 2004, The labor market consequences of experience in

self-employment, Labour Economics 11, 575-598.

[6] Evans, D. and Leighton, L., 1989, Some Empirical Aspects of Entrepreneurship,

American Economic Review 79(3), 519-35.

[7] Fairlie, R. and Woodruff, C., 2005, Mexican Entrepreneurship: A Comparison Of

Self-Employment In Mexico And The United States, Working Paper 1157, NBER.

[8] Frederick, S., Lowenstein, G., O’Donoghue, T., 2002 Time Discounting and Time

Preference: A Critical Review, Journal of Economic Literature, Vol. 40, No. 2, 351-

401.

[9] Gentry, W.M. and Hubbard, R.G., 2004 Entrepreneurship and Household Saving,

Advances in Economic Policy and Analysis 4(1).

[10] Bahk, Byong-Hyong and Gort, Michael, 1993 Decomposing Learning by Doing in

New Plants, Journal of Political Economy 101(4): 561-583.

34

[11] Hall, Robert E., Woodward, Susan E., 2010 The Burden of the Nondiversifiable Risk

of Entrepreneurship, American Economic Review, 100(3): 1163-94.

[12] Hamilton, B.H. 2000 Does Entrepreneurship Pay? An Emprirical Analysis of the

Returns to Self-Enployment, Journal of Political Economy 108(3): 604-631.

[13] Fang, Hanming and Wang, Yang, 2009 Estimating Dynamic Discrete Choice Mod-

els with Hyperbolic Discounting, with an Application of Mammography Decisions,

Working Paper.

[14] Harris, C., Laibson, D., 2001, Dynamic Choices Of quasi-geometric Consumers,

Econometrica, 69(4): 935-57.

[15] Heaton, John and Lucas, Deborah, 2000 Asset Pricing and Portfolio Choice: the

Importance of Entrepreneurial Risk, Journal of Finance, 55, 1163-1198.

[16] Hopenhayn, Hugo A. and Vereshchagina, Galina M., 2009 Risk Taking by Entr-

preneurs, AER.

[17] Hurst, Erik, Geng Li, and Benjamin Pugsley, Forthcoming, Are Household Surveys

Like Tax Forms: Evidence from Income Underreporting of the Self Employed, Review

of Economics and Statistics.

[18] Hyytinen, A., Rouvinen, P., 2008, The labour market consequences of self-

employment spells: European evidence, Labour Economics 15, 246?271.

[19] Jovanovic, Boyan, 1994, Firm Formation With Heterogeneous Management and La-

bor Skills, Small Business Economics 6: 185-191.

[20] Kihlstrom, Richard and Laffont, Jean-Jacques 1979 A General Equilibrium of En-

trepreneurial Theory of Firm Formation Based on Risk Aversion, JPE

[21] Kocherlakota, Narayna R., 2001, Looking for Evidence of Time-Inconsistent Prefer-

ences in Asset Market Data, Federal Reserve Bank of Minneapolis Quarterly Review

Vol. 25, No. 3.

35

[22] Landier, Augustin and Thesmar, David, 2009 Financial Contracting with Optimistic

Entrepreneurs, Review of Financial Studies, 22(1): 117-150.

[23] Laibson, David, 1997, Golden Eggs And quasi-geometric Discounting, Quarterly

Journal Of Economics, 112(2): 443-447

[24] Laibson, D., Repetto, A., Tobacman, J., 2007, Estimating Discount Functions with

Consumption Choices over the Lifecycle, Working Paper

[25] Levine, Ross and Yona Rubinstein, 2013 Smart and Illicit: Who Becomes an En-

treprenerur and Does it Pay?, Working Paper.

[26] Lowenstein, G. and Thaler, R., 1989, Anomalies: Intertemporal Choice, Journal of

Economics Perspectives, 3: 181-193.

[27] Lucas, Robert E., 1978, On the Size Distribution of Business Firms, Bell Journal of

Economics, 9(2): 508-523.

[28] Manso, Gustavo, 2013, Experimentation and the Returns to Entrepreneurship, Work-

ing Paper.

[29] Moskowitz, T.J. and Vissing-Jorgensen, A. 2002 The Returns to Entrepreneurial

Investment: A Private Equity Premium Puzzle, American Economic Review 92(4):

745-778.

[30] Parker, Simon, 2009, The Economics of Entrepreneurship, Cambridge University

Press.

[31] Phelps, E.S. and R.A. Pollack, 1968, On Second-best National Saving and Game-

equilibrium Growth, Review of Economic Studies, 35: 185-199.

[32] Pissarides, Christopher A. and Guglielmo Weber, 1989, An Expenditure-Based Esti-

mate of Britain’s Black Economy, Journal of Public Economics, 39(1), 17-32.

[33] Puri, Manju and Robinson, David T., 2005, Optimism and Economic Choice, Work-

ing Paper.

36

[34] Sarada, 2013, The Unobserved Returns from Entrepreneurship, Working Paper.

[35] Shepherd, Dean, Johan Wiklund and Michael Haynie, 2009, Moving forward: Bal-

ancing the financial and emotional costs of business failure, Journal of Business

Venturing 24, 134?148.

[36] Sherwin Rosen, 1981, The Economics of Superstars, American Economic Review,

71(5): 845-858.

[37] Strotz, R.H. 1956, Myopia and Inconsistency in Dunamic Uitility Maximization,

Review of Economic Studies, 23: 165-180.

37

A Proofs

Without Committment

Let’s first prove the following lemma:

Lemma A.1. For all 1 < t ≤ T :

lt =αβδ

[1 +

∑T−ti=1 [αiδi]

](lt−1/w)α

1 + β∑T−t+1

i=1 αiδi

Proof by induction. Let’s show that the statement is true for time t = T , then we’ll

assume it’s true for t < T and show that it’s true for t− 1.

Step 1: The Statement is true for t = T

At date T − 1, the entrepreneur has to choose how much to re-invest in his firm for

period T production. He solves the following maximization problem:

maxlT

log[(lT−1/w)α − lT ] + βδ log[lαT ]

After taking a first order derivative we obtain:

lt =αβδ(lT−1/w)α

1 + αβδ

Step 2: If the Statement is true for t, t+ 1, ... , T , let’s show it’s true for t− 1

Suppose the statement is true for date t, t+ 1, ... , T .

In this case we have:

38

lt =αβδ

[1 +

∑T−ti=1 [αiδi]

](lt−1/w)α

1 + β∑T−t+1

i=1 αiδi

At date t− 2, the entrepreneur takes lt−2 as given and chooses lt−1 to maximize:

maxlt−1

log[(lt−2/w)α − lt−1] + βδ log[(lt−1/w)α − lt] + βδ2 log[(lt/w)α − lt+1]

+ . . .+ βδT−t+2 log[(lT/w)α]

Using the induction statement, it is now easy to see that for any two consecutive dates,

the argument of the Log function ( lt−1

w)α− lt can be re-written as a function of lt−1 only:

(lt−1

w

)α− lt =

1−αβδ

[1 +

∑T−ti=1 [αiδi]

]1 + β

∑T−t+1i=1 αiδi

( lt−1

w

)α

So, we can write the maximization problem as:

maxlt−1

log[(lt−2/w)α − lt−1] + βδ log[C1(lt−1/w)α] + βδ2 log[C2(lt/w)α]

+βδ3 log[C3(lt+1/w)α] + . . .+ βδT−t+2 log[CT−t+2(lT/w)α]

Since by hypothesis the statement is true for t up to T , we can easily write ln as a function

of lm for all n > m:

ln =

(1

w

)∑n−m−11 αi (

lmw

)αn−mαβδ

1 +∑T−n

i=1 αiδi

1 + β∑T−n+1

i=1 αiδi

Πn−m−1j=1

(αβδ(1 +

∑T−n+ji=1 αiδi)

1 + β∑T−n+j+1

i=1 αiδi

)αj

39

Using the equation above, we can in fact write ln as a function of lt−1 for all n > t− 1:

ln =

(1

w

)∑n−t1 αi (

lt−1

w

)αn−t+1

αβδ1 +

∑T−ni=1 αiδi

1 + β∑T−n+1

i=1 αiδi

Πn−tj=1

(αβδ(1 +

∑T−n+ji=1 αiδi)

1 + β∑T−n+j+1

i=1 αiδi

)αj

=C1n,T (lt−1)

αn−t+1

And:

(lnw

)α=C2

n,T (lt−1)αn−t+2

where C1 and C2 are constants that will drop out when we take the first order condition

of the maximization problem.

Since the statement is true for t, t + 1, ... , T , we can use it to replace all the lt, lt+1,

lt+2, etc in Equation (3), and solve for lt+1. The maximization problem now becomes:45

maxlt−1

log[(lt−2/w)α − lt−1] + βδ log[lαt−1] + βδ2 log[lα2

t−1]

+βδ3 log[lα3

t−1] + . . .+ βδT−t+2 log[lαT−t+2

t−1 ]

The solution to the maximization problem is lt−1 that solves:

−1(lt−2

w

)α− lt−1

+ βδαl−1t−1 + βδ2α2l−1

t−1 + βδ3α3l−1t−1 + . . .+ βδT−t+2αT−t+2l−1

t−1 = 0 (1)

Easily:

lt−1 =αβδ

[1 +

∑T−t+1i=1 [αiδi]

](lt−2/w)α

1 + β∑T−t+2

i=1 αiδi

45For notational convenience I dropped the constants.

40

This concludes the proof of Lemma A.2.

From Lemma A.2 we obtain that:

ln =

(1

w

)∑n−21 αi (

l1w

)αn−1

αβδ1 +

∑T−ni=1 αiδi

1 + β∑T−n+1

i=1 αiδi

Πn−2j=1

(αβδ(1 +

∑T−n+ji=1 αiδi)

1 + β∑T−n+j+1

i=1 αiδi

)αj

To prove Theorem 4.1 is now fairly easy as we need only to confirm the expression for l1

and l0: In Period 0, the entrepreneur must choose l0 and l1. The maximization problem is:

maxl0,l1

log[(l0/w)α − l1 − l0] + βδ log[(l1/w)α − l2] + βδ2 log[(l2/w)α − l3]

+ . . .+ βδT log[(lT/w)α]

The solution for l1, is l1 that solves:

−1(l0w

)α − l1 − l0 + βδαl−11 + βδ2α2l−1

1 + βδ3α3l−11 + . . .+ βδT−t+2αT−t+2l−1

1 = 0 (2)

Easily:

l1 =αβδ

[1 +

∑T−1i=1 [αiδi]

]((l0w

)α − l0)1 + β

∑Ti=1 α

iδi

41

So, for all t > 0 we can write:

lt =

(1

w

)∑t−11 αi ((

l0w

)α− l0

)αt−1

αβδ1 +

∑T−ti=1 α

iδi

1 + β∑T−t+1

i=1 αiδi

Πt−1j=1

(αβδ(1 +

∑T−t+ji=1 αiδi)

1 + β∑T−t+j+1

i=1 αiδi

)αj

=C3t,T

((l0w

)α− l0

)αt−1

Finally, replacing all the lts for t > 0 by their expression as a function of l0 and taking

a first order condition (where all the constants drop out again), l0 solves:

C4t,T

((α l0w

)α − l0)l0((

l0w

)α − l0) = 0

Easily:

l0 = w(αw

) 11−α

This concludes the proof of Theorem 4.1.

With Committment

Let’s first prove the following lemma:

Lemma A.2. For all 1 < t ≤ T :

lt =αδ[1 +

∑T−ti=1 [αiδi]

](lt−1/w)α

1 +∑T−t+1

i=1 αiδi

42

Proof by induction. Let’s show that the statement is true for time t = T , then we’ll

assume it’s true for t < T and show that it’s true for t− 1.

Step 1: The Statement is true for t = T

At date 0, the entrepreneur has to choose how much he will re-invest in his firm at

period T − 1 for period T production. He solves the following maximization problem:46

maxlT

βδT−1 log[(lT−1/w)α − lT ] + βδT log[lαT ]

After taking a first order derivative we obtain:

lT =αδ(lT−1/w)α

1 + αδ

Step 2: If the Statement is true for t, t+ 1, ... , T , let’s show it’s true for t− 1

Suppose the statement is true for date t, t+ 1, ... , T for t > 1

In this case we have:

lt =αδ[1 +

∑T−ti=1 [αiδi]

](lt−1/w)α

1 +∑T−t+1

i=1 αiδi

At date t− 2, the entrepreneur takes lt−2 as given and chooses lt−1 to maximize:47.

maxlt−1

βδt−2 log[(lt−2/w)α − lt−1] + βδt−1 log[(lt−1/w)α − lt] + βδt log[(lt/w)α − lt+1]

+ . . .+ βδT log[(lT/w)α]

Using the induction statement, it is now easy to see that for any two consecutive dates,

46I only write the part of the utility function involving lT for simplicity47Again I simplify the utility function to keep only the relevant parts

43

the argument of the Log function ( lt−1

w)α− lt can be re-written as a function of lt−1 only:

(lt−1

w

)α− lt =

1−αδ[1 +

∑T−ti=1 [αiδi]

]1 +

∑T−t+1i=1 αiδi

( lt−1

w

)α

So, we can write the maximization problem as:

maxlt−1

βδt−2 log[(lt−2/w)α − lt−1] + βδt− 1 log[Ct−1(lt−1/w)α] + βδt log[Ct(lt/w)α]

+βδt+1 log[Ct+1(lt+1/w)α] + . . .+ βδT log[CT (lT/w)α]

Since by hypothesis the statement is true for t up to T , we can easily write ln as a function

of lm for all n > m:

ln =

(1

w

)∑n−m−11 αi (

lmw

)αn−mαδ

1 +∑T−n

i=1 αiδi

1 +∑T−n+1

i=1 αiδi

Πn−m−1j=1

(αδ(1 +

∑T−n+ji=1 αiδi)

1 +∑T−n+j+1

i=1 αiδi

)αj

Using the equation above, we can in fact write ln as a function of lt−1 for all n > t− 1:

ln =

(1

w

)∑n−t1 αi (

lt−1

w

)αn−t+1

αδ1 +

∑T−ni=1 αiδi

1 +∑T−n+1

i=1 αiδi

Πn−tj=1

(αδ(1 +

∑T−n+ji=1 αiδi)

1 +∑T−n+j+1

i=1 αiδi

)αj

=C1n,T (lt−1)

αn−t+1

And:

(lnw

)α=C2

n,T (lt−1)αn−t+2

44

where C1 and C2 are constants that will drop out when we take the first order condition

of the maximization problem.

Since the statement is true for t, t + 1, ... , T , we can use it to replace all the lt,

lt+1, lt+2, etc in Equation (3), and solve for lt+1. The maximization problem is equivalent

to:48

maxlt−1

log[(lt−2/w)α − lt−1] + δ log[lαt−1] + δ2 log[lα2

t−1]

+δ3 log[lα3

t−1] + . . .+ δT−t+2 log[lαT−t+2

t−1 ]

The solution to the maximization problem is lt−1 that solves:

−1(lt−2

w

)α− lt−1

+ δαl−1t−1 + δ2α2l−1

t−1 + δ3α3l−1t−1 + . . .+ δT−t+2αT−t+2l−1

t−1 = 0 (3)

Easily:

lt−1 =αδ[1 +

∑T−t+1i=1 [αiδi]

](lt−2/w)α

1 +∑T−t+2

i=1 αiδi

This concludes the proof of Lemma A.2.

From Lemma A.2 we obtain that:

ln =

(1

w

)∑n−21 αi (

l1w

)αn−1

αδ1 +

∑T−ni=1 αiδi

1 +∑T−n+1

i=1 αiδi

Πn−2j=1

(αδ(1 +

∑T−n+ji=1 αiδi)

1 +∑T−n+j+1

i=1 αiδi

)αj

48For notational convenience I dropped the constants.

45

To prove Theorem 4.2 is now fairly easy as we need only to confirm the expression for l1

and l0: In Period 0, the entrepreneur must choose l0 and l1. The maximization problem is:

maxl0,l1

log[(l0/w)α − l1 − l0] + βδ log[(l1/w)α − l2] + βδ2 log[(l2/w)α − l3]

+ . . .+ βδT log[(lT/w)α]

The solution for l1, is l1 that solves:

−1(l0w

)α − l1 − l0 + βδαl−11 + βδ2α2l−1

1 + βδ3α3l−11 + . . .+ βδT−t+2αT−t+2l−1

1 = 0 (4)

This is the same problem as the case without commitment. Easily:

l1 =αβδ

[1 +

∑T−1i=1 [αiδi]

]((l0w

)α − l0)1 + β

∑Ti=1 α

iδi

So, for all t > 0 we can write:

lt =

(1

w

)∑t−11 αi ((

l0w

)α− l0

)αt−1[αβδ

1 +∑T−1

i=1 αiδi

1 + β∑T

i=1 αiδi

]αt−1

αδ1 +

∑T−ti=1 α

iδi

1 +∑T−t+1

i=1 αiδi

Πt−2j=1

(αδ(1 +

∑T−t+ji=1 αiδi)

1 +∑T−t+j+1

i=1 αiδi

)αj

=C3t,T

((l0w

)α− l0

)αt−1

Finally, replacing all the lts for t > 0 by their expression as a function of l0 and taking

a first order condition (where all the constants drop out again), l0 solves:

C4t,T

((α l0w

)α − l0)l0((

l0w

)α − l0) = 0

46

Easily:

l0 = w(αw

) 11−α

This concludes the proof of Theorem 4.2.

B Income Graphs

1000

3000

5000

Month

ly Incom

e

0 10 20 30 40 50Tenure

Worker Entrepreneur

Worker (fitted) Entrepreneur (fitted)

(a) Monthly wages of entrepreneurs and workersby tenure with smoothed fitted lines.

-.5

0.5

11.5

2

0 10 20 30 40 50Tenure

Wage Difference (fitted) Wage Difference (data)

Zero

(b) Difference in absolute monthly wages ofentrepreneurs and workers by tenure withsmoothed fitted lines.

Figure 8: Difference in monthly wages of entrepreneurs and workers by tenure.

47