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An Estimated Structural Model of
Entrepreneurial Behavior

WP 17-07

John Bailey Jones
Federal Reserve Bank of Richmond
Sangeeta Pratap
Hunter College and Graduate Center CUNY

An Estimated Structural Model of
Entrepreneurial Behavior
John Bailey Jones
Federal Reserve Bank of Richmond
john.jones@rich.frb.org
Sangeeta Pratap∗
Department of Economics
Hunter College & Graduate Center - CUNY
sangeeta.pratap@hunter.cuny.edu
May 8, 2017
Working Paper No. 17-07
Abstract
Using a rich panel of owner-operated New York dairy farms, we provide new
evidence on entrepreneurial behavior. We formulate a dynamic model of farms facing uninsured risks and ﬁnancial constraints. Farmers derive nonpecuniary beneﬁts
from operating their businesses.

We estimate the model via simulated minimum

distance, matching both production and ﬁnancial data. We ﬁnd that ﬁnancial factors and nonpecuniary beneﬁts are of ﬁrst-order importance. Collateral constraints
and liquidity restrictions inhibit borrowing and the accumulation of capital. The
nonpecuniary beneﬁts to farming are large and keep small, low-productivity farms
in business. Although farmers are risk averse, eliminating uninsured risk has only
modest eﬀects on capital and output.

JEL Classification: G31, G32, L26
∗

We are grateful to Marco Bassetto, Paco Buera, Cristina De Nardi, Lars Hansen, Boyan Jovanovic,
Todd Keister, Vincenzo Quadrini, Todd Schoellman, Fang Yang and seminar participants at Emory
University, the Federal Reserve Banks of Chicago and Richmond, LSU, University at Albany, Hunter
College, University of Connecticut, Universit´ Paris I, the Midwest Macro Meetings, the ITAM-PIER
e
conference on Macroeconomics and the CEPR Workshop on Entrepreneurship Economics for helpful
comments. Cathryn Dymond and Wayne Knoblauch provided invaluable assistance with the DFBS data.
Pratap gratefully acknowledges the hospitality of the Paris School of Economics. The opinions and
conclusions are solely those of the authors, and do not reﬂect the views of the Federal Reserve Bank of
Richmond or the Federal Reserve System.

Introduction

The Statistics of US Businesses show that almost 80 percent of US businesses are
single proprietorships, partnerships or S Corporations. What mechanisms best explain
their behavior? While many researchers have focused on the importance of ﬁnancial
constraints, others have highlighted the role of nonpecuniary returns from running one’s
own business. Still other research has emphasized factors that encourage entrepreneurs
to take risks. Assessing the relative importance of these mechanisms is a priority for
policymakers. Compared with larger ﬁrms, small businesses are often viewed as more
proﬁcient at innovation and job creation, and also more likely to be hampered by ﬁnancial
constraints. Small ﬁrms are therefore seen as worth subsidizing. The value of such
subsidies is less apparent, however, if entrepreneurs are driven mainly by nonpecuniary
considerations. (Hurst and Pugsley, 2011, 2014).
In this paper we evaluate the importance of all these mechanisms by building and
estimating a rich model of entrepreneurial behavior. Risk-averse entrepreneurs make borrowing, investment, production, dividend and liquidation decisions. They face uninsured
risk, along with collateral and liquidity constraints, but also receive nonpecuniary beneﬁts
from operating their businesses. Older entrepreneurs retire, and entrepreneurs of any age
can exit to wage work, albeit with liquidation costs. Limited liability and the outside option of wage work create incentives for risk-taking. We estimate the model’s parameters
using a detailed panel of family owned and operated dairy farms in New York State. Using a simulated minimum distance estimator, we match both the production and ﬁnancial
sides of the data. Counterfactual experiments allow us to quantify the eﬀects of relaxing
ﬁnancial constraints, eliminating nonpecuniary beneﬁts, and reducing or eliminating risk.
Our paper makes two key contributions. The ﬁrst contribution is that, to the best of
our knowledge, we are the ﬁrst to formulate and estimate a model of entrepreneurial behavior that simultaneously accounts for ﬁnancial constraints, nonpecuniary beneﬁts and
risk-taking. While (sometimes inconclusive) evidence exists for each of these mechanisms
separately, to the best of our knowledge, we are the ﬁrst to undertake a joint treatment.
This is important, because none of these mechanisms can be measured correctly in isolation. For example, borrowing constraints that aﬀect ﬁnancial returns and nonpecuniary
beneﬁts both determine whether an enterprise remains in business.
Researchers have looked for ﬁnancial constraints in a variety of settings. While Evans
and Jovanovic (1989) and Blanchﬂower and Oswald (1998) ﬁnd that wealthier individuals
are more likely to become entrepreneurs, Hurst and Lusardi (2004) argue that a positive
relationship between wealth and business creation exists only at the top of the wealth
2

distribution.1 Even if ﬁnancial constraints do not restrict entry, they may restrict the
operations of ongoing ﬁrms (Quadrini, 2009; Cagetti and De Nardi, 2006). The tendency
of entrepreneurs to save at higher rates and invest much of their wealth in their own
businesses suggests the presence of ﬁnancial constraints (Quadrini, 2000, 2009; Cagetti
and De Nardi, 2006; Herranz et al., 2015), as does evidence that inheritances improve
their chances of survival (Holtz-Eakin et al., 1994).
There is both direct and indirect evidence that entrepreneurs are also motivated by
nonﬁnancial considerations. Hurst and Pugsley (2011) tabulate responses from the Panel
Study on Entrepreneurial Dynamics and ﬁnd that nonpecuniary beneﬁts (ﬂexible hours,
being one’s own boss, etc.) are a primary motive behind business formation. In general, entrepreneurs report greater job satisfaction than wage workers, as documented by
Blanchﬂower (2000) for the OECD, Anderson (2008) for Sweden, and Benz and Frey
(2008) for Germany, UK and Switzerland.
Indirect evidence of nonpecuniary beneﬁts is provided by the gap between the earnings
of the self-employed and wage workers (Hamilton, 2000), and by the gap between the returns on undiversiﬁed entrepreneurial investment and publicly traded equities (Moskowitz
and Vissing-Jørgensen, 2002).2 Hall and Woodward (2010) ﬁnd an extreme dispersion in
the payoﬀs to startups, and consequently, a very small risk-adjusted return.
These income diﬀerentials, however, could be due to diﬀerences in risk-taking behavior between entrepreneurs and wage workers. Kihlstrom and Laﬀont (1979) construct a
model where risk-loving individuals select into entrepreneurship, while Vereshchagina and
Hopenhayn (2009) show that limited liability can encourage risk taking behavior among
entrepreneurs even without a risk premium, indicating that nonpecuniary beneﬁts cannot be measured accurately without accounting for attitudes for risk. More generally, as
we mentioned above, none of our three mechanisms can be measured accurately without
accounting for the others.
Our second main contribution is that, in contrast with most studies on entrepreneurship, we use detailed panel data to identify the dynamic mechanisms of the model. The
data contain comprehensive information on the farms’ real and ﬁnancial activities, including input use, revenue, investment, borrowing and equity. Because they are drawn
from a single region and industry, the data are less vulnerable to issues of unobserved
1

Buera (2009) shows that a dynamic model with borrowing constraints has the opposite implication:
the likelihood of becoming an entrepreneur is increasing in wealth at low levels of wealth, but decreasing
in wealth when wealth is high
2
Kartashova (2014) argues that this “private equity puzzle” does not exist outside the 1990s, a period
of unusually high public equity returns.

heterogeneity.3
Our dataset allows us to estimate ﬁnancial constraints from investment dynamics,
rather than cross-sectional relationships. This is a quite diﬀerent and arguably more
direct source of identiﬁcation. For example, using the parameters estimated by our model,
we calculate revenue productivity shocks for each farm. These shocks can in turn be
decomposed into a permanent farm-speciﬁc component, an aggregate shock tied closely
to the price of milk, and an idiosyncratic transitory component. We ﬁnd that periods
of high aggregate productivity are also periods of high aggregate investment. Because
aggregate productivity appears uncorrelated over time, this association suggests that high
milk prices promote investment by increasing cash ﬂow, as opposed to signalling higher
future productivity. Volatile revenue shocks also help us assess attitudes toward risk, while
farm exit decisions help us measure nonpecuniary beneﬁts. This sort of identiﬁcation is
not available in cross-sectional datasets, such as the Survey of Consumer Finances or the
Survey of Small Business Finances, on which most studies of entrepreneurship are based.
Our empirical methodology has many parallels with the literature on structural corporate ﬁnance (Pratap and Rendon, 2003; Hennessey and Whited, 2007; Strebulaev and
Whited, 2012), which analyzes the real and ﬁnancial decisions of large publicly traded
corporations. However, the ﬁrms in our data are all family owned and operated, and
hence not comparable to ﬁrms listed on the stock market. The closest counterparts to
our analysis are arguably in the development literature, where structural models of entrepreneurship have been estimated with ﬁrm-level panel data, often from Townsend’s
Thai surveys.4 Such ﬁrms are of course quite diﬀerent in scale and operate in a radically diﬀerent environment. Our work also resembles that of Caggese (2007, 2013), who
calibrates and assesses his models with detailed data on Italian manufacturing ﬁrms.
Our main ﬁnding is that the eﬀects of ﬁnancial constraints and nonpecuniary beneﬁts
are of ﬁrst-order importance but those of risk are not. Financial constraints exercise
a signiﬁcant inﬂuence along the intensive margin of operation, i.e., on investment and
output. Collateral constraints hinder the accumulation of capital, especially among highproductivity farms that are seeking to expand. Liquidity constraints that force businesses
to hold cash and divert resources from investment have similar eﬀects, although their
quantitative impact is much smaller. The eﬀects of nonpecuniary beneﬁts are manifested
3

On the other hand, the average revenues of all US ﬁrms, and their distribution, are similar to those
in our sample. We discuss these and other aspects of the external validity of our results in Section 6.
4
Townsend et al. (1997) and Samphantharak and Townsend (2010) provide a description of the data.
A recent study especially relevant to ours is Karaivanov and Townsend (2014), which also contains a
literature review.

along the extensive margin, i.e., the decision to continue operations. The least productive
farms in our data have very low ﬁnancial returns. In the absence of signiﬁcant nonﬁnancial rewards, their continued operation is hard to justify. Liquidation costs likewise
aﬀect the extensive margin by impeding the exit of unproﬁtable farms. We also ﬁnd that
liquidation costs amplify the eﬀect of nonpecuniary beneﬁts. Both work in the direction
of discouraging exit of small, marginal farms. When combined, their eﬀects are much
larger than each in isolation.
In contrast, risk appears to play a minor role. The parameter estimates show that
entrepreneurs are risk averse. Risk aversion, along with the nonpecuniary rewards to
farming, combine to mitigate the appetite for risk-taking, despite limited liability and the
ability to exit into wage work. On the other hand, while eliminating risk leads farms to
expand along the intensive margin, the eﬀect is quantitatively modest, especially compared to the eﬀect of relaxing ﬁnancial constraints. We reach similar similar conclusions
regarding recent changes in US dairy policy. The 2014 Farm Bill replaced the previous
system of price supports, which were too low to be eﬀective, with a system of margin
(milk prices net of feed costs) supports (Schnepf, 2014). We estimate what the impact
of the margin supports program would have been had it existed during our 2001-2011
sample period, and we ﬁnd that the insurance provided by the program would have had
only a minor eﬀect on farm operations. The premium charged for the margin supports
would have had much larger, and negative, consequences.
Taken together, our results caution against the blanket subsidization of small ﬁrms
and argue for a more nuanced approach. Low-productivity operations driven mainly by
nonpecuniary concerns exist alongside high-productivity operations hindered by ﬁnancial
constraints. Relaxing the borrowing constraints may not have an expansionary eﬀect
across the board. Policies that encourage the formation of businesses as a way to spur
innovation or growth may not achieve those results. Entrepreneurial policy may work
better by helping the most promising entrepreneurs expand.
The rest of the paper is organized as follows. In section 2 we describe our data
and perform some diagnostic reduced form exercises. Section 3 sets out the model and
section 4 describes our estimation procedure. Section 5 presents our parameter estimates,
assesses the model’s ﬁt, and discusses parameter identiﬁcation. Section 6 elaborates on
the mechanisms of the model, considering nonpecuniary beneﬁts, ﬁnancial constraints,
and their interactions. We also discuss the external validity of our results. Section 7
evaluates the eﬀects of uninsured risk in general and the eﬀects of the 2014 Farm Bill in
particular. Section 8 contains sensitivity analyses. We conclude in section 9.

Data and Descriptive Analysis

2.1

The DFBS

The Dairy Farm Business Summary (DFBS) is an annual survey of New York dairy
farms conducted by Cornell University. The data include detailed ﬁnancial records of
revenues, expenses, assets and liabilities. Physical measures such as acreage and herd
sizes are also collected. Assets are recorded at market as well as book value. The
data are extensively reviewed by the DFBS staﬀ, who also construct income statements,
balance sheets, cash ﬂow statements, and a variety of productivity and ﬁnancial measures
(Cornell Cooperative Extension, 2006; Cornell Cooperative Extension, 2015b; Karzes et
al., 2013). Participants can use these measures to compare their management practices
with those of their peers. These diagnostics are an important beneﬁt of participation
in the survey, which is voluntary (Cornell Cooperative Extension, 2015a). We therefore
expect the data to be of high quality, a supposition that is conﬁrmed by internal data
consistency checks. However, larger farms are over-represented, and the average farm in
the DFBS data is larger than the population average for New York State.5 As long as
our structural model is speciﬁed correctly, our parameter estimates should be consistent
even with a nonrepresentative sample. However, the results of the numerical experiments,
which aggregate over the DFBS sample, are best interpreted as qualitative.
Our dataset is an extract of the DFBS covering the calendar years 2001-2011. This is
an unbalanced panel of 541 distinct farms, with approximately 200 farms surveyed each
year. Since our model is explicitly dynamic, we eliminate farms with observations for
only one year. We also remove farms for which there is no information on the age of
the operators, as we expect retirement considerations to inﬂuence both production and
ﬁnance decisions on family operated farms. These ﬁlters leave us with a ﬁnal sample of
363 farms and 2,222 observations. During the same period, the number of dairy farms
in New York State fell from 7,180 to 5,240 (New York State Department of Agriculture
and Markets, 2012), so that our sample contains roughly 5 percent of all New York dairy
farms.
5

In 2011, average revenue in our sample was $2,887,000, compared to the state average of $520,000. In
the same year, the average herd size of New York State dairy farms was 209 cows, while our sample had an
average herd size of 505. However, the sample is very similar to state averages in terms of demographics:
the principal operator statewide has an average age of 51, as in our sample. All the farms in our sample
are family-owned, as are virtually all (97 percent) of the dairy farms in New York State. (New York State
Department of Agriculture and Markets, 2012.)

Variable

Mean

Median

Standard
Deviation

No. of Operators
Operator 1 Age
Youngest Operator Age
Herd Size (Cows)
Total Capital
Machinery
Real Estate
Livestock
Owned Capital
Machinery
Real Estate
Livestock
Owned/Total capital
Revenues
Total Expenses
Variable Inputs
Leasing and Interest
Total Assets
Cash
Total Liabilities
Net Worth
Dividends

1.79
51.36
43.12
374
3,329
753
1,715
861
2,693
562
1,296
835
0.84
1,778
1,510
1,389
121
3,242
548
1,508
1,733
72

2
51
43
186
1,932
483
966
432
1,594
360
756
424
0.87
822
675
613
58
1,869
257
786
874
46

0.87
10.83
10.69
434
3,622
760
1,949
1,030
2,934
576
1,509
980
0.12
2,202
1,900
1,769
154
3,579
721
1,709
2,314
176

Maximum

Minimum

6
87
74
3,656
28,247
5,335
15,161
9,027
26,478
3,776
14,196
9,027
1.00
16,929
15,685
15,203
1,255
31,414
5,689
11,423
21,278
4,058

1
16
12
20
212
13
0
39
83
3
0
39
0.26
47
43
36
0
103
6
0
-734
-2,327

Table 1: Summary Statistics from the DFBS
Notes: Financial variables are expressed in thousands of 2011 dollars.

Table 1 provides summary statistics. A detailed data description is provided in Appendix A. The median farm is operated by two individuals and more than 80 percent of
farms have two or fewer operators. The average age of the main operator is 51 years. For
multioperator farms, however, the relevant time horizon for investment decisions is the
age of the youngest operator, who will likely become the primary operator in the future.
On average, the youngest operator tends to be about eight years younger than the main
operator. In our analysis, we will consider the age of the youngest operator as the relevant
one for age-sensitive decisions.
Table 1 also shows that these are substantial enterprises: the yearly revenues of the
average farm are in the neighborhood of $1.8 million in 2011 terms. The distribution of
revenues is heavily skewed to the right, with median farm revenues equal to less than half
7

the mean. A large part of farm expenses are accounted for by variable inputs: intermediate
goods and hired labor. Of these labor expenses are relatively small, accounting for about
14 percent of all expenditures on variable inputs. The remainder consists of intermediate
goods and services such as feed, fertilizer, seed, pest control, repairs, utilities, insurance,
etc. We also report the amounts spent on capital leases and interest, which are less than
10 percent of total expenditures for the median farm.
Capital stock consists of machinery, real estate (land and buildings) and livestock,
of which real estate is the most valuable. Most of the capital stock is owned, but the
median farm leases about 13 percent of its capital, mostly real estate and machinery.
(See Appendix A.) Livestock is almost always owned. Capital is by far the predominant
asset, accounting for more than 85 percent of farm assets, while liquid assets (what we
call cash) accounts for the rest.
Farm liabilities include accounts payable, debt, and ﬁnancial leases on equipment and
structures. For the median farm, this accounts for about 70 percent of total liabilities.
Deferred taxes constitute the remainder. Combining total asssets and liabilities reveals
that the average farm has a net worth of $1.7 million. Only 28 (or 1.3 percent) of all
farm-years report negative net worth.
Farms generate relatively little disposable income. The average dividend remitted to
the farm’s owners is $72,000, while the median is $46,000. Because the farms’ owner/operators
also supply most of the labor, these returns are quite modest.
Much more of the farm’s operating income is invested. The DFBS reports net investment for each type of capital. It also reports depreciation, allowing us to construct
a measure of gross investment. Following the literature, we focus on investment rates,
scaling investment by the market value of owned capital at the beginning of each period.
Table 2 describes the distribution of investment rates. Cooper and Haltiwanger (2006)
use data from the Longitudinal Research Database (LRD) to show that plant-level investment often occurs in large increments, suggesting a prominent role for ﬁxed investment
costs. For reference, Table 2 reproduces the statistics for gross investment rates shown in
their Table 1. Investment spikes are much less frequent in the DFBS than in the LRD.
The average investment rate is also a bit lower. Although the inaction rate is marginally
higher in our sample, the comparison as a whole suggests that ﬁxed investment costs are
less important in the DFBS, and in the interest of tractibility we omit them from our
structural model.
Farm technologies can be divided into two categories: stanchion barns and and milking
parlors, the latter considered the newer and larger-scale technology. About 60 percent of

DFBS
Average Investment Rate
Inaction Rate (< abs(0.01))
Fraction of Observations < 0
Positive Spike Rate (> 0.2)
Negative Spike Rate (< -0.2)
Serial Correlation

LRD

0.087
0.093
0.086
0.077
0.003
0.106

0.122
0.081
0.104
0.186
0.018
0.058

Table 2: Investment Rates
Notes: Column DFBS summarizes gross investment to owned capital ratios in the Dairy Farm Business
Survey. Column LRD, taken from Cooper and Haltiwanger (2006, Table 1), shows corresponding
statistics from the Longitudinal Research Database.

Stanchion
No. of Farms
No. of Operators
Total Capital
Herd Size (Cows)
Output/Capital
Intermed Goods/Capital
Investment/Capital
Debt/Assets
Cash/Assets

Parlor

146
1
597
52
0.37
0.27
0.04
0.43
0.11

217
2
1,710
181
0.51
0.40
0.07
0.49
0.16

Table 3: Medians by Technology
Notes: Capital and herd size measured per operator.

the farms in our sample are parlor operations. Table 3 displays summary statistics for
each technology group. Stanchions are smaller than parlors, both in herd size and capital
stock per operator. They invest less, have lower debt-to-asset ratios, and hold less cash.
Interestingly, they also have lower output to capital ratios, and use fewer intermediate
goods per unit of capital. These diﬀerences are consistent with both heterogeneous production functions and with heterogeneous exposure to ﬁnancial constraints that distort
the mix of inputs. Our model and estimation will allow for both possibilities.

2.2
2.2.1

Productivity
Our Productivity Measure

We assume that farms utilize a Cobb-Douglas production function
γ
Yit = zit Miα Kit Nit 1−α−γ ,

where we denote farm i’s gross revenues at time t by Yit and its entrepreneurial input,
measured as the time-averaged number of operators, by Mi .6 Kit denotes the capital
stock; Nit represents expenditure on all variable inputs, including hired labor and intermediate goods; and zit is a stochastic revenue shifter reﬂecting both idiosyncratic and
systemic factors.7 With the exception of operator labor, all inputs are measured in dollars. Although this implies that we are treating input prices as ﬁxed, variations in these
prices can enter our model through changes in the proﬁt shifter zit .
In per capita terms, we have
yit =

Yit
γ
= zit kit nit 1−α−γ .
Mi

In this formulation, returns to scale are 1 − α < 1, with α measuring an operator’s “span
of control”(Lucas, 1978).
Following Alvarez et al. (2012) and based on the descriptive statistics in Table 3,
we allow for two production technologies. Using the structural estimation procedure
described below, we ﬁnd that for stanchion operations α = 0.135 and γ = 0.174 and for
milking parlors α = 0.107 and γ = 0.122. In other words, parlor operators have higher
returns to scale but lower returns to capital than stanchions. These estimates allow us to
calculate total revenue productivity as
zit =

yit
.
γ
kit nit 1−α−γ

(1)

We assume that the resulting productivity measure can be decomposed into the individual
ﬁxed eﬀect µi , a time-speciﬁc component, common to all farms, ∆t , and the idiosyncratic
6

More than two-thirds of our farms display no change in family size.
The assumption of decreasing returns to scale in nonmanagement inputs is consistent with the literature. Tauer and Mishra (2006) ﬁnd slightly decreasing returns in the DFBS. They argue that while
many studies ﬁnd that costs decrease with farm size: “Increased size per se does not decrease costs—it is
the factors associated with size that decrease costs. Two factors found to be statistically signiﬁcant are
eﬃciency and utilization of the milking facility.”
7

i.i.d component εit :
ln zit = µi + ∆t + εit .

(2)

A Hausman test rejects a random eﬀects speciﬁcation. Regressing zit on farm and time
dummies yields estimates of all three components. The ﬁxed eﬀect has a mean of 0.812
but ranges from 0.072 to 1.273 with a standard deviation of 0.14, implying signiﬁcant
time-invariant diﬀerences in productivity across farms. The time eﬀect ∆t is constructed
to be zero mean. This series is eﬀectively uncorrelated and has a standard deviation of
0.059. The idiosyncratic residual εit can also be treated as uncorrelated, with a standard
deviation of 0.070.
As a measure of revenue productivity, zit captures variation in prices as well as productivity. Because dairy farmers are by and large price takers, however, price variation
should aﬀect only the aggregate component ∆t . Figure 1 plots ∆t against real milk prices
in New York State (New York State Department of Agriculture and Markets, 2012). The
aggregate shock follows milk prices very closely – the correlation is over 90 percent –
which gives us conﬁdence in our interpretation.8 We thus feel comfortable assuming that
the transitory shock εit and especially the ﬁxed eﬀect µi measure physical productivity.9
In Figure 1 we also plot the average value of the cash ﬂow (net operating income
less estimated taxes) to capital ratio. Aggregate cash ﬂow is also closely related to our
aggregate productivity measure. Cash ﬂow varies quite signiﬁcantly, indicating that farms
face signiﬁcant ﬁnancial risk.
2.2.2

Productivity and Farm Characteristics

How are productivity and farm performance related? Figures 2 and 3 illustrate how
farm characteristics vary as a function of the time-invariant component of productivity, µi .
We divide the sample into high- and low-productivity farms, splitting around the median
value of µi , and plot the evolution of several variables. To remove scale eﬀects, we either
express these variables as ratios, or divide them by the number of operators. Ninety-eight
8

The series in Figure 1 are consistent with the belief that that milk prices follow a three-year cycle.
Nicholson and Stephenson (2014) ﬁnd a stochastic cycle lasting about 3.3 years. While Nicholson and
Stephenson report that in recent years a “small number” of farmers appear to be planning for cycles,
they also report (page 3) that: “the existence of a three-year cycle may be less well accepted among
agricultural economists and many ... forecasts ... do not appear to account for cyclical price behavior.
Often policy analyses ... assume that annual milk prices are identically and independently distributed[.]”
9
Our assumption of perfect competition rules out the ﬁrm-level diﬀerences in market power or product
demand emphasized by Foster et al. (2008). Hsieh and Klenow (2009, Appendix I) show that their
framework for measuring productivity, with constant returns to scale in production and monopolistic
competition (“diminishing returns in utility”), is isomorphic to our framework, with decreasing returns
to scale in production and perfect competition.

Aggregate Patterns in the DFBS: 2001-11
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3

TFP
Real Milk Price
Cash Flow/Capital

0.2
0.1
2000

2002

2004

2006

2008

2010

2012

Year

Figure 1: Aggregate Productivity, Real Milk Prices, and Cash Flow

percent of the stanchions are low-productivity farms, and almost eighty percent of the
low-productivity farms are stanchions. eighty-two percent of the parlor operations are
high-productivity farms, and almost all (ninety-eight percent) of the high-productivity
farms are parlors.
Our convention will be to use thick solid lines to represent high-productivity farms
and thinner dashed lines to represent low-productivity farms. Figure 2 shows output
(revenues) and input choices. The top two panels of this ﬁgure show that high-productivity
farms operate at a scale several times larger than that of low-productivity farms.10 This
size advantage is increasing over time: high-productivity ﬁrms are growing while lowproductivity ﬁrms are static. These diﬀerences will prove crucial to identifying our model.
The bottom left panel shows that high-productivity farms lease a larger fraction of
their capital stock (18 percent vs. 8 percent). The leasing fractions are all small and
stable, however, implying that farms expand primarily through investment. In our model
we will assume that all capital is owned. The bottom right panel of Figure 2 shows
that the ratio of variable inputs – feed, fertilizer, and hired labor – to capital is also
higher for high-productivity farms (40 percent vs. 30 percent). This could be due to
diﬀering production functions between stanchion and parlor farms, given that most highproductivity farms are parlors. Another explanation for this diﬀerence in input use could
be ﬁnancial constraints on the purchase of variable inputs. We will account for both
possibilities in our model.
10

Using the 2007 US Census of Agriculture, Adamopoulos and Restuccia (2014) document that farms
with higher labor productivity are indeed substantially larger.

Median Output by TFP Group: Data

Median Capital Stock by TFP Group: Data
5500

Capital Stock (000s of 2011 dollars)

Output (000s of 2011 dollars)

3500

3000

2500

2000

1500

1000

500

0
2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

5000
4500
4000
3500
3000
2500
2000
1500
1000
500
2001

2011

2002

2003

2004

2005

Year

2006

2007

2008

2009

2010

2011

2010

2011

Year

Median Leased/Total Capital Ratio by TFP Group: Data

Median Int. Goods/Capital Ratio by TFP Group: Data

0.2

0.5

Int. Goods/Capital Ratio

Leased/Total Capital Ratio

0.18

0.16

0.14

0.12

0.1

0.45

0.4

0.35

0.3
0.08

0.06
2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

0.25
2001

2011

Year

2002

2003

2004

2005

2006

2007

2008

2009

Year

Figure 2: Production and Inputs by Productivity and Calendar Year
Notes: Thick, solid lines refer to high-productivity farms. Thinner, dashed lines refer to low-productivity
farms.

Figure 3 shows ﬁnancial variables. The top two panels contain median cash ﬂow and
gross investment. These variables are positively correlated in the aggregate; for example,
cash ﬂow and investment both fell during the recession of 2009. Given that the aggregate
shocks are not persistent, high current productivity (and cash ﬂow) does not indicate
high expected productivity. The correlation of cash ﬂow and investment thus suggests
ﬁnancial constraints, which are relaxed in periods of high output prices. However, the
interpretation of cash ﬂow regressions – much less simple correlations – is notoriously
diﬃcult (see for example Erickson and Whited, 2000, and Gomes, 2001, among others).
We will formally assess the importance of ﬁnancial constraints using our structural model.
The middle row shows investment rates and dividend payments. High-productivity
farms invest at a higher rate and pay larger dividends to their owner/operators, than low
productivity farms. Dividends and cash ﬂow are strongly correlated for both types of
farms. In general, dividend ﬂows are quite modest, especially for low-productivity farms.

The bottom left panel of Figure 3 shows debt/asset ratios.11 Although high-productivity
farms begin the sample period with more debt, over the sample period they decrease their
leverage, even as they expand their capital stocks.
In a static frictionless model, the optimal capital stock for a farm with productivity
∗
level µi is given by ki = [κ exp(µi )]1/α , where κ is a positive constant.12 The bottom
∗
right panel of Figure 3 plots median values of the ratio kit /ki , showing the extent to
which farms operate at their eﬃcient scales. The median low-productivity farm holds
close to the optimal amount of capital throughout the sample period. In contrast, the
capital stocks of high-productivity farms are initially almost half their optimal size, but
grow rapidly. This diﬀerence can explain why large ﬁrms are more indebted and invest
at higher rates.
While we do not seek a quantitative measure of allocative ineﬃciency, the bottom
right panel of Figure 3 does suggest that ﬁnancial constraints hinder the optimal allocation of capital. Midrigan and Xu (2014) ﬁnd that ﬁnancial constraints impose their
greatest distortions by limiting entry and technology adoption. To the extent that highproductivity farms are more likely to utilize new technologies, such as robotic milkers
(McKinley, 2014), our results are consistent with their ﬁndings. Our results also comport with Buera, Kaboski and Shin’s (2011) argument that ﬁnancial constraints are most
important for large-scale technologies.13
On the other hand, our ﬁndings seem at odds at with the evidence from the corporate
ﬁnance literature that large ﬁrms are less ﬁnancially constrained (Kaplan and Zingales,
1997; Whited and Wu, 2006; Hadlock and Pierce, 2010). This apparent discrepancy may
be due to age and/or vintaging eﬀects. Although larger farms appear more constrained,
they also carry more debt. Because these farms are more likely to use parlor technologies,
which are newer, they may simply have had less time to accumulate capital. It is also
possible that many smaller farms cannot acquire the ﬁnancing needed to install newer
technologies and are thus eﬀectively more constrained than the larger farms.14
11

To ensure consistency with the model, and in contrast to Table 1, we add capitalized values of leased
capital to both assets and liabilities.
γ 1−α−γ
12
This expression can be found by maximizing E(zit )kit nit
− nit − (r + δ − )kit . In contrast to
the construction of capital stock described in Appendix A, here we use a single user cost for all capital.
α+γ

1−α−γ

γ
Standard calculations show that κ = r+δ−
(1 − α − γ)
E(exp(∆t εit )).
13
Protracted capital stock growth could also reﬂect capital adjustment costs, which we rule out by assumption. Capital adjustment costs cannot generate, however, the observed positive correlation between
aggregate cash ﬂow and aggregate investment. Because the aggregate productivity shocks appear to be
serially uncorrelated, we do not believe the latter relationship is caused by current cash ﬂow acting as a
signal for future productivity.
14
We are grateful to an anonymous referee for this insight.

Median Gross Investment by TFP Group: Data

Median Cashflow by TFP Group: Data
400

Gross Investment (000s of 2011 dollars)

Cashflow (000s of 2011 dollars)

600

500

400

300

200

100

0
2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

350
300
250
200
150
100
50
0
2001

2011

2002

2003

2004

2005

Median Gross Inv./Capital Ratio by TFP Group: Data

2007

2008

2009

2010

2011

2009

2010

2011

2010

2011

Median Dividends by TFP Group: Data
110

0.1

100

Dividends (000s of 2011 dollars)

0.11

Gross Inv./Capital Ratio

2006

Year

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
2001

90
80
70
60
50
40
30
20

2002

2003

2004

2005

2006

2007

2008

2009

2010

10
2001

2011

2002

2003

2004

2005

Year

2006

2007

2008

Year

Median Debt/Asset Ratio by TFP Group: Data

Median Capital/Optimum Ratio by TFP Group: Data

0.7

1.5
1.4
1.3

Capital/Optimum Ratio

Debt/Asset Ratio

0.65

0.6

0.55

0.5

1.2
1.1
1
0.9
0.8
0.7
0.6

0.45

0.5
0.4
2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

0.4
2001

2011

Year

2002

2003

2004

2005

2006

2007

2008

2009

Year

Figure 3: Investment and Finances by Productivity and Calendar Year
Notes: Thick, solid lines refer to high-productivity farms. Thinner, dashed lines refer to low-productivity
farms.

Model
Consider a farm family seeking to maximize expected lifetime utility at “age” q:
Q

β h−q [u (dh ) + χ · 1{farm operating}] + β Q−q+1 VQ+1 (aQ+1 ) ,

Eq
h=q

where: q denotes the age of the principal (youngest) operator; dq denotes farm “dividends”
per operator; the indicator 1{farm operating} equals 1 if the family is operating a farm and
0 otherwise, and χ measures the psychic/nonpecuniary gains from farming; Q denotes the
retirement age of the principal operator; a denotes assets; and Eq (·) denotes expectations
conditioned on age-q information. The family discounts future utility with the factor
0 < β < 1. Time is measured in years. Consistent with the DFBS data, we assume
that the number of family members/operators is constant. We further assume a unitary
model, so that we can express the problem on a per-operator basis. To simplify notation,
throughout this section we omit “i” subscripts.
The ﬂow utility function u(·) and the retirement utility function VQ+1 (·) are specialized
as
1
(c0 + d)1−ν ,
1−ν
1
a 1−ν
VQ+1 (a) =
θ c1 +
,
1−ν
θ
u(d) =

with ν ≥ 0, c0 ≥ 0, c1 ≥ c0 , and θ ≥ 1. Given our focus on farmers’ business decisions, we
do not explicitly model the farmers’ personal ﬁnances and saving decisions. We instead
use the shift parameter c0 to capture a family’s ability to smooth variations in farm
earnings through outside income, personal assets, and other mechanisms. The scaling
parameter θ reﬂects the notion that upon retirement, the family lives for θ years and
consumes the same amount each year.
Before retirement, farmers can either work for wages or operate a farm. While working
for wages, the family’s budget constraint is
aq+1 = (1 + r)aq + w − dq ,

(3)

where: aq denotes beginning-of-period ﬁnancial assets; w denotes the age-invariant outside
wage; and r denotes the real risk-free interest rate. Workers also face a standard borrowing

constraint:
aq+1 ≥ 0.
Turning to operating farms, recall that gross revenues per operator are
γ
yq = zqt kq nq 1−α−γ ,

(4)

where kq denotes capital, nq denotes variable inputs, and zqt is a stochastic income shifter
reﬂecting both idiosyncratic and systemic factors. These factors include weather and
market prices, and are not fully known until after the farmer has committed to a production plan for the upcoming year. In particular, while the farm knows its permanent
productivity component µ, it makes its production decisions before observing the transitory eﬀects ∆t and εq .
A farm that operated in period q − 1 begins period q with debt bq and assets aq . As
˜
a matter of notation, we use bq to denote the total amount owed at the beginning of
age q: rq is the contractual interest rate used to deﬂate this quantity when it is chosen
at age q − 1. Expressing debt in this way simpliﬁes the dynamic programming problem
when interest rates are endogenous. At the beginning of period q, assets are the sum of
undepreciated capital, cash, and operating proﬁts:
aq ≡ (1 − δ +
˜

)kq−1 +

q−1

+ yq−1 − nq−1 ,

(5)

where: 0 ≤ δ ≤ 1 is the depreciation rate;
is the capital gains rate, assumed to be
constant; and q−1 denotes liquid (cash) assets, chosen in the previous period.
Because the farm enjoys limited liability, it may be able to void some of its debt via
liquidation. This leads to enforceability problems of the sort described in Kehoe and
Levine (1993). A family operating a farm thus makes an occupational decision at the
beginning of each period. The family has three options: continued operation with full
debt repayment, continued operation with reorganization, or liquidation followed by wage
work.
If the family decides to repay its debt and continue operating, it will have two sources
of funding: net worth, eq ≡ aq − bq ; and the age-q proceeds from new debt,
˜
bq+1 /(1 + rq+1 ). (We assume that all debt is one-period.) It can spend these funds in
three ways: purchasing capital; issuing dividends, dq ; or maintaining its cash reserves:
eq +

bq+1
bq+1
= aq − b q +
˜
= kq + dq +
1 + rq+1
1 + rq+1
17

(6)

Combining the previous two equations yields
iq−1 = kq − (1 − δ +

)kq−1

= [yq−1 − nq−1 − dq ] + [

q−1

−

bq+1
− bq .
1 + rq+1

(7)

Equation (7) shows that investment can be funded through three channels: retained
earnings (dq is the dividend paid after yq−1 is realized), contained in the ﬁrst set of
brackets; withdrawals from cash reserves, contained in the second set of brackets; and
additional borrowing, contained in the third set of brackets.
Operating farms face three ﬁnancial constraints:
ψbq+1 ≤ kq

(8)

nq ≤ ζ q ,

(9)

dq ≥ −c0 ,

(10)

with ψ ≥ 0 and ζ ≥ 1. The ﬁrst of these constraints, given by equation (8), is a
collateral constraint of the sort introduced by Kiyotaki and Moore (1997). Larger values
of ψ imply a tighter constraint, with farmers more dependent on equity funding. The
second constraint, given by equation (9), is a cash-in-advance or working capital constraint
(Jermann and Quadrini, 2012). Larger values of ζ imply a more relaxed constraint, with
farmers more able to fund operating expenses out of contemporaneous revenues. Because
dairy farms receive income throughout the year, in an annual model ζ is likely to exceed 1.
The third constraint, given by equation (10), limits the farm’s ability to raise funds by
issuing new equity. We also require that
bq+1 ≥ 0.

(11)

However, we allow farms to build up buﬀer stocks in the form of excess cash.
As alternatives to continued operation and full repayment, a farm can reorganize or
liquidate. If it chooses the second option, reorganization, some of its debt is written
down.15 The debt liability bq is replaced by ˆq ≤ bq , and the restructured farm continues
b
to operate. Finally, if the family decides to exit – the third option – the farm is liquidated
and assets net of liquidation costs (up to debt outstanding) are handed over to the bank.
15

Most farms have the option of reorganizing under Chapter 12 of the bankruptcy code, a special
provision designed for family farmers. Stam and Dixon (2004) review the bankruptcy options available
to farmers.

The family then exits to wage work, with assets aq where:
aq = max {(1 − λ)˜q − bq , 0} .
a
We assume that the information/liquidation costs of default are proportional to assets,
with 0 ≤ λ ≤ 1. Liquidation costs are not incurred when the family (head) retires at
age Q or if an ongoing operation decides to reduce its capital stock. While we allow the
family to roll over debt (bq can be bigger than aq ), Ponzi games are ruled out by requiring
˜
all debts to be resolved at retirement:
bQ+1 = kQ+1 = 0;

aQ+1 ≥ 0.

The interest rate realized on debt issued at age q − 1, rq = rq (sq , rq ), depends on the
state vector sq (speciﬁed below) and the contractual interest rate rq . If the farmer chooses
to repay his debt in full, rq = rq . If the farmer chooses to default,
rq =

min {(1 − λ)˜q , bq }
a
min {(1 − λ)˜q , bq }
a
− 1 = (1 + rq )
− 1.
bq /(1 + rq )
bq

The return on restructured debt is rq = (1 + rq )ˆq /bq − 1. We assume that loans are
b
supplied by a risk-neutral competitive banking sector, so that
Eq−1 (rq (sq , rq )) = r,

(12)

where r is the risk-free rate.
The decision to default or renegotiate is best expressed recursively. To simplify matters, we assume that the decision to work for wages is permanent, so that the worker’s
only decision is how much to save. The value function for a worker is thus
VqW (aq ) =

max

0≤dq ≤(1+r)aq +w

W
u(dq ) + βVq+1 (aq+1 ),

s.t. equation (3).
A family that has decided to fully repay its debt and continue farming chooses how
much income to withdraw from the farm (dq ) and how much new debt (bq+1 ) to issue. It
then allocates its ﬁnancial resources between capital (kq ) and cash ( q ). The cash, along
with revenues earned as the year proceeds, are used to buy intermediate goods (nq ). The

resulting value function is
VqF (eq , µ) =

max

{dq ,bq+1 ,nq ≥0,kq ≥0}

u(dq ) + χ + βEq (Vq+1 (˜q+1 , bq+1 , µ)) ,
a

s.t. equations (4) - (6), (8) - (12),
where Vq (·) denotes the continuation value prior to the age-q occupational choice:
Vq (˜q , bq , µ) = max
a

{V F ,V W }

VqF (˜q − min{bq , ˆq }, µ), VqW (max {(1 − λ)˜q − bq , 0}) .
a
b
a

The renegotiated debt level, ˆq , can then be expressed as
b
ˆq = max{b∗ , (1 − λ)˜q },
b
a
q
VqF (˜q − b∗ , µ) ≡ VqW (max {(1 − λ)˜q − bq , 0}),
a
a
q
so that ˆq = ˆq (sq ), with sq = {˜q , bq , µ}. The ﬁrst line of the deﬁnition ensures that
b
b
a
ˆq is incentive-compatible for lenders: the bank can always force farms into liquidation,
b
bounding ˆq from below at (1 − λ)˜q . However, if operators ﬁnd liquidation suﬃciently
b
a
unpleasant, the bank may be able to extract a larger value of b∗ . The second line ensures
q
∗
that such a payment is incentive-compatible for farmers, i.e., bq is set so that farmers are
indiﬀerent between continued operation with equity level aq − b∗ and liquidation followed
˜
q
by wage work. (We assume that once a farm chooses to renegotiate its debt, the bank
holds all the bargaining power.) Holding initial net worth ﬁxed, aq − bq , b∗ will be largest
˜
q
– and renegotiation most preferable to liquidation – when the farm is highly productive
(µ is large), or when the liquidation cost λ˜q is large.
a
ˆq (sq ) allows us to express the ﬁnance/occupation indicator I B ∈ {continue,
Solving for b
q
B
restructure, liquidate} as the function Iq (sq ). We can then divide the farm’s optimal
repayment amount by the loan’s face value and compare the realized return on the loan,
rq (sq , rq ), to the contractual interest rate rq :
ˆq (sq )
b
1 + rq (sq , rq )
B
B
= 1{Iq (sq ) = continue} + 1{Iq (sq ) = restructure} ·
1 + rq
bq
min{(1 − λ)˜, bq }
a
B
+ 1{Iq (sq ) = liquidate} ·
.
bq
Inserting this result into equation (12), we can calculate the equilibrium contractual rate

as16
1 + rq = [1 + r] / Eq−1

1 + rq (sq , rq )
1 + rq

(13)

A key feature of our model is limited liability. Dividends are bounded below by −c0 ,
the estimated value of which is small, limiting the amount the bank can extract through
renegotiation. If the farm liquidates, the bank at most receives (1 − λ)˜q . Coupled
a
with the option to become a worker, limited liability will likely lead the continuation
value function, Vq (·), to be convex over the regions of the state space where farming and
working have similar valuations (Vereshchagina and Hopenhayn, 2009).

Econometric Strategy

We estimate our model using a form of Simulated Minimum Distance (SMD). In
brief, this involves comparing summary statistics from the DFBS to summary statistics
calculated from model simulations. The parameter values that yield the “best match”
between the DFBS and the model-generated summary statistics are our estimates.
Our estimation proceeds in two steps. Following a number of papers (e.g., French,
2005; De Nardi, French and Jones, 2010), we ﬁrst calibrate or estimate some parameters
outside of the model. In our case there are four parameters. We set the real rate of return
r to 0.04, a standard value. We set the outside wage w to an annual value of $15,000,
or 2,000 hours at $7.50 an hour. As we show in Section 8, the choice of w is largely a
normalization of the occupation utility parameter χ. From the DFBS data we estimate
the capital depreciation rate δ to be 5.55 percent and the appreciation rate to be 3.56
percent, as described in the appendix. The liquidation loss, λ, is set to 35 percent. This
is at the upper range of the estimates found by Levin, Natalucci and Zakrajˇek (2004).
s
Given that a signiﬁcant portion of farm assets are site speciﬁc, high loss rates are not
implausible. We discuss a speciﬁcation with λ = 0.175 in Section 8.
In the second step, we estimate the parameter vector Ω = (β, ν, c0 , χ, c1 , θ, α, γ, n0 , λ,
ζ, ψ) using the SMD procedure itself. To construct our estimation targets, we sort farms
along two dimensions, age and size. There are two age groups: farms where the youngest
operator was 39 or younger in 2001; and farms where the youngest operator was 40 or
16

1+r (s ,r )

q
q q
The previous equation shows that the ratio
is independent of the contractual rate rq .
1+rq
Finding rq thus requires us to calculate the expected repayment rate only once, rather than at each
potential value of rq , as would be the case if debt incurred at age q − 1 were denominated in age-q − 1
terms. (In the latter case, bq would be replaced with (1 + rq )bq−1 .) This is a signiﬁcant computational
advantage.

older. This splits the sample roughly in half. We measure size as the time-averaged herd
size divided by the time-averaged number of operators. Here too, we split the sample in
half: the dividing point is between 91 and 92 cows per operator. As Section 2 suggests,
this measure corresponds closely to the ﬁxed productivity component µi . Then for each
of these four age-size cells, for each of the years 2001 to 2011, we match the following
sample moments:
1. The median value of capital per operator, k.
2. The median value of the output-to-capital ratio, y/k.
3. The median value of the variable input-to-capital ratio, n/k.
4. The median value of the gross investment-to-capital ratio.
5. The median value of the debt-to-asset ratio, b/˜
a
6. The median value of the cash-to-asset ratio, /˜.
a
7. The median value of the dividend growth rate, dt /dt−1 .17
We match medians rather than means so that extreme realizations of ﬁrm-speciﬁc ratios,
due mostly to small denominator values, do not distort our targets.
For each value of the parameter vector Ω, we ﬁnd the SMD criterion as follows. First,
we use α and γ to compute zit for each farm-year observation in the DFBS, following
equation (1). We then decompose zit according to equation (2). This yields a set of
ﬁxed eﬀects {µi }i and a set of aggregate shocks {∆t }t to be used in the model simulations
and allows us to estimate the means and standard deviations of µi , ∆t , and εiq for use in
ﬁnding the model’s decision rules. Using a bootstrap method, we take repeated draws
from the joint distribution of si0 = (µi , ai0 , bi0 , qi0 , ti0 ), where ai0 , bi0 and qi0 denote the
assets, debt and age of farm i when it is ﬁrst observed in the DFBS, and ti0 is the calendar
year it is ﬁrst observed. At the same time we draw ϑi , the complete set of dates that farm
i is observed in the DFBS.
Discretizing the asset, debt, equity and productivity grids, we use value function iteration to ﬁnd the farms’ decision rules. We then compute histories for a large number
of artiﬁcial farms. Each simulated farm j is given a draw of sj0 and the shock histories
17

Because proﬁtability levels, especially for large farms, are sensitive to total returns to scale 1 − α,
we match dividend growth, rather than levels. Both statistics measure the desire of farms to smooth
dividends, which in turn aﬀects their ability to fund investment through retained earnings.

{∆t , εjt }t . The residual shocks {εjt }jt are produced with a random number generator,
assuming a normal distribution and using the standard deviation of εiq described immediately above. The aggregate shocks are those observed in the DFBS. Combining these
shocks with the decision rules allows us to compute that farm’s history. We then construct summary statistics for the artiﬁcial data in the same way we compute them for
the DFBS. Let gmt , m ∈ {1, 2, ..., M }, t ∈ {1, 2, ..., T }, denote the realization of summary
statistic m in calendar year t, such as median capital for young, large farms in 2007.
∗
The model-predicted value of gmt is gmt (Ω). We estimate the model by minimizing the
∗
squared proportional diﬀerences between {gmt (Ω)} and {gmt }.
Because the model gives farmers the option to become workers, we also need to match
some measure of occupational choice. We do not attempt to match observed attrition,
because the DFBS does not report reasons for nonparticipation, and a number of farms
exit and re-enter the dataset. In fact, when data for a particular farm-year are missing in
the DFBS, we treat them as missing in the simulations, using our draws of ϑi . However,
we also record u = u(Ω), the fraction of farms that exit in our simulations but not in the
¯ ¯
data. We then add to the SMD criterion the penalty Ψ(¯).18
u
Our SMD criterion function is
M

m=1 t=1

∗
gmt (Ω)
−1
gmt

+ Ψ(¯).
u

Our estimate of the “true” parameter vector Ω0 is the value of Ω that minimizes this
criterion. Appendix B contains a detailed description of how we calculate standard errors.

Parameter Estimates, Goodness of Fit, and Identiﬁcation

5.1

Parameter Estimates

Column (1) of Table 4 displays parameter estimates and asymptotic standard errors
for the baseline speciﬁcation. The estimated values of the discount factor β, 0.991, and
the risk aversion coeﬃcient ν, 4.14, are both within the range of previous estimates
(see, e.g., the discussion in De Nardi et al., 2010). The retirement parameters imply
that farms greatly value post-retirement consumption; in the period before retirement,
18

We found that the penalty function (36¯)2 produced parameter estimates with reasonably low levels
u
of counterfactual exit.

farmers consume only 1.1 percent of their wealth and save the rest.19 The nonpecuniary
beneﬁt of farming χ, is expressed as a consumption decrement to the nonfarm wage w.
Mechanically, we set
χ=

1
1
(c0 + w)1−ν −
(c0 + w − χC )1−ν
1−ν
1−ν

and estimate (and report) χC . This quantity can be interpreted as the equivalent variation
for a switch from farming to wage work: how much consumption would a farmer surrender
to avoid this switch? With w equal to $15,000, the estimates imply that the psychic beneﬁt
from farming would oﬀset a $5,360 (35.7 percent) drop in consumption. Given the high
estimated value of ν, this translates into a large drop in utility.20 Even though the outside
wage is modest, the income streams of low-productivity farms are so small and uncertain
that some operators would exit if they did not receive signiﬁcant psychic beneﬁts.
The returns to management and capital are both fairly small, implying that the returns
to intermediate goods, 1 − α − γ, are between 69 and 77 percent. Table 1 shows that
variable inputs in fact equal about 78 percent of revenues. The collateral constraint
parameter ψ is 1.06, implying that each dollar of debt must be backed by a roughly
equivalent amount of capital. The liquidity constraint parameter ζ is estimated to be
about 2.83, implying that farms need to hold liquid assets equal to about four months
of expenditures. Although these two constraints together signiﬁcantly reduce the risk of
insolvency, farms with adverse cash ﬂow may ﬁnd themselves extremely illiquid.

This can be found by solving for optimal retirement wealth in the penultimate period of the operator’s

economically active life, ar (x) = argmax
ar ≥0

1
1−ν

1−ν

(c0 + x − ar )

θ
+ β 1−ν c1 +

ar (1+r)
θ

1−ν

, and calcu-

lating ∂ar (x)/∂x|ar (x)>0 . A derivation based on a similar speciﬁcation appears in De Nardi et al. (2010).
20
In previous drafts of this paper, we expressed the nonpecuniary beneﬁt as a consumption increase.
Given the curvature of the utility function, this results in a much higher value of consumption. In the
case at hand, the utility lost by decreasing consumption by $5,360 exceeds the utility gained by increasing
consumption by $100,000.

Speciﬁcation
Baseline λ = 0.175 w = $30K
(1)
(2)
(3)

Parameter Description
Discount factor

Risk aversion

Consumption utility shifter (in $000s) c0
c1

Retirement utility intensity
25

Retirement utility shifter (in $000s)

Nonpecuniary value of farming
(consumption decrement in $000s)
Returns to management: stanchions

χC

Returns to management: parlors

Returns to capital: stanchions

Returns to capital: parlors

Strength of collateral constraint

Degree of liquidity constraint

χ=0
(4)

ψ=0
(5)

No Renegotiation
(6)

0.991
(0.009)
4.143
(0.089)
4.561
(0.097)
20.89
(0.97)
92.96
(3.94)
5.365
(0.557)
0.135
(0.005)
0.107
(0.002)
0.174
(0.004)
0.122
(0.003)
1.062
(0.036)
2.827
(0.113)

0.996
(0.010)
4.806
(0.096)
3.239
(0.097)
16.99
(0.15)
63.03
(0.96)
0
(N.A.)
0.159
(0.003)
0.184
(0.004)
0.161
(0.003)
0.102
(0.002)
0.763
(0.003)
2.867
(0.060)

0.996
(0.007)
5.095
(0.035)
3.229
(0.080)
16.53
(0.15)
63.80
(0.67)
2.96
(0.122)
0.157
(0.0007)
0.183
(0.0007)
0.162
(0.0008)
0.103
(0.002)
0
(N.A.)
2.882
(0.054)

0.991
(0.013)
4.115
(0.078)
4.492
(0.016)
20.76
(0.10)
94.04
(1.35)
7.215
(0.275)
0.134
(0.0003)
0.107
(0.002)
0.173
(0.004)
0.122
(0.002)
1.064
(0.045)
2.830
(0.056)

0.991
(0.102)
4.113
(0.417)
4.546
(1.299)
20.77
(1.593)
92.93
(17.92)
6.156
(1.013)
0.135
(0.004)
0.108
(0.007)
0.174
(0.002)
0.122
(0.016)
1.063
(0.135)
2.831
(0.208)

0.991
(0.112)
4.130
(1.776)
4.571
(4.199)
20.77
(12.91)
93.38
(18.39)
21.87
(1.884)
0.135
(0.022)
0.107
(0.042)
0.174
(0.031)
0.122
(0.049)
1.063
(0.372)
2.833
(2.281)

Table 4: Parameter Estimates
Notes: Standard errors in parentheses.

5.2

Goodness of Fit

Figures 4 and 5 compare the model’s predictions to the data targets. To distinguish
the younger and older cohorts, the horizontal axis measures the average operator age of
a cohort at a given calendar year. The ﬁrst observation on each panel starts at age 30:
this is the average age of the youngest operator in the junior cohort in 2001. Observations
for age 31 correspond to values for this cohort in 2002. When ﬁrst observed in 2001, the
senior cohort has an average age of 48. As before, thick lines denote large farms, and thin
lines denote smaller farms. For the most part the model ﬁts the data well, although it
understates the capital holdings of older farms and overstates dividend growth. However,
it captures many of the diﬀerences between large and small farms and much of the yearto-year variation.
Median Capital: Data (Solid) vs. Model (Dashed)

Median N/K Ratio: Data (Solid) vs. Model (Dashed)
0.5

2500

0.45

2000

0.4

N/K Ratio

Capital (000s of 2011 dollars)

3000

1500

0.35

1000

0.3

500

0.25

0
28

0.2
28

Age

Median Output/Capital Ratio: Data (Solid) vs. Model (Dashed)
0.65

Output/Capital Ratio

0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
28

Age

Figure 4: Model Fits: Production Measures
Notes: Solid lines refer to DFBS data, dashed lines to model simulations. Thicker black lines refer to
farms with large herds, thinner green lines refer to farms with small herds.

Our estimation criterion includes a penalty for “false exits,” simulated farms that exit
when their data counterparts do not. False exit is uncommon, with a frequency of 2.1
26

percent. We also assess the model’s ﬁt of a number of untargeted cross-sectional moments.
As shown in Appendix C, the model does a satisfactory job along this dimension as well.

5.3

Identiﬁcation

Although all of the simulated moments depend on multiple model parameters, for
some parameters identiﬁcation is straightforward. The production coeﬃcients α and
γ are identiﬁed by expenditure shares and the extent to which farm size varies with
productivity. The cash constraint ζ is identiﬁed by the observed cash/asset ratio. The
parameter χ is identiﬁed by the counterfactual exit that would occur in its absence. The
parameter ψ, measuring the strength of the collateral constraint, is identiﬁed by two
features of the data: (1) high-productivity farmers expand their capital stock steadily
over time, rather than all at once; and (2) for farms of all sizes, years of high income are
years of high investment.
Median GI/K Ratio: Data (Solid) vs. Model (Dashed)

Median Dividend Growth: Data (Solid) vs. Model (Dashed)

0.16

1.1

0.14

Dividend Growth

GI/K Ratio

0.12
0.1
0.08
0.06

0.9

0.8

0.7

0.6
0.04
0.5

0.02
0
28

0.4
28

Age

Median Debt/Asset Ratio: Data (Solid) vs. Model (Dashed)

Median Cash/Asset Ratio: Data (Solid) vs. Model (Dashed)

0.7

0.16

0.65

0.15
0.14

Cash/Asset Ratio

0.6

Debt/Asset Ratio

Age

0.55
0.5
0.45
0.4
0.35

0.12
0.11
0.1
0.09
0.08

0.3
0.25
28

0.13

0.07
30

0.06
28

Age

Figure 5: Model Fits: Financial Measures
Notes: Solid lines refer to DFBS data, dashed lines to model simulations. Thicker black lines refer to
farms with large herds, thinner green lines refer to farms with small herds.

Fraction
Operating∗

Assets

Debt

Debt/
Assets†

Cash/
Assets†

Capital N/K †

Invest- Dividend
ment/
Growth
Capital
Rate
Optimal
(%)†
(%)‡
Capital

Baseline Model

1.000

1,964

1,052

0.536

0.114

1,593

0.378

6.59%

4.31%

2,833

(2)
(3)
(4)
(5)
(6)
28

(1)

β = 0.95
ν = 0.0
ν = 6.0
c0 = 200
c0 = 0

1.002
0.928
1.002
0.992
1.002

1,890
2,610
1,886
2,069
1,949

1,044
1,268
982
1,203
1,040

0.552
0.486
0.521
0.581
0.534

0.112
0.115
0.114
0.113
0.114

1,521
2,218
1,514
1,694
1,578

0.378
0.365
0.381
0.381
0.379

6.25%
7.86%
6.50%
6.60%
6.55%

3.46%
N.A.
4.12%
5.67%
4.29%

2,838
3,080
2,839
2,903
2,839

0.920
0.965
0.780

2,104
1,959
2,302

1,140
1,059
1,282

0.542
0.541
0.557

0.114
0.116
0.117

1,704
1,607
1,895

0.381
0.379
0.386

6.75%
6.40%
6.51%

4.36%
3.77%
3.30%

3,070
2,751
3,327

(10) ψ = 0.5
(11) ψ = 1.5

1.021
1.000

2,405
1,515

1,482
636

0.616
0.420

0.115
0.118

2,006
1,157

0.366
0.419

2.84%
8.43%

3.36%
4.75%

2,827
2,833

(12) ζ = 1
(13) ζ = 6

0.995
1.001

1,671
2,089

809
1,158

0.484
0.554

0.244
0.060

1,111
1,819

0.363
0.377

7.30%
6.08%

4.79%
4.20%

2,845
2,838

(14) No Aggregate Shocks
(15) No Transitory Shocks

1.000
1.000

1,988
2,043

1,089
1,169

0.548
0.572

0.114
0.116

1,616
1,671

0.381
0.386

6.58%
6.59%

4.43%
4.22%

2,833
2,833

(7) χ = 0
(8) λ = 0
(9) λ = χ = 0

∗ Relative

to baseline case. † Ratios of averages. ‡ Mean growth rates for annual averages. N.A. indicates negative initial dividends.

Table 5: Comparative Statics

The identiﬁcation of other parameters is more complicated and best illustrated through
comparative statics. Table 5 shows averages of model-simulated data over the 11-year
(pseudo-) sample period. Row (1) shows the statistics for the baseline model associated
with the parameters in column (1) of Table 4, while subsequent rows show the statistics
that arise as we vary diﬀerent parameters or features of the model.
Row (2) shows the averages that result when the discount factor β is lowered to 0.95.21
Farms hold less capital and invest less, as they place less weight on future returns. They
take on more debt, for the same reason. Dividends grow more slowly, with the average
growth rate falling from 4.31 to 3.46 percent.
Row (3) shows the eﬀects of setting the curvature parameter ν to zero, so that preferences are linear in dividends. Linearity leads farmers to invest more aggressively in
capital, as they are less concerned about uncertain returns and more willing to defer consumption. The average investment rate rises signiﬁcantly, while the average capital stock
increases from $1.59 to $2.22 million. Farms pay for this additional capital in several
ways. Dividends are initially negative, as the farmers raise funds internally. Farms also
raise more funds by borrowing. However, as Figure 6 shows, farms also deleverage more
quickly in later years. This is because the borrowing rate r = 0.04 exceeds the discount
rate of 0.009, and the opportunity cost of retained earnings – unsmoothed dividends – is
zero.22
Row (4) shows the eﬀects of raising ν from 4.14 to 6. Relative to the baseline case,
the dividend growth rate is lower, as farms withdraw more funds up front. Debt is also
lower, perhaps for precautionary reasons. The result is that the capital stock is lower in
every period: the average stock falls to $1.51 million.
Row (5) shows the eﬀects of increasing the utility shifter c0 to 200.23 In addition to
serving as a preference parameter, c0 limits the ability of farms to raise funds from equity
injections. A value of c0 of 200 thus allows farmers to inject up to $200,000 of personal
funds into their farms each year. Capital and assets increase. Average dividends grow
more quickly, as equity injections lead initial dividends to be low. Finally, increasing c0
reduces risk aversion, encouraging farms to take on more debt. Row (6) shows the eﬀects
of changing c0 in the opposite direction, to 0. Most variables move in the directions
opposite to those in row (5).
21

We restrict β to lie in the (0, 1) interval.
Row (3) also shows that the number of operating farms fall. This is an artifact of how we calculate
the nonpecuniary beneﬁt. When preferences are linear in consumption, the utility value of a $5,360 drop
in consumption is much smaller than in the baseline speciﬁcation. With small nonpecuniary beneﬁts,
unproductive farms have less incentive to operate.
23
We also increase the retirement shifter c1 by an equivalent amount.
22

Median Debt/Asset Ratio: Experiment (Solid) vs. Baseline (Dashed)
0.7
0.65

Debt/Asset Ratio

0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
28

Age

Figure 6: The Eﬀects of Imposing Linear Utility on the Debt/Asset Ratio
Note: Solid lines refer to model simulations with ν = 0, dashed lines to baseline simulations. Thicker
lines refer to farms with large herds, thinner lines refer to farms with small herds.

Because the discount factor β, the risk coeﬃcient ν, and the utility shifter c0 aﬀect
similar summary statistics – dividend growth, capital, and debt – identiﬁcation occurs
jointly. It is informative to consider the simple unconstrained Euler Equation
(c0 + dt )−ν = β(1 + ιt+1 )(c0 + dt+1 )−ν ,
where ι denotes the returns the farm enjoys on its capital expenditures. An important
feature of the data is that the average dividend growth rate is modest for both small
and large farms. Moreover, because large farms are expanding while small farms are not,
the model requires that the marginal product of capital be higher in large operations, to
justify the diﬀerent capital stock trajectories. This means that large farms have higher
values of ιt+1 . In order for both types of farms to have ﬂat dividend trajectories, ν must
be large and c0 must be small.24 Given this requirement, the parameters further adjust
to help the model match capital and debt. Another useful distinction is that raising c0 ,
which increases the scope for equity injections, is more eﬀective in allowing farmers to
acquire capital up front than is lowering ν. While the capital stock rises relative to the
24

One shortcoming of the model is that it does not match the high volatility of dividend growth observed
in the data. This is because values of ν that lead to low average dividend growth rates also dampen
dividend growth variation. An interesting extension that we do not pursue here would be to introduce
preferences (e.g., Epstein-Zin, 1989) that uncouple risk aversion and intertemporal substitutability.

baseline in both rows (3) and (5), the investment rate rises signiﬁcantly in row (3) but
rises only slightly in row (5).
The retirement parameters c1 and θ are identiﬁed by life-cycle variation not shown in
Table 5. As θ goes to zero and retirement utility vanishes, older farmers will have less
incentive to invest in capital, and their capital stock will fall relative to that of younger
farmers. Setting θ to zero also increases indebtedness, as farmers raise their average
dividend (not shown) by over 40 percent.

6
6.1

Model Mechanisms
Nonpecuniary beneﬁts

Our estimates imply that the nonpecuniary beneﬁt from farming is equivalent to the
ﬂow utility lost by decreasing consumption from $15,000 to $9,640. The parameter χ is
identiﬁed by occupational choice, namely the estimation criterion that farms observed in
the DFBS in a given year also be operating and thus observed in the simulations. Row (7)
of Table 5 shows the eﬀects of setting χ to zero. In the absence of psychic beneﬁts more
farms liquidate, so that the average number of operating farms drops by 7 percent. Not
surprisingly, it is the smaller, low-productivity farms that exit: the survivors in row (7)
have more assets, debt and capital. Their optimal capital stock (recall Figure 3) rises from
$2.83 to $3.07 million. Hamilton (2000) and Moskowitz and Vissing-Jørgensen (2002)
ﬁnd that many entrepreneurs earn below-market returns, suggesting that nonpecuniary
beneﬁts are large. (Also see Quadrini, 2009; and Hall and Woodward, 2010.) Similarly,
Figure 3 shows that many low-productivity farms have dividend ﬂows around the outside
salary of $15,000. Moreover, these ﬂows are uncertain, while the outside salary is not.
This is consistent with a high value of χ.
The high value of χ may reﬂect other considerations, such as eﬃciencies in home production or tax advantages.25 It may also be the case that farm income is under-reported
(Herrendorf and Schoellman, 2015). Furthermore, although $15,000 is roughly equivalent to the Federal Poverty Line for a two-person household, it may overstate the outside
earnings available to farmers. Poschke (2012, 2013) documents that the probability
of entrepreneurship is “U-shaped” in an individual’s prior nonentrepreneurial wage, and
argues that many low-productivity entrepreneurs start and maintain their businesses because their outside options are even worse. Herrendorf and Schoellman (2016) conclude
25

For example, farmers may be able to report (or misreport) personal consumption, such as the use of
a farm vehicle, as operating expenditures. We are indebted to Todd Schoellman for this point.

that the low wages of agricultural workers reﬂect low levels of human capital.
On the other hand, there is considerable direct evidence suggesting that farming,
and entrepreneurship in general, provides large nonpecuniary rewards. Recent surveys of
national well-being, published by the Oﬃce for National Statistics in the U.K., show that
the levels of life satisfaction of farmers and farm workers rank among the highest for all
occupations and are substantially higher than the levels of life satisfaction reported by
individuals in occupations with similar incomes such as construction and telephone sales
(O’Donnell et al., 2014). Looking across all sectors, Hurst and Pugsley (2011) ﬁnd that
nonpecuniary considerations “play a ﬁrst-order role in the business formation decision”
and that many small businesses have “no desire to grow big.” Both attitudes appear
consistent with the behavior of the small farms in our sample.

6.2

Financial Constraints

Our model contains three important ﬁnancial frictions: liquidation costs, collateral
constraints and liquidity constraints. We consider the eﬀects of each element on assets,
debt, capital, investment, and exit.
6.2.1

Liquidation Costs

Row (8) shows the eﬀects of setting the liquidation cost λ to zero. Eliminating the
liquidation cost reduces the number of operating farms by allowing farmers to retain more
of their wealth after exiting.26 Liquidation costs thus provide another explanation of why
entrepreneurs may persist despite low ﬁnancial returns. The eﬀect of setting λ = 0 is in
many ways similar to that of eliminating the psychic beneﬁt χ. This lack of identiﬁcation
is one reason why we calibrate rather than estimate λ. Row (9) shows that nonpecuniary
beneﬁts and liquidation costs reinforce each other; setting λ = χ = 0 leads over 20 percent
of the farms to exit.
Comparing row (8) to row (1) shows that the farms that remain after removing the
liquidation cost have more capital and assets, but are not more productive. This suggests
that the collateral and liquidity constraints devalue some productive farms. However,
comparing row (9) to row (7) shows that in the absence of nonpecuniary beneﬁts, the
farms that remain after removing the liquidation costs are signiﬁcantly more productive.
Liquidation costs can thus generate ﬁnancial ineﬃciency by discouraging the reallocation
of capital and labor to more productive uses.
26

Recall that we assume that liquidation costs are not imposed upon retiring farmers.

Median Capital: Experiment (Solid) vs. Baseline (Dashed)

Capital (000s of 2011 dollars)

3500

3000

2500

2000

1500

1000

500

0
28

Age

Figure 7: The Eﬀects of Relaxing the Collateral Constraint on the Capital Stock
Note: Solid lines refer to model simulations with ψ = 0.5, dashed lines to baseline simulations. Thicker
lines refer to farms with large herds, thinner lines refer to farms with small herds.

6.2.2

Collateral Constraints

Row (10) shows the eﬀects of setting the collateral constraint ψ to 0.5, allowing each
dollar of capital to back up to $2 of debt. Farms respond to the relaxed constraint by
borrowing more and acquiring more capital, with mean capital rising from $1.59 million to
$2.01 million. Much of this additional capital is purchased up front; the investment rate
falls from 6.6 percent to 2.8 percent. Figure 7, which compares capital stock trajectories,
shows that the increase in initial capital is concentrated in the large/high-productivity
farms, suggesting again that borrowing constraints are causing capital to be misallocated
across farms.
Row (11) of Table 5 shows the eﬀects of the opposite experiment, setting ψ to 1.5.
Tightening the constraint this much leads farms to drastically reduce their capital stock,
by 27 percent of its baseline value. Farms now accumulate their capital through retained
earnings. With capital more diﬃcult to fund, farms use more intermediate goods, so that
the fall in output, 19 percent, is smaller than the fall in capital. All of these changes
make farming less proﬁtable, and fewer farms remain in operation.

6.2.3

Liquidity Constraints

Rows (12) and (13) of Table 5 illustrate the eﬀects of the liquidity constraint, given by
equation (9). Row (12) shows what happens when we tighten this constraint by reducing
ζ to 1. Even though fewer farms remain in business, the average scale of operations
declines. While total assets fall by around 15 percent, capital falls by over 30 percent,
and the cash/asset ratio jumps from 0.114 to 0.244. Rather than holding their assets
in the form of capital, farms are obliged to hold it in the form of liquid assets used to
purchase intermediate goods. Output falls by over 30 percent.
Loosening the liquidity constraint (ζ = 4) allows farms to hold a larger fraction of
their assets in productive capital, raising the assets’ overall return. Total assets rise from
their baseline value by 6.4 percent, while capital rises even more, by 14.2 percent.

6.3

Overview

To sum up: our estimates and comparative statics exercises indicate that ﬁnancial
factors play an important role in farm outcomes both at the intensive and extensive
margin. The collateral and liquidity constraints hinder capital investment, reduce output
and assets, and sometimes drive farms out of business.27 Liquidation costs impede the
exit of low productivity farms, by reducing the wealth they can carry into their new
occupation.
Our analysis also reveals that nonpecuniary beneﬁts are a signiﬁcant motivating force.
They keep farms in operation, despite low and uncertain revenue ﬂows, reinforcing the
eﬀects of the liquidation costs. When both mechanisms are in place, only a few highly
unproductive farms choose to exit.

6.4

External Validity

The questions of external validity that plague most empirical studies also apply to
structural analyses of ﬁrm behavior, where nonlinear speciﬁcations and cross-industry
heterogeneity are the norm (Strebulaev and Whited, 2012). In this section we explain
why we believe our ﬁndings can be generalized beyond our dataset and beyond the dairy
farm industry.
The Statistics of US Businesses (SUSB) generated by the US Census Bureau reveal
27

Similar borrowing constraints have been shown to play an important role in ﬁnancial crises in Latin
America and East Asia (see for example Pratap and Urrutia, 2012; or Mendoza, 2010).

Distribution of Firm Revenues
0.60

0.40

0.20

10.0 to 15.0

7.5 to 10.0

5.0 to 7.5

Our Sample

More than 15

All US

2.5 to 5.0

1.0 to 2.5

0.5 to 1.0

0.1 to 0.5

Less than 0.1

0.00

Figure 8: Distribution of Firm Revenues

important similarities between our data and all US businesses.28 In 2012, 78 percent
of all US ﬁrms (excluding government and nonproﬁt enterprises) were single-proprietor
operations, partnerships or S Corporations. This is a group for which ﬁnancial data are
not readily available. Since all the farms in our sample are in this category, we can shed
some light on an often neglected set of ﬁrms.
The revenues of the average ﬁrm in the US are not dissimilar to revenue of the average
farm in our dataset.29 Among single proprietorships, partnerships and S Corporations,
average ﬁrm receipts were $2.76 million in 2012.30 The farms in our dataset had average
receipts of $2.89 million in 2011, the last year for which we have data.
The distribution of revenues in the US is also comparable with our data, as Figure 8
shows. Since the data are not separated by ownership type, it is not surprising that our
sample under-represents ﬁrms receiving more than $15 million annually. It also underrepresents very small enterprises, with less than $100,000 in annual revenue. However, it
is quite similar in the middle of the distribution.
Our results also comport, while diﬀering in sensible ways, with results from earlier
studies. One of our key ﬁndings is that ﬁnancial constraints, particularly collateral con28

These data come from the publicly available datasets on the Census Bureau’s website.
http://www.census.gov/data/tables/2012/econ/susb/2012-susb-annual.html
29
Since the Census Bureau does not publish further details about establishments in the SUSB we are
unable to compare ﬁrms along other dimensions.
30
Publicly available data for receipts of all US ﬁrms exists only for the Economic Census years, the last
of which was 2012.

straints, inhibit investment. Recall that our estimate of the constraint parameter ψ in
equation (8) is 1.06, implying that each dollar of debt must be backed by slightly more
than a dollar of capital. Earlier studies report more stringent constraints. Looking at
the decision to become an entrepreneur in the National Longitudinal Survey of Young
Men, Evans and Jovanovic (1989) ﬁnd ψ to be at least 2.37, implying that each dollar of
debt be backed by over $2 of capital. Buera (2009) studies the same issue in the Panel
Survey of Income Dynamics and ﬁnds a value for ψ of 1.26 in the constrained version of
his estimates, and 101 in the unconstrained version. Working with the Survey of Small
Business Finances (SSBF), Herranz et al. (2014) calibrate ψ to 6.31
Given that the ﬁrms in our sample are established businesses, it is not surprising that
we ﬁnd looser constraints than those estimated on ﬁrms that are starting up. While SSBF
respondents are established ﬁrms, many are quite small and likely have less access to debt
than our farms. Our conclusion that these constraints are important should be robust.
The other aspect of ﬁrm behavior our analysis highlights is the supplemental utility
from self-employment. Such beneﬁts are often referred to as “procedural utility,” i.e., utility gained from procedures, rather than outcomes (Benz and Frey, 2008). As we discussed
above, there is substantial indirect evidence of the nonpecuniary beneﬁts of entrepreneurship from the studies on the returns to the self-employed relative to other groups. There
is also a large body of direct evidence from survey literature where entrepreneurs report
high levels of job satisfaction relative to wage work. These estimates all suggest that
nonpecuniary beneﬁts of entrepreneurship are widespread across industries and countries
and not a special feature of agriculture.
Farmers willing to participate in the DFBS may well receive higher nonpecuniary
beneﬁts from their occupation than nonparticipants. Evidence of heterogeneity in the
procedural utility from self-employment is found by Fuchs-Schuldeln (2009), who shows
that job satisfaction from entrepreneurship is the largest for individuals who value independence. Binder and Coad (2013) ﬁnd that self-employed individuals who moved from
wage work experience larger changes in happiness levels than those who moved from
unemployment. Our ﬁndings nonetheless accord with a large body of evidence.
To sum, our data are drawn from a set of ﬁrms that are similar to a large fraction of
the ﬁrms in the United States, and our ﬁndings on ﬁnancial constraints and nonpecuniary
beneﬁts are consistent with many earlier, albeit piecemeal, analyses. We therefore expect
our results to be relevant across industries and regions.
31

Evans and Jovanovic (1989) and Buera (2009) do not estimate ψ itself, but the parameter constraint
k ≤ φa. Assuming that capital is the sum of debt and assets, k = a + b, and we have ψ = φ/(φ − 1).
Herranz et. at. (2014) work with the constraint b ≤ υa, with ψ = υ/(1 + υ).

The Eﬀects of Uninsured Risk

As discussed in Sections 5 and 5.3, our estimates suggest that our entrepreneurs are
very risk averse, with a coeﬃcient of relative risk aversion (ν) of about 4.14. This parameter is identiﬁed by the low observed dividend growth rates, as higher values of ν make
entrepreneurs less willing to substitute dividends across time. Table 5 shows that linear
utility would lead farmers to choose negative dividends at the beginning of the estimation
period, resulting in counterfactually high dividend growth.
Section 2 showed that farmers face signiﬁcant uninsured risk, which can be decomposed into an aggregate and an idiosyncratic component. Aggregate risk is in turn closely
related to ﬂuctuations in milk prices. This suggests a potentially useful role for government programs that insure farmers against milk price ﬂuctuations, as envisaged by
various dairy support programs.32 Before considering the speciﬁc provisions of the latest
program, as formulated in the 2014 Farm Bill, we ﬁrst examine the eﬀects of aggregate
and idiosyncratic risk in general.

7.1

Full Insurance

Comparing row (14) of Table 5 to the baseline model in row (1) shows the eﬀects of
shutting down aggregate risk, keeping mean productivity constant. Such a change can
be viewed as the introduction of complete insurance against aggregate shocks. Farms
expand operations by increasing debt, and using it to ﬁnance purchases both ﬁxed and
variable inputs. The average capital stock increases by about 1.4 percent. This increase
is modest, as the farms are still subject to idiosyncratic shocks and operators are risk
averse.
Like the aggregate shock, the idiosyncratic shock is also i.i.d and has a similar standard
deviation (6 percent compared to 7 percent for the aggregate shock). Row (15) shows that
the eﬀects of eliminating all transitory shocks (aggregate and idiosyncratic) are larger, but
qualitatively similar, to the results shown in row (14). The average operation increases
its capital stock by almost 5 percent and its debt by over 11 percent.
It is worth noting that the elimination of transitory risk, aggregate or idiosyncratic,
has little eﬀect on the extensive margin. The fraction of farms operating is virtually
unchanged, as is their time-invariant productivity level, as measured by the optimal capital
stock.
Rows (14) and (15) suggest that risk discourages investment, as reducing risk leads to
32

Within the DFBS, the use of derivatives to reduce milk price risk is limited.

16
12

Milk Margins
8
10

19
Milk Price

2014

2012

2010

2008

2006

2004

2002

2000

2014

2012

2010

2008

2006

2004

2002

2000

14
9

Year

Figure 9: Milk Prices and Margins
higher capital. In addition to risk aversion, ﬁnancial incentives induce farms to behave
this way. Although our model includes limited liability and occupational choice, most
low-productivity farms enter our sample with low levels of indebtedness and (relative to
the productivity ﬁxed eﬀect µi ) high capital stocks (see Figures 2 and 3). In such circumstances the risk-taking incentives described by Vereshchagina and Hopenhayn (2009)
are less likely to apply. Vereshchagina and Hopenhayn also argue that patient ﬁrms are
less likely to seek risky investment projects; our estimated discount rate is 0.9 percent.
Our data do not reject Vereshchagina and Hopenhayn’s proposed mechanisms, but they
suggest an environment where the mechanisms are unlikely to arise. Our results are more
in line with Caggese (2012), who ﬁnds that risk discourages entrepreneurial innovation.
On the other hand, risk discourages investment and production only modestly. As the
previous section shows, collateral and liquidity constraints have much larger eﬀects.

7.2

The Farm Bill of 2014

The dairy provisions of the Farm Bill of 2014 replaced a largely defunct dairy price
support program. Although the former program guaranteed a statutory price for milk,
either through direct purchase or through the purchase of other dairy products,33 the support price of $9.90 per hundred pounds (cwt.) of milk was widely considered inadequate.
As the left panel of Figure 9 shows, by 2000 the national average milk price, to which
the support price was indexed, was always substantially higher than $9.90 per cwt., while
still volatile.
This situation, coupled with an increase in feed costs, provided the impetus for a policy
33

Our description of the dairy provisions of the 2014 Farm Bill and its predecessors borrows heavily
from the discussion in Schnepf (2014).

Capital

Debt

Debt/
Assets

∗

1,596

1,054

0.536

0.113

6.61%

2,833

Margin Support, No Premium
Margin Support, Full Premium

1,623
1,509

1,080
978

0.542
0.523

0.113
0.112

6.72%
6.41%

2,832
2,457

Baseline Model

∗

Cash/
Assets

Investment/
Capital (%)

Optimal
Capital

Shock process revised to accommodate Farm Bill experiments. See Appendix D.

Table 6: Operational Eﬀects of the 2014 Farm Bill
change toward margin support, rather than price support. The milk margin is deﬁned as
the diﬀerence between the price of milk and the weighted average of the prices of corn,
soybean and alfalfa. The program oﬀers a baseline margin support of $4.00 per cwt. for
all participants and higher support levels in exchange for a premium. The right panel of
Figure 9 shows nominal milk margins, as calculated by Schnepf (2014), between January
2000 and September 2014. Margin supports in the range of $4 to $8 would have kicked
in several times in our sample period.
The close correlation between milk prices and the aggregate productivity shock, along
with the assumption of constant input prices, imply that within our framework, the aggregate shock acts as a margin shock. We therefore model margin ﬂoors by eliminating the
left-hand tail of the aggregate shock distribution. As Schnepf (2014) notes, the premium
structure is intended to encourage farmers to choose a margin support level of $6.50 per
cwt. This translates into truncating the aggregate shock distribution at the 9th percentile.
Appendix D provides more details on the margin support program, the calculation of the
truncation level, and the computational mechanics of truncating the distribution.
Table 6 shows the eﬀects of a margin support program of $6.50. The ﬁrst row of the
table shows the no-farm bill benchmark.34 The second row shows the eﬀect of a $6.50
margin support that is provided to farmers for free. The eﬀects of the margin support are
modest. The capital stock increases by about 1.7 percent and debt by 2.5 percent. The
extensive margin – the number of farms operating, not reported in the table – is essentially
unchanged. Recalling row (14) of Table 5 shows that the eﬀects of the margin support,
which eliminates the worst downside risk, are similar to those of completely eliminating
aggregate risk.
34
Assessing the Farm Bill requires that we use a discretized shock process with many more grid points
than in the baseline speciﬁcation used to estimate the model. See Appendix D. Using a ﬁner shock grid
signiﬁcantly increases computation time. However, the simulations generated by this case diﬀer little
from those of the baseline speciﬁcation.

Median
Margin Support, No Premium
Margin Support, Full Premium

Mean

Min.

4.06
4.92
0.18
-16.20 -18.75 -42.70

Correlation
Max.
with µ
24.08
-0.91

0.63
-0.58

Table 7: Welfare Eﬀects of the 2014 Farm Bill
Notes: Welfare measured as the increase in equity needed in the baseline model to achieve the welfare
level found in each experiment. All numbers expressed as percentages of initial equity.

Row (3) of Table 6 shows the eﬀects of coupling the margin support with the legislated
premium (for large volumes) of $0.29 per cwt. Even though this premium equals only
about 1.6 percent of the average milk price, it signiﬁcantly reduces the scale of operations.
Under our speciﬁcation, returns to scale per operator are given by 1 − α, with α estimated
to be between 0.107 and 0.135 (see Table 4). This means that in the absence of frictions,
the optimal scale of operations is quite sensitive to productivity, so that even a small fee
can generate a signiﬁcant contraction. The ﬁnal column of Table 6 shows that the margin
support premium reduces the optimal capital stock by more than 13 percent. The actual
capital stock falls by about 5.5 percent, however, as most farms are below their eﬃcient
size.
The small (or negative) eﬀects of the Farm Bill on production may be consistent with
large increases in the farmers’ welfare, given how risk averse they are. To study welfare
eﬀects, we compute the additional equity each farm would need in the baseline model to
achieve the lifetime utility that it would get in the model with margin supports. This
supplement is then expressed as a fraction of the farm’s initial equity.
Table 7 summarizes the welfare eﬀects of the margin support program, ﬁrst without
and then with premia. The ﬁrst row of Table 7 shows that receiving free margin supports is
equivalent to a once and for all 4 percent increase in equity for the median farm, equivalent
to about $14,400. There is some variance in these beneﬁts, with the largest being about
24 percent. Margin supports thus increase welfare, with the largest gains accruing to highproductivity farms, as shown by the positive correlation in the last column. However, the
negative eﬀects of the premium on production are matched by negative eﬀects on welfare.
The net eﬀect of the Farm Bill with a premium is equivalent to a 16 percent fall in equity
for the median farm. The average fall is about 19 percent, similar in magnitude to the
13 percent fall in optimal capital, whereas the largest losses are as much as 43 percent of
initial equity. High-productivity farms face the largest losses.

In short, we ﬁnd that the negative eﬀects of the margin support premium outweigh
the small positive eﬀects of the support itself. It is possible that our model, which is
calibrated to an annual frequency, understates the eﬀects of the margin support program,
which operates at a two-month frequency. Schnepf (2014) shows that milk margins change
signiﬁcantly from month to month. Moreover, the cash holdings observed in the DFBS,
recorded at the beginnings and ends of calendar years, may understate farms’ actual liquidity needs, which tend to be highest in the summer.35 The combination of seasonal cash
shortages and monthly margin ﬂuctuations may enhance the impact of margin support.
Finally, the distribution of aggregate shocks is based on data from our sample period of
2001-2011. As Figure 9 shows, milk margins fell below the ﬂoor in 2012. If the risk of low
margins is greater than that found in our sample, margin supports will be more valuable
than our estimates suggest.

Sensitivity Analyses

As we discussed above, nonpecuniary beneﬁts (χ) and liquidation costs (λ) both discourage the exit of low-productivity farms, and are thus diﬃcult to identify simultaneously. We also argued that the nonpecuniary beneﬁt is positively related to the outside
wage w: a low-productivity farm facing a high outside wage will remain in operation only
if the nonpecuniary beneﬁt to farming is also high. In this section we formally explore
the sensititivity of our results to alternative values of λ, χ, and w, by changing these
parameters and re-estimating the model. We also estimate versions of the model with no
nonpecuniary beneﬁts, no collateral constraint or no renegotiation. The parameter estimates for these alternative speciﬁcations can be found in rows (2)-(6) of Table 4. Table
8 compares predictions of a few key variables.

8.1

Liquidation Costs

Our benchmark estimate of λ, a loss rate of 0.35, is somewhat higher than those reported in the ﬁnance literature (see, e.g., Andrade and Kaplan, 1998; or Hennessy and
Whited, 2007). When λ is cut in half, to 0.175, the estimated value of the nonpecuniary beneﬁt χC increases by about 15 percent, from $5,365 to $6,155, so that the model
continues to match the participation observed in the data. The other parameters of the
model are essentially unaﬀected, and row (2) of Table 8 shows that the model-predicted
35

Conversation with Wayne Knoblauch, April 20, 2015.

moments are unaﬀected as well.

8.2

Value of the Outside Wage

Doubling the outside wage w to $30,000 (while leaving liquidation costs at their baseline value) also increases the estimated nonpecuniary beneﬁt, by $16,500, roughly the
increase in w. As in the previous exercise, higher nonpecuniary beneﬁts are needed to
match the extensive margin, but the other parameters are unaﬀected.

8.3

Nonpecuniary Beneﬁts

How do our results change if we eliminate nonpecuniary beneﬁts? To answer this
question, we re-estimate our model restricting χ to be zero, while keeping λ and w at
their baseline values. The new parameter estimates are shown in the fourth column of
Table 4. The parameters change in two notable ways: ﬁrst, the value of the collateral
constraint ψ falls from 1.06 to 0.76; and second, the returns to capital (γ) and the returns
to scale (1 − α) both fall below their baseline values.
In the absence of nonpecuniary beneﬁts, farmers need more pecuniary rewards to
continue operating. A looser borrowing constraint allows them to attain their optimal
capital stock more quickly – by borrowing more upfront – and to better smooth their
dividends. Dividends both rise on average and become signiﬁcantly less volatile: farming
becomes safer and more proﬁtable. Because improved access to funds would in isolation
lead to counterfactually larger operations, the production parameters adjust to reduce
the operations’ optimal sizes.
The fourth row of Table 8 shows that these changes lead the investment rate to fall
in half. This signiﬁcantly worsens the ﬁt of the model with respect to its investment
targets, and the SMD criterion increases by 50 percent. In other words, without nonpecuniary beneﬁts, our model cannot match investment rates and the extensive margin
simultaneously.
These results suggest that nonpecuniary beneﬁts are important to explaining the dynamics we observe in the data. While we would expect the value of these beneﬁts to vary
across industries, as a result of heterogeneous liquidation costs or outside options, our
estimation shows that it is hard to deny their existence.

Fraction
Operating∗
(1)
(2)
(3)
(4)
(5)
(6)

Baseline Model
λ = 0.175
w = $30, 000
χ=0
ψ=0
No Renegotiation

∗ Relative

Capital

Debt/
Assets†

Investment/
Capital
(%)†

1.000
1.001
1.001
1.001
1.010
0.975

1,593
1,588
1,584
1,616
1,630
1,619

0.536
0.534
0.534
0.557
0.560
0.543

6.59%
6.59%
6.61%
3.27%
3.23%
6.45%

Dividend
Growth
Rate
Optimal
(%)‡
Capital
4.31%
4.29%
4.29%
2.09%
2.05%
3.69%

2,833
2,811
2,820
1,770
1,790
2,730

to baseline case. † Ratios of averages. ‡ Mean growth rates for annual averages.

Table 8: Robustness Exercises

8.4

Collateral Constraint

Setting the collateral constraint parameter ψ to 0.76 eﬀectively eliminates the collateral
constraint. This can be seen by comparing the fourth row of Table 8 to the ﬁfth row,
where ψ is set to zero. The two speciﬁcations generate very similar outcomes. In neither
case does the model ﬁt its investment targets well. Table 4 shows that the parameter
estimates for the two speciﬁcations are also very similar.

8.5

Debt Renegotiation

In our baseline speciﬁcation indebted farms can renegotiate their loans. This is consistent with the DFBS, where farms with negative net worth sometimes continue to operate.
The bottom row of Table 8 shows results from a speciﬁcation with no renegotiation, where
farms with negative net worth must liquidate. The eﬀects of this change are modest. The
fraction of farms operating is only 2.5 percent smaller, and most other variables are similarly close to their baseline values. The most notable diﬀerence is that the optimal capital
stock is $0.1 million smaller, implying that the exiting farms are not from the bottom
of the productivity distribution. Renegotiation thus plays a role in keeping productive
farms alive. In most other respects, however, its eﬀects are minor. The last column of
Table 4 shows that the estimated value of the nonpecuniary beneﬁt is higher in a model
with no renegotiation. When farms with negative net worth are forced to exit, the model
overstates exit rates. A higher nonpecuniary beneﬁt encourages more farms to stay in
business.

Conclusions

Although a wide range of policy measures are aimed at promoting entrepreneurial
activity, there is still considerable debate about the forces that drive it. In this paper
we use a dynamic model to assess how ﬁnancial constraints, nonpecuniary beneﬁts and
risk jointly aﬀect entrepreneurs. We build a life-cycle model that incorporates all three
considerations – to our knowledge the ﬁrst of its kind – and estimate it with a rich panel
of owner-operated dairy farms in New York State. Using a simulated minimum distance
estimator, we ﬁt the model to real variables such as input use, capital and revenues, and to
ﬁnancial variables such as debt, dividends and cash holdings. Matching both production
and ﬁnancial variables allows us to disentangle the eﬀects of real and ﬁnancial factors.
Our principal ﬁnding is that the eﬀects of ﬁnancial constraints and nonpecuniary beneﬁts are of ﬁrst-order importance, but those of risk are not. Collateral constraints on
investment and liquidity constraints on the purchase of intermediate goods restrict, sometimes signiﬁcantly, capital holdings, input purchases and output. Nonpecuniary beneﬁts
and liquidation costs discourage low-productivity operators from exiting the industry. In
contrast, eliminating aggregate risk has very modest eﬀects on farm decisions. The insurance provided by the milk margin support program of the 2014 Farm Bill also provides
limited beneﬁts.
We ﬁnd that much of the variation in farm productivity can be attributed to permanent
idiosyncratic diﬀerences. While high-productivity farms grow steadily over the sample
period, low-productivity farms appear to have been close to their optimal size throughout.
This suggests that rather than maximizing the number of entrepreneurs, many of whom
operate for nonpecuniary reasons, entrepreneurial policy may work better by helping the
most promising entrepreneurs expand.
One reason our results are valuable is that detailed joint real and ﬁnancial data for
small ﬁrms are rarely available. But even though our data are especially well-suited to
our approach, the ﬁrms that generate them are similar to many other US ﬁrms. We thus
expect that our methodology and ﬁndings can be extended to a variety of settings.

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For Online Publication: Appendices
A

Data Construction

Sample Selection: The table below describes the ﬁlters used to construct the sample.
Since we are interested in the dynamic behavior of the enerprises, we also eliminate farms
with only observation. Finally, we drop farms with missing information on the age of the
youngest operator.
# of Farms # of Observations
Original Sample
Drop farms with one observation
Drop farms with missing age

541
385
363

2461
2305
2222

Owned Capital: The sum of the beginning of period market value of the three categories of capital stock owned by a farm: real estate and land, machinery and equipment,
and livestock.
Depreciation and Appreciation Rates: The depreciation rate δj for each type of
capital stock j is calculated as
δj =

1
T

Depreciationjit
i Market Value of Owned Capitaljit
i

where i indexes farms and T = 11 (2001-2011).
Analogously, the appreciation rate j for each type of capital stock j is given by

1
T

Appreciationjit
i Market Value of Owned Capitaljit
i

The actual depreciation rate δ is the weighted average of each δj , the weights being the
average share of each type of capital stock in owned capital. The appreciation rate
is
calculated similarly.
Leased Capital: The market value of leased capital (M V LK) of type j for farm i at
time t is calculated as
M V LKjit =

Leasing Expendituresjit
r + δj − j + π
52

where r is the risk-free rate of 4 percent, and π is the average inﬂation rate through the
period. The market value of all leased capital is the sum of the market value of all types
of leased capital.
Total Capital : Owned Capital + Leased Capital
Investment: Sum of net investment and depreciation expenditures in real estate, livestock and machinery.
Total Output: Value of all farm receipts.
Total Expenditure: Expenses on hired labor, feed, lease and repair of machinery and
real estate, expenditures on livestock, crop expenditures, insurance, utilities and interest.
Variable Inputs: Total expenditures less expenditures on interest payments and leasing
expenditures on machinery and real estate.
Total Assets: Beginning of period values of the current assets (cash in bank accounts
and accounts receivable), intermediate assets (livestock and machinery) and long term
assets (real estate and land).
Total Liabilities: Beginning of period values of current liabilities (accounts payable
and operating debt), intermediate and long-term liabilities.
Dividends: Net income (total receipts-total expenditures) less retained earnings and
equity injections.
Cash: Total assets less owned capital.

Econometric Methodology

We estimate the parameter vector Ω = (β, ν, c0 , χ, c1 , θ, α, γ, n0 , λ, ζ, ψ) using a
version of Simulated Minimum Distance. To construct our estimation targets, we sort
farms along two dimensions, operator age and size (cows per operator). Along each
dimension, we divide the sample in half. Then for each of these four age-size cells, for
each of the years 2001 to 2011, we match:
53

1. The median value of capital per operator, k.
2. The median value of the output-to-capital ratio, y/k.
3. The median value of the variable input-to-capital ratio, n/k.
4. The median value of the gross investment-to-capital ratio.
5. The median value of the debt-to-asset ratio, b/˜
a
6. The median value of the cash-to-asset ratio, /˜.
a
7. The median value of the dividend growth rate, dt /dt−1 .
Let gmt , m ∈ {1, 2, ..., M }, t ∈ {1, 2, ..., T }, denote a summary statistic of type m
in calendar year t, such as median capital for young, large farms in 2007, calculated
∗
from the DFBS. The model-predicted value of gmt is gmt (Ω). We estimate the model by
∗
minimizing the squared proportional diﬀerences between {gmt (Ω)} and {gmt }. Because the
model gives farmers the option to become workers, we also need to match some measure of
occupational choice. Let u = u(Ω) denote the fraction of farms that exit in our simulations
¯ ¯
but not in the data. We add to the SMD criterion the penalty Ψ(Ω) = (36¯(Ω))2 , a
u
function that in estimation delivered reasonably low levels of counterfactual exit.
Suppose we have a sample of I conditionally (on the aggregate shocks) independent
farms. Our SMD criterion function is
M

QI (Ω) =
m=1 t=1

∗
gmt (Ω)
−1
gmt

+ Ψ(Ω).

(14)

It bears noting that the counterfactual exit penalty Ψ(Ω) is subject to sampling variation.
The estimated productivity processes we feed into the model, as well as the initial values
of the state vectors in the simulations, reﬂect sampling variation as well.
Our estimate of the “true” parameter vector Ω0 is the value of Ω that minimizes the
criterion function QI (Ω). Let ΩI denote this estimate. Our approach for calculating the
variance-covariance matrix of Ω0 follows standard arguments for extremum estimators.
Suppose that
√
√ ∂QI (Ω0 )
IDI (Ω0 ) ≡ I
N (0, Σ),
∂Ω
so that the gradient of QI (Ω) is asymptotically normal in the number of cross-sectional observations. With this and other assumptions, Newey and McFadden (1994, Theorem 7.1)

show that

√

I (ΩI − Ω0 )

N (0, H −1 ΣH −1 ),

(15)

where

∂QI (Ω0 )
.
∂Ω∂Ω
We estimate H as HI , the numerical derivative of QI (Ω) evaluated at ΩI , setting
the step size for each parameter equal to 0.1 percent of that parameter’s absolute value.
Unfortunately, there is no analytical expression for for the limiting variance Σ. We instead
ﬁnd Σ via a bootstrap procedure. In particular, we create S = 49 artiﬁcial samples of
size I, each sample consisting of I bootstrap draws from the DFBS. Each draw contains
the entire history of the selected farm. In this respect, we follow Kapetanios (2008), who
argues that this is a good way to capture temporal dependence in panel data bootstraps.
Our bootstrap procedure does not account for variation in the aggregate shocks, and our
standard errors are thus biased downward. For each bootstrapped sample s = 1, 2, . . . , S,
we generate the artiﬁcial criterion function Qs (ΩI ), in the same way we constructed Qs (ΩI )
using the DFBS data. The function Qs (ΩI ) is then run through a numerical gradient
procedure to ﬁnd Ds (ΩI ), using the same step sizes as in the calculation of HI . The
estimated parameter vector ΩI is used for every s, but the simulations used to construct
Qs (·) employ diﬀerent random numbers, to incorporate simulation error.36 Finding the
variance of Ds (ΩI ) across the S subsamples yields ΣS , an estimate of Σ/I (not Σ).
An alternative, interpretation of our approach is to treat it as an approximation to a
one-step bootstrap, where Ω is re-estimated for each artiﬁcial sample s.37 Let GI denote
the vector containing all the summary statistics used in QI (·). The ﬁrst-order condition
for minimizing QI (·) implies that
H = plim

DI (ΩI ) = D(ΩI , GI ) = 0,
Implicit diﬀerentiation yields
∂ΩI
−1 ∂D(ΩI , GI )
≈ −HI
.
∂GI
∂GI
Because the mapping from GI to ΩI – the minimization of QI (Ω) – is too time consuming
∗
Because gmt () is found via simulation rather than analytically, the variance Σ must account for
simulation error. In most cases the adjustment involves a multiplicative adjustment (Pakes and Pollard,
1989; Duﬃe and Singleton, 1993; Gouri´roux and Monfort, 1996). Because each iteration of our bootstrap
e
employs new random numbers, no such adjustment is needed here.
37
See Andrews (2002). We are grateful to Lars Hansen for this suggestion.
36

Medians
Data Model
Capital
1,114
Y /K
0.47
0.35
N/K
I/K
0.06
0.54
Debt/Assets
Cash/Assets 0.12
0.93
Dividend
growth
∗ Standard

970
0.48
0.36
0.04
0.55
0.11
1.01

Means
Data Model
1,823
0.48
0.36
0.09
0.55
0.13
0.69

Std. Deviation† Autocorrelations†
Data
Model Data
Model

1,593 1,919
0.52 0.15
0.37 0.13
0.10 0.20
0.48 0.20
0.11 0.05
1.03 4.28

1,576
0.36
0.21
0.19
0.23
0.04
0.09

0.98
0.88
0.88
0.10
0.90
0.86
0.03

0.99
0.86
0.86
0.30
0.85
0.53
0.11

deviations and autocorrelations use deviations from annual means.

Table 9: Data and Baseline Model Moments
to replicate S times, we replace it with its linear approximation:
V (ΩI ) ≈

∂ΩI
∂ΩI
∂D(ΩI , GI ) −1
−1 ∂D(ΩI , GI )
= HI
HI
V (GI )
V (GI )
∂GI
∂GI
∂GI
∂GI

−1
−1
≈ HI ΣS HI .

Goodness of Fit

Tables 9 and 10 compare data moments from the DFBS with moments for data generated by the baseline model. The criterion function targets yearly medians directly, but
the model means and standard deviations are very similar to those in the data. A notable
exception is the variablility of the growth rate of dividends, which is much larger in the
data than in the model. This is partly due to the presence of outliers: dropping the top
and bottom 1 percent of the sample gives us a standard deviation of 1.97, while keeping
the mean virtually unchanged. However, as discussed in the main text (see footnote 24),
our CRRA utility function cannot generate the low degree of interremporal substitutability implied by the low observed dividend growth rate and also generate the low degree
of risk aversion implied by the high observed dividend volatility. Using Epstein-Zin type
preferences would be an interesting extension.
The model also does a reasonable job in matching the autocorrelations found in the
data. The cross correlations between the real and ﬁnancial variables are mostly captured
by the model as well.

Capital Y /K
Data
Capital
Y /K
N/K
I/K
Debt/Assets
Cash/Assets
Dividend Growth
Model
Capital
Y /K
N/K
I/K
Debt/Assets
Cash/Asset
Dividend Growth

N/K

I/K

Debt/ Cash/
Assets Assets

Dividend
Growth

1.00

0.33
1.00

0.36
0.95
1.00

0.04
0.23
0.20
1.00

0.13
0.22
0.20
0.12
1.00

0.19
0.57
0.53
0.10
0.00
1.00

0.01
0.07
0.04
0.01
-0.01
0.06
1.00

1.00

0.02
1.00

0.07 -0.14
0.97 0.46
1.00 0.46
1.00

0.21
0.27
0.30
0.19
1.00

0.12
0.23
0.32
0.28
0.00
1.00

-0.05
0.30
0.27
0.50
0.05
0.33
1.00

Table 10: Data and Baseline Model Cross Correlations
Notes: Correlations calculated using deviations from annual means.

Modelling the 2014 Farm Bill

The 2014 Farm Bill program replaces the previous dairy support program, which
guaranteed a minimum price for milk, with a guarantee of the milk margin, the diﬀerence
between the price of milk and a weighted average of the prices of corn, soybean and alfalfa.
As described by Schnepf (2014), the program provides a baseline margin support of $4.00
per cwt. for all participants. Higher support levels (in $0.50 increments) can be acquired
in exchange for a premium. For the ﬁrst 4 million lbs of output, the premium ranges from
$0.01 per cwt. for a margin of $4.50 to $0.475 for a $8.00 margin. For additional output
the premium ranges from $0.02 to $1.36 per cwt.
As discussed in the main text, we model margin ﬂoors by eliminating the left-hand
tail of the aggregate shock distribution. Schnepf (2014) notes that the premium structure
encourages farmers to choose a margin support level of $6.50 per cwt. In the implementation of the margin support program, the milk margin is computed and compared to the
margin ﬂoor every two months. At this frequency, during our sample period of 2001 to
2011 nominal milk margins fell beneath $6.50 about 13.6 percent of the time. Real milk
57

margins, measured in September 2014 dollars, fell beneath the ﬂoor about 7.6 percent
of the time. At the annual frequency used in our model, both nominal and real milk
margins fell below the $6.50 ﬂoor once, in 2009, a rate of 1/11 = 9.09 percent. We choose
this annual ﬁgure as our truncation level.
Following standard practice, when solving the model numerically we replace the continuous processes for the productivity shocks with discrete approximations. We use the
approach developed by Tauchen (1986) for discretizing Markov chains, simpliﬁed here to
i.i.d. processes. Under Tauchen’s approach one divides the support of the underlying
continuous process into a ﬁnite set of intervals. The values for the discrete approximation come from the interiors of the intervals (demarcated by the midpoints, except at
the upper and lower tails), and the transition probabilities are based on the conditional
probabilities of each interval. To impose the cutoﬀ, we construct the discretization for the
aggregate shock so the bottom two states/intervals have a combined probability of 9.09
percent. We then set the truncated value for these states to equal the 9.09th percentile
of the underlying continous distribution.
When ﬁnding the decision rules, we combine the aggregate and idiosyncratic shocks
into a single transitory shock. In the baseline speciﬁcation, which we use when estimating the model, we approximate the sum of the two-shock processes with an eight-state
discretization. To model the elimination of certain shocks, we simply change the standard deviation of this combined process. However, to capture the Farm bill, we need to
alter the distribution of the aggregate shock in isolation. We thus model the aggregate
shock with its own eight-state discretization. To get the joint shock, we approximate
the idiosyncratic shock with a four-state distribution, and convolute the aggregate and
idiosyncratic shocks into a 32-state distribution. We construct the four-state discretization so that the standard deviation of the convoluted 32-state process is the same as the
standard deviation of the eight-state shock used in the baseline model. Switching between the two discrete approximations aﬀects the model’s results only modestly (compare
the ﬁrst lines of Tables 5 and 6). Under either approximation, the shocks used in the
simulations are a combination of idiosyncratic shocks from a random number generator
and the aggregate shocks estimated from data. To simulate the Farm Bill, we simply
truncate the aggregate shock for 2009 to the 9.09th percentile.
For large volumes, the premium for a margin support of $6.50 is $0.290 per cwt.
This equals about 1.58 percent of the average real milk price over the 2001-2011 interval
($18.34). We impose the premium by incrementing the logged productivity process by
ln(0.9842), that is, by reducing the productivity level by 1.58 percent.

Full text of Working Papers (Federal Reserve Bank of Richmond) : An Estimated Structural Model of Entrepreneurial Behavior, Working Paper 17-07

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