Imperfect Information about Entrepreneurial
Productivity, Financial Frictions, and Aggregate
Productivity
(Click here for current version)
Ashique Habib
∗
January 17, 2017
Abstract
Households face uncertainty about their entrepreneurial productivity which can
be discovered by entrepreneurship. I propose that an important channel through
which financial frictions adversely impact aggregate productivity is by hindering
the discovery of productive entrepreneurs. I develop a model where households
have imperfect information about the quality of their business idea. I then show
how financial frictions arising from weak contract enforcement systematically reduce
access to capital for poor households with good ideas, which undermines their incentive to learn. After calibrating the model to US data, I find that with imperfect
information, TFP falls by 23% when contract enforcement is lowered to developing
country levels, compared to 12% with perfect information. Half of the productivity
loss in the economy with imperfect information is due to financial frictions hindering the discovery of good ideas by poor households. I find that these losses can be
substantially mitigated by subsidizing young entrepreneurs. Even with no financial
frictions, moving from perfect information to no information about productivity reduces TFP by 56%. Hence, differences in the degree of imperfect information can
be an important independent source of cross-country TFP differences.
∗
I would like to thank Diego Restuccia, Ronald Wolthoff, and Xiaodong Zhu for their guidance and
support. This project has benefited from discussions with Francisco Buera, Chaoran Chen, Kevin Donavan, Sebastian Dyrda, Burhan Kuruscu, Vincenzo Quadrini, Joseph Steinberg, the participants at the
University of Toronto Macroeconomics Brownbag, the Canadian Economics Association 2016 meeting
and the North American Productivity Workshop 2016. I would also like to thank the Social Sciences and
Humanities Research Council (SSHRC) and the Ontario Graduate Scholarship (OGS) programs for their
financial support. All errors are my own. Contact: Department of Economics, University of Toronto, 150
St. George Street, Toronto, Ontario, Canada, M5S 3G7. Email: [email protected].
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1
Introduction
Underdeveloped financial markets misallocate resources and distort entry by productive
entrepreneurs. A large literature has tried to understand how such misallocation hinders
economic development by lowering aggregate productivity.1 I argue that an important
negative impact of financial underdevelopment is that it impedes the discovery of productive entrepreneurs. To evaluate this hypothesis, I consider a model of a market economy
where implementing high-quality entrepreneurial ideas and allocating capital efficiently
across operating firms are the critical determinants of productivity. Entrepreneurs with
new ideas can only discover its productivity by operating a firm. I find that financial
frictions systematically distort the incentives of households with potentially good ideas
to learn about its quality, which substantially lower productivity.
My study is motivated by recent evidence that entry by new entrepreneurs is a key
driver of economic growth. For example, Haltiwanger et al. [2013] document that in the
US, new firms account for a disproportionate share of employment growth. However,
young firms face greater uncertainty about the viability of their idea. They gradually
learn about their idea’s productivity by actually observing its performance in the market (Kerr et al. [2014]). The literature also documents that young firms have difficulty
accessing credit because of their project’s uncertainty (Lerner [2009]). The relationship
between the financing environment and experimentation by young entrepreneurs is further emphasized by Kerr and Nanda [2009], Calvino et al. [2016] and others who find that
financing conditions have a strong influence on the entry, growth and survival of young
firms.
In this paper, I study the joint impact of financial and information frictions using
a general equilibrium model with heterogenous production units. My model builds on
Buera et al. [2011], which is a standard model to study how financial frictions impact the
aggregate economy by distorting both capital allocation and the selection of households
into entrepreneurship, as in Lucas [1978]. I introduce imperfect information about productivity following Jovanovic [1982]. In this framework, highly productive entrepreneurs
initially have low expected productivity but gradually discover their high productivity
over time. Conversely, low productivity entrepreneurs learn their firms’ productivity is
low and exit. This learning mechanism is an empirically relevant channel for explaining
age-dependent patterns in firm dynamics (e.g. Arkolakis et al. [2014]).
Let me review the critical features of my model and how they relate to aggregate productivity: All households are equally productive as workers but differ in the productivity
of their entrepreneurial ideas, as in Lucas [1978]. Therefore aggregate productivity is max1
See Restuccia and Rogerson [2013] and Buera et al. [2015] for surveys of the misallocation and
financial friction literatures respectively.
2
imized if the households with high entrepreneurial productivity sort into entrepreneurship.
The productivity of a household’s idea changes from time to time, and therefore re-sorting
is necessary. Households with new ideas need to actually implement their idea to learn
something about its productivity, which I consider a form of experimentation. They have
an incentive to experiment because households with very good ideas can earn large incomes. The allocation of capital across the operating entrepreneurs is also important for
productivity.
I introduce financial frictions by assuming contracts are not perfectly enforced and
there is a fixed cost of intermediation. Lenders in countries with weak legal systems have
difficulty enforcing contracts, and this is an often cited form of financial underdevelopment. The fixed intermediation cost captures screening and administrative costs. My
paper is the first to investigate how these frictions impact the discovery of productive
entrepreneurs.
When contracts cannot be perfectly enforced, lenders are willing to lend only to entrepreneurs who can credibly promise to repay. Lenders seize a defaulting entrepreneur’s
assets and recoup whatever income they can through the legal system. Households with
little assets and low-expected productivity either face tight credit limits or, if their assets
and expected productivity are very low, cannot access credit markets at all.
I show that households with new ideas are systematically more likely to face tight
financing conditions. First, these households are likely to have been workers in the recent
past and therefore have few assets to collateralize loans. Second, their expected productivity is lower than its true value because of imperfect information.2 Because they have
low assets and low expected productivity, young entrepreneurs in financially underdeveloped economies face an increased probability that they will not have access to external
credit at all. If they do have access, they are likely to face tight credit limits.
These tight financing conditions reduce the scale at which households with new ideas
can operate, lowering the net benefit of learning about their idea. As a result, many
poor households with new ideas forgo experimentation altogether. In contrast, if highly
productive households had perfect information about their productivity, these same tight
financing conditions would have little impact on their decision to operate. Knowing they
can earn very high incomes, these households would undo credit constraints by accumulating assets (Moll [2014], Midrigan and Xu [2014]). Therefore imperfect information limits
the ability of high-productivity households with new ideas to overcome credit constraints
through saving.
This paper contributes to a recent literature on the importance of imperfect informa2
Highly productive ideas are rare, and therefore households with new ideas rationally discount very
high initial signals as possibly due to noise.
3
tion on productivity. Greenwood et al. [2010] and Steinberg [2013] highlight the importance of cross-country differences in lenders’ ability to learn about borrowers. David et al.
[2016] shows that imperfect information about transitory shocks increase static misallocation and lower TFP.3 To my knowledge, my paper is the first to study the impact of
financial frictions on the discovery of entrepreneurial talent.4
I quantify the impact of financial and information frictions on TFP by calibrating
the model to US data, assuming the US has imperfect information and perfect contract
enforcement. My calibration strategy uses data on exit rates of firms with age to infer
the amount of imperfect information in the US economy. In the model, all entrepreneurs
face the same probability that their ideas die. Old firms have a very accurate assessment
of their productivity and therefore exit only when their idea dies. Young firms are less
informed about their productivity, and exit either if their idea dies or if they learn their
idea is not worth implementing any further. I use the difference between the exit rates of
young and old firms to identify the amount of imperfect information in the US economy.
I then use the calibrated model to evaluate the extent to which differences in imperfect information and contract enforcement can explain cross-country TFP differences.
I benchmark my results to an economy with both perfect enforcement and perfect information, because this economy not only has the highest productivity but is also the
standard benchmark used in the literature. I find that my calibrated US economy, which
has imperfect information and perfect enforcement, has a TFP about 3% lower than the
benchmark, suggesting that imperfect information alone lowers TFP.
In my first counterfactual experiment, I hold imperfect information to US levels and
weaken contract enforcement to developing country levels. I find that both TFP and GDP
per capita fall monotonically as financing conditions worsen, and at the lowest level of
contract enforcement, TFP is 23% lower than the benchmark economy. To understand
whether interaction between financial frictions and imperfect information play any role, I
repeat the same experiment of weakening contract enforcement, assuming households have
perfect information about their productivity. I find that TFP falls by 12% in the worst
case scenario. Taking into account that imperfect information directly lowers TFP by
3%, these exercises suggest that about 7% of the TFP loss in the economy with imperfect
information and low contract enforcement is due to the interaction between financial and
information frictions.
In order to further understand the interaction of the two frictions, I decompose the
the total TFP loss into several components including the portions due to capital misal3
This paper focuses on learning from stock prices and abstracts from financial frictions.
Learning about productivity or demand is an empirically relevant explanation for age-dependent
firm life-cycle dynamics (e.g. Arkolakis et al. [2014], Eaton et al. [2014], and Foster et al. [2016]). This
literature abstracts from financial frictions and the implications for aggregate productivity.
4
4
location and to the distorted selection of entrepreneurs. I find that the importance of
distorted selection differs markedly between the perfect information and imperfect information cases. In the economy with imperfect information when contract enforcement is
at its lowest level, distorted selection accounts for about half of the total productivity loss
of 23%. In contrast, in the economy with perfect information when contract enforcement
is at its lowest, distorted selection lowers TFP by about 2% (out of a total loss of 12%).
To provide further support of the importance of my mechanism, I also investigate how
weak contract enforcement impacts the exit rate of firms and young firms’ access to credit.
I find that my model’s predictions are consistent with cross-country differences in young
firm exit rates and access to credit, as recently documented by Hsieh and Klenow [2014]
and Chavis et al. [2011] respectively. In economies with weak contract enforcement, I
find that the exit rate of young firms is lower than in the US, consistent with Hsieh and
Klenow [2014]’s finding for India. Access to credit for young firms also falls substantially.
Weak contract enforcement alone is unable to generate these facts.
Having identified a novel channel through which financial frictions can reduce TFP,
I investigate whether a government policy to subsidize young entrepreneurs can correct
some of the distortions. I find that a relatively simple subsidy scheme financed by lumpsum taxes can go a long way to correcting the selection of entrepreneurs and increasing
TFP.
Finally, I explore whether higher levels of imperfect information about entrepreneurial
productivity might be an independent cause for productivity differences across countries.
My model allows me to evaluate the full range of information regimes, from perfect information to an environment with no learning. While holding contract enforcement at
the US level, I increase the amount of imperfect information. I find that TFP monotonically decreases as the ability to learn declines. Relative to the full information economy,
TFP in the economy with no learning is 56% lower. Although we do not have data on
how imperfect information about entrepreneurial productivity and the learning process
varies across countries, the literature does document that other forms of uncertainty are
generally higher in developing countries. My experiment suggests that further exploring
differences in imperfect information across countries is a fruitful channel for explaining
cross country income differences.
The rest of the paper is organized as follows: Section 2 presents the model, section 3
presents the calibration strategy and quantitative exercises. Section 4 concludes.
5
2
Model
I will use a simple, stylized model to illustrate how financial frictions reduce a household’s
incentive to learn about their idea. I will then present a framework where households
will face the same tradeoffs as in the simple model, but that is more appropriate for
quantitative assessment.
2.1
Stylized Model
In this simple model, households are risk-neutral and live for two periods. Each household
has an idea, but they do not initially know its productivity. The idea’s productivity (x)
takes either a high or a low value (i.e. x ∈ {xL , xH }). A fraction p of households have
high-quality ideas. In order to use their idea to produce output, households must first
implement it at an implementation cost w ≥ 0 and also use capital as an input.
The low productivity ideas produce no output (xL = 0), and therefore are never worth
implementing. If a household has productivity x and uses k units of capital, then their
net output is xk − w. There is a maximum scale k uc above which employing additional
capital produces no additional output. I capture the scale reducing effects of financial
frictions by assuming that each household draws the amount of capital they own from a
distribution at birth, and cannot adjust it afterwards.
New households observe a noisy signal which they use to update the probability that
their idea’s productivity is high. The updated probability is p̂. If a household implements
their idea, then they learn the exact productivity by observing the output.
In order to make experimentation both costly and potentially worthwhile, I assume
that if a high productivity idea is operated at the unconstrained scale, then the realized output is strictly greater than the implementation cost. However, high productivity
households are relatively rare: If a household’s expected probability their productivity is
high is the same as the population probability (p̂ = p), then the expected income from
implementing the project is negative. In particular, the assumptions are:
xH k U C − w > 0
pxH k U C − w < 0
(High quality ideas should be implemented)
(High quality ideas are rare)
In each period, households only implement their ideas if doing so maximizes their
expected lifetime income. Figure 1 presents the timeline of events and decisions each
household faces. Let me work backwards to characterize the implementation decisions.
6
Figure 1: Timeline (stylized model)
Second period, perfect information. A household that implemented their idea in
the first period knows their productivity exactly. They will never implement the idea in
the second period if their productivity is xL . If their productivity is xH then they will
only implement their idea if they can operate it at a sufficiently large scale. The threshold
scale (k2 ) above which households with productivity xH implement their idea is:
k2 =
w
xH
Second period, imperfect information. If the household did not implement the
project in the first period, then they still face imperfect information in the second period.
They will implement the project only if their expected output is greater than the implementation cost (p̂xH k ≥ w). For households with capital k, there is a threshold expected
probability p̂2 (k) above which they will implement their idea.
p̂2 (k) =
w
kxH
First period, imperfect information. In addition to the expected income in the first
period, there is a real option value of implementing the project because it reveals the
idea’s productivity and allows the household to implement it in the second period only
7
if it is high-productivity. A household with capital k implements the project if and only
their expected probability of having a high productivity is above a threshold p̂1 (k).
p̂1 (k) =
w
2xH k − w
Because of the value of learning about the idea’s quality, the threshold expected probability in period 1 is strictly less than the threshold expected probability in period 2 (i.e.
p̂1 (k) < p̂2 (k)).
Define the expected benefit of experimentation as B(p̂, k) = p̂(xH k − w) and the
expected cost as C(p̂, k) = w − p̂xH k. The expected benefit is the net income in the
second period if the productivity is high, multiplied by the probability that it might be
so. The cost is the first period net income, accounting for the probability that output
might be zero. In figure 2 I plot the cost and benefit functions for a household with
low capital and a household with high capital. Higher capital allows the household to
operate at a higher scale, reducing the expected cost and increasing the expected benefit
of operating in the first period. Therefore, households with more capital have a lower
threshold expected probability p̂1 (k) above which they will experiment.
Figure 2: Expected benefit and cost of operating in first period
In figure 3, I plot the threshold p̂1 as a function of capital k. The left panel illustrates
two key ideas. First, even if a household has perfect information and is certain they have
high productivity (p̂ = 1), they will not operate if their capital k is below a certain threshold (labeled ‘do not operate (perfect)’). This is the standard way financial frictions distort
entry into entrepreneurship. Second, if households are uncertain about their productivity
8
(p̂ < 1), they require a higher minimum scale to operate (labeled ‘threshold (imperfect)’).
Hence, imperfect information amplifies the distortion to entry from financial frictions.
Figure 3: Threshold belief for operating first period
Implementation cost. I illustrate the impact of the implementation cost w in the right
panel. The implementation cost w plays an important role in determining the threshold
p̂2 (k): In the right panel of figure 3 I show that the threshold for implementing the
project in period 1 is increasing in the implementation cost w. A higher implementation
cost increases both the cost of experimentation C(p̂, k) = w − p̂xH k and lowers the benefit
B(p̂, k) = p̂ (xH − w).
I now present the main model, which endogenizes the scale of operations and the
implementation cost, and allows long-lived households to increase the scale of operation
over time by accumulating assets.
2.2
Main model
There is a measure 1 of infinitely-lived households. Each household has an idea, the
productivity of which varies across the population. Households implement their ideas by
setting up and running a firm, and the quality of the idea determines the firm’s average
productivity.
Let me begin by describing the distribution of ideas and how households learn about
the quality of their ideas.
9
Distribution of ideas and learning process. Each household has an entrepreneurial
idea. The quality of the idea, X, is log-normally distributed across the population, i.e.
x = log(X) ∼ N (µx , σx2 ). Each period, the household’s current idea dies with probability
1−ρ,5 in which case the household draws a new idea from the population distribution. The
probability of carrying over the same idea into the next period is ρ ∈ (0, 1).6 Households
know when their idea has died, but do not directly observe the quality of their next draw.
i) Initial signal about new idea. Immediately after losing an idea and drawing a new
one, households observe a noisy signal s. This signal is normally distributed, with mean
equal to the idea’s quality (i.e. E(s|x) = x) and variance σs2 , that is s ∼ N (x, σs2 ). Since
all households draw ideas from the same population distribution, their prior about their
new idea’s quality is the population distribution for ideas N (µx , σx2 ). The household uses
the signal to update their beliefs and form their posterior distribution x̂1 ∼ N (µ1 , σ12 ).
The mean and variance µ1 and σ12 of the posterior distribution are given by equations 1
and 2.
µ1 =
σx2 s + σs2 µx
σx2 + σs2
(1)
σs2 σx2
σs2 + σx2
(2)
σ12 =
I include this initial signal to encapsulate the perfect information economy as a special
case of my general model. In particular, if σs2 = 0 then the signal immediately reveals the
new idea’s type. I will use this feature of the model to conduct counterfactual exercises.
ii) Subsequent learning. I capture the idea that discovering a business idea’s quality
requires observing its performance (Kerr et al. [2014]) by assuming that households can
only learn more about their idea’s quality by setting up a firm and observing its total
productivity. Households that do not implement their idea in a given period do not learn
anything new. Operating the firm requires all of the household’s time, and as a result
it precludes working in the labor market. Therefore, the cost of learning is the foregone
wage minus any income from actually operating the firm.
A household implementing a new idea does not immediately learn its quality from
the firm’s productivity because operating firms experience idiosyncratic, transitory pro5
We can think of the “death” of an idea as a permanent adverse shift in the demand for the good the
firm was producing. The firm only continues to operate if its next product is sufficiently profitable.
6
Since households are infinitely-lived, each one will shift into and out of entrepreneurship over time
based on the quality of their ideas.
10
ductivity shocks each period.7 I think of these shocks as transitory changes in market
conditions. The transitory shocks are also log-normally distributed with mean 0 and variance σe2 , i.e. e ∼ N (0, σe2 ). The total productivity of a firm with idea quality X hit by
transitory shock E is Z = XE = exp {x + e}.
If a household begins the period having observed j signals and their prior distribution
based on these signals has mean µ and variance σj2 , then after observing z and updating,
the mean and variance of their posterior distribution are given in equations 3 and 4.
µ+1 =
σj2 z + σe2 µ
σj2 + σe2
(3)
σj2 σe2
σe2 + σj2
(4)
2
σj+1
=
Terminology. I will refer to x as an idea’s quality and the first moment of the posterior
distribution µ as the idea’s expected quality. I call E(Z) the expected productivity of an
operating firm. Of course, with imperfect information, the expected productivity of a
firm depends on the expected quality of the underlying idea.
This completes the description of the learning process. I assume the amount firms
learn from operating for a period does not depend on the scale of their operation. If
learning depended on the scale of operation, then financial frictions could have a larger
impact by distorting the speed of learning. I leave exploring this channel for future work.
I will now describe the rest of the environment.
Preferences. All households have identical preferences, which are defined over a homogenous consumption good. Their period utility function is u(c) and their discount rate
is β. They maximize their lifetime expected utility by choosing among the sequences of
consumption {ct }∞
t=0 that satisfy their budget constraints, i.e.,
E
∞
X
s=0
β t u(ct ),
u(c) =
c1−γ
,
1−γ
γ > 1,
β ∈ (0, 1)
(5)
Production technology. Goods are produced by competitive firms, each run by an
entrepreneurial household. Each firm operates a technology that is specific to the en7
There is a large literature documenting firm-level idiosyncratic, transitory shocks, e.g. Castro et al.
[2009]
11
trepreneurial household.8 The technology takes the entrepreneur’s time as a fixed input,
and capital (k) and labor (l) as variable inputs. The output of a firm with log-total
productivity z as a function of capital k and labor l is:
ω̃(k, l; z) = ez k α lθ ,
α+θ <1
(6)
Since the above production function has decreasing returns to scale in the variable
inputs, the most productive entrepreneur does not completely dominate production. Instead, a distribution of firms operate in equilibrium. As alluded to earlier, the main
contribution of this study is to show how financial and information frictions distort the
distribution of operating firms.
Markets. Factors of production are traded in competitive markets. The labor market is
frictionless. The wage W equates labor supplied by worker households to labor demanded
by entrepreneurial ones.
The credit market consists of lenders, owned by the households, who intermediate
funds within the period. Near the beginning of the period, they take deposits from
households and lend to entrepreneurs at rental rate R. Near the end of the period, they
collect payments from entrepreneurs and repay households their deposits with interest r.
Financial frictions. This intermediation process is affected by two sources of financial
frictions: a lump-sum intermediation cost per loanand imperfect contract enforcement.
i) Intermediation cost. For every loan issued, lenders incur a fixed cost ψ ≥ 0. Following Dabla-Norris et al. [2015] and Arellano et al. [2012], this fixed cost is a reduced
form way to capture the administrative costs of intermediation, such as screening borrowers and operating bank branches. The lenders pass this cost on to the borrowers.
Borrowers can either pay the cost up front if they have sufficient wealth at the beginning
of the period, or at the end of the period from their wealth after production. Borrowers
who cannot pay the cost up front must credibly commit to pay the cost at the end of the
period.
I include this cost to introduce the possibility that some entrepreneurs will not be able
to access external credit. In particular, borrowers who cannot pay the cost up front or
8
The assumption that the technology requires the particular household’s managerial time as an input
is a standard way to rule out poor households selling their ideas to rich ones. In this environment, if a
poor household sells their good idea to a rich household, the poor household remains the monopoly seller
of a necessary input (his managerial effort) and would extract all the rents.
12
credibly commit to pay at the end of the period will not be able to access credit markets.
Instead, they must completely self-finance their project using their own assets.9 I will
show that young firms will be more likely to not have access.
ii) Imperfect enforcement. Borrowers can default and refuse to give the lender the
contracted payment if it is optimal for them to do so. Lenders will therefore offer loans
they know the borrower will actually honor. In this economy, all loan contracts are shortterm, and defaulting borrowers have full access to capital markets in subsequent periods
despite their default history. Therefore, lenders cannot impose any dynamic penalties.
Instead, they will seize any assets the borrower puts up as collateral, and take the borrower
to court to recoup as much of the borrower’s post-production wealth as possible.
Lenders can seize a fraction φ ∈ [0, 1] of the end of period wealth. The parameter
φ captures the full range of legal enforcement institutions.10 On one hand, contract
enforcement is perfect if φ = 1. In this case, borrowers have no incentive to default
since lenders can seize everything. On the other hand, if φ = 0 then lenders cannot seize
anything other than assets. In this case, loans must be fully collateralized by assets.
I will defer the formulation and solution to the contracting problem momentarily, and
instead first present the recursive formulation of the household’s problem.
Recursive formulation of household’s problem. At the beginning of each period,
households’ state variables are assets (a), and their beliefs about their entrepreneurial
idea which is summarized by the expected quality µ and the variance σj2 . To emphasize
the connection between learning and the entrepreneurs’ age, I will replace σj2 with j which
is the number of periods the agent has ran a firm based on this idea.
Each period, households first decide whether to be a worker or an entrepreneur. Then
they divide their wealth between consumption (c) and savings (a0 ). Entrepreneurial
households make additional decisions, which I describe momentarily. Let V W (a, µ, j),
V E (a, µ, j) and V (a, µ, j) be the household’s expected payoff from working, from entrepreneurship, and from optimally choosing their occupation respectively.
If a household works, then their wealth after production is y W (a) = (1 + r)a + W .
Their expected payoff is:
9
I prove this in proposition 3.
In practice, lenders utilize both formal and informal methods to collect payments. Allen et al. [2012]
documents in their study of firm financing in India that lenders also use social pressure and arbitration
by business partners.
10
13
V
W
0
0
u(c) + β ρV (a , µ, j) + (1 − ρ) E V (a , µ1 (s), 1)
(a, µ, j) = max
0
c,a ≥0
x,s
subject to:
(7)
c + a0 ≤ y W (a) = (1 + r)a + W
The worker’s value function V W depends on their beliefs about their entrepreneurial
productivity (µ, j) because they might choose to implement their idea in the future. Since
workers learn nothing new about their productivity this period, conditional on keeping
their current idea, their beliefs are still summarized by (µ, j) in the next period. With
probability ρ they keep their idea, in which case their expected payoff from taking a0
assets to the next period is V (a0 , µ, j). However, with probability 1 − ρ the household will
lose their current idea, draw a new one from the population distribution, and observe a
noisy signal s. The expected value of having a new idea and assets a0 is summarized by
E V (a0 , µ1 (s), 1).
x,s
An entrepreneur’s value function is:
0
0
V (a, µ, j) = max
u(c) + β ρEV (a , µ+1 , j + 1) + (1 − ρ) E V (a , µ1 (s), 1)
0
E
c,a ≥0
subject to:
z
x,s
(8)
c + a0 ≤ y E (a, µ, j)
An entrepreneur’s value function is similar to the worker’s, with two key differences.
First, by operating this period they learn something more about their idea’s quality. If
they get to keep their current idea next period, then their belief about its quality will
be (µ+1 , j + 1). Second, their end of period wealth after production is y E (a, µ, j). I will
characterize this variable when I solve the entrepreneurs’ problem.11
Remark. I assume the households choose their saving a0 prior to production. This assumption only matters for entrepreneurs12 , since they must choose a0 prior to observing
their current productivity and income. Entrepreneurs can choose a0 without worrying
about potentially violating their budget constraint because they will find it optimal to
insure their output against the transitory shock, and therefore will be able to characterize
their deterministic post-production wealth as a function of state variables. I will discuss
the insurance mechanism and what it buys me later.
Households choose the occupation that gives them the highest payoff. Let o(a, µ, j)
11
The interested reader can refer to proposition 5.
Since workers’ income is deterministic and their beliefs are the same before and after production,
when they choose a0 does not matter.
12
14
equal 1 if the household chooses to be an entrepreneur and 0 if they choose to be a worker.
Their occupational choice maximizes:
V (a, µ, j) =
max
(1 − o(a, µ, j))V W (a, µ, j) + o(a, µ, j)V E (a, µ, j)
(9)
o(a,µ,j)∈{0,1}
I now describe the contracting problem between lenders and entrepreneurs, which will
determine the income of entrepreneurs y E (a, µ, j).
2.2.1
Entrepreneurs’ and lenders’ problems
Entrepreneurs make financing, input and default choices based on the set of loan contracts
offered to them by lenders. On the other hand, lenders anticipate entrepreneurs’ behaviour
when determining the set of contracts to offer each type of entrepreneur. Therefore, the
entrepreneur’s and the lender’s problems must be solved jointly. Figure 4 presents the
timeline of events and decisions faced by an entrepreneur.
Figure 4: Timeline for an entrepreneur
Entrepreneurs make all choices before observing their total productivity (z). They
first choose whether to access external finance (f ) and the amount of capital to use.
If the entrepreneur accesses credit markets, then they deposit assets as interest-bearing
15
collateral, and choose the optimal quantity of capital given credit limits. If they do
not access external finance, then their capital choice is restricted by their assets. These
decisions are made at point (1) in figure 4.
Their remaining tasks are to decide whether to default (d), how many workers to hire
(l), and whether to insure their output against the transitory shock. Next, the transitory
shock e is realized and production takes place. Insurance contracts pay out and workers
are paid.
Finally, lenders enforce their contracts. Non-defaulting entrepreneurs make the required payments and receive back their collateral with interest. Defaulting entrepreneurs
lose their collateral, and the lender seizes a fraction φ of their remaining end of period
wealth.
All of the entrepreneur’s choices after choosing capital (points (2) onward in figure 4)
can be perfectly anticipated based on financing choices (f, k) and state variables (a, µ, j).
I will therefore solve the entrepreneur’s and lender’s problems in three steps: first, I will
solve the entrepreneur’s insuranc,e labor demand, and default decision taking financing
decisions (f, k) as given. Second, I will solve for the set of contracts lenders are willing
to offer this entrepreneur. Third, I will solve the entrepreneurs’ choice of capital and
accessing credit markets.
Insurance against the transitory shock. Since entrepreneurs choose inputs prior
to observing z, their realized resources might be less than their obligations to lenders
and workers and they might be forced to default. I abstract from equilibrium default
by assuming entrepreneurial households have access to a competitive insurance market
that opens after the decision to default has been made but prior to the realization of the
transitory shock.13 As proposition 1 shows, they choose an insurance contract that gives
them the expected output for all realizations of total productivity z.
Proposition 1 (Optimal insurance). An entrepreneur with inputs (k, l) and beliefs (µ, j)
finds it optimal to purchase an insurance contract that pays their expected output:
σj2 + σe2
k α lθ
E (ω̃(k, l; z)|µ, j) = exp µ +
z
2
(10)
for all realizations of total productivity z.
13
I abstract from lenders providing the insurance because if they could, they would condition payments
on the default decision and relax credit constraints. Furthermore, I assume the entrepreneur cannot
pre-commit to not insure himself if he defaults in order to reduce his expected payoff of doing so.
16
Optimal labor demand. The optimal choice does not depend on the decision to default. Given capital k, the optimal choice of labor and the output remaining after labor
is compensated is:
1
1−θ
σj2 + σe2
θ
l(k; µ, j) = exp µ +
kα
2
W
1
θ # 1−θ
σj2 + σe2
θ
kα
π̃(k; µ, j) = (1 − θ) exp µ +
2
W
(11)
"
(12)
Default decision. An entrepreneur’s end of period wealth if they do not default (y N D )
and if they do default (y D ) are:
y N D (k; a, µ, j) = π̃(k; µ, j) + (1 − δ)k + (1 + r)a − (1 + r)(k + ψ)
(13)
y D (k; a, µ, j) = (1 − φ) [π̃(k; µ, j) + (1 − δ)k]
(14)
If the entrepreneur does not default, they get the output after labor is paid (π̃(k; µ, j)),
the depreciated capital ((1−δ)k), their assets plus interest ((1+r)a), minus the payments
to the lender ((1 + r)(k + ψ)). If they do default, they get a fraction (1 − φ) of their
output after labor is paid and the depreciated capital.
Because there are no dynamic penalties, the entrepreneur defaults if it maximizes expected end-of-period wealth. Proposition 2 characterizes the default decision.
Proposition 2 (Optimal default decision). An entrepreneur with capital k and state
variables (a, µ, j) defaults if and only if it maximizes their end-of-period wealth. Their
default decision is:
d(k; a, µ, j) =

1
if y D (k; a, µ, j) > y N D (k; a, µ, j)
0
if otherwise
17
(15)
The lender’s problem. Lenders take into account an entrepreneur’s default decision
(proposition 2) when determining the set of contracts to offer. When an entrepreneur
defaults, their lender earns negative profit. Therefore, lenders will only offer contracts
which the entrepreneurs will honor. The capital lent must satisfy the following incentivecompatibility constraint:
π̃(k; µ, j) + (1 − δ)k + (1 + r)(a − ψ) − (1 + r)k ≥ (1 − φ)(π̃(k; µ, j) + (1 − δ)k)
(16)
We can re-arrange the IC constraint to isolate the role of assets on the left-hand side
(LHS) and the role of beliefs about productivity and the loan size on the right-hand size
(RHS).
(1 + r)(a − ψ) ≥ −φπ̃(k; µ, t) + (1 + r − φ(1 − δ))k
(17)
The entrepreneur will not be able to borrow if the incentive-compatibility constraint
17 cannot be satisfied for any value of k. Proposition 3 shows that entrepreneurs with
assets less than ψ and expected productivity below a certain level will not be able to
borrow.
Proposition 3 (Access to external finance). Entrepreneurs who cannot afford to pay the
access cost up front (a < ψ) can only access credit if the expected quality of their idea is
above a threshold µ̂(a, j). Entrepreneurs who can afford to pay the access cost up front
can access credit for all expected productivity. Define the constant C1 :
θ
σe2
− (1 − α − θ) log(1 + r)
C1 = − + α log α + θ log
2
W
The threshold µ̂(a, j) is:



C + (1 − θ) log φ + α log(1 + r − φ(1 − δ))

 1
σ2
µ̂(a, j) =
+(1 − α − θ) log(ψ − a) − 2j



−∞
if a < ψ
if a ≥ ψ
Let me now characterize the effective credit limits faced by entrepreneurs with state
variables (a, µ, j).
Lemma 1. Let k ∗ be the values of k for which the incentive-compatibility constraint
(equation 16) binds.
18
k ∗ (a, µ, j) = {k : (1 + r)(a − ψ) + φπ̃(k; µ, j) − (1 + r − φ(1 − δ))k = 0}
If there are two solutions, then let k L (a, µ, j) and k U (a, µ, j) be the lower and upper
ones.
Proposition 4 characterizes the incentive-compatible loan contracts that lenders offer
to entrepreneurs based on their state variables (a, µ, j).
Proposition 4 (Incentive-compatible loan contracts). The set of loans that lenders are
willing to extend to an entrepreneur with state variables (a, µ, j) is given by the interval
K(a, µ, j) ≡ [k(a, µ, j), k(a, µ, j)].
The lower limit and upper limits k(a, µ, j) and k(a, µ, j) are:


0


k(a, µ, j) = k L (a, µ, j)



0
if a < ψ,
µ < µ̂(a, j)
if a < ψ,
µ ≥ µ̂(a, j)
if a ≥ ψ



0


k(a, µ, j) = k U (a, µ, j)



k ∗ (a, µ, j)
if a < ψ,
µ < µ̂(a, j)
if a < ψ,
µ ≥ µ̂(a, j)
if a ≥ ψ
The entrepreneur’s wealth maximization problem. I can now solve for the entrepreneur’s wealth y E (a, µ, j). They choose whether to access external finance (f ∈
{0, 1}) and how much capital to use in production to maximize their expected wealth:
Proposition 5 (Entrepreneur’s wealth maximization problem). An entrepreneur with
state variables (a, µ, j) maximizes wealth by choosing whether to access external finance:
f ∈ {0, 1}, and how much capital to use given feasible sets. The problem is:
y E (a, µ, j) = max {π̃(k; µ, j) + (1 − δ)k + (1 + r)a − (1 + r)k − f ψ}
{f,k}
Subject to:
19

k ∈ K(a, µ, j)
k ≤ a
if f = 1
if f = 0
Where π̃(k; µ, j) is the expected output net of the wage bill as defined in equation 12.
K(a, µ, j) is the set of loan contracts lenders are willing to offer this entrepreneur, as
characterized in proposition 4.
2.2.2
Definition of stationary equilibrium
The stationary equilibrium consists of three prices r, R and W , household policy functions
for saving a0 (a, µ, j) and occupational choice o(a, µ, j), entrepreneur’s policy functions for
accessing external credit f (a, µ, j), capital demand k d (a, µ, j) and labor demand ld (a, µ, j),
lower and upper bounds of feasible contracts k(a, µ, j) and k(a, µ, j), and the stationary
distribution of the population over the state-space G(a, x, µ, j). These satisfy the following
properties:
i) Given prices, o(a, µ, j) and a0 (a, µ, j) solve the household’s problem.
ii) Given prices, f (a, µ, j), k d (a, µ, j) and ld (a, µ, j) solves the entrepreneur’s problem.
iii) Given prices, k(a, µ, j) and k(a, µ, j) solve the lender’s problem.
iv) Labor market clears:
s
L =
=
∞ Z
X
j=1
∞ Z
X
(1 − o(a, µ, j))G(da, dx, dµ, j)
ld (a, µ, j)o(a, µ, j)G(da, dx, dµ, j) = Ld
(18)
j=1
v) Capital market clears:
s
K =
=
∞ Z
X
j=1
∞ Z
X
aG(da, dx, dµ, j)
k d (a, µ, j)o(a, µ, j)G(da, dx, dµ, j) = K d
j=1
iv) Intermediaries make zero profit: R = r + δ
20
(19)
vi) Distribution G(a, x, µ, j) is the fixed point given the transition rules for a, x, µ and
j.
2.3
Properties of the model
Before turning to the quantitative analysis, I will highlight some important properties of
the learning process and how they interact with financial frictions. I will also describe
some properties of the model that will help us to discipline the learning process and
decompose the role of imperfect information and financial frictions.
2.3.1
Properties of the learning process
Although there is a stationary distribution of households’ beliefs over the state variables
(x, µ, j), this distribution does not have an analytical characterization because of selection.14 I will therefore shut down selection to characterize the evolution of beliefs for
different types of entrepreneurs.
I can characterize the distribution of beliefs of households with new ideas after they
observe the first signal. The key takeaway is that households with high-quality new ideas
on average have an expected quality (µ) that is lower than the true quality, and their
expected quality gradually increases towards its true value as they observe more signals.
They also on average have an expected productivity less than their true productivity,
which is relevant for accessing capital, credit limits, and income from entrepreneurship.
Consider a household with a new idea of quality x. This household uses the initial
signal s to update their expected quality µ1 (calculated according to equation 1). Given
2
2
s µx
and
an underlying type x, the distribution of µ1 is normal with mean E (µ1 |x) = σxσx+σ
2 +σ 2
x
s
2 2
x
variance V (µ1 |x) = σ2σ+σ
σs2 . The term E(µ1 |x) is the average expected quality of
2
x
s
households with idea of quality x.
Property 1 expresses the average expected quality as a deviation from the true quality. Households with ideas above (below) the population average (µx ) have an average
expected quality below (above) the true quality.
Property 1. The average expected quality of a new idea with underlying quality x is:
E (µ1 |x) = x −
σs2
σx2 + σs2
14
[x − µx ]
(20)
As discussed in section 2.2, all households with new ideas observe an initial signal. They learn more
only if they operate a firm, and therefore the distribution is affected by selection.
21
Households with idea quality x, above the population mean µx , on average have an
expected quality µ1 below the true value. Households with idea quality below the population
average have an expected quality above the true value.
Property 1 is intuitive. Households account for the possibility of noise when updating
their expected quality. If the signal s = x is greater than the prior expected quality µx ,
then they consider the possibility that the signal is upward biased by noise by partially
adjusting their expected quality. If the signal was precise (σs2 = 0), then the household’s
expected quality immediately jumps to equal their true quality x. If the signal is completely noisy (σs2 = ∞), then their expected quality does not change at all and remains
µx .
If the household with the average expected quality (µ1 = E(µ1 |x)) implements their
idea and operates a firm, then their firm’s expected productivity is:
E
Z|E(µ1 |x), σ12
σ12 + σe2
= exp E(µ1 |x) +
2
(21)
Although I can characterize the distribution of expected productivity, a more informative exercise is to compare the average expected productivity (equation 21) with the
expected productivity under perfect information. The expected productivity under perfect information is:
σ2
E (Z|x) = exp x + e
2
(22)
Property 2 compares the expected productivity of the household with the median
signal, under imperfect and perfect information.
Property 2. The difference in the log-expected productivity under imperfect and perfect
information, for an entrepreneur with new idea of quality x that has the average expected
quality E (µ1 |x) is:
log E
Z|E (µ1 |x) , σ12
) − log (E (Z|x)) = −
Households with an idea quality x above µx +
ductivity under imperfect information.
σs2
σx2 + σs2
σx2
2
σx2
x − µx +
2
(23)
have a lower average expected pro-
Property 2 shows that households takes into account that uncertainty raises expected
productivity by increasing the possibility of a high x. However, for households whose
quality x is actually high, the overall effect is to drive down their expected productivity.
22
In lemma 2, I extend property 2 to households with additional signals. Households
receiving signals corresponding to their true quality each period gradually update their
2
expected quality toward the true value. For ideas with quality x > µx + σ2x , the expected
productivity is less than the expected productivity with perfect information.
Lemma 2. The difference in log-expected productivity under imperfect and perfect information, for an entrepreneur who in the j + 1 period of operation and has observed the
mean signal (s = z1 = · · · = zj = x) so far is:
2
σj+1
σj2
σj2
E(µj+1 |x) +
−x =
−x
E(µj |x) +
2
σj2 + σe2
2
(24)
I will show next that imperfect information, by lowering expected productivity makes
financing conditions tighter and decreases the net benefit of experimentation.
2.3.2
Interaction between financial frictions and imperfect information
In lemma 3, I show that conditional on having access to external credit, an entrepreneur’s
credit limits relax with higher expected quality µ and higher assets a. The reason is that
an entrepreneur with either higher assets or higher expected quality loses more if they
default. Therefore, more capital can be lent to them while ensuring repayment.
Lemma 3. [Relaxing credit limits] For entrepreneurs who can access credit, (i.e. µ ≥
µ̂(a, j)), their credit limits relax if either their assets or their expected quality increase.
∂k
≤ 0,
∂µ
∂k
≤ 0,
∂a
∂k
>0
∂µ
∂k
>0
∂a
For a highly productive entrepreneur, lemmas 2 and 3 imply that if they repeatedly
receive the median signal, then their credit limit relaxes over time. In figure 5 I illustrate
this by taking a highly productive entrepreneur and evaluate their credit limit if they
receive the mean signal (s = z1 = x) for the first two periods. I also plot what their credit
limit would be if they had perfect information, and show that it is higher for all asset
levels.
23
Figure 5: Credit limit with age for high-quality entrepreneur
Access to finance. In proposition 3, I defined a threshold expected quality µ̂(a, j)
that entrepreneurs who cannot pay the intermediation cost up front must have to access
credit. Proposition 6 shows the minimum expected productivity necessary to access credit,
and how this threshold expected productivity changes with the contract enforcement
parameter φ. This threshold depends on an entrepreneurs assets but not their beliefs,
and is therefore a common threshold faced by all entrepreneurs.
Proposition 6. An entrepreneur with assets less than the intermediation cost has access
σ2
to credit if their expected productivity µ + 2j is greater than χ(a). χ(a) solves:
χ(a) = C1 − (1 − θ) log φ + α log (1 + r − φ(1 − δ)) + (1 − α − θ) log (ψ − a) ,
∀a < ψ
Where C1 is a constant defined in proposition 3. For assets a < ψ, the threshold
χ(a) → ∞ as φ → 0+ .15
Lemma 2 and proposition 6 suggest that for a highly productive entrepreneur who
is asset poor, the probability of having access increases with age. I illustrate this by
taking a high-productive entrepreneur and calculating their probability of having access
by integrating over all realizations of the initial signal s and the first period productivity
z1 . I also plot their probability of having access if their productivity was known.
15
The prices r and W depend on φ but are bounded (W ≥ 0, r ≥ −δ). The direct effect on χ(a) of
small values of φ dominates any indirect effect through the prices.
24
Figure 6: Access to credit with age for high-quality entrepreneur
These results suggest that imperfect information tightens financing conditions for
young firms when contract enforcement is weak. However, if high-productivity households operate, they gradually discover their idea’s quality and their financing conditions
relax.16 These tight financing conditions have a much bigger impact by reducing the scale
of operations, reduce the net expected benefit of learning about the idea’s quality and
leading some households to forego learning altogether. The stylized model in section 2.1
illustrates the intuition.
2.3.3
Other model properties
Capital allocation with perfect enforcement. Households have no incentive to default since they lose all their output to the lender. As a result, they are able to borrow
the amount that maximizes their profit, and is independent of wealth.
k uc (µ, j) = arg max {π̃(k; µ, j) − Rk}
(25)
k
The associated profit is π uc (µ, j). In this economy, expected marginal product of
capital is equated across operating units.
All entrepreneurs have access with zero intermediation cost. Weak contract
enforcement alone does not affect access rates.
16
They also accumulate assets, which helps relax credit limits further. The speed of accumulation is
lowered by imperfect information, since expected productivity is lower.
25
Lemma 4 (Zero intermediation cost). If the fixed intermediation cost ψ = 0, then all
entrepreneurs have access to credit.
Occupation choice under perfect information and enforcement. Households
have perfect information if the variance of the initial signal σs2 = 0 and s perfectly reveals
their type. Proposition 7 shows that the household’s occupational choice does not depend
on the number of signals j or their assets a.
Proposition 7 (Occupational choice with perfect information and enforcement). If households have perfect information, their beliefs are (µ, σj2 ) = (x, 0). With perfect enforcement,
the household’s occupational choice is determined by static income maximization. Households are entrepreneurs if:
π uc (x, 0) ≥ W
(26)
Isolating the role of imperfect information. The initial signal and the insurance
scheme means I can eliminate any direct impact of the transitory shocks. This is a useful
tool, because it will allow us to isolate the impact of imperfect information.17
2
Proposition 8 (Transitory shocks have no direct effects). If σs2 = 0 and µx = − σ2e , then
the transitory shocks have no impact on household decisions or aggregates.
3
Quantitative Analysis
I calibrate the stationary equilibrium of my model with perfect contract enforcement
(φ = 1) and imperfect information (σs2 , σe2 > 0) to moments from US data. I then study
how weaker contract enforcement and higher imperfect information impacts productivity
and income per capita.
3.1
Calibration
The full set of parameters in this model are: {γ, δ, α, θ, σx2 , σe2 , µx , σs2 , ρ, β, ψ}. I calibrate
these parameters to match moments from US data, mostly following the approaches in
Buera et al. [2011] and Ranasinghe and Restuccia [2016]. I set the risk-aversion parameter
γ = 1.5 and the depreciation rate δ = 0.06, which are standard values from the literature.
2
I normalize the mean log-productivity by setting µx = − σ2e . Furthermore, the initial
17
See Alfaro et al. [2016] for an example of how higher uncertainty and financial frictions interact.
26
signal s is a modelling tool to encapsulate the perfect information economy. I therefore
assume that the variance of the initial signal is equal to the variance of the transitory
shocks, which is what I propose determines the speed of learning (i.e. σs2 = σe2 ).
Although the remaining 6 parameters are calibrated jointly, for each parameter there
is a moment that is particularly informative. The fraction of income going to capital
and labor α + θ is chosen such that the fraction of the population that are entrepreneurs
α
= 0.3.
equals 7.5% (Cagetti and Nardi [2006]). Capital income share is then set to α+θ
The discount rate β is set to target an equilibrium real interest rate of r = 0.04. DablaNorris et al. [2015] report that the fraction of firms with access to credit is 95%, which
disciplines the intermediation cost ψ.18 The variance of the productivity distribution (σx2 )
is disciplined by the fraction of income going to the top 5% of households.
A key feature of my model is that with perfect information and perfect contract
enforcement, the exit rate of firms does not vary with firm age (proposition 7). However,
with imperfect information, the exit rate of young firms is higher than the exit rate
of old firms. I therefore exploit the difference in the exit rate between young and old
firms documented by Hsieh and Klenow [2014] to discipline the amount of imperfect
information. Old firms know their productivity and therefore exit only if their productivity
changes and their new idea is not worth implementing. Therefore their exit rate of 6% is
informative of the persistence of the productivity process ρ. Firms younger than 5 years
exit at a higher rate: From Hsieh and Klenow [2014], I calculate that after 5 years, only
52% of the initial entrants remain in operation. I use the higher exit rate for young firms
to discipline the variance of the transitory shocks, σe2 . Figure 7 illustrates the argument.
18
I map this moment to the data as the fraction of firms who can, conditional on paying the intermediation cost, face a non-zero credit limit (see proposition 3). Of course, this is different from the fraction
of firms who choose to access external credit.
27
Figure 7: Disciplining imperfect information parameters
Table 1 reports the values of the calibrated parameters.
Data
Fraction entrepreneurs = 0.075
Income share of top 5% = 0.3
Fraction exiting within 5 years = 0.48
Exit rate amongst old firms = 0.06
Real interest rate = 0.04
Fraction with external finance = 0.95
Model
0.075
0.3
0.48
0.06
0.04
0.95
Parameters Calibrated value
α+θ
0.81
2
σx
0.31
2
σe
0.04
ρ
0.92
β
0.92
ψ
0.16
Table 1: Calibrated parameters
There are two additional moments of interest: the calibrated economy has a capital
to output ratio of 3.1 which is similar to the US data. The external credit to GDP ratio
is 1.88, which is less than the 2.4 in the US (Beck et al. [2000]). However, this value is
well within the range of estimates for developed countries.19
Benchmark economy. I use the economy with perfect enforcement (φ = 1) and perfect
information (σs2 = 0) as the benchmark and present results for all other economies relative
19
When calibrating, I found that matching the US external credit to GDP ratio requires changing the
entrepreneurial share of income (1 − α − θ) and the variance of entrepreneurial productivity (σx2 ) in such
a way that the fraction of households that are entrepreneurs fall sharply. As the probability of having an
idea worth implementing is a critical determinant of the incentive to experiment, I maintain the fraction
of entrepreneurs as a target.
28
to outcomes in this one. This economy has the highest productivity and GDP per capita,
and also corresponds to the benchmark used in the literature. Table 2 compares the
benchmark economy to the calibrated US economy.
TFP
GDP per capita
capital-to-output
external credit to GDP
Benchmark
US
100.0%
96.8%
100.0%
96.7%
3.14
3.12
1.89
1.88
Table 2: Benchmark and calibrated (US) economies
Imperfect information lowers both productivity and GDP per capita by about 3%.
This productivity loss is due mainly to firms choosing inputs under imperfect information
about their productivity, and secondly to misallocation of some households who should
be entrepreneurs as workers.20
3.2
Weakening contract enforcement
I study what happens to productivity if contract enforcement is weakened, by reducing
φ. In figure 8, I plot TFP and GDP per capita against relative external credit to GDP.21
20
See figure 9 for a decomposition of TFP losses due to various sources.
I report the results against relative external credit to GDP (i.e. relative to external credit to GDP
in the benchmark economy) because it is monotonically increasing with φ, and is actually an observable.
For example, India has an external credit to GDP of about 0.3, which translates to 0.13 relative external
credit to GDP in figure 8.
21
29
Figure 8: TFP and GDP per capita for different levels of φ
In table 2, I reported that even with perfect contract enforcement, imperfect information lowers TFP by about 3%. This corresponds to the gap between the perfect and
imperfect information plots in the left panel, when relative external credit to GDP is 1.
Under both perfect and imperfect information, TFP falls monotonically as contract
enforcement is weakened. Income per capita also falls monotonically, falling by about 30%
in the worst-case economy with imperfect information. The drop is larger than the drop
in productivity because incentives to accumulate capital is also reduced and the capital
to output ratio is lower than in the perfect-enforcement case.
With imperfect information, TFP falls by 22.9% in the worst case scenario. With
perfect information, TFP falls by 12.6% in the worst-case scenario. After accounting for
the 3% TFP loss due to the direct effect of imperfect information, the steeper rate of
TFP loss under imperfect information suggest the two frictions interact. I will explore
this interaction below.
Remark on TFP loss under perfect information. The TFP loss from weakening
contract enforcement under perfect information is near the lower end of the range of
estimates in the literature. The reason is that my persistence parameter ρ is higher than
in comparable models. My persistence is higher because while many papers target the
average exit rate, I decompose the exit rate between that of young and old entrepreneurs,
and discipline the persistence by the exit rate of old firms, which is lower than the average.
With higher persistence and perfect information, highly productive firms have a stronger
incentive and more time to overcome credit limits by saving (Moll [2014]).
30
Decomposing the TFP losses. To understand how weak contract enforcement and
imperfect information interact to reduce TFP, I decompose the total TFP loss into several
components. For any given value of φ, I start from the equilibrium and move toward
the perfect information, perfect enforcement benchmark economy by gradually undoing
misallocation. I implement the following steps to both the imperfect and the perfect
information economies.
i) Holding aggregate capital K, the number and distribution of entrepreneurs constant,
ia) under imperfect information, reallocate capital to equate expected marginal
product of capital (reallocate capital)
ib) reveal quality (x) to entrepreneurs, then reallocate capital to equate expected
marginal product of capital (reveal type, imperfect information economies only) 22
ii) Holding aggregate capital K and the number of entrepreneurs constant, choose the
most productive households as entrepreneurs, and then under perfect information,
reallocate capital to equate expected marginal product of capital (reallocate talent)
iii) Allow capital and the number of entrepreneurs to adjust to benchmark economy
levels (i.e. solve all problems and aggregate at benchmark economy prices)
In the first step, I reallocate inputs, taking aggregate resources and the productivity
distribution of operating firms as given. This exercise is similar in spirit to Hsieh and
Klenow [2007], who calculate for the US, China and India the productivity gains from
optimally reallocating factors while taking each country’s productivity distributions as
given. However, because I have imperfect information, I equate expected marginal product
of capital rather than the realized marginal product of capital.
In step ii, I correct for distorted selection, by choosing the households with the most
productive ideas into entrepreneurship. I should emphasize that I do this exercise while
holding the underlying distribution of productivity from which entrepreneurs are selected
to the calibrated US productivity distribution.23
I report the results of this decomposition exercise in figure 9, with the results for
economies with imperfect information in the left panel and those for the economies with
perfect information in the right panel. The key insight is that weak contract enforcement
distorts the selection of entrepreneurs much more under imperfect information than under
22
I am equating expected MPK because, even with perfect information about x, entrepreneurs still
choose inputs before observing the transitory shock e.
23
Maintaining the log-Normal assumption, productivity distributions in other countries can differ in
the two moments µx and σx2 . Differences in µx can be normalized away. We have little evidence on how
the second moment σx2 varies across countries (Buera and Shin [2013]).
31
perfect information. In the imperfect information case, the loss due to distorted selection
is the gap between the line labeled ‘reallocate talent’ and the line labeled ‘reveal type’.
In the perfect information case, the loss due to distorted selection is the gap between
‘reallocate talent’ and ‘reallocate capital’.
Figure 9: Decomposing TFP losses from weaker contract enforcement
Table 3 summarizes the loss in the worst case:
Capital misallocation (imperfect)
Capital misallocation (perfect)
Talent misallocation
Number of entrepreneurs
Total loss
Imperfect
Perfect
6.1% (6.1%)
2.5% (8.6%)
7.8% (7.8%)
11.6% (20.2%) 2.2% (10.0%)
2.7% (22.9%) 2.4% (12.4%)
22.9%
12.4%
Table 3: Decomposing TFP loss
In each column, I report the loss due to each source of misallocation and the cumulative
loss in brackets. Standard capital misallocation is somewhat larger under imperfect information (8.6% vs 7.8%). However, the information regimes have starkly different losses
from distorted selection (talent misallocation). With perfect information, distorted selection lowers TFP by 2.2% (17.5% of the total loss) whereas with imperfect information,
distorted selection lowers TFP by 11.6% (50.6% of the total).
Under perfect information, there is very little TFP loss due to talent misallocation
because highly productive households always enter. They then accumulate assets to relax
32
credit limits.24 The larger TFP loss due to distorted selection is consistent with my hypothesis. Under imperfect information, many households facing tight financing conditions
forgo learning all together. Having foregone entrepreneurship, they have no incentive to
accumulate assets to overcome credit credit constraints.
3.2.1
Other effects of the weak contract enforcement
Access to credit with age. Chavis et al. [2011] document from the World Bank enterprise survey that younger firms in developing countries have lower access to finance. They
find that in developing countries, the fraction of young firms (younger than 5 years) with
access to credit is particularly low (∼ 35%) and rises with age. As discussed previously,
imperfect information, by introducing uncertainty about output, tightens access to credit
for poor young firms, and the impact is magnified as contract enforcement weakens.
Figure 10 reports the fraction of firms with access to credit in economies with different levels of contract enforcement under different information regimes, for ‘young’ (≤ 5
years) and ‘old’ (≥ 20 years) firms respectively.25 Access for young firms fall much more
sharply under imperfect information than under perfect information. This is because
under perfect information, although young firms are poorer than old firms, they can use
their high productivity to access credit. Under imperfect information, using expected
output becomes harder for young firms.
Notice that there is not much difference across the two information regimes for old
firms. This is because these firms are relatively wealthy and, in the case of imperfect
information have observed many signals so effectively know their idea’s quality.
24
The small loss due to selection under perfect information is similar to what Buera et al. [2011] finds
for the sector without fixed costs (services) in their study.
25
Ability to access credit depends on both expected productivity and assets here.
33
Figure 10: Access to credit by age
The model overstates access to credit for young firms in the US (Robb and Robinson
[2012]), because with a common intermediation cost, imperfect information alone is not
able to generate a large gap in the access probability if lenders can seize all of young
entrepreneurs’ output (perfect enforcement). However, as we see by comparing the left
and the right panel, the imperfect information can help explain why young firms are
particularly affected when contract enforcement is weak.26
Firm exit rate Hsieh and Klenow [2014] document that firm exit rate profile with age
is both lower and flatter in India than in the US. Since I use the US exit rate profile to
calibrate imperfect information in my model, I ask what happens to the exit rate when
I reduce contract enforcement to Indian levels. Figure 11 presents the exit rates in the
model version of the two economies.
26
Matching the access rates in the US would require imposing a higher intermediation cost on young
firms, which can be justified by appealing to higher screening costs for these firms (Lerner [2009]). This
would just amplify the productivity losses presented here.
34
Figure 11: Exit rate with age
Consistent with Hsieh and Klenow [2014], the exit rate profile is flatter in the economy
with Indian level of financial development. This is because poor entrants enter only if
their initial signal is very high (less experimentation) and therefore are less likely to drop
out later. The outside option is also lower, so wealthy, low quality entrants are less likely
to drop out.
These results go only part way to explaining the large differences in the exit rates
between India and the US. However, with perfect information the relative slopes of the
exit rates would be counterfactual: the exit rate for the US would be flat, while India’s
has a slight negative slope initially. Therefore, the exit rate profile in India would be
steeper than the one in the US.
3.3
‘Increasing’ imperfect information
I found that the calibrated US economy has lower TFP than the benchmark economy due
to the direct effects of imperfect information. Although we do not know how imperfect
information about entrepreneurial productivity and the speed of learning varies across
countries, the literature finds that other sources of uncertainty as well as the variance of
transitory shocks are generally higher in developing countries (e.g. Koren and Tenreyro
[2007], David et al. [2016]). My model can capture all possible levels of imperfect information. Therefore I use it to investigate how increasing imperfect information affects
TFP. I implement this experiment with perfect contract enforcement (φ = 1). I increase
2
the variance of the transitory shocks σe2 , adjusting σs2 (σs2 = σe2 ) and µx (µx = − σ2e )
appropriately.
35
As the variance σe2 increases, the speed of learning falls and in the limit is zero. The
intuition is that as the variance increases, the households attribute much of the realized
productivity z to transitory shocks, and therefore adjust beliefs slowly. In the limit as
σe2 → ∞, for any number of signals j, the moments of the posterior distribution approaches
the population mean and variance: (µj , σj2 ) → (µx , σx2 ). Figure 12 presents the results.
Figure 12: TFP as imperfect information increases
TFP declines monotonically with higher imperfect information and drops by 54% in
the limiting economy with no learning.27
Clearly, no country has a transitory shock distribution with variance 500× the US one.
However, the experiment shows the maximum possible loss, and shows that the fall in TFP
is steepest for small multiples of the US variance (right panel). What are plausible guesses
for how much higher the variance might be in developing countries? David et al. [2016]
finds that the variance of the particular shocks they study is up to 1.8 times higher in India
relative to the US. A multiple of 2 in the right panel would reduce TFP by an additional
3%. A multiple of 3 times would bring TFP down by 10% in total, a magnitude that
would compare to other direct explanations for TFP differences. Therefore, identifying
cross-country differences in imperfect information might be a fruitful way to explain crosscountry productivity differences.
27
As σe2 → ∞, all households behave as if they had the unconditional population productivity which is
completely persistent. Heterogeneity no longer plays a role and the economy collapses to a representative
agent one.
36
3.4
Policy intervention
In section 3.2, I quantified the impact of a novel channel through which weak contract
enforcement impacts TFP: I showed that weak contract enforcement substantially lowers
TFP by discouraging experimentation by poor households. Countries with weak contract
enforcement can increase productivity by reforming their legal institutions. By relaxing
financing conditions for new entrepreneurs, more households will experiment and more
high productivity entrepreneurs will be discovered (Kerr and Nanda [2009]).
While many countries have made progress improving contract enforcement (e.g. Campello
and Larrain [2015]), legal institutions are generally thought to be deeply dependent on
history (e.g. Djankov et al. [2007]) and therefore difficult to change.
As an alternate tool, many governments run programs to support business development, targeting either small or young businesses who they believe are credit constrained (Lerner [2009]).28 I investigate whether subsidy schemes targeted towards new
entrepreneurs can support experimentation and the discovery of new ideas.
Cost of experimentation. The cost of experimentation is static. It is the foregone
wage W minus the income from operating the firm. In particular, 29
C(a, µ, j) = max {0, W − π(a, µ, j)}
(27)
Subsidy program. The government gives a subsidy ξ to new entrepreneurs for the first
four years of their time in operation, as in Arkolakis et al. [2014]. A new entrepreneur
is defined as a household that has a new idea (j = 1) and was not an entrepreneur in
the previous period. I assume that the government knows that the household’s firm is
new and is implementing a new idea. The government learns this information because
households have to register their business and describe what they intend to produce in
order to get the subsidy.30 However, I assume the government does not screen based on
assets or expected quality µ.
The program is financed by a lump-sum tax T on all households, regardless of their
occupation. The government balances its budget in each period. Therefore, if a fraction
n of households are receiving the subsidy, then the government’s budget constraint must
satisfy:
28
Lerner [2009] also highlights the complicated incentive problems faced by the government when
designing any business support program and reviews the mixed results. To highlight the potential gains,
I abstract from these issues in these experiments.
29
I bound the cost at 0 because for entrepreneurs whose profit π(a, µ, j) > W , there is no cost of
experimentation as they would operate just to maximize income
30
I assume the government can verify the idea is new to eliminate the incentive for households to switch
into and out of entrepreneurship to collect the subsidy.
37
ξn = T
(28)
The subsidy benefits new entrepreneurs in several ways: first, the government pays
the subsidy to the lender if the household defaults, and therefore it relaxes credit limits.
Second, conditional on not defaulting, the household keeps the subsidy and therefore is
compensated for part of the cost of experimentation. Finally, the additional income due to
the subsidy allows the household to save more and relax credit constraints in the future.31
In general, the subsidy lowers the cost of experimentation.
C(a, µ, j; ξ) = max {0, W − π(a + ξ, µ, j) − ξ}
≤ max {0, W − π(a, µ, j)}
= C(a, µ, j; 0)
Scope of program. In this investigation, I am mainly interested in exploring how a
subsidy scheme can support the discovery of new ideas. I therefore limit the size of the
subsidy schemes I consider to focus on this channel. Financial frictions also introduce
static misallocation, which other programs could help correct (Buera et al. [2014]).
I explore the benefits of such a subsidy scheme using two experiments. In the first
experiment, I take an economy with a level of financial development similar to the nonOECD average and evaluate the impact of subsidizes of various sizes on TFP and income
per capita. Notice that if all households could earn at least the equilibrium wage as an
entrepreneur, then the cost of experimentation would be zero. I consider subsidies that
are various fractions of the pre-subsidy equilibrium wage. In particular,
˜ 0,
ξ = ξW
ξ˜ ∈ [0, 1]
(29)
In the second exercise, I set the size to half of the pre-subsidy equilibrium wage in
economies with varying levels of financial development, and study the effect of this subsidy
across different economies. In particular,
1
ξ(φ) = W 0 (φ)
2
(30)
Exercise 1: Subsidizing new firms in a representative developing country. I
choose the economy which, prior to the subsidy, has an external credit to GDP ratio of
0.3432 Figure 13 shows the impact on TFP and income per capita as I vary the size of the
31
32
By relaxing credit constraints in the future, it indirectly also increases the benefits of experimentation.
Beck et al. [2000].
38
subsidy.
Figure 13: TFP and GDP per capita as subsidy size varies
In figure 13, at ξ = 0 there is no subsidy and the TFP and GDP per capita correspond
to TFP and GDP per capita in 8 at a relative external credit to GDP of 0.16. Both TFP
and GDP per capita increases substantially as we increase the size of the subsidy, though
the rate of increase declines. For example, TFP increases by 8% when the subsidy is
at the largest value considered. The cost of the subsidy program is increasing with the
˜ The largest cost is about 1.82%. The reason why this program can have a
fraction ξ.
large impact is because once new entrepreneurs learn what their productivity is, they can
optimally choose to save out of the collateral constraints.
I should note that this subsidy is correcting both capital misallocation by relaxing
credit constraints for operating entrepreneurs and supporting experimentation. However,
the above exercise suggests that correcting the external margin can have a relatively large
impact on productivity.
Exercise 2: Impact of subsidy in different economies. To investigate the impact
of the subsidy across economies, I set the level of the subsidy to half the pre-subsidy equilibrium wage and study the impact on productivity. I decompose the TFP loss following
the scheme outlined in section 3.2. Figure 14 reports the results.
39
Figure 14: Decomposing TFP gains from subsidy
The left panel in figure 14 is the decomposition of TFP loss under imperfect information, without the subsidy. In the right panel is the decomposition with the subsidy. In the
economy with perfect enforcement, the subsidy lowers TFP a small amount by bringing
in extra entrepreneurs but this is canceled by the higher number of entrepreneurs. As I
lower the level of contract enforcement, TFP falls in both economies but by much less in
the economy with the subsidy. The distortion to the selection process is much smaller
with the subsidy.
Discussion. I present this subsidy scheme to highlight that supporting the discovery of
entrepreneurial talent can have a large impact on productivity. In economies with weak
contract enforcement, this scheme can be very helpful because once the quality of the idea
has been discovered, households can save their way out of credit constraints. A subsidy
scheme based on the firm’s age might also be harder for entrepreneurs to game than a
size-dependent subsidy. Furthermore, as the program does not target particular firms, it
might not be subject to the capture as identified in Buera et al. [2012].
4
Conclusion
In this paper, I investigate the impact of financial frictions arising from weak contract
enforcement on aggregate productivity. I highlight an important determinant of productivity, the discovery of productive entrepreneurs, and show that weak contract enforcement
distorts this discovery process. The impact of financial frictions is significantly amplified,
by up to 1.5 times.
40
I also find that imperfect information alone can account for potentially large TFP losses
in developing countries. Given the much larger estimates of other measures of uncertainty
in developing countries, it is plausible that discovering entrepreneurial productivity takes
longer in these countries as well. Disciplining this hypothesis with data is a promising
avenue for explaining cross-country productivity differences.
41
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44
5
Appendix
5.1
Proofs
Proof of proposition 1. I present the proof for an entrepreneur who has accessed external
finance and then defaulted (f = 1, d = 1). The proof for entrepreneurs who self-finance
(f = 0) or access external finance but do not default (f = 1, d = 1) are similar.
The entrepreneur chooses the sequence {ω(z)}z∈R , the insurance payments for all realizations of the total productivity z. Because the insurer must break even in expectation,
the sequence must satisfy:
Z
∞
ω(z; k, l)F̂ (dz|µ, j) = E [ω̃(k, l; z)|µ, j]
z
−∞
If the entrepreneur assigns y(z) to realization z, then their end of period resources will
be:
y(z) = max {0, (1 − φ)(ω(z) − W l + (1 − δ)k)}
Let Ṽ E (y(z), a0 , µ, j) be the post-production value function for an entrepreneur defined
over their end of period resources.
Ṽ E (y(z), a0 , µ, j) = u(y(z) − a0 ) + β[ρV (a0 , µ+1 (z), j + 1) + (1 − ρ)EV (a0 , µ1 , 1)]
It is obvious that the household will never choose y(z) ≤ a0 as that will give them zero
consumption. Therefore y(z) > a0 ≥ 0. We can therefore restrict attention to allocations
ω(z) that yield positive y(z). The household’s optimization problem is:
Z
L = max
{y(z)}
E
Z
0
Ṽ (y(z), a , µ, j)F̂ (dz|µ, j) + λ
[E [ω̃(k, l; z)|µ, j] − y(z)] F̂ (dz|µ, j)
z
z
Point-wise optimization gives:
y(z1 ) = y(z2 ),
∀z1 , z2 ∈ R
So far, I have implicitly assumed that it is possible to satisfy y(z) > a0 for all z, which
is equivalent to assuming:
(1 − φ)[E[ω̃(k, l; z)|µ, j] + (1 − δ)k − W l] > a0
If the above assumption is not satisfied, the household has a non-zero probability of
45
having zero consumption. Therefore, no matter the insurance allocation their expected
payoff is equal to −∞. The household will never choose (a0 , k, l, f, d) such that their
expected payoff is −∞. Since the particular allocation in these off-equilibrium paths has
no impact on the payoff (always −∞), WLOG I assume they insure.
The only difference between the above case and the cases where the entrepreneur does
not default or does not access finance is the components of their post-production wealth.
y(z) =

max {0, ω(z) − W l + (1 − δ)k + (1 + r)(a − k − ψ)}
if f = 1, d = 0
max {0, ω(z) − W l + (1 − δ)k + (1 + r)(a − k)}
if f = 0
Proof of proposition 2. Default is only a consideration if the lender has accessed external
finance (f = 1). When deciding to default, the entrepreneur has also already chosen k,
l, and a0 . In proposition 1, I showed that the entrepreneur will always insure themselves
from the transitory shock, whether or not they defaulted. We can then characterize the
resources available to the entrepreneur based on their default decision using equations 13
and 14.
y N D (k; a, µ, j) = π̃(k; µ, j) + (1 − δ)k + (1 + r)a − (1 + r)(k + ψ)
y D (k; a, µ, j) = (1 − φ) [π̃(k; µ, j) + (1 − δ)k]
Since a0 has already been chosen, by defaulting the entrepreneur can only affect his
consumption c = y − a0 . The entrepreneur will therefore choose the option that gives
them the highest payoff. To make this explicit,
i) If a0 ≤ min y N D , y D , Ṽ E (y, a0 , µ, j) is increasing in y, and therefore is maximized
by maximizing income.
ii) If a0 ∈ [min y N D , y D , max y N D , y D ), then the entrepreneur should choose the
option that maximizes income (choosing the other option gives a payoff of −∞)
iii) If a0 ≥ max y N D , y D , then either choice will give a payoff of −∞. WLOG, the
household should choose the one that gives higher income.
The above breakdown shows that no matter the choice of a0 , the household’s default
decision maximizes expected income. Therefore the household cannot adjust a0 to reduce
their incentive to default.
46
Proof of lemma 4. An entrepreneur with state variables (a, µ, j) has access to credit if
there is some loan k for which the incentive-compatibility constraint is satisfied:
1 + r − φ(1 − δ)
k
(1 + r)a ≥ −φ π̃(k; µ, j) −
φ
The left-hand side is non-negative because a ≥ 0. On the right-hand side, we can
replace π̃(k; µ, j) by equation 12, n
minimize with
o respect k to find that the right-hand side
σj2 +σe2
is always negative as long as exp µ + 2
> 0.
Proof of lemma 3. The results follow immediately from inspecting the incentive compatibility constraint.
For the upper bound k(a, µ, j),
(1 + r)(a − ψ) = −φπ̃(k; µ, j) + (1 + r − φ(1 − δ))k
Increasing either µ or a slackens the constraint, and k needs to increase for it to bind
again.
If the lower bound k(a, µ, j) = 0, then increasing assets or the expected quality µ has
no effect. If the lower bound k(a, µ, j) > 0, then increasing a or µ slackens the constraint
which can be made to bind again by lower k.
5.2
Modified environment where wealth smoothing is optimal
I present an environment in which entrepreneurs will find it optimal to insure themselves
against the transitory shocks. In particular, I show that they will choose an insurance
scheme that gives them the same wealth for all realizations of z. This result is obtained
because entrepreneurs create a plan for how to divide their expected wealth between consumption and savings for the next period prior to seeing their realized productivity z
this period, and after observing z cannot profitably deviate from this plan. Therefore the
decision to insure reduces to a decision regarding how to best allocate consumption today
across possible states.
Remark. If agents were allowed to choose assets for the next period after observing
z, then they would choose to redistribute more than (less than) the expected value to
low- (high-) realization states. The reason is that low realizations of z mean both less
income today and less expected income in the future. Therefore, additional funds in these
states are especially valuable. A separate issue is that the continuation value V is locally
47
convex in assets at points where V W and V E intersect. The environment presented below
overcomes these issues.
Figures 15 and 16 present the timelines for an entrepreneur and a worker, respectively.
In these figures, the bold elements are decisions and events from the timeline presented
in the main model, while the other elements are there to support the insurance mechanism.
choose l,
choose
D/ND
plan
(c, a’)
producer
sells a’
to retailer
for i(p)
z realized.
production
shopper/
producer
meet
shopper/
producer
separate
1
3
choose
occupation
(W/E)
choose
deposits,
capital (k),
access (f)
7
5
4
2
clear
balance
with retailer
6
choose
insurance
plan
9
10
8
insurance
pays out.
labor paid
shopper
buys a’
from
retailer
for i(s)
Figure 15: Timeline for entrepreneurs
48
11
consume
clear
balance
with
retailer
producer
sells a’
to retailer
for i(p)
plan
(c, a’)
shopper/
producer
meets
shopper/
producer
separate
1
3
4
2
choose
occupation
(W/E)
7
5
6
11
10
8
paid wage
choose
deposits
9
shopper
buys a’
from
retailer
for i(s)
consume
Figure 16: Timeline for workers
Let me now discuss the new elements in the environment.
Shoppers and Producers. Each household is composed of two members: a shopper
and a producer. The black elements are dealt with by the whole household, the blue
are dealt with by producers only, and the red are dealt with by shoppers only. At the
beginning of the period, the two members jointly choose the occupation, plan how much
to save and to consume, and then go to two separate islands. The producer takes all the
assets with him, as his island contains all the financial institutions. The two members
reunite at the end of the period right before consumption takes place.
Insurers. A competitive insurance market opens after producers have chosen inputs
and whether to default, but before the realization of total productivity z. The insurers
are owned by households, have no operating costs, and can observe the producers’ state
variables related to beliefs about productivity (µ, j), the input choices (k, l), and the
actual realization of z. They sell contracts that allow each producer to transfer wealth
across states (realizations of z) as they wish, conditional on the insurer breaking even.
49
Retailers. I assume that the consumption good produced by firms cannot be carried
over as-is into the next period without completely spoiling. However, a technology exists
that can convert units of the consumption good into units of a one-period storeable good.
The conversion rate is 1-to-1. This technology is operated by a continuum of competitive
firms, whom I call retailers, and they are owned by the households. In particular, the
conversion technology is:
ga (a) = a
Retailers have rational expectations about demand for storeable goods. They first go
to the producers’ island to buy the appropriate quantity of the consumption good, and
transforms it into the one period storable good. Then, they go to the shoppers’ island
and sell the storeable good to shoppers.
The retailers’ transactions with the shoppers and producers is based on short-term
credit, as in each transaction there is one party that has nothing immediately at hand
to give to the counterparty. The retailer issues to a producer who sells a0 units of a the
consumption good ip (a0 ) = a0 > 0 units of credits. The producers can use these credits to
cover purchases of the storable good from the retailer, within the period. Since shoppers
have no funds at hand, they purchase the saving good a0 on credit is (a0 ) = a0 < 0. When
the two members of the household reunite, they clear their balance with the retailer by
sending the IOU’s issued to the producer.
Issued credit expires at the beginning of the next period, and therefore excess amounts
cannot be used for intertemporal saving. If the household cannot clear the balance by the
end of the period, the retailers seize all their consumption.
5.2.1
Solution to the worker’s problem
Let me solve the worker’s problem first, since it is simpler and will help clarify the environment. After choosing the occupation (working), the two members jointly plan for the
period. As a worker household, their income ((1 + r)a + W ) is known, and they learn
nothing new in this period. Since their income is fixed, there is no insurance problem to
solve. However, they must still plan how much of the storable good the shopper must
acquire on her island and how much credit (ip ) the producer must acquire on his island,
so that the balance clears at the end of the period. The optimal choice of a0 is given by
maximizing the worker’s problem (WP):
50
V
W
0
0
u(c) + β ρV (a , µ, j) + (1 − ρ) E V (a , µ1 (s), 1)
(a, µ, j) = max
0
x,s
c,a
s.t.
c + a0 ≤ (1 + r)a + W
Neither party has any incentive to deviate from this plan. If one party did deviate and
as a result the household ended up with excess credit (is + ip > 0), then these would be
useless and the household would be strictly worse off. On the other hand, if a deviation
resulted in having not enough credit to cancel out the shopper’s obligations (ip + is < 0),
then they would default on the retailer and lose all consumption in the period and get
V = −∞.
5.2.2
Solution to the entrepreneur’s problem
Relative to the timeline presented in the main model, the only new components are that
the entrepreneur finds it optimal to stick to a savings plan chosen prior to realizing z and
that they choose to smooth wealth across states. I therefore focus on proving these two
parts. I will begin by solving the insurance problem. At the point where the producer
must choose the insurance contract, she has already chosen the asset for next period a0 ,
financing and capital (f , k), labor (l), and whether to default (d). Take any values for
these choices.
Let’s define a couple of useful objects. First, to evaluate the value of wealth, let
Ṽ (y, µ, j) be the value function defined over wealth y, right before consumption takes
place. Let ω̃(z; k, l) = ez k α lθ be the output generated by a firm with realized productivity
z employing capital k and labor l. Finally, let F̂z (z|µ, j) be the prior distribution of z for
an agent whose expected ability is µ and who has observed j signals.
The expected output of this agent is:
Z
∞
Eω(k, l, µ, j) = E (ω̃(z; k, l)|µ, j) =
ω̃(z; k, l)F̂z (dz|µ, j)
−∞
Let {ω(z; k, l)}z∈R be an allocation of insurance payments across states.33 Then the
break even condition (BE) is:
Z
∞
Eω(k, l, µ, j) =
ω(z; k, l)F̂ (dz|µ, j)
(BE)
−∞
The components of the entrepreneur’s end of period wealth depends on whether they
accessed external financing (f ∈ {0, 1}) and if they did access external finance, whether
33
The no-insurance allocation sets ω(z; ·) = ω̃(z, ·) for all values of z ∈ R.
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they defaulted (d ∈ {0, 1}). I will only present in detail the derivation of the optimal
insurance contract for the case where the entrepreneur has accessed external credit and
then defaulted. The solution for the other two possible cases are analogous.
Agent has defaulted. If the agent chooses insurance payment ω(z, ·) when total
productivity is z, then the resulting wealth is:
y(z; k, l, a) = max {0, (1 − φ) [ω(z, k, l) + (1 − δ)k − W l]}
The minimum value is zero because entrepreneurs have limited liability. Agents will
never want to have zero consumption, and therefore any allocation of wealth across states
must satisfy:
(1 − φ) [ω(z, k, l) + (1 − δ)k − W l] > a0 ≥ 0
The agent’s problem then is to choose the optimal allocation of wealth {y(z)}z∈R across
realizations of z. This is:
Z
∞
0
max
Ṽ (y(z), µ (z), j + 1)F̂ (dz|µ, j)
{y(z)}z∈R
−∞
s.t. (BE) and
y(z) > a0
(non-zero consumption)
Ignoring the non-zero consumption constraint for now, the Lagrangian and the first
order conditions from point-wise optimization is:
Z
∞
0
L = max
Ṽ (y(z), µ (z), j + 1)F̂ (dz|µ, j)
{{y(z)}z∈R ,λ} −∞
Z ∞
+λ [Eω(k, l, µ, j) + (1 − δ)k − W l] −
y(z, k, l)F̂ (dz|µ, j)
−∞
dṼ
dy
Envelope
=
1
= λ,
(y(z) − a0 )γ
∀z ∈ R
Equating first-order conditions across states, we find that the optimal wealth choice
is the same for every realization of z. The insurer’s break even condition implies that the
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expected wealth is the solution.
y D (k, l; µ, j) = (1 − φ) [Eω̃(k, l; µ, j) + (1 − δ)k − W l]
We can verify that the non-negative consumption constraint is satisfied as long as
a < (1 − φ) [Eω̃(k, l; µ, j) + (1 − δ)k − W l]. This restriction is without loss of generality,
as households will never choose a0 above their expected income. 34
0
No default. If the agent did not default and chose insurance scheme {ω(z, ·)}z∈R ,
then their realized wealth is:
y(z; k, l, a) = max {0, ω(z, k, l) − W l + (1 − δ)k + (1 + r)a − (1 + r)(k + ψ)}
The optimal insurance scheme assigns the expected wealth to each state.
y N D (k, l, µ, j) = Eω̃(k, l; µ, j) + (1 − δ)k + (1 + r)a − W l − (1 + r)(k + ψ)
Default decision based on expected incomes. The entrepreneur optimally defaults if:
E Ṽ (y D (k, l; µ, j), µ, j) > E Ṽ (y N D (k, l; µ, j), µ, j)
z
z
After some expanding and rearranging terms, we can rewrite the problem as:
Z
∞
[Ṽ (y D (k, l; µ, j), µ0 (z), j + 1) − Ṽ (y N D (k, l, µ, j), µ0 (z), j + 1)]F̂ (dz|µ, j) > 0
−∞
For all values of µ0 (z), the value function defined over wealth is strictly increasing in
wealth. Therefore, the above holds true iff y D (k, l, µ, j) > y N D (k, l, µ, j).
We can solve for the producer’s other decisions (f, k, l) exactly as in the main model.
Although the producer learns new information z which changes the value of carrying over
assets to the next period, he has no way to coordinate an adjustment to the plan with
the shopper. Knowing that the shopper will stick to the plan, the producer knows that
bringing back excess credit will be useless and bringing back less than the planned amount
Precisely, if they did choose such an a0 greater than their expected income, they will face a non-zero
probability of zero consumption
34
53
will lead to zero consumption. Therefore, they find it optimal to stick to the plan.
The optimal choice of a0 is given by:
0
0
0
0
V (a, µ, j) = max
u(c) + β ρE (V (a , µ (z), j + 1)|µ, j) + (1 − ρ) E (V (a , µ1 (s), 1))
0
E
c,a
s.t.
z
x,s
c + a0 ≤ y E (a, µ, j)
This is exactly the same problem solved in the main model, under the assumption
that agents insure against the transitory shock and assets are chosen prior to observing
z.
54