Imperfect Information about Entrepreneurial Productivity, Financial Frictions, and Aggregate Productivity (Click here for current version)

Imperfect Information about Entrepreneurial Productivity, Financial Frictions, and Aggregate Productivity (Click here for current version) Ashique Habib November 8, 2016 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: ashique.habib@mail.utoronto.ca. 1

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 max- 1 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 informa- 2 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 misal- 3 This paper focuses on learning from stock prices and abstracts from financial frictions. 4 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

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 {x L, x H }). 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 (x L = 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 ˆp. 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 (ˆp = p), then the expected income from implementing the project is negative. In particular, the assumptions are: x H k UC w > 0 px H k UC 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

draw (x,k) observe signal, update p hat operate discover idea quality period 2 operate/ do not operate do not operate period 2 no additional information operate/ do not operate 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 x L. If their productivity is x H then they will only implement their idea if they can operate it at a sufficiently large scale. The threshold scale (k 2 ) above which households with productivity x H implement their idea is: k 2 = w x H 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 (ˆpx H k w). For households with capital k, there is a threshold expected probability ˆp 2 (k) above which they will implement their idea. ˆp 2 (k) = w kx H 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 ˆp 1 (k). ˆp 1 (k) = w 2x H 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. ˆp 1 (k) < ˆp 2 (k)). Define the expected benefit of experimentation as B(ˆp, k) = ˆp(x H k w) and the expected cost as C(ˆp, k) = w ˆpx H 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 ˆp 1 (k) above which they will experiment. Benefit and cost of implementing in the first period threshold probability as k Cost, k UC Benefit, k UC Cost, k<k UC Benefit, k<k UC benefit as k cost as k 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability that productivity is high Figure 2: Expected benefit and cost of operating in first period In figure 3, I plot the threshold ˆp 1 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 (ˆp = 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

(ˆp < 1), they require a higher minimum scale to operate (labeled threshold (imperfect) ). Hence, imperfect information amplifies the distortion to entry from financial frictions. Threshold for operating Change in implementation cost (w) 1 1 0.9 0.9 Expected probability of x H (p hat ) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 do not operate do not operate (perfect info) operate (perfect info) threshold (imperfect info) Scale (k) operate 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 w Scale (k) w 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 ˆp 2 (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(ˆp, k) = w ˆpx H k and lowers the benefit B(ˆp, k) = ˆp (x H 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 (µ x, σx). 2 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 σs, 2 that is s N (x, σs). 2 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 (µ x, σx). 2 The household uses the signal to update their beliefs and form their posterior distribution ˆx 1 N (µ 1, σ1). 2 The mean and variance µ 1 and σ1 2 of the posterior distribution are given by equations 1 and 2. µ 1 = σ2 xs + σ 2 sµ x σ 2 x + σ 2 s (1) σ 2 1 = σ2 sσ 2 x σ 2 s + σ 2 x (2) I include this initial signal to encapsulate the perfect information economy as a special case of my general model. In particular, if σ 2 s = 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 pro- 5 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 σe, 2 i.e. e N (0, σe). 2 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 σj 2, then after observing z and updating, the mean and variance of their posterior distribution are given in equations 3 and 4. µ +1 = σ2 j z + σ 2 eµ σ 2 j + σ2 e (3) σ 2 j+1 = σ2 j σ 2 e σ 2 e + σ 2 j (4) 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 {c t } t=0 that satisfy their budget constraints, i.e., E β t u(c t ), s=0 u(c) = c1 γ, γ > 1, β (0, 1) (5) 1 γ Production technology. Goods are produced by competitive firms, each run by an entrepreneurial household. Each firm operates a technology that is specific to the en- 7 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: ω(k, l; z) = e z k α l θ, 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 σ 2 j. To emphasize the connection between learning and the entrepreneurs age, I will replace σ 2 j 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 (a ). households make additional decisions, which I describe momentarily. Entrepreneurial 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. 10 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. 13

{ [ ]} V W (a, µ, j) = max u(c) + β ρv (a, µ, j) + (1 ρ) E V (a, µ 1 (s), 1) c,a 0 x,s subject to: c + a y W (a) = (1 + r)a + W (7) 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 a assets to the next period is V (a, 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 a is summarized by E V x,s (a, µ 1 (s), 1). An entrepreneur s value function is: { [ ]} V E (a, µ, j) = max u(c) + β ρev (a, µ +1, j + 1) + (1 ρ) E V (a, µ 1 (s), 1) c,a 0 z x,s subject to: c + a 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 a prior to production. This assumption only matters for entrepreneurs 12, since they must choose a prior to observing their current productivity and income. (8) Entrepreneurs can choose a 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. 12 Since workers income is deterministic and their beliefs are the same before and after production, when they choose a does not matter. 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. choose default (d) purchase insurance insurance pays out 1 3 5 7 2 4 6 choose deposits, capital (k), access (f) choose labor (l) z realized. production lenders enforce contracts 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: { E ( ω(k, l; z) µ, j) = exp µ + σ2 j + σ 2 } e k α l θ (10) z 2 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: l(k; µ, j) = [ { exp µ + σ2 j + σe 2 } ( )] θ 1 k α 1 θ 2 W (11) π(k; µ, j) = (1 θ) [ exp { µ + σ2 j + σe 2 } ( ) ] 1 θ 1 θ θ k α 2 W (12) Default decision. An entrepreneur s end of period wealth if they do not default (y ND ) and if they do default (y D ) are: y ND (k; a, µ, j) = π(k; µ, j) + (1 δ)k + (1 + r)a (1 + r)(k + ψ) (13) y D (k; a, µ, j) = (1 φ) [ π(k; µ, j) + (1 δ)k] (14) If the entrepreneur does not default, they get the output after labor is paid ( π(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: 1 if y D (k; a, µ, j) > y ND (k; a, µ, j) d(k; a, µ, j) = 0 if otherwise (15) 17

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: π(k; µ, j) + (1 δ)k + (1 + r)(a ψ) (1 + r)k (1 φ)( π(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) φ π(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 ˆµ(a, j). Entrepreneurs who can afford to pay the access cost up front can access credit for all expected productivity. Define the constant C 1 : [ ( ) ] C 1 = σ2 e θ 2 + α log α + θ log (1 α θ) log(1 + r) W The threshold ˆµ(a, j) is: C 1 + (1 θ) log φ + α log(1 + r φ(1 δ)) ˆµ(a, j) = +(1 α θ) log(ψ a) σ2 j if a < ψ 2 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) + φ π(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 if a < ψ, µ < ˆµ(a, j) k(a, µ, j) = k L (a, µ, j) if a < ψ, µ ˆµ(a, j) 0 if a ψ 0 if a < ψ, µ < ˆµ(a, j) k(a, µ, j) = k U (a, µ, j) if a < ψ, µ ˆµ(a, j) k (a, µ, 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: Subject to: y E (a, µ, j) = max { π(k; µ, j) + (1 δ)k + (1 + r)a (1 + r)k fψ} {f,k} 19

k K(a, µ, j) if f = 1 k a if f = 0 Where π(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 a (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 l d (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 a (a, µ, j) solve the household s problem. ii) Given prices, f(a, µ, j), k d (a, µ, j) and l d (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: L s = = j=1 j=1 (1 o(a, µ, j))g(da, dx, dµ, j) l d (a, µ, j)o(a, µ, j)g(da, dx, dµ, j) = L d (18) v) Capital market clears: K s = = j=1 j=1 ag(da, dx, dµ, j) k d (a, µ, j)o(a, µ, j)g(da, dx, dµ, j) = K d (19) iv) Intermediaries make zero profit: R = r + δ 20

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 µ 1 (calculated according to equation 1). Given an underlying type x, the distribution of µ 1 is normal with mean E (µ 1 x) = σ2 xx+σsµ 2 x and σx 2+σ2 s ) 2 variance V (µ 1 x) = σ 2 s. The term E(µ 1 x) is the average expected quality of ( σ 2 x σ 2 x +σ2 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 (µ x ) 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: ( σ 2 E (µ 1 x) = x s σx 2 + σs 2 ) [x µ x ] (20) 14 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 µ x, on average have an expected quality µ 1 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 µ x, then they consider the possibility that the signal is upward biased by noise by partially adjusting their expected quality. If the signal was precise (σ 2 s = 0), then the household s expected quality immediately jumps to equal their true quality x. If the signal is completely noisy (σ 2 s = ), then their expected quality does not change at all and remains µ x. If the household with the average expected quality (µ 1 = E(µ 1 x)) implements their idea and operates a firm, then their firm s expected productivity is: E ( Z E(µ 1 x), σ 2 1 } ) = exp {E(µ 1 x) + σ2 1 + σe 2 2 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: { } E (Z x) = exp x + σ2 e 2 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 (µ 1 x) is: (21) (22) log ( E ( Z E (µ 1 x), σ 2 1 ( ) [ ( )] )) σ 2 ) log (E (Z x)) = s x µ σx 2 + σs 2 x + σ2 x 2 (23) Households with an idea quality x above µ x + σ2 x 2 have a lower average expected productivity under imperfect information. 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 expected quality toward the true value. For ideas with quality x > µ x + σ2 x 2, 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 = z 1 = = z j = x) so far is: [ ] ( E(µ j+1 x) + σ2 j+1 σ 2 ) [ ] j x = 2 σj 2 + E(µ j x) + σ2 j σ2 e 2 x (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 to credit, (i.e. µ ˆµ(a, j)), their credit limits relax if either their assets or their expected quality increase. k µ 0, k µ > 0 k a 0, k a > 0 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 = z 1 = 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

7 6 5 Age = 1, Imperfect information Age = 2, Imperfect information Age 1, Quality known Credit limit 4 3 2 1 0 0 Assets For those with access, limit increases with more information Figure 5: Credit limit with age for high-quality entrepreneur Access to finance. In proposition 3, I defined a threshold expected quality ˆµ(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 to credit if their expected productivity µ + σ2 j 2 is greater than χ(a). χ(a) solves: χ(a) = C 1 (1 θ) log φ + α log (1 + r φ(1 δ)) + (1 α θ) log (ψ a), a < ψ Where C 1 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 z 1. 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