Entrepreneurship, Business Wealth, and Social Mobility

Transcription

1 Entrepreneurship, Business Wealth, and Social Mobility Gabriel Basaluzzo UT Austin / ITAM November 2006 Preliminary and Incomplete Abstract I analyze the quantitative implications of introducing the value of businesses as part of entrepreneurial wealth on the wealth distribution, the portfolio composition and social mobility of rm owners. This value is determined as the reservation price that leaves the entrepreneur indi erent between keeping the rm and selling it to become a wage worker. Results show that business wealth explains the high degree of social mobility experienced by entrepreneurs at the moment of entry and exit, their high levels of wealth-income ratios and their low degree of portfolio diversi cation. Total wealth in the economy increases by 26%, falling primarily in the hands of business owners with the more productive projects. The corresponding increase in the Gini coe cient would be equivalent to a 100% relaxation of the borrowing constraint, if wealth of entrepreneurs consisted only of liquid assets. Finally I also study the feasibility and discuss the welfare implications of considering a market for entrepreneurial rms. 1

2 1 Introduction Several empirical studies of income and wealth distribution show that household wealth is highly concentrated and substantially more concentrated than the distribution of income. (See, for example,wol (1995)). Early work by Hubbard, Skinner, and Zeldes,( 1994,1995); Aiyagari, (1994); Huggett, (1996) and, more recently, DeNardi (2004) showed that standard life-cycle models augmented with precautionary saving motives captured a large part of the heterogeneity in saving among U.S. households. However, Quadrini and Ríos-Rull (1997) and Carroll (1998) showed that these models fail to explain why some families at the top of the wealth distribution accumulated such high levels of wealth. Castañeda, Díaz-Giménez, and Ríos-Rull (2003) provided a potential answer through a dynastic model with idiosyncratic shocks, after constructing an exogenous income process that matched the earnings dispersion observed in the data. This process implied very large earnings risk for the highest income earners and was the driving force, for precautionary reasons, behind the larger saving rates chosen by wealthier households. Using data from the Panel Study of Income Dynamics and the Survey of Consumer Finances, Gentry and Hubbard (2004) and Quadrini (1999) were the rst to establish a link between entrepreneurial activities and wealth concentration. They noticed that a disproportionate fraction of business owners were located at the top of the wealth distribution, holding a signi cant portion of their net worth in the form of privately held business assets. They also found that the saving patterns of entrepreneurs were di erent from those 2

3 of non-entrepreneurial households, with wealth-income ratios being higher for the former. They therefore concluded that entrepreneurial career decisions and entrepreneurs saving decisions were key to explaining the puzzle. Quadrini (2000), Cagetti and DeNardi (2006), and Buera (2006) proved them right, providing models of entrepreneurial choice capable of matching the entire wealth distribution. They essentially endogenized the Castañeda, Díaz-Giménez, and Ríos-Rull (2003) income process, giving households random access to business opportunities, some of them with very high rates of return. In their models, two features were behind the change in saving behavior after an entrepreneurial activity was undertaken. The rst one was the existence of borrowing constraints, which made households with pro table business opportunities face high rates of returns to saving, as they could operate larger rms and increase their pro ts. The second one as in Castañeda, Díaz-Giménez, and Ríos-Rull (2003) was the greater uninsurable income risk encountered by enterprising households, whose patterns of saving became more conservative for precautionary reasons. It is particularly surprising at this point that, having determined the key role played by entrepreneurial activities and business wealth in accounting for the degree of wealth inequality, no quantitative model has ever considered the value of entrepreneurial rms when computing household net worth. When looking at the Survey of Consumer Finances, where most of the data on household wealth is obtained, individuals are asked to provide a market price for their businesses 1. An assessment which contains, among other things, 1 Question 14 in the business section of the questionnaire asks, in reference to each privately owned 3

4 the asset value of the rm. This last component can be orders of magnitude larger than the value of all physical assets invested in the rm 2, becoming a key variable in accounting for wealth inequality, social mobility and household portfolio composition. For this reason, in this paper I study the role played by business wealth in shaping the dynamics and composition of entrepreneurial net worth, and their implications for the wealth distribution. I use a standard model of entrepreneurial choice with borrowing constraints and add the value of business opportunities to the wealth of entrepreneurs, measured as the monetary compensation that would leave them indi erent between continuing to operate their rms and becoming workers. Despite its simplicity, a model like this allows to study as well another feature of the data that has been interpreted by many as a result of the combination between entrepreneurial activities and borrowing constraints. The fact that the portfolio composition of entrepreneurs, especially for the richest households, is very undiversi ed: business wealth constitutes a large share of the entrepreneur s total wealth, being the entrepreneur s own private assets often used as collateral for business nancing. ( Cagetti and DeNardi (2006) and Moskowitz and Vissing-Jørgensen (2004) ). Moreover, as Gentry and Hubbard (2004) and Quadrini (1999) showed, the portfolios of continuing entrepreneurs grow less diversi ed over time, suggesting that the lack of diversi cation is not just related to downpayment business: "What could you sell it for? What is your share worth?" 2 This is particularly noticeable in the dot com industry. Firms like Google or YouTube were sold for $2.7 billion and $1.6 billion dollars respectively, while the value of their physical assets were certainly way below those numbers. 4

5 constraints for starting a business. Carroll (1998) found this result particularly puzzling, especially for entrepreneurial households at the top one percent of the wealth distribution. For them, almost half of all income comes directly from a single business, in which the household has on average 80% of his total entrepreneurial net worth, implying that the failure of the business would wipe out not only his business-derived income but also a large fraction of his wealth, making the riskiness of business ownership even greater than what appear from the high share of business wealth in total net worth. Existing models of entrepreneurial choice are silent on this issue, as businesses have no intrinsic value and all assets invested by entrepreneurs in their own rms are non-speci c liquid assets, that can be easily converted into consumption and that do not loose any value if the business opportunity fails. Incorporating business wealth as part of entrepreneurial wealth may help to show this linkage between net worth growth and the low degree of entrepreneurial portfolio diversi cation, as the former is a direct consequence of the latter. The only remaining question would then be why can t entrepreneurs just sell parts of their businesses to cashin some of that business wealth and lower their exposure to business risk. Answering this question requires to take into consideration a dynamic model of partnerships, something that exceeds the purpose of this paper, and that I set aside for future research 3. In the third place, incorporating business wealth explicitly into entrepreneurial net worth is also relevant if, as in Gentry and Hubbard (2004) or Quadrini (1999), one is 3 In an earlier paper, Basaluzzo (2006) considers the formation of partnerships, but in his model there are no actual equity purchases being made. A partnership is simply a one period capital sharing agreement in order to surmount borrowing constraints. 5

6 also interested in the dynamic aspects of the wealth distribution, that is, on the movement of households among wealth classes or socioeconomic mobility. The empirical work of Quadrini (1999) showed that entrepreneurs experienced greater upward wealth mobility than other agents, and that such fact was not merely a consequence of their higher incomes, since entrepreneurs experienced greater upward mobility in the ratio of wealth to income as well. Gentry and Hubbard (2004) attributed it to the higher cost of external nancing and the high returns to savings faced by entrepreneurs in the presence of borrowing constraints. These views were captured in Quadrini s (2000) model of entrepreneurial choice, who reproduced the general stylized facts about social mobility observed in the data. However, the model had two shortcomings: on the one hand, it was not capable of explaining the large shifts observed in household wealth after agents entered into and exited from entrepreneurship as all wealth changes in the model were gradual, mainly the result of savings/dissavings processes ; on the other, it could not replicate the wealth dynamics of entrepreneurial stars, those who make it to the top 1% of the wealth distribution within a generation. In this respect, studies by Cagetti and DeNardi (2006) and Smith (2001) show that from all the people contained in Forbes magazine richest individuals in the United States, approximately 75% of them were self-made, being only the remaining 25% recipients of an inheritance. As Quadrini (2000) pointed out, in these models households need a long period of time to save at very high rates in order to reach the top one percent of the distribution, making it necessary to consider more than one generation. The 4 While this is a very restricted sample, it certainly focuses on the rich. 6

7 intergenerational life cycle model of Cagetti and DeNardi (2006) also shares this feature, emphasizing the importance of bequests for wealth accumulation. Although the inclusion of business wealth as part of entrepreneurial net worth seems promising to help life cycle models account for the wealth dynamics of stars, the task is left for further research. In this paper the model is dynastic as in Quadrini (2000); and the focus is centered on the dynamics of wealth after entry and exit from entrepreneurship, on how the business value component of net worth commoves with changes in business pro tability over time, leading to the observable changes in total wealth. Finally, excluding business wealth from total entrepreneurial net worth introduces a bias in the calibration procedures aimed at measuring the magnitude of borrowing constraints by matching some measure of wealth inequality, as in Cagetti and DeNardi (2006). As the addition of business wealth makes it easier for entrepreneurs to accumulate wealth, estimations that exclude business wealth must underestimate the importance of nancial constraints if they want to replicate, say, the Gini coe cient in the wealth distribution. This bias would underestimate the welfare gains stemming from policies oriented to improve credit conditions for small business owners. In this paper, I assess the magnitude of the bias in the context of the particular model under consideration. With regards to the existing literature, Holmes and Schmitz (1990,1995), Silveira and Wrigth (2005) and Chari, Golosov and Tsyvinsky (2004) also consider economies where businesses have an intrinsic value, but their focus is centered on the e ciency properties of the economy in the presence of two types of agents: one more e cient at coming up with 7

8 business ideas (entrepreneurs) and another more e ective at implementing them (managers). In such a scenario, it is optimal to see entrepreneurs specializing in the production of new ideas and managers implementing them, requiring a market for businesses to transfer ideas from one group to the other. Therefore, any failures in this market has important aggregate consequences for the allocation of productive resources. In this respect, Holmes and Schmitz (1990,1995) study the general properties of such market for businesses i.e. which businesses are traded, at what price and at which frequency. In turn, Silveira and Wright (2005) focus their attention to the role played by liquidity constraints on the pricing and acquisition of businesses, as well as on the e ciency costs for the economy of such constraints. Finally, Chari, Golosov and Tsyvinsky (2004) consider the costly lock-in welfare e ects that occur when capital gains taxes are introduced in this type of economy, making entrepreneurs willing to hold on to their businesses rather than selling them to managers. This paper departs from said literature in two ways. In the rst place, and most importantly, it deals with the interaction between capital accumulation, business wealth and total wealth. Its main focus is centered on how business valuation a ects the wealth distribution, portfolio composition and social mobility of entrepreneurs. In the second place, in this paper businesses are assumed to be non transferable. Despite this assumption, every business has a value for the entrepreneur implementing it, measured as the monetary compensation that would leave him indi erent between continuing to operate his rm 8

9 and becoming a worker 5. The reason for this is that I want to isolate the e ects on the wealth distribution, portfolio composition and social mobility stemming purely from the implementation of a business idea, from all the other e ects on business transfers and the decision to become an entrepreneur that will appear if businesses can be sold. If businesses were transferable among households, the incentives to start a new rm would change, as prospective business owners would not only be interested in the potentially higher incomes resulting from business ventures, but also in the proceedings from an eventual sale of the rm. Even in the case where the majority of businesses had initially a low return, as claimed by Moskowitz and Vissing-Jørgensen (2004), the possibility of future large capital gains may pull households into business ownership anyway. Moreover, the decision to exit and the price of a business become also dependent on wealth and on the trade-o faced by the entrepreneur between continuing to receive a high income and securing the business value with a transfer. Even the saving patterns of entrepreneurs would change, as business owners internalize the riskiness of their business wealth. Finally, the transferability of ideas in the presence of borrowing constraints would also have welfare implications, as the possibility for poorer entrepreneurs to cash-in the value of a good idea would facilitate the entrepreneurial technology to be transferred to a wealthier and less borrowing constrained household class, increasing total output in the economy. All these considerations are set aside for the moment and left for future research. 5 Essentially, it would be the same price that would be observed in a market where potential buyers make a take ir or leave it o er to the business owner. 9

10 The paper is organized as follows: Section 2 shows some stylized facts on wealth inequality, social mobility and portfolio composition of entrepreneurs, and the role played in them by business wealth. Section 3 presents a standard model of entrepreneurial choice subject to borrowing constraints, incorporating business wealth to be explicitly accounted as wealth. Section 4 contains the parametrization strategy and Section 5 the results. Finally Section 6 concludes and provides some directions for further research. 2 On Entrepreneurs, Business Wealth and Social Mobility In this section, I present the main stylized facts relating entrepreneurial activities, wealth inequality, social mobility and the portfolio composition of entrepreneurs. To this end, I follow Gentry and Hubbard (2004) de nition of an entrepreneur : someone who combines up-front business investments with entrepreneurial skills to obtain the chance of earning economic pro ts. Speci cally, a household is considered to be an entrepreneur if it reports owning one or more active businesses with a total market value of at least $5000. With regards to the data used, Gentry and Hubbard (2004), Cagetti and DeNardi (2006), Carroll (1994) and Quadrini (2000), resort to the cross-section of households in the 1989 Federal Reserve Board Survey of Consumer Finances and the panel of households spanning the 1983 and the 1989 Surveys of Consumer Finances. Because the SCF attempts to describe the wealth characteristics of the population, it oversamples higherincome households. The 1989 SCF contains data on 3,143 households. The 1983 to

11 panel component of the SCF includes a subsample of 1,479 households in the 1983 and 1989 cross-sectional surveys. The data include population weights which allow the calculation of estimates of population statistics. 2.1 Entrepreneurs and the wealth distribution In table 1, I show the percentage of net worth and income owned by the top percentiles in each distribution for the U.S., as found by Quadrini (2000) using the 1989 wave of the SCF. From the table, it is easy to see that the wealth distribution is heavily concentrated, much more than the income distribution, especially when we look at the top percentiles. top percentile Gini 1% 5% 10% 20% 30% Index Wealth Income Table 1: Distribution of U. S. household wealth and income. SCF Source Quadrini (2000) For example, according to the 1989 SCF, the percentages of total wealth and income owned by the top 1 percent of families were 35.7 and 16.9 respectively. Quadrini (2000) shows that these proportions are relatively stable over time, indicating that they are true properties of the wealth and income distributions. To determine the link between entrepreneurial activities and the distribution of wealth we can take two approaches. First, we can look at the prevalence of entrepreneurs along the wealth distribution. According to table 2, entrepreneurs are disproportionately more 11

12 prevalent as we move up along the wealth distribution, indicating that richer households are more likely to be business owners. top percentile 1% 5% 10% 20% % of entrepreneurs Table 2: Prevalence of entrepreneurs among the rich SCF Source Cagetti and DeNardi (2006) Second, we can assess how concentrated is total wealth in the hands of entrepreneurs. In this respect, following Gentry and Hubbard (2004) and according to the 1989 SCF, it can be seen that only 8.7 percent of U.S. households t the de nition of entrepreneurs, and that they own 39.0 percent of total net worth. Similar ndings are reported by Quadrini (2000) from a PSID sample. In the sample, approximately 14 percent of all families are business families; they earn about 22 percent of the total income and they own 40 percent of the total wealth. This evidence con rms, that this relatively small group of households plays a major role in aggregate household wealth accumulation, which is not purely explained by the concentration of income among these families. 2.2 Portfolio composition of entrepreneurs Having established that entrepreneurship is an activity closely related to wealth concentration, the purpose of this section is to show to what extent the accumulation of business 12

13 wealth measured as the market value of the rm may be behind this degree of concentration. One way to check at this is to look at the degree of portfolio diversi cation by entrepreneurs, more precisely the share of business wealth to total net worth. 1% 5% 10% 20% > 50% > 75% > 90% Table 3: Each row reports the fraction of entrepreneurs with a business wealth to total net worth ratio higher than the given percentage, among those that are in the top percent (column) of the wealth distribution. In this regard, as reported by Cagetti and DeNardi (2006), in the SCF the median ratio of business wealth to net worth is 48%, the third quartile is 77%, and the top decile is 96%. As Table 3 shows, these shares are high for all quantiles of the wealth distribution, where approximately half of the net worth is constituted by business wealth for entrepreneurs both at the top and at the bottom of the distribution. Using data from the 1995 SCF, Carroll (1998) shows a similar picture. Over 70% of households at the top 1% of the wealth distribution own privately held businesses, holding on average 40% of their total net worth in them the average is computed counting all the households in the top 1%, implying that the exposure of those with business assets is even larger ; and, among those who own privately held rms, Carroll reports that they keep nearly 80% of that equity in a single entrepreneurial venture. 13

14 Something relevant at this point to determine is whether such poor portfolio diversi - cation corresponds to the entrepreneur reinvesting pro ts in his own rm, or if it is just the result of capital gains. Using the SCF Panel, Gentry and Hubbard (2004) show that most of the changes in portfolio composition occur at the time of entry and exit from entrepreneurship. Taking the group of households that were not entrepreneurs in 1983 but owned a business in 1989 ("entrants"), they nd that the share of net worth comprised by their businesses increased from 0% to 42.5%. However, for those households that were running business rms both in 1983 and 1989 ("continuing entrepreneurs"), their share of business wealth increased much less, going from 41.8% to 47.9%. To conclude, they also report that those who exited from entrepreneurship in the same time period ("exiting entrepreneurs"), saw their share of business wealth dropping from 47.7% down to 0%. In principle these facts should not be surprising as, by de nition, a person who enters or exits from entrepreneurship should have his share of business wealth going from 0% to some other percentage, and viceversa. However, we also know that these portfolio changes are matched with wealth variations observed upon entry and exit from entrepreneurship Gentry and Hubbard (2004). These facts, combined with the ndings by Hurst and Lusardi (2004), who show that entrants do not require on average a large amount of initial capital, suggest that the wealth and portfolio composition changes observed after the birth or death of a rm may be more closely related to the capital gains attached to the market value of the business opportunity, rather than to some gradual process of capital accumulation. 14

15 2.3 Wealth changes and Social Mobility So far, we have established that entrepreneurial activities are closely related to wealth accumulation, and that the portfolios of business owners are heavily biased towards their own business. This section now explores how entry and exit from entrepreneurship a ects entrepreneurial wealth holdings, moving households among di erent wealth classes. Table 4: Social Mobility of Entrepreneurs. PSID Source: Quadrini (2000) In this respect, table 4 shows the net wealth transition matrices corresponding to four types of households, for the period ( Quadrini (2000) ). The rst group is made of staying workers, that is, households who did not own a business in either 1984 or The second group is composed of switching workers, that is, agents that owned a business in 1989 but not in The third subsample is comprises switching entrepreneurs, that is, households that owned a business in 1984 but not in And nally, the fourth 15

16 subsample is made of staying entrepreneurs, that is, households that owned a business in both 1984 and Households in each group were assigned into one of three wealth classes, both in 1984 and These classes were determined according to the 1984 and 1989 household wealth distribution, each class containing one-third of all households. Each cell (i; j) of the matrices in table 4 speci es the probability that a household located in class i in 1984 were found in class j in From the transition matrices, Quadrini (2000) concluded that the probability of upward social mobility has been larger for households that were entrepreneurial in 1989; and that such group also faced a lower probability of downward social mobility. Median W/Y Wealth in 1982 Stay Entrep. Switch Worker Switch Entrep. Stay Worker Overall 1st Quintile 0.357* 0.423* n/a* nd Quintile n/a* n/a* rd Quintile 0.257* 0.445* * th Quintile th Decile th Decile Table 5: Source: Gentry and Hubbard (2004). Calculations based on data from 1,479 households from the Panel of the Survey of Consumer Finances. * denotes cells with fewer than 10 households. Similarly, Gentry and Hubbard (2004) use the SCF Panel and report the median change in wealth W;expressed as a fraction of income Y, experienced by the same four groups see table 5, controlling for household wealth in Notice that, in 16

17 line with Quadrini (2000), the average ratio W Y is systematically larger and positive for households that ended the period as entrepreneurs. Moreover, among these households, the larger W Y ratios were observed for switching workers, and the largest drops in the W Y ratio were observed for rich households that went out of business, a suggestive indicator of the role played by capital gains. Table 6: Transition matrices for wealth - income ratio, PSID Source Quadrini (2000) In order to show that the upward mobility experienced by entrepreneurs was not only a consequence of higher incomes earned by entrepreneurs, Quadrini (2000) and Gentry and Hubbard (2004) report the transition matrices for the ratio of wealth to income and the median wealth-income ratio for entrepreneurs and workers tables 6 and 7 respectively. As can be seen from the tables, the same patterns found for household s wealth are also found for the wealth-to-income ratio, indicating that the undertaking of an entrepreneurial 17

18 Median W/Y By Income All households Entrepreneurs Workers 1st Quintile * 0.4 2nd Quintile rd Quintile th Quintile th Decile th percentile th percentile th percentile Table 7: Median wealth to income ratio. Source: Gentry and Hubbard (2004). * denotes cells with fewer than 10 households. activity is an important way for households to switch to higher classes of wealth. 3 A model of entrepreneurs and business valuation The economy is inhabited by a continuum of in nitely lived households (agents) of total measure one. There is a single good, the numeraire, used for both consumption and investment. There are two production technologies. One is managed by households entrepreneurs, and has decreasing returns to scale in capital and labor. The other technology is used by a representative rm the corporate sector with constant returns to scale. In both cases, working capital is assumed to depreciate after production at rate : Such characterization of the production sector is standard in the entrepreneurship literature 6, and serves a dual purpose. In the rst place, it eliminates from the problem 6 See for example Quadrini (2000), De Nardi (2006), Buera (2006). 18

19 having to take into consideration the size distribution of large rms. The assumption of constant returns to scale allows to aggregate them into a single representative rm. In the second place, it makes possible to incorporate into the model some distinctive features between small and large rms. In order to properly determine the business value of a rm for a given household, it is necessary to introduce the limited availability of working capital faced in general by small rms. In the model, the nancial friction is introduced by assuming limited commitment on the part of households to repay their loans, coupled with limited enforceability on the part of the legal system to fully recover the lenders funds 7. The timing of the economy is as follows. All agents enter every period with an occupation. They can be either workers or single entrepreneurs. At that moment, uncertainty about entrepreneurial and working abilities is revealed. With that information, rms access capital and labor markets to produce the nal good. Capital markets work in the model through a representative nancial intermediary, with zero marginal costs, who collects the savings from households and makes short term loans to rms. After production has taken place, the nancial intermediary collects the proceedings from the loans and in turn pays back the returns on deposits. Households then decide how much to consume and how much to save. Finally, they have to decide their occupation for the following period. 7 For the purposes of this paper, the nature of the friction is actually irrelevant. This particular choice simply follows the literature on endogenous borrowing constraints, as Albuquerque and Hopenhayn (2002), Cagetti and De Nardi (2006) among others.. 19

20 3.1 The Household sector Preferences Households are assumed to have identical preferences over consumption streams fc t g 1 t=0, with per period utility function u (c t ) ; u (:) : R ++! R, u (:) 2 C 2 ; u 0 (:) > 0; u 00 (:) < 0, and satisfying Inada conditions lim c!0 u 0 (c) = 1 and lim c!1 u 0 (c) = 0: Preferences are assumed to admit a time separable, von Newmann - Morgenstern expected utility representation " 1 # X E 0 u (c t ) t ; t=0 where the future is discounted at rate 2 (0; 1) and the expectation is taken conditional on current information Characteristics Households are endowed with two individual characteristics: an entrepreneurial idea or ability " 2 E = f" 1 ; " 2 ; :::; " N g and a working ability 2 H = f 1 ; 2 ; :::; M g, being E and H nite sets, " 1 < " 2 < ::: < " N and 1 < 2 < ::: < M : " 1 is assumed to be 0, implying the lack of entrepreneurial ability. Both " and follow rst order Markov processes with transition probabilities t+1 j t and (" t+1 j" t ; d t+1 ) respectively, where d t+1 is the occupation chosen by the household for period t + 1 ( 1 if an entrepreneur and 0 if a worker). Conditioning (:) on the occupa- 20

21 tional choice attempts to capture the learning-by-doing process related to entrepreneurial activities, as described in Jovanovic (1982) or Quadrini (2000). 3.2 Production Sector Entrepreneurial Technology When households operate a small rm, her abilities ("; ) ; working capital k and possibly additional labor n are combined to produce the nal good with technology " f (k; l) where l = n+, since it is assumed that once the agent decides to become an entrepreneur, she can only use her labor time into her own rm. The production function f : R +2! R is assumed to have decreasing returns to scale, to be increasing in both arguments and concave in each of them, with f (0; :) = f (:; 0) = 0; to be twice continuously di erentiable, strictly increasing and strictly concave in R ++2. f (k; l) is also assumed to satisfy Inada conditions lim k!0 f 1 (k; l) = lim l!0 f 2 (k; l) = 1 and lim k!1 f 1 (k; l) = lim l!1 f 2 (k; l) = 0 in R ++2 : For a given interest rate i, wage rate w,working capital k and total labor l; pro ts for the rm are given by = " f (k; l) wl (i + ) k: 21

22 3.2.2 Corporate sector technology The representative rm in this sector has a typical neoclassical production function F (K; L) with constant returns to scale, satisfying Inada conditions. 3.3 Financial intermediary and borrowing constraints In the economy there is a nancial intermediary that collects deposits from households and makes loans to entrepreneurs asking for funds and to the corporate sector. The borrowing and lending rates are equal to i 8 for both sectors of production. Households deposit their liquid assets a at the end of every period and collect the proceedings of those savings (1 + i) a after production has taken place the following period. Once uncertainty about ("; ) is revealed at the beginning of every period, households that the previous period chose to become entrepreneurs must go to the nancial intermediary to borrow funds for production. Loans are assumed to be short term and expected to be repaid once production has taken place, productive capital has depreciated; and wages have been paid 9. As in Albuquerque and Hopenhayn (2002), the nancial friction of the model comes from the assumption that in the economy there is limited commitment on the part of entrepreneurs to repay loans and limited enforceability of the legal system to recover the 8 Some authors impose a gap between the borrowing and lending rate in order to stimulate asset accumulation from entrepreneurs (Quadrini 2000 ). Since I want to center the discussion on the value of businesses, I chose to eliminate this margin from the model. 9 In this model there is no uncertainty at the time of investment, eliminating the possibility of bankruptcy for the entrepreneur. This assumption implies that only the asset value of a businesses is at stake if the rm fails. The e ect of bankruptcy las will be considered in future research. 22

23 funds. To capture the easier access that the corporate sector has to capital markets relative to the entrepreneurial sector, we will assume that it always honors its debt and therefore is not subject to nancial frictions. When households choose to repudiate a loan, the intermediary is allowed to resort to an existing legal system to recover the money. I set the maximum amount that an entrepreneur can be forced to return to be the sum between the entire proceedings on his savings (1 + i) a which can always act as a collateral, plus a fraction 2 (0; 1)of the business gross sales net of labor costs ["f (k; n + ) wn] excluding the entrepreneur s compensation : I will further assume that the consequences of the default decision do not go beyond the current period, which in turn implies that households will not be able to borrow for consumption purposes a 0 : A single entrepreneur with abilities ("; ) and assets a,who borrows k from the nancial intermediary and hires n additional units of labor would prefer to pay back the loan if the following incentive compatibility condition is met: (1 ) [" f (k; n + ) wn] + k (1 ) " f (k; n + ) wn (i + ) k + (1 + i) a: The nancial intermediary will therefore set a borrowing constraint b = b ("; ; a) on loans so that there is no default in equilibrium. From the incentive compatibility constraint, the 23

24 function b ("; ; a) will be a solution of the equation b = a + [" f (b; n ("; ; b) + ) w n ("; ; b)] (1 + i) where n ("; ; b) is the optimal amount of labor that the entrepreneur hires if she has abilities ("; ) and capital b: 3.4 Households problem in recursive form Given the timing of the economy, subdividing every period into two subperiods simpli es the characterization of the households problem. The rst subperiod begins once idiosyncratic uncertainty is revealed. The second starts after consumption and savings decisions have been made, just before agents make their occupational choices for the following period. In what follows, value functions, state variables and measures contain a superindex i 2 f1; 2g indicating the corresponding subperiod. As is customary, variables with a prime indicate that they correspond to the following period. All agents begin every period characterized by a state variable z 1 = ("; ; d; a) in Z = E H f0; 1g R + containing his current entrepreneurial and working abilities ("; ) ; current occupational status d, and amount of net assets a: In a steady state of this economy, the aggregate state is fully described by an invariant measures de ned on a space (Z; Z) ; with Z being an appropriate algebra of Z. At the end of a generic period t, every single agent has to decide her occupational choice 24

25 d 0 for the following period, based on her current abilities ("; ) and the level of assets a 0 she has already decided to save for the next period: Letting v 2 z 2 and v 1 z 10 be her value functions at the end of period t and at the beginning of period t + 1 respectively, with z 2 = ("; ; d; a 0 ) and z 10 = (" 0 ; 0 ; d 0 ; a 0 ) the agent occupational choice problem can be written as 10 v 2 z 2 X X = max v 1 z 10 d 0 2f0;1g " j " 0 j"; d 0 ; (1) and its solution is given by a function d z 2 : Once a new period begins, uncertainty about ("; ) is revealed, every single agent is characterized by her current individual state z 1 = ("; ; d; a). At that point, the agent enters the production stage of the period, collects her income and decides how much to consume, c; and how much to save for the following period, a 0. The agent programming problem is v 1 z 1 = max c;a 0 0 u (c) + v2 z 2 (2) subject to 10 The value functions also depend on the aggregate state of the economy, characterized by the invariant measure : Since we are considering a steady state economy, I omit writing it explicitly as an argument to economize notation. 25

26 a 0 + c y z 1 + w + (1 + i) a z 2 = "; ; d; a 0 with 8 >< y z 1 0 if d = 0 = >: ("; ; a) if d = 1 and ("; ; a) = max 0 k b(";;a) n0 " f (k; n + ) w (n + ) ( + i) k (3) where prices (w; i) are taken as given. The solution to this problem are policy functions a z 1 ; c z 1 ; k z 1 and n z 1 : Business Wealth and total net worth Total wealth is measured at the end of the rst subperiod, right before households choose a new occupation, and is computed as the sum of liquid assets a 0 ; plus the value q of the business opportunity if the household is an entrepreneur in the current period. Business wealth is measured as the minimum non-negative monetary compensation x that makes the entrepreneur preferring to become a worker rather than staying as an entrepreneur, that is 26

27 q ("; ; a 0 ; d) = min x0 x s.t. v 2 (" 1 ; ; d; a 0 + x) v 2 ("; ; d; a 0 ) 0 that when a business owner sells his rm, his business idea becomes " 1 ; implying that he will choose to become a worker the following period. Also notice that only liquid assets were allowed to be used in the individual budget constraint for consumption purposes and as collateral, since businesses are assumed to be non transferable. According to this, total wealth is therefore given by tw = a 0 + q; and it is useful merely for accounting purposes as I want in this paper. Since households cannot sell their rms, they do not base their decisions on their total wealth, but rather on their liquid asset level a: So this model introduces the minimum possible distortion to household behavior with respect to existing models of entrepreneurial behavior, while at the same time allowing to look at the wealth distributional and dynamic issues of interest. 3.5 Equilibrium A steady state recursive competitive equilibrium is characterized by a set of value functions f v 1 (:) ; v 2 (:)g, policy functions fc (:) ; a (:) ; k (:) ; n (:) ; d (:)g; a set of prices fi; wg, a pair of input demand functions from the corporate sector fl c (w; i) ; K c (w; i)g ;and a measure such that: 1. Decision rules fd (:) ; c (:) ; a (:) ; k (:) ; n (:) g solve the household problem, as described in (1),(2) and (3), with v 2 (:) and v 1 (:) the associated value functions. 27

28 2. Factor demands fl c (w; i) ; K c (w; i)g ; maximize pro ts for the corporate sector. 3. Prices fi; wg are competitive: 4. Capital and labor markets clear Z k z 1 dz 1 Z + K c = Z e Z a dz 1 Z n z 1 + Z 1 dz 1 s + Lc = 1 Z e Z s dz 1 with Z e = f~z 2 Z s : d = 1g : 5. Measure is stationary and its transitional operator T (:) so that 0 = T () is consistent with individual behavior. 4 Parametrization In this section, we specify the functional forms used and the procedure performed to pin down the di erent parameters in the model. The duration of each time period is assumed to be one year. Given that the general structure of the model is similar to that used in the past to model entrepreneurial behavior Quadrini (2000), Fernández-Villaverde and Galdon-Sanchez (2003) and Cagetti and De Nardi (2003) many parameters are calibrated or determined using similar criteria. 28

29 4.1 Preferences In the model, agents maximize their expected lifetime utility " 1 # X E 0 u (c ) ; =0 where the per-period utility is assumed to be of the relative risk aversion form u (c) = c1 1 1 The risk aversion coe cient is assumed to be 2.0. This is a standard choice in the literature see for example Quadrini (2000) and Fernández-Villaverde and Galdon-Sanchez (2003). The discount factor is calibrated so that in equilibrium, the annual interest rate on deposits i equals 3.5%, obtained by averaging Mehra and Prescott (1985) reported returns on government bonds (0.5%) and risky nancial assets (6.5%) during the postwar period. This reference is the target also used by Quadrini (2000) A similar return (4.0%) is reported by McGrattan and Prescott (2000), obtained as the average values of the risk-free rate on in ation-protected bonds issued by the U.S. Treasury. In the rst quarter of 2000, the average return on 5-year in ation protected bonds was 3.99 percent, and the average return on 30-year in ation-protected bonds was 4.19 percent 29

30 4.2 Technology The corporate sector is assumed to have a standard neoclassical production function F (K; L) = A c K K L 1 K ; where K is set to 1=3 in order to match the distribution of income between capital and labor according to the shares measured by NIPA of 0.33 and 0.67 respectively. The productivity coe cient A c is calibrated in order to match the fraction of capital allocated in the corporate sector, reported by Quadrini (2000) to be approximately 60%. The entrepreneurial sector is assumed to possess a Cobb Douglas production function of the form " f (k; l) = " k a l 1 a ; where it is assumed that a = 1=3: This assumption again distributes income across capital and labor according to the shares measured by NIPA. The degree of returns to scale < 1 is calibrated to match the percentage of income earned by entrepreneurs. Quadrini (2000) reports that, according to PSID data, such percentage is 22%. The depreciation rate is set at 0.062, a standard choice in the literature that results from solving the steady state condition = I=Y K=Y 30

31 where K=Y = 2:65 12 and I=Y = 0:16 for the average period Financial Intermediation Sector The only parameter relevant for this sector is the appropiability factor : As in Fernández- Villaverde (2003), we set = 0:6; based on evidence reported by Moody s Investor Service Carty and Lieberman (2000). In such report, the average recovery rate from defaulted bank loans is approximately 60%, ranging from 34% to 71%, depending on whether the default is on multiple loans or not. 4.4 Labor abilities We parameterize the stochastic idiosyncratic labor abilities as in Cagetti and De Nardi (2003) and Fernández-Villaverde and Galdon-Sanchez (2003). They rst use the results of Storesletten et al. (1999), who build a rotating panel from the Panel Study of Income Dynamics (PSID) and estimate the stochastic part u it = ln ( it ) of the labor income process, after eliminating the component of the data accounted by a mincerian regression. The process for household i at time t is modelled as u it = z it + " it z it = z it 1 + it 12 This paper does not have a government sector, so a consistent de nition of aggregate capital should exclude public capital and consider only private tangible assets. This makes the K=Y ratio to be somewhat smaller than 3; the number generally reported in other studies. 31

32 where " it ~ N 0; 2 " and it ~ N 0; 2 are innovation processes, with point estimates = 0:935; 2 " = 0:017 and 2 = 0:061:Finally, they approximate this AR (1) process with a discrete Markov chain, using the method proposed by Tauchen and Hussey (1991). In this paper, we use a three point Markov chain, leading to the corresponding normalized state vector H and transition matrix : H = f0:57; 0:93; 1:51g 0 1 0:75 0:24 0:01 = 0:19 0:62 0:19 B A 0:01 0:24 0:75 As usual, the stationary distribution of labor abilities H is obtained as the eigenvector of with eigenvalue 1, giving H = [0:31; 0:38; 0:31] : 4.5 Entrepreneurial abilities With respect to the entrepreneurial abilities vector E; we follow the procedure described in Quadrini (2000), who uses data on the household distribution of businesses from the 1989 and 1992 waves of the SCF 13. From the decile distribution of business wealth among families reporting a net value of the business greater than zero, he classi es businesses in three categories: small, medium and large projects. Consequently, E is de ned as a four 13 Results are identical using data on assets for all waves of the Survey of Small Business Finances. 32

33 element vector f" 1 ; " 2 ; " 3 ; " 4 g ; with " 1 = 0 as already mentioned:in order to approximate the skewness in the distribution of business capital, Quadrini assigns 60 percent of all entrepreneurial ideas to the small, 30 percent to medium size and 10 percent to the large projects. Computing the ratios among the average values q i of business wealth for each group i, he nds that medium size projects are 10 times more valuable than small ones (q 3 =q 2 = 10), and that large projects are also an order of magnitude more valuable than medium ones ( q 4 =q 3 = 10 ). Approximating the average value of a businesses in each group to be proportional to the unconstrained level of pro ts for each possible project 14 we have q i " i 1 1 and normalizing " 2 = 1; we nd that " 3 = 1:26 and " 4 = 1:68: With respect to the transition matrices 0 i;j = (" ij" j ; d = 0) and 1 i;j = (" ij" j ; d = 1) the lack of good micro data makes their calibration somewhat problematic. In principle, we have two 4x4 matrices to calibrate, a total of 24 parameters when we take into account that each of their rows have to add up to one. With respect to 1 i;j ; we will assume that entrepreneurs learn from current ongoing projects and evolve according to a ladder of opportunities, as in Quadrini (2000). This ladder only allows entrepreneurs to move from 14 A rigorous calibration on this part would require to incorporate in the value of a business the expected capital gains or losses coming from transitions to better / worse projects. Alternatively, f" ig can be obtained nding the values for which, in equilibrium, the average value of each type of business matches the suggested ratios in the text. At this preliminary stage, I decided to take a shortcut and use this approximation. 33

34 one idea " j to another immediately adjacent to it: " j 1 or " j+1 : As a result 1 becomes a band matrix, with 1 i;j = 0 if ji jj > 1 and 6 undetermined elements. Parameter 1 1;2 is set at 0.03, to match the unconditional probability of entry into entrepreneurship holding over the rst nine deciles of the wealth distribution, as reported by Hurst and Lusardi (2004) from PSID. 1 2;1 is set at 0.30, to match the average exit rate of entrepreneurs with less than three years of experience as reported by Quadrini (2000) using PSID data 15. Imposing the restriction that 1 3;2 = 1 3;4, we set the remaining diagonal elements of 1 ; 1 2;2 ; 1 3;3 and 1 4;4 so that the stationary distribution of entrepreneurial ideas other from " 1 is E = [0:6; 0:3; 0:1], as previously mentioned. The resulting transition matrix is 0 1 = 0:97 0:03 0:00 0:00 0:30 0:63 0:07 0:00 0:00 0:15 0:70 0:15 0:00 0:00 0:41 0:59 1 C A With respect to 0 i;j, we will assume that 0 i;j = 1 1;j8i; j; implying that an agent that has been a worker in the current period, has to go to the bottom of the ladder of opportunities to draw a new idea for the subsequent period. In other words, previous ideas not implemented are lost forever. 15 Similar estimates are found from CPS by Evans and Leighton (1989), with entry and exit rates being 2.5% and 21% respectively. 34

35 5 Results Table 8 shows the calibration results, with all targets matched within the 5% error tolerance imposed. As expected, values for the discount factor and the degree of homogeneity in the entrepreneurial technology were found to be similar with those reported in earlier studies by Quadrini (2000) and De Nardi (2003). Parameter Value Target variable Target value Value obtained 0.93 i K A c 0.75 c K inc: entrep 0.9 inc: work + inc: entr: Table 8: Calibration targets and results 5.1 Determination of business wealth With regards to the determination of business values, the monotonicity of the value function v 2 (z) with respect to liquid assets a and entrepreneurial ability, implies that the reservation values of entrepreneurs should also be monotonically increasing in those variables. First of all, gures 1 and 2 show the value functions v 2 (" 1 ; 2 ; 1; a 0 ) in purple and v 2 (" i ; 2 ; 1; a 0 ) in blue for " i = " 2," 3 respectively. Notice that the scale for liquid assets is logarithmic, to show in more detail what occurs at lower levels of wealth. The value of a business of type " i for a household of liquid assets a is therefore measured has the horizontal gap x such that v 2 (" i ; 2 ; 1; a) = v 2 (" 1 ; 2 ; 1; a + x) : 35

36 V2(z) liquid assets Figure 1: Type " 2 businesses value determination, for every level of liquid assets of the owner. The labor ability is kept constant at 2 : V2(z) liquid assets Figure 2: Type " 3 businesses value determination, for every level of liquid assets of the owner. The labor ability is kept constant at 2 : 36