Housing as collateral, financial constraints, and small businesses

Accepted Manuscript Housing as Collateral, Financial Constraints, and Small Businesses

Taejun Lim

PII: DOI: Reference:

S1094-2025(18)30136-4 https://doi.org/10.1016/j.red.2018.03.001 YREDY 866

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Review of Economic Dynamics

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Please cite this article in press as: Lim, T. Housing as Collateral, Financial Constraints, and Small Businesses. Review of Economic Dynamics (2018), https://doi.org/10.1016/j.red.2018.03.001

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Housing as Collateral, Financial Constraints, and Small Businesses∗ Taejun Lim† Dongguk University-Seoul, South Korea

Abstract What drove the synchronization between the small business sector and the housing market during the U.S. housing market boom and bust in the 2000s? I develop an occupational choice model in which house price fluctuations affect the entry-exit and expansion-contraction decisions of small business owners through the housing collateral channel. I perform an experiment in which a sequence of housing shocks constructed to replicate the observed housing market conditions in the 2000s are plugged into the calibrated model. The experiment shows that the synchronization is due mainly to the housing collateral effect.

JEL classification: J24, L26, R31 Keywords: Housing collateral; small business; entrepreneurship. ∗

This paper is based on a chapter in my Ph.D. dissertation at the University of Rochester. I thank the editor, V. Quadrini, and two anonymous referees for their insightful comments and suggestions. I also thank Y. Bai, J.-H. Hong, Y. Chang, M. Bils, G. Alessandria, J. Kim, J. Woo, A. Davis, and A. Kim for their comments and supports. This work was supported by the Dongguk University Research Fund of 2018. † E-mail: [email protected]

1

1

Introduction

One remarkable feature of the U.S. housing market boom and bust in the 2000s is the synchronization between the small business sector and the housing market.1 As Figure 1 illustrates, house prices started to soar in 2000, hitting a historical peak in 2006.2 In just 6 years after the initiation of the housing market boom, house prices increased by 80 percent. However, they started to fall in 2007 and plummeted by more than 30 percent during the recent U.S. recession. Notably, the size of small business sector changed in step with house prices: the correlation between the housing price index and the number of small businesses for the period from 2000 to 2010 was 0.98, and that between the housing price index and small business employment 0.76.3 Figure 1: House prices and the size of small business sector (a) Number of small businesses

(b) Small business employment

Data: the Census Bureau’s Statistics of U.S. Businesses

This paper investigates the role of housing collateral in accounting for the synchronization between the small business sector and the housing market. Specifically, I propose a heterogeneous agent model that links a household’s housing choice (homeowner vs. renter) and occupational choice (small business owner or entrepreneur vs. worker). The key model components are i) households make an occupational choice given their asset portfolio, their entrepreneurial productivity, and their labor productivity, ii) there are two types of loans available to small business owners, the collateralized loan and the non-collateralized loan, which differ in their costs, and iii) a house is the only asset that can be used as collateral. Our model is motivated by the recent survey evidence on the importance of housing assets 1

I define small businesses as establishments with fewer than 500 employees, following the U.S. Small Business Administration. 2 I use the Case-Shiller national-composite home price index for house prices. 3 For the same period, the correlation between the growth rate of the housing price index and the growth rate of the number of small businesses (small business employment) was 0.88 (0.73).

2

as collateral to small business owners: according to the 2014 Small Business Credit Survey, (i) the most cited reason for credit denial in 2013 was insufficient collateral (43%), (ii) half of the credit applicants were requested to provide collateral to secure their business loans, and (iii) the most used type of collateral to secure business loans was personal real estate (50%).4 When the borrowing constraint binds in our model, a change in house prices affects a household’s occupational choice between becoming an entrepreneur and earning wages as an employee. For instance, an increase in house prices during the housing market boom implies an increase in collateral value, which enables homeowners, among potential entrepreneurs, to rely less on costly non-collateralized loans. The increased profitability due to the improvement in their borrowing cost structure induces more entries among marginal entrepreneurs who could not have entered the small business sector otherwise. The model also encompasses the investment decisions of incumbent entrepreneurs, in relation to the changes in their home equity values; an increase in house prices should be positively correlated with the investment decision of an incumbent small business owner by the same reasoning as for a potential entrepreneur’s entry decision. The model provides, therefore, a mechanism by which changes in house prices result in shifts in the size of the small business sector at the aggregate level by affecting individual decisions on entering or exiting entrepreneurship, or expanding or reducing business establishments. I use the model to study the role of housing collateral in accounting for the synchronization between the small business sector and the housing market. I first calibrate the steady state of the model to match salient features of the 2000 U.S. economy, such as the homeownership rate, the percentage of small business owners, the share of home equity in homeowner’s net worth, and so on.5 To provide justification for the eligibility of the model as a tool to quantify the impact of house price fluctuations on small businesses via the housing collateral channel, I check whether the steady state of the calibrated model is consistent with empirical findings. As observed in the data, the model shows a higher percentage of entrepreneurs among homeowners than among renters. Importantly, the model is consistent with the data in that homeowners show a higher entry rate and a lower exit rate from small business ownership than renters. The model also quantitatively captures the difference between entry/exit rate of homeowners and renters. Using the calibrated model, I analyze the degree to which the housing collateral channel 4

The survey is a national collaboration of the 12 Reserve Banks of the Federal Reserve System that aims to capture the perspectives of small businesses on an annual basis with an elaborated focus on their financing needs and choices, and borrowing experiences. The detailed introduction and results for the survey are available at https://www.newyorkfed.org/medialibrary/interactives/spring2014/spring2014/pdf/full-report.pdf. 5 Given that the main goal of this paper is to quantify the housing collateral effect from changes in house prices on small businesses, assuming that the 2000 U.S. economy was in a steady state is justifiable in that house prices started to fluctuate significantly only after the late 1990s .

3

can account for the synchronization between the small business sector and the housing market for the period from 2000 to 2009. I perform a simulation by feeding a series of housing shocks, constructed to replicate the observed changes in house prices from 2001 to 2009, into the steady state of the model (a model-counterpart to the 2000 U.S. economy).6 Methodologically speaking, I reverse the widely used technique for quantifying the impact of an unexpected shock of a given magnitude on the transition path of an economy in that my simulation is based upon the calculation of the magnitude of each year’s housing shock, which enables me to replicate the observed annual growth rates of house prices in the data. The simulation results suggest that the housing collateral effect alone can mostly account for the strong correlation between house prices and small business activities as measured by the numbers of businesses and of employees.

1.1

Motivating Evidence and Related Literature

Table 1 serves as anecdotal evidence that a change in housing prices has a disproportionately large impact on the employment of small businesses relative to large businesses. Following Fort et al. (2013) closely, I regress the differential growth rate of employment between small and large businesses on the growth rate of the real Federal Housing Finance Agency (FHFA) housing price index at the state level for the period from 2000 to 2010:7 ΔSBEs,t − ΔLBEs,t = α + β1 ΔHPIs,t + β2 ΔUNEMPs,t + β3 ΔRPIs,t + γs + θt + s,t where Δ denotes the annual growth rate of each variable; s and t denote state and year, respectively; SBE (LBE) denotes small (large) business employment; HPI denotes housing price index; ΔUNEMP and ΔRPI denote the changes in unemployment and real personal income, which are included as the cyclical indicators; and γs and θt denote the state and the year fixed effects, respectively. Table 1 shows that regardless of the types of cyclical indicators included in the regressions, the estimated coefficient for the housing price growth rate is positive and statistically significant at 1 percent level.

In line with this regression result, there is a growing empirical

literature that suggests the crucial role of housing collateral to entrepreneurs by analyzing the linkage between home equity and entrepreneurial activity. For example, using the topological elasticity of housing supply as an instrumental variable for home equity, Corradin 6

In particular, I assume that the housing boom and bust in the 2000s resulted from shifts in household preferences as a reduced-form modeling approach. This assumption can be justified by an observation that house prices and homeownership rates move in the same direction during this period. 7 The data, which Fort et al. (2013) is based upon, is taken from the website of Teresa Fort (http://faculty.tuck.dartmouth.edu/teresa-fort/data). It is compiled from the Census Bureau’s Business Dynamics Statistics and Longitudinal Business Database.

4

Table 1: State-level evidence on the relationship between small business employment and housing price

ΔHousing price index ΔUnemployment

(1)

(2)

(3)

0.084**

0.086**

0.073**

(0.026)

(0.028)

-0.039**

-0.036**

(0.012)

(0.012)

ΔReal personal income N

(0.028)

561

0.121*

0.072

(0.071)

(0.072)

561

561

The dependent variable is the differential growth rate of employment between small and large businesses. Standard errors are in parentheses. *p<0.05, **p<0.01

and Popov (2013) found a positive elasticity for individual entry into entrepreneurship with respect to home equity values. Fairlie and Krashinsky (2012) and Harding and Rosenthal (2017) exploited the 1993-2004 matched Current Population Surveys and the 1985-2011 American Housing Survey panel, respectively, to draw the same result. Jensen et al. (2014) and Schmalz et al. (2017) delivered similar arguments using micro-level data from France and Denmark, respectively. Unlike these papers that focused explicitly on the individual’s self-employment decision, Adelino et al. (2012) reported a high correlation between crossregional house price increases and rises in small business employment, and Mehrotra and Sergeyev (2016) exploited the Metropolitan Statistical Area (MSA) level variations to find the relationship between housing prices and job flows. Based on a panel VAR analysis, Fort et al. (2013) provided evidence that the large decline in house prices is an important factor for the disproportionately large decline in employment growth of young/small businesses relative to old/large businesses during the Great Recession. There is another body of literature that identifies distinct factors, other than the housing collateral channel, that affect entrepreneurial activities. This literature discusses how house price fluctuations have real effects through changes in demand. Campbell and Cocco (2007), for instance, argued that rising house prices stimulate consumption through wealth effects. Mian and Sufi (2011, 2014) elaborated a similar idea by dividing households into low and high credit groups; their findings suggest that home equity-based borrowing among low credit households is linked to an increased level of consumption. Mian et al. (2013) also showed a large elasticity of consumption with respect to housing net worth using a unique data at the county level. Mian et al. (2015) demonstrated the negative impact of foreclosures on residential investment over the recession of 2007 to 2009.

5

Challenged by this literature, Adelino et al. (2015) revisited the importance of the housing collateral channel for small business employment by identifying the causal effect of rising house prices on the growth of small business employment, separate from the effects of changes in demand, during the housing boom from 2002 to 2007. They confirmed that small businesses in the MSAs with rising house prices and mortgage volume experienced stronger growth in employment than large businesses in the same area and industries. Importantly, to rule out the alternative hypothesis (the demand effect), they exploited the variations in the level of start-up capital across industries. They showed that the industries for which the low level of capital is required for start-up and thus housing collateral is more crucial are more likely to experience the disproportionately large employment increase in small businesses relative to large businesses when house prices soar up. Thus, Adelino et al. (2015) provide evidence that our regression result is not just anecdotal but is a result of a causal effect of the housing collateral channel. This paper builds on empirical works that provide evidence on the role of financial constraints on entrepreneurship. Cetorelli and Strahan (2006), Kerr and Nanda (2009), and Greenstone et al. (2014) explored the impact of local banking conditions on entrepreneurship. Boeri et al. (2012) presented empirical evidence that financial shocks played an important role in employment adjustment at the firm level during the recent recession. More closely related to my work, Siemer (2014) showed that financial constraints could explain a significant portion of the disproportionately large reduction in employment growth of small firms relative to large firms during the Great Recession. There is also abundant theoretical literature exploring the effects of financial constraints in the dynamics of real variables. Jermann and Quadrini (2012) built a quantitative model to estimate financial shocks from the data on debt, capital, and output to claim that financial shocks significantly contribute to the dynamics of real variables. Banerjee and Newman (1993), Quadrini (2000), Li (2002), Cagetti and Nardi (2006), and Kitao (2008) are are among many papers that developed occupational choice models to address the financial constraints as a main barrier to deter individuals from being entrepreneurs.8 Their arguments are based on financial market imperfections: costs related to information asymmetries between financial institutions and business owners are reduced with provision of sufficient collateral (e.g., Bernanke and Gertler (1989), Bernanke et al. (1999), and Kiyotaki and Moore (1997)).9 Related to my paper, Buera et al. (2015) claimed that a sudden tightening of collateral constraints could cause the disproportionately large reduction in employment growth of small firms relative to large firms during the Great Recession. 8

See Quadrini (2009) for a comprehensive survey on entrepreneurship. Since the prospects of future profits for small businesses are much more nebulous than for large businesses, financial institutions should be more reluctant in lending to small establishments, (e.g., Stiglitz and Weiss (1981)) and thus, collateral value becomes a crucial determinant for startups. 9

6

Decker (2015) is the closest to my own research in that both papers develop an occupational choice model with financial constraints in which fluctuations in house prices affect entrepreneurial activities by changing the housing collateral value. Decker (2015) performed a quantitative exercise to compare the impacts of the housing crisis and the financial crisis and claimed that only the housing collateral effect could explain significant drops in entrepreneurial activity during the Great Recession. Although Decker (2015) and this paper deliver a very similar message, our approaches to this topic are quite different. While Decker (2015) relied on the comparative statics based on the steady states of experimental economies, my result is based on the construction of a series of housing preference shocks that allows the transition path of the model economy to accurately replicate the observed housing market conditions in the 2000s. The rest of the paper is organized as follows. Section 2 presents the model and Section 3 describes the calibration of the model. Section 4 analyzes the quantitative results and Section 5 concludes the paper.

2

Model

2.1

Preferences

Household preferences are expressed in terms of discounted, expected utility over sequences of consumption and housing status, (ct , ht ), as given in (1): U (c, h) = E u(ct , ht ) =

∞ 

t=0 1−σ ct

1−σ

 t

β u(ct , ht )

(1)

− κ |1 − ht |

where β is the discount factor, σ is the coefficient of relative risk aversion, and h is an indicator for housing status, taking a value of one for owning a house and zero otherwise. Households suffer disutility from renting a house compared to owning it. The parameter κ governs the extent to which households resent not owning a house. The expectation operator E is taken with respect to the stochastic processes for idiosyncratic shocks (labor and entrepreneurial productivity). I assume, following Hatchondo et al. (2015), that all households reside in a house, either renting or owning it, that they do not own more than one house, and that those who own a house do not have an option to sublease. For simplicity, housing transaction costs are ignored and a choice for the size of the house, which is assumed to be one, is abstracted from.

7

2.2

Occupational Choice

Following the literature on entrepreneurship, the model includes an occupational choice for individual households: to become either a worker or an entrepreneur. As in Cagetti and Nardi (2006), each household has two types of productivities: a labor productivity, θ, and an entrepreneurial productivity, z. These productivities are stochastic, positively correlated over time, and uncorrelated with each other. In each period, once the productivities are realized, households make a decision on whether to become a worker or an entrepreneur. For simplicity, I assume that entrepreneurs are not considered as part of the overall labor supply. As (1) suggests, there is no labor/leisure choice for individual households; those who become a worker supply their labor inelastically.

2.3

Technology

As in Quadrini (2000), Kitao (2008), and Cagetti and Nardi (2006), there are two production sectors within an economy, a corporate sector and a small business sector. They differ in terms of technology and ownership structure. The corporate sector consists of a publiclyowned representative firm with constant returns to scale technology. The small business sector consists of small firms, each of which is owned by an entrepreneurial household with a single project with decreasing returns to scale. Firms in both sectors must rent capital from the financial market prior to production. Corporate Sector

The representative firm in the corporate sector produces consumption

goods in accordance with constant returns to scale: Y = AK α L1−α

(2)

where K and L denote the aggregate capital and labor supplied in the sector, A is the aggregate productivity, and α denotes the capital share. There is no capital accumulation within the sector. Instead, capital is supplied through the financial market at the rental rate r. With the depreciation rate of δ, only part of the capital, K (1 − δ), is returned after production. Each factor price is competitively determined by its marginal productivity. Small Business Sector

Each small firm produces consumption goods according to de-

creasing returns to scale technology due to the owner’s limited span of control introduced in Lucas (1978): y = zk ν1 nν2

8

(3)

where z, k, and n denote entrepreneurial productivity, capital, and the efficiency units of labor employed, respectively. It is assumed in order to capture decreasing returns to scale that ν1 + ν2 < 1. Capital has to be rented from the financial market and is depreciated as in the corporate sector.

2.4

Financial Market

The financial market is incomplete in that entrepreneurs – small business owners10 – are faced with a higher capital rental cost than the representative firm in the corporate sector unless they can provide sufficient collateral. In each period households’ financial assets are channeled into the corporate and small business sectors via the financial market.11 The representative firm in the corporate sector can rent as much capital as it needs at the rental rate r, which is competitively determined in equilibrium. In contrast, firms in the small business sector can rent capital at the same rental rate r as the representative firm in the corporate sector only if the level of capital is less than the owner’s net worth. If the desirable level of capital exceeds their financial assets, however, then they need to take loans. In addition, I assume that only entrepreneurs can gain access to these loans, but not workers. The details of each loan are described below. Collateralized Loans

I assume that a house is the only asset that can be used as col-

lateral. So, only home-owning entrepreneurs are able to access collateralized loans. The maximum amount of collateralized loans is given as: LC  ηH

ph 1+r−δ

(4)

where p denotes the house price. The parameter ηH takes a value in the range of [0, 1]: home-owning entrepreneurs cannot get collateralized loans more than a fraction ηH of the present value of their house. Non-collateralized Loans

Some home-owning entrepreneurs who find the sum of their

financial assets and collateralized loans insufficient to fund the desirable level of capital, along with all home-renting entrepreneurs whose financial assets are not enough to selffinance their project, must resort to non-collateralized loans. Unlike collateralized loans, non-collateralized loans are available without a borrowing limit. The financial intermediary, however, charges an extra fee of γ on top of the rental rate r. 10

In this paper, I use “entrepreneurs” and “small business owners” interchangeably. I assume that financial assets are in the form of consumption goods and can be freely converted to/from capital. 11

9

In the model, not only do households accumulate financial assets for smoothing consumption, but potential and incumbent entrepreneurs save to lower the need for non-collateralized loans. Dependency on the costly non-collateralized loans leads to reduced profits, and in some cases it even precludes households from becoming entrepreneurs.

2.5

Housing Market

Housing Demand

Given a house price, p, households make a housing choice at the end

of each period. The disutility of not owning a house provides an incentive to be homeowners. Housing Supply

I take a simple form of an iso-elastic housing supply function: H S = μp

(5)

where  denotes the elasticity of housing supply and μ is a scaling parameter. Note that homeownership rate maps into the quantity of housing at the equilibrium due to the assumption that the size of a house is one.

2.6

Household’s Problem

Timing of Events

Households start each period with a pre-determined housing status,

h, and hold financial assets, a, that yield the real rate of return, r − δ. At the beginning of each period, the two idiosyncratic shocks, entrepreneurial productivity, z, and labor productivity, θ, are realized. After observing the realized productivities, households make an occupational decision. To install capital before production begins, entrepreneurs, as discussed above, rent capital from the financial market in the form of collateralized or/and non-collateralized loans. Workers provide labor to a firm either in the corporate or small business sector.12 After production ends, entrepreneurs earn profits after paying wages and capital rental fees, and workers receive wages. Finally, households make a housing choice and a saving decision for the next period. Optimization Problem

Once households observe their realized productivities (z, θ),

they make their occupational choice, to become a worker or an entrepreneur: 

V (z, θ, a, h) = max V W (z, θ, a, h), V E (z, θ, a, h)



(6)

where V denotes the value function of households before the occupational choice is made. V W and V E denote the value functions as a worker and an entrepreneur, respectively. 12

It does not matter which sector workers provide labor to since wages are equalized across these sectors in equilibrium.

10

Let o(z, θ, a, h) denote the occupational choice of a household, which takes E (W ) if the household chooses to be an entrepreneur (a worker). Depending on the occupational choice, households make different choices within the period. Workers do not make any production decisions but inelastically provide firms with their labor. Entrepreneurs need to choose the levels of labor and capital they will employ, and decide how to fund the capital. Worker’s Problem 

V W (z, θ, a, h) = max u (c, h) + βE V (z  , θ , a , h )|z, θ  



(7)

c,a ,h

subject to



c + p h − h + a = wθ + (1 + r − δ) a c, a



(8)

 0

(9)

Facing the constraints (8) and (9), workers choose consumption for the current period, and financial assets and housing status for the next period, in order to maximize expected lifetime utility which is discounted by the discount factor of β. The budget constraint (8) shows that total income, which is the sum of wage income and the net return on financial assets, is used to consume, accumulate financial assets, and adjust housing status. Entrepreneur’s Problem V E (z, θ, a, h) =



max

c,a ,h ,k,n,LN C ,LC

u (c, h) + βE V (z  , θ , a , h )|z, θ



(10)

subject to 1. If a  k,



c + p h − h + a = zk ν1 nν2 − wn − rk + (1 + r − δ)a LC 

c, a

(11)

= LN C = 0

(12)

 0

(13)

2. If a < k,



 





c + p h − h + a = zk ν1 nν2 − wn − r a + LC + (r + γ) LN C + (1 + r − δ) a

11

(14)

a + LC + LN C

= k

(15)

ph  ηH 1+r−δ  0

LC c, a

(16) (17)

Entrepreneurs not only make decisions on the same choices workers are faced with, but also decide the quantity of each factor of production to employ and a funding method for their capital. If their financial assets are greater than the quantity of capital input (a  k), entrepreneurs can self-finance it without any loans, and the capital rental cost is r. On the contrary, if their financial assets are smaller than the quantity of capital input, (a < k), entrepreneurs need to get loans from the financial market to fill the gap, k − a. Home-owning entrepreneurs first get collateralized loans at the rate r, up to the limit given in (16). Beyond the limit, they can utilize non-collateralized loans at a higher rate r + γ. Home-renting entrepreneurs must directly resort to non-collateralized loans to fund capital beyond their financial assets. Thus, the cost of capital depends on an entrepreneur’s homeownership status and financial assets. In the current model setting, the entrepreneur’s decisions can be divided into two: (i) an intertemporal decision on the levels of consumption and financial assets and the housing status and (ii) a static decision on the production factors and funding sources. The static decision can be solved independently of the intertemporal decision; the Kuhn-Tucker conditions give an analytic solution for the optimal decisions on the production factors and funding sources, which can be summarized as follows: 1. If 0 ≤ a < k  (z; w, r) − ηH

ph , 1+r−δ

k(z, a, h; w, r, p) = k





(z; w, r) ≡

ν1ν2 −1 ν2−ν2 wν2 (r + γ)1−ν2 z

w n(z, a, h; w, r, p) = zν2 {k  (z; w, r)}ν1 ph LC (z, a, h; w, r, p) = ηH 1+r−δ LN C (z, a, h; w, r, p) = k  (z; w, r) − a − ηH

1/(ν1 +ν2 −1)

1/(ν2 −1)

ph 1+r−δ

where k  (z; w, r) denotes the level of capital at which the marginal productivity of capital is equalized to the cost of non-collateralized loan, r + γ.

12

2. If k  (z; w, r) − ηH

ph ph ≤ a < k (z; w, r) − ηH , 1+r−δ 1+r−δ

ph 1+r−δ  1/(ν2 −1) w n(z, a, h; w, r, p) = zν2 k(z, a, h; w, r, p)ν1 ph LC (z, a, h; w, r, p) = ηH 1+r−δ k(z, a, h; w, r, p) = a + ηH

LN C (z, a, h; w, r, p) = 0 where k  (z; w, r) ≡





ν1ν2 −1 ν2−ν2 w ν2 r 1−ν2 /z

 1/(ν1 +ν2 −1)

denotes the level of capital at

which the marginal productivity of capital is equalized to the cost of of collateralized loan, r. 3. If k  (z; w, r) − ηH

ph ≤ a < k (z; w, r), 1+r−δ

k(z, a, h; w, r, p) = k  (z; w, r) 

n(z, a, h; w, r, p) =

w  zν2 {k (z; w, r)}ν1

1/(ν2 −1)

LC (z, a, h; w, r, p) = k  − a LN C (z, a, h; w, r, p) = 0

4. If a ≥ k (z; w, r), k(z, a, h; w, r, p) = k  (z; w, r) 

n(z, a, h; w, r, p) =

w  zν2 {k (z; w, r)}ν1

1/(ν2 −1)

LC (z, a, h; w, r, p) = 0 LN C (z, a, h; w, r, p) = 0

Figure 2 illustrates the entrepreneur’s optimal decisions on the production factors and funding sources derived from the Kuhn-Tucker conditions. For the purpose of illustration, let a and a denote k  −ηH ph/ (1 + r − δ) and k  −ηH ph/ (1 + r − δ), respectively. When an entrepreneur’s financial assets lie in [0, a ), the optimal quantity of capital input is fixed at k  regardless of the amount of financial assets. To fund k  , the entrepreneur uses his entire financial assets and all possible collateralized loans, resorting to non-collateralized loans 13

Figure 2: Optimal quantity of capital input and funding sources (a) Home-renting entrepreneur

(b) Home-owning entrepreneur

<Ύ

<Ύ

<

<

ĂΎΎ where a ≡ k − ηH

>E

>E

>

Ă

ϰϱŽ

<ΎΎ

Ă

<

<

<ΎΎ

ĂΎ

Ă

ϰϱŽ

ĂΎΎ

Ă

ĂΎ

ph ph and a ≡ k − ηH . 1+r−δ 1+r−δ

only for the shortage. As the entrepreneur’s financial assets increase, non-collateralized loans are substituted for by financial assets, and when the amount of financial assets reaches a , k  can be funded without the use of non-collateralized loans. For any amount of financial assets greater than a , non-collateralized loans are not utilized. If an entrepreneur’s financial assets lie in [a ,a ), the optimal quantity of capital input is a + ηH ph/ (1 + r − δ), which is the sum of the entrepreneur’s entire financial assets and the maximum amount of allowable collateralized loans. In this range of financial assets, the optimal quantity of capital input increases in proportion to financial assets. Once the entrepreneur’s financial assets hit a , the optimal capital input becomes k  and it no longer increases with a further increase in financial assets. The use of collateralized loans decreases with an increase in financial assets until they become unnecessary. Figure 3 illustrates how home-owning entrepreneurs change the optimal quantity of capital input and the funding resources according to house prices. In Figure 3, the baseline economy with a house price p0 is denoted by subscript 0, and the economy with a higher house price than the baseline economy, say p1 (> p0 ), is denoted by subscript 1. A higher house price and thus a higher housing collateral value translate into a greater amount of collateralized loans that home-owning entrepreneurs can utilize. Its impact on home-owning  entrepreneurs are as follows: first, those having held financial assets less than a 0 (≡ k −

ηH p0 / (1 + r − δ)) and thus having relied on non-collateralized loans as well as collateralized loans to fund k  can reduce the use of costly non-collateralized loans while maintaining the same level of capital as before, and accordingly, their profits increase; those having held 14

Figure 3: Entrepreneur’s optimal choices for different house prices

<Ύ

<

<ΎΎ

ĂϭΎΎĂϬΎΎ

ĂϭΎ ĂϬΎ <Ύ

Ă E Ϭ

>

<Ϭ

 Ϭ

>

<ϭ

>ϭE

>ϭ

The entrepreneur’s optimal choices for the baseline case are denoted by subscript 0; those for the case where house prices are higher are denoted by subscript 1.  − η p / (1 + r − δ)) and a are no longer in need of financial assets between a H 1 1 (≡ k 0

the non-collateralized loans and they can hire a greater amount of capital input than k  .

  Second, those having held financial assets between a 0 and a1 (≡ k − ηH p1 / (1 + r − δ)),

and thus having been able to determine the level of capital above k  (yet below k  ) without even using non-collateralized loans, can hire an even greater amount of capital input than before and evidently their profits are improved. Third, those having held financial assets between a1 and a0 (≡ k  − ηH p0 / (1 + r − δ)) can increase their capital input to k  thanks to the increased limit of collateralized loans, which makes their production scales more efficient and thus improves their profits. Thus, Figure 3 implies that when house prices are higher, all else being equal, home-owning households are more likely to be entrepreneurs, and incumbent home-owning entrepreneurs are less likely to quit and more likely to expand their businesses.13

2.7

Stationary Competitive Equilibrium

Let s denote a quadruple of state variables (z, θ, a, h). Stationary equilibrium consists of   a set of policy rules

o (s) , c (s) , a (s) , h (s) , k (s) , n (s) , LN C (s) , LC (s) , a triple of 



prices (r, w, p), a pair of quantities K Cop , N Cop , and a distribution Φ (s) such that 13

The same logic can be applied to the case for lower house prices: home-owning households are less likely to become entrepreneurs, and incumbent home-owning entrepreneurs are more likely to quit and less likely to expand their businesses.

15

1. Given the prices (r, w, p), the policy rules solve the household’s maximization problems in 2.6 for each state s. 



2. Given the factor prices (r, w), the quantities K Cop , N Cop satisfy the F.O.C’s of the corporate sector: 

αA K Cop /LCop

α−1



(1 − α) A K Cop /LCop

α

= r

(18)

= w

(19)

3. The distribution Φ (s) is invariant over time. 4. Capital and labor markets clear:  

a (s) dΦ (s) = K

Cop



+

1 {o (s) = W } · θ dΦ (s) = N Cop +



k (s) dΦ (s)

(20)

n (s) dΦ (s)

(21)

where 1 {·} denotes an indicator function which takes a value of one if the argument inside the parentheses holds true and zero otherwise. 5. Housing market clears:14 





h (s) dΦ (s) =

h dΦ (s)

(22)

6. Goods market clears: ¯ N C = C + δK Y − γL ¯ N C denote aggregate output, consumption, financial assets, where Y , C, K, and L and usage of non-collateralized loans, respectively. That is, 

Y

= 

C = K = ¯NC L 14

=

 



zk(s)ν1 n(s)ν2 dΦ (s) + A K Cop

α 

N Cop

1−α

c (s) dΦ (s) a (s) dΦ (s) LN C (s) dΦ (s) .

Note that housing stocks do not change in the stationary competitive equilibrium.

16

3

Calibration

A period in the model is set to one year. For the purpose of calibration, I take the 2000 U.S. economy as a base steady state.15 I divide the model parameters into two groups, one that can be externally calibrated or has been estimated in many previous studies, and one that has to be calibrated strictly within the frame of the model. Table 2 lists the seven parameters that are externally calibrated. The coefficient of relative risk aversion, σ, is set to 2, which is a common estimate in the business cycle literature. The capital share, α, of the corporate sector is set to 0.36 which is again within the range often found in the literature. The depreciation rate, δ, is 0.06 as in Kitao (2008). Following Kitao (2008), I assume that the stochastic process for labor productivity θ follows a first order autoregressive process in logarithms: log θt+1 = λθ log θt + θ,t+1 

(23)



2 . Both the persistence parameter λ and the variance of labor where θ,t ∼ i.i.d.N 0, σ,θ θ

2 are taken from Kitao (2008), which are 0.94 and 0.02, respectively. Followproductivity σ,θ

ing Tauchen (1986), I discretize the labor productivity process with seven grid points. The fraction of the discounted present value of a house against which entrepreneurs can borrow, ηH , is set to 0.8 in accordance with SBA Guideline 7(a).16 Lastly, in order to determine the parameter γ which measures difference in interest rates between collateralized and noncollateralized loans, I refer to a 2013 report of Accion. Accion is a non-profit organization that lends exclusively to small businesses who lack access to standard types of loans due to their insufficient collateral. According to the report, γ is in the range [0.16, 0.21].17 In particular, I use the lower bound of this range, 0.16. With a higher value for γ, the impact of a change in house prices on small businesses would increase. Therefore, the results in the following section demonstrate the smallest possible model impact for a given fluctuation in house prices. The other set of parameters has to be calibrated strictly within the frame of the model. I list the eight parameters belonging to this group in Table 3. I set the subjective discount factor β as 0.907 to match the capital-output ratio of 2.30.18 I assume that the ratio of labor 15

While it is hard to say that the economy of any year was in steady state, the U.S. housing market did not start to fluctuate significantly before 2000. Given that the main goal of this paper is to analyze the collateral effect of housing price fluctuations on small businesses, it is acceptable to base the model on 2000. 16 SBA 7(a) is the primary loan program of the U.S. Small Business Administration (SBA) for helping start-ups and incumbent small businesses. SBA does not directly make loans, but guarantees loans made by participating financial institutions. SBA Guideline 7(a) is a document that provides detailed information on the procedure for a small business to obtain a loan guarantee, which also includes how SBA calculates collateral values by asset class. 17 To see the report, visit the website at http://www.accionusa.org. 18 In calculating the ratio, I use the sum of fixed assets and consumer durable goods net of residential fixed

17

Table 2: Externally calibrated parameters Parameter

Value

Source

σ

2

Literature

α

0.36

Literature

δ

0.06

Kitao(2008)

λθ

0.94

Kitao(2008)

2 σ,θ

0.02

Kitao(2008)

ηH

0.80

SBA 7(a) Guideline

γ

0.16

Accion

θ:

share to capital share in the small business sector is the same as the ratio in the corporate sector. Then, (3) can be replaced by y = zk να nν(1−α) , where ν determines the severity of the decreasing returns to scale (ν < 1). I set it as 0.776 to match the small businesses’ share of total payroll in 2000, 0.45. Next, I assume that the stochastic process for entrepreneurial productivity, z, follows a first order autoregressive process in logarithms: log zt+1 = λz log zt + z,t+1 

(24)



2 . The annual exit rate of entrepreneurs and the fraction of enwhere z,t ∼ N 0, σ,z

trepreneurs are calculated at 0.12 and 0.11 from the 2001 SIPP.19 By targeting these two

moments jointly, I set the persistence parameter and variance of entrepreneurial productivity as

2 λz , σ,z

= (0.944, 0.022). I use Tauchen (1986) to discretize entrepreneurial

productivity with thirteen grid points. I calibrate the aggregate productivity of the corporate sector, A, to match the small businesses’ share of GDP in 2000, 0.50. Due to the omission of a mean in the specification of entrepreneurial productivity, z (see (24)), the parameter A essentially governs the relative aggregate productivity between the corporate sector and the small business sector. I set it as 0.953. There are three parameters associated with the household’s housing choice: the disutility from renting a house, κ, the scaling parameter for housing supply, μ, and the elasticity of housing supply, . I calibrate κ to match the homeownership rate in 2000, 0.67. With μ, I target an average ratio of home equity value to net worth among homeowners, calculated at 0.36 based on the 2001 Survey of Income and Program Participation (SIPP). It is important to closely match these moments because i) in the model, a household’s housing status influences occupational choice through different loan prices, and also ii) the relative assets owned by households for the level of capital. The average capital-output ratio from 1997 to 2000 was 2.30. 19 I define entrepreneurs as households with their own business with a positive number of employees.

18

Table 3: Internally calibrated parameters Parameter

Value

Target

Data (Model)

β

0.907

Capital-output ratio, NIPA 2000

2.30 (2.30)

ν

0.776

Small businesses’ share of total payroll, 2000

0.45 (0.44)

λz

0.944

Annual exit rate of small businesses, SIPP 2001

0.12 (0.11)

2 σ,z

0.022

% of entrepreneurs, SIPP 2001

0.11 (0.10)

A

0.953

Small businesses’ share of GDP, 2000

0.50 (0.50)

κ

0.285

Homeownership rate, 2000

0.67 (0.67)

μ

0.642

% of home equity in homeowners’ net worth, SIPP 2001

0.36 (0.37)



0.034

Explained in the main text

-

z:

size of home equity to net worth essentially determines the leverage ratio associated with collateralized loans. κ and μ are set to 0.285 and 0.642, respectively. It is noteworthy that for an arbitrarily chosen value of , μ and κ can be pinned down as explained above. If the analysis of the steady state economy is the sole purpose of this paper, it is okay to pin down the values of κ and μ for any value of . For the analysis of the transition path, however, the elasticity of housing supply, , plays a key role in the quantitative exercise since it governs how much homeownership rate relative to housing price responds to housing demand (preference) shocks. To calibrate , I consider an experiment in which there is an unexpected change in housing preference at t = 2001, a period after the initial steady state t = 2000 (say, the disutility from renting a house is unexpectedly changed to κ ˆ at t = 2001). I set the value of  (= 0.034), for which I can pin down the value of κ ˆ such that the ratio of the change in homeownership rate to the change in housing price for the period between t = 2000 and t = 2001 on the perfect-foresight transition path toward a new steady state is equal to the data counterpart for the years between 2000 and 2001.

4

Quantitative Analysis

Changes in house prices affect home-owning entrepreneurs’ borrowing capacity and cost of capital by shifting the limit of collateralized loans, which in turn influences the entry-exit and expansion-contraction decisions of small business owners. In this section, I quantify the impact of house price fluctuations, through the housing collateral channel, on the size of the small business sector over the U.S. housing market boom and bust period from 2000 to 2009. This section is constructed as follows. First, I analyze the steady state results, focusing on how heterogeneous households make different occupation, production, and loan choices depending on their housing status, financial assets, labor productivity, and

19

entrepreneurial productivity. Second, I study how successfully the model can generate the strong correlation between the size of the small business sector and house prices – or, the synchronization of the small business sector and the housing market – observed during the housing market boom and bust period when a sequence of housing shocks is plugged into the model to replicate the observed housing market conditions.

4.1

Steady State

In this subsection, I look into the entrepreneur’s optimal decisions on the quantity of each factor of production and capital funding sources and the occupational choice of heterogeneous households.20 Figure 4 illustrates how the entrepreneur’s optimal choices on capital input and usage of loans vary according with the amount of financial assets. Figure 4 (a) and (b) display the difference between the home-renting entrepreneurs’ and the home-owning entrepreneurs’, both of which present the optimal choices of the heterogeneous entrepreneurs differing in the level of entrepreneurial productivity: low (z = z9 ), middle (z = z11 ), and high (z = z13 ) productivities.21 According to Figure 4, there are two commonalities between home-renting entrepreneurs and home-owning entrepreneurs. First, for any given level of financial assets, the higher is the entrepreneurial productivity, the more capital and non-collateralized loans are used.22 All else being equal, a higher entrepreneurial productivity implies a higher marginal productivity of capital and thus a higher profitability, which justifies more usage of costly non-collateralized loans. Second, for any given level of entrepreneurial productivity, capital input (weakly) increases in financial assets. This property derives from the incompleteness of financial market that disallows productivity-based loan contracts. The detailed explanation is provided in 2.6. The difference in the pattern of the optimal choices between homeowners and renters derives from the difference in the accessibility to collateralized loans. A home-owning entrepreneur depends less on more costly non-collateralized loans than a home-renting entrepreneur with the same levels of financial assets and entrepreneurial productivity thanks to the ability to access to collateralized loans, thereby resulting in higher levels of capital input and profit. 20

See Appendix A.1 for a detailed description on the computational algorithm. I use thirteen grid points in applying Tauchen (1986) to discretize the stochastic process of the entrepreneurial productivity. Let a higher subscript number denote a higher level of entrepreneurial productivity: z1 < z2 < · · · < z12 < z13 . No household whose entrepreneurial productivity belongs to {zi }8i=1 chooses to be an entrepreneur in the steady state. Among the top five levels of entrepreneurial productivity on which positive measures of households are observed to be an entrepreneur, I display the results only for three different levels of entrepreneurial productivity {z9 , z11 , z13 }. 22 An entrepreneur with a higher entrepreneurial productivity stops using non-collateralized loans at a higher level of financial assets owing to the increasing marginal benefit of non-collateralized loans with 21

20

Figure 4: Optimal quantity of capital and amount of each loan (a) Home-renting entrepreneur Low Productivity

45

k

LNC

k

40

Middle Productivity

45

LNC

k

40 35

35

30

30

30

25

25

k

25

k

20

20

15

15

15

10

10

10

5

5

5

0

0

0

-5

-5

-5

0

5 10 15 20 25 30 35 40 45

LNC

k

40

35

20

High Productivity

45

0

5 10 15 20 25 30 35 40 45

a

0

5 10 15 20 25 30 35 40 45

a

a

(b) Home-owning entrepreneur Low Productivity

45

k

40

k

L

NC

Middle Productivity

45 L

C

k

40

L

NC

L

35

35

30

30

30

25

25

20

20

LC

20

15

15

10

10

10

5

5

5

0

0

0

-5

-5

-5

5 10 15 20 25 30 35 40 45

LNC

25

k

15

0

k

40

35

k

High Productivity

45 C

0

5 10 15 20 25 30 35 40 45

a

a

0

5 10 15 20 25 30 35 40 45

a

The horizontal and vertical axes represent the amount of financial assets and the quantity of capital, respectively; LN C and LC denote the quantities of capital input funded by the non-collateralized and collateralized loans, respectively.

Next, I decompose the entire population into four subgroups by occupation (entrepreneur and worker) and homeownership status (homeowner and renter) and compare it with the data (the 2001 SIPP). Table 4 shows that the model accurately captures the population distribution over homeowners and renters as well as entrepreneurs and workers, especially in terms of the difference in the entrepreneurship ratio between homeowners and renters: home-owning entrepreneurs and home-renting entrepreneurs comprise 8.7 and 2.5 percent of the total population in the data, and 9.6 and 0.9 percent in the model, respectively.23 respect to the level of entrepreneurial productivity. 23 Table 4 implies that the model overestimates the role of housing collateral in the household’s occupational choice. In the model, I adopt the simplified assumption that home-owning entrepreneurs can get access to collateralized loans without any extra costs other than the rental fee itself. The overestimated role of housing collateral, thus, can be resolved by positing a more realistic assumption such that getting collateralized loans

21

Table 4: Population subgroups by occupation and homeownership status: data vs. model Data

Model

%

E

W

Total

%

E

W

Total

H

8.7

58.3

67.0

H

9.6

57.5

67.1

R

2.5

30.5

33.0

R

0.9

32.0

32.9

Total

11.2

88.8

100.0

Total

10.5

89.5

100.0

H and R denote homeowners and renters, respectively; E and W denote entrepreneurs and workers, respectively. All values are shown in percent. (Data source: the 2001 SIPP)

Table 5: Entry and exit rates by homeownership status: data vs. model Entry rate

Exit rate

%

H

R

H-R

%

H

R

H-R

Data

3.37

3.06

0.32

Data

9.99

18.84

-8.86

Model

1.51

0.57

0.94

Model

9.37

17.86

-8.49

H and R denote homeowners and renters, respectively; H-R denotes the difference in the entry/exit rate between homeowners and renters with negative signs indicating a higher rate for renters. All values are shown in percent. (Data source: the 2001 SIPP)

Note that this similarity is not a result of calibration. While I target only the homeownership rate and the fraction of entrepreneurs (unconditional probability) in calibrating the model, the model also succeeds in matching the probability of becoming an entrepreneur conditional on homeownership status (conditional probability) observed in the data. Lastly, I examine whether the entry and exit rates of entrepreneurs by homeownership status are comparable between the data (the 2001 SIPP) and the model. The results are presented in Table 5. Table 5 shows that not only is the model consistent with the data in that the entry (exit) rate of homeowners is higher (lower) than that of renters, but the model also closely matches the observed difference in the entry/exit rate between homeowners and renters in the data. In the data, the entry rate of homeowners is 0.32 percentage point higher than renters while the exit rate of homeowners is 8.86 percentage points lower than renters. Such differences between homeowners and renters are closely simulated in the model; the entry rate and the exit rate of homeowners in the model are 0.94 percentage point higher and 8.49 percentage points lower than renters, respectively.

22

Table 6: Comparative statics κ

0.285

0.420

Housing price

1.000

1.401

% of home equity in total wealth

0.371

0.465

Homeownership rate

0.671

0.679

Fraction of entrepreneurs

0.104

0.110

Small businesses’ share of GDP

0.503

0.523

Small businesses’ share of employment

0.440

0.460

Housing price is normalized to one for the case of κ = 0.285 (the benchmark economy). κ = 0.42 is chosen so that housing price increases by 40 percent compared to the benchmark economy.

Comparative Statics Table 6 summarizes how the individual entrepreneurs’ optimal responses to housing market conditions are aggregated. In particular, I compare the steady states of the benchmark economy corresponding to the calibrated model (κ = 0.285) and the economy with stronger housing demand (κ = 0.42).24 It shows that when housing price increases by 40 percent, homeownership rate increases from 67.1 to 67.9 percent and the share of home equity in total wealth increases from 37.1 to 46.5 percent. Importantly, Table 6 confirms that the small business sector expands during a housing market boom: the fraction of entrepreneurs increases from 10.4 to 11.0 percent; the small businesses’ share of GDP (employment) increases from 50.3 (44.0) to 52.3 (46.0) percent.

4.2

Transition Path: the Housing Market Boom and Bust in the 2000s

In this subsection, I perform a quantitative exercise to analyze to what degree changes in the housing collateral value can explain the synchronization of the small business sector and housing market during the U.S. housing market boom and bust in the 2000s. For the exercise, I construct a series of housing preference shocks and plug it into the model so that the changes in house prices on the transition path can replicate the data counterpart for the years from 2001 to 2009. Construction of Housing Preference Shocks Suppose that the economy starts at its steady state in the initial period t = 2000, where the value of housing preference parameter, say κ2000 , is set to 0.285 as in the previous requires an extra cost which is yet lower than the cost associated with non-collateralized loans. 24

κ = 0.42 is chosen so that housing price increases by 40 percent compared to the calibrated model.

23

subsection and that there is an unexpected change in housing preference at t = 2001; the value of housing preference parameter is unexpectedly changed from κ2000 to κ2001 .25 Starting from t = 2001, the economy moves along its perfect-foresight transition path toward the new steady state. I set the value of κ2001 so that the change in housing price for the period between t = 2000 and t = 2001 is close enough to the data counterpart for the years between 2000 and 2001. Now, consider the economy is at t = 2002 on the transition path and there is another unexpected change in housing preference from κ2001 to κ2002 . The economy then adjusts its transition path toward the “new” steady state. Again, the value of κ2002 can be selected so that the change in housing price for the period between t = 2001 and t = 2002 is equal to the data counterpart for the years between 2001 and 2002. By iterating this procedure, a sequence of housing preference shocks, {κt }t=2009 t=2001 , can be constructed so that the changes in housing price for the period from t = 2000 to t = 2009 can closely follow the data counterpart for the U.S. housing market boom and bust in the 2000s.26 Methodologically, this approach is the reverse of the widely used method in the literature: I calculate the magnitude of each year’s housing preference shock to replicate the observed changes in house prices in the data, sequentially from 2001 to 2009; on the contrary, in the literature, the transition path is explored when the size of the unexpected shock is given. Boom (2000-2007) To quantify the housing collateral effect on small businesses, it is important to check the model’s ability to replicate the changes in house prices and homeownership rate observed in the data. The reason that matching the change in homeownership rate is important to this exercise is as follows. In the model, households can hold assets only in two forms: financial assets or a house. Since financial assets are the only asset directly transformable into capital goods through the financial market, individual households’ decisions on their housing status collectively determine the level of aggregate capital supply and thus the interest rate at equilibrium. Thus, the failure to match the observed homeownership rates distorts the housing collateral effect through a general equilibrium effect. Figure 5 (a) and (b) illustrate the changes in house prices and homeownership rates, respectively, over the period. The dotted lines and the solid lines denote the data and the model, respectively. I maintain this notation unless mentioned otherwise. Figure 5 (a) and (b) confirm that the model-generating changes in house prices and homeownership rate 25

As a reduced-form modeling approach to capture the observed comovement between house prices and homeownership rates over the period, I assume that changes in the household’s preference over housing status, κ, lead to changes in housing market conditions. 26 See Appendix A.2 for a detailed description of the computational algorithm.

24

Figure 5: Transition path from 2000 to 2009 (a) House price

(b) Homeownership rate

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25

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Bust (2007-2009) Figure 5 (a) presents the drop in house prices of more than 30 percent since 2007, which is also well-captured in the model transition path. Figure 5 (b) shows that homeownership rate fell by 1.0 percentage point from 2007 to 2009 in the model which is comparable to the data. In the data, according to Figure 5 (c) and (d), the number of small businesses fell by 4.5 percent and the small business employment dropped by 6.0 percent from 2007 to 2009. For the corresponding period, the model yields a 1.9 percent drop in the number of small businesses and a 5.6 percent drop in small business employment. Housing Collateral Effects To gauge the housing collateral effect on the size of small business sector along the transition path of the model, I decompose the quantity of aggregate capital in the small business sector by funding sources: financial assets, non-collateralized loans, and collateralized loans. The results are displayed in Figure 6 with the normalization of the quantity of capital input in 2000 being set to 100. As seen in Figure 6, 85.0 percent of aggregate capital in the small business sector in 2000 is funded by entrepreneurs’ own financial assets; 12.7 percent by collateralized loans and 2.2 percent by non-collateralized loans. With a consistent increase over the housing market boom period, the quantity of aggregate capital in the small business sector reaches its peak in 2007, 12.4 percent above the 2000 value, and then decreases by 7.4 percent between 2007 and 2009. More importantly, while the uses of financial assets and non-collateralized loans for funding capital barely change throughout the whole period, the use of collateralized loans moves parallel to the quantity of aggregate capital in the small

26

business sector, which confirms that the housing collateral effects are mainly responsible for the synchronization between the small business sector and housing market in the model.

5

Conclusion

Since the housing market crash of 2007 a growing literature has examined the crucial role of housing collateral on an entrepreneur’s entry-exit decision and the expansion-contraction of owned business. Most studies provide empirical evidence on the effect of housing collateral at an individual level while closer examination at an aggregate level has been neglected. Not surprisingly, there is a strong correlation between house prices and small business activities at the aggregate level. I attempt to fill this gap by showing how individual decisions on entering or exiting entrepreneurship, or expanding or reducing small business establishments would result in shifts in the size of the small business sector at the aggregate level. The model in this paper links a household’s housing and occupational choices through the housing collateral channel, which allows quantification of the housing collateral effect on the small business sector at the aggregate level. Specifically, by juxtaposing the data and the model for the housing market boom and bust in the 2000s, I find that the housing collateral channel can mainly account for the observed fluctuations in the small business activities. Last but not least, an important caveat to the interpretation of the paper is that I assume that there is no production sector for housing in the model. Intuitively, if there is a housing production sector, a positive housing demand shock puts upward pressure on housing price, which functions as an incentive for firms in the housing production sector to increase housing supply. Importantly, it pushes up the prices of production factors, by which the increased entrepreneurs’ profits from the housing collateral effect are somewhat offset. Thus, the housing collateral effect that boosts up the small business activities should be diminished, which implies that the model overpredicts the housing collateral effect.

27

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A

Appendix: Computational Method

A.1

Steady State

There are three markets (labor, capital, and housing) to be cleared in equilibrium. Adoption of a special form (Cobb-Douglas) for the production function of the corporate sector reduces the number of markets to be cleared to two, enabling me to deal only with the ratio of capital to labor for labor and capital markets – instead of dealing with the wage and interest rate separately. A stationary competitive equilibrium for the steady state can be achieved by executing the following algorithm: 1. Let i denote the number of loop iterations for this algorithm and take a value of one for  now. Pick  initial values for the capital-to-labor ratio and house price, which are denoted Cop K by and pi , respectively. LCop i 2. A pair of factor prices, (ri , wi ), can be calculated as follows:



ri = FK



wi = FL

K Cop LCop

K Cop LCop

 −1 i

 −1 i

where F (K, L) ≡ AK α L1−α 3. Given a triple of prices (ri, wi , pi ), solve the household’s problems described in the subsection 2.6 and obtain an invariant distribution using the policy functions derived from the household’s problems.



4. Given the policy functions and invariant distribution, calculate

K Cop LCop



and pnew new

that satisfy clearing (20), (21),

 the market   conditions   and(22). Cop Cop K K 5. If , pnew is equal to , pi , then a stationary competitive Cop L LCop i new equilibrium is achieved. Stop the process. Otherwise, update the capital-to-labor ratio and house price as follows, and go back to Step 2:



K Cop LCop





, pi+1 i+1



=τ·

K Cop LCop





, pi i

where τ ∈ (0, 1)

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+ (1 − τ )

K Cop LCop





, pnew new

A.2

Transition Path

In the following algorithm, I explain how I obtain the 2001 house price which enables me to match the actual growth rate of house prices between 2000 and 2001 in the data. 1. Choose a higher value for the housing demand parameter, κ, than the 2000 value, rather arbitrarily.27 2. With the current value of κ, calculate a new steady state following the algorithm introduced in A.1. Assume that the new steady state is a counterpart to the 2300 U.S. economy.28 3. Take an initial conjecture on the path of the capital-to-labor ratio in the corporate sector for the period from 2000 to 2300. Make sure that the initial value on the path is equal to the capital-to-labor ratio in the old steady state (2000) and the final value is equal to the capital-to-labor ratio in the new steady state (2300). Similarly, take an initial conjecture on the path of house prices. 4. With the value functions for the new steady state (2300) obtained from Step 2, and the conjectured paths of the capital-to-labor ratio in the corporate sector and house prices, solve the household’s problems backward, starting from 2300 all the way back to 2001. 5. With the distributions in the the old steady state (2000) and the policy functions derived from Step 4, obtain new paths for the capital-to-labor ratio in the corporate sector and house prices. In doing so, follow Step 4 in A.1. 6. If the newly obtained paths from Step 5 are equal to the previously conjectured paths, the transition path associated with a given value of κ from Step 1 is achieved. Proceed to Step 7. If the newly obtained paths are not equal to the previously conjectured paths, update the paths for the capital-to-labor ratio in the corporate sector and house prices. Take a weighted average between the previously conjectured paths and the newly obtained ones with a higher weight on the former.29 Go back to Step 4. 7. Calculate the annual housing price growth rate from the 2000 and 2001 house prices taken from the path obtained in Step 5, and compare it with the data. If they happen to coincide each other, the goal is achieved; stop this process. If the annual housing price growth rate of the model is lower (higher) than that of the data, then take a higher (lower) value for the housing demand parameter, κ. Go back to Step 2. The above algorithm is designed to obtain the 2001 house price (model) that matches the observed housing price growth between 2000 and 2001. In order to derive an entire series 27

Note that for a given value of κ, there is a corresponding series of house prices which is determined in equilibrium on the transition path. I choose a higher value of κ to generate a transition path along which house prices are higher than in the initial period. To generate a transition path corresponding to the housing market bust, choose a lower value of κ than the 2000 value. 28 For my model, three hundred years are more than enough for the convergence of the transition path. 29 I used 0.99 for the weight. The higher the weight, the more stable is the algorithm.

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of house prices for the period from 2000 and 2009, keep iterating on the above algorithm. There are two things to be modified in the iteration process: (a) depending on the sign of the growth rate of house prices, determine whether to increase or decrease the value of κ; increase it for the boom and decrease it for the bust period; (b) the old steady state is no longer 2000; when solving for the house price of 2004, for example, the old steady state corresponds to 2003.

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