Small business survival and inheritance: evidence from Germany

Abstract

This paper investigates whether small businesses face financial constraints that affect their survival. A model of moral hazard is developed in which financial constraints arise endogenously. The model predicts that higher private assets relax financial constraints and have a positive effect on the firm’s probability of survival. The empirical analysis confirms that the entrepreneur has a higher propensity to stay in business when she inherits capital. This effect is particularly strong for entrepreneurs who switch from self-employment into wage employment.

Appendix

1.1 Proof of Proposition 1

For A = 0 both project qualities yield equal profit if

$$ p_h X- I = p_l Y- \frac{p_l}{p_h} I. $$

(4)

This equality is only satisfied with I = 0 for the benchmark type $\hat{\theta}$. If quality h were realized and 〈R _h I〉 were offered the benchmark type $\hat{\theta}$ would face perfect financial constraints. Now, consider projects for which Assumption 1 is satisfied. Given that $\theta_i > \hat{\theta}$ and I = 0 the left hand side of (4) increases. Note that an increase in I decreases the left hand side more than the right hand side. Thus, for $\theta_i > \hat{\theta}$, equality in (4) can be restored with some $I=\hat{I} > 0$. Since ∂E _h (θ _i , ·)/∂θ _i  > 0, the crucial amount of I that satisfies (4) increases if θ _i increases: $\partial\hat{I}/\partial \theta_i > 0$. The monotonicity of the relationship between $\hat{I}$ and θ _i implies that there exists a $\theta_i=\tilde{\theta}$ for which the loan granted in equilibrium approaches I ^*, and financial constraints vanish. The second step of the proof takes into account the possibility that the bank could alternatively offer only contract 〈R _l I ^* _l 〉. Note that the optimal I in case of 〈R _l I ^* _l 〉, which satisfies

$$ \frac{\partial E_l}{\partial I}= \frac{\partial p_l(I,\bar{\theta})}{\partial I} Y-1=0 $$

(5)

is smaller than the optimal I in case of 〈R _h I ^*〉, which satisfies

$$ \frac{\partial E_h}{\partial I}= \frac{\partial p_h(I, \theta_i)}{\partial I}X-1=0. $$

Thus, financial constraint C = I ^*−I ^* _l arises if banks, in order to avoid losses, offer only 〈R _l I ^* _l 〉. Recall that $E_h(\hat{I})$ increases monotonically if quality h becomes better. Moreover, $E_l(I_l^\ast, \bar{\theta}) > E_h(\hat{I},\hat{\theta})$ in the benchmark case. Both properties imply that there exists a crucial level $\theta_i=\breve{\theta}$ such that $E_l(I_l^\ast, \bar{\theta} )=E_h(\hat{I}, \theta_i)$. For a rather small difference between l and h, that is, ability is in the range of $\theta_i\in ( \hat{\theta}, \breve{\theta})$, the active financial constraint is C = I ^*−I ^* _l since the bank only offers 〈R _l I ^* _l 〉. For all $\theta_i\in (\breve{\theta}, \tilde{\theta})$ the active financial constraint is $C=I^\ast-\hat{I}$.   q.e.d.

1.2 Proof of Proposition 2

The profit for all $\theta_i > \hat{\theta}$ is given by

$$ E_h = p_h X - I - (1-p_h)(1-\beta)A $$

and

$$ E_l = p_l Y - \frac{p_l}{p_h} I - (1-p_l) \left({1 -\frac{1-p_h}{1-p_l} \frac{p_l}{p_h}\beta}\right)A $$

for quality h and l, respectively. Consider a given type θ with $\theta_i\in (\hat{\theta}, \tilde{\theta})$. For symplicity we assume β = 1. Recall that without private assets the firm is constrained by (4). If pledgable assets are available, the profit function for quality l is lowered but the profit function for quality h remains unchanged. This feature, in combination with (4), immediately implies

$$ p_h X - I - (1-p_h)(1-\beta)A=p_l Y - \frac{p_l}{p_h} I - (1-p_l) \bigg(1 -\frac{1-p_h}{1-p_l} \frac{p_l}{p_h}\beta\bigg)A $$

only for $I=\hat{I}(A > 0) > \hat{I}(A=0).$ The lowering of E _l induces $E_h(\hat{I}(A > 0), A) > E_h(\hat{I}, A=0)$ for all $\hat{I}(A > 0)\in(\hat{I}(A=0),I^\ast)$. With β < 1, both profit functions are lowered if debt is secured. However, because of (1−p _h )p _l /(1−p _l )p _h  < 1, the decrease for profit E _l is always larger than the decrease for E _h . This feature, in combination with the fact that $E_h(\hat{I}(A > 0), A)$ increases with $\hat{I}(A > 0)$ for β = 1, guarantees that $E_h(\hat{I}(A > 0), A)$ also increases for β = 1−δ, where δ is not too large. Thus, the pledging of assets is compatible with C2. It increases profits as it allows a greater I. However, for types $\theta_i \in (\hat{\theta}, \breve{\theta})$, the increase in profits has to be large enough to exceed E _l (R _l I ^* _l ) if assets were to ease financial constraints.   q.e.d.

1.3 Proof of Proposition 3

Consider S = 0. In this case E ^S _h (I ^*, S = 0) > E _h  = E _l since investment incentives are not distorted in case of the screening option, and the pledging of assets and the reduced investment in the non screening case lowers profit E _h as the entrepreneur has to bear the dead-weight cost. Differentiation of E ^S _h yields

$$ \frac{\partial E_h^S}{\partial I}= \frac{\partial p_h}{\partial I} X-1\quad \rightarrow \quad \frac{\partial p_h(I^\ast_S)}{\partial I}=\frac{1}{X} $$

(6)

$$ \frac{\partial E_h^S}{\partial S}= -1 < 0. $$

(7)

The first derivative (6) indicates that the optimal investment level I ^* is independent of S. The second derivative (7) shows that the maximal profit decreases monotonically with S. For a given type, the amount of pledged assets A determines investment and profit in the non-screening scenario. Both properties and E ^S _h (I ^*, S = 0) > E _h  = E _l ensure that, for each given amount of pledged assets, there exists a level $\bar{S}$ such that C5 is satisfied for all $S < \bar{S}$. In this array of S, the available amount of assets will not be pledged. Thus, assets have no influence on the firms’ success probability. If $S > \bar{S}$, C5 is not satisfied. Assets secure the debt and Proposition 2 applies.    q.e.d.