The consequences of Bankruptcy

This sub-section analyses in detail the problem of bankruptcy and how this aects the credit and the investment.

The evolution of the entrepreneurial net worth can be written as:

N_t+1^j =

Equation (S1) describes the case of non-defaulted entrepreneur, the net worth at the beginning of the next period is given by the wage plus the net return of capital. The second equation (S2) describes the defaulted entrepreneur. In this case the entrepreneur looses all the invested capital and her net worth in the next period will be equal to the wage.

Supposing the case of entrepreneur defaults, the bank obtains (1−µ)ω^jtR^k_tQt−1K_t^j,

12For example, see the analysis in economic psychology of Kahneman et al. (1991).

this can be rewritten as:

(1 − µ)ω_t^jR^k_t

B_t+1^j + N_t^j

= (1 − µ)ω^j_tR^k_t(V_t−1^j + W_t−1^e + B_t^j),

therefore the lender registers a net loss in t. Supposing that the bank does not take any strategical action after the rst default of the rm, in the following period the defaulted entrepreneur has access to credit and therefore she could not be able to full the debt contract once again. In this case, the nancial intermediary records loan return lower than expected in both periods, by den-ition:

(1−µ)ω^j_tR^k_t(V_t−1^j +W_t−1^e +B_t^j)+(1−µ)ω_t+1^j R^k_t+1(W_t^e+B_t+1^j ) < Z_tB_t^j+Z_t+1B^j_t+1. (2.31) Hence, it seems quite reasonable to expect that, after a default, the bank changes its credit policy towards the bankrupt entrepreneur. This aspect is completely absent in the BGG model, indeed in that framework the bank di-versies his risk among entrepreneurs and does not take any action against the insolvent entrepreneurs, i.e. the bank has not memory of the past.

Conversely, this ABM version of the nancial accelerator introduces in the

nancial contract an additional cost to external funds for the defaulted entre-preneurs. This premium increases the cost of credit and alters consequently the amount of available funds. As described in equation (2.31), when an entrepren-eur defaults the bank suers some losses. Therefore, in the following period, the available credit to the default entrepreneur will be at higher cost. This because for the defaulted entrepreneur the bank will set an interest rate on loan able to recover the losses of the past period.

The nancial contract for the bankrupt entrepreneur becomes:

Z_t+1^j = R_t+

In equation (2.32), the interest rate on loans is a mark-up on the risk free interest rate and the nancial soundness of the bank, as in equation (2.26), plus a premium equal to the losses due to non performing loans of the previous period. For the sake of tractability, the nancial contract considers a temporal horizon of two periods, i.e. for each defaulted borrower the bank expects to smooth its losses of t in the next period.

The new optimization problem of the entrepreneur is

max Solving the maximization problem, the rst order conditions may be written as:

It can be easily seen that if the entrepreneur is not defaulted, Losst = 0, hence equation (2.33) becomes equation (2.29). However, the new nancial contract is designed with the aim to take into account the possibility of stop lending. This happens when the spread between the expected return and the mark-up on risk free interest rate is lower or equal to zero. This could have two dierent reasons. The rst occurs if the expectation are too low due to bad entrepreneur's past performances whereas the second takes place if the amount of past losses is huge.

At this point, supposing that the nancial intermediary stops lending to the entrepreneur, Bt^j= 0, and it registers a negative variation in its wealth for the unexpected losses. From equation (S2), the amount of available funds that the bankrupt entrepreneur may invest in the following period consist only in her wage:

QtK_t+1^j = N_t+1^j = W_t^e.

The new participation constraint of the entrepreneur does not take into account the external nance cost but concerns the ratio between the expected realization of investment and the risk-free rate:

E_tR^k_t+1 W_t^e= R_t+1W_t^e. (2.34)

The entrepreneur invests her capital only if her expectation on capital return is such that:

EtR^k_t+1 Rt+1

≥ 1. (2.35)

Since the risk-free rate is given, the entrepreneur's choice depends on the expected return on capital, therefore the expectation formation mechanism plays a crucial role in the investment decisions. If Equation (2.35) is not veried, the entrepreneur does not invest and uses her wealth for consumption.

Through these variations, the nancial accelerator model is able to explain the fraction of entrepreneurs which does not receive loans, i.e. the share of them whose total wealth is given by the wage. Within this share of entrepreneurs there is a portion γ that leaves the market. With the aim to hold o the problem that after few periods there are no more entrepreneurs in the market, dierent types of mechanisms could be set. Instead of supposing a mechanism à la Gertler

and Kiyotaki (2010) which settles the number of the entrepreneurs entering in the market as a parameter maintaining constant the number of rms, this

rst version of the model does not x any inactivity period for the bankrupt entrepreneurs before to obtain again access to the credit market. In other words, if an entrepreneur defaults and drops out of the market in period t she can borrow in the following period.