# Exercise 13.1 JT: Improved governance

The setup in this exercise:

* Two period model, $t=0,1$.

* Continuum of entrepreneurs with initial wealth $(A)$ distributed with the CDF $G(A)$ with support on $[0,I]$.

* At time $t=0$: Entrepreneurs can invest $I$ and borrow $(I-A)$ or invest in other firms.

* Entrepreneurs are risk neutral and thus have utility $u_e=c_0+c_1$.

* As before: Receive $R$ is investment is successful, with probability $p_H$ if entrepreneur behaves or $p_L$ if he shirks (misbehaves). Shirking yields private benefits of $B$.

* Now add consumers: Behavior results in an increasing savings function $S(r)$, with $S(r)=0$ for $r<0$.


## Q1: When are projects financed?

Consider the entrepreneur again (similar to the exercises above): 

* He behaves if (IC) holds:

$$\begin{align}
    (\Delta p)R_b\geq B. \tag{IC}
\end{align}$$

* If (IC) holds profits are given by:

$$\begin{align}
    p_H\left(R-\dfrac{B}{\Delta p}\right). \tag{IC, $\pi$}
\end{align}$$

* Lenders will finance project as long as profits are larger than alternative cost:

$$\begin{align}
    (1+r)(I-A)\leq p_H\left(R-\dfrac{B}{\Delta p}\right), \tag{IR}
\end{align}$$

where the LHS is the profit from saving (alternative cost w. rent r) and the RHS is the profit from investing with a loan contract such that the (IC) constraint holds with equality.


* As we saw earlier the solution depends on the level $A$. All $A>A^*$ such that (IR) holds are financed.

## Q2: Give the expression determining the interest rate r. How does it change if investor protection improves (interpreted as fall in B)?

The **supply** side of the financial market consists of consumers with savings function $S(r)$ and entrepreneurs who do not get extended credit to carry out the project, i.e. all $A<A^*$. Thus:

$$\begin{align}
    \tilde{S}(r) = S(r)+\int_0^{A^*(r)}A\mbox{ }dG(A)
\end{align}$$

The **demand** side of the financial market consists of entrepreneurs with $A\geq A^*$:

$$\begin{align}
    D(r) = \int_{A^*(r)}^I (I-A)\mbox{ }dG(A)
\end{align}$$


Equilibrium balances demand and supply, i.e. $D(r)=\tilde{S}(r)$.


**Investor protection improves:** ($B$ falls)

* Motive for not behaving drops for all entrepreneurs. Thus entrepreneurs that were before marginally unprofitable to invest in, now increase the *pledgeable income* they can promise investors. In other words; the IC constraint is easier to satisfy, $R_b$ is lower.

* This lowers the threshold $A^*$, i.e. more entrepreneurs are extended credit. Demand increases.

* For supply to increase and adjust to the higher demand, the interest rate increaes. 




**An extra comment on the model, The balance-sheet channel:**
Note that in this model, a marginal change in r has the effect on supply:

$$\begin{align}
    S'(r)+A^*(r)g(A^*(r))\dfrac{\partial A^*(r)}{\partial r}>0.
\end{align}$$
If $r$ increaess, less investment projects are profitable $\Rightarrow$ supply increases. The effet on demand is given by 

$$\begin{align}
    -(I-A^*(r))dg(A^*(r))\leq 0
\end{align}$$

The implication is that *shocks that shift demand upwards (downwards) and supply downwards (upwards) can lead to large fluctuations in investment, without significant changes in the interest rate (r)*. Large fluctuations in investment, but small changes in r, is a stylized fact; something that we actually observe.

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MIT
ChampionApe/FinancialFriction
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eb5be29c7951871972b55fd863c89b83bb50d295
MIT
ChampionApe/FinancialFriction
PS1/PS1_4.ipynb
eb5be29c7951871972b55fd863c89b83bb50d295
MIT
ChampionApe/FinancialFriction
PS1/PS1_4.ipynb