# Exercise from PS: Adverse selection with heterogenous ability

## The setup:

* Continuum of risk-neutral entrepreneurs (same initial $A$)

* Investment cost of cost $I$ (where $I>A$) yields $R$ in return w. probability $p$. Otherwise $0$ return.

* Agents differ in the probability of success $p$. Is distributed on $[0,1]$ according to the pdf function $f(p_i)$. (As usual $F(p)$ is the share of agents w. $p_i\leq p$. $f(p)$ is the marginal change in that probability around $p_i=p$.)

* Total supply of loanable funds is simply presented by the increasing function $S(r)$. 

* This $S(r)$ is supplied by a competetive market (e.g. banks).

* Assume perfect information.

* Compared to before: No assumption of private benefits from not behaving.

## Q1: Individual rationality constraint, given that the agent has the choice between investment $I$ or depositing $A$. 

This is straightforwardly given by:

$$\begin{align}
    p(R-R_b(p)) \geq (1+r)A \tag{IR}
\end{align}$$

Note that we assume that the payment $R_b$ may depend on $p$, as we assume **perfect information** for now. Note here:

* The LHS denotes profits from the investments after paying $R_b(p)$ on loan if successful. Otherwise zero. 

* The RHS denotes the value of depositing $A$ at interest rate $r$.

## Q2: Write equations determining the riskless rate $r$, the identity of the marginal investor, and the zero (marginal) profit for the bank.

Let us start by imposing **zero profits** for the banks:

$$ \begin{align}
pR_b(p) = (1+r)(I-A), && \forall p \tag{ZP}
\end{align}$$

where $pR_b(p)$ is the payment if the project is successful, and the cost is the alternative cost of the loan: $(1+r)(I-A)$. Denote for now $B\equiv I-A$ the loan given by the bank. Note once again that the perfect information assumption implies that the zero profits hold for all $p$. 

The **marginal investor** is given by the $\bar{p}$, where the IR constraint exactly holds. Imposing the zero profits condition we then have:

$$\begin{align} 
  \bar{p}R-\underbrace{(1+r)(I-A)}_{\equiv \bar{p}R_b(\bar{p})\text{ from (ZP)}}=(1+r)A && \Rightarrow && \bar{p}(r) = \dfrac{1+r}{R}I
\end{align}$$

Finally the **riskless rate (r)** (it is riskless, due to the perfect information assumption) is determined by the equilibrium condition; i.e. that demand equals supply. With the threshold $\bar{p}$ all entrepreneurs who have a larger probability of success than the level $\bar{p}$, where profits are zero and (IR) holds with equality.

Recall that $f(p)$ denoted the distribution of the entrepreneurs. Thus demand for credit is extended to all entrepreneurs with $p\geq \bar{p}$, or:

$$\begin{align}
    D(r) = \int_{\bar{p}(r)}^1(I-A)f(p)\mbox{ }dp
\end{align}$$

Supply is given by the *aggregate supply function* $S(r)$, as well as the initial wealth $(A)$ from the entrepreneurs who do not obtain credit to carry out their investment. Define this new *total* supply function $\tilde{S}(r)$:

$$\begin{align}
    \tilde{S}(r) = S(r) + \int_0^{\bar{p}(r)} A f(p)\mbox{ }dp
\end{align}$$

## Q3: Confirm that only projects with positive net present value are carried out.

This is given, as banks have a zero profits condition, and entrepreneurs have the IR condition.

## Q4: Assume p is not observable. Compute equilibrium lending contract, the idendity of the marginal investor and show that there is overinvestment.

Now that the banks cannot observe $p$, they cannot make the contracts contingent on $p$. Thus they can only propose one contract, offering $R_b^*$ if the project is successful. Let $p^*$ denote the marginal investor now. From the IR constraint (with equality) we get:

$$\begin{align} 
    p^*(R-R^*_b)=(1+r)A \tag{IR*}
\end{align}$$

Zero profits for the bank:

$$\begin{align}
    \int_{p^*}^1 p R_b^*f(p)\mbox{ }dp = \left(1- F(p^*)\right)(1+r)(I-A). 
\end{align}$$

The LHS gives the payout to successful projects. The RHS is the alternative cost of the loan contracts. Note that $1-F(p^*)$ is the share of entrepreneurs with $p\geq p^*$. 


Note that we can define this as the **conditional expected value**:

$$\begin{align}
    \dfrac{\int_{p^*}^1 p R_b^*f(p)\mbox{ }dp}{1- F(p^*)} \equiv R_b^* \mathbb{E}\left[p|p\geq p^*\right] = (1+r)(I-A).  \tag{ZP*}
\end{align}$$

Combining the zero profits, and the marginal investor identity we then have:

$$\begin{align}
        (1+r)A&= p^*R-p^*R_b^* \\
              &= p^*R-R_b^* \mathbb{E}\left[p|p\geq p^*\right] \dfrac{p^*}{\mathbb{E}\left[p|p\geq p^*\right]} \\ 
              &= p^*R-(1+r)*(I-A)*\dfrac{p^*}{\mathbb{E}\left[p|p\geq p^*\right]}
\end{align}$$

where the last line uses $(IR^*)$.

Rearranging this expression we get:

$$\begin{align}
    p^*R = I (1+r)\dfrac{p^*}{\mathbb{E}\left[p|p\geq p^*\right]}+A(1+r)\left[1-\dfrac{p^*}{\mathbb{E}\left[p|p\geq p^*\right]}\right]\leq (1+r)I \tag{Marginal*}
\end{align}$$


The RHS is a linear combination of $(1+r)I$ and $(1+r)A$. Compared to this the marginal investor was defined by $\bar{p}R=(1+r)I$ previously. As $I>A$ this establishes the inequality in (Marginal$^*$). Finally this entails directly that $p^*\leq \bar{p}$.

This inequality entails **overinvestment**:

* When $p$ is observed by the bank, the marginal investor is identified, such that *marginal* profits are zero.

* When $p$ is not observed by the bank, the marginal investor is identified, such that *average* profits are zero. 

* Thus credit is extended to types that are unprofitable, $p^*\leq \bar{p}$.

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