The Economics of Banking

Solutions to selected exercises

DOI: 10.4324/9780429356773-18

Chapter 2

1. EXERCISE 1

   Proof: It’s known that $-\frac{cu″\left(c\right)}{u\prime \left(c\right)}>1$, this is equivalent to $\frac{\partial \left[cu\text{'}\left(c\right)\right]}{\partial c}<0$. Together with $c_1^{*}<c_2^{*}$ and $R>1$, it implies that $1\cdot u\prime (1)>Ru\prime (R)$. Further, $(c_1^{*},c_2^{*})$ satisfies resource constraint $c_2^{*}=\frac{\left(1-p{c}_1^{*}\right)R}{1-p}$ and optimality condition $u\prime \left(c_1^{*}\right)=Ru\prime \left(c_2^{*}\right)$. Resource constraint implies either (a) $c_1^{*}>1$ and $c_2^{*}<R$, or (b) $c_1^{*}\le 1$ and $c_2^{*}\ge R$.

   • If (a) is true, $1\cdot u\prime \left(1\right)>c_1^{*}u\prime \left(c_1^{*}\right)>c_2^{*}u\prime \left(c_2^{*}\right)>Ru\prime \left(R\right)$, it goes through;
   • If (b) is true, $u\prime \left(c_1^{*}\right)=Ru\prime \left(c_2^{*}\right)\le Ru\prime \left(R\right)<1\cdot u\prime \left(1\right)$, which implies $c_1^{*}>1$. A contradiction.
2. EXERCISE 2 RISK SHARING AND FINANCIAL INTERMEDIATION
   1. (2a) $c_1^{*}=\frac{R}{\left(1-\pi \right)R^{\frac{1}{\gamma }}+\pi R}$, $c_2^{*}=\frac{R^{1+\frac{1}{\gamma }}}{\left(1-\pi \right)R^{\frac{1}{\gamma }}+\pi R}$, and ${\text{lim}}_{\gamma \to +\infty }\frac{c_2^{*}}{c_1^{*}}=R^{\frac{1}{\gamma }}=1$.
3. EXERCISE 3 COMPLETE MARKET AND CONSTRAINED EFFICIENCY, ...