Exercises: Decision Making under Uncertainty (Solutions)

Worked Answers for 5 Exercises

1 Solutions Overview

This document contains detailed solutions to the 5 problems in the companion exercises.

2 Easy

2.1 Exercise 1 (Expected utility and CE)

Lottery: 100 with prob 0.4, 400 with prob 0.6; utility \(u(y)=\sqrt{y}\).

  • Expected utility: \[\mathbb{E}[u(\tilde{y})] = 0.4\sqrt{100} + 0.6\sqrt{400} = 0.4\cdot 10 + 0.6\cdot 20 = 16.\]

  • Certainty equivalent CE solves \(\sqrt{\text{CE}} = 16\) so \(\text{CE} = 256\).

  • Expected value \(= 0.4\cdot 100 + 0.6\cdot 400 = 280\). Risk premium \(= 280 - 256 = 24\).

2.2 Exercise 2 (Jensen approximation of risk premium)

Let \(m=\mathbb{E}[\tilde{X}]\), \(\sigma^2=\mathrm{Var}(\tilde{X})\). Expand \(u(\tilde{X})\) at \(m\):

\[u(\tilde{X}) \approx u(m) + u'(m)(\tilde{X}-m) + \tfrac12 u''(m)(\tilde{X}-m)^2.\]

Taking expectations yields \(\mathbb{E}[u(\tilde{X})] \approx u(m) + \tfrac12 u''(m)\sigma^2\).

Let CE solve \(u(\text{CE}) = \mathbb{E}[u(\tilde{X})]\). With \(\text{CE} = m - \pi\) and linearizing, \(u(\text{CE}) \approx u(m) - \pi u'(m)\), so

\[u(m) - \pi u'(m) \approx u(m) + \tfrac12 u''(m)\sigma^2 \;\Rightarrow\; \pi \approx -\tfrac12 \frac{u''(m)}{u'(m)}\sigma^2 = \tfrac12 R_A(m)\,\sigma^2.\]

2.3 Exercise 3 (CARA vs CRRA)

  • CARA \(u(x)= -\tfrac{1}{\nu} e^{-\nu x}\): \(u'(x)=e^{-\nu x}\), \(u''(x)=-\nu e^{-\nu x}\).

    • \(R_A(x)=\nu\) (constant), \(R_R(x)=\nu x\) (increasing in \(x\)).

  • CRRA \(u(x)= \tfrac{x^{1-\gamma}}{1-\gamma}\) or \(u(x)=\ln x\):

    • \(u'(x)=x^{-\gamma}\), \(u''(x)=-\gamma x^{-\gamma-1}\); hence \(R_A(x)=\gamma/x\), \(R_R(x)=\gamma\) (constant). For \(u=\ln x\): \(R_A(x)=1/x\), \(R_R(x)=1\).

  • Interpretation: CARA keeps absolute risk aversion constant (additive risks), CRRA keeps relative risk aversion constant (multiplicative risks).

3 Intermediate

3.1 Exercise 4 (Mean–variance portfolio rule)

Given \(y_0=100\), \(r_f=0.05\), \(\mu=0.10\), \(\sigma=0.20\), \(\gamma=3\) with CRRA \(u(y)= y^{1-\gamma}/(1-\gamma)\).

  • \(Y = y_0(1+r_f) = 105\) and \(R_A(Y)=\gamma/Y = 3/105 \approx 0.028571\).

  • Use \(w^* \approx (\mu - r_f)/\big(R_A(Y)\,\sigma^2\big) = 0.05 / (0.028571\cdot 0.04) \approx 43.75\).

  • Under CRRA, \(R_A(Y)=\gamma/Y\) so \(w^* \propto Y \propto y_0\). The elasticity of \(w^*\) with respect to \(y_0\) is 1.

4 Advanced

4.1 Exercise 5 (Optimal insurance with log utility)

States: loss \(L=40\) with prob \(p=0.1\), no loss otherwise. Coverage \(I\in[0,L]\) with per-unit premium \(\pi\). Utility \(u(c)=\ln c\).

  • Budget outcomes:

    • No loss: \(c_{NL} = y - \pi I\).

    • Loss: \(c_L = y - L - \pi I + I = y - L + (1-\pi) I\).

  • Expected utility:

    \[U(I) = (1-p)\,\ln(y - \pi I) + p\,\ln\big(y - L + (1-\pi) I\big).\]

  • FOC for interior optimum:

    \[U'(I) = (1-p)\frac{-\pi}{y - \pi I} + p\frac{1-\pi}{y - L + (1-\pi)I} = 0.\]

  • Solve for \(I^*\):

    \[(1-p)(-\pi)\big(y - L + (1-\pi)I^*\big) + p(1-\pi)\big(y - \pi I^*\big) = 0.\]

    Rearranging gives

    \[I^* = \frac{p y - (1-p)(y - L)\frac{\pi}{1-\pi}}{\pi + (1-\pi)\frac{\pi}{1-\pi}} = \frac{p(y) - (1-p)\frac{\pi}{1-\pi}(y - L)}{\pi + \pi} = \frac{p(y) - (1-p)\frac{\pi}{1-\pi}(y - L)}{2\pi}.\]

    A cleaner route is to note that at \(\pi=p\) (actuarially fair), \(c_{NL}=c_L\) at \(I=L\), which equalizes marginal utilities and yields full insurance.

  • Premium threshold for full insurance \(I=L\):

    At \(I=L\), the FOC becomes \((1-p)\frac{-\pi}{y - \pi L} + p\frac{1-\pi}{y} = 0\).

    This simplifies to \(-\pi(1-p) + p(1-\pi) = 0 \Rightarrow \pi = p\).

  • Conclusion: Full insurance is optimal iff \(\pi=p\) (actuarially fair). For \(\pi>p\), partial insurance; for \(\pi<p\), corner may exceed \(I=L\) but constrained to \(I=L\).