What is the optimal amount of risky assets, Financial Management

Consider a world with two assets: a riskless asset paying a zero interest rate, and a risky asset whose return r can take values +10% or -8% with equal probability. An individual has preferences represented by the utility function u(x) = ln x and an initial wealth w0 = 10.

a) Solve the portfolio choice problem of the agent. What is the optimal amount z* of risky assets?

Assume now that the agent also faces an exogenous additive background risk ε, with a distribution independent of r, that can take values +4 or -4 with equal probability (additive means that the agent's final wealth x is given by the portfolio of assets plus ε).

b) Show that this background risk reduces the demand for risky assets.

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