 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-43/5/with-transaction-costs/i/solution

h_{x} (θ) = θ x - ε ∣ θ ∣

is concave when

ε \geq 0

. The utility function is increasing and concave, so

U (h_{x} (s θ + (1 - s) η)) \geq U (s h_{x} (θ) + (1 - s) h_{x} (η)) \geq s U (h_{x} (θ)) + (1 - s) U (h_{x} (η)) .

Take finite expectations. Thus $G is concave$ . The monotonicity of

U

is essential to composing it with the concave transaction-cost payoff.

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