Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 2 / e

Solution

Y

and

ε \geq 0

Y^{*} + ε Y

is a deflator. Minimality and right differentiation at zero give

E [u^{'} (Y_{1}^{*}) Y_{1}] + X_{0} Y_{0} \geq 0.

Taking

Y = Y^{*}

and varying the positive scalar multiple

(1 + ε) Y^{*}

in both directions around one gives equality.

ξ_{1} = - u^{'} (Y_{1}^{*})

and

ξ_{0} = X_{0}

. The preceding inequality says

E [ξ_{1} Y_{1}] \leq X_{0} Y_{0}

for every deflator, with equality at

Y^{*}

. Part c produces

H^{*}

with

H^{*} \cdot P_{0} = X_{0}

and

H^{*} \cdot P_{1} = ξ_{1}

. The inverse relation between conjugate derivatives gives

u^{'} (ξ_{1}) = Y_{1}^{*}

, so equality holds in part d:

E [u (H^{*} \cdot P_{1})] = E [u (Y_{1}^{*})] + X_{0} Y_{0}^{*} .

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