Question: (a) Amber has preferences captured by the utility function \(u^{A}\left(x^{A}, y^{A}\right)=\ln x^{A}+y^{A}\) where \(x^{A}\) denotes her consumption of apples and \(y^{A}\) denotes her consumption of pears. She has an endowment of \(X\) apples and no other income. If an apple exchanges for \(p\) pears, where \(p \geq 1 / X\), then explain why her demand for apples is \(1 / p\)

(b) Beryl has preferences captured by utility function \(u^{B}\left(x^{B} \cdot y^{B}\right)=\ln x^{B}+\ln y^{B}\) where \(x^{B}\) denotes her consumption of apples and \(y^{B}\) denotes her consumption of pears. She has an endowment of \(Y\) pears and no other income. If an apple exchanges for \(p\) pears then explain why her demand for apples is \(Y / 2 p\).

(c) If Amber and Beryl trade competitively then find the price \(p\) at which the apple market clears.

(d) Explain why this price also clears the market for pears.

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