
\subsection{Specific function results}

\subsubsection{Examples of utility functions}

\subsubsection{Cobb Douglas}

\(U=f(x,y)\)

\(U=x^\alpha y^\beta \)

\(MU_x=\alpha x^{\alpha -1}y^\beta \)

\(MU_x=\alpha \dfrac{y^\beta }{x^{1- \alpha }}\)

\(MU_y=\beta \dfrac{x^\alpha}{y^{1- \beta }}\)

So:

\(\dfrac{MU_x}{MU_y}=\dfrac{\alpha \dfrac{y^\beta }{x^{1- \alpha }} }{\beta \dfrac{x^\alpha}{y^{1- \beta }}}\)

\(\dfrac{MU_x}{MU_y}=\dfrac{\alpha }{\beta }\dfrac{y}{x}\)

\subsubsection{Constant elasticity of substitution}

\(U=f(x,y)\)

\(U=(ax^b+(1-a)y^b)^{\dfrac{1}{b}}\)

\(MU_x=\dfrac{1}{b}abx^{b-1}(ax^b+(1-a)y^b)^{\dfrac{1}{b}-1}\)

\(MU_y=\dfrac{1}{b}(1-a)by^{b-1}(ax^b+(1-a)y^b)^{\dfrac{1}{b}-1}\)

So:

\(\dfrac{MU_x}{MU_y}=\dfrac{\dfrac{1}{b}abx^{b-1}(ax^b+(1-a)y^b)^{\dfrac{1}{b}-1}}{\dfrac{1}{b}(1-a)by^{b-1}(ax^b+(1-a)y^b)^{\dfrac{1}{b}-1}}\)

\(\dfrac{MU_x}{MU_y}=\dfrac{ax^{b-1}}{ (1-a)y^{b-1}}\)

\subsubsection{Constant elasticity of substitution}

\(U=f(x,y)\)

\(U=ax+by\)

\(MU_x=a\)

\(MU_y=b\)

So

\(\dfrac{MU_x}{MU_y}=\dfrac{a}{b}\)

\subsubsection{Constant elasticity of substitution}

\(U=f(x,y)\)

\(U=\alpha .ln(x)+\beta .ln(y)\)

\(MU_x=\dfrac{\alpha }{x}\)

\(MU_y=\dfrac{\beta }{y}\)

So:

\(\dfrac{MU_x}{MU_y}=\dfrac{\alpha }{\beta }\dfrac{y}{x}\)

This is the same as Cobb Douglas. This reflects that the logarithmic function preseres positive inflections.

