Constant Elasticity of Substitution (CES) Utility Function

Budget Constraint

where $m$ is the consumer's budget, $p_1$ is the price of good 1, $p_2$ is the price of good 2, $q_1$ is the quantity of good 1, and $q_2$ is the quantity of good 2.

CES Utility

where $U$ is utility, $q_1$ is the quantity of good 1, $q_2$ is the quantity of good 2, $\alpha_{1}$ is the share parameter of good 1, $\alpha_{2}$ is the share parameter of good 2, and $\rho$ characterises substitution between $q_1$ and $q_2$.

Since we only have two goods and $\alpha_{1}$ and $\alpha_{2}$ are the respective share parameters, we require $\alpha_{1}+\alpha_{2}=1$. The elasticity of substitution, which we use as a parameter below, is given by $\sigma=\frac{1}{1-\rho }$. Note that this value is constant.

CES ndifference curve

where $k$ is a constant. Hence, the indifference curve is given by:

Note that by requiring $U(q_{1},q_{2})=k$ the total differential of utility $dU=0$, i.e. there is no change in utility along an indifference curve.