Given a continuous preference relation $\succeq$ over $X=\mathbb{R}^2_{+}$ where all sets: $$ I_x\equiv\{y\in X:y\sim x\} $$ are lines on $X,\forall x\in X$, and are parallel to $I_y,\forall y\notin I_x$.
How can I show that $\succeq$ has a linear representation?
It seems intuitive that if every indifference curve is a line, then the utility itself must be a line, but I'm not sure how to go back to it. Would someone be willing to help me?
Thanks! Any helpful tips are appreciated! :D
The indifference curves are constructed by viewing the utility function as an equation (for a fixed utility index value per curve). So from
$$U = U(x_1,x_2)$$
where the left side is just a symbol, we move to
$$\bar U = U(x_1,x_2)$$ where now the left side is a specific number.
Take the total differential on both sides to obtain
$$0 = U_1dx_1 + U_2dx_2 \implies \frac {dx_2}{dx_1} = -\frac {U_1}{U_2}$$
Any straight line in the two-dimensional plane has a constant slope so also
$$\frac {dx_2}{dx_1} =c$$
Same for $y$-bundle. Etc.
