# Indifference curve - corner point - Q about notation

I wonder if someone can help me interpret the vertical bar notation used in the picture. From the graph, it is apparent that the consumer will consume only good $$x_1$$, since the indifference curve is steeper than the budget line at $$x^*$$. I assume that is what eq. C.5 is expressing? I'm hoping that C.6 and C.7 will become apparent if I first understand C.5.

Appreciate any help!

The vertical bar notation is used to denote conditions/restrictions applied to the expression to its left. For example, $$\frac{\mathrm dx_2}{\mathrm dx_1}\bigg\vert_{\text{u constant}}$$ reads:
The derivative of $$x_2$$ with respect to $$x_1$$, holding $$u$$ constant.
This apparently refers to the slope of the indifference curve, holding utility constant at $$I_1$$. Similarly, the RHS of [C.5] refers to the slope of the budget line. The inequality holds at $$x^*$$, where the slope of the indifference curve is smaller (i.e. more negative) than the slope of the budget line.
• @TomasR: Yes, that's my educated guess based on the snapshot you provide. The RHS has a restriction on income $M$ which usually occurs in the budget equation (e.g. $M=p_1x_1+p_2x_2$), while the LHS has a restriction on utility level, which naturally brings to mind indifference curves. Given utility function $u(x_1,x_2)$, the slope of the an indifference curve is given by $\frac{\mathrm dx_2}{\mathrm dx_1}=-\frac{\partial u(\cdot)/\partial x_1}{\partial u(\cdot)/\partial x_2}$ according to the implicit function theorem. Jul 4 at 16:25