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!

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1 Answer 1


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.

  • $\begingroup$ Thanks - this makes sense. The one confusing thing is that the inequality has the same dx2/dx1 fraction on both sides. But, if I understand you correctly, it is implicit that the RHS fraction refers to the budget line because the bar notation refers to the budget line? $\endgroup$
    – Tomas R
    Jul 4 at 15:47
  • 1
    $\begingroup$ @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. $\endgroup$
    – Herr K.
    Jul 4 at 16:25

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