Geometric intuition behind the point where the indifference curve meets the budget line So, this isn't to solve any kind of problem, but rather about the intuition behind the concept. I was wondering if at the optimal point where the indifference curve is tangent to the budget line and utility is maximized, at say, some point A. If you were to draw a vertical line from A down to the horizontal axis, and then draw a horizontal line from A to the vertical axis, would the resulting rectangle have a larger area than any other rectangle created by the other points on the curve? Kind of like the intuition behind least squares regression, but trying to maximize area? Thanks in advance for any help.
No. This is not in general true.
Maximizing the rectangle formed in the described manner would maximize the product $x_1 \times x_2$. In the special case where your utility function takes the form $u(x_1,x_2)=x_1x_2$, utility maximizing implies you pick a point on your budget constraint where the rectangle you described is maximized.
Suppose however, that instead your utility function is given by $v(x_1,x_2)=x_1^{1/3}x_2^{2/3}$ And suppose you are currently maximizing $x_1 \times x_2$ on your budget constraint. Suppose that $p_1=p_2=1$, then maximizing $x_1 \times x_2$ implies $x_1^*=x_2^*$. However maximizing $v$ with these prices gives: $x_1^*=\frac{1}{2}x_2^*$. Thus not maximizing the rectangle subject to your constraint, but still defining a point of tangency between the indifference curve and budget constraint.