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  • Calculus-Based Methods for Analyzing Indifference Curves

Verifying Diminishing MRS with the Second Derivative

To mathematically prove that the Marginal Rate of Substitution (MRS) is diminishing along an indifference curve, the second derivative is used. This involves differentiating the indifference curve's equation a second time with respect to the good on the horizontal axis (for example, free time, tt). A positive result for this second derivative (i.e., d2cdt2>0\frac{d^2c}{dt^2} > 0) confirms that the indifference curve is convex to the origin. This mathematical convexity implies that the curve's slope gets less steep as tt increases, meaning the MRS, which is the absolute value of the slope, is diminishing.

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