Changing Hearts and Minds

Often, a utility function is defined by some parameter that economists use to match it to some kind of observed behavior.

Each of the following questions asks you to draw two indifference curves to illustrate someone changing their preferences. The purpose is to get a sense as to how changing the parameters of a utility function affect the shape of the corresponding indifference curves. In each case, draw the two indifference curves in different colors and use arrows to show the shift in the curves.

Specifically, in each case, draw the indifference curve passing through the bundle (10, 10) before and after the change in preferences. Verify your answer using the graphs below.

Then, calculate the MRS at the bundle (10, 10) before and after the change. Finally, explain intuitively why the change in the utility function has that effect on the indifference curve and the MRS. (Don’t write a novel; but try to briefly explain in words the story that’s being told by the indifference curves.)

1. Eloise’s preferences over tea (good 1) and biscuits (good 2) are represented by the “perfect complements” utility function

   \[u(x_1, x_2) = \min\left\{\frac{x_1}{\alpha}, \frac{x_2}{1 − \alpha}\right\}\]

   Show the effect of a decrease in α from $0.7$ to $0.3$.

2. Carson’s preferences over peanut butter (good 1) and celery (good 2) are represented by the “Constant Elasticity of Substitution” utility function. \[u(x_1, x_2) = (x^r_1 + x^r_2)^\frac{1}{r}\] Show the effect of an change in r from $−1$ to $-3$.