Question

Suppose your indifference curves are all described by equations of the form x y ¼ constant, with a different constant for each indifference curve. a. Show that for any point P ¼ (x, y), the indifference curve through P has slope −y/x at P. (This requires calculus. If you don’t know enough calculus, you can just pretend you’ve solved this part and go on to part (b).) b. Suppose that your income is $40, the price of X is $1, and the price of Y is $1. How much X do you buy? Hint: The problem is to find your optimal basket (x, y). First, write down an equation that says (x, y) is on the budget line. Next, write down an equation that says the slope of the indifference curve at (x, y) is equal to the slope of the budget line at (x, y). (Remember that you have a formula for the slope of the budget line from part (a), and that you can compute the slope of the budget line from the prices of X and Y.) Then solve these two equations simultaneously. c. Suppose your income and the price of Y remain as above, but the price of X rises to $4. Now how much X do you consume? (Use the same hint as in part b.) d. Based on your answers to parts (b) and (c), draw two points on your demand curve for X. e. After the price of X rises from $1 to $4, suppose that your income rises by just enough to bring you back to your original indifference curve. Now how much X do you buy? (Hint: The problem is to find the basket (x, y) where the compensated budget line is tangent to the original indifference curve.) First, write down the equation of the original indifference curve (remember that it is of the form xy ¼ constant, and you can figure out the constant because you already know the coordinates of one point on that curve). Next, write down an equation that expresses the condition that the slope of the indifference curve must equal the slope of the compensated budget line. Then solve these two equations simultaneously. f. When the price of X rises from $1 to $4, how much of the change in your consumption is due to the substitution effect? How much is due to the income effect?

Answer 1