Suppose you have $400,000 saved up and purchase a medium-sized house f

Suppose you have $400,000 saved up and purchase a medium-sized house for $200,000. Consider the following 2 scenarios:

i. The very next day, the prices of all houses, including the one you have just bought, double.
ii. The very next day, the prices of all houses, including the one you bought, fall by half.
Show that both scenarios increase your utility.

ANSWER

In both scenarios, you are still able to consume your original bundle — the selected house plus $200,000 of other goods. If the price of housing drops, the substitution effect allows you to reoptimize by selecting a larger house and spending less on other goods. If the price of housing increases, you can select a smaller house and spend more on other goods. In both cases, the re-optimization leads you to a higher utility level.