Harry's visit to Gringotts (Consumer Theory)
The Weasleys had just taken Harry to the Gringotts Wizarding Bank for the first time,
where Harry discovers that he has received a large inheritance of m = 1,900,000
galleons (the currency of the wizarding world) from his parents. Harry must decide
how much to spend currently and how much to save for later. We model this problem
as Harry deciding on the consumption of two goods: consumption today $x_1$ and
consumption in the future $x_2$. Harry's preferences are
$u = log x_1 + 0.9 cdot log x_2$.
We can interpret that as saying that Harry has utility of consumption of
$log$ (consumption) in any given time period, but that he values the future at only 90%
of what he values the present ($log$ stands for the natural log). For each $100 that Harry
saves today, he gets 10% interest, so that he can consume $110 in the future. This
means that Harry's budget constraint is
$x_1 + frac{1}{1.1}x_2 le 1,900,000$.
a) Find Harry's optimal consumption choices.
b) Harry finds out about a better investment, so that the interest rate he gets on his
money is now 20% instead of 10%. What is Harry's new budget constraint?
c) What are Harry's consumption choices after he finds out about the new
investment opportunity?
d) Solve for Harry's optimal consumption choice if Harry has the same 10% return as
before, but he discovers that he has actually inherited 1,800,000 galleons, not
1,900,000 galleons as he had thought before. In addition, now Harry has become
more impatient, so that his preferences are
$u = log x_1 + 0.8 cdot log x_2$.
