1. Craig is still consuming apples and bananas. His utility function is $U(x_A, x_B) = x_A x_B$. We want to
find his demand function for apples, $x_A(p_A, p_B, m)$, and his demand function for bananas,
$x_B(p_A, p_B, m)$. His MRS is $-\frac{x_B}{x_A}$.
a) When the prices are $p_A$ and $p_B$ and Craig's income is $m$, the equation for Craig's budget line is
$p_A x_A + p_B x_B = m$. What is the slope of Craig's indifference curve at the bundle $(x_A, x_B)$?
What is the slope of Craig's budget line? Write the equation that needs to be satisfied for
Craig's indifference curve to be tangent to his budget line at the point $(x_A, x_B)$.
b) You now have two equations, the budget equation and the tangency equation, that must be
satisfied by the bundle demanded. Solve these two equations to find the optimal $x_A$ and $x_B$.
c) In general, the demand for both commodities will depend on the price of both commodities
and on income. But for Craig's utility function, the demand function for apples depends only
on income and the price of apples. Similarly, the demand for bananas depends only on income
and the price of bananas. Craig always spends the same fraction of his income on bananas.
What fraction is this?
