Suppose that you have two consumption choices: good X, and good Y. An indifference curve is the set of consumption choices with a CONSTANT utility. For example if $X=10$ and $Y=6$ gives the same utility as consuming $X=11$ and $Y=5$, than these are both points on the same indifference curve. An indifference map is the set of all indifference curves for EVERY given utility.

The Cobb-Douglas utility function gives a simple indifference map:
$U = X^{\alpha}Y^{\alpha-1}$ , where $0 \le \alpha \le 1$ .

A budget curve gives the set of possible consumption choices with a given income. If you have an income of \$196 and the price of good X is given by $p_x$, and the price of good Y given by $p_y$. The equation for the budget line is given by: $196 = p_x X + p_y Y$.

A utility maximizing combination of goods X and Y occurs when the budget line is tangent to an indifference curve.

Find X as a function of its price, where $\alpha = .5$.
(If Y represents all other goods, than this function is just a demand curve for X).

$X =$
(Use px for $p_x$)

Let $X_0$ be the value for X when $p_x = 7$ and $p_y = 14$.
$X_0 =$

(you will lose 25% of your points if you do)

You can earn partial credit on this problem.