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. Let your income be given by c, 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: $c = p_x X + p_y Y$.

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

Find X and Y as a function of $c$, where $\alpha = 0.5$,$p_x = 0.5$ and $p_y = 0.5$

$X =$
$Y =$

Write Utility as a function of c
$U(c) =$
$\frac{dU}{dc} =$

The easiest way to solve this question is to use Lagrange multiplier, which allows you to maximize functions subject to constraint.

Suppose we want to maximize $f(x,y)$ subject to $g(x,y) = 0$. We define the Lagrenge Function as:
$\Lambda(x,y,\lambda) = f(x,y) - \lambda g(x,y)$.

$f(x,y)$ will be maximized when all the partial derivatives of the Lagrenge Function are equal to zero.

Find $\lambda$ as a function of c. (It is possible that $\lambda$ could just be a constant as well).
$\lambda =$