Consumer Choice and General Equilibrium: Practice Problems

Suppose a consumer's utility function is given by $U(x_1, x_2) = x_1^{0.4}x_2^{0.6}$. If the prices of the goods are $p_1 = 2$ and $p_2 = 3$, and the consumer's income is $m = 60$, what are the optimal quantities of $x_1$ and $x_2$?

Suppose a consumer's utility function is given by $U(C, L) = C^{0.6}L^{0.4}$, where $C$ is consumption and $L$ is leisure. The wage rate is €20 per hour, and the consumer has 24 hours to allocate between labor and leisure. There is no non-labor income. What is the optimal amount of leisure the consumer will choose?

In a two-person, two-good exchange economy, suppose the aggregate excess demand function for good X is given by $E_X(p_X, p_Y) = 2p_X - p_Y$. According to Walras' Law, what must be the aggregate excess demand function for good Y, $E_Y(p_X, p_Y)$?