Problem

乙、計算、證明題：共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。

3. Goods 1 and 2 are available at prices (in dollars) of $p_1$ per unit of good 1 and $p_2$ per unit of good 2. A utility function $U(x_1,x_2)$ is a function representing the utility or benefit of consuming $x_j$ units of good $j$. The marginal utility of the $j$th good is $\partial U/\partial x_j$, the rate of increase in utility per unit increase in the $j$th good. Prove the following law of economics:

Given a budget of $L$ dollars, utility is maximized at the consumption level $(a,b)$ where the ratio of marginal utility is equal to the ratio of prices:

$$\frac{\text{Marginal utility of good 1}}{\text{Marginal utility of good 2}} = \frac{\partial U/\partial x_1}{\partial U/\partial x_2} = \frac{p_1}{p_2}.$$