### 날짜 : 2024-03-25 13:16 ### 주제 : Optimization Rule #economics ---- The optimization rule in consumer choice theory is based on the concept of equating the ratio of the marginal utilities of two goods to the ratio of their prices. This rule dictates that consumers allocate their income in a way that maximizes their total utility. Here's the standard formulation of the optimization rule: For two goods, \(X\) and \(Y\), with prices \(P_X\) and \(P_Y\) respectively, and marginal utilities \(MU_X\) and \(MU_Y\), a consumer maximizes utility by consuming the goods in quantities such that: \[ \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y} \] In words, the consumer should spend their money such that the last dollar spent on each good provides the same amount of additional satisfaction (marginal utility). When this balance is achieved, any deviation would lead to a lower total utility, which means it would not be an optimal consumption bundle. ### Example of the Optimization Rule Let's illustrate this with a simple example involving apples and bananas. Imagine you really like both apples and bananas, and you're trying to decide how many of each to buy with a limited budget. Apples cost $1 each, and bananas cost $0.50 each. #### Step 1: Assessing Marginal Utility Suppose the first apple gives you a marginal utility of 100 utils, and the first banana gives you 50 utils. If you've got exactly $1 to spend, you could either buy one apple or two bananas. How should you decide what to buy? #### Step 2: Calculating Marginal Utility Per Dollar You calculate the marginal utility per dollar spent on each fruit: - For apples: \[ \frac{MU_{\text{apples}}}{Price_{\text{apples}}} = \frac{100}{1} = 100 \frac{\text{utils}}{\text{dollar}} \] - For bananas: \[ \frac{MU_{\text{bananas}}}{Price_{\text{bananas}}} = \frac{50}{0.5} = 100 \frac{\text{utils}}{\text{dollar}} \] Both apples and bananas give you the same utility per dollar spent. Hence, you are indifferent to which one to buy, because spending your last dollar on either fruit would give you the same marginal utility. #### Step 3: Examining Changes in Marginal Utility and Recalculating After eating a few fruits, the marginal utility you derive from consuming additional apples or bananas changes due to diminishing marginal utility. Let's say after consuming two apples, the marginal utility of another apple decreases to 80 utils, and after eating two bananas, the marginal utility of another banana falls to 30 utils. Recalculate the marginal utility per dollar: - For an additional apple: \[ \frac{MU_{\text{apples}}}{Price_{\text{apples}}} = \frac{80}{1} = 80 \frac{\text{utils}}{\text{dollar}} \] - For an additional banana: \[ \frac{MU_{\text{bananas}}}{Price_{\text{bananas}}} = \frac{30}{0.5} = 60 \frac{\text{utils}}{\text{dollar}} \] #### Step 4: Making a Choice Based on Higher Marginal Utility Per Dollar Given this change, your next dollar should be spent on apples, not bananas, because the marginal utility per dollar you'd get from apples is higher (80 utils per dollar, compared to 60 utils per dollar for bananas). This example epitomizes the rule of utility maximization: continue redistributing your budget until the marginal utility per dollar spent on each good is equal. When you have no more money to reallocate and the marginal utility per dollar is the same for all products, you've reached an <mark style="background: #FFB86CA6;">optimal bundle</mark> given your budget. At this point, there is no way to rearrange your spending to gain more total satisfaction.