The concept of marginal rate of substitution (MRS) and its relationship to the price ratio is fundamental in consumer theory within economics. Let's break down why the MRS between two goods must equal the ratio of their prices for a consumer to achieve maximum satisfaction (or utility):
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Marginal Rate of Substitution (MRS): The MRS measures how much of one good a consumer is willing to give up in order to obtain an additional unit of another good while keeping their overall utility constant. Mathematically, it's the absolute value of the slope of the indifference curve at a given point.
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Budget Constraint: A consumer has limited resources (income) that can be allocated among different goods. The budget constraint represents all possible combinations of goods that a consumer can afford given their income and the prices of the goods.
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Utility Maximization: Consumers aim to maximize their utility given their budget constraint. The optimal consumption point occurs where the consumer can no longer increase their satisfaction by reallocating their spending between the two goods.
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Equilibrium Condition: For a consumer to maximize their utility, the rate at which they are willing to trade off one good for another (MRS) should equal the market's rate of trade-off, which is represented by the price ratio (P1/P2). This means:
\[ \text{MRS} = \frac{P_1}{P_2} \]
where \(P_1\) is the price of good 1 and \(P_2\) is the price of good 2.
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Intuition Behind the Equality:
- If MRS > (P1/P2): The consumer is willing to give up more of good 2 for good 1 than what the market requires. Thus, the consumer should buy more of good 1 and less of good 2 to increase their utility.
- If MRS < (P1/P2): The consumer values good 2 more than the market does relative to good 1. Therefore, the consumer should buy more of good 2 and less of good 1.
- At the point where MRS = (P1/P2), the consumer is in equilibrium and cannot increase their utility by changing their consumption choices, as the rate at which they are willing to trade off the goods equals the rate at which they can trade them in the market.
In summary, for maximum satisfaction under budget constraints, the consumer must equate their personal valuation of the goods (MRS) to the market valuation (price ratio). This ensures that they are optimizing their consumption given their preferences and resources.