Why Marginal Rate of Substitution must to equal the ratio of price of 2 given goods for the consumer to achieve maximum satisfaction?
OCW Scholar - Principles of Microeconomics
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If your rate of substitution is different from your ratio of prices, then the relative benefits from the goods is different from the relative costs.
Suppose the consumer had chosen a bundle at which his MRS was higher than the market
price ratio. This means that he is willing to give up more units of c2 for a little more c1
than the market requires him to – so he should use the markets to trade some of his c2 for
c1 because he would be made unambiguously better off! Now suppose he has traded
himself in this way all the way to the point where his MRS equals the price ratio. Should
he trade yet more units of c2 to obtain a little more c1 ? The answer is no, because doing
so would now mean having to give up more units of c2 than he willing to for a little more c1.
Thus, once he has traded his way to the bundle at which his MRS equals the price
ratio, he can do no better – he has arrived at his optimal consumption choice, the one that
maximizes u(c1, c2) subject to his budget constraint.
The MRS describes the rate at which the consumer is willing to trade one good for another to maintain the same level of satisfaction. The ratio of prices describes the trade-off that the market is willing to make between the same two goods. The tangency of the indifference curve with the budget line represents the point at which the trade-offs are equal and consumer satisfaction is maximized. If the MRS between two goods is not equal to the ratio of prices, then the consumer could trade one good for another at market prices to obtain higher levels of satisfaction.