Proof of Marginal Rate of Substitution (MRS) at Consumer's Optimum Point

Indifference Curve:
- An indifference curve represents different combinations of two goods that give the consumer the same level of satisfaction.
- Consumers aim to reach the highest possible indifference curve to maximize their utility.

Consumer's Optimum Point:
- The consumer's optimum point is where the budget line is tangent to the indifference curve.
- At this point, the consumer achieves the highest level of satisfaction given their budget constraint.

Marginal Rate of Substitution (MRS):
- MRS is the rate at which a consumer is willing to give up one good in exchange for another while maintaining the same level of satisfaction.
- MRS is calculated as the negative ratio of the marginal utility of good x to the marginal utility of good y.

Ratio of Prices:
- The ratio of prices is the relative price of the two goods in the market.
- It represents the amount of one good that can be exchanged for another in the market.

Equality of MRS and Price Ratio:
- At the consumer's optimum point, the MRS must be equal to the ratio of prices for the consumer to maximize utility.
- If MRS is not equal to the price ratio, the consumer can still reallocate their consumption to increase satisfaction.
- Therefore, the consumer will adjust their consumption until the MRS equals the price ratio to reach the optimum point.

In conclusion, at the consumer's optimum point on an indifference curve, the marginal rate of substitution is equal to the ratio of prices as this equality allows the consumer to maximize utility within their budget constraint.