Cobb-Douglas and Perfect Substitutes

A commonly used utility function in economics with the form U(x, y) = x^α y^(1-α), where α represents the proportion of income spent on good x.

The values of x and y that maximize a Cobb-Douglas utility function, given prices (px and py) and income (I).

The rate at which a consumer is willing to trade one good for another, while maintaining the same level of utility. In Cobb-Douglas, it's decreasing in x/y.

The proportion of income spent on a specific good (x or y), calculated by multiplying the good's price (px or py) by the optimal quantity (x* or y*) and dividing by income (I).

A parameter in the Cobb-Douglas utility function representing the proportion of income spent on good x. It directly reflects the consumer's preference for good x.

In a perfect substitutes utility function, the consumer can substitute one good for another with a fixed rate, regardless of consumption amounts.

The Marginal Rate of Substitution (MRS) in perfect substitutes is constant, indicating a consistent rate of substitution between goods.

When the price ratio is greater than the MRS, the consumer will only consume the good with the higher benefit per unit price.

The consumer will consume only the good with the higher benefit per unit price when the price ratio and MRS are not equal.

A curve representing all combinations of two goods that provide the consumer with the same level of satisfaction or utility.

The rate at which a consumer is willing to trade one good for another while maintaining the same level of satisfaction.

A line representing all possible combinations of two goods that a consumer can afford given their income and prices.

Goods that are consumed together in fixed proportions. Utility is only derived when both goods are consumed together.

A mathematical operation that returns the smallest value among the given inputs.

Preferences where the consumer prefers a mix of goods to having all of one good or all of another.

In a Cobb-Douglas utility function, changing the price of a good affects the quantity demanded but not the consumer's spending share on that good.

A utility function where the consumer can substitute one good for another at a constant rate, regardless of consumption amounts, represented by U(x, y) = Ax + By.

The price ratio reflects the actual trading rate between goods, whereas the MRS reflects the consumer's willingness to trade. The consumer will consume only the good with the higher benefit per unit price when the price ratio is greater than the MRS.

The MRS reflects a consumer's willingness to trade, while the price ratio reflects the actual trade rate in the market. If the MRS is greater than the price ratio, the consumer will consume only the good with a higher benefit per unit price.

The indifference curves for perfect substitutes are linear, reflecting the constant MRS. The budget constraint will either be steeper, flatter, or parallel to the indifference curves. The optimal consumption point occurs where the budget constraint intersects the indifference curve at a corner.

A mathematical function used in perfect complements, returning the smallest value between two inputs, representing the utility derived from consuming the limited good.

A right-angled shape, reflecting that increasing the quantity of one good beyond a certain point does not increase utility, as the other good is limited.

U(x, y) = min(Ax, By), where A and B are constants representing the fixed proportion of consumption.

The Marginal Rate of Substitution cannot be applied because the min function is not differentiable, making it impossible to calculate the rate of substitution.

In convex preferences, as you consume more of one good (x), the rate at which you're willing to trade it for another good (y) decreases.

Preferences where the consumer's optimal consumption bundle remains proportional to their income, even as income changes.

Convex indifference curves indicate convex preferences, reflecting the value of variety in consumption.

Convex preferences ensure that a unique, well-defined solution exists for the consumer's optimization problem. This means there's a single point where utility is maximized given the budget constraint.