Cobb-Douglas and Perfect Substitutes

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the correct expression for Marshallian demand x* in terms of I, px, and py?

$x^* = \frac{I}{p_y + p_x}$

$x^* = \frac{I}{p_x + I p_y}$

$x^* = \frac{I}{p_x + p_y}$

$x^* = \frac{I}{B p_x + A p_y}$ (correct)

What characterizes a consumer with convex preferences?

They derive more utility from mixtures of goods. (correct)

Their marginal rate of substitution (MRS) is increasing.

They prefer consuming all of one good over mixed goods.

Their preferences are perfectly elastic.

In the context of preferences, what does homothetic mean?

Preferences are independent of income.

Utility functions are linear with respect to goods consumed.

Marginal utility is constant regardless of consumption.

Preferences maintain proportional changes in consumption with changes in income. (correct)

What is the main significance of the convexity of indifference curves?

It simplifies the utility maximization problem. Signup and view all the answers

Which of the following conditions must a Marginal Rate of Substitution (MRS) satisfy for convex preferences?

MRS must decrease as the consumer substitutes good x for good y. Signup and view all the answers

What is the general form of a Cobb-Douglas utility function?

U(x, y) = x^{α} y^{1-α} Signup and view all the answers

What does the parameter α in the Cobb-Douglas utility function represent?

The fraction of income spent on good x Signup and view all the answers

How is the Marginal Rate of Substitution (MRS) calculated in the Cobb-Douglas utility function?

MRS = (α * y) / ((1 - α) * x) Signup and view all the answers

If the utility function is U(x, y) = x^{1/3} y^{2/3} and the prices are px = 2, py = 4, and income I = 120, what is the optimal consumption of good x?

20 Signup and view all the answers

What is a property of the Cobb-Douglas utility function regarding expenditure allocation?

The fraction of expenditure on good x is determined by the parameter α. Signup and view all the answers

What is the total spending on good x if each unit costs 2 and it represents one third of the income?

40 Signup and view all the answers

In the case where the price of good x increases to 4, how does this affect the total spending on x?

It remains the same Signup and view all the answers

If the consumer's MRS is 4 and good x costs 2 while good y costs 1, what can the consumer do regarding their consumption choices?

Trade one unit of x for two units of y Signup and view all the answers

When will a consumer prefer to consume x over y in the context of perfect substitutes?

When the benefit of consuming x is greater than the cost Signup and view all the answers

What characterizes the indifference curves of a perfect substitutes utility function?

They have constant slopes Signup and view all the answers

What does the general form for utility of perfect complements represent?

U(x, y) = min(Ax, By) Signup and view all the answers

How are the indifference curves for perfect complements typically characterized?

They form a rectangular shape, being L-shaped curves. Signup and view all the answers

In the context of indifference curves, what happens when one of the goods is increased while holding the other constant?

Utility remains constant at the minimum value. Signup and view all the answers

What is the significance of the condition where MRS is less than the price ratio in indifference curves?

It suggests the consumer should consume only good x. Signup and view all the answers

Why can't the usual marginal rate of substitution (MRS) equal price ratio condition be applied to perfect complements?

Perfect complements are not differentiable functions. Signup and view all the answers

What happens to the quantity demanded of good x when its price increases to 4, considering the initial total spending is maintained?

The quantity demanded of good x decreases. Signup and view all the answers

In the utility function U(x, y) = Ax + By for perfect substitutes, what does the term MRS represent?

The rate at which a consumer is willing to trade x for y. Signup and view all the answers

When a consumer prefers to consume only good x over good y, what condition must hold regarding the benefits and costs?

The benefits of consuming x must exceed the costs of consuming y. Signup and view all the answers

In the example with coefficients A = 4 and B = 1, how many units of good y does a consumer need to substitute for one unit of good x?

4 units of y. Signup and view all the answers

What is a key characteristic of the indifference curves for perfect substitutes?

They have a constant slope regardless of the quantities of goods. Signup and view all the answers

What happens when the Marginal Rate of Substitution (MRS) is greater than the price ratio (ppxy)?

The consumer spends all income on good x. Signup and view all the answers

What is the outcome when MRS is equal to the price ratio (ppxy)?

The consumer will be indifferent to combinations of x and y on the budget line. Signup and view all the answers

In the scenario where MRS is less than the price ratio, what does the consumer's demand look like?

The consumer will spend all income on good y. Signup and view all the answers

What characteristic do the indifference curves exhibit when dealing with perfect substitutes?

They are linear. Signup and view all the answers

If the budget line intersects with the indifference curves at multiple points, which case does it represent?

MRS is equal to the price ratio. Signup and view all the answers

What does the utility function U(x, y) = min(Ax, By) represent for perfect complements?

The combined utility from both goods when consumed together. Signup and view all the answers

How does increasing one good while holding the other constant affect utility for perfect complements?

It has no effect on utility. Signup and view all the answers

In the context of perfect complements, what is the effect of a flatter indifference curve compared to the budget constraint?

It means the consumer is not optimizing their consumption choices. Signup and view all the answers

What utility level corresponds to Ū = 50 when using the function U(x, y) = min(10x, 5y)?

Any combination of x and y such that either term is equal to 50. Signup and view all the answers

Why is the usual MRS equal to price ratio condition not applicable for perfect complements?

Due to the min function which is not differentiable. Signup and view all the answers

What is the relationship between convex preferences and consumer choices?

Consumers prefer a balanced mix of different goods over extremes of one good. Signup and view all the answers

How can one check the convexity of a utility function using the Marginal Rate of Substitution (MRS)?

By analyzing if the MRS decreases when the ratio of good x to good y is increased. Signup and view all the answers

In the context of Cobb-Douglas preferences represented by U = x^{1/2} y^{1/2}, how does a consumer's utility change with a balanced consumption bundle compared to an unbalanced bundle?

Utility is higher with a balanced bundle than with an unbalanced bundle. Signup and view all the answers

What does the term 'homothetic preferences' imply about utility functions?

Utility mixed in proportions that change with scale but maintain the same MRS. Signup and view all the answers

What effect does the convexity of indifference curves have on the solution to a consumer's maximization problem?

It guarantees the existence of a point where the consumer's utility is maximized. Signup and view all the answers

Study Notes

Special Utility Functions

Utility functions represent consumer preferences.
Cobb-Douglas utility function (U(x,y) = x^αy^1-α) is widely used.
α's value has an economic meaning (e.g. fraction of income spent on good x).
Example: U(x, y) = x^1/2y^1/2, where α = 1/2.
The Marginal Rate of Substitution (MRS) for Cobb-Douglas is decreasing in x/y, dependent solely on x/y ratio.
Marshallian demands can be found for Cobb-Douglas functions.
Fractional income spent on x is α, and on y is 1 − α.

Perfect Substitutes

Perfect substitutes: consumers are indifferent between goods.
General form: U(x, y) = Ax + By (coefficients A and B show substitution possibilities).
MRS is constant, unaffected by x and y quantities.
Indifference curves are straight lines.
Consumers choose the good with a better price ratio.
Example: A = 4, B = 1; MRS = 4. Given prices Px = 2, Py =1, the consumer would prefer good x.

Perfect Complements

Perfect complements: goods consumed in fixed proportions.
General form: U(x, y) = min(Ax, By).
Indifference curves are L-shaped.
MRS is either 0 or undefined.
Example: left and right shoes.

Properties of Preferences

Convex Preferences: Indifference curves are convex; consumers prefer a mix of goods.
Homothetic Preferences: MRS depends only on the ratio x/y; not on the absolute amounts.

Corner Solutions

Corner solutions occur when optimal bundles are on the boundary of feasible set.
Concave indifference curves may lead to spending all income on one good.
Increasing MRS suggests spending all income on one good.
Optimal bundle for concave preferences is at a boundary; MRS=price ratio not used.
Always check corner points first (consuming all of one good) for concave preferences.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz explores special utility functions, focusing on the Cobb-Douglas and perfect substitutes. Understand the implications of parameters, marginal rates of substitution, and how consumer preferences are modeled. Test your knowledge of consumer theory in economics through this targeted quiz.