Verifying Diminishing MRS with the Second Derivative
To mathematically prove that the Marginal Rate of Substitution (MRS) is diminishing along an indifference curve, the second derivative is used. This involves differentiating the indifference curve's equation a second time with respect to the good on the horizontal axis (for example, free time, ). A positive result for this second derivative (i.e., ) confirms that the indifference curve is convex to the origin. This mathematical convexity implies that the curve's slope gets less steep as increases, meaning the MRS, which is the absolute value of the slope, is diminishing.
0
1
Tags
Science
Economy
CORE Econ
Social Science
Empirical Science
Economics
Introduction to Microeconomics Course
The Economy 2.0 Microeconomics @ CORE Econ
Ch.3 Doing the best you can: Scarcity, wellbeing, and working hours - The Economy 2.0 Microeconomics @ CORE Econ
Related
Small Increments and the Condition for Constant Utility
Activity: Algebraic Verification of Convexity for Karim's Preferences
Verifying Diminishing MRS with the Second Derivative
Marginal Rate of Substitution as the Ratio of Marginal Utilities
Activity: Analyzing a Quasi-Linear Indifference Curve via Direct Differentiation
Calculating the Slope of an Indifference Curve
A consumer's preferences for two goods, x and y, are represented by the utility function U(x, y) = x^α * y^β, where α and β are positive constants. Using calculus, what is the expression for the slope (dy/dx) of the indifference curve for this utility function?
Consumer's Trade-off Calculation
A consumer's preferences are represented by a utility function U(x, y), where x and y are two goods. To find the slope of the indifference curve (dy/dx) at any point by holding the level of satisfaction constant, you must follow a specific sequence of mathematical operations using implicit differentiation. Arrange the following steps in the correct logical order.
Error Analysis in Indifference Curve Calculation
Evaluating Calculus Methods for Indifference Curve Analysis
A consumer's preferences for two goods,
xandy, can be represented by different mathematical forms of a utility function,U(x, y). Each form results in an indifference curve with a characteristic slope (dy/dx). Match each utility function form to the general expression or description for the slope of its indifference curve.A consumer's level of satisfaction from consuming two goods, Good X and Good Y, is held constant along a curve. At their current consumption bundle, the rate at which their satisfaction increases from one additional unit of Good X is 6. The rate at which their satisfaction increases from one additional unit of Good Y is 2. To maintain the exact same level of satisfaction, if this consumer decides to consume one less unit of Good X, approximately how many units of Good Y must they consume?
True or False: For a utility function U(x, y), if the marginal utility of good x is always a constant multiple of the marginal utility of good y (i.e., MUx = k * MUy, where k is a positive constant), then the slope of the indifference curves will change as the consumer moves along any given curve.
A consumer's preferences for goods
xandyare described by the utility functionU(x, y) = 2x + ln(y). At any point on an indifference curve for this consumer where their consumption of goodyis 10 units, the absolute value of the slope of the curve at that point is ____.Consumer's Trade-off Calculation
A consumer's preferences are represented by a utility function U(x, y), where x and y are two goods. To find the slope of the indifference curve (dy/dx) at any point by holding the level of satisfaction constant, you must follow a specific sequence of mathematical operations using implicit differentiation. Arrange the following steps in the correct logical order.
Learn After
Mathematical Proof of Diminishing MRS
A consumer's preferences for consumption (c) and free time (t) are described by the utility function U(c, t) = c * t^(1/2). To determine if the marginal rate of substitution (MRS) is diminishing, one can analyze the second derivative of the indifference curve equation, d²c/dt², where consumption is expressed as a function of free time for a constant utility level. Based on this mathematical test for the given utility function, what can be concluded about the shape of the indifference curves and the behavior of the MRS?
Analyzing Consumer Preferences for Convexity
For an indifference curve where consumption (c) is expressed as a function of free time (t), a finding that the second derivative
d²c/dt²is negative confirms that the curve is convex and the marginal rate of substitution is diminishing.An indifference curve expresses consumption (c) as a function of free time (t). Match each mathematical condition for the second derivative of this function (d²c/dt²) with its corresponding economic interpretation regarding the shape of the curve and the behavior of the Marginal Rate of Substitution (MRS).
A consumer's preferences for consumption (c) and free time (t) are represented by the utility function U(c, t) = ln(c) + t. To verify if the indifference curves for these preferences are convex to the origin, we analyze the second derivative of consumption with respect to free time, d²c/dt², holding utility constant. The sign of this second derivative is always ____.
A microeconomist wants to mathematically confirm that a consumer's preferences, represented by a utility function, exhibit a diminishing Marginal Rate of Substitution (MRS). Arrange the following steps in the correct logical order to conduct this proof using the second derivative test.
Connecting Calculus to Economic Intuition
An economics student is analyzing a consumer's preferences represented by the utility function U(c, t) = c + t², where c is consumption and t is free time. The student performs the following steps to determine if the Marginal Rate of Substitution (MRS) is diminishing:
- Express c as a function of t for a constant utility level U_bar: c = U_bar - t².
- Calculate the first derivative: dc/dt = -2t.
- Calculate the second derivative: d²c/dt² = -2.
- Conclude: "Since the second derivative is negative, this confirms a diminishing MRS."
Which of the following statements best identifies the fundamental error in the student's analysis?
Mathematical Verification of Preference Convexity