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General Utility Function for Consumption and Free Time
Utility Function as a 3D Surface
A utility function that depends on two variables, such as consumption and free time (), can be graphically represented as a surface in a three-dimensional space. However, because 3D diagrams can be complex to interpret and manage, economists typically use a two-dimensional representation instead.
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Property of Utility Functions: More is Better
Utility Function as a 3D Surface
Calculating Marginal Utilities Using Partial Differentiation
An individual's preferences for consumption (represented by
c) and free time (represented byt) are captured by the utility functionU(t, c) = t * c. Which of the following combinations of free time and consumption provides the individual with the highest level of utility?Interpreting a Utility Function
An individual's preferences are modeled using the general utility function
U(t, c), where preferences depend on free time (t) and consumption (c). Match each component of this model to its correct description.In a model where an individual's preferences are described by the function U(t, c), where 't' is free time and 'c' is consumption, the value of U represents the combined monetary cost of the consumption and the free time.
Modeling Preferences with Utility Functions
In a model of preferences for consumption (
c) and free time (t), the functionU(t, c)assigns a numerical value to each combination of these two goods. This numerical value represents the individual's level of __________.The Role of the Utility Function in Economic Modeling
An individual uses a utility function,
U(t, c), to model their preferences between different combinations of free time (t) and consumption (c). Arrange the following steps in the logical order required to use this function to compare two different combinations.An economist is modeling the preferences of two individuals for consumption (
c) and free time (t). Individual A's preferences are represented by the functionU_A(t, c) = t + c, while Individual B's preferences are represented byU_B(t, c) = t * c. What key difference in their underlying preferences do these two functional forms reveal?An individual describes their preferences for free time (
t, measured in hours) and consumption (c, measured in dollars) as follows: 'I only derive satisfaction when I have them in a specific combination. For every 1 hour of free time I enjoy, I must have $20 worth of consumption. Having more of one without a proportional amount of the other adds no extra utility to me.' Which of the following utility functions,U(t, c), best captures this individual's preferences?An individual's preferences for consumption (represented by
c) and free time (represented byt) are captured by the utility functionU(t, c) = t * c. Which of the following combinations of free time and consumption provides the individual with the highest level of utility?Interpreting a Utility Function
In a model of individual choice, the function U(t, c) is used to calculate the exact monetary value an individual places on their combined satisfaction from free time (t) and consumption (c).
In a model of individual choice, preferences are represented by the function
U(t, c). Match each component of this model to its correct description.Analyzing the Structure of a Utility Function
In a model where an individual's preferences are described by the function U(t, c), the variables
tandcrepresent the quantities of free time and consumption, respectively. The output of this function,U, represents the individual's level of ________.Modeling Preferences with a Utility Function
A microeconomic model is built to represent an individual's preferences for two desirable items: consumption and free time. Arrange the following statements into the logical order that describes how these preferences are formally represented as a mathematical relationship.
An economist proposes the function
U(t, c) = 2t + 5cto model an individual's preferences for free time (t) and consumption (c). A critic argues this function is a poor representation of how most people value these two items. Which of the following statements best supports the critic's argument?Consider an individual whose satisfaction depends solely on their level of consumption (
c) and is completely unaffected by the amount of free time (t) they have. The functionU(t, c) = t + caccurately represents this individual's preferences.
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Indifference Curves as Contours of a Utility Surface
Imagine a three-dimensional graph where the two horizontal axes represent the quantity of daily free time and the quantity of goods consumed, respectively. The vertical axis represents the level of satisfaction, or utility, derived from different combinations of free time and consumption. If Point A on this 3D surface is located directly above the combination (18 hours free time, $50 consumption) and is vertically higher than Point B, which is located directly above the combination (16 hours free time, $60 consumption), what can be concluded?
Analyzing Trade-offs on a Utility Surface
Interpreting the Shape of a Utility Surface
On a three-dimensional graph where two horizontal axes represent quantities of two different goods and the vertical axis represents the level of satisfaction (utility), any two points that lie on the same horizontal plane must represent combinations of the two goods that provide the consumer with an identical level of satisfaction.
In the context of a three-dimensional representation of a utility function with two goods, match each geometric component or movement with its correct economic interpretation.
Evaluating the 3D Representation of Utility
Interpreting a Cross-Section of a Utility Surface
Consider a three-dimensional graph where the two horizontal axes represent quantities of two different goods (Good X and Good Y), and the vertical axis represents the level of satisfaction (utility). If a consumer's preferences follow the principle that 'more is always better' for both goods, what must be true about the shape of this 3D utility surface?
Limitations of 3D Utility Models
In a three-dimensional graphical representation where the two horizontal axes represent the quantities of two different goods a person might consume, the vertical height of the surface at any given point represents the level of __________ achieved from that combination of goods.