Simplification of the Utility Equivalence Equation in Figure E3.4
The primary equation for utility equivalence, , is simplified to make it easier to solve. By taking the square root of both sides, the expression reduces to the economically meaningful relationship . This simplified form is then used to substitute the expressions for and to solve for the hypothetical income, J.
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Point C as the Hypothetical Choice in Figure E3.4
Simplification of the Utility Equivalence Equation in Figure E3.4
An individual's daily well-being is modeled by the expression
w × t², wherewis their hourly wage andtis the hours of free time they choose. Initially, their wage is $20 per hour. After a promotion, their wage increases to $25 per hour, and they are observed to choose 16 hours of free time per day. An economist wants to determine the level of well-being at the new wage and then find the hypothetical choice of free time (t_H) that would provide this same level of well-being if the indiFormulating a Utility Equivalence Equation
Calculating Hypothetical Free Time for Utility Equivalence
An individual's satisfaction from their daily choices is modeled by the expression
w × t², wherewis their hourly wage andtis the hours of free time they enjoy. This individual's wage increases from $50/hour to $72/hour. At the new, higher wage, they are observed to choose 10 hours of free time. To find the hypothetical amount of free time (t_H) that would provide this same level of satisfaction if they were still earning their original wage, the following equation must be solved: `50An analyst is studying a person whose daily satisfaction is represented by the formula
w × t², wherewis the hourly wage andtis hours of free time. The person's wage increases from $49/hour to $64/hour, and at the new wage, they choose 7 hours of free time. The analyst states that to find the hypothetical amount of free time (t_H) that would have provided this same level of satisfaction at the original wage, one must solve the equation:$64 × t_H² = $49 × 7². Is the analyst's statAn economist is studying how a wage change affects an individual's choice of free time. The individual's daily satisfaction is modeled by the expression
U = w × t², wherewis the hourly wage andtis hours of free time. The wage increases fromw_initialtow_final, and the individual is now observed choosingt_finalhours of free time. The economist sets up the following equation to solve for a hypothetical amount of free time,t_hypothetical:`w_initial × (t_hypothetical)² = w_f
An individual's daily satisfaction is modeled by the expression
U = w × t², whereUis the satisfaction level,wis the hourly wage, andtis hours of free time. This person's wage increases from $36/hour to $64/hour. At the new, higher wage, they choose to have 9 hours of free time. To analyze this change, an economist sets up the following equation to find a hypothetical choice of free time (t_H):$36 × t_H² = $64 × 9². Match each mathematical expression from the equation to the coAn economist wants to analyze the impact of a wage increase on an individual's choices. The individual's daily satisfaction is modeled by the expression
U = w × t², wherewis the hourly wage andtis hours of free time. The wage increases fromw_initialtow_final, and at the new wage, the individual choosest_finalhours of free time. The economist needs to find the hypothetical amount of free time (t_hypothetical) that would provide the same level of satisfaction as the finalEvaluating an Economic Interpretation
An economist is analyzing an individual's response to a wage change, where daily satisfaction is modeled by the expression
U = w × t²(w= hourly wage,t= hours of free time). After the individual's wage increased to $72/hour, they were observed to choose 10 hours of free time. The economist calculated that to achieve this same level of satisfaction at the original wage, the individual would have needed 12 hours of free time. Based on this analysis, what was the individual's original hoAn individual's satisfaction from their daily choices is modeled by the expression
w × t², wherewis their hourly wage andtis the hours of free time they enjoy. This individual's wage increases from $50/hour to $72/hour. At the new, higher wage, they are observed to choose 10 hours of free time. To find the hypothetical amount of free time (t_H) that would provide this same level of satisfaction if they were still earning their original wage, the following equation must be solved: `50
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Rejection of the Negative Root in the Utility Equivalence Simplification
In an economic model, two scenarios are found to provide equal utility to an individual when the equation
32X^2 = 50Y^2is satisfied. Here, X and Y represent consumption levels, which must be non-negative. To make this relationship easier to work with, it should be simplified to its most basic linear form. What is the correct simplified equation?Simplifying an Economic Relationship
In an economic model comparing two investment strategies, the point of equal value is described by the equation
48X^2 = 75Y^2, where X and Y represent positive investment amounts. To make the relationship easier to analyze, this equation is simplified to its most basic linear form. Complete the simplified equation:4X = ___.In an economic model comparing two choices, the point of indifference is represented by the equation
64X^2 = 100Y^2, where X and Y are positive values. A student simplifies this equation to the linear form32X = 50Y. This simplification is a valid and correct step for making the equation easier to work with.Analyzing a Flawed Economic Calculation
In various economic models, a state of equal value or utility between two positive quantities, X and Y, is often represented by a quadratic equation. To facilitate further analysis, these equations are simplified to a linear form. Match each quadratic equation to its correct simplified linear form.
Justifying a Mathematical Simplification in Economic Modeling
An economic model establishes that two choices provide equal value when
96X^2 = 150Y^2, where X and Y are positive quantities. To use this relationship in further calculations, it must be simplified to a linear form. Arrange the following steps into the correct logical sequence to perform this simplification.Critiquing an Economic Analysis
Evaluating Simplification Strategies
Equivalence of a $1,560 Unearned Income and a Wage Rise to $150 in Utility Terms (Figure E3.4)