Rejection of the Negative Root in the Utility Equivalence Simplification
When simplifying the equation $96t_C^2 = 150t_D^2 by taking the square root, two algebraic solutions arise. The solution $4t_C = -5t_D is rejected because it has no logical economic interpretation. Since free time () must be a positive quantity, a relationship where one positive value equals a negative one is impossible in this context.
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Rejection of the Negative Root in the Utility Equivalence Simplification
Equivalence of a $1,560 Unearned Income and a Wage Rise to $150 in Utility Terms (Figure E3.4)
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
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An economic model establishes a relationship between the quantities of two goods, Good A and Good B, represented by variables 'a' and 'b' respectively. The model's core equation is
16a^2 = 49b^2. A key assumption of the model is that 'a' and 'b' must be non-negative, as they represent physical quantities. When solving this equation for the relationship between 'a' and 'b', two mathematical solutions arise:4a = 7band4a = -7b. Which of the following statements provides the most accurate economic analysis of these solutions?Economic Interpretation of Mathematical Solutions
Evaluating a Junior Analyst's Model
An economic model describes the relationship between the quantity of corn produced (C) and the quantity of wheat produced (W) with the equation
100C^2 = 64W^2. A researcher, solving for the relationship between C and W, states that one possible valid economic outcome is10C = -8W. Is this statement a valid interpretation within the context of the economic model?Critique of a Mathematical Solution in an Economic Context
Justifying Model Simplification in Economics
An economic model relates the production of solar panels (S) and wind turbines (W) with the equation
25S² = 81W². Both S and W represent physical quantities and therefore must be non-negative. A student solves this and concludes that two possible relationships exist:5S = 9Wand5S = -9W. The student decides to include both relationships in their final report. What is the primary analytical error in the student's conclusion?An economist is analyzing models that relate two economic quantities, X and Y, which must be positive values. Match each derived mathematical relationship with its correct economic interpretation.
An economic model relates a firm's investment in technology (T) and its investment in labor (L), both of which must be positive values, using the equation
4T^2 = 9L^2. The only economically meaningful relationship that can be derived from this equation is2T = ___ L.An economist is modeling the relationship between two variables, capital (K) and labor (L), which must both be positive quantities. The model yields the equation
49K² = 144L². Arrange the following steps in the correct logical order to arrive at the single, economically meaningful conclusion from this equation.An economic model describes the relationship between the quantity of corn produced (C) and the quantity of wheat produced (W) with the equation
100C^2 = 64W^2. A researcher, solving for the relationship between C and W, states that one possible valid economic outcome is10C = -8W. Is this statement a valid interpretation within the context of the economic model?