Deriving the Pareto Efficiency Curve
Consider an economic scenario where an individual's production of goods (g) is determined by their hours of free time (t). The total available time is 48 hours, so hours worked (h) is defined as h = 48 - t. The production function is given by g = (48h - h^2)/40. The individual's preferences over goods (g) and free time (t) are represented by a Cobb-Douglas utility function: U(t, g) = t^α * g^(1-α). Derive the equation for the Pareto efficiency curve, which expresses the relationship between g and t for all Pareto-efficient allocations. Your final answer should be an equation for g in terms of t and α.
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CORE Econ
Introduction to Microeconomics Course
The Economy 2.0 Microeconomics @ CORE Econ
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Specific Parameter for Cobb-Douglas Utility in Pareto Efficiency Example (α = 8/13)
Figure E5.7 - Indifference Curves and Feasible Frontier for Cobb-Douglas Preferences
Characteristics of the Pareto Efficiency Curve in the Cobb-Douglas Example
Deriving the Pareto Efficiency Curve
In an economic model, an individual's preferences for grain (g) and hours of free time (t) are represented by a Cobb-Douglas utility function. The amount of grain produced depends on hours of work (h), where h = 24 - t. The production function is given by g = (48h - h^2)/40. Which of the following conditions must be met to identify the set of all Pareto-efficient allocations of grain and free time?
In an economic model where an individual's utility is represented by a Cobb-Douglas function for grain (g) and hours of free time (t), and the production of grain is determined by the function g = (48h - h^2)/40, with h being the hours of work (h = 24 - t), the marginal rate of transformation (MRT) between free time and grain is directly proportional to the hours of free time.
Evaluating the Efficiency of an Allocation
You are tasked with finding the set of Pareto-efficient allocations in an economic model. The production of a good (g) is determined by the hours of work (h) according to the function g = (48h - h^2)/40. An individual's utility depends on their consumption of the good (g) and their hours of free time (t), where t = 24 - h. Their preferences are represented by a Cobb-Douglas utility function. Arrange the following steps in the correct logical order to derive the equation for the Pareto efficiency curve.
Impact of Utility Function Form on Pareto Efficiency
In an economic model, an individual's utility is a function of their consumption of grain (g) and hours of free time (t). The production of grain is determined by the hours of work (h), where h = 24 - t, according to the production function g = (48h - h^2)/40. Match each economic concept to its correct mathematical representation or definition within this specific model.
In an economic model, the production of grain (g) is described by the function g = (48h - h^2)/40, where h is the number of hours worked. The total available time per day is 24 hours, so free time is t = 24 - h. For any Pareto-efficient allocation, the Marginal Rate of Substitution (MRS) between grain and free time must equal the Marginal Rate of Transformation (MRT). At a Pareto-efficient point where 10 hours are dedicated to work, the value of both the MRS and MRT is ______. (Round your answer to one decimal place).
In an economic model, an individual's production of grain (g) is determined by their hours of work (h) according to the function g = (48h - h^2)/40. Their preferences for grain and free time (t = 24 - h) are represented by a Cobb-Douglas utility function. Consider an allocation where the individual's Marginal Rate of Substitution (MRS) of grain for free time is 1.5, and the Marginal Rate of Transformation (MRT) of free time into grain is 1.2. Which of the following describes a potential Pareto-improving change?
Evaluating Economic Efficiency of Time Allocation