Characteristics of the Pareto Efficiency Curve in the Cobb-Douglas Example
In the Angela-Bruno model, when a Cobb-Douglas utility function is assumed, the resulting Pareto efficiency curve has specific properties. Mathematically, it is an upward-sloping quadratic function of free time () that passes through the origin. Economically, every point along this curve satisfies the Pareto-efficiency condition where the slope of the indifference curve (MRS) equals the slope of the feasible frontier (MRT). Points on this curve that are below the feasible frontier represent efficient allocations where the total output is divided between Angela and Bruno.
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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
Learn After
Figure E5.8 - The Pareto Efficiency Curve for Cobb–Douglas Preferences
Point P1 as an Endpoint on the Cobb-Douglas Pareto Efficiency Curve
Point P2 as a Shared-Surplus Allocation on the Cobb-Douglas Pareto Efficiency Curve
Point P0 as an Endpoint on the Cobb-Douglas Pareto Efficiency Curve
Consider a situation where a farmer's output depends on her free time. The total output is then divided between the farmer and a second party. At a specific allocation, the farmer's marginal rate of substitution (the rate she is willing to trade free time for goods) is greater than the marginal rate of transformation (the rate at which free time can be technologically converted into goods). What can be concluded about this allocation?
Assessing Allocative Efficiency
Evaluating an Economic Allocation
In a two-person model where total output is a function of one person's labor, consider the set of all efficient allocations. If an allocation is changed to give the laborer more free time, their own consumption of the output must necessarily decrease for the new allocation to also be efficient.
Consider a scenario where an individual's labor produces a good, and the total output is divided between this individual and another party. The set of all efficient allocations is represented by a curve showing the individual's consumption of the good (c) for each level of their free time (t). If the individual's preferences are of the Cobb-Douglas type, which of the following equations best describes the shape of this efficiency curve?
Analyzing the Properties of Efficient Allocations
Interpreting Economic Efficiency
In a model where one person's labor generates an output that is then divided between them and another party, the set of all efficient allocations is shown on a graph with the laborer's free time on the x-axis and their consumption on the y-axis. Match each described point or condition on the graph with its correct economic interpretation.
Deriving Conditions for Economic Efficiency
Evaluating Economic Efficiency of Allocations