Differentiating the Feasible Frontier Using the Chain Rule
To determine the slope of the feasible frontier, it is necessary to differentiate the frontier's equation with respect to free time, . Because the feasible frontier is expressed as a composite function, such as , this mathematical operation requires the use of the composite function rule, which is also commonly referred to as the chain rule in calculus.
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A farmer's grain output (
y) is determined by the number of hours they work (h) according to the production functiony = 8√h. The farmer has 24 hours per day to allocate between work (h) and free time (t). Which of the following equations correctly represents this farmer's feasible frontier, which shows the maximum possible output for any given amount of free time?True or False: If a production technology shows constant returns to labor (meaning each additional hour of work adds the same amount to total output), the corresponding feasible frontier relating output to free time will be a straight line.
The Shape of the Feasible Frontier
Impact of Technological Improvement on Production Possibilities
An individual has 24 hours per day to divide between work (
h) and free time (t). Their output (y) is determined by a production technology that relates output to hours worked. Match each production technology on the left with its corresponding feasible frontier equation on the right, which expresses output as a function of free time.Analyzing the Link Between Production and Feasible Choices
An individual's daily output of goods (
y) is determined by the number of hours they work (h), according to the functiony = 10 * h^(1/2). The individual has 24 hours available per day, which they can divide between work and free time. If they decide to have 8 hours of free time, the maximum output they can produce is ____.You are given a production function that describes the relationship between an individual's hours of work (
h) and their total output (y). You are also told that the individual has a total of 24 hours per day to allocate between work and free time (t). Arrange the following steps in the correct logical order to derive the feasible frontier, which shows the maximum output for any given amount of free time.An individual's feasible frontier, showing the relationship between free time and maximum output, is a straight line with a negative slope. Assuming the individual values free time and consumption, what does this imply about the relationship between work hours and output?
An economist is studying two self-sufficient farmers, Farmer A and Farmer B. Each farmer has 16 hours per day to allocate between work (
h) and free time (t). Farmer A's production of grain (y) is given by the functiony = 4h. Farmer B's production is given byy = 12√h. Which of the following statements accurately compares the feasible frontiers (the relationship between free time and maximum grain output) for the two farmers?
Learn After
Verification of Feasible Frontier Properties using Differentiation
MRT as the Marginal Product of Labor
A student's final grade (G) is a function of the hours they study (h), represented by the equation G = 10√h. The hours of study are determined by the hours of free time (t) in a 24-hour day, where h = 24 - t. What is the derivative of the grade with respect to free time (dG/dt)?
Calculating and Interpreting the Rate of Transformation
A student's final grade (G) in a course is a function of the hours they study (h), represented by the equation G = 10√h. The hours of study are determined by the hours of free time (t) in a 24-hour day, where h = 24 - t. To find the rate at which the grade changes with respect to free time (dG/dt), one must break the problem into parts. Match each component of this calculation with its correct mathematical expression.
A student's grade (G) in a course is determined by their hours of study (h), according to the function G = 8√h. The hours they can study are constrained by the 24 hours in a day, such that h = 24 - t, where 't' is hours of free time. To find the rate at which the grade changes for each additional hour of free time (dG/dt), you must use the rule for differentiating a composite function. Arrange the following steps into the correct logical sequence for finding this rate.
Farmer's Production and Leisure Trade-off
A student's potential exam score (y) is a function of the hours they study (h), given by the equation y = 5h². The hours they study are constrained by the 24 hours in a day, such that h = 24 - t, where t represents hours of free time. A student incorrectly claims that the rate of change of the score with respect to free time (dy/dt) is 10h.
Rationale for a Differentiation Method in an Economic Model
A student's potential exam score (S) is a function of the hours they study (h), represented by the equation S = f(h). The hours they study are constrained by the 24 hours in a day, such that h = 24 - t, where t represents hours of free time. The rate at which the score changes with respect to free time (dS/dt) can be found by multiplying the rate of change of the score with respect to study hours (dS/dh) by the rate of change of study hours with respect to free time (dh/dt). Given this information, the numerical value for dh/dt is ____.
Analyzing a Manager's Profit Calculation
A firm's daily revenue (R) is a function of the number of widgets (q) it sells, described by the function R = f(q). The number of widgets sold is, in turn, a function of the daily advertising budget (a), described by q = g(a). To understand the impact of advertising on revenue, the firm needs to find the rate of change of revenue with respect to the advertising budget (dR/da). Which of the following statements correctly describes how to calculate this rate of change by breaking it down into its intermediate components?