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General Production Function of a Farmer (y=g(h))
A farmer's production technology is modeled by the function , where represents non-negative daily work hours and is the resulting grain output. This function is characterized by being increasing and strictly concave, properties which can be confirmed through algebraic verification. It is also assumed that production is zero with zero work hours, so . For analytical purposes in economic models, this function is frequently expressed in a simple algebraic form.
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Ch.5 The rules of the game: Who gets what and why - The Economy 2.0 Microeconomics @ CORE Econ
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Increasing Nature of the Production Function
Fixed Inputs and Diminishing Marginal Product
General Production Function of a Farmer (y=g(h))
A production function describes the relationship between a variable input (X) and the resulting total output (Y), for non-negative values of X. For this relationship to be economically plausible, it must satisfy two key conditions: 1) Zero input results in zero output, and any positive input yields a positive output. 2) The function must be consistently increasing, meaning more input always leads to more output. Based on these conditions, which of the following mathematical expressions could NOT represent a plausible production function?
Plausibility of a Production Model
A relationship between a single input (X) and total output (Y) must meet two core conditions to be considered an economically plausible production model: 1) Output is zero if input is zero, and positive for any positive input. 2) Output consistently increases as input increases. Analyze each mathematical function below (assuming X ≥ 0) and match it to the description that correctly explains its plausibility.
Plausibility of a Proposed Production Function
Consider a production process where the relationship between a single input (X, where X ≥ 0) and total output (Y) is described by the equation Y = -X² + 10X. This equation represents a plausible production function for all positive values of input X because it shows that adding more input initially leads to a significant increase in output.
Evaluating a Production Model's Plausibility
Economic Rationale for Production Model Properties
Critique of a Proposed Production Model
A production process is described by the relationship between a single input (X) and the total output (Y), where X ≥ 0. For this relationship to be considered a viable model of production, it must satisfy two fundamental properties: 1) no input produces no output, and any positive amount of input produces a positive amount of output; 2) the total output must consistently rise as more input is used. Given these properties, which of the following equations represents a plausible production function for all positive values of the input?
Critiquing and Correcting a Production Model
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Parameter Values for the Algebraic Form of a Farmer's Production Function
Algebraic Verification of the Properties of the Production Function g(h)
A Specific Concave Production Function (y = 10h^0.4)
Analysis of a Farmer's Production Data
A farmer's production of grain (y) is described by the function y = g(h), where h is the number of hours worked per day. This function is known to be increasing and strictly concave for all h > 0. Which of the following statements must be true?
A farmer's production of grain (y) is modeled by the function y = g(h), where h represents non-negative daily work hours. Match each mathematical property of this function to its correct economic interpretation.
Interpreting Production Function Properties
Data Center's Environmental Impact
A factory's production process releases pollutants into a river, harming a downstream fishing business. To address this, a regulator considers two options, both designed to reduce the factory's output to the efficient level: Policy A requires the factory to pay the fishing business an amount equal to the damages caused. Policy B imposes a tax on the factory equal to the damages caused, with the revenue going to the government. Which statement best analyzes the financial outcomes for the fishing business under these two policies?
A farmer's production technology, which relates daily work hours (
h) to grain output (y), is described by a functiony = g(h). This function is known to be increasing for allh ≥ 0, strictly concave for allh > 0, and to haveg(0) = 0. Based on these properties, which of the following algebraic forms could plausibly represent this production function?A farmer's production of grain (y) is modeled by a function y = g(h), where h represents non-negative daily work hours. This function is known to be increasing and strictly concave for all h > 0. Based on these properties, the following statement is true: 'The amount of additional grain produced by working the tenth hour is greater than the amount of additional grain produced by working the first hour.'
Comparing Production Scenarios
Evaluating a Production Strategy