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Implicit Differentiation in Economics
Implicit differentiation is a calculus technique often employed in economic models. Its method involves analyzing how one variable, such as consumption, would need to change in response to a small increase in another variable, like free time, in order to keep a third value, such as utility, constant.
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Social Science
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CORE Econ
Economics
Economy
Introduction to Microeconomics Course
The Economy 2.0 Microeconomics @ CORE Econ
Ch.3 Doing the best you can: Scarcity, wellbeing, and working hours - The Economy 2.0 Microeconomics @ CORE Econ
Related
Marginal Rate of Substitution (MRS)
Introduction to Mathematical Extensions for Economic Analysis
Variable Slope of a Curve
Marginal Utility
Implicit Differentiation in Economics
Composite Function Rule (Chain Rule)
Delta (Δ) Symbol
A coffee shop is considering staying open for an additional hour, from 4 PM to 5 PM. The table below shows their total revenue and total costs for different hours of operation. Based on an analysis of the change from the 4th to the 5th hour, what is the most logical decision and why?
Hours Open Total Revenue Total Cost 4 $200 $120 5 $240 $150 Production Decision at a Small Firm
Decision on Studying
A software company has invested $5 million in developing a new product. Due to unexpected competition, the projected future revenue from the product has dropped to $3 million. The company estimates it will cost an additional $1 million to complete the product. Based on a rational analysis of incremental changes, the company should abandon the project because the total cost ($6 million) will exceed the total revenue ($3 million).
Marginal vs. Average Analysis in Decision-Making
A rational decision-maker thinks at the margin by comparing the additional benefits and additional costs of a small change. For each scenario below, match it with the correct marginal comparison that should guide the decision.
Rational individuals make decisions at the margin. This means they will choose to take an action only if the additional, or marginal, benefit of the action is at least as great as the marginal ____.
A firm is considering whether to produce one additional unit of a good. Arrange the following steps in the logical order a rational decision-maker would follow when thinking at the margin.
Fertilizer Application Decision
A company's total cost of producing widgets is shown in the table below. What is the marginal cost of producing the 3rd widget?
Quantity of Widgets Total Cost 0 $10 1 $15 2 $22 3 $32 4 $45 Numerical Approximation of a Derivative
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Calculus-Based Methods for Analyzing Indifference Curves
A consumer's satisfaction from consuming two goods, Good X and Good Y, is described by the function U(X, Y) = X^(1/3) * Y^(2/3). If the consumer's level of satisfaction must remain constant, what expression represents the rate at which the consumption of Good Y must change for a one-unit increase in the consumption of Good X?
Production Input Substitution
Choosing the Right Analytical Method
Consider a firm's production process represented by a function where total output depends on the amounts of capital and labor used. The mathematical technique of implicit differentiation is correctly applied to determine how much total output will increase if the firm adds one more unit of labor while keeping the amount of capital unchanged.
A consumer's preferences are represented by a utility function U(X, Y), where X and Y are two different goods. To find the rate at which the consumer is willing to substitute Good Y for Good X while maintaining the same level of satisfaction, one must find the slope of the indifference curve (dY/dX). Arrange the following steps in the correct logical order to derive this slope using the appropriate calculus technique.
The Role of Implicit Differentiation in Economic Modeling
An economist is analyzing a consumer's choices between two goods, X and Y, using a utility function U(X, Y) = k, where k is a constant level of satisfaction. To find the rate at which the consumer is willing to trade Y for X (dY/dX) while keeping satisfaction constant, the economist uses a specific calculus technique. Match each mathematical component from this process to its correct economic interpretation or role.
A manufacturing firm's output is determined by the function
Q = K^(1/2) * L^(1/2), where Q is the total units produced, K is units of capital, and L is units of labor. The firm currently operates at a point where it uses 4 units of capital and 25 units of labor, keeping its output level constant. To maintain this constant output, the rate at which capital must change for a one-unit change in labor (dK/dL) at this specific point is ____.Evaluating a Policy Recommendation
A consumer's satisfaction from two goods, food (F) and clothing (C), is represented by a utility function where satisfaction is held constant along a curve. Consider two points on this curve: Point A, where the consumer has a large quantity of food and a small quantity of clothing, and Point B, where the consumer has a small quantity of food and a large quantity of clothing. Using the mathematical technique for finding the rate of substitution that keeps satisfaction constant, how does the consumer's willingness to give up food to obtain one additional unit of clothing compare between these two points?