Solving Constrained Choice Problems via Substitution
One mathematical approach to solving a constrained choice problem is the substitution method. This process involves using the constraint equation to express one variable in terms of the other (e.g., expressing consumption as a function of free time ). This expression is then substituted into the utility function, transforming it from a function of two variables, , into a function of a single variable, . This simplification makes the optimization problem solvable using single-variable calculus.
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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
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Solving Constrained Choice Problems via Substitution
Marginal Rate of Transformation (MRT) as the Wage Rate (w)
Calculating an Optimal Consumption Bundle
Optimal Consumption Bundle Calculation
A consumer is choosing between two goods, food (F) and clothing (C). At their current consumption bundle, their marginal rate of substitution of food for clothing (MRS_FC) is 4. The price of food is $10 per unit, and the price of clothing is $2 per unit. To maximize their utility, this consumer should:
A consumer is choosing between apples (A) and bananas (B). At their current consumption bundle on their budget line, their marginal rate of substitution of apples for bananas (MRS_AB) is 3. The price of an apple is $2 and the price of a banana is $1. This consumer is currently maximizing their utility.
A consumer aims to maximize their satisfaction from consuming two goods, X and Y, subject to a limited income. Arrange the following mathematical steps in the correct logical order to find the consumer's optimal consumption bundle using the tangency condition.
A consumer seeks to maximize their satisfaction from consuming two goods, X and Y, given their prices (Px, Py) and income (I). Match each mathematical expression with its corresponding economic concept in this constrained choice problem.
Evaluating Solution Methods for Consumer Optimization
For an interior solution to a consumer's utility maximization problem, the optimal consumption bundle is found on the budget line at the point where the marginal rate of substitution is equal to the ratio of the two goods' ____.
Setting Up a Utility Maximization Problem via Substitution
A student is solving for a consumer's optimal bundle of two goods, X and Y. The consumer's utility is given by U(X,Y) = XY, the price of X is $2, the price of Y is $4, and their income is $80. The student's work is shown below:
- Budget Constraint: 2X + 4Y = 80
- Marginal Rate of Substitution (MRS): Y/X
- Condition for Optimum: Y/X = 80
Which statement best identifies the conceptual error in the student's approach?
A consumer is choosing between apples (A) and bananas (B). At their current consumption bundle on their budget line, their marginal rate of substitution of apples for bananas (MRS_AB) is 3. The price of an apple is $2 and the price of a banana is $1. This consumer is currently maximizing their utility.
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Finding an Optimum for a Single-Variable Function using First and Second-Order Conditions
An individual's preferences for consumption (c) and free time (t) are represented by the utility function u(t, c) = t * c. The individual has 24 hours available per day, which can be divided between work and free time. They earn a wage of $20 per hour and spend all their income on consumption. By using the substitution method to incorporate the time and income constraint, what is the resulting utility function expressed solely in terms of free time (t)?
Logic of the Substitution Method in Constrained Choice
An individual aims to find the optimal consumption bundle that maximizes their utility, subject to a budget constraint. To achieve this, they use the substitution method to convert the two-variable optimization problem into a single-variable problem. Arrange the following steps in the correct logical order to execute this method.
Analyzing a Flawed Optimization Setup
When using the substitution method to solve a constrained choice problem, the solution that maximizes the transformed single-variable utility function must be checked separately to ensure it also satisfies the original constraint equation.
Evaluating the Substitution Method in Constrained Choice
To solve a constrained choice problem, a utility function of two variables,
u(t, c), can be transformed into a function of a single variable by substituting the constraint into the utility function. For each combination of a utility function and a constraint provided, match it to the correctly substituted utility function that is expressed solely in terms of free time,t.An individual's utility is derived from consumption (c) and free time (t), represented by the function u(t, c) = 4t + c. The individual's choices are limited by a constraint, which can be expressed as c = 10(24 - t). To find the optimal choice, the first step is to substitute the constraint into the utility function. This transforms the utility function into an expression of a single variable, t. The transformed utility function is u(t) = _________.
Setting Up a Constrained Choice Problem for Substitution
Interpreting a Transformed Utility Function