Isolating a Choice Variable by Substituting the Budget Constraint into the Optimality Condition
A common method to solve the system of simultaneous equations for an optimal choice is substitution. This involves using the budget constraint (the second equation) to express one variable, such as consumption (), in terms of the other, like free time (). This expression is then substituted into the optimality condition equation (MRS = MRT), which is the first equation. This action reduces the system to a single equation with only one unknown variable (for example, optimal free time ), which can then be solved algebraically.
0
1
Tags
Science
Economy
CORE Econ
Social Science
Empirical Science
Economics
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
Isolating a Choice Variable by Substituting the Budget Constraint into the Optimality Condition
Expressing Optimal Utility in Terms of a Single Variable
Optimal Choice as a Function of Wage and Unearned Income
Calculating an Optimal Consumption Bundle
Optimal Consumption Choice Calculation
A consumer's preferences for two goods, X and Y, can be represented by the utility function U(X, Y) = X * Y. The price of good X is $2, the price of good Y is $4, and the consumer has an income of $80 to spend. To maximize their satisfaction subject to their budget, what combination of goods X and Y should the consumer purchase?
A consumer is considered to be making an optimal choice as long as their marginal rate of substitution between two goods is equal to the ratio of the prices of those goods, regardless of whether they spend their entire budget.
A consumer is considered to be making an optimal choice as long as their marginal rate of substitution between two goods is equal to the ratio of the prices of those goods, regardless of whether they spend their entire budget.
A consumer aims to maximize their satisfaction by choosing a bundle of two goods, given their income and the prices of the goods. To find the exact quantities of each good in the optimal bundle, a specific analytical procedure is followed. Arrange the steps of this procedure in the correct logical sequence.
Evaluating a Consumption Decision
The Two Conditions for an Optimal Choice
A consumer seeks to find their optimal consumption bundle of two goods, Good A and Good B. This involves setting up and solving a system of two equations. Match each conceptual component of this system to its correct description.
A consumer is choosing between two goods, pizza (P) and soda (S). At their current consumption bundle, their marginal rate of substitution of soda for pizza (MRS_sp) is 3. This means they are willing to give up 3 sodas to get one more pizza. The price of a pizza is $10 and the price of a soda is $2. Based on this information, what should the consumer do to increase their total satisfaction, assuming they stay within their budget?
Learn After
Solving for the Second Optimal Variable via Back-Substitution
A student is determining their optimal amount of consumption (c) and free time (t). Their choice must satisfy two conditions simultaneously:
- Optimality Condition: c / t = 15
- Budget Constraint: c = 15(24 - t)
To find the value of the single variable 't', the budget constraint must be substituted into the optimality condition. Which of the following equations correctly shows the result of this specific substitution?
An individual's optimal choice between consumption (c) and free time (t) is determined by a system of two related equations. To find the optimal value for free time (t), the system must be reduced to a single equation and solved. Given the initial system below, arrange the subsequent equations into the correct logical sequence that solves for
t.Initial System:
c = 10tc + 5t = 120
Solving for a Choice Variable via Substitution
Guiding a Decision-Making Process
When solving a consumer's choice problem, the standard procedure is to substitute the budget constraint into the optimality condition to solve for one variable. An alternative procedure of isolating a variable in the optimality condition and substituting it into the budget constraint would lead to an incorrect final optimal choice.
To find a single optimal variable, like free time (t), in a choice problem, an expression for the other variable, consumption (c), is taken from the budget constraint and substituted into the optimality condition. Match each system of equations (representing an optimality condition and a budget constraint) to the correct resulting equation that contains only the variable 't'.
The Logic of Substitution in Consumer Choice
Solving for an Optimal Variable with an Unarranged Constraint
A person is solving for their optimal mix of daily consumption (
c) and free time (t). They correctly combine their optimality condition (c = 20t) and their budget constraint (c = 10(24 - t)) into a single new equation:10(24 - t) = 20t. What is the primary analytical advantage of deriving this new equation?