Solving for the Second Optimal Variable via Back-Substitution
After the optimal value for the first choice variable (e.g., free time, ) has been determined, its value is substituted back into one of the original simultaneous equations, such as the budget constraint. This step, known as back-substitution, allows for the algebraic calculation of the optimal value for the second choice variable (e.g., consumption, ).
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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?
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An individual's optimal choice between two goods, X and Y, is described by the following pair of equations:
4X + 3Y = 50Y = 2X
After solving this system, the optimal quantity for good X was found to be
X* = 5. What is the optimal quantity for good Y (Y*)?Calculating Optimal Consumption
Calculating Optimal Consumption via Back-Substitution
Calculating Optimal Consumption via Back-Substitution
A student is solving for an optimal choice between two goods, x and y. They have correctly determined the following system of equations:
- Budget Constraint:
10x + 5y = 100 - Optimality Condition:
y = 4x
The student's work to find the optimal quantities (x*, y*) is shown below. In which step does the first error occur?
Step 1: Substitute the optimality condition into the budget constraint.
10x + 5(4x) = 100Step 2: Solve for the optimal quantity of x (x*).
10x + 20x = 10030x = 100x* = 10/3Step 3: Substitute the value of x* back into the budget constraint to find y*.
10(10/3) + 5y = 100Step 4: Solve for the optimal quantity of y (y*).
100/3 + 5y = 1005y = 100 + 100/35y = 400/3y* = 80/3- Budget Constraint:
Choosing an Efficient Back-Substitution Path
A consumer's optimal choice between goods
xandyis determined by a system of two equations: a budget constraint and an optimality condition. Arrange the following steps in the correct logical order to solve for the optimal quantities of both goods,x*andy*.A student is determining their optimal choice between consumption (
c) and leisure time (t). After setting up and solving part of the problem, they have correctly found that the optimal amount of leisure ist* = 10. They have the following two original equations available to find the optimal amount of consumption (c*):c + 20t = 400c = 20t
Which statement best evaluates the next step to find
c*?A consumer's optimal choice is found by solving a system of two equations: a budget constraint and an optimality condition. Once the optimal quantity of the first good has been determined, its value must be substituted back into the original budget constraint to find the optimal quantity of the second good. Using the optimality condition for this substitution step will yield an incorrect result.
A student is solving for a consumer's optimal bundle of two goods. They have already used the optimality condition and the budget constraint to find the optimal quantity of the first good,
x*. They then substitute this value,x*, back into the budget constraint equation to solve for the optimal quantity of the second good,y*. What does this final step of substituting back into the budget constraint specifically ensure about the resulting optimal bundle (x*,y*)?