An individual's optimal choice between two goods, X and Y, is described by the following pair of equations:

1. 4X + 3Y = 50
2. Y = 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:

1. Budget Constraint: 10x + 5y = 100
2. 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) = 100

Step 2: Solve for the optimal quantity of x (x*). 10x + 20x = 100 30x = 100 x* = 10/3

Step 3: Substitute the value of x* back into the budget constraint to find y*. 10(10/3) + 5y = 100

Step 4: Solve for the optimal quantity of y (y*). 100/3 + 5y = 100 5y = 100 + 100/3 5y = 400/3 y* = 80/3

Choosing an Efficient Back-Substitution Path

A consumer's optimal choice between goods x and y is 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* and y*.

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 is t* = 10. They have the following two original equations available to find the optimal amount of consumption (c*):

1. c + 20t = 400
2. c = 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*)?