In Problem 1, if Charlie's utility function were X⁶_AX_B, if apples cost 40 cents each, and if bananas cost 10 cents each, Charlie's budget line would be tangent to one of his indifference curves whenever A) X_B = X_A. B) 6X_B = 4X_A. C) X_B = 6X_A. D) X_A = 6X_B. E) 40X_A + 10X_B = M.

Question

Quizplus · Accepted Answer

The budget line can be represented as 40A + 10B = M, where M is Charlie's income. To find the point of tangency between the budget line and one of Charlie's indifference curves, we need to find the optimal combination of apples and bananas that maximizes his utility subject to the budget constraint. This can be represented as the following optimization problem:

Max XS1U1P16S1S1P0S1U1B1AS1U1B0XS1U1B1BS1U1B0
subject to 40A + 10B = M

Using Lagrange multipliers, we can form the following equation:

16S1S1P0XS1U1B1AS1U1B0 - S1U1B1BS1U1B0 = λ40
16S1S1P0XS1U1B1BS1U1B0 - S1U1B1AS1U1B0 = λ10
40A + 10B = M

Solving this system of equations, we get:

XS1U1B1AS1U1B0 = 6XS1U1B1BS1U1B0
40A + 10B = M

Substituting the first equation into the budget constraint, we get:

40A + 10(6XS1U1B1AS1U1B0) = M

Simplifying, we get:

4A + XS1U1B1AS1U1B0 = 0.25M

This is the equation of the budget line. The slope of the budget line is -4/XS1U1B1AS1U1B0, so the point of tangency occurs when:

-4/XS1U1B1AS1U1B0 = -40/10

Solving for XS1U1B1AS1U1B0, we get:

XS1U1B1AS1U1B0 = 6XS1U1B1BS1U1B0

Substituting this into the budget constraint, we get:

40A + 10(6XS1U1B1BS1U1B0) = M

Simplifying, we get:

4A + XS1U1B1BS1U1B0 = 0.25M

This is the equation of the budget line when it is tangent to one of Charlie's indifference curves. Thus, the answer is B.

In Problem 1, If Charlie's Utility Function Were X6AXB, If

In Problem 1, If Charlie's Utility Function Were X⁶_AX_B, If