Question

1. Consider the marginal rate of substitution (MRS) as it applies to Cletus, who consumes the 2 goods x₁ and x₂. Assume in your answers below that MRS is always defined to be a negative number (given that both x₁ and x₂ are goods), and that x₁ is on the horizontal axis and x₂ is on the vertical axis.

1. (4 points) How is the MRS defined, mathematically, in terms of the utility Cletus receives from x₁ and x₂? Be sure to explain, in words, the context in which this mathematical expression applies; that is, for example, if something is a slope, what exactly is it the slope of?
2. (4 points) When we talk about there being diminishing MRS, what exactly does that concept entail? Be as specific as possible.
3. (8 points) If Cletus’s utility function is U = x₁^0.5x₂^0.5, calculate his MRS. Is his MRS diminishing?
4. (8 points) If Cletus’s utility function is U = 3x₁ + 2x₂ calculate his MRS. Is his MRS diminishing?
5. (14 points) Suppose Cletus has the utility function U = x₁^0.5x₂^0.5 (as in part (c)). Suppose also that he faces a typical and simple budget line based on given values of P₁ and P₂ (the prices of x₁ and x₂, respectively) and of I (nominal income). Is it possible that he maximizes utility at a corner solution (where he chooses either x₁* = 0 or x₂* = 0)? Is it possible that he maximizes utility at an interior solution (where x₁* > 0 and x₂* > 0)?
6. (14 points) Suppose Cletus has the utility function U = 3x₁ + 2x₂ (as in part (d)). Suppose also that he faces a typical and simple budget line based on given values of P₁ and P₂ (the prices of x₁ and x₂, respectively) and of I (nominal income). Is it possible that he maximizes utility at a corner solution (where he chooses either x₁* = 0 or x₂* = 0)? Is it possible that he maximizes utility at an interior solution (where x₁* > 0 and x₂* > 0)?
7. (8 points) How does MU_x1/P₁ compare with MU_x2/P₂ in (e)? In (f)?

[Note: MU_x1 is the marginal utility of x₁ and MU_x2 is the marginal utility of x₂.]

Answer 1