Question

A consumer’s preferences are represented by the following utility function: u(x, y) = lnx + 1/2 lny

1. Recall that for any two bundles A and B the following equivalence then holds

A ≽ B ⇔ u(A) ≥ u (B)
Which of the two bundles (xA,yA) = (1,9) or (xB,yB) = (9,1) does the consumer prefer?

2. Take as given for now that this utility function represents a consumer with convex preferences. Also remember that preferences ≽ are convex when for any three bundles A,B, and C, if B ≽ A and C ≽ A then αB + (1 − α)C ≽ A for any α ∈ [0, 1].

   Use this information and your answer to part (1) to determine which of the two bundles (xC , yC ) = (5, 5) or (xB , yB ) = (9, 1) the consumer prefers. Verify your answer.

3. Derive an equation for the marginal rate of substitution. Interpret the MRS. What is the MRS at the point (xC,yC) = (5,5)?

4. Derive an equation for the indifference curve through the bundle (xB , yB ) = (9, 1)

Answer 1

1) U(1,9) = ln(1) + .5*ln(9) = .5*ln9

U(9,1)= ln(9)+.5*ln(1) = ln9

So higher utility is whith (9,1)

(XB,YB) is preferred

2) U(5,5) = 1.5*ln5 = 2.414 & is preferred over (9,1)

3) MRS = MUx/MUy = 2y/x

MRS imply that in order to increase x by 1 unit, & keeping Utility level unchanged, then MRS units of y should be given up.

at (5,5), MRS = 2

so to increase x by 1 unit, & keeping on same IC , 2 units of y should be given up.

4) U= ln9 + .5*ln1

lnx + .5lny = ln9

X*√Y = 9

Y = 81/X^{​​​​​​2}