Question

Please give detailed steps. Thank you.

3.4 Utility IV (Challenging) Consider an agent whose utility function u(x,X2- + x1x2 + expresses her a) Compute the marginal rate of substitution. b) Draw two or more of the agent's indifference curves

Answer 1

u = x1² + x1x2 + x2²

(a) MRS = MU1/MU2

MU1 = $\dpi{80} \partial$ u/$\dpi{80} \partial$x1 = 2x1 + x2

MU2 = $\dpi{80} \partial$ u/$\dpi{80} \partial$x2 = x1 + 2x2

MRS = (2x1 + x2) / (x1 + 2x2)

(b) Indifference curves are as follows. The farther away from origin the indifference curve is, the higher the total utility.

(c) Preference is convex if $\dpi{80} \partial$ ²u/$\dpi{80} \partial$x1² > 0 and $\dpi{80} \partial$ ²u/$\dpi{80} \partial$x2² > 0.

$\dpi{80} \partial$²u/$\dpi{80} \partial$x1² = $\dpi{80} \partial$ /$\dpi{80} \partial$x1($\dpi{80} \partial$u/$\dpi{80} \partial$x1) = $\dpi{80} \partial$ /$\dpi{80} \partial$x1(2x1 + x2) = 2 > 0

$\dpi{80} \partial$²u/$\dpi{80} \partial$x2² = $\dpi{80} \partial$ /$\dpi{80} \partial$x2($\dpi{80} \partial$u/$\dpi{80} \partial$x2) = $\dpi{80} \partial$ /$\dpi{80} \partial$x2(x1 + 2x2) = 2 > 0

Therefore preferences are convex.