Question

Bob’s utility function is u(x1, x2) = 3x1 + x2, and he has income m = 10. The price of good 2 is p2 = 1. Let p1 denote the initial price of good 1, and let p 0 1 denote a new lower price of good 1, so p 0 1 < p1. (a) For what values (if any) of p1 and p 0 1 is the substitution effect on good 1 equal to zero? (b) For what values (if any) of p1 and p 0 1 is the income effect on good 1 equal to zero?

Answer 1

At eqm, MRS = P1/P2

MRS = MU1/MU2 = 3, constant

So two goods are perfect Substitutes

corner solution, where at eqm only one good is consumed, based on price ratio

if MRS > P1/P2 , IC is steeper than budget constraint

so only X1 is consumed, X1*= m/P1, X2*= 0

if MRS < P1/P2,

X1= 0, X2 = m/P2

So demand curve of X1

X1 = { 0, if 3< P1, P1>3

= { m/P1, 3 > P1, P1<3

a) only income effect

before price change , if X1 is Consumed, then after price change, again X1 is consumed,

we don't shift to other good, due to lower prices, effective income rises and more of X1 is consumed while zero amount of good 2 is still consumed

so P1< 3 & P 01 < 3

B) only Substitution effect :

initially X2 is consumed, P1> 3, & price fall in a way that complete shift to good 1 . P01 < 3