Answer to Question #265189 in Microeconomics for umair

Question

1. utility function u = x1^1/3 x2^2/3. and income = $60.

a) Solve for MU, and MU2 and use these to determine the MRS. Now use the tangency condition MRS = -p1/p2 together with budget line to solve for the demand functions for x1 and x2 for this consumer.

b) Initially we have p1 = 2 and p2 = 1, but then p1 falls to 1. Use your demands to solve for points A and C (the optimal points pre and post price change). Show these points on a clear well-labelled graph

c) Now determine the Slutsky demand by computing the income that would make point A just affordable with the new prices. Plug this hypothetical income and the new prices into your demands to solve for point B, as done in class. Show both the hypothetical budget line and point B on either your graph in a) or a new graph. Show the substitution and income effects on your graph, and compute them.

d) Graphically, do the same analysis using the Hicks decomposition method ( show it on a graph). Show the income and substitution effects on your graph.

Answer 1

(a)

$U=x_1^\frac{1}{3} x_2^\frac{2}{3}$

Income=$60.

Budget constraint: $P_1x_1+P_2x_2\le m$

m= Income.

$P_1,P_2-$ market prices of the goods.

Maximize $U(x_1^\frac{1}{3} x_2^\frac{2}{3})$

s.t $P_1x_1+P_2x_2\le m$

At the optimal bundle:

$(x_1^*,x_2^*), MRS=-\frac{P_1}{P_2}$

where $MRS=\frac{MU_1}{MU_2}$

Get the marginal utility functions respectively:

Given $U(x_1,x_2)=x_1,x_2$

$MU_1=\frac {∆U}{∆x_1}=x_2$

$MU_2=\frac {∆U}{∆x_2}=x_1$

Then $MRS=-\frac{MU_1}{MU_2}=-\frac{x_2}{x_1}$

To fulfill tangency condition,

$-\frac{x_2}{x_1}=\frac{P_1}{P_2}$

(b)

At the optimal bundle $(x_2^*,x_1^*)$ , to fulfill budget exhaustion condition,

$P_1x_1^*+P_2x_2^*=m$

$x_2^*=(\frac{P_1}{P_2})x_1^*$

Optimal level for good 1:

$P_1x_1^*+P_2(\frac{P_1}{P_2})x_1^*=m$

$P_1x_1^*+P_1x_1^*=m$

$2P_1x_1^*=m$

$x_1^*=\frac{m}{2P_1}$

Optimal level for good 2:

$x_2^*=(\frac{P_1}{P_2})x_1^*$