Question

Consider a consumer with utility function u(x1, x2) = α_1x_1^( 2) + α_2x_2^( 2) where α1 > 0 and α2 > 0. Assume that p1, p2 > 0. (a) Show that the utility function represents strongly monotone preferences (b) Draw indifference curves passing through points (1, 2), (3, 3) and (0, 3). What properties of the preference relation can you derive from these indifference curves? (c) State the expenditure minimization problem and derive the Hicksian demand. Does the EMP problem have a unique solution at every price vector p >> 0? (d) Derive expenditure function e(p, u). Verify that it is homogeneous of degree 1 in p and increasing in u. (e) Using expenditure function and Hicksian demand, calculate Walrasian demand and indirect utility 

Answer 1

$Given U=α_1X_1^2+α_2X_2^2\\ Where\\ α_1>0 \space and\space α_2>0$

Part a)

For Strictly monotone preference shows that consumer consumes more of one good and marginal utility of the good is positive. and total utility is increasing.

$U=α_1X_1^2+α_2X_2^2$

$\frac{∂U}{ ∂X_1} =2α_1X_1$ −−−− It is positive When α1>0

$\frac{∂U}{ ∂X_2} =2α_2X_2$ −−−− It is positive When α2>0

That means consumer is ready to consume more of the 2 producs as it's

 marginal productivity is positive.

For Strictly monotone preference we need to prove that MRS should be

 positive as it shows that indifference curve is downward sloping

It  shows that utility function is showing increasing monotone prefernce.

$MRS=\frac{MUX_1}{MUX_2}\\MRS=\frac{2α_1X_1}{2α_2X_2}−−−−− When α1>0 and α2>0$

Part b)

Given Quantity of X1and X2=(1,2),(3,3),(0,3)

We will substitute values in utility function

If it gives same utility means consumer is indifferent between all combination but if it gives different utility then consumer has preference relations.

$U=α_1X_1^2+α_2X^2\\ suppose \\α_1=0.5 \\and\\ α_2=0.6\\Then\space For\space X_1=1\space and\space X_2=2\\U=0.5\times (1)^2+0.6(2)^2\\U=2.9\\Then \space For\space X_1=3 \space and\space X2=3\\U=0.5\times (3)^2+0.6(3)^2\\U=9.9\\Then \space For\space X_1=0\space and\space X_2=3\\U=0.5\times (0)^2+0.6(3)^2\\U=5.4$

We can observe that all the point are giving different utility so consumer is not indifferent between these indifference curves

We can observe that IC₁ and IC₃ are convex to the origin and and it fulfills the highest indifference curve shows higher utility.

IC₂ has the corner solution 

IC₃>IC₂ >IC1 is the relation of preference among the indifference curves.

Part c)

 Minmization of expenditure

Here constraint is constant utility

$U=α_1X_1^2+α_2X_2^2$

$Min E=P_1X_1+P_2X_2+μ(U−α_1X_1^2−α_2X_2^2)\\\frac{∂E}{∂X_1}=P_1−μ_2α_1X_1−−−−(1)\\\frac{∂E}{∂X_2}=P_2−μ_2α_2X_2−−−−(2)\\ using \space 1 \space and \space 2\\ X_1=\frac{α_2P_1X_2}{α_1P_2}−−−−−(3)$

By substituting X1 value in Constraint

$U=α_1(\frac{α_2P_1X_2}{α_1P_2})^2+α_2X_2^2\\X_2=\sqrt{\frac{U}{(\frac{α_2P_1}{α_1P_2})−α_2}}⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯−−−$  Hicksian demand curve for X₂

By substituting X2 into equation 3

$X_1= (\frac{α_2P_1}{α_1P_2})\sqrt{\frac{U}{(\frac{α_2P_1}{α_1P_2})−α_2}}⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯$ Hicksian demand curve for X₂−−Hicksian demand curve for X₁

We can conclude that Every price vector XI and X2 values will be unique solution.