Question

Let X = (−∞,∞) × R L + and assume that the preferences are strictly convex (so that Walrasian demand is signle-valued) and quasilinear. Normalize price of good 1 to be p1 = 1 (a) Show that the Walrasian demand functions for goods 2, . . . , L are independent of wealth w. [Hint: you have to show that when x(p, w) is Walrasian demand at prices p and wealth w, x(p, w) + αe1 is the Walrasian demand at prices p and wealth w + α for any α > −w]. (b) Argue that the indirect utility can be written in the form v(p, w) = w + γ(p) for some function γ(·). [Hint: Recall from the previous assignment that quasilinear preferences can be represented by a utility function that takes the form u(x1, . . . , xL) = x1 + φ(x2, . . . , xL) for some function φ(·).

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.