Question

An MRS & Elasticity of substitution question. Consider the following utility function : U_a(x₁, x₂) = x₁² x₂²

(a) Calculate MRS(x₁, x₂)

(b) Find the Elasticity of Substitution

(c) Is the preference homothetic? Is the preference quasilinear?

(d) Sketch the indifference curve with x₁ on the horizontal and x₂ on the vertical axis

Answer 1

Solution:

U(x1,x2) = x1²x2²

a) Marginal rate of substitution, MRSx1,x2 = Marginal utility of x1 (MUx1)/ marginal utility of x2 (MUx2)

MUx1 = = 2*x1*x2²

MUx2 = = 2*x1²*x2

So, MRSx1,x2 = (2*x1*x2²)/(2*x1²*x2) = x2/x1

b) Elasticity of substitution is the percentage change in ratio of two goods demanded with respect to percentage change in their relative price. Also, at optimum, we know that MRSx1,x2 = relative price ratio = p1/p2 (where p1 is price of good x1 and p2 is price of good 2)

So, at optimum, we have x2/x1 = p1/p2

So, in relative terms, it is simply X = p (where X is good ratio and p is the price ratio)

Elasticity of substitution, e =

Since X = p, so, dX/dp = 1, this becomes: e = 1*(p/p) = 1

c) Preference is homothetic: if I increase both goods by factor t, new utility = (t*x1)²*(t*x2)² = t⁴(x1²x2²)

Since the given utility function is simply monotonic transformation, the preference is homothetic.

A quasilinear preference is one when MRS of the utility function is independent of any 1 good. Since here MRS depends in ratio of goods (that is, both goods come in its formulation) given preference is not quasilinear.

d) Following is the required graph: indifference curves are well defined has they follow a perfect 'convex to origin' shape.