Question

An individual's utility function is

U= (X+1)0.5 (Y+2)0.5

(a) By implicit differentiation, find the slope of any indifference curve and show that it is given by the ratio of marginal utilities of the two goods.

(b) Find the indifference curve for U= 6 and show that it is negatively sloped and convex. Does the indifference curve cut the axes? What does this imply about this individual's tastes?

Answer 1

a)

Now we have the slope of the indifference curve, we term as the marginal rate of substitution (MRS) as,

 

MRS = MUx/MUy

         = 0.5(X+1)-½(Y+1)½/0.5(X+1)½(Y+1)-½

 

Therefore we have,

 

MRS = (Y+1)/(X+1)

 

 

b)

Now we know that,

MRS = - dY/dX = (Y+1)/(X+1)

So, dY/dX = - (Y+1)/(X+1)

 

Hence is negatively sloped.

 

Now we have,

d²Y/dX² = (Y+1)/(X+1)² > 0

 

Hence is convex and here if X = 0 we have Y ≠0 and vice-versa which implies the fact that these two goods aren't complements.

a) MRS = (Y+1)/(X+1)

b) Hence is negatively sloped.