Question

Consider utility function u(x1,x2) =1/4x₁² +1/9x₂². Suppose the prices of good 1 and

good 2 are p1 andp2, and income is m.

1. Do bundles (2, 9) and (4, radical54) lie on the same indifference curve?
2. Evaluate the marginal rate of substitution at (x1,x2) = (8, 9).
3. Does this utility function represent convexpreferences?
4. Would bundle (x1,x2) satisfying (1) MU1/MU2 =p1/p2 and (2) p1x1 + p2x2 =m be an optimal choice? (hint: what does an indifference curve look like?)

Answer 1

The given utility function is

The prices are p1 and p2 of x1 and x2 respectively. The income is m.

(a) We know that, the utilities are same along an indifference curve. If we take any two points on an indifference curve, we will get the same utilities. Now for the given two points (2,9) and (4,√54), if we find that

u(2,9)=u(4,√54)

We can say that, the points lie on the same indifference curve.

Now, from the utility function, we get

Hence,

or, u(2,9) = (1/4)×4 + (1/9)×81

or, u(2,9) = 10

And,

or, u(4,√54) = (1/4)×16 + (1/9)×54

or, u(4,√54) = 10

Hence, the utilities are same at the two points.

Hence, the bundles (2,9) and (4,√54) lie on the same indifference curve.

(b) The Marginal Rate of Substitution or MRS is the ratio of Marginal Utilities of x1 and x2.

Hence, Marginal Utility of x1 is

MU1 = du/dx1 = (1/4).(2.x1)

or, MU1 = (1/2).x1........(1)

And, the Marginal Utility of x2 is

MU2 = du/dx2 = (1/9).(2.x2)

or, MU2 = (2/9).x2.........(2)

Hence, Marginal Rate of Substitution at (x1,x2) is

MRS = MU1/MU2

or, MRS = (9/4).(x1/x2).......(3)

Now, when (x1,x2) = (8,9)

Then, MRS = (9/4)×(8/9)

or, MRS = 2

Hence, marginal rate of substitution at (8,9) is 2.

(c) The preferences are convex, when the MRS or Marginal Rate of Substitution is diminishing as x1 increases. In other words, when

d(MRS)dx1 < 0

We will say that the utility function represents convex preferences.

Here, MRS = (9/4).(x1/x2)

Hence, dMRS/dx1 = (9/4).(1/x2)

Here, x2>0 and (9/4)>0. Hence,

dMRS/dx1 > 0

The MRS is not diminishing. MRS increases as x1 increases. Hence, this can not represent a conves preference.

Hence, the utility function does not represent convex preferences.

(d) A bundle (x1,x2) satisfying (1) MU1/MU2 =p1/p2 and (2) p1x1 + p2x2 =m will be optimal bundle, only when the preferences are convex. Here, the preferences are not convex. Let us look at the preference pattern in a diagram.

We can see that, at point E (x1,x2), MRS= p1/p2 and also p1x1+p2x2 = m.

But, the preferences are concave here. The consumer can afford point F which is on the budget set and it also given higher utility i.e. u2(>u1) to the consumer. Hence, he will prefer to consume at F. But, at point F

Which means that, the slope of the indifference curve is not equal to the slope of the budget line.

Hence, the statement is not TRUE.

The bundle (x1,x2) satisfying (1) MU1/MU2 =p1/p2 and (2) p1x1 + p2x2 =m, will not be an optimal choice.

Hope the solutions are clear to you my friend.