Question

1. Consider the preferences of an individual over two goods, x and y, with prices p_x and p_y and income I.

(a) If the individual's preferences can be represented by the utility function u(x,y) =2x^1/2 + y, what is the marginal rate of substitution? What does this MRS imply about how this consumer would trade y for x? Are the underlying preferences homothetic (explain)? Graphically illustrate a typical indifference curve and explain how you know the shape.

(b) If the individual's preferences can be represented by the utility function u(x,y) =5x+2y, what is the marginal rate of substitution? What does this MRS imply about how this consumer would trade y for x? Are preferences strictly convex, weakly convex, or non-convex? Graphically illustrate the indifference curve for u=10 .

(c) If the individual's preferences can be represented by the utility function u(x,y) =min(5x,2y), what is the marginal rate of substitution? What does this MRS imply about how this consumer would trade y for x? Are preferences strictly monotonic, weakly monotonic, or non-monotonic? Graphically illustrate the indifference curve for u=10 .

Answer 1

(a)

U(x,y) = 2x^1/2 + y

Marginal rate of substitution is calculated as;

MRS = MUx/MUy
= d/dx (2x^1/2 + y) / d/dy (2x^1/2 + y)
= 2/2x^1/2+ 0 / 0 + 1
= 1/x^1/2 / 1
MRS = 1/x^1/2

MRS = 1/x^1/2 means that consumer is willing to give up 1/x^1/2 units of good y in order to gain 1 more unit of good x.

Homothetic preferences : It means consumers with different incomes but facing the same prices and having identical preferences.

As we know,

MRS = 1/x^1/2

keeps on changing with units of x.

Therefore, this utility function is not homothetic.

Let, U = 10

U = 2x^1/2 + y

10 - 2x^1/2 = y

This case is for substitute goods, which means x and y are substitute goods and thus the shape of IC will be downward sloping line.

(b)

U(x,y) = 5x + 2y

Marginal rate of substitution is ;

MRS = MUx/MUy
= d/dx (5x + 2y) / d/dy (5x + 2y)
=5+ 0 / 0 + 2
MRS = 5/2

MRS = 5/2 means that consumer is willing to give up 5/2 units of good y in order to gain 1 more unit of good x.

The preferences of the utility function is not convex because this utility function is of perfect substitute goods.

For U = 10;

U(x,y) = 5x + 2y

10 = 5x + 2y

(c)

The utility function ;

U (x,y) = min(5x,2y)

The above utility function is the case for perfect complement goods. The MRS will be;

Horizontal fragment : MRS = 0
Vertical fragment : MRS =

As x and y are perfect complements, consumer will not give up any units of y in order to gain x because he/she will consume both the goods together.

Monotonic preferences : Monotonicity means that consumer will always prefer more of the bundle.

However, as given in the question the goods are perfect complements so consumer will always consume both the goods together and if will try to increase consumption of one good it wont be possible.

Given;

U = 10

U (x,y) = min(5x,2y)

5x = 10
x = 2

2y = 10
y = 5

The indifference curve for perfect complement goods will be L-shaped curve as we can see above, for utility = 10 consumer will consume 2 units of x and 5 units of y together.