1. The utility function is given as

U(x1, x2) = 2.√(x1.x2)

Hence, MRS or Marginal Rate of Substitution is

MRS = MU1/MU2

Marginal Utility of x1 is

MU1 = dU/dx1 = (2√x1)/(2√x2)

or, MU1 = √(x1/x2)

Marginal Utility of x2 is

MU2 = dU/dx2 = (2√x2)/(2√x1)

or, MU2 = √(x2/x1)

Hence, MRS = MY1/MU2

or, MRS = x2/x1

Now, prices are, p1 = 1 and p2 = 2. Incone is m = 100.

Hence, budget constraint is

p1.x1 + p2.x2 = m

or, x1 + 2.x2 = 100..........(BL)

When consumer chooses the best bundle, we can write,

MRS = p1/p2

or, x2/x1 = p1/p2 = 1/2

or, x1 = 2.x2

Putting this in the budget constraint (BL) we get,

2.x2 + 2.x2 = 100

or, x2* = 25

Hence, x1* = 2.x2* = 2×25

or, x1* = 50

Consumer's best choice is

(x1*, x2*) = (50, 25).

(2) If p1 increases to p1' = 2, hence new budget constraint is

p1'.x1 + p2.x2 = m

or, 2.x1 + 2.x2 = 100

At best choice of the consumer,

MRS = p1'/p2

or, x2/x1 = 2/2

or, x1 = x2

Putting this in the budget constraint we get,

2.x1 + 2.x1 = 100

or, x1'* = 25

And, x2'* = x1'* = 25

New consumption bundle is

(x1'*, x2'*) = (25, 25).

The following diagram shows the new consumption bundle.

(3)

• Budget Line: Budget line is the combination of goods which a consumer can afford for given prices and income.

• Indifference Curves: Indifference curves are the locus of all consumption bundles for which the utility of the consumer remains constant.

The optimal consumption bundle is found at that point where, budget line is tangent to a indifference curve.

The following diagram shows the optimal consumption bundle.

PLEASE SEE THE ATTACHED FILE FOR THE COMPLETE SOLUTION.