A utility function is given by u(x1, x2) = min{x1 + 2x2, 3x1 + x2}

a)

MRS = Price ratio (is the minimum function)

The calculation of MRS is not done by the function of utility.

The consumption of Goods in the fixed ratio are the equation

MRS= Ratio of Consumption

MRS = (x1 + 2x2)/(3x1 + x2) = p1/p2 = 1

(x1 + 2x2) = (3x1 + x2)

x2 = 2x1

The budget line =

30 = x1 + x2

30 = x1 + 2x1

x1* = 10, x2*

= 20.

Optimal situation = (10, 20)

b)

If price of the goods 2 increase to p2 is equal to 3

A new set of condition are p1 = 1, p2 = 2, I = 20 and p2 which rising to the 3

At,

p1 = 1,

p2 = 2,

20 = x1 + 2x2

20 = x1 + 2*2x1

20 = x1 + 4x1

x1* = 4,

x2* = 8

The another optimal situation is (4,8)

If

p1 = 1,

p2 = 3,

20 = x1 + 3x2

20 = x1 + 3*2x1

20 = x1 + 6x1

x1* = 20/7,

x2* = 40/7

The another optimal situation is (20/7, 40/7)

c)

Calculate the utility P1 = 1, P2 = 1 and income I= 30

30 = x1 + x2

30 = x1 + 2x1

x1* = 10,

x2* = 20

u(x1, x2) = min{x1 + 2x2, 3x1 + x2}

= min{10 + 2*20, 3*10 + 20}. = 50

The consumption utility is least giving rising of 50 utility is P1 = 1, P2 = 1, I = 30

30 = x1 + x2

50 = min{x1 + 2x2, 3x1 + x2}

Substitute the value

x1 = 30 - x2

50 = min{30 - x2 + 2x2, 90 - 3x2 + x2}

50 = min{30 + x2, 90 - 2x2}

50 = 30 + x2 or

50 = 90 ? 2x2

x2 = 20, x1 = 10. (This situation is consumption bundle)