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Homework answers / question archive / A utility function is given by u(x1, x2) = min{x1 + 2x2, 3x1 + x2}
A utility function is given by u(x1, x2) = min{x1 + 2x2, 3x1 + x2}.
(a) Graphically determine the optimal consumption for the following prices and income: p1 =1, p2 =1,I =30.
(b) Suppose the price of good 2 increases to p2 = 3. Determine graphically the change in consumption of goods 2 and 3. p1 = 1, p2 = 2, I = 20.
(c) Determine the least costly consumption bundle which gives the consumer the same utility as in (b) when prices are p1 = 1, p2 = 1, i.e., Determine the Hicksean demand.
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)
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)
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)
