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Income Effect and Income Consumption Curve

Illustration 25

Utility function of an individual is given by U = f (a,b) = a^¾ b^¼. Ascertain the optimal quantities of the two commodities of using Lagrange method if it is provided that price of commodity ‘a’ is $12 per unit, price of commodity is $6 per unit and income of the individual (1) is equal to $240.

Solution

Provided U = a^¾ b^¼

Pa = 12, Pb = 6 and I = 240.

Lagrange expression for the above question is as follows:

L = a^¾ b^¼ + λ (240 – 12a – 6b)

Differentiating the Lagrange function (L) with respect to a, b, λ and setting them equal to null or zero we procure,

dL = ¾ a^–¼ b^ ¼ - 12 λ = 0 ….Equation (1)
da

dL = ¼ a ^¾ b^–¾ - 6 λ = 0 ….Equation (2)
db

dL = 240 – 12a – 6b = 0 ….Equation (3)
dλ

Through restructuring, we procure

¾ a^–¼ b^¼ = 12λ ….Equation (4)

¼ a ^¾ b^– ¾ = 6λ ….Equation (5)

12a + 6b = 240 ….Equation (6)

To solve for a we divide the equation (4) by equation (5), therefore,

¾ a^–¼ b^¼ = 12λ
¼ a ^¾ b^– ¾ 6λ

6a ^ -1 * b = 12
6

Or 6b = 2 or 6b = 3b
a 2

Substituting the value of a = 3b in the budget equation (6), we procure,

12 * 3b + 6 b = 240

36b + 6b = 240

42b = 240

b = 240 / 42 = 5.71

Now substituting the value of b = 5.71 in the budget equation (6) we procure,

12a + 6 * 5.71 = 240

12a = 205.74

a = 205.74 / 12 = 17.15

Therefore, the maximum quantities the value of and b which optimises utility are a = 17.15 and b = 5.71.

Illustration 26

There are products a1 and a2 on which consumer expends his absolute income in a day. He has utility function U = √ a1a2. ascertain the maximum quantities of a1 and a2 if prices of a1 and a2 are $10 and $4 correspondingly and his day to day income parities $1000.

Solution

Provided utility function:

U = √ a1a2

U = a1^½ a2^½

And with Pa1 = 10 and Pa2 = 4 and I = 1000, budget equation is

1000 = 10a1 + 4a2

Differentiating the provided utility function with respect to a1 and a2 we procure

MUa1 = dU = ½ a1 ^ -½ a2 ^½ …Eq (1)
da1

MUa2 = dU = ½ a1 ^ ½ a2 ^-½ …Eq (2)
da2

Dividing Equations (1) by (2)

MRSa1a2 = MUa1 = ½ a1 ^ -½ a2 ^½
MUa2 ½ a1 ^ ½ a2 ^-½

= a1 ^ -1 a2 = a2
a1

For optimisation of utility it is necessary that

MRSab = Pa1
Pa2

a2 = 10 = 5
a1 4 2

a2 = 5a1
2

Substituting the value of a2 in the budget equation, we procure

I = a1Pa1 + a2Pa2 (Budget Equation)

1000 = a110 + 5 a1*4
2

1000 = a110 + 10a1

1000 = 20a1

a1 = 1000 / 20 = 50

Substituting the value of a1 in the budget equation,

1000 = 50 * 10 + 4a2

1000 – 500 = 4a2

500 / 4 = a2

a2 = 125

Therefore, the maximum quantities of a1 and a2 are 50 and 125 correspondingly.

Illustration 27

Let us consider the general utility U = ab is provided. I stands for income and Pa and Pb refers to prices of the two products a and b. Ascertain the mathematical derivation of demand function using Lagrange method.

Solution

Optimise U = ab …..Equation (1)

Subject to Pa.a + Pb.b = 1 …..Equation (2)

Lagrange expression for the above problem is

L = ab + λ (I - Pa.a + Pb.b)

For optimising utility it is necessary that

dL = dU – λ Pa = 0 …..Equation (3)
da db

dL = dU – λ Pb = 0 …..Equation (4)
db da

dL = I - Pa.a + Pb.b = 0 …..Equation (5)
dλ

Note that derivative dU of the utility function U = ab is equal to b and derivative dU
da db
of the utility function U = ab is equal to a.

Moreover make a note that these derivatives denote marginal utilities of the commodities.

Restructuring the equation (3) and (4), we procure

MUa = λ and MUb = λ
Pa Pb

Or MUa = MUb …..Equation (6)
Pa Pb

Since in the given utility function MUa = b and MUb = a, we procure

b = a …..Equation (7)
Pa Pb

b = a Pa …..Equation (8)
Pb

Putting the value of b into the budget equation (2) we procure,

Pa.a + Pb. Pa / Pb .a = 1

Pa. 2a = 1

a = 1 / 2pa …..Equation (9)

Similarly, for the function b = 1/2Pb …..Equation (10)

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