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Income Effect And Income Consumption Curve

 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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