given prices px = 1, py = 10 and income m = 30 (a) find the marginal rate of sub

Question

given prices px = 1, py = 10 and income m = 30 (a) find the marginal rate of substitution, (b) find the interior solution to the consumer's. Utility maximization problem, (c) find the corner solution to the consumer's utility maximization problem, (d) find the optimal consumption bundle of x and y. (e) if the price of good y is py = 2, how does your answer to part d change? 2. A consumer has the following utility function. U(x,y) = ln(x) + y Given prices px-1, p,- (a) find the marginal rate of substitution, (b) find the interior solution to the consumer's utility maximization problem, (c) find the corner solutions to the consumer's utility maximization problem, 10 and income m-30: (d) find the optimal consumption bundle of r and y (d) find the optimal consumption bundle of c and y. (e) If the price of good y is py-2, how does you answer to part d change?

Explanation / Answer

A. Given, utility function :                          U (x,y) = ln(x) + y

                                     Price line :              px.x + py.y = m  

           or,      x + 10y = 30                               [ Given px = 1, py = 10 , m = 30]

a) Marginal rate of substitution (MRS)

MRSxy = MUx / MUy

MUx = d [U (x,y)] / dx = d [ln(x) + y] / dx = 1/ x

MUy = d [U (x,y)] / dy = d [ln(x) + y] / dy = 1

MRSxy = 1/ x

b) The interior solution to the consumer's utility maximization problem can be found out by;

                                      Maximizing utility ;   U (x,y) = ln(x) + y

                                                        subject to , x + 10y = 30

To solve for an interior solution lets take a Langrangian Multiplier, L, and for a new equation for maximization, be Z, subject to the budget contraint or price line,

Z = U (x,y) + L ( 30 - x - 10y) = ln(x) + y + L ( 30 - x - 10y)

The first order condition for maximization , be derived as follows,

dz / dx = 1/x - L= 0    ;       or L = 1/x ............i.

dz / dy = 1 - L.10 = 0      ;   L = 1/ 10 ..................ii.

dz / dL = 30 - x - 10y = 0 ...................... iii

Now deriving for the second order condition for maximization,

d2Z / d x2 = -1 /x2 < 0

d2Z / d y2 = 0

Now to solve for the interior solution, we equate L ,

1/x = 1/ 10

or, x = 10

To find out the interior solution , lets put the value of X = 10 in equn. iii,

30-10-10y = 0

or, 10y = 20

or, y = 2

These are the interior solutions for X = 10 and y = 2 of the above problem.

Moreover, we can also find the interior solution where ,

MUx / Px = MUy / Py

or, 1/ x    = 1 / 10

or , x = 10 ,

and thus putting this value in the price line we get, y = 2.

c) The corner solution to the consumer's utility maximization problem can be found out by putting zero for the alternative commodity in the price line,

Finding corner solution for y , we put x = 0 and get ,

30 = 0 + 10y

or, y = 3

Similarly finding corner solution for x , we put y = 0 and get ,

30 = x +0

or x = 30

Thus the corner solutions for utility maximixations would be for,

x = 30

and y = 3.

d) Th optimum consumption bundle of ( x , y ) be (10 , 2) respectively, from the interior solution, where the indifference curve is tangent to the price line. The tangency provides us theinterior solution (10, 2).

e) Now, if py = 2 , then

MUx / Px = MUy / Py   changes,

i.e,

1/ x = 1/2

or , x = 2

and then putting this value of x = 2 in our new budget line , 30 = x + 2 y , we get a new ' y ' ,

2y = 30 - 2

or y = 28 /2 = 14

therefore, the new optimal consumption bundle changes from answer (d) to a new bundle, of

(x * , y* ) of   ( 2 ,14).

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