Question

Consider a consumer whose income is 100 and his preference is given by U-10x04yo6. If PX-Py-1, what is the optimal consumption bundle by the consumer? (Please write out the constraint utility maximization problem completely, including the budget function.) Derive the demand of Good X and Y by this consumer. (The result should be a function giving you the amount of X he will buy at every given price level Px, and a function for good Y as well.) a. b. C. The consumer will consume according to MRS Px/Py. (Please check your 203 notes in case you can no longer remember). Please derive the relative demand function. (So the demand of X/Y as a function of relative price Px/Py.) Draw the function in a graph where the relative price is shown in y-axis and the relative quantity in x-axis.

Answer 1

The consumer�s problem in to maximize u = 10 times x^0.4 y^0.6 s t 100 = P-x x + P-x y = x + y x, y greaterthanorequalto 0 The required Lagrange function is L = 10x^0.4 y^0.6 + lambda [100 - x - y] The Focs are partial differential L/ partial differential = 4 (y/x)^0.6 - lambda = 0 -(1) partial differential L/ partial differential y = 6 (x/y)^0.4 - lambda = 0 -(2) partial differential L/ partial differential lambda = 100 - x - y = 0 ---(3) (1) (2) Rightarrow 4/6 y/x = 1 or, 2y = 3x -(4) substituting (4) in (3) 100 = 2/3 y + y = 5/3 y }Therefore t* = 60 Therefore x* = 2/3 times 60 = 49} b The consumers problem is to maximum u = 10x^0.4 y^0.6 s t p_x x + p_y y = 100 x, y greaterthanorequalto 0 The required Lagrange function is L = 10x^0.4 y^0.6 + lambda [100 - P_x x - P_y y]

c)

The marginal rate of substitution between X and Y is

$MRS=\frac{MU_X}{MU_Y}=\frac{\frac{\partial U}{\partial X}}{\frac{\partial U}{\partial Y}}$

$\therefore MRS=\frac{4\left ( \frac{Y}{X} \right )^{0.6}}{6\left ( \frac{X}{Y} \right )^{0.4}}=\frac{2Y}{3X}$

At equilibrium:

$MRS=\frac{P_X}{P_Y}$

$\therefore \frac{2Y}{3X}=\frac{P_X}{P_Y}$

$\therefore \frac{X}{Y}=\frac{2}{3\left (\frac{P_X}{P_Y} \right )}$............... (1)

The figure depicting the relative demand in equation (1) below