Question

Q2) (based on textbook problem 3.19) Consider the utility function 2Vx +3v/y a) Find MU and MU b) Is the assumption that more is better satisfied for both goods? c) Does the marginal utility of each good diminish, remain constant, or increase as the consumer buys more? d) What is MR.Sa,y? Is it diminishing, constant, or increasing as the consumer sub- stitutes x for y along an indifference curve? e) On a graph with r on the horizontal axis and y on the vertical axis, draw a typical indifference curve. (It need not be exactly to scale, but the major points like how slope changes along the curve should be clear). Also indicate whether the indifference curve will intersect either or both axes. Label this curve U f) Draw a second curve U2 such that U2 > Ui

Answer 1

2. The utility function is given as $\dpi{80} u = 2x^{1/2} + 3y^{1/2}$ .

(a) We have $\dpi{80} MU_x = \frac{\partial u}{\partial x}$ or $\dpi{80} MU_x = \frac{\partial }{\partial x}(2x^{1/2} + 3y^{1/2})$ or $\dpi{80} MU_x = x^{-1/2}$ , and $\dpi{80} MU_y = \frac{\partial u}{\partial y}$ or $\dpi{80} MU_y = \frac{\partial }{\partial y}(2x^{1/2} + 3y^{1/2})$ or $\dpi{80} MU_y = 1.5y^{-1/2}$ .

(b) For the bundle (x1,y1) and (x2,y1), where x2 > x1, we have $\dpi{80} u_1 = 2x_1^{1/2} + 3y_1^{1/2}$ and $\dpi{80} u_2 = 2x_2^{1/2} + 3y_1^{1/2}$ . Thus, $\dpi{80} u_2 - u_1 = 2x_2^{1/2} + 3y_1^{1/2} - (2x_1^{1/2} + 3y_1^{1/2})$ or $\dpi{80} u_2 - u_1 = 2x_2^{1/2} - 2x_1^{1/2}$ or $\dpi{80} u_2 - u_1 = 2(x_2^{1/2} - x_1^{1/2})$ , and for $\dpi{80} x_2 > x_1$ , we would have $\dpi{80} x_2^{1/2} > x_1^{1/2}$ , and hence $\dpi{80} u_2 - u_1 = 2(x_2^{1/2} - x_1^{1/2}) > 0$ or $\dpi{80} u_2 > u_1$ . In this case, more is better assumption is satisfied.

Similarly, for the bundle (x1,y1) and (x1,y2), where y2 > y1, we have $\dpi{80} u_1 = 2x_1^{1/2} + 3y_1^{1/2}$ and $\dpi{80} u_2 = 2x_1^{1/2} + 3y_2^{1/2}$ . Thus, $\dpi{80} u_2 - u_1 = 2x_1^{1/2} + 3y_2^{1/2} - (2x_1^{1/2} + 3y_1^{1/2})$ or $\dpi{80} u_2 - u_1 = 3y_2^{1/2} - 3y_1^{1/2}$ or $\dpi{80} u_2 - u_1 = 3(y_2^{1/2} - y_1^{1/2})$ , and for $\dpi{80} y_2 > y_1$ , we would have $\dpi{80} y_2^{1/2} > y_1^{1/2}$ , and hence $\dpi{80} u_2 - u_1 = 3(y_2^{1/2} - y_1^{1/2}) > 0$ or $\dpi{80} u_2 > u_1$ . In this case also, more is better assumption is satisfied.

(c) We have $\dpi{80} MU_x = x^{-1/2}$ , and hence $\dpi{80} \frac{\partial }{\partial x}(MU_x) = (-1/2)x^{-3/2} < 0$ for all good x. Hence, as the consumption of x increases, the marginal utility of x decreases.

Also, $\dpi{80} MU_y = 1.5y^{-1/2}$ , and hence $\dpi{80} \frac{\partial }{\partial y}(MU_y) = (-1.5/2)y^{-3/2} < 0$ for all y. Hence, as the consumption of y increases, the marginal utility of y decreases.

(d) The MRS would be $\dpi{80} MRS = -\frac{MU_x}{MU_y}$ or $\dpi{80} MRS = -\frac{x^{-1/2}}{1.5y^{-1/2}}$ . As can be seen, MRS<0. Also, as the consumer increases x for y, MU of x decreases, while MU of y increases, and that would increase the $\dpi{80} MRS = -\frac{MU_x}{MU_y}$ as it would tend to zero. Hence, the MRS is diminishing if consumer substitutes x for y.

(e) The graph is as below.

20 1:5 1.0 20

The points are marked where the indifference curve touches the x and y axis.

(f) The graph is as below.