Question

2. Zhixiu has the following linear preferences over coffee (x) and candy (y): u(x, y) = 2x+4y (a) Graph the indifference curves for Zhixiu that represent each of the following utility levels: 10, 20, 40, and 60. (You should be drawing 4 indifference curves, make sure to label with one represents which utility level) Make sure to show your calculations for how you identified the equation of the indifference curves.(4 points) (b) Setup Zhixiu's utility maximization problem with a general budget constraint (2 points)

Answer 1

Answer:

linear preferences over coffee (x) and candy (y)

u(x , y) = 5 x + 4 y

a) utility levels : 20 ,40, 60 , 80

5 x + 4 y = 20

5 x + 4 y = 40

5 x + 4 y = 60

5 x + 4 y = 80

the indifference curve of the consumer are represented in the adjacent diagram

21 - 2 , x paob CamScanner Scanned with (С, (A 240) 1cs (4 - Go) (98 - 7 )һ»).

b) Setup Zhixiu's utility maximization

Let the price of

Good X is P_x and price of good Y =p_y

and income of the consumer be M , thus , general budget constraint - X .P_x + Y . P_y = M

Maximize 5 x + 4 y

subject to X . P_x + Y. P_y  = M

c) Yes, the budget constraint is active as it is needed to solve the utility maximization problem .

No , in this case the interier solution sufficient condition is not satisfied because this is the case of perfect

substitutes or linear utility function and budget.

Constraint cannot be tangent to utility function here. Thus , boundary point are the optimum solutions in case of substitute goods.

d) m = 100 , p_x = 6 : slope of budget line =$\frac{P_{x}}{P_{y}} = \frac{6}{P_{y}}$

MRS = $\frac{\frac{\partial v}{\partial x}}{\frac{\partial v}{\partial y}} = \frac{5}{4}> \frac{6}{P_{y}}$

then only X* = $\frac{100}{P_{x}= 6}$ = 16.67 and Y^* = 0

MRS = 5/4 < 6 / P _y$\rightarrow$ X^* = 0 , Y^* = 100 / p _y

MRS= 6 / p _y , then the person is indifferent between good x and good y