Question

(a) 1.2 (10 mks each). In parts a) and b) below, assume px = $1, Py = $5, I = income = $21. Solve the U-max problem for each of the following two utility functions: U= xy?, x, y 2 0; (b) U = x1/3y2/3, x, y = 0; now, let px = P, Py = $5, I = $21, find the u-max solution for U = xy?, x, y 2 0; let px = 1, Py =p, I = $21, find the u-max solution for U = xy?, x, y 2 0; (e) show that the u-max solution for U = xy?, given (px, py, I), is x = fpx, px, 1) = 1/(3px), and y = g(px, py, I)= 21/(3py). Hint: Use equal MU/$ method. Note that part a) is identical” to the example solved in class. (c) (d)

Answer 1

Solution:: In order to solve the problem, we need to find MRS_xy & Budget Constraint Line

a) U = xy² & p_x= 1, p_y= 5 , I = 21

To find MRS , Differentiate U function w.r.t x and then w.r.t y

U Max eqn is MUx / MUy = Px/Py

MRS = Differentiation of x / Differentiation of y

MUx = y²

MUy = 2yx

MRS = y²/2yx

Therefor, MRS = y/2x

Now set MRS with price ratio

price ratio = px/py

Price ratio = 1 / 5

MRS = p_x / p_y

Y / 2x = 1/5 ( Putting value of MUx, MUy and Px, Py )

y = 2x / 5 ( Substitute value of y into budget constraint)

Budget constratint = P_x + P_y = I

1.x + 5.y = 21

x + 5y = 21

Since y = 2x/5,

x + 5(2x/5) = 21

3x = 21

x = 7 ( Substitute this value into budget constraint )

Since x = 7 , Now Budget Constraint

7 + 5y = 21

Solving this give y = 2.8

b) U = x ^1/3 y^2/3 & p_x= 1, p_y= 5 , I = 21

MRS = p_x / p_y

MU_x = 1/3 x^-2/3y^2/3

MU_y = 2/3 x^1/3 y ^-1/3

Solving these 2 equations we get,

y/2x = 1/5

5y = 2x

y= 2x / 5

Putting these value into budget constraint, we get

x + 5.(2x/5 ) = 21

3x = 21

x = 7 ( Substitute this value into budget constraint )

Since x = 7 , Now Budget Constraint

7 + 5y = 21

5y = 21-7

y = 14/5

Solving this give y = 2.8

Subtituting value of y gives value of

1.x + 5.y = 21

x + 5(2.8) = 21

x = 7

c) U = xy² & p_x= P , p_y= 5 , I = 21

MUx = y²

MUy = 2yx

MUx/MUy = px / Py

y²  / 2yx = P / 5

y / 2x = P / 5

5/2(y) = Px

Substitute value of px into budget constraint

Budget constraint = xP + 5y = 21

5/2y + 5y = 21

15y = 42

y = 42 / 15

y = 2.8

therefor px = 7

x = 7/p

d) U = xy² & p_x= 1, p_y= P, I = 21

MRS = Slope of Price

MUx = y2

Muy = 2yx

Px/py = 1/p

y/2x = 1/p

yp = 2x

Putting this value into Budget Constraint

X+py = 21

X + 2x = 21

3x = 21

x = 7

putting this value into budget constraint we get

7+py = 21

py = 21-7

py = 14

y = 14 /p