Question

Maximize U(x,y)=xy+x+2y subjected to Px=,Py=5 and M=51

Answer 1

Solution:

Utility function U(x,y) = xy + x + 2y

The utility-maximizing condition: $\frac{MUx}{MUy} = \frac{Px}{Py}$

First, derive MRS:

MRS = $\frac{MUx}{MUy}$

MUx = $\frac{\partial U} {\partial X} = y + 2y = 3y$

MUy = $\frac{\partial U} {\partial Y} = x + x = 2x$

MRS = $\frac{MUx}{MUy} = \frac{3y}{2x}$

Set MRS equal to $\frac{Px}{Py}$ to derive the utility-maximizing bundle:

Px = 2

Py = 5

$\frac{3y}{2x} = \frac{2}{5}$

y = $\frac{4x}{15}$

Plug Y into the budget constraint to derive X:

Budget constraint: M = PxX + PyY

51 = 2X + 5Y

51 = 2X + 5($\frac{4x}{15})$

51 = 2X + $\frac{20x}{15}$

51 = 2X + $\frac{4x}{3}$

Multiple all sides by 3

153 = 6X + 4X

153 = 10X

X = 15.3

Plug this into Y equation:

Y = $\frac{4x}{15} = \frac{(4\times 15.3)}{15} = \frac{61.2}{15} = 4.08$

Y = 4.08

Utility maximizing bundle (Ux,y) = (15.3, 4.08)