Question

A person has utility function u(x, y) = 100xy + x + 2y. Suppose that the price per unit of x is $2, and that the price per unit of y is $4. The person receives $1 000 that all has to be spent on the two commodities x and y. Solve the utility maximization problem

Answer 1

Answer:

x = 250

y = 125

u(x,y) = 3125500

Step-by-step explanation:

As given,

The utility function u(x, y) = 100xy + x + 2y

$P_{x}$ = 2 , $P_{y}$ = 4

Now,

Budget constraint -

$P_{x}$ x + $P_{y}$ y = 1000

⇒2x + 4y = 1000

So,

Let v(x, y) = 2x + 4y - 1000

Now,

By Lagrange Multiplier

Δu = Δv

⇒< 100y + 1, 100x + 2 > = < 2, 4 >

By comparing, e get

100y + 1 = 2         ........(1)

100x + 2 = 4        .........(2)

Divide equation (2) to equation (1) , we get

$\frac{100y + 1}{100x + 2} = \frac{1}{2}$

⇒2(100y+1) = 1(100x+2)

⇒200y + 2 = 100x + 2

⇒200y = 100x

⇒2y = x

Now,

As 2x + 4y = 1000

⇒2x + 2(2y) = 1000

⇒2x + 2x = 1000

⇒4x = 1000

⇒x = 250

Now,

As 2y = x

⇒2y = 250

⇒y = $\frac{250}{2}$ = 125

∴ we get

x = 250

y = 125

Now,

u(250, 125) = 100(250)(125) + 250 + 2(125)

                   = 3125000 + 250 + 250

                  = 3125000 + 500

                 = 3125500

⇒u(250, 125) = 3125500