Uxyx5y income m1 Price Px2 Py3 Whats optimal consumption bun

U(x,y)=x^5*y, income m=1, Price Px=2, Py=3, What\'s optimal consumption bundle?

Solution

Utility function is as follows -

U(x, y) = x⁵ y

Income (m) = 1

Price of Good x (P_x ) = 2

Price of Good y (P_y) = 3

Bydget line is as follows -

P_X * x + P_y * y = m

2x + 3y = 1

Calculate marginal utility of Good x -

MUx = d[U(x, y)]/dx = d[x⁵ y]/dx = 5yx⁴

Calculate marginal utility of Good y ^-

MUy = d[U(x, y)]/dy = d[x⁵ y]/dy = x⁵

Calculate marginal rate of substitution (MRS) -

MRS = -MU_x / MU_y = -(5yx⁴)/x⁵ = -5y/x

Condition for optimal consumption bundle is as follows -

MRS_xy = -(P_x/P_y)

-5y/x = -(2/3)

y = 2x/15

Putting value of y in budget line -
2x + 3y = 1

2x + 3*(2x/15) = 1

2x + (2x/5) = 1

x = 5/12

y = 2x/15 = 2*(5/12)/15 = 1/18

So, x = 5/12 units and y = 1/18 units is the optimal consumption bundle.