Question

Anna’s utility function is U(x,z) = x? + z?. Solve for her optimal values of good x and good z as a function of the price of good x, px, the price of good z, pz, and income, Y . For simplicity, assume that pz = 1. step by step

Answer 1

We have been given the following information

Utility function = U(x,z) = x? + z?

x = Good x

z = Good z

px = Price of Good x

pz= price of Good z = 1 (given)

Y = Income

Budget constraint = Y = px(x) + pz(z)

Using lagrangian multiplier

µ = x? + z? + ?(Y – px(x) – pz(z))

?µ/?x = ? – ?px = 0

?µ/?z = ? – ?pz = 0

?µ/?? = Y – px(x) – pz(z) = 0

? = ?/px

? = ?/pz

?/px = ?/pz

px = pz

pz = 1 given so,

Y – px(x) – pz(z) = 0

Y = x – z

Demand function of x = x = Y + Z

Demand function of z = z = x – Y