Question

Diego’s utility function is U(x, z) = 100x 0.9 z 0.1 . Solve for her optimal values of good x and good z as a function of the price of good x, px = 2, the price of good z, pz = 2, and income, Y = 800.

Answer 1

Px = 2

Pz = 2

U = 100 * X^.9 * Z ^.1 --------------------- (1)

Differentiation of equation 1 w.r.t. X will give MUx.

MUx = 100*.9*(Z/X)^.1

Differentiation of equation 1 w.r.t. Z will give MUz.

MUz = 100* .1*(X/Z)^.9

To maximize the utility,

MUx/Px = MUz/Pz

(100*.9*(Z/X)^.1)/2 = (100* .1*(X/Z)^.9)/2

9Z = X

Now, as per the budget line,

800 = Px*X + Pz*Z

800 = 2X + 2Z

400 = X + Z

400 = 9Z + Z

Z = 400/10 = 40 units

X = 9*40 = 360 units

So in optimum bundle, X = 360 units and Z = 40 units is for Diego.