Question

2) A consumer's utility function is a(x,y) = (a) Find the consumer's optimal choice for x as a function of income I and prices px,Py' (b) Sketch the demand curve for x as a function of its own price Pz when py = 10, 1 = 100. (It may be easiest to plot a few points.)

Answer 1

u = - (1/3x³) - y = - [(1/3)x^-3] - y

Budget line: I = x.px + y.py

(a) Utility is maximized when MUx/MUy = px/py

MUx = u/x = 3 x (1/3) x x^-4 = x^-4

MUy = u/y = - 1

MUx / MUy = x^-4 / 1 = px/py

x⁴ = py/px

x = (py/px)^1/4

Substituting in budget line,

I = [(py/px)^1/4].px + y.py

I = [(py)^1/4(px)^3/4] + y.py

y.py = I - [(py)^1/4(px)^3/4]

y = (I - [(py)^1/4(px)^3/4]) / py

(b) Demand function for x is independent of I. When py = 10,

y = (10/px)^1/4

When px = 1, x = (10/1)^1/4 = 1.78

When px = 5, x = (10/5)^1/4 = 1.19

When px = 10, x = (10/10)^1/4 = 1

In following graph, x is plotted as function of px.

lo ざご丨.19 el.78 0 0