Question

1. Suppose an individual’s utility function is u=x₁^1/2, x₂^1/2. Let p₁=4, p₂=5, and income equal $200.
   1. With a general equation and general prices, derive the equal marginal principle. Graphically illustrate equilibrium and disequilibrium conditions and how consumers can reallocate their consumption to maximize utility.
   2. What is the optimal amount of x₁ consumed?
   3. What is the optimal amount of x₂ consumed?
   4. What is the marginal rate of substitution at the optimal amounts of x₁ and x₂?
   5. As functions of p₁, p₂, and m, derive x₁’s demand curve?
   6. As functions of p₁, p₂, and m, derive x₂’s demand curve?

Answer 1

u = x1^1/2x2^1/2

General Case:

Budget line: M = p1.x1 + p2.x2

Utility is maximized when MU1/MU2 = p1/p2

MU1 = $\partial$ u/$\partial$x1 = (1/2).(x2/x1)^1/2

MU2 = $\partial$ u/$\partial$x2 = (1/2).(x1/x2)^1/2

Marginal rate of substitution (MRS) = MU1/MU2 = [(1/2).(x2/x1)^1/2] / [(1/2).(x1/x2)^1/2] = x2 / x1 = p1 / p2

p1.x1 = p2.x2

Plugging into budget line,

M = p1.x1 + p1.x1 = 2p1.x1

x1 = M / (2p1)

Similarly,

M = p2.x2 + p2.x2 = 2p2.x2

x2 = M / (2p2)

In following graph, utility is maximized at point E where budget line AB is tangent to indifference curve U0 with optimal bundle being (x1*, x2*).

22 A E xa VO B x

Specific case:

When p1 = 4, p2 = 5 & M = 200,

x1 = 200 / (2 x 4) = 25

x2 = 200 / (2 x 5) = 20

MRS = 20/25 = 0.8