MicroAssn4

1) Derive the demand function for a consumer with the utility function u(x) = 12x^(1/3) y^(2/3) . Also derive the indirect utility function.

u(x) = 12x^(1/3) y^(2/3) sb. px +qy = M Lagrangian function L = 12x^(1/3) y^(2/3) + λ(M−px−qx2) ∂Λ/∂x = 0 : 4 y^(2/3) / x^(2/3) - λ p = 0 - (1) ∂Λ/∂y = 0 : 8 x^(1/3) / y^(1/3) - λ q = 0 - (2) ∂Λ/∂λ = 0 : M - px1 - qx2 = 0 - (3) λ = 4 y^(2/3) / x^(2/3) / p = 8 x^(1/3) / y^(1/3) / q y = 2px/q => px+qy = M x=M/(3p), y=2M/(3q) V(p, q, M) = u[M/(3p), 2M/(3q)] = 12 * 2^(2/3) M / [3 p^(1/3) q^(2/3)]

2) Let someone have a utility function defined by u(x1, x2 ) =5lnx1 + 4lnx2 with an income of 100. Suppose the current price of the two goods are p1 = 2 and p2 = 1.

a) What is their demand function for both goods? (Note: This demand function is known as Walrasian demand or Marshallian demand)

Lagrangian function L = 5lnx1 + 4lnx2 + λ(100−2x1−x2) ∂Λ/∂x1 = 0 : 5/x1 - 2λ = 0 - (1) ∂Λ/∂x2 = 0 : 4 / x2 - λ = 0 - (2) ∂Λ/∂λ = 0 : 100 - 2x1 - x2 = 0 - (3) λ = 5/(2x1)=4/x2 x2=8x1/5 2x1+x2=100 => x1=500/18~27.78 x2=8x1/5~44.44

b) What can you say about the proportion of income spent on the two goods.

px1=227.78~55.56 - /100 100% -> 55.56% qx2=144.44~44.44 - /100 100% -> 44.44%

c) What is the maximum utility this person could get from consuming these goods.

u(27.78, 44.44) = 31.79

3) The individual has a utility function u(x1, x2) = min{4x1, 5x2} and faces prices p1 = 2 and p2 = 1. We know they consume 20 units of x2 and spend all their income. What is the demand for x1 ? What is the individual’s income?

since it’s not differeciable, we cannot use lagrange use x, y, p, q here to present x1, x2, p1, p2 M = px + qy =20y=20 4x=5y p5y/4 + qy = M -> y(p, q, M) = 4M/ (5p + 4q) -> x(p, q, M) = 5M/ (5p + 4q) => M = 20 p=1 q=2 -> y* = 80/14=40/7, x* = 100/14 = 50/7 px+qy=40/7 + 100/7=140/7=20

4) Let the utility function b e of the form u(x1, x2) = 5lnx1 + x2. Find the relevant demand functions. What do you find unique about this kind of utility functions? For what values of M (income) would we consume only good 1?

Lagrangian function L = 5lnx1 + x2 + λ(M−px1−qx2) ∂Λ/∂x1 = 0 : 5/x1 - λp = 0 - (1) ∂Λ/∂x2 = 0 : 1 - λq = 0 - (2) ∂Λ/∂λ = 0 : M - px1 - qx2 = 0 - (3) λ=5/(px1)=1/q x1=5q/p - px1+qx2 = M -> x2 = M/q -5 x1=5q/p, x2=M/q -5 What is unique about this utility function is that the demand for x1 is independent of income M. The quantity demanded for x1 only depends on the relative prices p and q , while the demand for x2 increases linearly with income.

The marginal utility of x2 is constant (because it’s linear in x2), while the marginal utility of x1 decreases as more of x1 is consumed (logarithmic in x1).

5) Let’s say the utility function is given by the CES utility function u(x) = [x1^ρ + x2^ρ]^(1/ρ). Compute the Walrasian/Marshallian demand function.

Lagrangian function L = [x1^ρ + x2^ρ]^(1/ρ) + λ(M−px1−qx2) x1p + x2q = M ∂Λ/∂x1 = 0 : (x1^(ρ-1)) / (x1^ρ + x2^ρ)((1-ρ)/ρ) - λp = 0 - (1) ∂Λ/∂x2 = 0 : (x2^(ρ-1)) / (x1^ρ + x2^ρ)((1-ρ)/ρ) - λq = 0 - (2) ∂Λ/∂λ = 0 : M - px1 - qx2 = 0 - (3) (x1 / x2)^(ρ-1) = p / q x1 = x2 * (p / q)^(1/(1-ρ)) x2 * (p * (p / q)^(1/(1-ρ)) + q) = M x2 = M / (p^(1/(1-ρ)) * q^((ρ-1)/(1-ρ)) + q) x1 = (M * (p / q)^(1/(1-ρ))) / (p^(1/(1-ρ)) * q^((ρ-1)/(1-ρ)) + q)

6) Derive the demand functions for a consumer with the utility function u(x) = U(x,y) = 2x + 3y. Also find the indirect utility function for this consumer.

( H i n t : You saw one non-standard utility function in class today for perfect complements. The utility function here relates to perfect substitutes. Do not blindly use the formula from class to solve for demand functions)

px+qy=M Lagrangian function L = 2x + 3y + λ(M−px−qx2) ∂Λ/∂x = 0 : 2 - λ p = 0 - (1) ∂Λ/∂y = 0 : 3 - λ q = 0 - (2) ∂Λ/∂λ = 0 : M - px - qy = 0 - (3) λ = 2/p = 3/q p=2q/3 x=M/(2p), y=M/(3q) V(p, q, M) = 2M/(2p) + 3M/(3q) = Mx/p + My/p