Reference no: EM132881184

ENGM90011 Economic Analysis for Engineers - University of MeLbourne

Question 1. Let there be two goods with quantities x = (x₁, x₂) ∈ R₊² ie x₁ ≥ 0, x₂ ≥ 0. Let prices p = (p₁, p₂) ∈ R₊₊² ie p₁ > 0, p₂ > 0. Let the consumer's preferences be represented by the CES utility function U(x₁, x₂ ) = (x₁^ρ + x₂^ρ ⁄ where ρ < 1, ρ ≠ 0. Note: parameter ρ in the utility function is the Greek Letter rho not the price vector p. Derive the expenditure function e(p₁, p₂, U).

Question 2. Let there be two factors (inputs), L and K, (L, K) ∈ R₊² , with factor prices w and r, respectively, (w, r) ∈ R₊₊² . Let the firm's production function be y = L_αK_β where α, β > 0.
a. Find the long run total cost function C(w, r, y).
b. Let capital be fixed at K = K¯ in the short run. Find the short run total cost function C(w, r, y; K = K¯).
c. Let p be the price of y. If this is a competitive firm, find how much would it produce in the short run.

Question 3. Let the firm's output be y ∈ R₊. Let the output price be p ∈ R₊₊. Let the quantities of the m inputs be x = (x₁, ... , x_m) ∈ R_m. Let input prices be w = (w₁, ... , w_m) ∈ R_m . The firm takes input prices as given ie they are parameters. Prove that the profit function Π(p, w) is non-decreasing in p.

Question 4. Concavity and quasiconcavity

a. Let f(x) = x^ρ for x ∈ R₊₊. Notice that x here is a single variable, x ∈ R₊₊, not a vector.

Using calculus, determine for which values of the parameter ρ the function x^ρ is a concave function over the convex set R₊₊.
b. Let f and g be concave functions over a convex set S ⊆ Rⁿ. Let a, b ∈ R₊ ie a ≥ 0, b ≥ 0. Define the function af + bg over S to be (af + bg )(x) = af(x) + bg(x).

Prove that the function af + bg is a concave function over S.

GeneraL case: If f¹, f², ... , f^k are concave functions over the convex set S ⊆ Rⁿ and a = (a₁, a₂, ... , a_nk) ∈ R₊^k then a₁f¹ + a₂f² + a_kf^k is a concave function over the set S.

Don't need to prove.

Here f², f³, etc just means a second function, third function, etc, not the square of a function, cube, etc.

Similar results apply to convex functions.

c. Concave functions in R^m and in Rⁿ, m < n.

i. Let x ∈ R and g(x) = 1 - x². Show g is a concave function over the convex set R.

ii. Now Let us embed g into R². Let x = (x₁, x₂) ∈ R2 and f(x₁, x₂) = 1 - x₁². (f does not depend on x₂). Show f is a concave function over the convex set R².

In generaL, for n > m, if g(x_1, ... , x_m) is concave in (x₁, ... , x_m) then f(x₁, ... , x_m, ... , x_n) = g(x₁, ... , x_m) is concave in (x₁, ... , x_n).

d. Let f be a concave function over a convex set S ⊆ Rn. Prove that f is aLso a quasiconcave function over S.

e. Let f be a quasiconcave function over a convex set S ⊆ Rn. Let F be a strictly increasing function. Prove that a monotonic transformation of a quasiconcave function over S is a quasiconcave function over S ie prove that g = F ° f is a quasiconcave function over S .

Hint: ∀a, b ∈ R, F(min{a, b}) = min{F(a), F(b)}.

f. Let ( α₁, ... , α_n) ∈ R₊₊ⁿ and x = (x₁, ... , x_n) ∈ R₊₊ⁿ . Let A > 0, μ > 0, ρ ≠ 0.

Define the generalised CES function f over the convex set Rn .

∀x ∈ R₊₊ⁿ , f(x) = A(α₁x₁^ρ + α₂x₂^ρ + ....... α_nx_n^ρ )^μ⁄ρ

Using the resuLts from this question, show that this generaLised CES function f is a quasiconcave function over R₊₊ⁿ when ρ ∈ (0, 1].

Note: Need only question 2c,4c2,4e,4f