Problem set 1 Micro Theory S. Wang Question1.1. Show that“f(X)=f(x),Vx∈R,A>1” implies“f(A)= Af(x),Vx∈R,A>0.” estion 1.2. Use a Lagrange function to solve c(w1, w2, y) for the following problem c(w1, w2, y)= min 1-1+w2.C2 x1,x2 Question 1.3. Use a graph to solve the cost function for the following problem y)≡ Inin w1. I1+2x t +bC2 Question 1. 4. Find the cost function for the following problem c(1,2,y)≡minn11+22 t mIn Question 1.5. Consider the factor demand system (2,2,y)=|b1+b2( n2(,m2,y)=|h+a2 where b11, b12, b21, b22>0 are parameters. Find the condition(s)on the parameters so that this demand system is consistent with cost minimizing behavior. What is the cost function then corresponding to the above factor demand system?Problem Set 1 Micro Theory, S. Wang Question 1.1. Show that “ f(λx) = λf(x), ∀ x ∈ Rn +, λ > 1 ” implies “ f(λx) = λf(x), ∀ x ∈ Rn +, λ > 0. ” Question 1.2. Use a Lagrange function to solve c(w1, w2, y) for the following problem: ⎧ ⎪⎪⎨ ⎪⎪⎩ c(w1, w2, y) ≡ min x1, x2 w1x1 + w2x2 s.t. xρ 1 + xρ 2 = yρ Question 1.3. Use a graph to solve the cost function for the following problem: ⎧ ⎪⎪⎨ ⎪⎪⎩ c(w1, w2, y) ≡ min x1, x2 w1x1 + w2x2 s.t. y = ax1 + bx2 Question 1.4. Find the cost function for the following problem: c(w1, w2, y) ≡ min x1, x2 w1x1 + w2x2 s.t. y = min {ax1, bx2} Question 1.5. Consider the factor demand system: x1(w1, w2, y) = % b11 + b12 w2 w1 1 2 & y, x2(w1, w2, y) = % b22 + b21 w1 w2 1 2 & y where b11, b12, b21, b22 > 0 are parameters. Find the condition(s) on the parameters so that this demand system is consistent with cost minimizing behavior. What is the cost function then corresponding to the above factor demand system? 1