Question: A perfectly competitive, profit maximizing competitive firm has the production function

\[f\left( K,L \right)=\log \left( 1+L \right)+\log \left( 1+K \right)\]

a) Solve this problem

Max pf(K,L) – wL – vK

K>=0 , L>=0

From the first order conditions find the long run input demand functions

L* = L (w ,v, p), K * = K (w ,v ,p)

Where p, w, and v are the prices of output, labor and capital, respectively. What assumptions on prices do you need to make in order to guarantee interior solutions, i.e. to guarantee that K*>0 , L*>0? Be sure to check second order conditions. Determine the supply function, i.e. find y* = f (K*, L*). Then find the profit function, i.e. find –

∏* = ∏ (w,v,p) = py* - wL* - vK*

b) Given the solutions in A, determine all “comparative statistics results” that is, determine all possible partials like Partial L* and “sign” them

Partial v

c) Show that

Partial ∏ = -L*

Partial w

Partial ∏ = - K*

Partial v