Production Function - Suppose we have a production function

Production Function

Suppose we have a production function, 

q = K0.25 L0.25 S0.5

where S refers to the seating capacity of a restaurant which in the short run is fixed at S=100.

a. What is the demand function for labor, L(q,v,w) and for capital, K(q,v,w)?

b. What is the total cost function, C(v,w,q) ?

c. What is the supply function, q (P,v,w)? 

d. What is the profit function, P (P,v,w)?

e. What is the demand function for labor, L(P,v,w) and for capital, K(P,v,w)?

f. Find the homogeneity degree of profit function in prices?

g. Show the profit function is convex in prices?

h. Show the profit function is non-decreasing in price and non-increasing in each qi.