When firms rent capital from household at real interest rate

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Under perfect competition firms take prices as given. In that case, and under constant returns to scale, firms make 0 profits in equilibrium. Show that this is true for the Cobb Douglas production function, when firms rent capital from households at a real interest rate r and hire workers at a real wage w. And what is the returns to scale for a. F(a,b) = min(a, b) b. F(K,L) = max(K,L) c. F(x,y,z) = xαy βz γ where α,β,γ>0 are parameters d. F(x,y) = ax + by where a,b>0 are parameters e. F(x1,x2) = (x1 + x2)^1/2 f. F(x1,x2) = (x1^2 + x2^2 )^1/2.

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