Question - A firm has two variable factors and a production function, f(x₁, x₂) = x₁^½ x₂^¼. The price of its output is 4. Factor 1 receives a wage of w₁ and factor 2 receives a wage of w₂.

(a) Write an equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1 2x₁^-½ x₂^¼ = w₁ and an equation that says that the value of the marginal product of factor 2 is equal to the wage of factor 2. x₁^½ x₂^-¾  = w₂. Solve two equations in the two unknowns, x₁ and x₂, to give the amounts of factors 1 and 2 that maximize the firm's profits as a function of w₁ and w₂. This gives x₁ = 8/(w₁³w₂) and x₂ = 4/(w₁²w₂²).