A firm has two variable factors and a production function, f(x₁, x₂) = x₁^1/2x₂^1/4 . 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 _____________ and an equation that says that the value of the marginal product of factor 2 is equal to the wage of factor 2. __________. 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 _________ and x₂ = ________. You could use the first equation to solve for x₁ as a function of x₂ and of the factor wages. Then substitute the answer into the second equation and solve for x₂ as a function of the two wage rates. Finally use your solution for x₂ to find the solution for x₂.)

(b) If the wage of factor 1 is 2, and the wage of factor 2 is 1, how many units of factor 1 will the firm demand ________. How many units of factor 2 will it demand ________. How much output will it produce ___________. How much profit will it make ___________?