1. A perfectly competitive firm has a production function (x₁, ₂) = ₁₂^1/2

2. Input prices are given by ₁ = 10 and ₂ = 5.

a. Does this production function exhibit decreasing, constant, or increasing returns to scale? (1 pt)

b. In the short run, input factor 2 is fixed at ₂ = 1 and the firm chooses the optimal input quantity ₁ to minimize the cost of producing output = 27. Derive ₁ . (1 pt)

c. Calculate the costs _s associated with the above short-run solution. (1 pt)

d. In the long run, the firm chooses the optimal input quantities ₁ and ₂ to minimize the cost of producing output = 27.

Write the Lagrangian function for this firm's long-run cost minimization problem. (1 pt)

e. Derive the first-order conditions for this firm's long-run cost minimization problem. (3 pt) 4

f. Solve the above first-order conditions to derive the optimal long-run input quantities ₁ and ₂ .