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A perfectly competitive firm has a production function f(X1, X2) - X1 1/3X2^1/3. Suppose that output and input prices are p = 6, w1=1, and w2=1. a. Suppose that in the short run the quantity of factor 2 is fixed at X2-1/8, Solve the firms short-run profit maximization problem to derive the optimal input quantity X1. b.Use the information in part a. to derive the optimal short-run output level y* and the corresponding profits *. c. Now consider the long run in which both factors are variable. Solve the firms long-run profit maximization problem to derive the optimal input quantities X1* and X2* d.Use the information in part c. to derive the optimal long run output level y* and the corresponding profits 2. A perfectly competitive firm has a production function f(X1, X2)=X1 1 2X2. Suppose that input prices are Wi= 1 and W2= 2. The firm wants to find the cheapest way of producing y=8. a.Suppose that in the short run the quantity of factor 2 is fixed at X2-2. Solve the firms short-run: cost minimization problem to derive the optimal input quantity X17 b.Derive the corresponding costs c* of producing y=8 in the short ron. c. Now consider the long run, in which both factors are variable. Set up the Lagrangian for the firms tong run cost minimizetion problem and derive the first order conditions. d.Solve the above first-order conditions to derive the optimal input quantities X1= end X2-2. A perfectly competitive firm has a Cobb-Douglas production function f(x1, X2) = xx2. Suppose that input prices are w, = 1 and W2 = 2. The firm wants to find the cheapest way of producing y = 8. a. Suppose that in the short run the quantity of input factor 2 is fixed at X2 = 2. Solve the firm's short-run cost minimization problem to derive the optimal input quantity xi. (1 pt) b. Derive the corresponding costs Cs of producing y - 8 in the short run. (1 pt) c. Now consider the long run, in which both input factors are variable. Set up the Lagrangian for this firm's long-run cost minimization problem and derive the first-order conditions. (3 pt) d. Solve the above first-order conditions to derive the optimal input quantities x; and X2. (2 pt) e. Derive the corresponding costs c' of producing y = 8 in the long run. (1 pt)A perfectly competitive firm has a Cobb-Douglas production function f (X1, X2) = X,X2- Suppose that input prices are W, = 1 and W2 = 1. The firm wants to find the cheapest way of producing y = 32. a. Suppose that in the short run the quantity of input factor 2 is fixed at X2 = 8. Solve the firm's short-run cost minimization problem to derive the optimal input quantity xi . b. Derive the corresponding costs Cs of producing y = 32 in the short run. c. Now consider the long run, in which both input factors are variable. Set up the Lagrangian function for this firm's long-run cost minimization problem. d. Derive the first-order conditions of the above long-run cost minimization problem. e. Solve the above first-order conditions to derive the optimal input quantities x, and X2. f. Derive the corresponding costs c' of producing y = 32 in the long run.Question 1: Consider an industry with only two firms. The industry inverse demand curve is given by: P = 120 - Q where P is the market price and Q is total industry output. Let each firm have identical marginal cost of $20 and produce identical products. In addition, there are no fixed costs of production. Suppose the firms engage in Cournot strategic competition. Find the Cournot equi- librium price P and industry output Q. Determine the profits for each firm in the Cournot equilibrium