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function? (ii) What are firm's down production point for this firm? [ill ) Derive the formula for its MC. (iv) What is the short-run shut b. In the long run a firm produces q = 100 using L and K which are imperfect substitutes. (i) Show the cost minimization equilibrium point of this firm using diagram. (ii) Using calculus/equation show that at this point marginal rate of technical substitution is equal to input price ratio. (iii) On the same diagram show the equilibrium for the second firm, which uses more K and less L (than the first firm) to produce the same level of output. Question 3 a. Suppose the profit of the firm that produces output q is given by (q), the revenue is given by R(q) and the cost function is given by C(q). (i) Using calculus, derive the profit maximizing condition for this firm? For a perfectly competitive firm, how can we write this condition? (ii) Using diagram show the short run equilibrium when the firm is earning abnormal profit. (iii) Show the long-run equilibrium in the diagram. b. Given that a competitive firm's short-run cost function is C(q) = 100q - 4q2 + 0.2q3 + 200, what is the firm's short-run supply curve? If the price P = 115, how much output does the firm produce? estion 4 a. Using downward slopping demand and upward slopping supply curves of a product, competitive equilibrium, (i) show the measure of consumer surplus (CS), producere) (12 points) Graph these curves: a. profit-maximizing labor demand: part d (w on vertical axis, L on horizontal axis) b. short-run supply: part b (P on vertical axis, q on horizontal axis) c. profit: from above (T on vertical axis, P on horizontal axis) The graphs do not need to be super-accurate but pay attention to the shape of the curves and where they intersect the axes (if they do) () (12 points) Suppose P = P,. On the 3 graphs from part e, show what happens when w increases from w, to w2- Explain how and why labor, supply, and profit change. 2. (54 points) Short-run costs. Suppose w = 1, r = 10 and K = 20. a) (5 points) We have TC = w (*) q3 + rk = ()q3 + 200 On one graph (with q on the horizontal axis), graph the Total Cost, Variable Cost, and Fixed Cost functions. Pay attention to the shape of the curves, where they intercept the axes and each other (if they do), and the position of the curves relative to each other. b) (9 points) Using the graph from part a, show how AC, AVC, and MC can be shown when q = 20. c) (5 points) From the TC function in part a, find Marginal Cost, Average Cost, and Average Variable Cost. d) (8 points) Use the function you found in part c. On one graph (with q on the horizontal axis), graph the AC, AVC, and MC functions. Pay attention to the shape of the curves, where they intercept the axes and each other (if they do), and the position of the curves relative to each other. e) (6 points) Calculate the values of AC, AVC, and MC when q = 20. Show these points on the graph from part d. Relative to each other, are the values for AC, AVC, and MC at q = 20 consistent with the results in part b? f) (6 points) Calculate the price at which this firm would break even (zero profit). What is the optimal output at this price? g) (15 points) Suppose the market price is P = 20. What is the firm's optimal output and profit? With respect to profitability, what is the firm's optimal course of action? Explain.a) the marginal product of the 6" unit of labor? b) at what level of labor APL and MPL equal? c) the total cost of producing 455 units of output 3. Assume a firm with a production function (=12404