Production function

1. Consider the following production function 2qKL= a. With w = $20 and r = $40, find the cost minimizing input combination to produce 100 units of output. b. Assume w = $20 and r = $40 as in part (a) but now treat q as a variable. Derive the firm’s demand for labor and capital and use them to find the long run total cost, long run average cost, and long run marginal cost as functions of output q. c. Again let w = $20 and r = $40, but now assume K = 5 in the short run. Derive the short run total cost, short run average cost, and short run marginal cost as functions of output q. d. In the same diagram, draw the graph of short run and long run total cost functions you found in parts (b) and (c). For what level of output are the two costs the same? Verify your answer and show this level of output on the diagram.

3. Consider a competitive market in which the market demand for the product is expressed as P = 75 – 1.5Q, and the supply of the product is expressed as P = 25 + 0.50Q.Price, P, is in dollars per unit sold, and Q represents rate of production and sales in hundreds of units per day. The typical firm in this market has a marginal cost of MC = 2.5 + 10q. a. Determine the equilibrium market price and output. b. Determine the level of output of the typical firm, given your answer to part (a) above. c. If the market demand were to increase to P = 100 – 1.5Q, what would the new price and output in the market be? What would the new level of output for the typical firm be? d. If the original supply and demand represented a long-run equilibrium condition in the market, would the new equilibrium (c) represent a new long-run equilibrium for the typical firm? Explain.

4. There are currently 100 identical firms in the perfectly competitive gadget manufacturing industry, each having short-run total costs given by STC= 0.5q^2 +10q +5where q is the output of gadgets per day. a. What is the short-run supply curve for each firm in the industry? What is the short-run supply curve for the industry as a whole, Qs? b. Suppose the market demand for gadgets is given by Qd = 1100-50pWhat is the equilibrium price and quantity in this market? What will each firm’s total short-run profits be? c. Graph the market demand and supply curves and compute total producer surplus. d. Show that the total producer surplus you calculated in part (c) is equal to total industry profits plus industry short-run fixed costs. e. Suppose the government imposed a tax of $3 on each gadget sold. How would this tax change the market equilibrium? f. How would the burden of this tax be shared between sellers and buyers of gadgets. g. Calculate the total loss in producer surplus as a result of the tax on gadgets. Show that this loss equals the change in total short-run profits in the gadget manufacturing industry.

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