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Econ302-hw4-summer09-solution - Econ302 Homework Assignment...

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Econ302 Homework Assignment 4 Solutions 1. There are currently 100 identical firms in the perfectly competitive gadget  manufacturing industry, each having short-run total costs given by 2 0.5 10 5 STC q q = + + where   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, S Q ? The short-run supply curve of the firm is that portion of the short-run  marginal cost curve that lies above the average variable cost curve. So we  need to find the short-run marginal cost curve, SMC: ( ) 10 d STC SMC q dq = = + To find the firm’s supply curve as a relationship between price and quantity  supplied, we notice that  in perfect competiton, the condition for profit  maximization is  P  =  SMC . Plug in the expression for  SMC   and write  q  as a  function of  P : 10 10 P SMC q q P = = + = - The industry consists of 100 identical firms. So the industry short-run supply  curve is: 100 100( 10) 100 1000. s Q q P P = = - = - b. Suppose the market demand for gadgets is given by 1100 50 . D Q P = -  What is  the equilibrium price and quantity in this market? What will each firm’s  total short-run profits be? 1
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2 2 1100 50 100 1000 14 1100 50(14) 400 400 Each firm's output is 4. 100 100 Each firm's profit is Revenue-Cost (0.5 10 5) 14 4 [0.5(4) 10(4) 5] 3. D S Q Q P P P Q Q q pq q q = - = - = = - = = = = = = - + + = - + + = c. Graph the market demand and supply curves and compute total producer  surplus. Producer Surplus =  ½ (14-10)*400 = 800. d. Show that the total producer surplus you calculated in part (c) is equal to  total industry profits plus industry short-run fixed costs. Industry profits = 100 * firm profit = 100*3 = 300. Form the equation of the cost curve we have  firm’s fixed  equal to 5. So Industry fixed cost = 100 * 5 = 500. Therefore, Industry profits + industry fixed cost = 300 + 500 = 800 = Producer  Surplus. P.S. 10 1 14 22 Q P Q s Q D 400 2
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e. Suppose the government imposed a tax of $3 on each gadget sold. How  would this tax change the market equilibrium? Let be the price that buyers pay and let be the price that sellers receive. We now have 3. The market demand and supply equations can be written as: 1100 50 100 1000 To find equilibr b s b s D b S s P P P P Q P Q P = + = - = - ium with tax set and substitute for in terms of : 1100 50 100 1000 1100 50( 3) 100 1000 150 1950 13 13 3 16 100(13) 1000 300. D S b s b s s s s s b D S Q Q P P P P P P P P P Q Q = - = - - + = - = = = + = = = - = f. How would the burden of this tax be shared between sellers and buyers of  gadgets. Buyres were paying $14 before tax but now they are paying $16 with tax.  Therefore, the tax burden on buyers is 2 / 3 = 67%.
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