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# 410 Misztal Spring 2010 Homework #5 You must justify each answer. Unless otherwise specied, assume we are operating in the short run where the...

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Econ. 410 Misztal Spring 2010 Homework #5 You must justify each answer. Unless otherwise speciﬁed, assume we are operating in the short run where the number of ﬁrms is ﬁxed. Also assume each market is perfectly competetive. 1. The bolt industry currently consists of 20 producers, all of whom operate with the identical cost curve C ( q ) = 16 + q 2 where q is the annual output of the ﬁrm. The annual market demand for bolts is D ( p ) = 110 - p . (a) What is a single ﬁrm’s short-run supply function? (b) What is the market supply function? (c) What is the equilibrium price and quantity in the market? (d) For the rest of the problem, assume we are concerned with the long run where ﬁrms can enter / exit the market. With n identical ﬁrms, what is the market supply? (e) In terms of n , what is the equilibrium price? (f) In terms of n , how many bolts will each ﬁrm produce at this price? (g) In terms of n , how much proﬁt will each ﬁrm make at this equilibrium price? (h) Verify n = 25.5 will make this proﬁt equal zero. This is your long-run number of ﬁrms in the market. (i) Use n = 25.5 to determine the actual long-run equilibrium price and market quantity. 2. The current recession has sparked a debate concerning the automotive industry. In an extremely simpliﬁed framework (with completely made-up cost curves), we will look at a few of the concerns being discussed. Assume there are two automobile manufacturers in a perfectly competetive environment (they are price takers but don’t treat this like a regular perfect competition problem.) The ﬁrst manufacturer has high costs. Debate exists whether these high costs are due to bad management, strategic failures, unions, unfair trade practices, or some combination of these and other issues. Regardless, assume the high cost ﬁrm has a cost curve of C ( q ) = 60 + 2 q 2 where q is a speciﬁc class of automobile. Assume the second manufacturer has low costs of the form C ( q ) = 40 + q 2 for an identical class of automobile. The market demand for an automobile in this particular class is D ( p ) = 35 - p . (a) What is each ﬁrm’s short-run supply function? (b) What is the market supply function? (c) What is the equilibrium price and quantity in the market? (d) Will both ﬁrms stay in the market in the short-run if the ﬁxed costs are sunk? Hint: Solve for each ﬁrm’s proﬁt using the equilibrium price and quantity and compare it to the cost they will incur by leaving the market. (e) Will both ﬁrms be making a positive proﬁt in the short-run? (f) Now consider the long-run where sunk costs can be avoided. Will both ﬁrms stay in the market? (g) Assume you are now an American senator. The high cost manufacturer represents an American auto manufacturer. The low cost manufacturer represents a foreign auto manufacturer. You must now decide whether the American auto manufacturers should be allowed to fail. Given our current assumptions, should the ﬁrm stay in business? In particular, will it lose money in both the short and long-run? If so, for what other reasons might we want to prevent the ﬁrm from shutting down?
(h) If the American auto manufacturer can demonstrate that it is enacting cost-cutting measures that will change its long-run cost-curve to C ( q ) = 60 + q 2 , would there be a role for “bailing the ﬁrm out”? In other words, might it be wise to give the auto manufacturer money in the short-run so that it can offset its short-run losses while it takes appropriate long-run cost-cutting measures? 3. Two individuals exist in a market. The ﬁrst individual’s preferences are Cobb-Douglas of the form U 1 ( x 1 , x 2 ) = x 1 / 4 1 x 3 / 4 2 . He has an income of \$16. The second individual’s preferences are also Cobb-Douglas, but are of the form U 2 ( x 1 , x 2 ) = x 1 / 2 1 x 1 / 2 2 . Her income is \$20. Two ﬁrms also exist in the market. They produce x 1 . The ﬁrst ﬁrm’s production technology for x 1 is Cobb-Douglas of the form x 1 = f 1 ( n 1 , n 2 ) = n 1 / 4 1 n 1 / 4 2 (where n is a necessary input good). The second ﬁrm’s production technology is Cobb-Douglas of the form x 1 = f 2 ( n 1 , n 2 ) = n 1 / 3 1 n 1 / 3 2 . The costs of the input goods are both \$1. (a) What are ﬁrm 1’s conditional factor demand functions? (b) What are ﬁrm 2’s conditional factor demand functions? (c) Using parts (a) and (b), what is ﬁrm 1’s cost function? (d) Using parts (a) and (b), what is ﬁrm 2’s cost function? (e) Set up the proﬁt maximization problem for ﬁrm 1 using its cost function. (f) Set up the proﬁt maximization problem for ﬁrm 2 using its cost function. (g) Solve the proﬁt maximization problem for ﬁrm 1 to determine its supply function. (h) Solve the proﬁt maximization problem for ﬁrm 2 to determine its supply function. (i) Combine the individual supply functions to determine the market supply function for x 1 . (j) In terms of p 1 , how many units of x 1 will the ﬁrst individual demand? Hint: Cobb-Douglas shortcut. (k) In terms of p 1 , how many units of x 1 will the second individual demand? (l) Combine the individual demand functions to determine the market demand function of x 1 . (m) At the equilibrium price, p * 1 , the market demand will equal the market supply. Using your results, show that p * 1 = \$4.36 is the equilibrium price. Note there might be a very slight rounding error. 4. Consider a perfectly competetive market with n identical ﬁrms. Assume each ﬁrm faces the cost function, C ( q ) = q 2 + 1. (a) What is the supply function for each ﬁrm? (b) What is the market supply function? Hint: If you add a term, A , to itself n times, you end up with nA . In other words, A + A + A + ··· + A where A is repeated n times equals nA . (c) Assume the market demand is D ( p ) = 10 - p . What is the equilibrium price? Hint: Your answer will be in terms of n . (d) How many units of output will each ﬁrm supply at this price? Hint: Your answer will be in terms of n . (e) For the rest of the problem, assume we are concerned with the long run where ﬁrms can enter / exit the market. How many ﬁrms will enter the market? Hint: Use the zero proﬁt condition and solve for n . (f) What is the long run equilibrium price? Hint: Your answer will NOT be in terms of n . (g) How many units of output will each ﬁrm produce? Hint: Your answer will NOT be in terms of n . 5. Consider a monopolist facing a linear demand function, q = 10 - 2 p , and a cost function C ( q ) = 2 + q 2 . (a) What is the inverse demand function the monopolist is facing? (b) Set up the monopolist’s proﬁt maximization problem. (c) Solve the monopolist’s proﬁt maximization problem to determine the number of units he will sell, q * and their price, p * .
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