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Exercise 1: For the production function from the previous homework: y=KL Assume that the firm is in the long run (so you can use the long-run cost function you found in the previous homework) to answer the following: a. Consider the case of =1/(3+A), where A is the last digit of your ASU ID#. Also, assume that the market operates under perfect competition. Obtain long-run quantity produced and profits for the firm (as a function of parameters w1,w2 and p ). Also obtain the (long-run) unconditional input demands (as a function of the same parameters). b. For the case of =1/2 obtain the long-run output price p (as a function of input prices w1 and w2 ) that implies zero profits for the firm. Argue why a higher price is not consistent with perfect competition and a lower one would induce the firm to shut down. c. Consider the case of =(2+B)/(3+B), where B is the second-to-last digit of your ASU ID#. Also assume that the firm has a monopoly. The inverse demand function is: p=y(6+3B3+2B) Obtain the long-run profit-maximizing quantity and price (as a function of parameters w1 and w2 ). Exercise 3 (Long-run equilibrium with free entry): Assume every firm in an industry has the following cost function: c(y)={25+(10+20B)y+y2,0,ify>0ify=0 where B is the second-to-last digit of your student ID#. The industry demand function is: Y=270+20AP where A is the last digit of your student ID\#. a. Assume there is only one firm in the market. The firm is a monopolist. Graph average costs and marginal costs as a function of how much is produced. Obtain price charged, quantity produced and profits. b. A second firm sees it profitable to enter the market. Assume that once the second firm enters, both firms take the price of output as given (a perfectly competitive market). First, obtain the individual supply functions. Second, obtain the industry supply function by adding up both individual supply functions. Third, obtain the equilibrium price by equating industry supply to demand. Fourth, obtain quantities produced by each firm and profits for each firm. c. If two firms are making positive profits then it may be profitable for a third firm to enter the market. Use the same procedure to see if the industry supports a third firm. d. Assume there are N firms in the market (the general case). Use the same procedure as in b and c to find equilibrium price (now as a function of the number of firms), quantities produced and profits (all as a function of the number of firms). To know the number of firms that the market will end up having, find N such that firms are making zero or positive profits, but if you add one more firm they start making negative profits (Hint: There is an easy and a difficult way of solving for N ). Exercise 4 (Economic Rent): You are required by the city of New York to review the taxi medallion system (a medallion gives the owner the right to provide a taxi-cab service - one carfor life). Obtaining operating costs of providing taxi-cab services is not easy, since owners don't want to give away that information. You have gathered the following facts: 1. The (annual) cost function, after including all possible costs (i.e: paying for a driver, gas, repairs, etc.) is the following: C(y)={ay2+by+c,0,ify>0ify=0 where output y is measured in miles and c is the quasi-fixed cost of operating the cab. 2. The minimum efficient scale of operation is 40,000 miles driven per year. 3. The Quasi-fixed cost is $10,000 4. There are currently only 12,000 medallions (taxis) operating in NY. 5. Aggregate demand for cab services (again, in miles per year) in NY is the following: Yd(P)=3,840,000,000960,000,000P 6. The equilibrium price of a cab ride is P=$3 (per mile). Do the following: a. Obtain the individual supply function Graph it. (Hint: use facts 1,2 and 3 to obtain supply as a function of price P and parameter b ) b. Determine whether medallion owners are operating at their minimum efficient scale or not (Hint: use facts 2,4,5 and 6 ) c. Obtain total costs and average costs per year of operating a taxi-cab in NY (Hint: to do this you will have to use the individual supply function and the equilibrium condition to back out the value of parameter b; only then will you be able to obtain total and average costs) d. Obtain total profits per year. e. If the interest rate is 3% per year, obtain the value of owning a medallion (Hint: the value is equal to the profits per year divided by the interest rate). f. Give the mayor of NY a dollar estimate of how much better off society would be if the taxi-cab industry was de-regulated (Hint: Calculate the deadweight loss of having the medallion system). Show your result in a graph (with aggregate demand and supply). Also determine how many cabs would be operating if the industry was de-regulated. Exercise 1: For the production function from the previous homework: y=KL Assume that the firm is in the long run (so you can use the long-run cost function you found in the previous homework) to answer the following: a. Consider the case of =1/(3+A), where A is the last digit of your ASU ID#. Also, assume that the market operates under perfect competition. Obtain long-run quantity produced and profits for the firm (as a function of parameters w1,w2 and p ). Also obtain the (long-run) unconditional input demands (as a function of the same parameters). b. For the case of =1/2 obtain the long-run output price p (as a function of input prices w1 and w2 ) that implies zero profits for the firm. Argue why a higher price is not consistent with perfect competition and a lower one would induce the firm to shut down. c. Consider the case of =(2+B)/(3+B), where B is the second-to-last digit of your ASU ID#. Also assume that the firm has a monopoly. The inverse demand function is: p=y(6+3B3+2B) Obtain the long-run profit-maximizing quantity and price (as a function of parameters w1 and w2 ). Exercise 3 (Long-run equilibrium with free entry): Assume every firm in an industry has the following cost function: c(y)={25+(10+20B)y+y2,0,ify>0ify=0 where B is the second-to-last digit of your student ID#. The industry demand function is: Y=270+20AP where A is the last digit of your student ID\#. a. Assume there is only one firm in the market. The firm is a monopolist. Graph average costs and marginal costs as a function of how much is produced. Obtain price charged, quantity produced and profits. b. A second firm sees it profitable to enter the market. Assume that once the second firm enters, both firms take the price of output as given (a perfectly competitive market). First, obtain the individual supply functions. Second, obtain the industry supply function by adding up both individual supply functions. Third, obtain the equilibrium price by equating industry supply to demand. Fourth, obtain quantities produced by each firm and profits for each firm. c. If two firms are making positive profits then it may be profitable for a third firm to enter the market. Use the same procedure to see if the industry supports a third firm. d. Assume there are N firms in the market (the general case). Use the same procedure as in b and c to find equilibrium price (now as a function of the number of firms), quantities produced and profits (all as a function of the number of firms). To know the number of firms that the market will end up having, find N such that firms are making zero or positive profits, but if you add one more firm they start making negative profits (Hint: There is an easy and a difficult way of solving for N ). Exercise 4 (Economic Rent): You are required by the city of New York to review the taxi medallion system (a medallion gives the owner the right to provide a taxi-cab service - one carfor life). Obtaining operating costs of providing taxi-cab services is not easy, since owners don't want to give away that information. You have gathered the following facts: 1. The (annual) cost function, after including all possible costs (i.e: paying for a driver, gas, repairs, etc.) is the following: C(y)={ay2+by+c,0,ify>0ify=0 where output y is measured in miles and c is the quasi-fixed cost of operating the cab. 2. The minimum efficient scale of operation is 40,000 miles driven per year. 3. The Quasi-fixed cost is $10,000 4. There are currently only 12,000 medallions (taxis) operating in NY. 5. Aggregate demand for cab services (again, in miles per year) in NY is the following: Yd(P)=3,840,000,000960,000,000P 6. The equilibrium price of a cab ride is P=$3 (per mile). Do the following: a. Obtain the individual supply function Graph it. (Hint: use facts 1,2 and 3 to obtain supply as a function of price P and parameter b ) b. Determine whether medallion owners are operating at their minimum efficient scale or not (Hint: use facts 2,4,5 and 6 ) c. Obtain total costs and average costs per year of operating a taxi-cab in NY (Hint: to do this you will have to use the individual supply function and the equilibrium condition to back out the value of parameter b; only then will you be able to obtain total and average costs) d. Obtain total profits per year. e. If the interest rate is 3% per year, obtain the value of owning a medallion (Hint: the value is equal to the profits per year divided by the interest rate). f. Give the mayor of NY a dollar estimate of how much better off society would be if the taxi-cab industry was de-regulated (Hint: Calculate the deadweight loss of having the medallion system). Show your result in a graph (with aggregate demand and supply). Also determine how many cabs would be operating if the industry was de-regulated

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