# Homework 5 – Let’s review some easy differentiations

Homework 5Submission: November 18th before my classAs told before, you can make groups of upto 3 people. You should submit one HW per group,and clearly mention the name of the group members on top. (Now off course you can free ride onthe other group members and get a good score, BUT if you actually do this HW by yourself, youdon’t need study much more for the second Midterm)Question 1Let’s review some easy differentiations, and decompose the total cost functions. Use theformulas from the slides the decomposition. For finding the Marginal Cost, use the derivativeformulas forandfrom the “Some Calculus.pdf”.Example: If ( )( ) ( , Then) ( ) ( ) ( ) Find the Fixed Cost (FC), Variable Cost (VC), Average Total Cost (ATC) and the Marginal Cost(MC) for the following Total Cost Functions,a)b) ( )( ) c) ( ) d) ( ) Question 2A firm used Labor (L) and Machines (M) according to the production functionmarginal products are ( ) . The ( ) . The cost of Labor is $40 per unit and the cost of using a Machine is $10.a) Suppose the Firm wants to produce at the cheapest way. Find the number of machines itwould use per worker.b) If it wants to produce 40 units of output, how many L and M will it employ to minimizecosts? What will be this cost?Suppose now that this firm is a MONOPOLY and it faces a Demand Function . c) Find the Marginal Cost function from the Total Cost function in part b. Find the MarginalRevenue function from the Demand Function given above.d) If the Firm is maximizing profits, what is it’s profit maximizing price ( ) and output( )e) How much profit/loss is it earning at this price and output?Question 3Suppose a competitive firm has the following short-run cost function: ( )a) Find the firm’s Marginal Cost (MC) and the Average Variable Cost (AVC).Let’s say that these two curves look like this, Redraw this in your answer sheet and answer thefollowing, b) Mark the supply curve for the competitive firm.c) Find the Shutdown price and mark it on the graph (Hint: remind yourself what happensgraphically at the shutdown price and then solve it mathematically)Question 4Let’s consider a real world scenario of price wars. This is a situation when in a competitivemarket, a group of firm keeps on reducing the prices to capture the whole demand. Suppose eachof these perfectly competitive firms face a cost function ( ),a) Let’s say at the beginning of the price-war, the Market price. At this price findeach firm’s output and their Profits.b) Now one of the firms starts the war by reducing the price to 40. As you know in PerfectCompetition when one firm reduces the price, the whole market has to reduce the price.So now the market price is. At this new price how much does each firmproduce? And what are their profits/Loss? In the Short Run are they going to keep onproducing or shutdown? c) Some other firm now reduces the market price to 30 (). Calculate the outputand the profits/loss of each firm. Are they going to produce/ Shutdown in the short run?d) Can you find the price at which each firm shuts down in the short run?e) Suppose if such a thing happened in the Long Run, can you find a price when the firmswill stop production?Question 5Professor Dope has just written a book called Up Your Isoquant. Market research shows that thedemand for such a book is. It will cost $1000 to set the book in type. Thissetup cost is necessary before any copies can be printed. In addition to the setup cost, there is amarginal cost of $4 per every book printed,a) Write the Profit function for Prof. Dope production. (Remember profit is a function of y,not p)b) Find the Marginal Revenue function and Marginal Cost and hence the profit maximizingnumber of books. How much profit does Prof. Dope make?c) Try to draw the Demand Function, the MR and MC on a graph. Point the Profit Maxoutput and price. Shade the region that shows the Prof’s profits.d) The Government doesn’t like the contents of the Prof’s book. So to reduce it’s sales, ProfDope is charged a lump-sum tax of $500. How does this affect Prof’s output and profits?Question 6Suppose you are the owner of an Amusement Park. You have gone into business to extract everylast penny from the society. There are no more theme parks in the area. Suppose your Costfunction is ( ), being the number of rides in the park. It is easy to see that you havea constant Marginal Cost of 100. You also know that your demand is given by some function( ).a) To start, do the same thing you have been doing. Find the profit maximizing number ofrides in your park and the price for each ride.b) Draw the situation and mark the rectangle showing the Revenue and the triangle showingthe CS.c) Now, though you have a profit, you can see that the Consumers are attaining some levelof surplus. Since you are out to extract every last bit of surplus from the society, try andcome up with a way you can take away this consumer surplus and transform it into yourprofits. (This part of the question has no maths. I want to see how innovative you can beto come up with real world solutions.)