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 Subject: marginal analysis and economics Category: Business and Money > Economics Asked by: tomandsharonz-ga List Price: \$10.00 Posted: 10 Sep 2005 19:38 PDT Expires: 10 Oct 2005 19:38 PDT Question ID: 566631
 ```I simply am not good at math. Could somebody please explain to me (really, like I'm a 5-year-old) how to "do the Calculus" in economic marginal analysis? Here's the problem I am facing: Suppose a firm's inverse demand curve is given by P = 120 - .5Q (what does this mean? Is P, price and Q, quantity?) and it's cost equation is C = 420 +60Q +Qsquared (what does this mean?) a. find the firm's optimal quantity, price, and profit (1) by using the profit and marginal profit equations and (2) by setting MR equal to MC (I understand the concept of marginal costs and revenues, just don't know how to solve for it? I really don't understand Calculus). b. Suppose instead that the firm can sell any and all of its output at the fixed market price P = 120. Find the firm's optimal output. Thanks! Tom p.s. Today is Saturday, 9/10/2005. I need to know quickly. Monday at the very latest. Thanks!```
 ```Hi!! The inverse demand function is the inverse function for the demand, that is if demand (Q) function is defined by Q = f(P), where P is the price; then the inverse demand function is the inverse function of f(P) and it is given by P = f^-1(Q), it defines P in function of Q. See the folowing page for further reference: http://www.econ.ucsb.edu/~rabbit/econ100b/studentquestion1 The cost equation defines the cost in function of the quantity Q produced, in other words gives us how much cost to produce certain quantity Q of products, in this case to satisfy the demand. Now, the solution: a. find the firm's optimal quantity, price, and profit (1) by using the profit and marginal profit equations Revenue = P*Q = (120 - 0.5*Q)*Q = = 120*Q - 0.5*Q^2 Profit = Revenue - Cost = = (120*Q - 0.5*Q^2) - (420 + 60*Q + Q^2) = = 60*Q - 1.5*Q^2 - 420 Marginal Profit is the additional profit derived from the sale of one additional unit, and it is easy calculated as the derivative of the profit. The same is valid for all the marginal values for the purposes of this problem. For further references see: "Marginal Quantities": http://earthmath.kennesaw.edu/main_site/review_topics/marginal_quantities.htm Then: Marginal Profit = 60 - 3*Q If profit has a maximum it occurs when its derivative is zero, since Marginal Profit is the derivative of the profit, if profit has a maximum it occurs when marginal profit is zero: Marginal Profit = 60 - 3*Q = 0 Then Q = 60/3 = 20 ---> This is the optimal quantity Replacing this in the price function: P = 120 - 0.5*Q = = 120 - 0.5*20 = = 110 ---> this is the optimal price. Profit = 60*Q - 1.5*Q^2 - 420 = = 60*20 - 1.5*20^2 - 420 = = 1200 - 600 - 420 = = 180 (2) by setting MR equal to MC MR = derivative of Revenue = = MC = derivative of cost = = 60 + 2*Q MC = MR ==> 60 + 2*Q = 120 - Q ==> Q = (120 - 60)/3 = 20 Then P = 120 - 0.5*Q = 120 - 10 = 110 We find the same value using both methods. ------------------------------------------------------ I hope that this helps you. Feel free to request for a clarification if you need it. Regards, livioflores-ga``` Request for Answer Clarification by tomandsharonz-ga on 11 Sep 2005 14:27 PDT ```Could you please provide more detail on solving the math? 1. Are you saying that revenue equals = 120*Q - 0.5*Q^2 (^2 does this mean squared?... I'm just making sure...) 2. Are you saying that profit equals = 60*Q - 1.5*Q^2 - 420 3. How did you solve for MP? Specifically, how did you get the 3*Q? 4. How did you solve for Price = 110. And, Profit = 180. 5. I'm lost... Can you help? Thanks! Tom``` Request for Answer Clarification by tomandsharonz-ga on 11 Sep 2005 14:53 PDT ```Two more questions: 1. How would you graph MR and MC? 2. Suppose the instead that the firm can sell any and all of its output at the fixed market price P = 120. Find the firm's optimal output? Thanks! Tom and Chuck (he's lost,too)...``` Clarification of Answer by livioflores-ga on 11 Sep 2005 21:29 PDT ```Hi Tom!! 1. Are you saying that revenue equals = 120*Q - 0.5*Q^2 (^2 does this mean squared?... I'm just making sure...) Revenue is the amount of money that a firm actually receives from its activities, in this case for selling its products, and this amount is quantity sold (Q) times the product price (P): Revenue = P*Q = 120*Q - 0.5*Q^2 (and yes, ^2 means squared). 2. Are you saying that profit equals = 60*Q - 1.5*Q^2 - 420 Yes, profit is revenue less cost and the result of such subtraction is this. 3. How did you solve for MP? Specifically, how did you get the 3*Q? As I told you, the marginal profit is calculated as the derivative of the Profit formula, in this case we must to find the derivative (Profit)' of: Profit = 60*Q - 1.5*Q^2 - 420 Then Marginal Profit = (Profit)' = 60*1 - 1.5*2*Q - 0 = 60 - 3*Q Again take a look at the page "Marginal Quantities": http://earthmath.kennesaw.edu/main_site/review_topics/marginal_quantities.htm 4. How did you solve for Price = 110. And, Profit = 180. You want to maximize the profit, to do that you must equal the Marginal Profit to zero, that is: 0 = 60 - 3*Q ==> 3*Q = 60 ==> Q = 60/3 = 20 This value of Q is the optimal quantity, if you put this value in the Price formula you will get the optimal price: P = 120 - 0.5*Q = 120 - 0.5*20 = 120 - 10 = 110 Doing the same with the Profit formula and you will get the maximum profit: Maximum Profit = 60*Q - 1.5*Q^2 - 420 evaluated at Q = 20, that is Maximum Profit = 60*20 - 1.5*20^2 - 420 = = 1200 - 1.5*400 - 420 = = 1200 - 600 - 420 = = 180 I hope this helps you to understand this answer. Feel free to continue using the clarification feature if you need it. Regards, livioflores-ga``` Clarification of Answer by livioflores-ga on 11 Sep 2005 22:34 PDT ```Hi again!! You are asking two new questions that are related to this one but require additional work, the correct way in these cases is to open a new question in the forum pricing it as your criterion. I will try to answer these two additional questions for you, but if you have more questions to do please post them at Google Answers' forum as separated ones. 1. How would you graph MR and MC? MR is the marginal revenue and it is the derivative of the Revenue function: Revenue = 120*Q - 0.5*Q^2 ==> MR = 120 - Q The graph of MR as a function of Q is a straight line with slope 1 and when Q is zero MR = 120 and when Q = 120 then MR = 0. MC is the marginal cost and it is the derivative of the Cost function: Cost = 420 + 60*Q + Q^2 ==> MC = 60 + 2*Q The graph of MC as a function of Q is a straight line with slope 2 and when Q is zero MC = 60 and to get another point use Q = 30, then MC = 120. One more useful graph in these cases is the Demand curve, that is the graph of the Price function, in this case Price = 120 - 0.5*Q , again we have a straight line where Price is 120 when Q = 0 and Price is zero if Q = 240. To see and approximate graph follow this link: http://www.geocities.com/artistaflores/MC_MRgraph.jpg 2. Suppose the instead that the firm can sell any and all of its output at the fixed market price P = 120. Find the firm's optimal output? If Price is fixed at \$120 then: Revenue = P*Q = 120*Q ==> MR = derivative of (120*Q) = 120 Setting MC = MR we can find the optimal output: 60 + 2*Q = 120 ==> 2*Q = 120-60 ==> 2*Q = 60 ==> Q = 60/2 = 30 . For a fixed price of \$120 the firm's optimal output is Q = 30. ----------------------------------------------- I hope this helps you. Best regards, livioflores-ga```
 tomandsharonz-ga rated this answer: ```Excellent answer and follow-up... I'm very "overwhelmed" by much of this, but the answer allowed me to "fill in the blanks." Thank you.```