4. Suppose the total revenue function is 4x2 + 3x + 2. Find the average and marginal revenue. In addition, find the values which maximizes TR. Suppose, however, that the total revenue. is y 4x³+3x2+2x, find the average, marginal, the value which maximizes y.
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- If the inverse demand function is p=800-4Q what is the marginal revenue function? Draw the demand and marginal revenue curves. At what quantities do the demand and marginal revenue curves hit the quantity axis? (Hint: See Q&A 9.1)The total revenue curve of a firm is R(q) = 40q − 12q2 and its average cost A(q) =1/30 q2 − 12.85q + 20 + 400/q, where q is the firm's output. i. Is the rate of change of profit increasing or decreasing when theouput level of the firm is 10 units?ii. Determine the level of output for which the firmᇱs profit is maximized.iii. What is the firm's maximum profit?A firm produces a steel bar. When the price of the steel bar is $ 30,000, the quantity demanded is 8 metric tons, a 100% change in the price would change the quantity demanded by 25%. 1. How much would be the maximum total revenue of the firm? 2. What is the demand function of the firm? (Use the variable Q for the quantity demanded and the variable P for the price; no currency; use fraction form for the slope) 3. What is the marginal revenue function of the firm? (Use the variable Q for the quantity and the variable MR for the marginal revenue; no currency)
- Which of the following are true when Marginal Revenue (MR) < Average Revenue (Demand), and Marginal Revenue > 0? |e|= 1 Total Revenue is at a maximum Total Revenue is rising as Q rises MR is rising as Q rises1. Demand and Costs. Assume you are faced with the following demand curve,P = 20-0.5QWhere P is the dollar price per unit and Q is the number of units sold per month, and Q must be2 or more.a. Write the expression (definition) for this firm’s Total Revenue (TR)b. Write the expression for this firm’s marginal revenue (MR).c. What is Q when the P is zero?d. What is P when the Q is zero?e. Profit maximization occurs where MR=MC. If MC=$10 what is the profit maximizing levelof Q and P?f. As a business owner, you are thinking about lowering the price of this product. If youlower the price by 10% from your answer in e., will your TR rise or fall?7. A paint store sells 270 pails of paint per month at a price of $32 each. A customersurvey indicates that for each $1.20 decrease in price, sales will increase by 6pails of paint.(a) Determine the demand, or price, function.(b) Determine the revenue function (using (a)).(c) Determine the marginal revenue function.(d) Solve R’(x) = 0. Interpret the meaning of this value for this situation.(e) What is the price that corresponds to the value found in part (d)?
- The demand function for a particular product is given by D(x)=0.5x^2+3x+190‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√D(x)=0.5x^2+3x+190 dollars, where x is the number of units sold. What is the marginal revenue when 88 items are sold? Round your answer to 2 decimal places. D(x)=0.5x^2+3x+190‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾11.2 Your college newspaper, The Collegiate Investigator, sells for 90¢ per copy. The cost of producing x copies of an edition is given by C(x) = 70 + 0.10x + 0.001x2 dollars. (a) Calculate the marginal revenue R'(x) and profit P'(x) functions. HINT [See Example 2.] R'(x) = P'(x) = (b) Compute the revenue and profit, and also the marginal revenue and profit, if you have produced and sold 500 copies of the latest edition. revenue $ profit $ marginal revenue $ per additional copy marginal profit $ per additional copy Interpret the results. The approximate from the sale of the 501st copy is $ . (c) For which value of x is the marginal profit zero?x = copiesInterpret your answer. The graph of the profit function is a parabola with a vertex at x = , so the profit is at a maximum when you produce and sell copies.Solve only c and d 2. There are 2 groups with different demand in a market, as follows: ?!=40−?1 and ?"=100−2?2 a. Give the inverse demand curves and marginal revenue in each of these groups. b. If marginal cost is flat at $10, calculate the profit-maximizing quantities and prices associated with this market place. Are the prices for each group different? Comment on the outcomes. c. Calculate the producer surplus associated with the outcomes generated in part b. d. Now calculate the equilibrium price and quantity if the firm charges one price across all consumers. What is the producer surplus associated with this outcome and how does it differ to that calculated in part c?
- Suppose the revenue function y=R(x) is linear, and points (x,y) of (10,550) and (13,715) are on the line. a. What is the marginal revenue? b. Suppose also the total costs are given by C(X) = 150 + 15x +2x^2. Find the x-value that maximizes profit.Now assume that firm T faces a downward-sloping (straight-line) demand Fill in the columns for TR and MR in the table (Note that the figures for MR are entered between 0 and 1, 1 and 2, 2 and 3, etc.) The demand curve for the product of firm T Price (AR) (£) Quantity (Units) Total Revenue (TR) (£) Marginal Revenue (MR) (£) 20 18 16 14 12 10 8 6 0 1 2 3 4 5 6 7 ……… ……… ……… ……… ……… ……… ……… ……… ……… ……… ……… ……… ……… ……… ……… (b) What is the price elasticity of demand at P = £10?.................................................................... (c) Over what price range is demand price elastic?......................................................................... (d) Over what price range is demand price inelastic? ....................................................................................Which of the following statements about the relationship between marginal revenue, price, and the price elasticity of demand, is true? a) At the midpoint of a linear demand curve, the value of price elasticity of demand is zero. b) With a perfectly elastic demand curve, whatever output the firm produces, the marginal revenue is always greater than the market price. c) If the price elasticity of demand PED = -1 at all points on a demand curve, then the firm can choose any level of output, and earn the same revenue. d) If a firm has inelastic demand, then the marginal revenue curve will be upward sloping.