Suppose that a firm has a revenue function given as R(x, y) = px + qy and a total cost function given as C(x, y) = ax² + y² • What are the needed first order conditions for profit maximization? • What is the optimal x* and y*? • What is the maximum profit *?
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- The demand function for small high-definition televisions is given by p = 9,000 − 55x, where p is the price, in dollars, that x televisions can be sold. The cost function (in dollars) to produce televisions is C = 3,500x + 2,000. (Simplify your answers completely.) (a)Find the revenue function, R. R = (b)Find the profit function, P. P = (c)Find how many televisions should be sold to maximize profit. televisions (d)What is the maximum profit (in dollars)? $A two - product firm faces the following demand and cost functions: Q1 = 40 - 2P1 - P2 Q2 = 35 - P1 - P2 C = Q 2/1 + 2Q 2/2 + 10 (a) find the output levels that satisfy the first order condition for maximum profit. (use fractions) (b) check the second - order sufficient condition. Can you conclude that this problem possesses a unique absolute maximum? (c) What is the maximal profit?Suppose the cost function for a firm is given by C(Q) = 100 + Q2. If the firm sells output in a perfectly competitive market and other firms in the industry sell output at a price of $10, what level of output should the firm produce to maximize profits or minimize losses? What will be the level of profits or losses if the firm makes the optimal decision?
- a. Find the input demand (x), output supply (y), and profit function for the technology, y = ln(x), where y = 0 for x ≤ 1. b. Use Hotelling’s Lemma on the profit function in part (a) to find the input demand for x and the output supply for y.The total revenue curve of a firm is R(q) = 40q − 12q2 and its average cost A(q) =1/30q2 − 12.85q + 20 +400/q, where q is the firms output. Is the rate of change of profit increasing or decreasing when the ouput level of the firm is 10 units? Determine the level of output for which the firms profit is maximized. What is the firm's maximum profit?Suppose the cost function for a firm is given by C(Q) = 100 + Q2. If the firm sells output in a perfectly competitive market and other firms in the industry sell output at a price of $10,A) What level of output should the firm produce to maximize profits or minimize losses?B) What are the profits at the optimal output amount? C) Should the company produce this optimal amount or should it shut down?
- Tasty Inc. sells table salt to both retail grocery chains and commercial users (e.g., bakeries, snack food makers, etc.). The demand function for each of these markets is: Retail grocery chains: P1 = 180 − 8Q1 Commercial users: P2 = 100 − 4Q2 where P1 and P2 are the prices charged and Q1 and Q2 are the quantities sold in the respective markets. Tasty's total cost function (which includes a "normal" return to the owners) for salt is: TC = 50 + 20(Q1 + Q2) (a) Determine Tasty's total profit function. (b) Assuming that Tasty Inc. is effectively able to charge different prices in the two markets, what are the profit-maximizing price and output levels for the product in the two markets? What is Tasty's total profit under this condition?A firm's marginal revenue and marginal cost functions are given by MR=10-4Q and MC= 1. If fixed costs are 4, find the profit when Q=2.A two product firm faces the following demand and cost functions: Q1 = 40 - 2P1 - P2 Q2= 35 - P1 - P2 C = Q21 + 2 Q22 + 10 (a) Find the output levels that satisfy the first-order condition for maximum profit. (use fractions.) (b) Check the second-order sufficient condition. Can you conclude that this problem possesses a unique absolute maximum? (c) What is the maximal profit?
- Suppose the situation changes. JointJuice has its plant in Portland Oregon. The local government passes a new tax on businesses that raises JointJuice’s fixed cost by $25 per period. The cost function now becomes: C(q) = 300 + 6q + 0.1q^2 All the other firms in the market are in other states/cities that are not subject to the laws or taxes of Portland. Calculate JointJuice’s optimal output level and profits if the market price for the product stays the same as for part a. What will the firm do in the long run? Is it reasonable to assume the market price prevailing today will remain the same in the long run? If so why? If not, why not? How about the number of firms in the market?A business sells items according to the following Cost and Revenue functions: C(x)=10x+6100Cx=10x+6100 R(x)=−3.5x2+340xRx=-3.5x2+340x (a) Find the profit function: P(x)=Px= (b) How many items should be sold to break even? (Round to the nearest whole number. Enter your answer as a comma-separated list if necessary) (Enter the smaller number first) x=x= (c) What is the maximum profit for this business? (Round to two decimal places)Total profit is defined as total revenue, R(x), minus total cost, C(x), and is given by the function P(x)=R(x)-C(x). Given R(x)=59x-0.4x^(2) and C(x)=2x+15, find each of the ollowing. P(x) R(80),C(80), and P(80)